Memoirs on Differential Equations and Mathematical Physics

Volume 4. 1995

S. Kharibegashvili

GOURSAT AND DARBOUX TYPE PROBLEMS FOR LINEAR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS AND SYSTEMS

Abstract. In the present paper, for hyperbolic equations and systems in angular domains, we consider the formulations of problems representing natural continuation and further development of the well-known classical formulations of Goursat and Darboux type problems. For a wide class of linear normally hyperbolic equations and systems of second order, the dependence of unique solvability of the problems under consideration on the structure of an angular domain as well as on the weighted space in which the solution is sought, is established. Some correct multidimensional analogues of Goursat and Darboux type problems for hyperbolic equations are also considered.

Mathematical Subject Classification. 35L20.

Key words and Phrases. Hyperbolic equations and systems, problems of Goursat and Darboux type, characteristic problem, multidimensional analogues of Goursat and Darboux type problems, angular domain, weighted space.

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INTRODUCTION

The questions of searching for and investigation of correctly posed boundary value problems are of great interest in the theory of equations and systems of hyperbolic type. Among these problems boundary value problems for hyperbolic equations and systems representing natural continuation and further development of the well-known classical formulations of the Goursat and Darboux problems are especially interesting.

Unlike the multidimensional case, more simple structure of characteristic manifolds in the two-dimensional case allows one to obtain most complete results on the solvability of these problems for hyperbolic equations.

In the two-dimensional case for the equation of string oscillation written in terms of the characteristic variables

u_{xy} = 0,  (1)

the Goursat problem is formulated as follows [19, 24, 64, 75]: in a rectangular domain D_{0} : 0 < x < a, 0 < y < b find a regular solution u(x, y) of equation (1) of the class C(\overline{D}_{0}), satisfying on the segments of characteristics γ_{1} : y = 0, 0 ≤ x ≤ a and γ_{2} : x = 0, 0 ≤ y ≤ b the following boundary conditions

u|_{γ_{i}} = f_{i}, i = 1, 2,  (2)

where f_{i}, i = 1, 2, are given real functions satisfying the agreement condition f_{1}(O) = f_{2}(O) at the origin O(0, 0).

The solution u(x, y), continuous together with its partial derivatives u_{x}, u_{y} and u_{xy}, is called regular in the domain D_{0} solution of equation (1).

To solve the problem (1), (2) the use can be made of the well-known Asgeirsson’s mean value theorem [17] which in the case of equation (1) is formulated as follows: if Ω : a_{1} ≤ x ≤ a_{2}, b_{1} ≤ y ≤ b_{2} is a characteristic rectangle wholly contained in \overline{D}_{0}, then for any regular solution u(x, y) of equation (1) of the class C(\overline{D}_{0}), the equality

u(A) + u(C) = u(B) + u(K),  (3)

is valid, where A(a_{1}, b_{1}), B(a_{1}, b_{2}), C(a_{2}, b_{2}), K(a_{2}, b_{1}) are vertices of the rectangle Ω.

Let M(x, y) be an arbitrary point of the domain D_{0}, and let P_{1}(x, 0) ∈ γ_{1} and Q_{1}(0, y) ∈ γ_{2} be the points of intersection with γ_{1} and γ_{2} of the characteristics of equation (1) coming out of M(x, y). Then by virtue of (3) applied to the characteristic rectangle OP_{1}MQ_{1}, the regular solution u(x, y) of the Goursat problem (1), (2) of the class C(\overline{D}_{0}), for f_{1} ∈ C^{1}(0, a] ∩ C[0, a], f_{2} ∈ C^{1}(0, b] ∩ C[0, b] is given by the formula

u(M) = f_{1}(P_{1}) + f_{2}(Q_{1}) − f_{1}(O).  (4)

Let us now consider the Darboux problems [6, 19] for equation (1). Denote by D_{1} the domain lying at the angle x > 0, y > 0 and bounded by the characteristics μ_{1} : y = 0, 0 ≤ x ≤ a, l_{1} : y = b, kb ≤ x ≤ a, l_{2} : x = a, 0 ≤ y ≤ b of equation (1) and by a non-characteristic curve μ_{2} : x = ky, 0 ≤ y ≤ b, where a, b and k are positive constants with kb < a.

The first Darboux problem: find in D_{1} a regular solution u(x, y) of equation (1) of the class C(\overline{D}_{1}) satisfying the boundary conditions

u|_{μ_{i}} = f_{i}, i = 1, 2,  (5)

where f_{1} and f_{2} are given real functions belonging respectively to the classes C^{1}(0, a) ∩ C[0, a] and C^{1}(0, b) ∩ C[0, b], and satisfying f_{1}(O) = f_{2}(O).

If M(x, y) is an arbitrary point of D_{1}, then by P_{1}(x, 0) ∈ μ_{1} and Q_{1}(ky, y) ∈ μ_{2} we denote the points of intersection with the curves μ_{1} and μ_{2} of characteristics of equation (1) coming out of M(x, y). Let P_{2}(ky, 0) ∈ μ_{1} be the point of intersection with μ_{1} of the characteristic coming out of Q_{1}.

Applying equality (3) to the characteristic rectangle P_{2}Q_{1}MP_{1}, we obtain for the regular solution u(x, y) of the first Darboux problem (1), (5) the following formula

u(M) = f_{1}(P_{1}) + f_{2}(Q_{1}) − f_{1}(P_{2}).  (6)

Denote now by D_{2} the domain lying at the angle x > 0, y > 0 and bounded by the characteristics l_{3} : y = b, k_{2}b ≤ x ≤ a, l_{4} : x = a, k_{1}a ≤ y ≤ b of equation (1) and by non-characteristic curves σ_{1} : y = k_{1}x, 0 ≤ x ≤ a, σ_{2} : x = k_{2}y, 0 ≤ y ≤ b, where a, b and k_{i}, i = 1, 2, are positive constants satisfying k_{1}a < b and k_{2}b < a.

The second Darboux problem: find in D_{2} a regular solution u(x, y) of equation (1) of the class C(\overline{D}_{2}) satisfying on the curves σ_{1} and σ_{2} the boundary conditions

u|_{σ_{i}} = f_{i}, i = 1, 2,  (7)

where f_{i}, i = 1, 2, are given real functions belonging to the same classes as in the case of the problem (1), (5), and f_{1}(O) = f_{2}(O).

Remark. It is seen from the formulas (4) and (6) that the value of the solution u(x, y) of both the Goursat problem (1), (2) and the first Darboux problem (1), (5) at a point M(x, y) depends on the values of functions f_{1}, f_{2} at a finite number of points. At the same time, as it will be seen below, the value of the solution u(x, y) of the second Darboux problem (1), (7), if it exists, will depend on the values of functions f_{1}, f_{2} at an infinite number of points convergent to zero.

Let M_{0}(x_{0}, y_{0}) be an arbitrary point of D_{2}. By L_{1}(M_{0}) and L_{2}(M_{0}) we denote, respectively, the characteristics x = x_{0} and y = y_{0} of equation (1) passing through M_{0}. Let P_{1} ∈ σ_{1} and Q_{1} ∈ σ_{2} be the points of intersection of the characteristics L_{1}(M_{0}) and L_{2}(M_{0}) of equation (1) with the curves σ_{1} and σ_{2}. If the points P_{n−1} ∈ σ_{1} and Q_{n−1} ∈ σ_{2} are well determined, then by P_{n} ∈ σ_{1} and Q_{n} ∈ σ_{2} we denote the points of intersection of the characteristics L_{1}(Q_{n−1}) and L_{2}(P_{n−1}) with σ_{1} and σ_{2}, respectively. Continuing this process, we shall get the sequences P_{1}, P_{2}, …, P_{n}, … and Q_{1}, Q_{2}, …, Q_{n}, … of points lying respectively on σ_{1} and σ_{2} and tending for n → ∞ to the origin O.

Denote by M_{n} ∈ D_{2} the point of intersection of the characteristics L_{2}(P_{n}) and L_{1}(Q_{n}). Obviously, the sequence of points M_{n} also tends to the origin O for n → ∞. Without restriction of generality we can assume u(O) = f_{1}(O) = f_{2}(O) = 0, since otherwise the function v = u−f_{1}(O) is considered as a new unknown function.

Applying (3) to the rectangle M_{n−1}P_{n}M_{n}Q_{n}, we obtain

u(M_{n−1}) = f_{1}(P_{n}) + f_{2}(Q_{n}) − u(M_{n}), n = 1, 2, ….  (8)

From (8) we have

u(M_{0}) = ∑_{i=1}^{n}(−1)^{i+1}[f_{1}(P_{i}) + f_{2}(Q_{i})] + (−1)^{n}u(M_{n}).  (9)

If the problem (1), (7) is solvable, then passing in (9) to the limit for n → ∞ and taking into account that lim_{n→∞} u(M_{n}) = u(O) = f_{1}(O), we get that the series

I = ∑_{i=1}^{∞}(−1)^{i+1}[f_{1}(P_{i}) + f_{2}(Q_{i})]  (10)

converges. Thus the convergence of (10) is necessary and sufficient for the problem (1), (7) to be solvable in the class of regular solutions introduced above.

Passage in (9) to the limit for n → ∞ when f_{1} = f_{2} = 0 also shows that in the class of regular solutions the second Darboux problem cannot have more than one solution.

Now let us show that the series (10) converges not for all functions f_{1} and f_{2} from the above mentioned classes. For the sake of simplicity let a = b = 1, 0 < k_{1} = k_{2} = k < 1, f_{2} ≡ 0, and let x_{0} = y_{0} = 1 be the coordinates of M_{0}. As a function f_{1} = f_{1}(x) of the class C^{1}(0, 1) ∩ C[0, 1], we take

f_{1}(x) = \frac{cos(π \frac{ln x}{ln k})}{ln \frac{1}{2}x} .

In this case (10) takes the form

I = ∑_{i=1}^{∞} \frac{1}{ln \frac{1}{2}k^{i−1}} = ∑_{i=1}^{∞} \frac{1}{(i − 1) ln k + ln \frac{1}{2}} ,

and, obviously, diverges.

Since in (10)

lim_{n→∞} P_{n} = lim_{n→∞} Q_{n} = O,

to ensure convergence of this series we additionally require of the functions f_{1} and f_{2} to be regular in a neighborhood of O. For example, it suffices to require that

f_{1} ∈ C^{1}(0, a] ∩ C[0, a], f_{2} ∈ C^{1}(0, b] ∩ C[0, b]

and for some α, 0 < α = const < 1, the first order derivatives of these functions have integrable at O singularities of the type

|f_{1}^{(1)}(x)| ≤ \frac{C}{x^{α}} , |f_{2}^{(1)}(y)| ≤ \frac{C}{y^{α}} , C = const > 0.  (11)

In this case, the series (10) and that obtained from (10) by termwise differentiation with respect to x or y converge uniformly in D_{2} and the regular solution of the problem (1), (7) is given by the formula

u(M_{0}) = ∑_{i=1}^{∞}(−1)^{i+1}[f_{1}(P_{i}) + f_{2}(Q_{i})].

The solution and its partial derivatives with respect to x and y satisfy in a neighborhood of O the estimates

\begin{matrix}|u(x, y)| ≤ C_{1}(|x| + |y|)^{1−α}, |u_{x}(x, y)| ≤ \frac{C_{1}}{(|x| + |y|)^{α}} , \\ |u_{y}(x, y)| ≤ \frac{C_{1}}{(|x| + |y|)^{α}} , C_{1} = const > 0. \end{matrix}  (12)

Thus, to ensure the solvability of the second Darboux problem (1), (7), we have naturally come to the consideration of weighted spaces defined by inequalities (11) for the functions f_{1}, f_{2} and by inequalities (12) for the regular solutions of equation (1).

Chapter I of the present paper deals with the boundary value problems for equation (1) which are formulated more generally than the above-mentioned Goursat and Darboux type problems.

The results obtained for equation (1) are in a definite sense complete and simple by form and serve as a visual model for investigation of boundary value problems for second order hyperbolic systems with two independent variables.

Let γ_{1} : y = γ_{1}(x), 0 ≤ x ≤ x_{0}, and γ_{2} : x = γ_{2}(y), 0 ≤ y ≤ y_{0}, be the two simple smooth curves coming out of the origin O and lying wholly at the angle x ≥ 0, y ≥ 0. Below it is assumed that the functions γ_{1}(x) and γ_{2}(y) are monotonically non-decreasing, i.e., γ_{1}^{(1)}(x) ≥ 0, γ_{2}^{(1)}(y) ≥ 0, and γ_{1}(γ_{2}(y)) < y for 0 < y ≤ y_{0}. Denote by D the domain lying at the angle x > 0, y > 0 bounded by the curves γ_{1}, γ_{2} and the characteristics L_{1}(P_{0}) : x = x_{0} and L_{2}(P_{0}) : y = y_{0} coming out of the point P_{0}(x_{0}, y_{0}).

Consider the boundary value problem formulated as follows [29]: find in the domain D a regular solution u(x, y) of equation (1) satisfying on the curves γ_{1} and γ_{2} the following conditions

(M_{i}u_{x} + N_{i}u_{y})|_{γ_{i}} = f_{i}, i = 1, 2,  (13)

where M_{i}, N_{i}, f_{i}, i = 1, 2, are given real functions.

Remark. Note that the Goursat and Darboux type problems considered above are reduced to a problem of the type (1), (13) by differentiating the corresponding boundary conditions along the tangents of data carriers of these problems.

The solution of the problem (1),(13) is sought in the following weight space

C_{α}^{1,1}(\overline{D}) = \{u ∈ C(\overline{D}) : u_{x}, u_{y}, u_{xy} ∈ C(\overline{D}\O), u(0, 0) = 0,

sup_{z∈\overline{D}\O} |z|^{−α}|u_{x}(z)| < ∞, sup_{z∈\overline{D}\O} |z|^{−α}|u_{y}(z)| < ∞,

sup_{z∈\overline{D}\O} |z|^{−(α−1)}|u_{xy}(z)| < ∞\},

where z = x + iy, i = \sqrt{−1}, and α > −1 is a real parameter.

Obviously, if u ∈ C_{α}^{1,1}(\overline{D}), then sup_{z∈\overline{D}\O} |z|^{−(1+α)}|u(z)| < ∞.

If the solution u(x, y) of the problem (1), (13) is sought in the space C_{α}^{1,1}(\overline{D}), then we require of the boundary functions f_{1}, f_{2} that

f_{1}(x) ∈ C_{α}(γ_{1}) = {f_{1} ∈ C(0, x_{0}] : sup_{0<x≤x_{0}} |x^{−α}f_{1}(x)| < ∞},

f_{2}(y) ∈ C_{α}(γ_{2}) = {f_{2} ∈ C(0, y_{0}] : sup_{0<y≤y_{0}} |y^{−α}f_{2}(y)| < ∞}.

It is shown that the correctness of the problem (1), (13) in the class C_{α}^{1,1}(\overline{D}) depends essentially on the parameter α, as well as on the angle between the supports of boundary data γ_{1} and γ_{2} at the common point O and their configuration [29]. For example, if the curves γ_{1}, γ_{2} are not characteristics of equation (1), do not have a common tangent line at O, and M_{i}|_{γ_{i}} ≠ 0, N_{i}|_{γ_{i}} ≠ 0, i = 1, 2, then for α > −\frac{ln |σ|}{ln τ_{0}} the problem (1),

(13) is uniquely solvable in the class C_{α}^{1,1}(\overline{D}), while for α < −\frac{ln |σ|}{ln τ_{0}} the homogeneous problem corresponding to (1), (13) has an infinite number of linearly independent solutions, where σ = (M_{1}^{−1}N_{1}M_{2}N_{2}^{−1})(O), 0 < τ_{0} = γ_{1}^{(1)}(0)γ_{2}^{(1)}(0) < 1.

In the case where the curves γ_{1}, γ_{2} have the same tangent line at O, i.e. τ_{0} = γ_{1}^{(1)}(0)γ_{2}^{(1)}(0) = 1 and M_{i}|_{γ_{i}} ≠ 0, N_{i}|_{γ_{i}} ≠ 0, i = 1, 2, then for |σ| < 1 the problem (1), (13) is uniquely solvable in the class C_{α}^{1,1}(\overline{D}), while for |σ| > 1 the homogeneous problem corresponding to (1), (13) has an infinite number of linear independent solutions [29]. We should also note the work [53] in which sufficient conditions for unique solvability of the problem (1), (13) in the class C^{2}(\overline{D}) are obtained in the case where γ_{1} and γ_{2} are segments of non-characteristic straight lines coming out of the common point O. The case |σ| = 1 which corresponds to the case where the directions of differentiation operators \frac{∂}{∂l_{i}} = M_{i} \frac{∂}{∂x} + N_{i} \frac{∂}{∂y}, i = 1, 2, appearing in the boundary conditions (13) coincide at the point O, turned out to be more complicated. More interesting results in this direction are obtained by T. M. Makharadze [51, 52]. He has established that the correctness of formulation of the problem under consideration depends on the parameter α, the order of tangency of the curves γ_{1}, γ_{2} and the directions of differentiation operators \frac{∂}{∂l_{1}} , \frac{∂}{∂l_{2}} at O. The results of Firmani concerning the second Darboux problem in the case where the curves σ_{1} and σ_{2} have a common tangent line at O are also worth mentioning [20–22].

In the same chapter it is shown that when condition M_{1}(x, y) ≠ 0 or N_{2}(x, y) ≠ 0 is violated on the whole curve γ_{1} or γ_{2}, the existence of the lowest terms in this problem may affect the correctness of formulation of the problem (1), (13). The case where condition M_{1}(x, y) ≠ 0 or N_{2}(x, y) ≠ 0 violates at one point O only, is also considered. In this case, in the class C_{α}^{1,1}(\overline{D}) the homogeneous problem corresponding to (1), (13) has an infinite number of linearly independent solutions. At the same time, the functional space C_{α,χ}(\overline{D}) is determined such that the problem (1), (13) is uniquely solvable.

Additional difficulties arise when we pass to second order hyperbolic systems. This has been first shown by A. V. Bitsadze [7] who constructed examples of second order hyperbolic systems for which the corresponding homogeneous characteristic problem (the Goursat problem with data on the characteristics) has a finite or even an infinite number of linearly independent solutions. Characteristic problem for second order hyperbolic systems with two independent variables and constant leading coefficients has been investigated in the works of the author [30–32]. In particular, these works reveal new effects connected with the problems of smoothness of solutions and the possibility for the characteristic problem to have a non-zero finite index. Simple examples of second order hyperbolic systems in A. V. Bitsadze’s work [8] illustrate how the lowest terms affect the correctness of formulation of the characteristic problems.

S. L. Sobolev [68], V. P. Mikhaǐlov [58, 59] and L. A. Mel’tser [55] investigated some analogues of the Goursat type problem in the case of first order hyperbolic systems with two independent variables.

Chapter II deals with the boundary value problems for second order linear normal hyperbolic systems with variable coefficients of the type

Au_{xx} + 2Bu_{xy} + Cu_{yy} + A_{1}u_{x} + B_{1}u_{y} + C_{1}u = F

in the weighted spaces C^{∘}_{α}^{k}(\overline{D}) [33–37, 54]. Boundary conditions in these problems are determined by a first order differential operator, while the carrier of these conditions are the two arcs γ_{1} and γ_{2} with a common point at the origin. The sufficient conditions imposed both on the coefficients of the system and on the curves γ_{1}, γ_{2} ensuring correctness of the problems in

the spaces C^{∘}_{α}^{k}(\overline{D}) are also given in the same chapter. The structure of the domain of definition of the solution is determined depending on the location of data carriers with respect to the characteristics of the system.

Characteristic problems for second order linear hyperbolic systems of the types

y^{m}Au_{xx} + 2y^{\frac{m}{2}} Bu_{xy} + Cu_{yy} + au_{x} + bu_{y} + cu = F  (14)

and

Au_{xx} + 2y^{\frac{m}{2}} Bu_{xy} + y^{m}Cu_{yy} + au_{x} + bu_{y} + cu = F  (15)

with parabolic degeneration along the straight line y = 0 are studied in Chapter III. Boundary conditions in these problems are determined by means of Goursat type data, while the carrier of these conditions are the two arcs of adjoint characteristics of the system coming out of the point of parabolic degeneration. Under certain conditions imposed on the coefficients of the system and boundary operator, we prove theorems on the unique solvability of these problems in special weighted spaces determined with regard to the character of parabolic degeneration [38–40]. The condition obtained in this case and imposed on the lowest terms of the system is an exact analogue of the well-known Gellerstedt’s condition for one equation.

It should be noted that the characteristic problem with boundary conditions u|_{γ_{i}} = f_{i}, i = 1, 2, on segments of characteristics γ_{1} and γ_{2} coming out of the origin O(0, 0), has been investigated by L. Sh. Agababyan and A. B. Nersesyan [1–3] for one second order hyperbolic equation with parabolic degeneration of the type

y^{m}u_{xx} − u_{yy} + au_{x} + bu_{y} + cu = F

in a rectangle bounded by characteristics of that equation coming out of the points O(0, 0) and P (0, 1). The characteristic problem for the equation

y^{2}u_{xx} − u_{yy} + au_{x} = 0

has been studied by T. Sh. Kalmenov [27] in a triangular domain bounded by the segment [0, 1] of the axis x and by pieces of characteristics coming out of the points O(0, 0) and Q_{1}(1, 0). Note also the works of V. N. Vragov [76] and B. A. Bubnov [15] where, in particular, the characteristic problem in domains containing a segment of a line of degeneration is studied. The case when OP_{1} is a segment of a axis x and OP_{2} is that of a characteristic of one hyperbolic equation with parabolic degeneration for y = 0, has been studied in the works of V. N. Vragov [76] and A. M. Nakhushev [60–62], while for the systems of the type

K(y)u_{xx} − Eu_{yy} + au_{x} + bu_{y} + cu = F,

this case has been studied by M. Meredov [56, 57].

In this chapter, the class of hyperbolic systems of the type (14) and (15) for which characteristic problems are investigated, contains the systems with non-split principal parts and the higher term 2y^{\frac{m}{2}} Bu_{xy} different from zero.

The last Chapter IV concerns with certain multidimensional variants of Goursat and Darboux type problems for linear hyperbolic differential equations.

If in the two-dimensional case the problems of the Goursat and Darboux type for hyperbolic equations and systems are investigated with sufficient completeness, in the multidimensional case we have in this direction only individual results. One of the main reasons is probably the existence of a continual bundle of bicharacteristics of a hyperbolic equation, owing to which, in particular, to ensure the correctness of this or that problem, one should require definite orientation of data supports.

A multidimensional analogue of the Goursat problem (the Cauchy characteristic problem) when the solution of a second order hyperbolic equation is sought inside a characteristic conoid, has been studied by D’Adhemar [18], Hadamard [25], S. L. Sobolev [69], Riesz [67], Lundberg [50], A. A. Borgardt and D. A. Karnenko [14]. In the case when a second order hyperbolic system is split in its principal part, the same problem has been investigated by Cagnac [16] in the four-dimensional space.

It should be noted that the Cauchy characteristic problem for a non-split in the principal part second order hyperbolic system has not been studied so far. Here, alongside with technical difficulties, there arise principal algebraic difficulties connected with determination of geometric structure of a characteristic conoid in a vicinity of the vertex.

Certain multidimensional analogues of the first or the second Darboux problems are treated by C. L. Sobolev [70], Gårding [23], A. V. Bitsadze [9], V. N. Vragov [76], T. Sh. Kalmenov [28] and Rassias [65, 66] for the case where the solution of a second order hyperbolic equation is sought in a conic domain, one part of whose boundary is of time-type and the other is either characteristic or wholly of time-type. One variant of the second Darboux problem in a conic domain of time-type is studied by S. S. Kharibegashvili in the case where a second order hyperbolic system is nonsplit in its principal part and for one hyperbolic equation of higher order with constant coefficients at higher derivatives [41–43]. Note that for general hyperbolic equations and systems both variants of the Darboux problems in conic domains are not treated.

Other multidimensional analogues of the Goursat and Darboux problems for one second order hyperbolic equation in a bihedral angle when either both sides are characteristic or one side is characteristic and the other is a hypersurface of time-type, have been considered in the works of Beudon [5], Hadamard [25], Tolen [71] and S. S. Kharibegashvili [44–46]. The second Darboux problem when both sides are hypersurfaces of time-type is more complicated. This case is considered by S. S. Kharibegashvili in [47].

In Chapter IV we shall restrict ourselves to the statement of the results obtained in the course of investigation of multidimensional analogues of the Goursat and Darboux problems for the second order hyperbolic equation with the wave operator in its principal part in a bihedral angle of a quite definite orientation [44–47]. The final paragraph of this chapter concerns with a multidimensional variant of the second Darboux problem for a higher order hyperbolic equation with constant coefficients at higher derivatives in a conic domain located fully in the interior cone of rays [43].

CHAPTER I

BOUNDARY VALUE PROBLEMS FOR

A SECOND ORDER HYPERBOLIC EQUATION

§ 1. Statement of the Problem and Its Reduction to the Functional Equation

In the plane of variables x, y let us consider a second order hyperbolic equation of the type

u_{xy} + a_{1}u_{x} + b_{1}u_{y} + c_{1}u = F,  (1.1)

where a_{1}, b_{1}, c_{1}, F are given real functions and u is an unknown one.

Let γ_{1} : y = γ_{1}(x), 0 ≤ x ≤ x_{0}, and γ_{2} : x = γ_{2}(y), 0 ≤ y ≤ y_{0}, be two simple curves of the class C^{1} coming out of the origin O(0, 0) of the plane of variables x, y and located completely in the angle x ≥ 0, y ≥ 0.

Below we shall assume that γ_{1}(γ_{2}(y)) < y, 0 < y ≤ y_{0}, and each of the curves γ_{i}, i = 1, 2, either is a characteristic of equation (1.1) or it has characteristic direction at none of its points, except maybe O(0, 0). This implies that if γ_{1} (γ_{2}) is not a characteristic, then the function y = γ_{1}(x) (x = γ_{2}(y)) is strictly monotonically increasing. Denote by D the domain lying at the angle x > 0, y > 0, bounded by the curves γ_{1}, γ_{2} and the characteristics L_{1}(P_{0}) : x = x_{0} and L_{2}(P_{0}) : y = y_{0} of equation (1.1), coming out of the point P_{0}(x_{0}, y_{0}).

Consider the boundary value problem formulated as follows: in the domain D find a regular solution u(x, y) of (1.1) satisfying on γ_{1} and γ_{2}

(M_{i}u_{x} + N_{i}u_{y} + S_{i}u)|_{γ_{i}} = f_{i}, i = 1, 2,  (1.2)

where M_{i}, N_{i}, S_{i}, f_{i}, i = 1, 2, are given real functions.

The solution of the problem (1.1), (1.2) is sought in the weighted space

C_{α}^{1,1}(\overline{D}) = \{u ∈ C(\overline{D}) : u_{x}, u_{y}, u_{xy} ∈ C(\overline{D}\O), u(0, 0) = 0,

sup_{z∈\overline{D}\O} |z|^{−α}|u_{x}(z)| < ∞, sup_{z∈\overline{D}\O} |z|^{−α}|u_{y}(z)| < ∞,

sup_{z∈\overline{D}\O} |z|^{−(α−1)}|u_{xy}(z)| < ∞\},

where z = x + iy, i = \sqrt{−1}, α > −1 is a real parameter.

Obviously, if u ∈ C_{α}^{1,1}(\overline{D}), then sup_{z∈\overline{D}\O} |z|^{−(1+α)}|u(z)| < ∞.

When considering the problems (1.1), (1.2) in the space u ∈ C_{α}^{1,1}(\overline{D}), we require that a_{1}, b_{1}, c_{1} ∈ C(\overline{D}), M_{i}, N_{i}, S_{i} ∈ C(γ_{i}), i = 1, 2,

f_{1}(x) ∈ C_{α}(γ_{1}) = {f_{1} ∈ C(0, x_{0}] : sup_{0<x≤x_{0}} |x^{−α}f_{1}(x)| < ∞},

f_{2}(x) ∈ C_{α}(γ_{2}) = {f_{2} ∈ C(0, y_{0}] : sup_{0<y≤y_{0}} |y^{−α}f_{2}(y)| < ∞},

F (z) ∈ C_{α−1}(\overline{D}) = \{F ∈ C(\overline{D}\O) : sup_{z∈\overline{D}\O} |x^{−(α−1)}|F (z)| < ∞\}.

For the sake of simplicity, we shall restrict ourselves to the consideration of the equation of the string oscillation

u_{xy} = 0,  (1.3)

and in the boundary conditions (1.2) we shall assume S_{i} = 0, i = 1, 2, i.e.,

(M_{i}u_{x} + N_{i}u_{y})|_{γ_{i}} = f_{i}, i = 1, 2.  (1.4)

Denoting v = u_{x} and w = u_{y}, we can rewrite the problem (1.3), (1.4) equivalently in the form

v_{y} = 0,  (1.5)
w_{x} = 0,  (1.6)
u_{y} = w  (1.7)

with boundary conditions

(M_{i}v + N_{i}w)|_{γ_{i}} = f_{i}, i = 1, 2,  (1.8)
(u_{x} + γ_{1}^{(1)}u_{y})|_{γ_{1}} = (v + γ_{1}^{(1)}w)|_{γ_{1}}.  (1.9)

Indeed, if u(x, y) is a solution of the problem (1.3), (1.4), then it is clear that the system of functions u, v and w satisfies (1.5)–(1.9). Conversely, let u, v, w be a solution of the problem (1.5)–(1.9). Then, obviously, equalities u_{x} = v, w = u_{y} imply that u(x, y) is a solution of the problem (1.3), (1.4). Therefore, by virtue of (1.7) it suffices to prove that u_{x} = v.

Let g = v − u_{x}. Then owing to (1.5)–(1.7), we have

g_{y} = v_{y} − u_{xy} = 0 − (u_{y})_{x} = 0 − w_{x} = 0.

Hence g(x, y) = g(x), i.e.,

g(P) = g(P^{∗}) = (v − u_{x})|_{γ_{1}},  (1.10)

where P^{∗} is the projection of an arbitrarily taken point P (x, y) ∈ D on the curve γ_{1}, parallel to the axis Oy.

By (1.7) and (1.9) we have

(u_{x} + γ_{1}^{(1)}u_{y})|_{γ_{1}} = (u_{x} + γ_{1}^{(1)}w)|_{γ_{1}} = (v + γ_{1}^{(1)}w)|_{γ_{1}},

whence u_{x}|_{γ_{1}} = v|_{γ_{1}} and, according to (1.10), we get g ≡ 0 which means that u_{x} = v in D.

Denoting v|_{γ_{1}} = φ(x) and w|_{γ_{2}} = ψ(y), we rewrite the boundary conditions (1.8) as a system of two functional equations

M_{1}φ(x) + N_{1}ψ(γ_{1}(x)) = f_{1}(x), 0 < x ≤ x_{0},  (1.11)
M_{2}φ(γ_{2}(y)) + N_{2}ψ(y) = f_{2}(y), 0 < y ≤ y_{0},  (1.12)

with respect to the unknown functions (φ, ψ) ∈ C_{α}(γ_{1}) × C_{α}(γ_{2}).

Evidently, if φ(x) and ψ(y) are a solution of the system (1.11), (1.12), then the functions u, v and w of the problem (1.5)–(1.9) can be uniquely defined by the formulas

v(x, y) = φ(x), w(x, y) = ψ(y), u(x, y) = ∫_{OP} vdx + wdy,

where OP ⊂ D is a curve connecting the point P (x, y) ∈ D with the origin O(0, 0).

Below we shall assume that

M_{1}|_{γ_{1}} ≠ 0, N_{2}|_{γ_{2}} ≠ 0.  (1.13)

Excluding in the system (1.11), (1.12) the unknown function ψ(y), for φ(x) we obtain the functional equation

T φ ≡ φ(x) − a(x)φ(τ(x)) = f(x), 0 < x ≤ x_{0}.  (1.14)

Here

a(x) = M_{1}^{−1}(x)N_{1}(x)N_{2}^{−1}(γ_{1}(x))M_{2}(γ_{1}(x)),  (1.15)
τ(x) = γ_{2}(γ_{1}(x)),  (1.16)
f(x) = M_{1}^{−1}(x)f_{1}(x) − M_{1}^{−1}(x)N_{1}(x)N_{2}^{−1}(γ_{1}(x))f_{2}(γ_{1}(x)).

Remark. It is obvious that when the conditions (1.13) are fulfilled, the problem (1.3), (1.4) in the class C_{α}^{1,1}(\overline{D}) is equivalently reduced to one functional equation (1.14) with respect to the unknown function φ(x) of the class C_{α}(0, x_{0}].

§ 2. The Case Where the Curves γ_{1} and γ_{2} do not Have a Common Tangent Line at the Point O(0, 0)

Let τ_{0} = τ^{(1)}(0). If the curves γ_{1} and γ_{2} do not have a common tangent line at O(0, 0), then due to the requirements imposed on γ_{1} and γ_{2} we have 0 ≤ τ_{0} < 1, where τ_{0} = 0 if and only if one of the curves γ_{1} or γ_{2} has a characteristic direction at this point.

Lemma 1.1. If at least one of the curves γ_{1} or γ_{2} is a characteristic of equation (1.3), then equation (1.14) is uniquely solvable in the class C_{α}(0, x_{0}] for α > 0.

Proof. Obviously, in this case τ(x) ≡ 0. Moreover, since α > 0, from φ(x) (f(x)) ∈ C_{α}(0, x_{0}] we have φ(x) (f(x)) ∈ C[0, x_{0}] and φ(0) = 0 (f(0) = 0). Therefore in this case equation (1.14) takes the trivial form

φ(x) = f(x). ■

Let now γ_{1} and γ_{2} not be characteristics of (1.3). Then according to the requirements imposed on γ_{1} and γ_{2}, the continuously differentiable function τ(x) defined by (1.16) is strictly monotonically increasing on [0, x_{0}] and

τ(0) = 0, 0 < τ(x) < x for 0 < x ≤ x_{0}.  (1.17)

Therefore if τ_{k}(x) = τ(τ_{k−1}(x)), τ_{1}(x) = τ(x), 0 ≤ x ≤ x_{0}, then according to (1.17) the sequence {τ_{k}(x)}_{k=1}^{∞} on the interval [0, x_{0}] tends uniformly to zero, as k → ∞. Hence there exists a natural number n such that

τ_{k}(x) ≤ ε, 0 ≤ x ≤ x_{0}, k ≥ n.  (1.18)

Lemma 1.2. Let equation (1.14) be uniquely solvable on the interval (0, ε], 0 < ε = const < x_{0}, in the class C_{α}(0, ε]. Then equation (1.14) is likewise uniquely solvable on the whole interval (0, x_{0}] in the class C_{α}(0, x_{0}], and its solution φ(x) can be represented in the form

φ(x) = \begin{cases}φ_{0}(x), & 0 < x ≤ ε, \\ (Λ^{n}φ_{0})(x) + f(x) + ∑_{i=1}^{n−1}(Λ^{i}f)(x), & x > ε, \end{cases} (1.19)

where φ_{0}(x) is the solution of equation (1.14) on (0, ε] of the class C_{α}(0, ε], (Λ^{n}φ)(x) = a(x)a(τ(x)) ⋯ a(τ_{n−1}(x))φ(τ_{n}(x)), and the number n is chosen by inequality (1.18).

Proof of Lemma 1.2 is trivial.

The following lemma is obvious.

Lemma 1.3. In the class C_{α}(0, x_{0}], (1.14) is equivalent to the equation

ψ(x) − a(x)(\frac{τ(x)}{x})^{α}ψ(τ(x)) = g(x), 0 < x ≤ x_{0},  (1.20)

in the class C_{0}(0, x_{0}], where ψ(x) = x^{−α}φ(x) ∈ C_{0}(0, x_{0}], g(x) = x^{−α}f(x) ∈ C_{0}(0, x_{0}].

Lemmas 1.2 and 1.3 immediately yield

Lemma 1.4. Equation (1.14) is uniquely solvable in the class C_{α}(0, x_{0}] if and only if equation (1.20) is uniquely solvable for some ε, 0 < ε < x_{0}, in the class C_{0}(0, ε].

Lemma 1.5. Let the curves γ_{1}\O and γ_{2}\O not be characteristics of equation (1.3) and at least one of them have characteristic direction at the point O. Then equation (1.14) is uniquely solvable in the class C_{α}(0, x_{0}] for α > 0. If, however, −1 < α ≤ 0, then (1.14) is uniquely solvable in the class C_{α}(0, x_{0}] when the condition

\overline{lim}_{x→+0}|a(x)(\frac{τ(x)}{x})^{α}| < 1  (1.21)

is fulfilled.

Proof. By virtue of Lemma 1.4, it suffices to prove that for sufficiently small ε > 0 the operator

(T_{0}ψ)(x) = a(x)(\frac{τ(x)}{x})^{α}ψ(τ(x)),  (1.22)

appearing in (1.20) has in the space C_{0}(0, ε] the norm which is less than unity, i.e.,

∥T_{0}∥_{C_{0}(0,ε]→C_{0}(0,ε]} < 1.  (1.23)

Really, in this case the Neumann series

(I − T_{0})^{−1} = I + T_{0} + ⋯ + T_{0}^{n} + ⋯

for the operator T_{0} converges in the space C_{0}(0, ε] and the unique solution ψ(x) of (1.20) can be represented in the form

ψ = f + T_{0}f + ⋯ + T_{0}^{n}f + ⋯,

where I is an identical operator.

In the first case, when α > 0 and at least one of the curves γ_{1} or γ_{2} has the characteristic direction at O, we have τ_{0} = τ^{(1)}(0) = 0, and

lim_{x→+0} |a(x)(\frac{τ(x)}{x})^{α}| = lim_{x→+0} |a(x)| lim_{x→+0} (\frac{τ(x)}{x})^{α} =

= |a(0)|(τ^{(1)}(0))^{α} = 0.

Therefore, since the function a(x)(\frac{τ(x)}{x})^{α} is continuous in a vicinity of zero, there exists a sufficiently small number ε > 0 such that for 0 < x ≤ ε we have

max_{0<x≤ε} |a(x)(\frac{τ(x)}{x})^{α}| ≤ q = const < 1,

whence we get

∥T_{0}ψ∥_{C_{0}(0,ε]} = sup_{0<x≤ε} |a(x)(\frac{τ(x)}{x})^{α}ψ(τ(x))| ≤

≤ q sup_{0<x≤ε} |ψ(τ(x))| ≤ q sup_{0<x≤ε} |ψ(x)| = q∥ψ∥_{C_{0}(0,ε]},

i.e.,

∥T_{0}∥_{C_{0}(0,ε]→C_{0}(0,ε]} ≤ q < 1.

In the second case, when −1 < α ≤ 0 and (1.21) is fulfilled, the estimate (1.23) for the norm of the operator T_{0} defined by (1.22) can be proved analogously. ■

Let now the curves γ_{1} and γ_{2} not be characteristics of equation (1.3) and have no characteristic direction at O. In this case 0 < τ_{0} < 1. Put σ = a(0).

Lemma 1.6. Let the curves γ_{1}\O and γ_{2}\O not be the characteristics of equation (1.3) and have no characteristic direction at O. Then for α > −\frac{log |σ|}{log τ_{0}}, equation (1.14) is uniquely solvable in the class C_{α}(0, x_{0}].

Remark. In Lemma 1.6 for σ = 0, that is for N_{1}(0)M_{2}(0) = 0, one should assume −\frac{log |σ|}{log τ_{0}} = −∞, and in this case equation (1.14) is uniquely solvable for any α > −1.

Proof. It follows from the condition α > −\frac{log |σ|}{log τ_{0}} that

|σ|τ_{0}^{α} < 1,

whence we directly obtain (1.21)

\overline{lim}_{x→+0}|a(x)(\frac{τ(x)}{x})^{α}| = |σ|τ_{0}^{α} < 1

which, as is shown in Lemma 1.5, ensures the unique solvability of equation (1.14) in the class C_{α}(0, ε]. ■

Lemma 1.7. Let the curves γ_{1}\O and γ_{2}\O not be characteristics of equation (1.3) and have no characteristic direction at O. If N_{1}(0)M_{2}(0) ≠ 0, then for α < −\frac{log |σ|}{log τ_{0}} equation (1.14) is solvable in the class C_{α}(0, x_{0}], and the homogeneous equation corresponding to (1.14) has an infinite number of linearly independent solutions in this class.

Proof. Since N_{1}(0)M_{2}(0) ≠ 0, i.e. σ ≠ 0 and α < −\frac{log |σ|}{log τ_{0}}, there exists a positive number ε, ε < x_{0}, such that for 0 < x ≤ ε we have N_{1}(x) ≠ 0, M_{2}(x) ≠ 0, and

|a(x)(\frac{τ(x)}{x})^{α}| ≥ \frac{1}{q} = const > 1, 0 < x ≤ ε.  (1.24)

Since the function τ(x) is strictly monotone, for any x from the interval 0 < x < τ(ε) there exists a unique natural number n_{1} = n_{1}(x) satisfying

τ(ε) < τ^{−n_{1}}(x) ≤ ε.

Analogously, for any x satisfying ε_{1} < x ≤ x_{0} there exists a unique natural number n_{2} = n_{2}(x) such that

τ(ε) ≤ τ^{n_{2}}(x) < ε.

By virtue of Lemma 1.3, it suffices to prove that equation (1.20) is solvable in the class C_{0}(0, ε], and for the homogeneous equation corresponding to (1.20) there exists an infinite number of linearly independent solutions of this class.

Since the function τ(x) is strictly monotonically increasing, there exists a function inverse to τ(x) which we denote by τ^{−1}(x). It is easily seen that the operator T_{0} defined by (1.22) is invertible, and

(T_{0}^{−1}ψ)(x) = a^{−1}(τ^{−1}(x))(\frac{x}{τ^{−1}(x)})^{−α}ψ(τ^{−1}(x)).  (1.25)

It can be easily verified that every solution of (1.20) which is continuous in a half-interval 0 < x ≤ x_{0} is given by

ψ(x) = \begin{cases}ψ_{0}(x), & τ(ε) ≤ x ≤ ε, \\ (T_{0}^{−n_{1}(x)}ψ_{0})(x) − ∑_{i=1}^{n_{1}(x)} (T_{0}^{−i}g)(x), & 0 < x < τ(ε), \\ (T_{0}^{n_{2}(x)}ψ_{0})(x) + ∑_{i=0}^{n_{2}(x)−1} (T_{0}^{i}g)(x), & ε < x ≤ x_{0}, \end{cases} (1.26)

where ψ_{0}(x) is an arbitrary function of the class C[τ(ε), ε] satisfying the condition ψ_{0}(ε) − a(ε)(\frac{τ(ε)}{ε})^{α}ψ_{0}(τ(ε)) = g(ε).

Let us show that if g ∈ C_{0}(0, ε], the function ψ(x) given by (1.26) belongs to the class C_{0}(0, x_{0}] for any ψ_{0} ∈ C[τ(ε), ε], ψ_{0}(ε) − a(ε)(\frac{τ(ε)}{ε})^{α}ψ_{0}(τ(ε)) = g(ε). From this and owing to the the arbitrariness of the function ψ_{0}, we obtain the assertion of Lemma 1.7.

Obviously, in order to prove that ψ ∈ C_{0}(0, x_{0}], it suffices to show that the functions

(T_{0}^{−n_{1}(x)}ψ_{0})(x) and ∑_{i=1}^{n_{1}(x)}(T_{0}^{−i}g)(x)

are bounded in the interval 0 < x < τ(ε).

(1.24) and (1.25) yield

|(T_{0}^{−n_{1}(x)}ψ_{0})(x)| ≤ q^{n_{1}(x)} max_{τ(ε)≤x≤ε} |ψ_{0}(x)| < max_{τ(ε)≤x≤ε} |ψ_{0}(x)|,

| ∑_{i=1}^{n_{1}(x)} (T_{0}^{−i}g)(x)| ≤ ∑_{i=1}^{n_{1}(x)} q^{i} sup_{0<x≤x_{0}} |g(x)| < \frac{1}{1 − q} sup_{0<x≤x_{0}} |g(x)|. ■

Remark. One can prove that in the critical case where α = −\frac{log |σ|}{log τ_{0}}, equation (1.14) in the class C_{α}(0, ε] is not Hausdorff normally solvable, that is, the set of all right-hand sides f ∈ C_{α}(0, ε] for which (1.14) is solvable, is everywhere dense in C_{α}(0, ε] but not coinciding with it.

From the above proven lemmas it follows that the following theorems are valid.

Theorem 1.1. Let the conditions (1.13) be fulfilled and at least one of the curves γ_{1} or γ_{2} be characteristics of equation (1.3). Then the problem (1.3), (1.4) is uniquely solvable in the class C_{α}^{1,1}(\overline{D}) for α > 0.

Theorem 1.2. Let the conditions (1.13) be fulfilled, the curves γ_{1}\O and γ_{2}\O not be characteristics of equation (1.3) and at least one of them have characteristic direction at O. Then the problem (1.3), (1.4) is uniquely solvable in the class C_{α}^{1,1}(\overline{D}) for α > 0. If, however, −1 < α ≤ 0, then the problem (1.3), (1.4) is uniquely solvable in the class C_{α}^{1,1}(\overline{D}), when the condition

\overline{lim}_{x→+0}|a(x)(\frac{τ(x)}{x})^{α}| < 1

is fulfilled.

Theorem 1.3. Let the conditions (1.13) be fulfilled, the curves γ_{1}\O and γ_{2}\O not be characteristics of equation (1.3) and have no characteristic direction at O. If N_{1}(0)M_{2}(0) = 0, then the problem (1.3), (1.4) is uniquely solvable in the class C_{α}^{1,1}(\overline{D}) for α > −1.

Theorem 1.4. Let the conditions (1.13) be fulfilled, the curves γ_{1}\O and γ_{2}\O not be characteristics of equation (1.3) and have no characteristic direction at O. If N_{1}(0)M_{2}(0) ≠ 0, then for α > −\frac{log |σ|}{log τ_{0}} the problem (1.3),

(1.4) is uniquely solvable in the class C_{α}^{1,1}(\overline{D}), while for α < −\frac{log |σ|}{log τ_{0}} it is solvable in the class C_{α}^{1,1}(\overline{D}), and the homogeneous problem corresponding to (1.3), (1.4) has an infinite number of linearly independent solutions in this class.

Remark. Using Picard’s method of successive approximations, one can prove that the assertions of Theorems 1.1–1.3 and those of the first part of Theorem 1.4 are also valid for the problem (1.1), (1.2) in the class C_{α}^{1,1}(\overline{D}); moreover, the estimate

∥u∥_{C_{α}^{1,1}(\overline{D})} ≤ C( ∑_{i=1}^{2} ∥f_{i}∥_{C_{α}(γ_{i})} + ∥F ∥_{C_{α−1}(\overline{D})})

with a positive constant C not depending on f_{i} and F, is valid for the solution u(x, y).

Here

∥u∥_{C_{α}^{1,1}(\overline{D})} = sup_{z∈\overline{D}\O} |z|^{−α}|u_{x}(z)| + sup_{z∈\overline{D}\O} |z|^{−α}|u_{y}(z)| +

+ sup_{z∈\overline{D}\O} |z|^{−(α−1)}|u_{xy}(z)|,

∥f_{i}∥_{C_{α}(γ_{i})} = sup_{z∈γ_{i}\O} |z|^{−α}|f_{i}(z)|, ∥F ∥_{C_{α−1}(\overline{D})} = sup_{z∈\overline{D}\O} |z|^{−(α−1)}|F (z)|.

The assertion of the second part of Theorem 1.4 is likewise valid, but in this case instead of the solvability of the problem (1.1), (1.2) in the class C_{α}^{1,1}(\overline{D}) there takes place the Hausdorff normal solvability [6]. Note also that in the critical case where α = −\frac{log |σ|}{log τ_{0}}, the Hausdorff normal solvability of the problem (1.1), (1.2) in the class C_{α}^{1,1}(\overline{D}) will, generally speaking, be violated.

§ 3. The Case Where the Curves γ_{1} and γ_{2} Have a Common Tangent

Line at O(0, 0), But the Directions of Differentiation in the Boundary Conditions (1.4) do not Coincide at This Point

By virtue of the requirements imposed on the curves γ_{1} and γ_{2} in the case where they have a common tangent line at O(0, 0), we have τ_{0} = τ^{(1)}(0) = 1. The fact that |σ| = |a(0)| = |N_{1}M_{2}M_{1}^{−1}N_{2}^{−1}(0)| ≠ 1 means that the directions of differentiation in the boundary conditions (1.4) do not coincide at O(0, 0).

Repeating the same arguments as in §2, we can prove the validity of the following

Theorem 1.5. Let the conditions (1.13) be fulfilled, the curves γ_{1} and γ_{2} have a common tangent line at the point O(0, 0), but the directions of differentiation in the boundary conditions (1.4) not coincide at this point, i.e., |σ| ≠ 1. If N_{1}(0)M_{2}(0) = 0, then the problem (1.3), (1.4) is uniquely solvable in the class C_{α}^{1,1}(\overline{D}) for α > −1. If, however, N_{1}(0)M_{2}(0) ≠ 0, then in the case |σ| < 1 the problem (1.3), (1.4) is uniquely solvable in the class C_{α}^{1,1}(\overline{D}) for α > −1, while in the case |σ| > 1 the problem (1.3), (1.4) is solvable in the class C_{α}^{1,1}(\overline{D}) for α > −1; moreover, the homogeneous problem corresponding to (1.3), (1.4) has an infinite number of linearly independent solutions.

Note that in this case, the remark following after Theorem 1.4 of the previous paragraph is also valid.

§ 4. The Case Where the Tangents to the Curves γ_{1} and γ_{2} as Well as the

Directions of Differentiation in the Boundary Conditions (1.4) Coincide at the Point O(0, 0)

For the sake of simplicity we shall assume below that the curves γ_{1}, γ_{2} and the coefficients M_{i}, N_{i}, i = 1, 2, in the boundary conditions (1.4) belong to the class C^{∞}. In this case it is obvious that τ(x) ∈ C^{∞}[0, x_{0}], and the coefficient a(x) ∈ C^{∞}[0, x_{0}] in the functional equation (1.14).

Let γ_{1} and γ_{2} have a common tangent line at O(0, 0) and the order of tangency be equal to k. This, obviously, is equivalent to the conditions

τ_{0} = τ^{(1)}(0) = 1, τ^{(i)}(0) = 0, 1 < i ≤ k, τ^{(k+1)}(0) ≠ 0. (1.27)

Therefore the function τ(x) ∈ C^{∞}[0, x_{0}] can be represented in the form

τ(x) = x + \frac{τ^{(k+1)}(0)}{(k + 1)!}x^{k+1} + λ(x)x^{k+1},  (1.28)

where λ(x) = o(x) for x → 0, i.e., lim_{x→0} λ(x) = 0.

Since τ(x) < x for 0 < x ≤ x_{0}, (1.27) and (1.28) imply

c = −\frac{τ^{(k+1)}(0)}{(k + 1)!} > 0.  (1.29)

Taking into account (1.29), we rewrite (1.28) as

τ(x) = x − cx^{k+1} + λ(x)x^{k+1}.  (1.30)

Assume τ_{n}(x) = τ(τ_{n−1}(x)), τ_{1}(x) = τ(x), 0 < x ≤ x_{0}. As it is noted above, the monotonicity of the function τ(x) and the validity of the conditions (1.17) imply that the sequence of the functions {τ_{n}(x)}_{n=1}^{∞} vanishes uniformly on [0, x_{0}] for n → ∞, i.e., τ_{n}(x) ⇉ 0, n → ∞.

Below we shall concern ourselves with the asymptotics when the sequence x_{n} = τ(x_{n−1}), x_{1} = x ∈ (0, x_{0}] tends to zero with respect to n.

The following lemma holds.

Lemma 1.8 ([51]). The behavior of the sequence x_{n} = τ_{n}(x) for n → ∞ can be written by the formula

x_{n} = \frac{ξ_{n}}{\sqrt[k]{ckn}} ,  (1.31)

where the function ξ_{n} = ξ_{n}(x) tends uniformly on the segment 0 ≤ x ≤ x_{0} to unity as n → ∞, i.e., ξ_{n}(x) ⇉ 1, n → ∞.

Proof. Because of (1.30) and the well-known equality (1 + η)^{p} = 1 + pη + λ_{1}(η)η for p ≥ 0, where lim_{η→0} λ_{1}(η) = 0, we have

\frac{1}{x_{n}^{p}} = \frac{1}{[τ(x_{n−1})]^{p}} = \frac{1}{(x_{n−1} − cx_{n−1}^{k+1} + λ(x_{n−1})x_{n−1}^{k+1})^{p}} =

= \frac{1}{x_{n−1}^{p}} \frac{1}{(1 − cx_{n−1}^{k} + λ(x_{n−1})x_{n−1}^{k})^{p}} =

= \frac{1}{x_{n−1}^{p}} \frac{1}{(1 − pcx_{n−1}^{k} + λ_{2}(x_{n−1})x_{n−1}^{k})} =

= \frac{1}{x_{n−1}^{p}} (1 + pcx_{n−1}^{k} + λ_{3}(x_{n−1})x_{n−1}^{k}) =

= \frac{1}{x_{n−1}^{p}} + pcx_{n−1}^{k−p} + λ_{3}(x_{n−1})x_{n−1}^{k−p},  (1.32)

where lim_{η→0} λ_{i}(η) = 0, i = 2, 3.

Assuming p = k and n = i in (1.32), we find that

\frac{1}{x_{i}^{k}} = \frac{1}{x_{i−1}^{k}} + ck + λ_{3}(x_{i−1}).  (1.33)

Adding equalities (1.33) for i = 2, 3, …, n, we get

\frac{1}{x_{n}^{k}} = \frac{1}{x_{1}^{k}} + ck(n − 1) + ∑_{i=2}^{n} y_{i},

i.e.,

\frac{1}{cknx_{n}^{k}} = \frac{1}{cknx_{1}^{k}} + \frac{n − 1}{n} + \frac{1}{ck} \frac{∑_{i=2}^{n} y_{i}}{n} ,  (1.34)

where the sequence y_{i} = y_{i}(x) = λ_{3}(x_{i−1}) = λ_{3}(τ_{i−1}(x)) tends uniformly on the segment 0 ≤ x ≤ x_{0} to zero, i.e., y_{i}(x) ⇉ 0, n → ∞.

Since

lim_{n→∞} \frac{1}{cknx_{1}^{k}} = 0, lim_{n→∞} \frac{n − 1}{n} = 1, lim_{n→∞} y_{n} = 0

and hence

lim_{n→∞} \frac{1}{n} ∑_{i=2}^{n} y_{i} = 0,

we obtain finally from (1.34) that the sequence \frac{1}{ξ_{n}^{k}} = \frac{1}{cknx_{n}^{k}} tends uniformly on [0, x_{0}] to unity. ■

As already noted, coincidence of the directions of differentiation in the boundary conditions (1.4) means that |σ| = |a(0)| = 1. Let first a(0) = σ = 1. Then since a(x) ∈ C^{∞}[0, x_{0}], the representation

a(x) = 1 + dx^{m} + μ(x)x^{m},  (1.35)

where lim_{x→0} μ(x) = 0 and

a^{(i)}(0) = 0, 1 ≤ i ≤ m − 1, a^{(m)}(0) ≠ 0, d = \frac{a^{(m)}(0)}{m!} .

is valid.

Because of the fact that a(x) = \frac{K_{1}(x)}{K_{2}(x)}, where K_{1}(x) = \frac{N_{1}(x)}{M_{1}(x)}, K_{2}(x) =

\frac{N_{2}(x)(γ_{1}(x))}{M_{2}(γ_{1}(x))} we have K_{1}(x) − K_{2}(x) = O(x^{m}) for x → 0. Therefore, geometrically the value m − 1 can be interpreted as the order of tangency of the directions of differentiation at O(0, 0) in the boundary conditions (1.4).

We rewrite equation (1.4) in the form

(T φ)(x) ≡ φ(x) − (Λφ)(x) = f(x), 0 < x ≤ x_{0},  (1.36)

where (Λφ)(x) = a(x)φ(τ(x)).

From (1.36) we have

φ(x) = (Λ^{n}φ)(x) + ∑_{i=0}^{n−1}(Λ^{i}f)(x),  (1.37)

where Λ^{0} = I is the unit operator.

Lemma 1.9 ([51]). For m > k, equation (1.36) cannot have more than one solution in the class C_{α}(0, x_{0}], α > 0.

Proof. Let φ(x) be a solution of the homogeneous equation corresponding to (1.36) in the class C_{α}(0, x_{0}], α > 0. Then because of (1.37) the equality φ(x) = (Λ^{n}φ)(x) holds.

Equality (1.35) yields

|a(x)| ≤ 1 + d_{1}x^{m}, 0 ≤ x ≤ x_{0},  (1.38)

for some d_{1} = const > 0. Therefore

|(Λ^{n}φ)(x)| = |a(x)a(τ(x)) ⋯ a(τ_{n−1}(x))φ(τ_{n}(x))| ≤
≤ (1 + d_{1}x^{m})(1 + d_{1}τ^{m}(x)) ⋯ (1 + d_{1}τ_{n−1}^{m}(x))|φ(τ_{n}(x))|. (1.39)

As is known, the convergence of an infinite product ∏_{i=1}^{∞}(1 + η_{i}) is

equivalent to that of the series ∑_{i=1}^{∞} η_{i} if the values η_{i} have the same sign.

Therefore the convergence of the product

∏_{i=1}^{∞} (1 + d_{1}τ_{i}^{m}(x))

is equivalent to that of the series

∑_{i=1}^{∞} τ_{i}^{m}(x),

which, in its turn, is equivalent to the convergence of the series ∑_{i=1}^{∞} \frac{1}{i^{m/k}} in

virtue of (1.31). The series ∑_{i=1}^{∞} \frac{1}{i^{m/k}} converges for m > k. Therefore there

exists a number M = const > 0 such that for n ≥ 1 the equality

∏_{i=1}^{n} (1 + d_{1}τ_{i}^{m}(x)) ≤ M  (1.40)

is valid.

Inequalities (1.39) and (1.40) imply

|(Λ^{n}φ)(x)| ≤ M|φ(τ_{n}(x))|.  (1.41)

Since φ ∈ C_{α}(0, x_{0}] and α > 0, it is obvious that φ ∈ C[0, x_{0}] and φ(0) = 0. Therefore, since the sequence {τ_{n}(x)}_{n=1}^{∞} converges uniformly on the segment 0 ≤ x ≤ x_{0} to zero, we have

lim_{n→∞} φ(τ_{n}(x)) = 0, 0 ≤ x ≤ x_{0}.  (1.42)

By virtue of (1.41) and (1.42), passing in the equality φ(x) = (Λ^{n}φ)(x) to the limit for n → ∞, we finally obtain that φ(x) ≡ 0. ■

Lemma 1.10 ([51]). If f(x) ∈ C_{β}(0, x_{0}], β > k, then for m > k equation (1.36) has the solution in the class C_{α}(0, x_{0}], 0 < α < β − k.

Proof. It can be easily verified that the functional series

T^{−1}f = ∑_{i=0}^{∞} Λ^{i}f  (1.43)

is formally a solution of equation (1.36). Therefore, to prove that equation (1.36) is solvable, it is sufficient to show that the series (1.43) converges in the class C_{α}(0, x_{0}], α < β − k.

Since f ∈ C_{β}(0, x_{0}], β > k, the equality

|f(x)| ≤ M_{1}x^{α}x^{β_{1}}, 0 ≤ x ≤ x_{0},  (1.44)

where M_{1} = const > 0, is valid for β_{1} = k + ε, ε = β − k − α > 0. From (1.31), (1.40), (1.44) and because τ_{i}(x) ≤ x, we have

|(Λ^{i}f)(x)| = |a(x)||a(τ(x))| ⋯ |a(τ_{i−1}(x))||f(τ_{i}(x))| ≤
≤ MM_{1}[τ_{i}(x)]^{α}[ \frac{ξ_{i}}{\sqrt[k]{cki}}]^{β_{1}} ≤ MM_{1}ξ_{i}^{β_{1}}(ck)^{− \frac{β_{1}}{k}} x^{α} \frac{1}{i^{β_{1}/k}} . (1.45)

Since β_{1} > k, (1.45) implies the convergence of (1.43) in the class C_{α}(0, x_{0}]. ■

Remark. The fact that the solution φ(x) of equation (1.36) for f ∈ C_{β}(0, x_{0}], β > k, does not, in general, belong to the class C_{α}(0, x_{0}] for α > β − k, is seen from the following example. It is not difficult to see that the function φ_{0}(x) = x^{β−k} ∈ C_{β−k}(0, x_{0}]. By (1.30) and (1.35), we have

(T φ_{0})(x) = x^{β−k} − a(x)(τ(x))^{β−k} =
= x^{β−k} − (1 + dx^{m} + μ(x)x^{m})(x − cx^{k+1} + λ(x)x^{k+1})^{β−k} =
= x^{β−k} − (1 + dx^{m} + μ(x)x^{m})x^{β−k}(1 − c(β − k)x^{k} + λ̃(x)x^{k}) =
= c(β − k)x^{β} + μ̃(x)x^{β},

where lim_{x→0} λ̃(x) = lim_{x→0} μ̃(x) = 0. Hence the function f_{0}(x) = (T φ_{0})(x) ∈

C_{β}(0, x_{0}], and the function φ_{0}(x) = x^{β−k} itself which does not belong to the class C_{α}(0, x_{0}] for any α > β − k, is the unique solution of equation (1.36) for f(x) = f_{0}(x).

Note that the above proven lemmas are also valid in the case a(0) = σ = −1.

Owing to Lemmas 1.9 and 1.10, the following theorem is valid.

Theorem 1.6 ([51]). Let τ_{0} = 1, |σ| = 1 and m > k. Then the problem (1.3), (1.4) cannot have more than one solution in the class C_{α}(0, x_{0}], α > 0. If f_{i} ∈ C_{β}(γ_{i}), i = 1, 2, where β > k, then the problem (1.3), (1.4) has a unique solution in the class C_{α}^{1,1}(\overline{D}), 0 < α < β − k.

We shall give the following results from [52] without proofs.

Theorem 1.7. Let τ_{0} = 1, |σ| = 1 and m = k, σd > 0. Then for any f_{i} ∈ C_{β}(0, x_{0}], i = 1, 2, β > k + 2\frac{|d|}{c}, the problem (1.3), (1.4) is uniquely solvable in the class C_{α}^{1,1}(\overline{D}), where \frac{|d|}{c} < α < β − k − \frac{|d|}{c}.

Theorem below does not involve the dependence between m and k.

Theorem 1.8. Let τ_{0} = 1, |σ| = 1 and σd < 0. Then for any f_{i} ∈ C_{β}(0, x_{0}], i = 1, 2, β > k, the problem (1.3), (1.4) is uniquely solvable in the class C_{α}^{1,1}(\overline{D}), where 0 < α < β − k.

§ 5. Influence of the Lower Terms on the Correctness of Statement of the

Problem (1.1), (1.2) in the Case Where Conditions (1.13) are Violated on the Whole Curve γ_{1} or γ_{2}

As the example of the equation u_{xy} = 0 shows, the problem (1.1), (1.2) may appear to be ill-posed when the conditions (1.13) are violated. Below we shall show that the existence of lower terms in equation (1.1) and in the boundary conditions (1.2) may affect the correctness of the statement of the problem (1.1), (1.2).

For simplicity let M_{i} = const, N_{i} = const, S_{i} = const and |M_{i}| + |N_{i}| + |S_{i}| ≠ 0, i = 1, 2. Without loss of generality we may assume |M_{i}|+|N_{i}| ≠ 0, i = 1, 2, since, otherwise, this can be achieved by differentiating the corresponding boundary condition with respect to a tangent curve γ_{i}.

As γ_{1} and γ_{2} let us take the characteristic segments γ_{1} : y = 0, 0 ≤ x ≤ x_{0}, γ_{2} : x = 0, 0 ≤ y ≤ y_{0}.

Let the second condition in (1.13) be fulfilled, while the first one be violated on the whole segment γ_{1}, i.e.,

M_{1}|_{γ_{1}} = 0.  (1.46)

Below we shall restrict ourselves to consideration of the problem (1.1), (1.2) in the class

C^{∘}^{2}(\overline{D}) = \{u ∈ C^{2}(\overline{D}) : \frac{∂^{i+j}u(0, 0)}{∂x^{i}∂y^{j}} = 0, 0 ≤ i + j ≤ 2\}

and assume that a, b, c ∈ C^{2}(\overline{D}), F ∈ C^{1}(\overline{D}), F (0, 0) = 0, f_{i} = C^{∘}^{1}(γ_{i}) = {f_{i} ∈ C^{1}(γ_{i}) : f_{i}(0) = f_{i}^{(1)}(0) = 0}, i = 1, 2.

Denote by R(x, y; x_{1}, y_{1}) the Riemann function which, by definition, is the solution of the so-called conjugate equation [10]

R_{xy} − (aR)_{x} − (bR)_{y} + cR = 0  (1.47)

which on the characteristics x = x_{1}, y = y_{1} takes the values

\begin{matrix}R(x_{1}, y; x_{1}, y_{1}) = exp ( ∫_{y_{1}}^{y} a(x_{1}, η)dη), \\ R(x, y_{1}; x_{1}, y_{1}) = exp ( ∫_{x_{1}}^{x} b(ξ, y_{1})dξ), \end{matrix}  (1.48)

where (x_{1}, y_{1}) is an arbitrarily fixed point in the domain D_{1}.

Due to (1.47) and (1.48), the function R(x, y; x_{1}, y_{1}) satisfies the integral equation

R(x, y; x_{1}, y_{1}) − ∫_{x_{1}}^{x} b(ξ, y)R(ξ, y; x_{1}, y_{1})dξ −

− ∫_{y_{1}}^{y} a(x, η)R(x, η; x_{1}, y_{1})dη +

+ ∫_{x_{1}}^{x} dξ ∫_{y_{1}}^{y} c(ξ, η)R(ξ, η; x_{1}, y_{1})dη = 1.  (1.49)

It is known that equation (1.49) has the unique solution R(x, y; x_{1}, y_{1}) which, as it can be easily verified, possesses the following continuous derivatives

∂_{x,y}^{i,j}∂_{x_{1},y_{1}}^{i_{1},j_{1}}R(x, y; x_{1}, y_{1}) ∈ C(\overline{D} × \overline{D}),  (1.50)
0 ≤ i + j ≤ 1, 0 ≤ i_{1} + j_{1} ≤ 2,

where ∂_{x,y}^{i,j} = \frac{∂^{i+j}}{∂x^{i}∂y^{j}}, ∂_{x_{1},y_{1}}^{i_{1},j_{1}} = \frac{∂^{i_{1}+j_{1}}}{∂x_{1}^{i_{1}} ∂y_{1}^{j_{1}}} .

From (1.48) we have

\begin{matrix}R_{y}(x_{1}, y; x_{1}, y_{1}) − a(x_{1}, y)R(x_{1}, y; x_{1}, y_{1}) = 0, \\ R_{x}(x, y_{1}; x_{1}, y_{1}) − b(x, y_{1})R(x, y_{1}; x_{1}, y_{1}) = 0, \\ R(x_{1}, y_{1}; x_{1}, y_{1}) = 1, \\ R_{y_{1}}(x, y; x, y_{1}) + a(x, y_{1})R(x, y; x, y_{1}) = 0, \\ R_{x_{1}}(x, y; x_{1}, y) + b(x_{1}, y)R(x, y; x_{1}, y) = 0, \\ R(x, y; x, y) = 1. \end{matrix} \}  (1.51)

On account of (1.50), every solution u(x, y) of equation (1.1) of the class C^{2}(\overline{D}) can be represented in the form [10]

u(x, y) = R(x, 0; x, y)φ(x) + R(0, y; x, y)ψ(y) − R(0, 0; x, y)φ(0) +

+ ∫_{0}^{y} [a(0, η)R(0, η; x, y) − R_{y}(0, η; x, y)]ψ(η)dη +

+ ∫_{0}^{x} [b(ξ, 0)R(ξ, 0; x, y) − R_{x}(ξ, 0; x, y)]φ(ξ)dξ +

+ ∫_{0}^{x} dξ ∫_{0}^{y} R(ξ, η; x, y)F (ξ, η)dη,  (1.52)

as the solution of the Goursat problem

u(x, 0) = φ(x), u(0, y) = ψ(y), φ(0) = ψ(0),

where φ and ψ are given functions of the class C^{2}.

When considering the problem (1.1), (1.2) in the class C^{∘}^{2}(\overline{D}), one should assume that

φ^{(i)}(0) = ψ^{(i)}(0) = 0, i = 0, 1, 2.  (1.53)

From (1.52) and because of (1.53) we have

u_{x}(x, y) = (R_{x}(x, 0; x, y) + R_{x_{1}}(x, 0; x, y))φ(x) +

+R(x, 0; x, y)φ^{(1)}(x) + R_{x_{1}}(0, y; x, y)ψ(y) +

+ ∫_{0}^{y} [a(0, η)R_{x_{1}}(0, η; x, y) − R_{yx_{1}}(0, η; x, y)]ψ(η)dη +

+[b(x, 0)R(x, 0; x, y) − R_{x}(x, 0; x, y)]φ(x) +

+ ∫_{0}^{x} [b(ξ, 0)R_{x_{1}}(ξ, 0; x, y) − R_{xx_{1}}(ξ, 0; x, y)]φ(ξ)dξ +

+ ∫_{0}^{y} R(x, η; x, y)F (x, η)dη + ∫_{0}^{x} dξ ∫_{0}^{y} R_{x_{1}}(ξ, η; x, y)F (ξ, η)dη, (1.54)

u_{y}(x, y) = R_{y_{1}}(x, 0; x, y)φ(x) + (R_{y}(0, y; x, y) + R_{y_{1}}(0, y; x, y))ψ(y) +

+R(0, y; x, y)ψ^{(1)}(y) + [a(0, y)R(0, y; x, y) − R_{y}(0, y; x, y)]ψ(y) +

+ ∫_{0}^{y} [a(0, η)R_{y_{1}}(0, η; x, y) − R_{yy_{1}}(0, η; x, y)]ψ(η)dη +

+ ∫_{0}^{x} [b(ξ, 0)R_{y_{1}}(ξ, 0; x, y) − R_{xy_{1}}(ξ, 0; x, y)]φ(ξ)dξ +

+ ∫_{0}^{x} R(ξ, y; x, y)F (ξ, y)dξ + ∫_{0}^{x} dξ ∫_{0}^{y} R_{y_{1}}(ξ, η; x, y)F (ξ, η)dη. (1.55)

Assuming in equalities (1.54), (1.55) x = 0, y = 0 and taking into account (1.53), we obtain

u_{x}(0, y) = R_{x_{1}}(0, y; 0, y)ψ(y) + ∫_{0}^{y} [a(0, η)R_{x_{1}}(0, η; 0, y) −

−R_{yx_{1}}(0, η; 0, y)]ψ(η)dη + ∫_{0}^{y} R(0, η; 0, y)F (0, η)dη,  (1.56)

u_{y}(x, 0) = R_{y_{1}}(x, 0; x, 0)φ(x) + ∫_{0}^{x} [b(ξ, 0)R_{y_{1}}(ξ, 0; x, 0) −

−R_{xy_{1}}(ξ, 0; x, 0)]φ(ξ)dξ + ∫_{0}^{x} R(ξ, 0; x, 0)F (ξ, 0)dξ.  (1.57)

It easily follows from (1.51) that

R_{x_{1}}(0, y; 0, y) = −b(0, y), R_{y_{1}}(x, 0; x, 0) = −a(x, 0).  (1.58)

Substituting the expressions obtained in (1.56), (1.57) for u_{x} and u_{y} into the boundary conditions (1.2) and taking into consideration the equalities u(x, 0) = φ(x), u(0, y) = ψ(y) and (1.58), (1.46), we find that

−N_{1}a(x, 0)φ(x) + N_{1} ∫_{0}^{x} [b(ξ, 0)R_{y_{1}}(ξ, 0; x, 0) −

−R_{xy_{1}}(ξ, 0; x, 0)]φ(ξ)dξ + S_{1}φ(x) = f_{3}(x), 0 ≤ x ≤ x_{0}, (1.59)
−M_{2}b(0, y)ψ(y) +

+M_{2} ∫_{0}^{y} [a(0, η)R_{x_{1}}(0, η; 0, y) − R_{yx_{1}}(0, η; 0, y)]ψ(η)dη +

+N_{2}ψ^{(1)}(y) + S_{2}ψ(y) = f_{4}(y), 0 ≤ y ≤ y_{0},  (1.60)

where

f_{3}(x) = f_{1}(x) − N_{1} ∫_{0}^{x} R(ξ, 0; x, 0)F (ξ, 0)dξ,

f_{4}(y) = f_{2}(y) − M_{2} ∫_{0}^{y} R(0, η; 0, y)F (0, η)dη.

Obviously, the problem (1.1), (1.2) in the class C^{∘}^{2}(\overline{D}) is equivalent to the system of equations (1.59), (1.60) with respect to unknown functions

φ ∈ C^{∘}^{2}[0, x_{0}], ψ ∈ C^{∘}^{2}[0, y_{0}].

Let the condition

(S_{1} − aN_{1})|_{γ_{1}} ≠ 0.  (1.61)

be fulfilled. From (1.48), (1.49) we have

K_{1}(ξ, x) = b(ξ, 0)R_{y_{1}}(ξ, 0; x, 0) − R_{xy_{1}}(ξ, 0; x, 0) =

= (a_{x}(ξ, 0) + a(ξ, 0)b(ξ, 0) − c(ξ, 0)) exp ( ∫_{x}^{ξ} b(τ, 0)dτ), (1.62)
K_{2}(η, y) = a(0, η)R_{x_{1}}(0, η; 0, y) − R_{yx_{1}}(0, η; 0, y) =
= (b_{y}(0, η) + a(0, η)b(0, η) − c(0, η)) exp ( ∫_{y}^{η} a(0, τ)dτ). (1.63)

Let

ψ_{0}(y) = ψ^{(1)}(y), ψ(y) = ∫_{0}^{y} ψ_{0}(τ)dτ.  (1.64)

By virtue of (1.61)–(1.64) and owing to the condition N_{2}|_{γ_{2}} ≠ 0, the system of equations (1.59), (1.60) can be rewritten in the form

φ(x) + N_{1}λ(x) ∫_{0}^{x} K_{1}(ξ, x)φ(ξ)dξ = f_{5}(x), 0 ≤ x ≤ x_{0}, (1.65)

ψ_{0}(y) + μ(y) ∫_{0}^{y} ψ_{0}(τ)dτ +

+M_{2}N_{2}^{−1} ∫_{0}^{y} K_{3}(τ, y)ψ_{0}(τ)dτ = f_{6}(y), 0 ≤ y ≤ y_{0},  (1.66)

where λ(x) = (S_{1} − aN_{1})^{−1}(x, 0), μ(y) = N_{2}^{−1}(S_{2} − bM_{2})(0, y), K_{3}(τ, y) = ∫_{τ}^{y} K_{2}(η, y)dη, f_{5}(x) = λ(x)f_{3}(x), f_{6}(y) = N_{2}^{−1}f_{4}(y).

Since equations (1.65) and (1.66) are second order Volterra type integral equations, for equations (1.65) and (1.66) to be solvable, respectively, in the

classes C^{∘}^{2}[0, x_{0}] and C^{∘}^{1}[0, y_{0}], it is sufficient to require that

\begin{matrix}K_{1}(ξ, x) ∈ C^{1}(0 ≤ \begin{matrix}ξ \\ x \end{matrix} ≤ x_{0}), \\ \frac{∂^{2}K_{1}(ξ, x)}{∂x^{2}} ∈ C(0 ≤ \begin{matrix}ξ \\ x \end{matrix} ≤ x_{0}), f_{5} ∈ C^{∘}^{2}[0, x_{0}], \end{matrix}  (1.67)

\begin{matrix}K_{2}(η, y) ∈ C(0 ≤ \begin{matrix}η \\ y \end{matrix} ≤ y_{0}), \\ \frac{∂K_{2}(η, y)}{∂y} ∈ C(0 ≤ \begin{matrix}η \\ y \end{matrix} ≤ y_{0}), f_{6} ∈ C^{∘}^{1}[0, y_{0}]. \end{matrix}  (1.68)

Due to the requirements imposed on the coefficients a, b, c of equation (1.1) and the functions F, f_{1}, f_{2}, the condition (1.68) will obviously be fulfilled. However, for the condition (1.67) to be valid, one should additionally require that

f_{1} ∈ C^{2}(OP_{1}), f_{1}^{(2)}(0) − N_{1}F_{x}(0, 0) = 0.

Consider now the case where the condition (1.61) is violated, i.e.,

(S_{1} − aN_{1})|_{γ_{1}} = 0.  (1.69)

Since, by the assumption, |M_{1}| + |N_{1}| ≠ 0, M_{1}, N_{1}, S_{1} = const, we have on account of (1.46) and (1.69)

a|_{OP_{1}} = const .  (1.70)

When the condition (1.69) is fulfilled, equation (1.59) with respect to the unknown function φ(x) is an integral Volterra type equation of the first kind

∫_{0}^{x} K_{1}(ξ, x)φ(ξ)dξ = N_{1}^{−1}f_{3}(x), 0 ≤ x ≤ x_{0}.  (1.71)

Differentiating both parts of equation (1.71) with respect to x and taking into account (1.70), we get

(ab − c)(x, 0)φ(x) − b(x, 0) ∫_{0}^{x} K_{1}(ξ, x)φ(ξ)dξ = N_{1}^{−1}f_{3}^{(1)}(x). (1.72)

Similarly, when the condition

(ab − c)|_{γ_{1}} ≠ 0  (1.73)

is fulfilled, in order that equation (1.72) to be solvable in the class C^{∘}^{2}[0, x_{0}], we should require that

f_{1} ∈ C^{3}(OP_{1}), F ∈ C^{2}(OP_{1}), f_{1}^{(2)}(0) − N_{1}F_{x}(0, 0) = 0,
f_{1}^{(3)}(0) − N_{1}F_{xx}(0, 0) + N_{1}b(0, 0)F_{x}(0, 0) = 0.

If, however, the condition (1.73) is violated, i.e.,

(ab − c)|_{γ_{1}} = 0,

then, according to (1.62), (1.70), we have

K_{1}(ξ, x) ≡ 0.

In this case the left-hand side of equation (1.71) is equal identically to zero and the equality

f_{3}(x) = f_{1}(x) − N_{1} ∫_{0}^{x} ( exp ∫_{x}^{ξ} b(τ, 0)dτ)F (ξ, 0)dξ ≡ 0, 0 ≤ x ≤ x_{0},

is a necessary and sufficient condition for the problem (1.1), (1.2) to be solv-

able in the class C^{∘}^{2}(\overline{D}); moreover, the homogeneous problem corresponding to (1.1), (1.2) has an infinite number of linearly independent solutions which are given by

u(x, y) = R(x, 0; x, y)φ(x) +

+ ∫_{0}^{x} [b(ξ, 0)R(ξ, 0; x, y) − R_{x}(ξ, 0; x, y)]φ(ξ)dξ,

where φ(x) is an arbitrary function of the class C^{∘}^{2}[0, x_{0}].

Thus the following theorem is valid.

Theorem 1.9. Let the conditions M_{1} = 0, N_{2} ≠ 0 be fulfilled. Then for (S_{1} − aN_{1})|_{γ_{1}} ≠ 0, the problem (1.1), (1.2) is uniquely solvable in the class

C^{∘}^{2}(\overline{D}) if f_{1} ∈ C^{2}(γ_{1}), f_{1}^{(2)}(0)−N_{1}F_{x}(0, 0) = 0. If, however, (S_{1}−aN_{1})|_{γ_{1}} =

0, then for (ab−c)|_{γ_{1}} ≠ 0 the problem (1.1), (1.2) is uniquely solvable in the

class C^{∘}^{2}(\overline{D}) if f_{1} ∈ C^{3}(γ_{1}), F ∈ C^{2}(γ_{1}), f_{1}^{(2)}(0)−N_{1}F_{x}(0, 0) = 0, f_{1}^{(3)}(0)−

N_{1}F_{xx}(0, 0) + N_{1}b(0, 0)F_{x}(0, 0) = 0. In the case where (S_{1} − aN_{1})|_{γ_{1}} = 0 and (ab − c)|_{γ_{1}} = 0, for the problem (1.1), (1.2) to be solvable in the class

C^{∘}^{2}(\overline{D}), it is necessary and sufficient that

f_{1}(x) − N_{1} ∫_{0}^{x} exp ( ∫_{x}^{ξ} b(τ, 0)dτ)F (ξ, 0)dξ ≡ 0, 0 ≤ x ≤ x_{0};

moreover, the homogeneous problem corresponding to (1.1), (1.2) has an infinite number of linearly independent solutions which are given by

u(x, y) = R(x, 0; x, y)φ(x) +

+ ∫_{0}^{x} [b(ξ, 0)R(ξ, 0; x, y) − R_{x}(ξ, 0; x, y)]φ(ξ)dξ,

where φ(x) is an arbitrary function of the class C^{∘}^{2}[0, x_{0}], and R(x, y; x_{1}, y_{1}) is a Riemann function for equation (1.1).

The cases M_{1}|_{γ_{1}} ≠ 0, N_{2}|_{γ_{2}} = 0 and M_{1}|_{γ_{1}} = N_{2}|_{γ_{2}} = 0 can be considered in a similar manner.

§ 6. The Case Where the Conditions (1.13) are Violated at One Point O(0, 0) Only

For simplicity, below we shall assume that in the problem (1.3), (1.4)

γ_{1} : y = ρ_{1}x, 0 ≤ x ≤ x_{0}, γ_{2} : x = ρ_{2}y, 0 ≤ y ≤ y_{0},
ρ_{i} = const > 0, i = 1, 2, ρ_{1}x_{0} < y_{0}, ρ_{2}y_{0} < x_{0}.

Let N_{1}|_{γ_{1}} ≠ 0, M_{2}|_{γ_{2}} ≠ 0, and let the second condition of (1.13) be fulfilled, while the first one be violated only at one point O(0, 0) in the form

M_{1}(x) = x^{p}ω(x),

where ω(x) ≠ 0, 0 ≤ x ≤ x_{0}, p > 0 and ω(x) ∈ C[0, x_{0}].

It is known that every solution u(x, y) of equation (1.3) of the class C_{α}^{1,1}(\overline{D}), α > −1, can be represented uniquely as [6]

u(x, y) = φ̃(x) + ψ̃(y),

where φ̃(x) ∈ C[0, x_{0}], φ^{(1)}(x) ∈ C_{α}(0, x_{0}], ψ̃(y) ∈ C[0, y_{0}], ψ̃^{(1)}(y) ∈ C_{α}(0, y_{0}], φ̃(0) = ψ̃(0) = 0.

In the notations φ(x) = u_{x}(x, y) = φ^{(1)}(x), ψ(y) = u_{y}(x, y) = ψ̃^{(1)}(y), we rewrite the boundary conditions (1.13) in the form of a system of equations

x^{p}ω(x)φ(x) + N_{1}(x)ψ(ρ_{1}x) = f_{1}(x), 0 < x ≤ x_{0},  (1.74)
M_{2}(y)φ(ρ_{2}y) + N_{2}(y)ψ(y) = f_{2}(y), 0 < y ≤ y_{0},  (1.75)

with respect to unknown functions φ(x) ∈ C_{α}(0, x_{0}], ψ(y) ∈ C_{α}(0, y_{0}].

It is easily seen that the system of equations (1.74), (1.75) is equivalent to the system

x^{p}φ(x) − b_{1}(x)φ(τ_{0}x) = f_{3}(x), 0 < x ≤ x_{0},  (1.76)
ψ(y) = −(N_{2}^{−1}M_{2})(y)φ(ρ_{2}y) + (N_{2}^{−1}f_{2})(y), 0 < y ≤ y_{0}, (1.77)

where τ_{0}=ρ_{1}ρ_{2} < 1, b_{1}(x)=(ω^{−1}N_{1})(x)(N_{2}^{−1}M_{2})(ρ_{1}x), f_{3}(x)=(ω^{−1}f_{1})(x)− (ω^{−1}N_{1})(x)(N_{2}^{−1}f_{2})(ρ_{1}x).

The following lemma holds.

Lemma 1.11. The homogeneous equation corresponding to (1.76) has an infinite number of linearly independent solutions in the class C_{α}(0, x_{0}] for all α.

Proof. It can be easily verified that the function

χ(t) = t^{\frac{p}{2} ( \frac{log t}{log τ_{0}} −1)}

belongs to the class C^{∞}[0, ∞), tends to zero as t → +0 more rapidly than any power t^{m}, m > 0, χ(t) > 0 for t > 0 and strictly monotonically increases on the segment 0 ≤ t ≤ τ_{0}^{1/2}; moreover,

χ(τ_{0}t) = t^{p}χ(t).  (1.78)

Bearing in mind (1.78), after substitution φ(x) = χ(x)φ_{1}(x), the homogeneous equation corresponding to (1.76) takes with respect to the unknown function φ_{1} the form

φ_{1}(x) − b_{1}(x)φ_{1}(τ_{0}x) = 0, 0 < x ≤ x_{0}.  (1.79)

For simplicity, let b_{1}(x) = const ≠ 0. According to (1.26), every solution of (1.79), continuous in the half-interval 0 < x ≤ x_{0}, can be represented in the form

φ_{1}(x) = \begin{cases}φ_{1}^{0}(x), & τ_{0}x_{0} ≤ x ≤ x_{0}, \\ b_{1}^{−n_{1}(x)}φ_{1}^{0}(τ_{0}^{−n_{1}(x)}x), & 0 < x < τ_{0}x_{0}, \\ & n_{1}(x) = [\frac{log x}{log τ_{0}} ], \end{cases}  (1.80)

where [\frac{log x}{log τ_{0}} ] is an integer part of the number [\frac{log x}{log τ_{0}} ], while φ_{1}^{0} is an arbitrary function of the class C[τ_{0}x_{0}, x_{0}] satisfying φ_{1}^{0}(x_{0}) − b_{1}φ_{1}^{0}(τ_{0}x_{0}) = 0.

If |b_{1}| < 1, then

|b_{1}|^{−n_{1}(x)} ≤ |b_{1}|^{− \frac{log x}{log τ_{0}}} = x^{− \frac{log |b_{1}|}{log τ_{0}}} ,  (1.81)

and for |b_{1}| ≥ 1 we have

|b_{1}|^{−n_{1}(x)} ≤ |b_{1}|^{− \frac{log x}{log τ_{0}} +1} = |b_{1}|x^{− \frac{log |b_{1}|}{log τ_{0}}} .  (1.82)

From (1.80)–(1.82) we have

|φ_{1}(x)| ≤ c̃x^{− \frac{log |b_{1}|}{log τ_{0}}} ∥φ_{1}^{0}∥_{C[τ_{0}x_{0},x_{0}]},  (1.83)

where c̃ = max(1, |b_{1}|).

Since the function χ(x) along with all its derivatives vanishes for x → +0 more rapidly than any power x^{m}, m > 0, owing to (1.83) we have

lim_{x→+0} |x^{−α}χ(x)φ_{1}(x)| = 0

for any α. Therefore the function φ(x) = χ(x)φ_{1}(x), being the solution of equation (1.76), belongs to the class C_{α}(0, x_{0}].

Because the function φ_{1}^{0}(x) in (1.80) is arbitrary, equation (1.76) has in fact an infinite number of linearly independent solutions of the class C_{α}(0, x_{0}]. ■

By Lemma 1.11, when condition M_{1}|_{γ_{1}} ≠ 0 is violated at one point O(0, 0) only, the homogeneous problem corresponding to (1.3), (1.4) has an infinite number of linearly independent solutions in the class C_{α}^{1,1}(\overline{D}) for all α > −1. At the same time, we can find a functional space C_{α,χ}^{1,1}(\overline{D}) in which the problem (1.3), (1.4) is uniquely solvable.

Introduce into consideration the space

C_{α,χ}^{1,1}(\overline{D}) = \{u ∈ C(\overline{D}) ∩ C^{1}(\overline{D}\O) : u(0, 0) = 0, χ^{−1}(x)u_{x} ∈ C_{α}(\overline{D}\O),

y^{−p \frac{log ρ_{2}}{log τ_{0}}} χ^{−1}(y)u_{y} ∈ C_{α}(\overline{D}\O), u_{xy} ∈ C(\overline{D}\O)\},

where

C_{α}(\overline{D}\O) = \{u ∈ C(\overline{D}\O) : sup_{z∈\overline{D}\O} |z|^{−α}|u(z)| < +∞\}.

As it is shown above, the problem (1.3), (1.4) in the class C_{α,χ}^{1,1}(\overline{D}) is equivalently reduced to the system of equations (1.76), (1.77) with respect to the unknown functions φ(x) and ψ(y), where

χ^{−1}(x)φ(x) ∈ C_{α}(0, x_{0}], y^{−p \frac{log ρ_{2}}{log τ_{0}}} χ^{−1}(y)ψ(y) ∈ C_{α}(0, y_{0}].

The spaces consisting of the functions φ(x) and ψ(y) and satisfying these conditions we denote, respectively, by C_{α,χ}(0, x_{0}] and C_{α,y^{q_{1}} χ}(0, y_{0}], where q_{1} = p\frac{log ρ_{2}}{log τ_{0}} .

If φ(x) ∈ C_{α,χ}(0, x_{0}], then it is obvious that φ(ρ_{2}y) ∈ C_{α,y^{q_{1}} χ}(0, y_{0}]. Therefore by virtue of (1.77) we require that f_{2} ∈ C_{α,y^{q_{1}} χ}(0, y_{0}].

Since

x^{p}φ(x), φ(τ_{0}x) ∈ C_{α,x^{p}χ}(0, x_{0}]

and

f_{2}(ρ_{1}x) ∈ C_{α,y^{q_{2}} y^{q_{1}}} (0, y_{0}] = C_{α,y^{p}χ}(0, y_{0}],

where q_{2} = p\frac{log ρ_{1}}{log τ_{0}} and q_{1} + q_{2} = p owing to ρ_{1}ρ_{2} = τ_{0}, in equation (1.76) in order to f_{3} ∈ C_{α,x^{p}χ}(0, x_{0}] we require of the boundary function f_{1} that f_{1} ∈ C_{α,x^{p}χ}(0, x_{0}]. Therefore if we consider the problem (1.3), (1.4) in the class C_{α,χ}^{1,1}(\overline{D}), then we shall assume that in the boundary conditions (1.3)

f_{1} ∈ C_{α,x^{p}χ}(0, x_{0}], f_{2} ∈ C_{α,y^{q_{1}} χ}(0, y_{0}].

Let σ_{1} = b_{1}(0) = (ω^{−1}N_{2}^{−1}N_{1}M_{2})(0).

Lemma 1.12. For α > −\frac{log |σ_{1}|}{log τ_{0}} equation (1.76) is uniquely solvable in

the class C_{α,χ}(0, x_{0}], while for α < −\frac{log |σ_{1}|}{log τ_{0}} equation (1.76) is solvable in the class C_{α,χ}(0, x_{0}]; moreover, the homogeneous equation corresponding to (1.76) has an infinite number of linearly independent solutions in this class.

Proof. Because of (1.78), substituting in equation (1.76) φ(x) = χ(x)φ_{1}(x) for the unknown function φ_{1}(x), we obtain the equation

φ_{1}(x) − b_{1}(x)φ_{1}(τ_{0}x) = f(x),  (1.84)

where φ_{1}(x) ∈ C_{α}(0, x_{0}], if φ(x) ∈ C_{α,χ}(0, x_{0}] and f(x) = x^{−p}χ^{−1}(x)f_{3}(x) ∈ C_{α}(0, x_{0}].

It is now evident that Lemma 1.12 is a direct consequence of Lemma 1.7 applied to equation (1.84). ■

Thus the following theorem is valid.

Theorem 1.10. Let N_{i}|_{γ_{i}} ≠ 0, i = 1, 2, M_{2}|_{γ_{2}} ≠ 0 and M_{1}(x) = x^{p}ω(x), p > 0, ω(x) ∈ C[0, x_{0}], ω(x) ≠ 0, x ∈ [0, x_{0}]. Then for α > −\frac{log |σ_{1}|}{log τ_{0}} the problem (1.3), (1.4) is uniquely solvable in the class C_{α,χ}^{1,1}(\overline{D}), while for α < −\frac{log |σ_{1}|}{log τ_{0}} the problem (1.3), (1.4) is solvable in the class C_{α,χ}^{1,1}(\overline{D}); moreover, the homogeneous problem corresponding to (1.3), (1.4) has an infinite number of linearly independent solutions in this class.

CHAPTER II

BOUNDARY VALUE PROBLEMS FOR LINEAR SECOND ORDER NORMALLY HYPERBOLIC SYSTEMS

§ 1. Statement of the Problem

In the plane of variables x, y let us consider a system of linear differential equations of the type

Au_{xx} + 2Bu_{xy} + Cu_{yy} + A_{1}u_{x} + B_{1}u_{y} + C_{1}u = F,  (2.1)

where A, B, C, A_{1}, B_{1}, C_{1} are given real n × n-matrices, F is a given and u is an unknown n-dimensional real vector, respectively, and it is assumed that det C ≠ 0, n > 1.

Denote by p(x, y; ξ, η) the characteristic determinant of the system (2.1), that is,

p(x, y; ξ, η) = det Q(x, y; ξ, η),

where Q(x, y; ξ, η) = A(x, y)ξ^{2} + 2B(x, y)ξη + C(x, y)η^{2}.

Since det C ≠ 0, we have the representation

p(x, y; 1, λ) = det C ∏_{i=1}^{l} (λ − λ_{i}(x, y))^{k_{i}}, ∑_{i=1}^{l} k_{i} = 2n,

l = l(x, y), k_{i} = k_{i}(x, y), i = 1, …, l.

Obviously, the system (2.1) degenerates parabolically only at the point (x, y) in the case l = 1. The system (2.1) is said to be hyperbolic at (x, y) if l > 1 and all the roots λ_{1}(x, y), …, λ_{l}(x, y) of the polynomial p(x, y; 1, λ) are real numbers.

It can be easily verified that [6]

k_{i}(x, y) ≥ n − rank Q(x, y; 1, λ_{i}(x, y)), i = 1, …, l.

The hyperbolic system (2.1) is said to be normally hyperbolic at the point (x, y) if the equalities [6]

k_{i}(x, y) = n − rank Q(x, y; 1, λ_{i}(x, y)), i = 1, …, l,

are fulfilled.

Below we shall assume that at every point (x, y) of the domain of definition of the coefficients A, B, C the system (2.1) is normally hyperbolic, and the multiplicities k_{1}(x, y), …, k_{l}(x, y) of the roots λ_{1}(x, y), …, λ_{l}(x, y) of the characteristic polynomial p(x, y; 1, λ) do not depend on the variables x, y, i.e., k_{i} = const, i = 1, …, l.

Note that strictly hyperbolic systems, i.e. when l = 2n, k_{i} = 1, i = 1, …, 2n, form a subclass of normally hyperbolic systems.

Let γ_{i} : x = x_{i}(t), y = y_{i}(t), 0 ≤ t < ∞, i = 1, 2, be simple curves of the class C^{k}, k ≥ 2, coming out of the origin O(0, 0), having no common point at t > 0 and dividing the plane into two simply connected unbounded angles. Denote by D the angle between γ_{1} and γ_{2} whose size at the point O(0, 0) is less than π. In §2 under certain restrictions imposed on the curves γ_{1}, γ_{2} and characteristics of the system (2.1), we construct the domain D_{1} ⊂ D representing either a curvilinear quadrangle or a triangle (depending on the location of certain points P_{1}, P_{2} on γ_{1}, γ_{2}) with a vertex at O(0, 0) which is bounded by γ_{1}, γ_{2} and well-defined characteristics of the system (2.1), coming out of the points P_{1}, P_{2}. D_{1} is assumed to be a subdomain of the domain of definition of the system (2.1).

Consider the boundary value problem formulated as follows [6]: find in the domain D_{1} a regular solution u(x, y) of the system (2.1), satisfying on the segments OP_{1} and OP_{2} of the curves γ_{1} and γ_{2} the conditions

(M_{1}u_{x} + N_{1}u_{y} + S_{1}u)|_{OP_{1}} = f_{1},  (2.2)
(M_{2}u_{x} + N_{2}u_{y} + S_{2}u)|_{OP_{2}} = f_{2},  (2.3)

where M_{i}, N_{i}, S_{i}, i = 1, 2, are given real m_{i} × n-matrices, f_{i}, i = 1, 2, are given m_{i}-dimensional vectors and m_{1} and m_{2} are non-negative integers which will be defined below.

Introduce the functional spaces

C^{∘}^{k}(\overline{D}_{1}) = \{u ∈ C^{k}(\overline{D}_{1}) : ∂^{i,j}u(0, 0) = 0, 0 ≤ i + j ≤ k\},

∂^{i,j} = \frac{∂^{i+j}}{∂x^{i}∂y^{j}} ,

C^{∘}_{α}^{k}(\overline{D}_{1}) = \{u ∈ C^{∘}^{k}(\overline{D}_{1}) : max_{i+j=k} sup_{z∈\overline{D}_{1}\O} |z|^{−α}|∂^{i,j}u(z)| < ∞\},

− ∞ < α < ∞.

Obviously, C^{∘}_{α}^{k}(\overline{D}_{1}) = C^{∘}^{k}(\overline{D}_{1}) for α ≤ 0. Analogously we introduce the

weighted spaces C^{∘}_{α}^{k}(OP_{i}), i = 1, 2.

When considering problems (2.1)–(2.3) in the class C^{∘}_{α}^{k}(\overline{D}_{1}), k ≥ 2, α ≥ 0, we shall require that A, B, C ∈ C^{k}(R^{2}), where R^{2} is the plane of variables x, y; A_{1}, B_{1}, C_{1} ∈ C^{k−1}(\overline{D}_{1}); M_{i}, N_{i}, S_{i} ∈ C^{k−1}(OP_{i}), i = 1, 2;

f_{i} ∈ C^{∘}_{α}^{k−1}(OP_{i}), i = 1, 2; F ∈ C^{∘}_{α−1}^{k−1}(\overline{D}_{1}).

§ 2. Requirements Imposed on the Curves γ_{1}, γ_{2} and on the Characteristics of the System (2.1). Determination of Numbers m_{1} and m_{2}. Construction of Domains D_{1} and D_{P}

In §3 it will be shown that under certain assumptions made with respect to the coefficients A, B, C of the system (2.1), the roots of the characteristic polynomial p(x, y; 1, λ) at every point (x, y) ∈ R^{2} can be renumerated so that λ_{i}(x, y) ∈ C^{k}(R^{2}), i = 1, …, l.

Through every point (x, y) ∈ R^{2} there pass l characteristic curves L_{i}(x, y), i = 1, …, l, of the system (2.1), satisfying the ordinary differential equations

dx + λ_{i}(x, y)dy = 0, i = 1, …, l.

Let the location of the curves γ_{1}, γ_{2} on the plane be such that while moving towards O(0, 0) along γ_{2} and then along γ_{1}, the domain D bounded by γ_{1}, γ_{2} remains to the left. Renumerate the roots of the polynomial p(x, y; 1, λ) in such a way that the characteristic curves L_{1}(P_{1}), …, L_{l}(P_{1}) corresponding to the roots λ_{1}, …, λ_{l} and coming out of the point P_{1} into the domain {P ∈ D : |P − P_{1}| < ε} would turn out to be renumerated counter-clockwise if we count from L_{1}(P_{1}), where ε is a sufficiently small positive number.

If the curves γ_{1} and γ_{2} do not have a common tangent line at O(0, 0), then we denote by l_{0}, 0 ≤ l_{0} ≤ l, the number of different characteristics issued from O(0, 0) into the domain {(x, y) ∈ D : x^{2} +y^{2} < ε^{2}}. In the case where γ_{1}, γ_{2} have a common tangent line at O(0, 0), we assume l_{0} = 0.

Below we impose on the curves γ_{1}, γ_{2} and the characteristics L_{i}(P), P ∈ \overline{D}, i = 1, …, l, the following restrictions.

1. Each of the curves γ_{1}, γ_{2} either is a characteristic of the system (2.1) or it has characteristic direction at none of its point.

2. For i > l_{0} every characteristic L_{i}(P), P ∈ \overline{D}\O, extended maximally to either side in \overline{D} possesses one of the following properties:

a) it entirely coincides with one of the curves γ_{1} or γ_{2};

b) it intersects γ_{1} (γ_{2}) only at one point, when γ_{1} (γ_{2}) is a non-characte-ristic curve or γ_{1} (γ_{2}) is a characteristics of the system (2.1), not belonging to the family L_{i}.

If, however, 1 ≤ i ≤ l_{0}, then the characteristics L_{i}(O) divide D into two simply-connected unbounded angles and the characteristics L_{i}(P) intersect the curve γ_{1} or γ_{2} at one point only, depending on the location of the point P in \overline{D}\L_{i}(O).

3. The family of characteristics L_{i} is described in \overline{D} by the equation L_{i} : Ω_{i}(x, y) = const, 1 ≤ i ≤ l, where Ω_{i} ∈ C^{k}(\overline{D}) and | grad Ω_{i}||\overline{D} ≠ 0.

For the sake of simplicity, let the characteristics L_{i}(P_{1}), i = 1, …, l, issued from the point P_{1} into D not intersect the curve γ_{2} at the point P_{2}. We take the number m_{1} of boundary conditions in (2.2) to be equal to the number of characteristics, with regard for their multiplicities, issued from the point P_{1} into D and not intersecting with the closed segment OP_{2} ⊂ γ_{2}. Substituting the point P_{1} by P_{2} and the segment OP_{2} by OP_{1} ⊂ γ_{1}, we can determine analogously the value m_{2}. In particular, if m_{i} = 0, then the segment OP_{i} ⊂ γ_{i}, i = 1, 2, becomes completely free from the boundary conditions. It is clear that under such a choice the numbers m_{1} and m_{2} depend on the location of the points P_{1} and P_{2} on the curves γ_{1} and γ_{2}; moreover, 0 ≤ m_{i} ≤ 2n, i = 1, 2.

Let us introduce into the consideration the domains D_{1} and D_{P}, P ∈ \overline{D}\O. If m_{i} > 0, i = 1, 2, then let D_{1} be a curvilinear quadrangle with a vertex at the point O(0, 0), bounded by the curves γ_{1}, γ_{2}, L_{s_{0}}(P_{1}), L_{s_{1}}(P_{2}), where L_{s_{0}}(P_{1}) is the last (moving counter-clockwise) characteristic, coming out of the point P_{1} into the domain D and not intersecting the closed segment OP_{2}, while L_{s_{1}}(P_{2}) is the last (moving clockwise) characteristic coming out of the point P_{2} into the angle D and not intersecting with the closed segment OP_{1}. In this case D_{P} is a curvilinear quadrangle with a vertex at the point O(0, 0), bounded by the curves γ_{1}, γ_{2}, L_{s_{0}}(P) and L_{s_{1}}(P). Clearly, s_{1} = s_{0} + 1 for 0 < m_{1} < 2n and s_{1} = 1, s_{0} = l for m_{1} = 2n, l_{0} > 0, but in the case m_{1} = 2n, l_{0} = 0 the number m_{2} = 0. If, however, m_{1} = 0, then D_{1} and D_{P} are curvilinear triangles bounded, respectively, by the curves γ_{1}, γ_{2}, L_{1}(P_{2}) and γ_{2}, L_{1}(P), L_{l}(P). Similarly, for m_{2} = 0 the domains D_{1} and D_{P} are bounded, respectively, by the curves γ_{1}, γ_{2}, L_{l}(P_{1}) and γ_{1}, L_{1}(P), L_{l}(P).

§ 3. Reduction of the Problem (2.1)–(2.3) to the System of Integro-Functional Equations

Owing to normal hyperbolicity of the system (2.1), at every point (x, y) we have rank Q(x, y; 1, λ_{i}(x, y)) = n−k_{i}, 1 ≤ i ≤ l. Hence dim Ker Q(x, y; 1, λ_{i}(x, y)) = k_{i}, where Ker Q(x, y; 1, λ_{i}(x, y)) is a kernel of the matrix operator Q(x, y; 1, λ_{i}(x, y)) acting in R^{n}. Let {ν_{ij}}_{j=1}^{k_{i}} be a basis chosen arbitrarily in Ker Q(x, y; 1, λ_{i}(x, y)). It can be easily verified that at every point (x, y), the value λ_{i}(x, y), 1 ≤ i ≤ l, is an eigen-value, while the 2n-dimensional vectors (ν_{ij}, λ_{i}ν_{ij})(x, y) corresponding to λ_{i}(x, y) are eigenvectors of the matrix operator

A_{0}(x, y) = ∥\begin{matrix}0 & −E \\ C^{−1}A & 2C^{−1}B \end{matrix}∥ (x, y).

Note that if the 2n-dimensional vector (ν_{ij}^{1}, ν_{ij}^{2})(x, y) is a an eigen-vector of the operator A_{0} corresponding to the eigen-value λ_{i}(x, y), then ν_{ij}^{2}(x, y) = λ_{i}(x, y)ν_{ij}^{1}(x, y), and ν_{ij}^{1}(x, y) ∈ Ker Q(x, y; 1, λ_{i}(x, y)). Since the system (2.1) is normally hyperbolic, the vectors (ν_{ij}, λ_{i}ν_{ij}), i = 1, …, l, j = 1, …, k_{i}, form a complete system of eigen-vectors of the operator A_{0}(x, y), and hence diagonalizing the operator, A_{0} we obtain the equality

K^{−1}A_{0}K = D_{0}  (2.4)

at the point (x, y), where

K = (\begin{matrix}ν_{11} & ⋯ & ν_{1k_{1}} & ν_{21} & ⋯ & ν_{lk_{l}} \\ λ_{1}ν_{11} & ⋯ & λ_{1}ν_{1k_{1}} & λ_{2}ν_{21} & ⋯ & λ_{l}ν_{lk_{l}} \end{matrix}) ,

D_{0} = diag [ − λ_{1}, …, −λ_{1}, −λ_{2}, …, −λ_{2}, …, −λ_{l}].

Denote by Δ_{r} the square {(x, y) ∈ R^{2} : |x| < r, |y| < r}. Since the matrix operator A_{0} is diagonalizable, belongs to the class C^{k}(R^{2}) and the multiplicities k_{i} of the eigen-values λ_{i}, i = 1, …, l, do not depend on the variables x, y, owing to the results of [72], for any fixed r > 0 at every point (x, y) ∈ Δ_{r} we can renumerate the numbers λ_{i}(x, y), i = 1, …, l, and choose the basis vectors ν_{ij}(x, y), j = 1, …, k_{i}, in the space Ker Q(x, y; 1, λ_{i}(x, y)) such that λ_{i}(x, y) ∈ C^{k}(\overline{Δ}_{r}), i = 1, …, l, and ν_{ij}(x, y) ∈ C^{k}(\overline{Δ}_{r}), i = 1, …, l; j = 1, …, k_{i}. From this it is not difficult to see that we can choose the numbering of λ_{1}, …, λ_{l} such that λ_{i}(x, y) ∈ C^{k}(R^{2}), i = 1, …, l. Indeed, performing additional renumeration, we may assume that for any r > 0

λ_{1}^{r}(0, 0) < λ_{2}^{r}(0, 0) < ⋯ < λ_{l}^{r}(0, 0)  (2.5)

and λ_{i}^{r}(x, y) ∈ C^{k}(\overline{Δ}_{r}), i = 1, …, l. Now let us show that (2.5) implies the validity of the same inequalities at any other point (x, y) ∈ \overline{Δ}_{r}, i.e.,

λ_{1}^{r}(x, y) < λ_{2}^{r}(x, y) < ⋯ < λ_{l}^{r}(x, y).  (2.6)

If at a point (x_{0}, y_{0}) ∈ \overline{Δ}_{r} the inverse inequality λ_{i}^{r}(x_{0}, y_{0}) > λ_{j}^{r}(x_{0}, y_{0}) took place for i < j, then due to the continuity of the function g_{ij}(x, y) = λ_{i}^{r}(x, y)−λ_{j}^{r}(x, y) and because of the inequalities g_{ij}(0, 0) < 0, g_{ij}(x_{0}, y_{0}) > 0, one could find on the portion of the straight line connecting the points (0, 0) and (x_{0}, y_{0}), a point (x_{1}, y_{1}) ∈ \overline{Δ}_{r} such that g_{ij}(x_{1}, y_{1}) = 0, i.e., λ_{i}^{r}(x_{1}, y_{1}) = λ_{j}^{r}(x_{1}, y_{1}), but this equality contradicts the fact that at every point (x, y) all the numbers λ_{1}^{r}(x, y), …, λ_{l}^{r}(x, y) differ. Since inequalities (2.6) are valid for any r and for 0 < r_{1} < r_{2} the sets {λ_{1}^{r_{1}}(x, y), …, λ_{l}^{r_{1}}(x, y)} and \{λ_{1}^{r_{2}}(x, y), …, λ_{l}^{r_{2}}(x, y)\} coincide at every point (x, y) ∈ \overline{Δ}_{r_{1}}, we get

λ_{i}^{r_{1}}(x, y) = λ_{i}^{r_{2}}(x, y) for (x, y) ∈ \overline{Δ}_{r_{1}}, i = 1, …, l.  (2.7)

It follows from (2.7) that the functions

λ_{i}(x, y) = λ_{i}^{r}(x, y) for (x, y) ∈ Δ_{r}, i = 1, …, l,

belong to the class C^{k}(R^{2}).

Since the domain D_{1} constructed in §2 is bounded, D_{1} ⊂ Δ_{r} for some r > 0. Therefore, owing to the above arguments, the basis vectors ν_{ij}(x, y) will be assumed to be chosen in the space Ker Q(x, y; 1, λ_{i}(x, y)) such that ν_{ij}(x, y) ∈ C^{k}(\overline{D}_{1}), i = 1, …, l, j = 1, …, k_{i}.

Without loss of generality we may assume that the domain D_{P}, P (x_{0}, y_{0}) ∈ \overline{D}_{1}, constructed in §2 is located entirely in the half-plane y ≤ y_{0}; moreover, every characteristic L_{i}(x_{0}, y_{0}), 1 ≤ i ≤ l, of the system (2.1) issued from the point P (x_{0}, y_{0}) into the closed domain \overline{D}_{P} to the intersection with one of the curves γ_{1} or γ_{2} admits parametrization of the type L_{i}(x_{0}, y_{0}) : x = z_{i}(x_{0}, y_{0}; y) ∈ C^{k}, y = t. Otherwise, as it can be easily verified, because of the requirement 3 imposed on the characteristics L_{i}, this can be achieved by means of a non-degenerate transform of variables x̃ = J_{1}(x, y), ỹ = J_{2}(x, y), J_{1}(0, 0) = J_{2}(0, 0) = 0 which translates the families of characteristics L_{s_{0}}(x, y) and L_{s_{1}}(x, y) to those of straight lines ỹ + x̃ = const and ỹ − x̃ = const, respectively, while the domain D_{1} to a subdomain D̃_{1} of the half-plane ỹ ≥ 0. In the plane of variables x̃, ỹ, every characteristic L̃_{i}(x̃_{0}, ỹ_{0}), 1 ≤ i ≤ l, issued from the point P̃(x̃_{0}, ỹ_{0}) ∈ \overline{D̃}_{1} into the the domain \overline{D̃}_{P} to the intersection with the curve γ̃_{1} or γ̃_{2} will entirely lie in the quarter-plane ỹ+ x̃ ≤ ỹ_{0} + x̃_{0}, ỹ− x̃ ≤ ỹ_{0} − x̃_{0}, and hence at every point P̃(x̃_{0}, ỹ_{0}) ∈ \overline{D̃}_{1} the tangent to the characteristic L̃_{i}(x̃_{0}, ỹ_{0}) is not parallel to the axis ox̃. This, in its turn, implies that the portion of the characteristic L̃_{i}(x̃_{0}, ỹ_{0}) which is located in the domain \overline{D̃}_{1} admits a parametrization of the form x̃ = z̃_{i}(x̃_{0}, ỹ_{0}; t) ∈ C^{k}, ỹ = t.

Denote by ω_{i}(x_{0}, y_{0}) the ordinate of the point of intersection of the characteristic L_{i}(x_{0}, y_{0}), issued from the point P (x_{0}, y_{0}) ∈ \overline{D}_{1} into the domain \overline{D}_{P}, with one of the curves γ_{1} or γ_{2}. This curve depends both on the index i of the characteristic L_{i} and on the location of the point P in \overline{D}_{1} and we denote it by γ_{i(P)}. According to the requirements imposed on the characteristics L_{i} and the curves γ_{1}, γ_{2} we have ω_{i} ∈ C^{k}(\overline{D}_{1}), ω_{i}(x_{0}, y_{0}) ≤ y_{0}, (x_{0}, y_{0}) ∈ \overline{D}_{1}; moreover, L_{i}(P) ∩ \overline{D}_{P} : x = z_{i}(x_{0}, y_{0}; t) ∈ C^{k}, y = t, ω_{i}(x_{0}, y_{0}) ≤ t ≤ y_{0}.

Below we shall assume that a portion OP_{i} of the curve γ_{i} is described in terms of the equation x = γ_{i}(y), 0 ≤ y ≤ d_{i}, i = 1, 2. One can easily

verify that the problem (2.1)–(2.3) in the class C^{∘}_{α}^{k}(\overline{D}_{1}) can be equivalently rewritten in the form

v_{y} + A_{0}v_{x} + B_{0}v + C_{0}u^{0} = F^{0},  (2.8)
( − λ_{1} \frac{∂}{∂x} + \frac{∂}{∂y} )u = −λ_{1}v_{1} + v_{2},  (2.9)
(M_{1}v_{1} + N_{1}v_{2} + S_{1}u)|_{OP_{1}} = f_{1},  (2.10)
(M_{2}v_{1} + N_{2}v_{2} + S_{2}u)|_{OP_{2}} = f_{2},  (2.11)
(\frac{dγ_{i}}{dy} \frac{∂}{∂x} + \frac{∂}{∂y} )u|_{OP_{i}} = (\frac{dγ_{i}}{dy} v_{1} + v_{2})|_{OP_{i}}, i = 1, 2,  (2.12)

where

A_{0} = ∥\begin{matrix}0 & −E \\ C^{−1}A & 2C^{−1}B \end{matrix}∥ , B_{0} = ∥\begin{matrix}0 & 0 \\ C^{−1}A_{1} & C^{−1}B_{1} \end{matrix}∥ ,

C_{0} = ∥\begin{matrix}0 & 0 \\ C^{−1}C_{1} & 0 \end{matrix}∥ ,

v_{1} = u_{x}, v_{2} = u_{y}, u ∈ C^{∘}_{α}^{k}(\overline{D}_{1}), v = (v_{1}, v_{2}) ∈ C^{∘}_{α}^{k−1}(\overline{D}_{1}),

u^{0} = (u, 0), F^{0} = (0, C^{−1}F) and E is the unit n × n-matrix.

In the case l_{0} = 0, one should write instead of (2.12) the equality

(\frac{dγ_{1}}{dy} \frac{∂}{∂x} + \frac{∂}{∂y} )u|_{OP_{1}} = (\frac{dγ_{1}}{dy} v_{1} + v_{2})|_{OP_{1}}.

If u ∈ C^{∘}_{α}^{k}(\overline{D}_{1}) is a solution of the problem (2.1)–(2.3), then the system of vectors u, v_{1} = u_{x}, v_{2} = u_{y} will, obviously, give the solution of the problem (2.8)–(2.12). Conversely, let u ∈ C^{∘}_{α}^{k}(\overline{D}_{1}), v = (v_{1}, v_{2}) ∈ C^{∘}_{α}^{k−1}(\overline{D}_{1}) be a solution of the problem (2.8)–(2.12). Let us show that u is a solution of the problem (2.1)–(2.3), and v_{1} = u_{x}, v_{2} = u_{y}. For simplicity, let us assume that λ_{1} = const. It follows from the first n equations of the system (2.8) that v_{1y} = v_{2x}. Next, because of (2.9) we have

( − λ_{1} \frac{∂}{∂x} + \frac{∂}{∂y} )(u_{x} − v_{1}) =

= \frac{∂}{∂x}( − λ_{1} \frac{∂}{∂x} + \frac{∂}{∂y} )u − ( − λ_{1} \frac{∂}{∂x} + \frac{∂}{∂y} )v_{1} =

= \frac{∂}{∂x}(−λ_{1}v_{1} + v_{2}) + λ_{1}v_{1x} − v_{1y} =

= −λ_{1}v_{1x} + v_{2x} + λ_{1}v_{1x} − v_{1y} = v_{2x} − v_{1y} = 0.

Thus u_{x}−v_{1} ≡ 0, since, by requirements imposed both on the characteristics L_{i} and on the curves γ_{1}, γ_{2}, the system of equations (2.9), (2.12) is uniquely solvable with respect to u_{x} and u_{y} on the segments OP_{1} ⊂ γ_{1}, OP_{2} ⊂ γ_{2}. Moreover, (u_{x}−v_{1})|_{OP_{1}} = (u_{x}−v_{1})|_{OP_{2}} = 0, if l_{0} > 1 and (u_{x}−v_{1})|_{OP_{1}} = 0 for l_{0} = 0. Because of u_{x} = v_{1}, it follows from (2.9) that u_{y} = v_{2} and by (2.8), (2.10), (2.11) we easily obtain that u is a solution of the problem (2.1)–(2.3). In the case λ_{1}(x, y) ≢ const, we shall act as follows. Denote by λ̃(x, y) a function of the class C^{1}(\overline{D}_{1}) such that ∇_{1}λ̃ = ∇λ_{1} and λ̃ − λ ≠ 0 in \overline{D}_{1}, where ∇_{1} = −λ_{1} \frac{∂}{∂x} + \frac{∂}{∂y}, ∇_{2} = −λ̃\frac{∂}{∂x} + \frac{∂}{∂y}. By equalities v_{1y} = v_{2x}

and ∇_{1}λ̃ = ∇_{2}λ_{1}, we can easily verify that

∇_{1}∇_{2} = ∇_{2}∇_{1}, ∇_{2}(−λ_{1}v_{1} + v_{2}) = ∇_{1}(−λ̃_{1}v_{1} + v_{2}),

whence, taking into account (2.9), we get

∇_{1}(∇_{2}u − (−λ̃v_{1} + v_{2})) = ∇_{2}∇_{1}u − ∇_{2}(−λ_{1}v_{1} + v_{2}) =
= ∇_{2}(∇_{1}u − (−λ_{1}v_{1} + v_{2})) = 0.

From this, due to the unique solvability of the system of equations (2.9), (2.12) with respect to u_{x} and u_{y} on OP_{1} and OP_{2} and, as a consequence, of the equalities (∇_{2}u−(−λ̃v_{1}+v_{2}))|_{OP_{1}∪OP_{2}} = 0 or (∇_{2}u−(−λ̃v_{1}+v_{2}))|_{OP_{1}} = 0 for l_{0} > 0 and l_{0} = 0, respectively, we find that ∇_{2}u − (−λ̃v_{1} + v_{2}) = 0 in \overline{D}_{1}. Since λ̃ − λ ≠ 0 in \overline{D}_{1}, it follows from (2.9) and the obtained equality ∇_{2}u−(−λ̃v_{1} +v_{2}) = 0 that u_{x} = v_{1}, u_{y} = v_{2} which, in its turn, implies that u is a solution of the problem (2.1)–(2.3). To construct the function λ̃(x, y) with properties indicated above, we rewrite the equality ∇_{1}λ̃ = ∇_{2}λ_{1} in terms of the linear first order differential equation

( − λ_{1} \frac{∂}{∂x} + \frac{∂}{∂y} )λ̃ + λ_{1x}λ̃ = λ_{1y}.

Integrating this equation as an ordinary differential equation along the first characteristic L_{1} of the system (2.1) and taking as the initial Cauchy data sufficiently large absolute values λ̃ on OP_{1} ∪ OP_{2} for l_{0} > 0 or on OP_{1} for l_{0} = 0, we get the function λ̃ satisfying the conditions ∇_{1}λ̃ = ∇_{2}λ_{1} and λ̃ − λ_{1} ≠ 0 in \overline{D}_{1}.

Substitution of the unknown function v = Kw by (2.8)–(2.12) results in

w_{y} + D_{0}w_{x} = B_{2}w + C_{2}u^{0} + F^{1},  (2.13)
( − λ_{1} \frac{∂}{∂x} + \frac{∂}{∂y} )u = (−λ_{1}K_{1} + K_{2})w,  (2.14)
((M_{1}K_{1} + N_{1}K_{2})w + S_{1}u)|_{OP_{1}} = f_{1},  (2.15)
((M_{2}K_{2} + N_{2}K_{2})w + S_{2}u)|_{OP_{2}} = f_{2},  (2.16)
(\frac{dγ_{i}}{dy} \frac{∂}{∂x} + \frac{∂}{∂y} )u|_{OP_{i}} = (\frac{dγ_{i}}{dy} K_{1} + K_{2})w|_{OP_{i}}, i = 1, 2, (2.17)

where B_{2} = −K^{−1}(K_{y} + A_{0}K_{x} + B_{0}K), C_{2} = −K^{−1}C_{0}, F^{1} = K^{−1}F^{0}, and K_{1} and K_{2} are the matrices of order n × 2n composed, respectively, of the first and the last n rows of the matrix K.

Integrate the (q_{i} + j)-th equation of the system (2.13), where q_{1} = 0, q_{i} = k_{1} + ⋯ + k_{i−1}, j = 1, …, k_{i}, along the i-th characteristic L_{i}(x, y) coming out of P (x, y) ∈ \overline{D}_{1} into the domain \overline{D}_{P}, from P (x, y) to the point of intersection of L_{i}(x, y) with the curve γ_{1} or γ_{2}, depending both on the index i of the characteristic L_{i} and on the location of P in \overline{D}_{1}, and integrate equation (2.14) with respect to the first characteristic. We obtain

w_{q_{i}+j}(x, y) = w_{q_{i}+j}(γ_{i(P)}(ω_{i}(x, y), ω_{i}(x, y)) +
+ ∫_{ω_{i}(x,y)}^{y} ( ∑_{p=1}^{2n} a_{ijp}^{1}w_{p} + ∑_{p=1}^{n} b_{ijp}^{1}u_{p})(z_{i}(x, y; t), t)dt + F_{ij}^{2}(x, y), (2.18)
1 ≤ i ≤ l, 1 ≤ j ≤ k_{i},
u(x, y) = g(ω_{1}(x, y)) +
+ ∫_{ω_{1}(x,y)}^{y} ((−λ_{1}K_{1} + K_{2})w)(z_{1}(x, y; t), t)dt,  (2.19)

where a_{ijp}^{1}, b_{ijp}^{1}, F_{ij}^{2} are well-defined functions depending only on the coefficients and the right-hand side of the system (2.1); moreover, by (2.17) we have

g(ω_{1}(x, y)) = u(γ_{1(P)}(ω_{1}(x, y), ω_{1}(x, y)) =

= ∫_{0}^{ω_{1}(x,y)}(\frac{dγ_{1(P)}}{dy} K_{1} + K_{2})w(γ_{1(P)}(t), t)dt.

Let

φ_{q_{i}+j}^{1}(y) = w_{q_{i}+j}(γ_{1}(y), y), 0 ≤ y ≤ d_{1},
i = 1, …, s_{0}; j = 1, …, k_{i},
φ_{q_{i}+j}^{2}(y) = w_{q_{i}+j}(γ_{2}(y), y), 0 ≤ y ≤ d_{2},
i = 1, …, l_{0}; j = 1, …, k_{i},
φ_{q_{i}+j−k_{0}}^{2}(y) = w_{q_{i}+j}(γ_{2}(y), y), 0 ≤ y ≤ d_{2},
i = s_{0} + 1, …, l; j = 1, …, k_{i},

where k_{0} = ∑_{i=l_{0}+1}^{s_{0}} k_{i}, the numbers l_{0} and s_{0} are determined in §2, and the

number of components of the vector φ^{i}(y) is obviously equal to m_{i}, i = 1, 2.

Due to the requirements imposed on the curves γ_{1}, γ_{2} and L_{i}, we can see that

ω_{i}(γ_{1}(y), y) = \begin{cases}y & for i = 1, …, s_{0}, \\ τ_{i}^{1}(y) & for i = s_{0} + 1, …, l, \end{cases}

ω_{i}(γ_{2}(y), y) = \begin{cases}y & for i = 1, …, l_{0}, \\ τ_{i}^{2}(y) & for i = l_{0} + 1, …, s_{0}, \\ y & for i = s_{0} + 1, …, l, \end{cases}

where ω_{i}(x, y) ∈ C^{k}(\overline{D}_{1}), τ_{i}^{1}(y) ∈ C^{k}[0, d_{1}], i = s_{0} + 1, …, l, τ_{j}^{2}(y) ∈ C^{k}[0, d_{2}], j = l_{0} + 1, …, s_{0}, and τ_{l}^{1}(y) ≡ 0, if γ_{1} is a characteristic of the system (2.1). Analogously, τ_{l_{0}+1}^{2}(y) ≡ 0, if γ_{2} is a characteristic, and the remaining functions τ_{i}^{p}(y) satisfy the inequality τ_{i}^{p}(y) < y for 0 < y ≤ d_{p}, p = 1, 2.

Substituting the expressions for w(x, y) and u(x, y) from (2.18) and (2.19) into the boundary conditions (2.15) and (2.16), we get

\begin{matrix}G_{0}^{1}(y)φ^{1}(y) + ∑_{i=s_{0}+1}^{l} G_{i}^{1}(y)φ^{2}(τ_{i}^{1}(y)) + \\ +(T_{1}w)(y) + (T_{2}u)(y) = f_{3}(y), 0 ≤ y ≤ d_{1}, \\ G_{0}^{2}(y)φ^{2}(y) + ∑_{j=l_{0}+1}^{s_{0}} G_{j}^{2}(y)φ^{1}(τ_{j}^{2}(y)) + \\ +(T_{3}w)(y) + (T_{4}u)(y) = f_{4}(y), 0 ≤ y ≤ d_{2}, \end{matrix}  (2.20)

where G_{i}^{1}, G_{j}^{2}, i = s_{0} + 1, …, l; j = l_{0} + 1, …, s_{0} are well-defined matrices of the class C^{k−1}, and T_{i}, i = 1, …, 4, are linear integral operators.

Obviously, G_{0}^{i}, i = 1, 2, from (2.20) are the matrices of order m_{i} × m_{i} which can be represented as the product

G_{0}^{i} = Γ_{i} × V_{i}, i = 1, 2,

where Γ_{i} = (M_{i}, N_{i}), i = 1, 2, are rectangular m_{i} × 2n-matrices and V_{i}, i = 1, 2, are matrices of order 2n × m_{i} written in the form

V_{1} = (\begin{matrix}ν_{11} & ⋯ & ν_{1k_{1}} & ⋯ & ν_{s_{0}1} & ⋯ & ν_{s_{0}k_{s_{0}}} \\ λ_{1}ν_{11} & ⋯ & λ_{1}ν_{1k_{1}} & ⋯ & λ_{s_{0}}ν_{s_{0}1} & ⋯ & λ_{s_{0}}ν_{s_{0}k_{s_{0}}} \end{matrix} ) ,

V_{2} = (\begin{matrix}ν_{11} & ⋯ & ν_{l_{0}k_{l_{0}}} & ν_{s_{0}+1,1} & ⋯ & ν_{lk_{l}} \\ λ_{1}ν_{11} & ⋯ & λ_{l_{0}}ν_{l_{0}k_{l_{0}}} & λ_{s_{0}+1}ν_{s_{0}+1,1} & ⋯ & λ_{l}ν_{lk_{l}} \end{matrix}) .

Under the assumption that

det (Γ_{i} × V_{i})|_{OP_{i}} ≠ 0, i = 1, 2,  (2.21)

we can rewrite equation (2.20) in the form

\begin{matrix}φ^{1}(y) − ∑_{i=s_{0}+1}^{l} ∑_{j=l_{0}+1}^{s_{0}} G_{ij}^{3}(y)φ^{1}(τ_{ij}^{1}(y)) + \\ +(T_{5}w)(y) + (T_{6}u)(y) = f_{5}(y), 0 ≤ y ≤ d_{1}, \\ φ^{2}(y) − ∑_{i=l_{0}+1}^{s_{0}} ∑_{j=s_{0}+1}^{l} G_{ij}^{4}(y)φ^{2}(τ_{ij}^{2}(y)) + \\ +(T_{7}w)(y) + (T_{8}u)(y) = f_{6}(y), 0 ≤ y ≤ d_{2}, \end{matrix}  (2.22)

where τ_{ij}^{1}(y) = τ_{j}^{2}(τ_{i}^{1}(y)), τ_{ij}^{2}(y) = τ_{j}^{1}(τ_{i}^{2}(y)), G_{ij}^{3} and G_{ij}^{4} are matrices of orders m_{1}×m_{1} and m_{2}×m_{2}, and T_{5}, T_{6}, T_{7}, T_{8} are linear integral operators.

If γ_{1} or γ_{2} is a characteristic of the system (2.1), then we will have respectively τ_{lj}^{1}(y) = τ_{il}^{2}(y) ≡ 0, i, j = l_{0} + 1, …, s_{0}, and τ_{il_{0}+1}^{1}(y) = τ_{l_{0}+1j}^{2}(y) ≡ 0, i, j = s_{0} + 1, …, l. Therefore our discussion below will concern the remaining functions τ_{ij}^{1} and τ_{ij}^{2} which, as is easily verified, possess the following properties:

1) τ_{ij}^{p} ∈ C^{k}[0, d_{p}], τ_{ij}^{p}(0) = 0, p = 1, 2;

2) τ_{ij}^{p}, p = 1, 2, are strictly monotonically increasing functions;

3) τ_{ij}^{p}(y) < y, 0 < y ≤ d_{p}, p = 1, 2;

4) if the curves γ_{1} and γ_{2} do not have a common tangent line at the point O(0, 0), then

0 ≤ σ_{ij}^{p} = \frac{dτ_{ij}^{p}}{dy} (0) < 1, p = 1, 2,  (2.23)

or

σ_{ij}^{p} = \frac{dτ_{ij}^{p}}{dy} (0) = 1, p = 1, 2

otherwise.

The validity of property 1) is obvious. To prove the validity of the other properties, we shall give geometric interpretation of the functions τ_{ij}^{p}. Let a characteristic L_{i}(Q_{1}) be issued from Q_{1}(y, γ_{1}(y)) ∈ OP_{1} ⊂ γ_{1} to the intersection with γ_{2} at the point Q_{2}, and let a characteristic L_{j}(Q_{2}) be issued from Q_{2} to the intersection with γ_{1} at Q_{3}. It is easily seen that the ordinate of Q_{2} is equal to τ_{i}^{1}(y), while that of Q_{3} is equal to τ_{ij}^{1}(y) = τ_{j}^{2}(τ_{i}^{1}(y)). In a similar manner we can determine τ_{ij}^{2}(y) by interchanging the curves γ_{1} and γ_{2}. The validity of properties 2) and 3) follows directly from the geometrical meaning of the functions τ_{ij}^{p} if we take into account the requirements which have been imposed on the curves γ_{1}, γ_{2} and characteristics L_{i}.

Let us now prove the validity of property 4).

In a neighborhood V of O(0, 0) one can specify a family of characteristics L_{i} in the form of the equality L_{i} : μ_{i}(x, y) = const, where μ_{i} ∈ C^{k}(V), |∇μ_{i}||_{V} ≠ 0, i = 1, …, l. Since ∇μ_{i}(0, 0) = (\frac{∂μ_{i}}{∂x} , \frac{∂μ_{i}}{∂y})(0, 0) = c_{i}(1, λ_{i}(0, 0)), c_{i} = const ≠ 0, the Jacobian of transformation of the independent variables ỹ = μ_{i}(x, y), x̃ = μ_{j}(x, y) at the point O(0, 0) is different from zero for the fixed i and j, i ≠ j. Therefore, in a sufficiently small neighborhood V of the point O(0, 0) this mapping will be a diffeomorphism. In the plane of variables x̃, ỹ let us denote by γ̃_{i} the image of the curve γ_{i} ∩ V, i = 1, 2, under this mapping. By the assumptions on the curves γ_{1}, γ_{2} and characteristics L_{i}, L_{j}, the curves γ̃_{1}, γ̃_{2} are located in the angle x̃ ≥ 0, ỹ ≥ 0 and described by the equations γ̃_{1} : ỹ = γ̃_{1}(x̃), γ̃_{2} : ỹ = γ̃_{2}(x̃), 0 ≤ x̃ ≤ ε, ε > 0, where γ̃_{1}, γ̃_{2} ∈ C^{k}, 0 < γ̃_{1}(x̃) < γ̃_{2}(x̃) for 0 < x̃ ≤ ε and γ̃_{1}(0) = γ̃_{2}(0) = 0. Introduce into the consideration the function τ_{ij}^{1}(x̃), 0 ≤ x̃ ≤ ε, which under the above-mentioned transform corresponds to the function τ_{ij}^{1}(y). Let us draw the straight line parallel to the axis ox̃ from Q̃_{1}(x̃, γ̃_{1}(x̃)) ∈ γ̃_{1} to the intersection with the curve γ̃_{2} at Q̃_{2} and the straight line parallel to the axis oỹ from Q̃_{2} to the intersection with γ̃_{1} at Q̃_{3}. The value τ_{ij}^{1}(x̃) is equal to the abscissa of Q̃_{3}, and hence,

τ_{ij}^{1}(x̃) = γ_{2}^{−1}(γ̃_{1}(x̃)), 0 ≤ x̃ ≤ ε.

If γ_{1} and γ_{2} have a common tangent line at O(0, 0), then it is evident that γ_{1}^{(1)}(0) = γ_{2}^{(1)}(0), otherwise 0 ≤ γ_{1}^{(1)}(0) < γ_{2}^{(1)}(0). Consequently,

\frac{dτ_{ij}^{1}}{d_{x}} (0) = \frac{γ_{1}^{(1)}(0)}{γ_{2}^{(1)}(0)} = 1, if γ_{1} and γ_{2} have a common tangent line at O(0, 0),

and 0 ≤ \frac{dτ_{ij}^{1}}{d_{x}} (0) < 1, otherwise.

Let us now show that

\frac{dτ_{ij}^{1}}{dy} (0) = \frac{dτ_{ij}^{1}}{dx̃} (0) ,

which will imply the validity of property 4). As is easily seen, the functions τ_{ij}^{1}(y) and τ_{ij}^{1}(x̃) are connected by the relation

τ_{ij}^{1}(y) = χ_{2}(τ_{ij}^{1}(μ_{j}(γ_{1}(y), y)), γ̃_{1}(τ_{ij}^{1}(μ_{j}(γ_{1}(y), y))))

for sufficiently small y, where x = χ_{1}(x̃, ỹ), y = χ_{2}(x̃, ỹ) realize the mapping inverse to the given one, x̃ = μ_{j}(x, y), ỹ = μ_{i}(x, y).

Since

γ_{1}^{(1)}(0) = \frac{dμ_{i}(γ_{1}(y), y)}{dy}(\frac{dμ_{j}(γ_{1}(y), y)}{dy})^{−1}|_{y=0} =

= \frac{\frac{∂μ_{i}}{∂x} (0, 0)γ_{1}^{(1)}(0) + \frac{∂μ_{i}}{∂y} (0, 0)}{\frac{∂μ_{j}}{∂x} (0, 0)γ_{1}^{(1)}(0) + \frac{∂μ_{j}}{∂y} (0, 0)} ,

\frac{∂χ_{2}}{∂x̃} = \frac{−\frac{∂μ_{i}}{∂x}}{Δ} , \frac{∂χ_{2}}{∂ỹ} = \frac{\frac{∂μ_{j}}{∂x}}{Δ} ,

Δ = \frac{∂μ_{j}}{∂x} \frac{∂μ_{i}}{∂y} − \frac{∂μ_{i}}{∂x} \frac{∂μ_{j}}{∂y} ,

we have

\frac{dτ_{ij}^{1}}{dy} (0) = \frac{∂χ_{2}}{∂x̃} (0, 0)\frac{dτ_{ij}^{1}}{dx̃} (0)(\frac{∂μ_{j}}{∂x} (0, 0)γ_{1}^{(1)}(0) + \frac{∂μ_{j}}{∂y} (0, 0)) +

+ \frac{∂χ_{2}}{∂ỹ} (0, 0)γ_{1}^{(1)}(0)\frac{dτ_{ij}^{1}}{dx̃} (0)(\frac{∂μ_{j}}{∂x} (0, 0)γ_{1}^{(1)}(0) + \frac{∂μ_{j}}{∂y} (0, 0)) =

= \frac{dτ_{ij}^{1}}{dx̃} (0)(\frac{∂μ_{j}}{∂x} (0, 0)γ_{1}^{(1)}(0) + \frac{∂μ_{j}}{∂y} (0, 0)) ×

× ( − \frac{∂μ_{i}}{∂x} (0, 0) + \frac{∂μ_{j}}{∂x} \frac{\frac{∂μ_{i}}{∂x} (0, 0)γ_{1}^{(1)}(0) + \frac{∂μ_{i}}{∂y} (0, 0)}{\frac{∂μ_{j}}{∂x} (0, 0)γ_{1}^{(1)}(0) + \frac{∂μ_{j}}{∂y} (0, 0)})Δ^{−1} =

= \frac{dτ_{ij}^{1}}{dx̃} (0).

Now we can easily calculate the value

σ_{ij}^{1} = \frac{dτ_{ij}^{1}}{dy} (0) = \frac{dτ_{ij}^{1}}{dx̃} (0) = \frac{γ_{1}^{(1)}(0)}{γ_{2}^{(1)}(0)} =

= \frac{\frac{∂μ_{i}}{∂x} (0, 0)γ_{1}^{(1)}(0) + \frac{∂μ_{i}}{∂y} (0, 0)}{\frac{∂μ_{j}}{∂x} (0, 0)γ_{1}^{(1)}(0) + \frac{∂μ_{j}}{∂y} (0, 0)}( \frac{\frac{∂μ_{i}}{∂x} (0, 0)γ_{2}^{(1)}(0) + \frac{∂μ_{i}}{∂y} (0, 0)}{\frac{∂μ_{j}}{∂x} (0, 0)γ_{2}^{(1)}(0) + \frac{∂μ_{j}}{∂y} (0, 0)})^{−1} =

= \frac{(γ_{1}^{(1)}(0) + λ_{i}(0, 0))(γ_{2}^{(1)}(0) + λ_{j}(0, 0))}{(γ_{1}^{(1)}(0) + λ_{j}(0, 0))(γ_{2}^{(1)}(0) + λ_{i}(0, 0))} ,

since ∇μ_{i}(0, 0) = c_{i}(1, λ_{i}(0, 0)), c_{i} = const, i = 1, …, l. The case of the function τ_{ij}^{2} is considered in a similar way.

Remark. It is obvious that when conditions (2.21) are fulfilled, the prob-

lem (2.1)–(2.3) in the class C^{∘}_{α}^{k}(\overline{D}_{1}) is equivalent to the system of integrofunctional equations (2.18), (2.19), (2.22) with respect to unknown functions

u ∈ C^{∘}_{α}^{k}(\overline{D}_{1}), w ∈ C^{∘}_{α}^{k−1}(\overline{D}_{1}) and φ^{i} ∈ C^{∘}_{α}^{k−1}[0, d_{i}], i = 1, 2.

§ 4. Investigation of the System of Integro-Functional Equations (2.18), (2.19), (2.22)

Let us consider functional equations of the type

(K_{1p}φ)(y) = φ(y) − ∑_{i=s_{0}+1}^{l} ∑_{j=l_{0}+1}^{s_{0}} G_{ijp}^{3}(y)φ(τ_{ij}^{1}(y)) = g_{1}(y), (2.24)
0 ≤ y ≤ d_{1}, p = 0, 1, …, k − 1,
(K_{2p}ψ)(y) = ψ(y) − ∑_{i=l_{0}+1}^{s_{0}} ∑_{j=s_{0}+1}^{l} G_{ijp}^{4}(y)ψ(τ_{ij}^{2}(y)) = g_{2}(y), (2.25)
0 ≤ y ≤ d_{2}, p = 0, 1, …, k − 1,

where

G_{ijp}^{3}(y) = G_{ij}^{3}(y)(\frac{dτ_{ij}^{1}}{dy} (y))^{p}, G_{ijp}^{4}(y) = G_{ij}^{4}(y)(\frac{dτ_{ij}^{2}}{dy} (y))^{p},

and the values G_{ij}^{3}, G_{ij}^{4}, τ_{ij}^{1}, τ_{ij}^{2} are determined in equations (2.22).

Remark. As is easily seen, the expressions K_{1p}φ^{1} and K_{2p}φ^{2} for p = 0 coincide with the functional parts of equations (2.22). Moreover, if we differentiate p times the expression (K_{10}φ)(y) with respect to y, then in the expression obtained after differentiation the sum of those summands which involve the function φ(y) with the derivative φ^{(p)}(y), yields (K_{1p}φ^{(p)})(y). Similar remark holds also for the operator K_{2p}.

We shall consider equations (2.24) and (2.25) in the spaces C^{∘}_{k−1+α−p}[0, d_{1}]

and C^{∘}_{k−1+α−p}[0, d_{2}].

Denote by m̃_{1} the number of characteristics taking into account their multiplicities, issued from the point P_{1} into the domain D_{1} and intersecting an open segment OP_{2}. The number m̃_{2} can be defined in a similar manner by substituting the point P_{1} by P_{2} and OP_{2} by an open segment OP_{1}. It is easily seen that m̃_{1}m̃_{2} = 0 if, for example, either l_{0} = 2n or m_{1}m_{2} = 0.

Obviously, the columns 2n × m_{i} of the matrix V_{i}, i = 1, 2, are composed of the well-defined columns of the matrix K, where the matrices K, V_{1}, V_{2} have been introduced in §3. Denote by Ṽ_{i}, i = 1, 2, the matrix of order 2n × (2n − m_{i}), composed of the remaining columns of the matrix K, i.e., of the columns not belonging to the matrix V_{i}.

We have the following

Lemma 2.1. Let either m̃_{1}m̃_{2} = 0 or at least one of the equalities (Γ_{1} × Ṽ_{1})|_{OP_{1}} = 0 or (Γ_{2} × Ṽ_{2})|_{OP_{2}} = 0 hold. Then equations (2.24) and (2.25)

are uniquely solvable in the spaces C^{∘}_{k−1+α−p}[0, d_{1}] and C^{∘}_{k−1+α−p}[0, d_{2}] for all k ≥ 2, α ≥ 0.

The proof follows from the fact that under the conditions of Lemma 2.1 either all values τ_{ij}^{1} ≡ τ_{ij}^{2} ≡ 0 or all matrices G_{ij}^{3} ≡ G_{ij}^{4} ≡ 0. In both cases

the operators K_{1p} and K_{2p} are identical in the spaces C^{∘}_{k−1+α−p}[0, d_{1}] and

C^{∘}_{k−1+α−p}[0, d_{2}], i.e., K_{1p}φ = φ, K_{2p}ψ = ψ.

Consider the functions

h_{1}(ρ) = ∑_{i=s_{0}+1}^{l} ∑_{j=l_{0}+1}^{s_{0}}(σ_{ij}^{1})^{ρ−1}∥G_{ij}^{3}(0)∥, −∞ < ρ < ∞,

h_{2}(ρ) = ∑_{i=l_{0}+1}^{s_{0}} ∑_{j=s_{0}+1}^{l}(σ_{ij}^{2})^{ρ−1}∥G_{ij}^{4}(0)∥, −∞ < ρ < ∞,

where ∥ · ∥ is the norm of the matrix operator, acting from one Euclidean space of the other.

Assume that the curves γ_{1} and γ_{2} do not have a common tangent line at the point O(0, 0). If for some values of the indices i, j, ∥σ_{ij}^{1}G_{ij}^{3}(0)∥ and ∥σ_{ij}^{2}G_{ij}^{4}(0)∥ are different from zero, then by (2.23) the functions h_{1} and h_{2} are continuous and strictly monotonically decreasing on (−∞, ∞); moreover, lim_{ρ→−∞} h_{i}(ρ) = +∞ and lim_{ρ→+∞} h_{i}(ρ) = 0, i = 1, 2. Therefore there

exist unique real numbers ρ_{1} and ρ_{2} such that h_{1}(ρ_{1}) = 1 and h_{2}(ρ_{2}) = 1. If, however, all the values ∥σ_{ij}^{1}G_{ij}^{3}(0)∥ = 0, then we assume ρ_{1} = −∞. Similarly, assume ρ_{2} = −∞ if all the values ∥σ_{ij}^{2}G_{ij}^{4}(0)∥ = 0. It is evident that all these cases are realizable if either m̃_{1}m̃_{2} = 0 or at least one of the equalities (Γ_{i} × Ṽ_{i})(0) = 0, i = 1, 2, holds.

Lemma 2.2. Assume that the curves γ_{1}, γ_{2} do not have a common tangent line at the point O(0, 0), and m̃_{1}m̃_{2} ≠ 0, (Γ_{i} × Ṽ_{i})|_{OP_{i}} ≠ 0, i = 1, 2. Then for k + α > ρ_{0} the equations (2.24) and (2.25) are uniquely solvable

in the spaces C^{∘}_{k−1+α−p}[0, d_{1}] and C^{∘}_{k−1+α−p}[0, d_{2}], and the estimates

∥(K_{1p}^{−1}g_{1})(y)∥_{R^{m_{1}}} = ∥φ(y)∥_{R^{m_{1}}} ≤
≤ c_{1}y^{k−1+α−p}∥g_{1}∥_{C^{∘}_{k−1+α−p}[0,d_{1}]},  (2.26)
∥(K_{2p}^{−1}g_{2})(y)∥_{R^{m_{2}}} = ∥ψ(y)∥_{R^{m_{2}}} ≤
≤ c_{2}y^{k−1+α−p}∥g_{2}∥_{C^{∘}_{k−1+α−p}[0,d_{2}]},  (2.27)

hold, where c_{1}, c_{2} are positive constants not depending on g_{1}, g_{2}.

Proof. Condition k + α > ρ_{0} imples

h_{1}(k + α) = ∑_{i=s_{0}+1}^{l} ∑_{j=l_{0}+1}^{s_{0}}(σ_{ij}^{1})^{k−1+α}∥G_{ij}^{3}(0)∥ < 1.

Therefore, owing to the continuity of τ_{ij}^{1}, \frac{dτ_{ij}^{1}}{dy}, G_{ij}^{3} and equalities \frac{dτ_{ij}^{1}}{dy} (0) = σ_{ij}^{1} < 1, there exist positive numbers ε (ε < d_{1}), δ and β such that for 0 ≤ y ≤ ε the inequalities

∥G_{ij}^{3}(y)∥ ≤ ∥G_{ij}^{3}(0)∥ + δ,  (2.28)

\frac{dτ_{ij}^{1}}{dy} (y) ≤ σ_{ij}^{1} + δ, ∥G_{ijp}^{3}(y)∥ =

= ∥(\frac{dτ_{ij}^{1}}{dy} (y))^{p}G_{ij}^{3}(y)∥ ≤ (σ_{ij}^{1} + δ)^{p}(∥G_{ij}^{3}(0)∥ + δ),  (2.29)
τ_{ij}^{1}(y) ≤ (σ_{ij}^{1} + δ)y,  (2.30)
∑_{i=s_{0}+1}^{l} ∑_{j=l_{0}+1}^{s_{0}}(σ_{ij}^{1} + δ)^{k−1+α}(∥G_{ij}^{3}(0)∥ + δ) = β > 1.  (2.31)

are valid.

Since the functions τ_{ij}^{1} possess properties 1)–3) cited in §3, there exists a natural number q_{0} such that for q ≥ q_{0}

τ_{i_{q}j_{q}}^{1}(τ_{i_{q−1}j_{q−1}}^{1}(⋯ (τ_{i_{1}j_{1}}^{1}(y)) ⋯)) ≤ ε, 0 ≤ y ≤ d_{1},  (2.32)

where s_{0} + 1 ≤ i_{s} ≤ l, l_{0} + 1 ≤ j_{s} ≤ s_{0}, s = 1, …, q.

Because of the property 3), for the functions τ_{ij}^{1} and the inequalities (2.30) and (2.32) we have

τ_{i_{q}j_{q}}^{1}(τ_{i_{q−1}j_{q−1}}^{1}(⋯ (τ_{i_{1}j_{1}}^{1}(y)) ⋯)) =

= τ_{i_{q}j_{q}}^{1}(τ_{i_{q−1}j_{q−1}}^{1}(⋯ (τ_{i_{q_{0}} j_{q_{0}}}^{1} (τ_{i_{q_{0}−1}j_{q_{0}−1}}^{1}(⋯ (τ_{i_{1}j_{1}}^{1}(y)) ⋯))) ⋯)) ≤

≤ (σ_{i_{q}j_{q}}^{1} + δ)τ_{i_{q−1}j_{q−1}}^{1}( ⋯ (τ_{i_{q_{0}} j_{q_{0}}}^{1} (τ_{i_{q_{0}−1}j_{q_{0}−1}}^{1}(⋯ (τ_{i_{1}j_{1}}^{1}(y)) ⋯))) ⋯) ≤

≤ ⋯ ≤ (σ_{i_{q}j_{q}}^{1} + δ)(σ_{i_{q−1}j_{q−1}}^{1} + δ) ⋯ (σ_{i_{q_{0}+1}j_{q_{0}+1}}^{1} + δ) ×
×τ_{i_{q_{0}} j_{q_{0}}}^{1} (τ_{i_{q_{0}−1}j_{q_{0}−1}}^{1}(⋯ (τ_{i_{1}j_{1}}^{1}(y)) ⋯)) ≤
≤ [ ∏_{s=q_{0}+1}^{q}(σ_{i_{s}j_{s}}^{1} + δ)]y, 0 ≤ y ≤ d_{1}, q > q_{0}.  (2.33)

Introduce into the consideration the operators Λ_{1p}, K_{1p}^{−1} acting by the formulas

(Λ_{1p}φ)(y) = ∑_{i=s_{0}+1}^{l} ∑_{j=l_{0}+1}^{s_{0}} G_{ijp}^{3}(y)φ(τ_{ij}^{1}(y)),

K_{1p}^{−1} = I + ∑_{q=1}^{∞} Λ_{1p}^{q},

where I is the identical operator. Obviously, K_{1p}^{−1} is formally inverse to K_{1p}, i.e., K_{1p}K_{1p}^{−1} = K_{1p}^{−1}K_{1p} = I. Therefore it suffices for us to show that

K_{1p}^{−1} is continuous in the space C^{∘}_{k−1+α−p}[0, d_{1}].

It can be easily seen that the expression Λ_{1p}^{q}g_{1} is a sum consisting of the summands of the form

J_{i_{1}j_{1}···i_{q}j_{q}}(y) =
= G_{i_{1}j_{1}p}^{3}(y)G_{i_{2}j_{2}p}^{3}(τ_{i_{1}j_{1}}^{1}(y))G_{i_{3}j_{3}p}^{3}(τ_{i_{2}j_{2}}^{1}(τ_{i_{1}j_{1}}^{1}(y))) ⋯
⋯ G_{i_{q}j_{q}p}^{3}(τ_{i_{q−1}j_{q−1}}^{1}(τ_{i_{q−2}j_{q−2}}^{1}(⋯ (τ_{i_{1}j_{1}}^{1}(y)) ⋯))) ×
×g_{1}(τ_{i_{q}j_{q}}^{1}(τ_{i_{q−1}j_{q−1}}^{1}(⋯ (τ_{i_{1}j_{1}}^{1}(y)) ⋯))),

where s_{0} + 1 ≤ i_{s} ≤ l, l_{0} + 1 ≤ j_{s} ≤ s_{0}, s = 1, …, q.

Let

max_{s_{0}+1≤i≤l} max_{l_{0}+1≤j≤s_{0}} max_{0≤y≤d_{i}} ∥G_{ijp}^{3}(y)∥_{R^{m_{1}}} = η_{p}.

By virtue of (2.28)–(2.33) we have

∥J_{i_{1}j_{1}···i_{q}j_{q}}(y)∥_{R^{m_{1}}} ≤ ∥G_{i_{1}j_{1}p}^{3}(y)∥ ⋯

⋯ ∥G_{i_{q_{0}} j_{q_{0}} p}^{3}(τ_{i_{q_{0}−1}j_{q_{0}−1}}^{1}(τ_{i_{q_{0}−2}j_{q_{0}−2}}^{1}(⋯ (τ_{i_{1}j_{1}}^{1}(y)) ⋯)))∥ ×

×∥G_{i_{q_{0}+1}j_{q_{0}+1}p}^{3}(τ_{i_{q_{0}} j_{q_{0}}}^{1} (τ_{i_{q_{0}−1}j_{q_{0}−1}}^{1}(⋯ (τ_{i_{1}j_{1}}^{1}(y)) ⋯)))∥ ⋯

⋯ ∥G_{i_{q}j_{q}p}^{3}(τ_{i_{q−1}j_{q−1}}^{1}(⋯ (τ_{i_{1}j_{1}}^{1}(y)) ⋯)))∥ ×
×∥g_{1}(τ_{i_{q}j_{q}}^{1}(⋯ τ_{i_{1}j_{1}}^{1}(y)) ⋯))∥_{R^{m_{1}}} ≤

≤ η_{p}^{q_{0}}(σ_{i_{q_{0}+1}j_{q_{0}+1}} + δ)^{p}(∥G_{i_{q_{0}+1}j_{q_{0}+1}}^{3}(0)∥ + δ) ⋯

⋯ (σ_{i_{q}j_{q}}^{1} + δ)^{p}(∥G_{i_{q}j_{q}}^{3}(0)∥ + δ) ×

×|τ_{i_{q}j_{q}}^{1}( ⋯ (τ_{i_{1}j_{1}}^{1}(y)) ⋯)|^{k−1+α−p}∥g_{1}∥_{C^{∘}_{k−1+α−p}[0,d_{1}]} ≤

≤ η_{p}^{q_{0}}[ ∏_{s=q_{0}+1}^{q}(σ_{i_{s}j_{s}}^{1} + δ)^{p}(∥G_{i_{s}j_{s}}^{3}(0)∥ + δ)] ×

×[ ∏_{s=q_{0}+1}^{q}(σ_{i_{s}j_{s}}^{1} + δ)^{k−1+α−p}]y^{k−1+α−p}∥g_{1}∥_{C^{∘}_{k−1+α−p}[0,d_{1}]} =

= η_{p}^{q_{0}}[ ∏_{s=q_{0}+1}^{q}(σ_{i_{s}j_{s}}^{1} + δ)^{k−1+α}(∥G_{i_{s}j_{s}}^{3}(0)∥ + δ)]

×y^{k−1+α−p}∥g_{1}∥_{C^{∘}_{k−1+α−p}[0,d_{1}]}  (2.34)

for q > q_{0}, g_{1} ∈ C^{∘}_{k−1+α−p}[0, d_{1}], and

∥J_{i_{1}j_{1}···i_{q}j_{q}}(y)∥_{R^{m_{1}}} ≤

≤ η_{p}^{q_{0}} |τ_{i_{q}j_{q}}^{1}( ⋯ (τ_{i_{1}j_{1}}^{1}(y)) ⋯)|^{k−1+α−p}∥g_{1}∥_{C^{∘}_{k−1+α−p}[0,d_{1}]} ≤

≤ η_{p}^{q}y^{k−1+α−p}∥g_{1}∥_{C^{∘}_{k−1+α−p}[0,d_{1}]}  (2.35)

for 0 < q ≤ q_{0}.

Because of (2.34), (2.35) and (2.31) we have

∥(Λ_{1p}^{q}g_{1})(y)∥_{R^{m_{1}}} = ∥ ∑_{i_{1},j_{1},...,i_{q},j_{q}} J_{i_{1}j_{1}···i_{q}j_{q}}(y)∥_{R^{m_{1}}} ≤

≤ ( ∑_{i_{1},j_{1},...,i_{q_{0}} ,j_{q_{0}}} 1)^{q_{0}}η_{p}^{q_{0}}[ ∑_{i=s_{0}+1}^{l} ∑_{j=l_{0}+1}^{s_{0}}(σ_{ij}^{1} + δ)^{k−1+α} ×

×(∥G_{ij}^{3}(0)∥ + δ)]^{q−q_{0}}y^{k−1+α−p}∥g_{1}∥_{C^{∘}_{k−1+α−p}[0,d_{1}]} =

= c_{3}β^{q}y^{k−1+α−p}∥g_{1}∥_{C^{∘}_{k−1+α−p}[0,d_{1}]}  (2.36)

for q > q_{0}, and

∥(Λ_{1p}^{q}g_{1})(y)∥_{R^{m_{1}}} ≤ c_{4}y^{k−1+α−p}∥g_{1}∥_{C^{∘}_{k−1+α−p}[0,d_{1}]}  (2.37)

for 0 < q ≤ q_{0}, where

c_{3} = η_{p}^{q_{0}}β^{−q_{0}}( ∑_{i_{1},j_{1},...,i_{q_{0}} ,j_{q_{0}}} 1)^{q_{0}}, c_{4} = η_{p}^{q}( ∑_{i_{1},j_{1},...,i_{q},j_{q}} 1).

From (2.36) and (2.37) we finally obtain

∥(K_{1p}^{−1}g_{1})(y)∥_{R^{m_{1}}} = ∥φ(y)∥_{R^{m_{1}}} ≤

≤ ∥g_{1}(y)∥_{R^{m_{1}}} + ∑_{q=1}^{q_{0}} ∥(Λ_{1p}^{q}g_{1})(y)∥_{R^{m_{1}}} + ∑_{q=q_{0}+1}^{∞} ∥(Λ_{1p}^{q}g_{1})(y)∥_{R^{m_{1}}} ≤

≤ (1 + c_{4}q_{0} + c_{3} \frac{β^{q_{0}+1}}{1 − β} )y^{k−1+α−p}∥g_{1}∥_{C^{∘}_{k−1+α−p}[0,d_{1}]},

whence it follows that the operator K_{1p}^{−1} is continuous in the space

C^{∘}_{k−1+α−p}[0, d_{1}], and the estimate (2.26) is valid. The operator K_{2p}^{−1} is considered in a similar manner. ■

Lemma 2.3. Let the curves γ_{1}, γ_{2} have a common tangent line at the point O(0, 0), and m̃_{1}m̃_{2} ≠ 0, (Γ_{i} × Ṽ_{i})|_{OP_{i}} ≢ 0, i = 1, 2. Then for h_{i}(1) < 1, i = 1, 2, the equations (2.24) and (2.25) are uniquely solvable in the

spaces C^{∘}_{k−1+α−p}[0, d_{1}] and C^{∘}_{k−1+α−p}[0, d_{2}] for all k ≥ 2, α ≥ 0, and the estimates (2.26) and (2.27) take place.

The proof of Lemma 2.3 does not differ from that of Lemma 2.2 if in inequalities (2.28)–(2.31) we substitute the different from zero numbers σ_{ij}^{1} by unity.

It easily follows from Lemmas 2.1–2.3 that if either m̃_{1}m̃_{2} = 0 or at least one of the equalities (Γ_{1} × Ṽ_{1})(O) = 0 or (Γ_{2} × Ṽ_{2})(O) = 0 holds, then the assertion of Lemma 2.1 is valid for all k ≥ 2, α ≥ 0.

Theorem 2.1. Let the conditions (2.21) be fulfilled. If either m̃_{1}m̃_{2} = 0 or at least one of the equalities (Γ_{1} × Ṽ_{1})(O) = 0 or (Γ_{2} × Ṽ_{2})(O) = 0 holds,

then the problem (2.1)–(2.3) is uniquely solvable in the class C^{∘}_{α}^{k}(\overline{D}_{1}) for all k ≥ 2, α ≥ 0.

Theorem 2.2. Let the conditions (2.21) be fulfilled, and m̃_{1}m̃_{2} ≠ 0, (Γ_{i} × Ṽ_{i})(O) ≠ 0, i = 1, 2. If the curves γ_{1}, γ_{2} do not have a common tangent line at the point O(0, 0), then for k + α > ρ_{0} the problem (2.1)–(2.3) is uniquely

solvable in the class C^{∘}_{α}^{k}(\overline{D}_{1}).

Theorem 2.3. Let the conditions (2.21) be fulfilled, and m̃_{1}m̃_{2} ≠ 0, (Γ_{i} × Ṽ_{i})(O) ≠ 0, i = 1, 2. If the curves γ_{1}, γ_{2} have a common tangent line at the point O(0, 0), then for h_{i}(1) < 1, i = 1, 2, the problem (2.1)–(2.3) is

uniquely solvable in the class C^{∘}_{α}^{k}(\overline{D}_{1}) for all k ≥ 2, α ≥ 0.

Before passing to the proof of Theorems 2.1–2.3, let us make some remarks.

1. Since the 2n × m_{i}-matrix V_{i}, i = 1, 2, has a maximal rank equal to m_{i}, for any normally hyperbolic system (2.1) one can always indicate boundary conditions (2.2), (2.3) such that the conditions (2.21) are fulfilled when conditions cited in §2 hold.

2. The values ρ_{0} and h_{i}(1), i = 1, 2, in Theorems 2.2 and 2.3 depend only on the coefficients A, B, C, M_{i}, N_{i}, S_{i}, i = 1, 2, of the problem (2.1)–(2.3) and the direction of the tangents to γ_{1} and γ_{2} at the point O(0, 0).

3. When conditions of Theorems 2.1–2.3 are violated, as it has been shown in Chapter I for one equation of hyperbolic type, the problem (2.1)– (2.3) may turn out to be ill-posed. In particular, the homogeneous problem corresponding to (2.1)–(2.3) may have an infinite number of linearly independent solutions.

Proof of Theorems 2.1–2.3. Using the method of successive approximations we solve the system of equations (2.18), (2.19) and (2.2) with respect to

unknown functions u ∈ C^{∘}_{α}^{k}(\overline{D}_{1}), w ∈ C^{∘}_{α}^{k−1}(\overline{D}_{1}) and φ^{i} ∈ C^{∘}_{α}^{k−1}[0, d_{i}],

i = 1, 2.

Assume

u_{0}(x, y) ≡ 0, w_{0}(x, y) ≡ 0, φ_{0}^{i}(y) ≡ 0, i = 1, 2,

w_{q_{i}+j,m}(x, y) = φ_{q_{i}+j,m}^{i(P)}(ω_{i}(x, y)) + ∫_{ω_{i}(x,y)}^{y} ( ∑_{p=1}^{2n} a_{ijp}^{1}w_{p,m−1} +

+ ∑_{p=1}^{n} b_{ijp}^{1}u_{p,m−1})(z_{i}(x, y; t), t)dt + F_{ij}^{2}(x, y),  (2.38)
1 ≤ i ≤ l, 1 ≤ j ≤ k_{i},

u_{m}(x, y) = ∫_{0}^{ω_{1}(x,y)}(\frac{dγ_{1(P)}}{dy} K_{1} + K_{2})w_{m−1}(γ_{1(P)}(t), t)dt +

+ ∫_{ω_{1}(x,y)}^{y} ((−λ_{1}K_{1} + K_{2})w_{m−1})(z_{1}(x, y; t), t)dt,  (2.39)

where

φ_{q_{i}+j,m}^{i(P)}(ω_{i}(x, y)) = \begin{cases}φ_{q_{i}+j,m}^{i(P)}(ω_{i}(x, y)), & 1 ≤ i ≤ l_{0}, 1 ≤ j ≤ k_{i}, \\ φ_{q_{i}+j,m}^{1}(ω_{i}(x, y)), & l_{0} + 1 ≤ i ≤ s_{0}, 1 ≤ j ≤ k_{i}, \\ φ_{q_{i}+j−k_{0},m}^{2}(ω_{i}(x, y)), & s_{0} + 1 ≤ i ≤ l, 1 ≤ j ≤ k_{i}, \\ & k_{0} = ∑_{i=l_{0}+1}^{s_{0}} k_{i}, \end{cases}

The values φ_{m}^{1}(y) and φ_{m}^{2}(y) are determined from the equations

(K_{10}φ_{m}^{1})(y) + (T_{5}w_{m−1})(y) + (T_{6}u_{m−1})(y) = f_{5}(y)  (2.40)

and

(K_{20}φ_{m}^{2})(y) + (T_{7}w_{m−1})(y) + (T_{8}u_{m−1})(y) = f_{6}(y).  (2.41)

The operators K_{10} and K_{20} here act by formulas (2.24), (2.25) for p = 0.

We rewrite the system of equations (2.18), (2.19) in a more convenient form

w_{m}(x, y) = φ̃_{P,m}(x, y) +
+ ∑_{i=1}^{l} ∫_{ω_{i}(x,y)}^{y} (Ω_{1i}w_{m−1} + Ω_{2i}u_{m−1})(z_{i}(x, y; t), t)dt + F^{2}(x, y), (2.42)

u_{m}(x, y) = ∫_{0}^{ω_{1}(x,y)}Ω̃_{3}w_{m−1}(γ_{1(P)}(t), t)dt +

+ ∫_{ω_{1}(x,y)}^{y} Ω̃_{4}w_{m−1}(z_{1}(x, y; t), t)dt,  (2.43)

where the (q_{i} + j)-th component of the vector φ̃_{P,m}(x, y) is equal to φ_{q_{i}+j,m}^{i(P)}(ω_{i}(x, y)), 1 ≤ i ≤ l, 1 ≤ j ≤ k_{i}, and Ω_{1i}, Ω_{2i}, Ω̃_{3}, Ω̃_{4} are welldefined matrices.

It is easily seen that the operators T_{5}w_{m−1} + T_{6}u_{m−1} and T_{7}w_{m−1} + T_{8}u_{m−1} from (2.40) and (2.41) can be represented in the form

T_{9}(w_{m−1}, u_{m−1})(y) = (T_{5}w_{m−1} + T_{6}u_{m−1})(y) =

= ∑_{i=s_{0}+1}^{l} ∫_{τ_{i}^{1}(y)}^{y} (E_{1i}^{1}w_{m−1} + E_{2i}^{1}u_{m−1})(z_{i}(γ_{1}(y), y; t), t)dt +

+ ∑_{j=s_{0}+1}^{l} ∑_{i=l_{0}+1}^{s_{0}} ∫_{τ_{ji}^{1}(y)}^{τ_{j}^{1}(y)}(E_{3ij}^{1}w_{m−1} + E_{4ij}^{1}u_{m−1})(z_{i}(γ_{2}(τ_{j}^{1}(y)), τ_{j}^{1}(y); t), t)dt,

T_{10}(w_{m−1}, u_{m−1})(y) = (T_{7}w_{m−1} + T_{8}u_{m−1})(y) =

= ∑_{i=l_{0}+1}^{s_{0}} ∫_{τ_{i}^{2}(y)}^{y} (E_{1i}^{2}w_{m−1} + E_{2i}^{2}u_{m−1})(z_{i}(γ_{2}(y), y; t), t)dt +

+ ∑_{j=l_{0}+1}^{s_{0}} ∑_{i=s_{0}+1}^{l} ∫_{τ_{ji}^{2}(y)}^{τ_{j}^{2}(y)}(E_{3ij}^{2}w_{m−1} + E_{4ij}^{2}u_{m−1})(z_{i}(γ_{1}(τ_{j}^{2}(y)), τ_{j}^{2}(y); t), t)dt,

where E_{1i}^{p},E_{2i}^{p}, E_{3ij}^{p}, E_{4ij}^{p}, p = 1, 2, are well-defined matrices.

The following estimates hold:

∥u_{m+1}(x, y) − u_{m}(x, y)∥ ≤ M^{∗} \frac{M_{∗}^{m}}{m!} y^{m+k+α−1},  (2.44)
∥w_{m+1}(x, y) − w_{m}(x, y)∥ ≤ M^{∗} \frac{M_{∗}^{m}}{m!} y^{m+k+α−1},  (2.45)
∥φ_{m+1}^{1}(y) − φ_{m}^{1}(y)∥ ≤ M^{∗} \frac{M_{∗}^{m}}{m!} y^{m+k+α−1},  (2.46)
∥φ_{m+1}^{2}(y) − φ_{m}^{2}(y)∥ ≤ M^{∗} \frac{M_{∗}^{m}}{m!} y^{m+k+α−1},  (2.47)

where M_{∗} and M^{∗} are sufficiently large positive numbers not depending on m.

Due to the requirements imposed on f_{1}, f_{2} and F, we have f_{5} ∈C^{∘}_{α}^{k−1}[0, d_{1}],

f_{6} ∈ C^{∘}_{α}^{k−1}[0, d_{2}], F ∈ C^{∘}_{α}^{k−1}(\overline{D}_{1}). Therefore, it is obvious that the estimates

∥∂^{i,j}F^{2}(x, y)∥ ≤ Θ_{1}y^{k−1+α−(i+j)},  (2.48)
(x, y) ∈ \overline{D}_{1}, 0 ≤ i + j ≤ k − 1,
∥∂^{i}f_{4+j}(y)∥ ≤ Θ_{1+j}y^{k−1+α−i},  (2.49)
0 ≤ y ≤ d_{j}, j = 1, 2, 0 ≤ i ≤ k − 1,
Θ_{i} = const > 0, i = 1, 2, 3,

are valid since, by the assumption, D_{1} is such that for any point z = x + \sqrt{−1}y ∈ \overline{D}_{1} the two-sided estimate y ≤ |z| = \sqrt{x^{2} + y^{2}} ≤

(max_{i=1,2} max_{0≤y≤d_{i}} |γ_{i}^{(1)}(y)|)y is valid.

Since u_{0} ≡ w_{0} ≡ 0, φ_{0}^{1} ≡ φ_{0}^{2} ≡ 0 and under the conditions of Theorems 2.1–2.3 the estimates (2.26), (2.27) are valid for p = 0, we have from (2.40), (2.41) and (2.49) that

∥φ_{1}^{i}(y) − φ_{0}^{i}(y)∥ = ∥φ_{1}^{i}(y)∥ ≤ c_{3}Θ_{4}y^{k−1+α}, i = 1, 2  (2.50)
c_{3} = max(c_{1}, c_{2}), Θ_{4} = max(Θ_{2}, Θ_{3}).

In its turn, it follows from (2.50) that

∥φ̃_{P,1}(x, y) − φ̃_{P,0}(x, y)∥ = ∥φ̃_{P,1}(x, y)∥ =

= ∑_{1≤i≤l} ∑_{1≤j≤k_{i}} |φ_{q_{i}+j,1}^{i(P)}(ω_{i}(x, y))| ≤

≤ ∑_{1≤i≤l} ∑_{1≤j≤k_{i}} c_{3}Θ_{4}(ω_{i}(x, y))^{k−1+α} ≤ 2nc_{3}Θ_{4}y^{k−1+α},  (2.51)

since ∑_{1≤i≤l} ∑_{1≤j≤k_{i}} 1 = 2n, and as noted in §3, 0 ≤ ω_{i}(x, y) ≤ y, i = 1, …, l.

Now, by virtue of (2.48) and (2.51), from (2.42) and (2.43) we have

∥w_{1}(x, y) − w_{0}(x, y)∥ = ∥w_{1}(x, y)∥ ≤
≤ ∥φ̃_{P,1}(x, y)∥ + ∥F^{2}(x, y)∥ ≤
≤ 2nc_{3}Θ_{4}y^{k−1+α} + Θ_{1}y^{k−1+α} = (2nc_{3}Θ_{4} + Θ_{1})y^{k−1+α}, (2.52)
∥u_{1}(x, y) − u_{0}(x, y)∥ = ∥u_{1}(x, y)∥ = 0.  (2.53)

Under the assumption that the estimates (2.44)–(2.47) are valid for m, m > 0, let us prove their validity for m + 1 for sufficiently large M_{∗} and M^{∗}.

From (2.40) we have

(K_{10}(φ_{m+2}^{1} − φ_{m+1}^{1}))(y) = −T_{9}(w_{m+1} − w_{m}, u_{m+1} − u_{m})(y). (2.54)

Furthermore, for the right-hand side of equation (2.54) the estimate

∥T_{9}(w_{m+1} − w_{m}, u_{m+1} − u_{m})(y)∥ ≤

≤ ∑_{i=s_{0}+1}^{l} ∫_{τ_{i}^{1}(y)}^{y} (∥E_{1i}^{1}∥∥w_{m+1} − w_{m}∥ +

+∥E_{2i}^{1}∥∥u_{m+1} − u_{m}∥)(z_{i}(γ_{1}(y), y; t), t)dt +

+ ∑_{j=s_{0}+1}^{l} ∑_{i=l_{0}+1}^{s_{0}} ∫_{τ_{ij}^{1}(y)}^{τ_{j}^{1}(y)}(∥E_{3ij}^{1}∥∥w_{m+1} − w_{m}∥ +

+∥E_{4ij}^{1}∥∥u_{m+1} − u_{m}∥)(z_{i}(γ_{2}(τ_{j}^{1}(y)), τ_{j}^{1}(y); t), t)dt.  (2.55)

is valid.

The largest of the numbers max_{y,t} ∥E_{1j}^{p}(y, t)∥, max_{y,t} ∥E_{2j}^{p}(y, t)∥,

max_{y,t} ∥E_{3ij}^{p}(y, t)∥, max_{y,t} ∥E_{4ij}^{p}(y, t)∥, we denote by ξ_{p}, p = 1, 2. Since 0 ≤

τ_{ji}^{1}(y) ≤ τ_{j}^{1}(y) ≤ y and owing to (2.44) and (2.45), we have from (2.55) that

∥T_{9}(w_{m+1} − w_{m}, u_{m+1} − u_{m})(y)∥ ≤

≤ ξ_{1}M^{∗} \frac{M_{∗}^{m}}{m!} ( ∑_{i=s_{0}+1}^{l} ∫_{τ_{i}^{1}(y)}^{y} 2t^{m+k+α−1}dt +

+ ∑_{j=s_{0}+1}^{l} ∑_{i=l_{0}+1}^{s_{0}} ∫_{τ_{ij}^{1}(y)}^{τ_{j}^{1}(y)}2t^{m+k+α−1}dt) ≤

≤ 2ξ_{1}M^{∗} \frac{M_{∗}^{m}}{m!} ( ∑_{i=s_{0}+1}^{l} 1 + ∑_{j=s_{0}+1}^{l} ∑_{i=l_{0}+1}^{s_{0}} 1) ∫_{0}^{y} t^{m+k+α−1}dt ≤

≤ 2ξ_{1}M^{∗} \frac{M_{∗}^{m}}{m!} (l + l^{2})\frac{1}{m + k + α}y^{m+k+α} ≤

≤ 2(l + l^{2})ξ_{1}M^{∗} \frac{M_{∗}^{m}}{(m + 1)!}y^{m+1+k+α−1}.  (2.56)

Now (2.54), (2.56) and (2.26) imply

∥φ_{m+2}^{1}(y) − φ_{m+1}^{1}(y)∥ ≤
≤ 2(l + l^{2})c_{1}ξ_{1}M^{∗} \frac{M_{∗}^{m}}{(m + 1)!}y^{m+1+k+α−1}.  (2.57)

for p = 0.

Similarly, from (2.41), (2.44), (2.45) and (2.27) we find

∥φ_{m+2}^{2}(y) − φ_{m+1}^{2}(y)∥ ≤
≤ 2(l + l^{2})c_{2}ξ_{2}M^{∗} \frac{M_{∗}^{m}}{(m + 1)!}y^{m+1+k+α−1}.  (2.58)

Proceeding similarly as in deducing the estimate (2.51), we obtain

∥φ̃_{P,m+2}(x, y) − φ̃_{P,m+1}(x, y)∥ ≤ ξ_{3}M^{∗} \frac{M_{∗}^{m}}{(m + 1)!}y^{m+1+k+α−1}, (2.59)

where ξ_{3} = 4n(l + l^{2})c_{3}ξ̃_{2}, ξ̃_{2} = max(ξ_{1}, ξ_{2}).

Denote by η the largest of the numbers max_{\overline{D}_{1}} ∥Ω_{1i}∥, max_{\overline{D}_{1}} ∥Ω_{2i}∥, max_{\overline{D}_{1}} ∥Ω̃_{3}∥,

max_{\overline{D}_{1}} ∥Ω̃_{4}∥, where the matrices Ω_{1i}, Ω_{2i}, i = 1, …, l, Ω̃_{3}, Ω̃_{4} are determined

in (2.42), (2.43). By virtue of (2.59) we have from (2.42) and (2.43)

∥w_{m+2}(x, y) − w_{m+1}(x, y)∥ ≤ ∥φ̃_{P,m+2}(x, y) − φ̃_{P,m+1}(x, y)∥ +

+ ∑_{i=1}^{l} ∫_{ω_{i}(x,y)}^{y} (∥Ω_{1i}∥∥w_{m+1} − w_{m}∥ + ∥Ω_{2i}∥∥u_{m+1} − u_{m}∥)(z_{i}(x, y; t), t)dt ≤

≤ ξ_{3}M^{∗} \frac{M_{∗}^{m}}{(m + 1)!}y^{m+1+k+α−1} + 2lη ∫_{0}^{y} M^{∗} \frac{M_{∗}^{m}}{m!} t^{m+k+α−1}dt ≤

≤ (ξ_{3} + 2lη)M^{∗} \frac{M_{∗}^{m}}{(m + 1)!}y^{m+1+k+α−1},  (2.60)
∥u_{m+2}(x, y) − u_{m+1}(x, y)∥ ≤ 2ηM^{∗} \frac{M_{∗}^{m}}{(m + 1)!}y^{m+1+k+α−1}, (2.61)

since 0 ≤ ω_{i}(x, y) ≤ y, i = 1, …, l.

It immediately follows from (2.50), (2.52), (2.53), (2.57), (2.58), (2.60) and (2.61) that if we put

M^{∗} = 2nc_{3}Θ_{4} + Θ_{1}, M_{∗} = max (2(l + l^{2})c_{1}ξ_{1}, 2(l + l^{2})c_{2}ξ_{2}, ξ_{3} + 2lη),

the estimates (2.44)–(2.47) will be valid for any integer m ≥ 0.

Differentiating the equalities (2.40)–(2.47) with respect to x and y and using the obtained estimates (2.44)–(2.47) as well as the solvability of equations (2.24) and (2.25) and the estimates (2.26) and (2.27) for p = 1, we analogously obtain

∥\frac{∂}{∂x}(u_{m+1} − u_{m})(x, y)∥ ≤ M_{1}^{∗} \frac{M_{∗1}^{m−1}}{(m − 1)!}y^{m+k+α−2},

∥\frac{∂}{∂y} (u_{m+1} − u_{m})(x, y)∥ ≤ M_{1}^{∗} \frac{M_{∗1}^{m−1}}{(m − 1)!}y^{m+k+α−2},

∥\frac{∂}{∂x}(w_{m+1} − w_{m})(x, y)∥ ≤ M_{1}^{∗} \frac{M_{∗1}^{m−1}}{(m − 1)!}y^{m+k+α−2},

∥\frac{∂}{∂y} (w_{m+1} − w_{m})(x, y)∥ ≤ M_{1}^{∗} \frac{M_{∗1}^{m−1}}{(m − 1)!}y^{m+k+α−2},

∥\frac{∂}{∂y} (φ_{m+1}^{1} − φ_{m}^{1})(y)∥ ≤ M_{1}^{∗} \frac{M_{∗1}^{m−1}}{(m − 1)!}y^{m+k+α−2},

∥\frac{∂}{∂x}(φ_{m+1}^{2} − φ_{m}^{2})(y)∥ ≤ M_{1}^{∗} \frac{M_{∗1}^{m−1}}{(m − 1)!}y^{m+k+α−2}.

Continuing this process, we find that for m ≥ i + j, 0 ≤ i + j ≤ k − 1

\begin{matrix}∥∂^{i,j}(u_{m+1} − u_{m})(x, y)∥ ≤ M_{i+j}^{∗} \frac{M_{∗i+j}^{m−i−j}}{(m − i − j)!}y^{m+k+α−i−j−1}, \\ ∥∂^{i,j}(w_{m+1} − w_{m})(x, y)∥ ≤ M_{i+j}^{∗} \frac{M_{∗i+j}^{m−i−j}}{(m − i − j)!}y^{m+k+α−i−j−1}, \\ ∥\frac{∂^{i+j}}{∂y^{i+j}} (φ_{m+1}^{p} − φ_{m}^{p})(y)∥ ≤ M_{i+j}^{∗} \frac{M_{∗i+j}^{m−i−j}}{(m − i − j)!}y^{m+k+α−i−j−1}, \\ p = 1, 2, \end{matrix} \} (2.62)

where M_{i}^{∗}, M_{∗i}, i = 1, …, k − 1, are sufficiently large positive numbers not depending on m.

It follows from (2.62) that the series

u(x, y) = lim_{m→∞} u_{m}(x, y) = ∑_{m=1}^{∞} (u_{m}(x, y) − u_{m−1}(x, y)),

w(x, y) = lim_{m→∞} w_{m}(x, y) = ∑_{m=1}^{∞} (w_{m}(x, y) − w_{m−1}(x, y)),

φ^{p}(y) = lim_{m→∞} φ_{m}^{p}(y) = ∑_{m=1}^{∞} (φ_{m}^{p}(y) − φ_{m−1}^{p}(y)), p = 1, 2,

converge in the spaces C^{∘}_{α}^{k−1}(\overline{D}_{1}), C^{∘}_{α}^{k−1}[0, d_{p}], p = 1, 2, and on account of (2.40)–(2.43) the limit functions u, w, φ^{1}, φ^{2} satisfy the system of equations (2.18), (2.19), (2.22). Hence it follows that u_{x} = K_{1}w, u_{y} = K_{2}w, where K = \binom{K_{1}}{K_{2}} is the 2n × 2n-matrix from (2.4). Consequently, u_{x}, u_{y} ∈

C^{∘}_{α}^{k−1}(\overline{D}_{1}) since w ∈ C^{∘}_{α}^{k−1}(\overline{D}_{1}), K ∈ C^{k}(\overline{D}_{1}), and therefore u ∈ C^{∘}_{α}^{k}(\overline{D}_{1}).

Thus we have shown that the obtained function u(x, y) is a solution of the

problem (2.1)–(2.3) in the class C^{∘}_{α}^{k}(\overline{D}_{1}).

Let us now show that under the conditions of Theorem 2.1–2.3 the problem (2.1)–(2.3) has no other solution in the class C^{∘}_{α}^{k}(\overline{D}_{1}). Indeed,

if u(x, y) ∈ C^{∘}_{α}^{k}(\overline{D}_{1}) is a solution of the homogeneous problem corresponding to (2.1)–(2.3), then the corresponding functions u, w, φ^{1}, φ^{2} satisfy the homogeneous system of equations

\begin{matrix}w_{q_{i}+j}(x, y) = φ_{q_{i}+j}^{i(P)}(ω_{i}(x, y))+ \\ + ∫_{ω_{i}(x,y)}^{y} ( ∑_{p=1}^{2n} a_{ijp}^{1}w_{p} + ∑_{p=1}^{n} b_{ijp}^{1}u_{p})(z_{i}(x, y; t), t)dt, \\ 1 ≤ i ≤ l, 1 ≤ j ≤ k_{i}, \\ u(x, y) = ∫_{0}^{ω_{1}(x,y)}(\frac{dγ_{1(P)}}{dy} K_{1} + K_{2})w(γ_{1(P)}(t), t)dt+ \\ + ∫_{ω_{1}(x,y)}^{y} ((−λ_{1}K_{1} + K_{2})w)(z_{1}(x, y; t), t)dt, \\ (K_{10}φ^{1})(y) + (T_{5}w)(y) + (T_{6}u)(y) = 0, \\ (K_{20}φ^{2})(y) + (T_{7}w)(y) + (T_{8}u)(y) = 0. \end{matrix} \}  (2.63)

To the system of equations (2.63), let us apply the method of successive approximations taking u, w, φ^{1}, φ^{2} as zero approximations. Since these functions satisfy the system of equations (2.63), every next approximation will coincide with it, that is,

u_{m}(x, y) ≡ u(x, y), w_{m}(x, y) ≡ w(x, y),
φ_{m}^{p}(y) ≡ φ^{p}(y), p = 1, 2.

Taking into consideration that these functions satisfy the estimates of the type (2.48), (2.49), and arguing as in deducing the estimates (2.44)–(2.47), we obtain

∥u(x, y)∥ = ∥u_{m+1}(x, y)∥ ≤ M̃^{∗} \frac{M̃_{∗}^{m}}{m!} y^{m+k+α−1},

∥w(x, y)∥ = ∥w_{m+1}(x, y)∥ ≤ M̃^{∗} \frac{M̃_{∗}^{m}}{m!} y^{m+k+α−1},

∥φ^{1}(y)∥ = ∥φ_{m+1}^{1}(y)∥ ≤ M̃^{∗} \frac{M̃_{∗}^{m}}{m!} y^{m+k+α−1},

∥φ^{2}(y)∥ = ∥φ_{m+1}^{2}(y)∥ ≤ M̃^{∗} \frac{M̃_{∗}^{m}}{m!} y^{m+k+α−1},

whence in the limit as m → ∞, we find that

u ≡ w ≡ φ^{1} ≡ φ^{2} ≡ 0. ■

The particular case of the boundary value problem (2.1)–(2.3) is the problem of Goursat type, when the boundary conditions (2.2), (2.3) have the form

u|_{OP_{1}} = f_{1},  (2.64)

u|_{OP_{2}} = f_{2}.  (2.65)

Differentiating the equalities (2.64) and (2.65) with respect to the tangent to the curves γ_{1} and γ_{2}, we have

(\frac{dγ_{1}}{dy} u_{x} + u_{y})|_{OP_{1}} = f_{1}^{(1)},  (2.66)
(\frac{dγ_{2}}{dy} u_{x} + u_{y})|_{OP_{2}} = f_{2}^{(1)}.  (2.67)

Below we shall assume that all the requirements imposed on the curves γ_{1},γ_{2} and the characteristics of the system (2.1) quoted in §2, are fulfilled; moreover, the number l_{0} = 0 and the points P_{1} and P_{2} are located on the curves γ_{1} and γ_{2} such that m_{1} = m_{2} = n.

It is easily seen that in the class C^{∘}_{α}^{k}(\overline{D}_{1}), k ≥ 2, α ≥ 0, the problem (2.1), (2.64), (2.65) is equivalent to the problem (2.1), (2.66), (2.67).

Since the matrix coefficients for the problem (2.1), (2.66), (2.67) have in the boundary conditions (2.66), (2.67) the form

M_{i} = \frac{dγ_{i}}{dy} E, N_{i} = E, S_{i} = 0, i = 1, 2,

where E is the unit n × n-matrix, it is obvious that the conditions (2.21) are equivalent to the following ones

rank {ν_{ij}, 1 ≤ i ≤ s_{0}, 1 ≤ j ≤ k_{i}}|_{OP_{1}} = n,  (2.68)
rank {ν_{ij}, s_{0} < i ≤ l, 1 ≤ j ≤ k_{i}}|_{OP_{2}} = n.  (2.69)

In this case the equalities Ũ_{1} = U_{2} and Ũ_{2} = U_{1} are valid, the condition (Γ_{i} × Ũ_{i})(O) = 0 being fulfilled if and only if γ_{i} = L_{i_{0}}(O), 1 ≤ i_{0} ≤ l, and k_{i_{0}} = n.

From Theorems 2.1–2.3 we have the following assertions:

1. Let the conditions (2.68), (2.69) be fulfilled. If either m̃_{1}m̃_{2} = 0 or at least one of the equalities (Γ_{1} × U_{2})(O) = 0 or (Γ_{2} × U_{1})(O) = 0 holds, then

the problem (2.1), (2.64), (2.65) is uniquely solvable in the class C^{∘}_{α}^{k}(\overline{D}_{1}) for all k ≥ 2, α ≥ 0.

2. Let the conditions (2.68), (2.69) be fulfilled, and m̃_{1}m̃_{2} ≠ 0, (Γ_{1} × U_{2})(O) ≠ 0, (Γ_{2} × U_{1})(O) ≠ 0. If the curves γ_{1}, γ_{2} do not have a common tangent line at the point O(0, 0), then for k + α > ρ_{0} the problem (2.1),

(2.64), (2.65) is uniquely solvable in the class C^{∘}_{α}^{k}(\overline{D}_{1}).

3. Let the conditions (2.68), (2.69) be fulfilled, and m̃_{1}m̃_{2} ≠ 0, (Γ_{1} × U_{2})(O) ≠ 0, (Γ_{2} × U_{1})(O) ≠ 0. If the curves γ_{1}, γ_{2} do not have a common tangent line at the point O(0, 0), then for h_{i}(1) < 1, i = 1, 2, the problem

(2.1), (2.64), (2.65) is uniquely solvable in the class C^{∘}_{α}^{k}(\overline{D}_{1}) for all k ≥ 2, α ≥ 0.

Remark. Let D_{P}, P ∈ \overline{D}_{1}, be the domain constructed in §2 of the present chapter, and let γ_{iP} = γ_{i} ∩ ∂D_{P}, i = 1, 2. As is seen from the proofs of Theorems 2.1–2.3, when conditions of these theorems are fulfilled, the domain of dependence of the solution u(x, y) of the problem (2.1)–(2.3) for the point P ∈ \overline{D}_{1} is contained in the domain D_{P}, and for the solution u(x, y) the estimate

∥u∥_{C^{∘}_{α}^{k}(\overline{D}_{P})} ≤ c( ∑_{i=1}^{2} ∥f_{i}∥_{C^{∘}_{α}^{k−1}(γ_{iP} )} + ∥F ∥_{C^{∘}_{α−1}^{k−1}(\overline{D}_{P})}),

is valid, where c = const > 0 does not depend on F and f_{i}, i = 1, 2,

∥u∥_{C^{∘}_{α}^{k}(\overline{D}_{P})} = max_{i+j=k} sup_{z∈\overline{D}_{P} \O} |z|^{−α}|∂^{i,j}u(z)|, ∂^{i,j} = \frac{∂^{i+j}}{∂x^{i}∂y^{j}} .

The norms in the spaces C^{∘}_{α}^{k−1}(γ_{ip}) and C^{∘}_{α−1}^{k−1}(\overline{D}_{P}) are defined analogously.

§ 5. The Case of Hyperbolic Systems with Constant Coefficients

Let us consider a normally hyperbolic system with constant coefficients of the type

Au_{xx} + 2Bu_{xy} + Cu_{yy} = 0.  (2.70)

As the curves γ_{1} and γ_{2} let us take straight beams γ_{i} : x = γ_{i}^{0}y, y ≥ 0, γ_{i}^{0} = const, i = 1, 2, γ_{1}^{0} > γ_{2}^{0}. Denote by D the angle contained between the beams γ_{1} and γ_{2} and located in a half-plane y ≥ 0. On the beams γ_{1} and γ_{2} let us take arbitrarily the points P_{1} and P_{2} different from O(0, 0) and assume that the straight line passing through P_{1} and P_{2} is not a characteristic of the system (2.70). Because of the fact that γ_{1} and γ_{2} are the straight beams, and the characteristics L_{i} : x + λ_{i}y = const, λ_{i} = const, i = 1, …, l, of the system (2.70) are the straight lines, all the requirements of §2, imposed both on γ_{1}, γ_{2} and L_{i}, i = 1, …, l, will be fulfilled. In a similar way as in §2, we construct the domain D_{1} and determine the numbers m_{1} and m_{2}.

Introduce into the consideration the following spaces

C^{∘}_{α,β}^{k}(\overline{D}) =\{u ∈ C^{∘}^{k}(\overline{D}) : max_{i+j=k} sup_{0<|z|≤1,z∈\overline{D}} |z|^{−α}|∂^{i,j}u(z)| < ∞,

max_{i+j=k} sup_{|z|≥1,z∈\overline{D}} |z|^{−β}|∂^{i,j}u(z)| < ∞\}, k ≥ 2, α ≥ 0, β ≥ 0,

C^{∘}_{α}^{k}(\overline{D}) =\{u ∈ C^{∘}^{k}(\overline{D}) : max_{i+j=k} sup_{0<|z|≤1,z∈\overline{D}} |z|^{−α}|∂^{i,j}u(z)| < ∞\},
k ≥ 2, α ≥ 0.

The space C^{∘}_{α}^{k}(\overline{D}_{1}) has been introduced at the end of §1 in Chapter II.

Below the m_{i} × n-matrices M_{i}, N_{i}, i = 1, 2, appearing in the boundary conditions (2.2), (2.3) are assumed to be constant, and S_{i} = 0, i = 1, 2.

When considering the problem (2.70), (2.2), (2.3) in the spaces C^{∘}_{α}^{k}(\overline{D})

and C^{∘}_{α,β}^{k}(\overline{D}), we assume that equalities (2.2) and (2.3) take place respectively on the beams γ_{1} and γ_{2}.

When investigating the same problem in the above-mentioned spaces, the use will be made of the Bochner method of solution of functional equations which will be cited below.

When considering the problem (2.70), (2.2), (2.3) in the classes C^{∘}_{α}^{k}(\overline{D}),

C^{∘}_{α,β}^{k}(\overline{D}), C^{∘}_{α}^{k}(\overline{D}_{1}) it is required that f_{i} ∈ C^{∘}_{α}^{k−1}(γ_{i}), f_{i} ∈ C^{∘}_{α,β}^{k−1}(γ_{i}), f_{i} ∈

C^{∘}_{α}^{k−1}(OP_{i}), i = 1, 2, respectively, where f_{1} and f_{2} are the right-hand sides of equalities (2.2), (2.3).

Similarly, as in §3, the problem (2.70), (2.2), (2.3) in the class C^{∘}_{α}^{k}(\overline{D}_{1}) is reduced equivalently to the system of equations (2.20) in which G_{0}^{i} = Γ_{i}×V_{i},

φ^{i} ∈ C^{∘}_{α}^{k−1}[0, d_{i}], i = 1, 2, τ_{j}^{1}(y) = σ_{j}^{1}y, 0 ≤ σ_{j}^{1} = const, j = s_{0} + 1, …, l,

τ_{i}^{2}(y) = σ_{i}^{2}y, 0 ≤ σ_{i}^{2} = const, i = l_{0} + 1, …, s_{0}, T_{j} = 0, j = 1, …, 4, f_{3} = f_{1}, f_{4} = f_{2}.

After substitution φ̃(t) = (φ^{1}(d_{1}t), φ^{2}(d_{2}t)) we rewrite the obtained system of equations in the form of one equation

G_{0}φ̃(t) + ∑_{i=1}^{r} G_{i}φ̃(τ_{i}t) = f̃(t),  (2.71)

where φ̃ ∈ C^{∘}_{α}^{k−1}[0, 1], G_{i}, i = 0, …, r, are well-defined real constant (m_{1} +

m_{2})×(m_{1} +m_{2})-matrices; moreover, G_{0} = (\begin{matrix}G_{0}^{1} & 0 \\ 0 & G_{0}^{2} \end{matrix}), 0 < τ_{i} = const < 1,

i = 1, …, r, and f̃(t) ∈ C^{∘}_{α}^{k−1}[0, 1].

Analogously one can show that the problem (2.70), (2.2), (2.3) in the

classes C^{∘}_{α}^{k}(\overline{D}), C^{∘}_{α,β}^{k}(\overline{D}) is equivalent to the system of equations (2.71) with respect to an unknown function φ̃ belonging, respectively, to the spaces

C^{∘}_{α}^{k−1}[0, ∞) and C^{∘}_{α,β}^{k−1}[0, ∞).

Differentiating equation (2.71) (k − 1) times with respect to t, we get

(Gψ)(t) = G_{0}ψ(t) + ∑_{i=1}^{r} τ_{i}^{k−1}G_{i}ψ(τ_{i}t) = f(t),  (2.72)

where ψ(t) = φ^{(k−1)}(t), f(t) = f̃^{(k−1)}(t).

Obviously, equation (2.71) with respect to φ̃ ∈ C^{∘}_{α,β}^{k−1}[0, ∞) (C^{∘}_{α}^{k−1}[0, ∞),

C^{∘}_{α}^{k−1}[0, 1]) is equivalent to equation (2.72) with respect to ψ ∈ C^{∘}_{α,β}[0, ∞)

(C^{∘}_{α}[0, ∞), C^{∘}_{α}[0, 1]).

Denote by σ the set of all real numbers {σ_{0}, σ_{1}, …, σ_{i}, … } representable

in the form ∑_{i=1}^{r} n_{i} log τ_{i}, where n_{i} are arbitrary integers, and σ_{0} = 0, σ_{i} ≠ σ_{j}

for i ≠ j.

Let

Δ(s) = det (G_{0} + ∑_{i=1}^{r} τ_{i}^{k−1}G_{i}e^{s log τ_{i}}).

It is obvious that Δ(s) is an entire function represented as

Δ(s) = ∑_{i=0}^{m_{0}} η_{i}e^{_{σ_{i}}s}, σ̃_{i} ∈ σ,  (2.73)

where η_{i}, σ̃_{i} are certain real numbers, and σ̃_{m_{0}} < σ̃_{m_{0}−1} < ⋯ < σ̃_{0} ≤ 0.

We can easily see that in the case Δ(s) ≡ 0, the homogeneous problem corresponding to (2.72) has for any s a non-trivial solution of the type ψ(t) = c(s)t^{s}, ∥c(s)∥ ≠ 0. Evidently, if det G_{0} ≠ 0, then Δ(s) ≢ 0.

Below we shall assume that Δ(s) ≢ 0, and in this case one can suppose that η_{i} ≠ 0, i = 0, …, m_{0} in equality (2.73).

The set 𝔐 of real parts of all zeros of the entire function Δ(s) is a finite or countable bounded closed set; moreover, this set is empty if and only if Δ(s) = η_{0}e^{_{σ_{0}}s} [11]. The set 𝔐 divides the real axis of the plane of the variable s = Re s + i Im s into not more than a countable set of intervals Γ̃_{i}, i = 0, 1, 2, …, among which there are the half-lines (−∞ < Re s < b_{0}) = Γ̃_{0}, (a_{0} < Re s < ∞) = Γ̃_{1}.

It is shown in [12], [13] that the analytic almost-periodic function \frac{1}{Δ(s)}

expands in the strip Π_{i} = {s : Re s ∈ Γ̃_{i}} into an absolutely convergent series of the type

\frac{1}{Δ(s)} = ∑_{j=0}^{∞} γ_{ij}e^{σ_{j}s}, σ_{j} ∈ σ,  (2.74)

whose coefficients can be uniquely determined.

Since σ̃_{0} > σ̃_{j}, j = 1, …, m_{0}, we have

| ∑_{j=1}^{m_{0}} η_{0}^{−1}η_{j}e^{(_{σ_{j}}−_{σ_{0}})s}| < 1

for Re s > c_{0}, where c_{0} is a sufficiently large real number. Therefore for Re s > c_{0} there takes place an expansion

\frac{1}{Δ(s)} = [η_{0}e^{_{σ_{0}}s}(1 + ∑_{j=1}^{m_{0}} η_{0}^{−1}η_{j}e^{(_{σ_{j}}−_{σ_{0}})s})]^{−1} =

= η_{0}^{−1}e^{−_{σ_{0}}s}(1 + ∑_{i=1}^{∞}(−1)^{i}( ∑_{j=1}^{m_{0}} η_{0}^{−1}η_{j}e^{(_{σ_{j}}−_{σ_{0}})s})^{i}).  (2.75)

Due to the uniqueness theorem for analytic almost-periodic functions [49], the coefficients γ_{1j} of the series (2.74) in the strip Π_{1} can be defined from the expansion (2.75), and hence

γ_{1j} = 0 for σ_{j} > −σ̃_{0} ≥ 0.  (2.76)

Denote by Δ_{ij} the algebraic supplement of the element with the indices j, i of the determinant Δ(s),

Δ_{ij}(s) = ∑_{p=0}^{N_{0}} ξ_{ijp}e^{σ_{p}s}, i, j = 1, …, m_{1} + m_{2},

where N_{0} is a natural number and ξ_{ijp} are definite real numbers.

Denote by g_{p}^{ij} the element of the matrix τ_{p}^{k−1}G_{p} with indices i, j, where τ_{0} = 1, p = 0, …, r, i, j = 1, …, m_{1} + m_{2}.

Because of determinant properties, we can easily see that for Re s ∈ Π_{i_{0}}, i_{0} ≥ 0,

\frac{1}{Δ(s)} ∑_{j=1}^{m_{1}+m_{2}} ( ∑_{ρ′=0}^{r} g_{ρ′}^{ij}e^{s log τ_{ρ′}} )Δ_{jρ}(s) =

= ∑_{p=0}^{∞} ∑_{j=1}^{m_{1}+m_{2}} ∑_{ρ′=0}^{r} ∑_{q=0}^{N_{0}} g_{ρ′}^{ij}ξ_{jρq}γ_{i_{0}p}e^{(log τ_{ρ′} +σ_{p}+σ_{q})s} =

= ∑_{ν=0}^{∞} ( ∑_{(p,j,ρ′,q)∈J_{ν}} g_{ρ′}^{ij}ξ_{jρq}γ_{i_{0}p})e^{σ_{ν}s} = \begin{cases}1 & for i = ρ, \\ 0 & for i ≠ ρ, \end{cases}  (2.77)

where J_{ν} is the set of all collections (p, j, ρ′, q) of numbers p, j, ρ′, q for which log τ_{ρ′} + σ_{p} + σ_{q} = σ_{ν}.

From (2.77), due to the absolute convergence of the series (2.74) in the strip Π_{i_{0}}, and because of the uniqueness theorem for analytic almostperiodic functions, we obtain

∑_{(p,j,ρ′,q)∈J_{ν}} g_{ρ′}^{ij}ξ_{jρq}γ_{i_{0}p} = \begin{cases}1 & for i = ρ, ν = 0, \\ 0 & for i = ρ, ν ≥ 1, \\ & or i ≠ ρ, ν ≥ 0. \end{cases}  (2.78)

Analogous reasonings as in the case of the expression

\frac{1}{Δ(s)} ∑_{ρ=1}^{m_{1}+m_{2}} ( ∑_{ρ′=0}^{r} g_{ρ′}^{ρj}e^{s log τ_{ρ′}} )Δ_{iρ}(s)

result in the equalities

∑_{(p,ρ,ρ′,q)∈J_{ν}} g_{ρ′}^{ρj}ξ_{iρq}γ_{i_{0}p} = \begin{cases}1 & for i = j, ν = 0, \\ 0 & for i = j, ν ≥ 1, \\ & or i ≠ j, ν ≥ 0. \end{cases}  (2.79)

Let now G̃_{i_{0}} = (G̃_{i_{0}1}, …, G̃_{i_{0}m_{1}+m_{2}}) be the operator acting by the formula

(G̃_{i_{0}i}f)(t) = ∑_{p=0}^{∞} ∑_{ρ=1}^{m_{1}+m_{2}} ∑_{q=0}^{N_{0}} ξ_{iρq}γ_{i_{0}p}f_{ρ}(e^{σ_{p}+σ_{q}}t),  (2.80)
i = 1, …, m_{1} + m_{2}.

The lemma below is due to Bochner [11].

Lemma 2.4. The operator G defined by the formula (2.72) is invertible

in the space C^{∘}_{α,β}[0, ∞) and G^{−1} = G̃_{i_{0}} if

𝔐 ∩ I_{α,β} = ∅, I_{α,β} = [ min(α, β), max(α, β)] ⊂ Π_{i_{0}}.

In the spaces C^{∘}_{α}[0, ∞) and C^{∘}_{α}[0, 1] the following lemma takes place.

Lemma 2.5. The assertion of Lemma 2.4 is valid in the space C^{∘}_{α}[0, ∞)

for α > sup 𝔐 = sup_{x∈𝔐} x and in C^{∘}_{α}[0, 1] if det G_{0} ≠ 0 and α > sup 𝔐, in

both cases G^{−1} and G̃_{1} being equal.

To prove Lemma 2.5 we shall use the Bochner method [11]. α > sup 𝔐 implies that α ∈ Π_{1}, and hence, since the series (2.74) is absolutely convergent in Π_{1}, we have

c_{1} = ∑_{j=0}^{∞} |γ_{1j}|e^{σ_{j}α} < ∞.  (2.81)

Suppose p_{t}(f) = sup_{τ∈(0,t]} ∥τ^{−α}f(τ)∥_{R^{m_{1}+m_{2}}} .

By (2.76) the function G̃_{1i}f at the point t > 0 depends only on those val-

ues of f which it takes on the segment [0, t_{0}t], where t_{0} = e^{(−_{σ_{0}}+ max_{0≤q≤N_{0}} σ_{q})}. Therefore we have

p_{t}(G̃_{1}f) ≤ max_{1≤i≤m_{1}+m_{2}} p_{t}(G̃_{1i}f) ≤

≤ ∑_{p=0}^{∞} ∑_{ρ=1}^{m_{1}+m_{2}} ∑_{q=0}^{N_{0}} max_{1≤i≤m_{1}+m_{2}} |ξ_{iρq}e^{σ_{q}α}||γ_{1p}|e^{σ_{q}α}p_{t_{0}t}(f) ≤

≤ (m_{1} + m_{2})(N_{0} + 1)( max_{i,ρ,q} |ξ_{iρq}e^{σ_{q}α}|)c_{1}p_{t_{0}t}(f).  (2.82)

When deducing (2.82), the use has been made of (2.81) and the fact that p_{t}(f̃_{ρ}) ≤ e^{(σ_{p}+σ_{q})α}p_{t_{0}t}(f_{ρ}), where f̃_{ρ}(t) = f_{ρ}(e^{σ_{p}+σ_{q}}t). From (2.82) it

follows that the operator G̃_{1} is continuous in the space C^{∘}_{α}[0, ∞).

Let us check that GG̃_{1} = I, where I is the identity operator. If GG̃_{1} = ((GG̃_{1})_{1}, …, (GG̃_{1})_{m_{1}+m_{2}}), then by (2.78) and (2.80) we have

((GG̃_{1})_{i}f)(t) = ∑_{ρ′=0}^{r} ∑_{j=1}^{m_{1}+m_{2}} g_{ρ′}^{ij}(G̃_{1j}f)(τ_{ρ′}t) =

= ∑_{ρ′=0}^{r} ∑_{j=1}^{m_{1}+m_{2}} ∑_{p=0}^{∞} ∑_{ρ=1}^{m_{1}+m_{2}} ∑_{q=0}^{N_{0}} g_{ρ′}^{ij}ξ_{jρq}γ_{1p}f_{ρ}(e^{log τ_{ρ′} +σ_{p}+σ_{q}}t) =

= ∑_{ρ=1}^{m_{1}+m_{2}} ∑_{ν=0}^{∞} ( ∑_{(p,j,ρ′,q)∈J_{ν}} g_{ρ′}^{ij}ξ_{jρq}γ_{1p})f_{ρ}(e^{σ_{ν}} t) = f_{i}(t),

which proves the equality GG̃_{1} = I. In a similar way, using equality (2.79), we can easily check that G̃_{1}G = I. Thus G^{−1} = G̃_{1}, and Lemma 2.5 is

proved in the space C^{∘}_{α}[0, ∞).

Let now det G_{0} ≠ 0 and α > sup 𝔐. From (2.73) it follows that σ̃_{0} = 0 for det G_{0} ≠ 0. Therefore by (2.76) we have γ_{1j} = 0 for σ_{j} > 0. Since log τ_{i} < 0, i = 1, …, r, in the expansion

Δ_{ij}(s) = ∑_{p=0}^{N_{0}} ξ_{ijp}e^{σ_{p}s}

we have ξ_{ijp} = 0 for σ_{p} > 0, and thus ξ_{iρq}γ_{1p} = 0 or σ_{p} + σ_{q} > 0. Hence

the operator G̃_{1} defined by (2.80) acts from the space C^{∘}_{α}[0, 1] into itself. It

remains for us to note that the operator G in the space C^{∘}_{α}[0, ∞) is invertible for α > sup 𝔐, and G^{−1} = G̃_{1}. ■

Lemma 2.6. For α > sup 𝔐 and det G_{0} = 0 the equation (2.72) is solv-

able in the space C^{∘}_{α}[0, 1], and the homogeneous equation corresponding to

(2.72) has in the space C^{∘}_{α}[0, 1] an infinite number of linearly independent solutions.

Proof. If f ∈ C^{∘}_{α}[0, 1], then let f̃ be an arbitrary continuous extension of f

from the segment [0, 1] to [0, ∞). Clearly, f̃ ∈ C^{∘}_{α}[0, ∞), since f̃(t) = f(t) for 0 ≤ t ≤ 1. By Lemma 2.5 the equation Gψ = f̃ is uniquely solvable in

the space C^{∘}_{α}[0, ∞) for α > sup 𝔐. It is also clear that the vector function ψ̃(t) = ψ(t) = (G^{−1}f̃)(t) defined on the segment 0 ≤ t ≤ 1 belongs to the

space C^{∘}_{α}[0, 1] and is the solution of (2.72).

Let us show that dim Ker G = ∞. Since det G_{0} = 0, there exists a nondegenerate (m_{1} + m_{2}) × (m_{1} + m_{2})-matrix Ω such that the last q_{0} rows of the matrix G_{0}Ω are zero, where q_{0} = (m_{1} + m_{2}) − rank G_{0} > 0. Consider the operator G^{∗} defined by

(G^{∗}ψ)(t) = G_{0}Ωψ(t) + ∑_{i=1}^{r} G_{i}Ωψ(τ_{i}t), ψ ∈ C^{∘}_{α}[0, 1].

Assume τ̃_{0} = max_{1≤i≤r} τ_{i}, 0 < τ̃_{0} < 1. Let ψ̃ = (ψ̃_{1}, …, ψ̃_{m_{1}+m_{2}}) be an

arbitrary vector function of the class C^{∘}_{α}[0, 1] such that ψ̃_{i} ≡ 0 when i = 1, …, m_{1} + m_{2} − q_{0} and ψ̃_{i}(t) ≢ 0, ψ̃_{i}(t) = 0 for t ∈ [0, τ̃_{0}] when i = m_{1}+m_{2}−q_{0}+1, …, m_{1}+m_{2}. It can be easily seen that G^{∗}ψ̃ = 0, and hence Gψ = 0, where ψ = Ωψ̃. Therefore dim Ker G = ∞, since det Ω ≠ 0. ■

Remark. As the example of equation (2.72) with r = 1 shows, in the case where conditions of Lemmas 2.4–2.6 are violated, the unique solvability of the equation (2.72) may not hold. For r = 1 the following assertion is

valid: a) equation (2.72) in the space C^{∘}_{α,β}[0, ∞) for 𝔐 ∩ {α, β} ≠ ∅ as

well as in the spaces C^{∘}_{α}[0, ∞) and C^{∘}_{α}[0, 1] for α ∈ 𝔐 is not normally Hausdorff solvable; b) equation (2.72) is normally Hausdorff solvable in the

space C^{∘}_{α,β}[0, ∞) for 𝔐 ∩ I_{α,β} ≠ ∅, 𝔐 ∩ {α, β} = ∅, and for α < β we

have κ = d_{0} − d_{0}^{∗} = +∞, d_{0} = dim Ker G = ∞, d_{0}^{∗} = dim Ker G^{∗} = 0, while for α > β we have conversely κ = −∞, d_{0} = 0, d_{0}^{∗} = ∞; c) equation

(2.72) is normally Hausdorff solvable in the spaces C^{∘}_{α}[0, ∞) and C^{∘}_{α}[0, 1] for α ≤ sup 𝔐, α ∉ 𝔐; moreover, in both cases κ = +∞, d_{0} = ∞, d_{0}^{∗} = 0. This assertion can be proved by using the same method as we have used in proving Lemma 1.7 in §2 of Chapter I.

Recall that the condition det G_{0} ≠ 0 is equivalent to the fulfilment of the conditions (2.21); moreover, if det G_{0} ≠ 0, then the entire function Δ(s) ≢ 0. Denote by 𝔐_{0} the set of real parts of zeros of the entire function

Δ_{0}(s) = det (G_{0} + ∑_{i=1}^{r} G_{i}e^{(s−1) log τ_{i}}).

Since Δ(s) = Δ_{0}(s + k), we have 𝔐 = 𝔐_{0} − k = {x − k : x ∈ 𝔐_{0}}. From Lemmas 2.4–2.6 we have the following

Theorem 2.4. The problem (2.70), (2.2), (2.3) is uniquely solvable in: a)

the class C^{∘}_{α,β}^{k}(\overline{D}) for Δ_{0}(s) ≢ 0 and (𝔐_{0} − k) ∩ I_{α,β} = ∅; b) the class

C^{∘}_{α}^{k}(\overline{D}) for Δ_{0}(s) ≢ 0 and k + α > sup 𝔐_{0}; c) the class C^{∘}_{α}^{k}(\overline{D}_{1}) if the conditions (2.21) are fulfilled and k + α > sup 𝔐_{0}. In the case Δ_{0}(s) ≢ 0, k + α > sup 𝔐_{0}, if at least one of the conditions (2.21) is violated, then

the problem (2.70), (2.2), (2.3) is solvable in the class C^{∘}_{α}^{k}(\overline{D}_{1}), and the homogeneous problem corresponding to (2.70), (2.2), (2.3) has an infinite number of linearly independent solutions.

Remarks.

1. As noted above, when the conditions of Theorem 2.4 are violated in

the classes C^{∘}_{α,β}^{k}(\overline{D}), C^{∘}_{α}^{k}(\overline{D}), C^{∘}_{α}^{k}(\overline{D}_{1}) the problem (2.70), (2.2), (2.3) may turn out to be ill-posed.

2. If the set 𝔐_{0} is empty, then Δ_{0}(s) ≢ 0, and owing to Theorem 2.4,

the problem (2.70), (2.2), (2.3) is uniquely solvable in the classes C^{∘}_{α,β}^{k}(\overline{D}),

C^{∘}_{α}^{k}(\overline{D}) for all k ≥ 2, α ≥ 0, β ≥ 0, as well as in the class C^{∘}_{α}^{k}(\overline{D}_{1}) when the conditions (2.21) are fulfilled for all k ≥ 2, α ≥ 0. When the conditions of Theorem 2.1 are fulfilled, it is obvious that Δ_{0}(s) ≡ det G_{0} ≠ 0, and hence the set 𝔐 = ∅. Therefore, in the case of the problem (2.70), (2.2), (2.3) Theorem 2.1 is a direct consequence of the assertion b) of Theorem 2.4.

3. It can be easily verified that ρ_{0} ≥ sup 𝔐_{0}, where ρ_{0} is a number occurring in the condition k + α > ρ_{0} of Theorem 2.2 in the case of the problem (2.70), (2.2), (2.3). Therefore, the condition α > sup(𝔐_{0} − k) or, what is the same, the condition k + α > sup 𝔐_{0} in Theorem 2.4 is more exact than the condition k + α > ρ_{0} in Theorem 2.2.

4. In the case Δ_{0}(s) ≡ 0, one can easily verify that in the classes C^{∘}_{α,α}^{k}(\overline{D}),

C^{∘}_{α}^{k}(\overline{D}), C^{∘}_{α}^{k}(\overline{D}_{1}) for all k ≥ 2, α ≥ 0 the homogeneous problem corresponding to (2.70), (2.2), (2.3) has an infinite number of linearly independent solutions.

CHAPTER III

CHARACTERISTIC PROBLEMS FOR LINEAR

HYPERBOLIC SYSTEMS OF SECOND ORDER WITH

PARABOLIC DEGENERATION

§ 1. A Characteristic Problem for Hyperbolic Systems of Second Order with Non-Characteristic Line of Parabolic Degeneration

In the plane of variables x, y let us consider a system of linear differential equations of the type

y^{m}Au_{xx} + 2y^{\frac{m}{2}} Bu_{xy} + Cu_{yy} + au_{x} + bu_{y} + cu = F,  (3.1)

where A, B, C, a, b, c are given real n×n-matrices, F and u are, respectively, given and unknown n-dimensional vectors, m = const > 0, n > 1.

Below A, B, C are assumed to be constant matrices, det C ≠ 0, and the polynomial p_{0}(λ) = det(A+2Bλ+Cλ^{2}) is assumed to have only simple real roots λ_{1}, …, λ_{2n}. In this case the system (3.1) is strictly hyperbolic for y > 0, and the line of parabolic degeneration y = 0 is not a characteristic of the system (3.1). Under these conditions the numbers y^{\frac{m}{2}} λ_{1}, …, y^{\frac{m}{2}} λ_{2n} are the roots of the characteristic polynomial p(y; λ) = det(y^{m}A + 2y^{\frac{m}{2}} Bλ + Cλ^{2}) of the system (3.1), and the curves determined by the equations

L_{i}(P) : x + \frac{2λ_{i}}{m + 2}y^{\frac{m+2}{2}} = x_{0} + \frac{2λ_{i}}{m + 2}y0^{\frac{m+2}{2}} , i = 1, …, 2n, y_{0} > 0,

and passing through the point P (x_{0}, y_{0}) are characteristics of the system (3.1).

Denote by D a domain lying in the half-plane y > 0 and bounded by two adjoint characteristics

γ_{1} : x + \frac{2λ_{i_{1}}}{m + 2}y^{\frac{m+2}{2}} = 0, γ_{2} : x + \frac{2λ_{i_{2}}}{m + 2}y^{\frac{m+2}{2}} = 0, λ_{i_{1}} < λ_{i_{2}},

of the system (3.1), coming out of the origin O(0, 0). Take arbitrarily on γ_{1} a point P_{1} different from O(0, 0) and choose the numbering of characteristic curves L_{i}(P_{1}), i = 1, …, 2n, coming out of P_{1} into the angle D such that starting from L_{1}(P_{1}), they follow each other counter-clockwise. On the curve γ_{2} let us fix the point P_{2} lying strictly between the two points of intersection of characteristics L_{n}(P_{1}) and L_{n+1}(P_{1}) with the curve γ_{2}. Denote by D_{1} ⊂ D the characteristic quadrangle with a vertex at O(0, 0), bounded by the characteristics γ_{1}, γ_{2}, L_{n}(P_{1}) and L_{n+1}(P_{2}). Under these assumptions it is evident that

γ_{1} = L_{2n}(O) : x + \frac{2λ_{2n}}{m + 2}y^{\frac{m+2}{2}} = 0, γ_{2} = L_{1}(O) : x + \frac{2λ_{1}}{m + 2}y^{\frac{m+2}{2}} = 0.

For convenience we shall assume below that λ_{n} > 0 and λ_{n+1} < 0.

Consider the characteristic problem formulated as follows [38]: find in the domain D_{1} a regular solution u(x, y) of the system (3.1), satisfying on the segments OP_{i} of the characteristics γ_{i} the following conditions

u|_{OP_{i}} = f_{i}, i = 1, 2,  (3.2)

where f_{1}, f_{2} are given n-dimensional real vectors, f_{1}(O) = f_{2}(O).

Below we asume that a, b, c, F ∈ C^{1}(\overline{D}_{1}), f_{i} ∈ C^{2}(OP_{i}), i = 1, 2, and moreover, in the domain D_{1}

sup_{\overline{D}_{1}\O} ∥y^{(1− \frac{m}{2} )}a∥ < ∞, sup_{\overline{D}_{1}\O} ∥y^{(1− \frac{m}{2} )}a_{x}∥ < ∞,

sup_{\overline{D}_{1}\O} ∥y^{−(α+ \frac{m}{2} −1)}F ∥ < ∞, sup_{\overline{D}_{1}\O} ∥y^{−(α−2)}F_{x}∥ < ∞, α = const > 0,

f_{i}(O) = 0, sup_{OP_{i}\O} ∥y^{−(α+ \frac{m}{2} +1−j)}f_{i}^{(j)}∥ < ∞, i = 1, 2; j = 1, 2,

where ∥ · ∥ denotes the norm in R^{n}.

Since the roots λ_{1}, …, λ_{2n} of the polynomial p_{0}(λ) are simple, we can easily verify that dim Ker(A + 2Bλ_{i} + Cλ_{i}^{2}) = 1, i = 1, …, 2n. Let the vectors ν_{i} ∈ Ker(A + 2Bλ_{i} + Cλ_{i}^{2}) and ∥ν_{i}∥ ≠ 0, i = 1, …, 2n.

In §4 we shall prove the following

Theorem 3.1. Let the condition

rank{ν_{1}, …, ν_{n}} = rank{ν_{n+1}, …, ν_{2n}} = n  (3.3)

be fulfilled. Then there exists a positive integer α_{0} depending only on the coefficients A, B, C, α of the system (3.1) such that for α > α_{0} the problem (3.1), (3.2) is uniquely solvable in the class

\{u ∈ C^{2}(\overline{D}_{1}) : ∂^{i,j}u(0, 0) = 0,
sup_{\overline{D}_{1}\O} ∥y^{−(α+ \frac{m}{2} +1−( \frac{m}{2} +1)i−j)}∂^{i,j}u∥ < ∞, 0 ≤ i + j ≤ 2\},  (3.4)

∂^{i,j} = \frac{∂^{i+j}}{∂x^{i}∂y^{j}} .

It should be noted that the condition

sup_{\overline{D}_{1}\O} ∥y^{(1− \frac{m}{2} )}a∥ < ∞

for the lower coefficient a of the system (3.1) is a generalization of the wellknown Gellerstedt’s condition when (3.1) is a scalar equation (n = 1, A = −C = 1, B = 0).

§ 2. Some Structural Properties of the Hyperbolic System (3.1)

Consider the following 2n × 2n-matrices:

A_{0} = ∥\begin{matrix}0 & −E \\ C^{−1}A & 2C^{−1}B \end{matrix}∥ , Ã_{0} = ∥\begin{matrix}0 & −E \\ y^{m}C^{−1}A & 2y^{\frac{m}{2}} C^{−1}B \end{matrix}∥ ,

K = (\begin{matrix}ν_{1} & … & ν_{2n} \\ λ_{1}ν_{1} & … & λ_{2n}ν_{2n} \end{matrix}) , K̃ = (\begin{matrix}y^{− \frac{m}{2}} ν_{1} & … & y^{− \frac{m}{2}} ν_{2n} \\ λ_{1}ν_{1} & … & λ_{2n}ν_{2n} \end{matrix}) ,

where E is the unit n × n-matrix.

It can be easily verified that

K^{−1}A_{0}K = D_{0}, K̃^{−1}Ã_{0}K̃ = D̃_{0}.  (3.5)

Here D_{0} = diag[−λ_{1}, …, −λ_{2n}], D̃_{0} = diag[−y^{\frac{m}{2}} λ_{1}, …, −y^{\frac{m}{2}} λ_{2n}].

Assume K = colon(K_{1}, K_{2}), K^{−1} = (K_{1}^{0}, K_{2}^{0}), where K_{1}, K_{2} and K_{1}^{0}, K_{2}^{0} are matrices of orders n × 2n and 2n × n, respectively.

Obviously,

K̃ = colon (y^{− \frac{m}{2}} K_{1}, K_{2}), K̃^{−1} = (y^{\frac{m}{2}} K_{1}^{0}, K_{2}^{0}).  (3.6)

Owing to (3.6) we have

K̃_{y} = −\frac{m}{2} colon (y^{− \frac{m}{2} −1}K_{1}, 0), K̃^{−1}K̃_{y} = −\frac{m}{2y} K_{1}^{0} × K_{1}. (3.7)

If

B_{0} = ∥\begin{matrix}0 & 0 \\ C^{−1}a & C^{−1}b \end{matrix}∥ ,  (3.8)

then

K̃^{−1}B_{0}K̃ = \frac{1}{y} B̃_{0} + B̃_{1},  (3.9)

where B̃_{0} = y^{1− \frac{m}{2}} K_{2}^{0}C^{−1}aK_{1}, B̃_{1} = K_{2}^{0}C^{−1}bK_{2}. Since by the assumption

sup_{\overline{D}_{1}\O} ∥y^{(1− \frac{m}{2} )}a∥ < ∞, sup_{\overline{D}_{1}\O} ∥y^{(1− \frac{m}{2} )}a_{x}∥ < ∞,

we have

sup_{\overline{D}_{1}\O} ∥B̃_{0}∥ = sup_{\overline{D}_{1}\O} ∥y^{(1− \frac{m}{2} )}K_{2}^{0}C^{−1}aK_{1}∥ < ∞,

sup_{\overline{D}_{1}\O} ∥B̃_{0x}∥ = sup_{\overline{D}_{1}\O} ∥y^{(1− \frac{m}{2} )}K_{2}^{0}C^{−1}a_{x}K_{1}∥ < ∞.

(3.10)

§ 3. Reduction of the Problem (3.1), (3.2) to a System of Integro-Functional Equations

In the class (3.4) we can rewrite equivalently the problem (3.1), (3.2) in the form

v_{y} + Ã_{0}v_{x} + B_{0}v + C_{0}u^{0} = F^{0},  (3.11)
( − y^{\frac{m}{2}} λ_{1} \frac{∂}{∂x} + \frac{∂}{∂y} )u = −y^{\frac{m}{2}} λ_{1}v_{1} + v_{2},  (3.12)

\begin{matrix}( − y^{\frac{m}{2}} λ_{2n}v_{1} + v_{2})( − \frac{2λ_{2n}}{m+2}y^{\frac{m+2}{2}} , y) = f_{1}^{(1)}(y), 0 ≤ y ≤ d_{1}, \\ ( − y^{\frac{m}{2}} λ_{1}v_{1} + v_{2})( − \frac{2λ_{1}}{m+2}y^{\frac{m+2}{2}} , y) = f_{2}^{(1)}(y), 0 ≤ y ≤ d_{2}, \end{matrix} \} (3.13)

u( − \frac{2λ_{2n}}{m + 2}y^{\frac{m+2}{2}} , y) = f_{1}(y), 0 ≤ y ≤ d_{1},  (3.14)

where d_{i} is the ordinate of the point P_{i} ∈ γ_{i}, i = 1, 2, and the 2n × 2nmatrices Ã_{0}, B_{0} have been introduced in §2,

\begin{matrix}C_{0} = diag(0, C^{−1}c), u^{0} = (0, u), F^{0} = (0, F), \\ v_{1} = u_{x}, v_{2} = u_{y}, \\ v = (v_{1}, v_{2}), v_{1} ∈ C_{α, \frac{m}{2} +1,1}^{1}, v_{2} ∈ C_{α+ \frac{m}{2} , \frac{m}{2} +1,1}^{1}. \end{matrix}  (3.15)

Here

C_{α,p_{1},p_{2}}^{k} =\{u ∈ C^{k}(\overline{D}_{1}) : ∂^{i,j}u(0, 0) = 0,

sup_{\overline{D}_{1}\O} ∥y^{−(α−p_{1}i−p_{2}j)}∂^{i,j}u∥ < ∞, 0 ≤ i + j ≤ k\}.

In fact, if u is a solution of the problem (3.1), (3.2) from the abovementioned class, then it is obvious that u, v_{1} = u_{x}, v_{2} = u_{y} satisfy the problem (3.11)–(3.14), and (v_{1}, v_{2}) belongs to the class (3.15). Conversely, let u, v_{1}, v_{2} be solutions of the problem (3.11)–(3.14) for which (3.4), (3.15) hold. Let us show that u is a solution of the problem (3.1), (3.2) and v_{1} = u_{x}, v_{2} = u_{y}. From the first n equations of the system (3.11) we have that v_{1y} = v_{2x}. Furthermore, equation (3.12) yields

( − y^{\frac{m}{2}} λ_{1} \frac{∂}{∂x} + \frac{∂}{∂y} )(u_{x} − v_{1}) =

= \frac{∂}{∂x}( − y^{\frac{m}{2}} λ_{1} \frac{∂}{∂x} + \frac{∂}{∂y} )u − ( − y^{\frac{m}{2}} λ_{1} \frac{∂}{∂x} + \frac{∂}{∂y} )v_{1} =

= \frac{∂}{∂x}( − y^{\frac{m}{2}} λ_{1}v_{1} + v_{2}) + y^{\frac{m}{2}} λ_{1}v_{1x} − v_{1y} = v_{2x} − v_{1y} = 0.

which in its turn implies that u_{x} − v_{1} = 0, since because of (3.12)–(3.14) and the inequality λ_{1} ≠ λ_{2n} we have (u_{x} −v_{1})|_{OP_{1}\O} = 0, while at the point O(0, 0) the function (u_{x} − v_{1}) vanishes by the assumption u_{x}, v_{1} ∈ C^{1}(\overline{D}_{1}),

sup_{\overline{D}_{1}\O} ∥y^{−α}u_{x}∥ < ∞, sup_{\overline{D}_{1}\O} ∥y^{−α}v_{1}∥ < ∞ and α > 0. Since u_{x} = v_{1}, (3.12)

implies u_{y} = v_{2} and by (3.11)-(3.14) we can easily get that u is a solution of the problem (3.1)–(3.2).

Note that for the above converse assertion to be valid, it suffices to require of the unknown function u that u ∈ C_{α+ \frac{m}{2} , \frac{m}{2} ,0}^{1}. In this case one should consider the differential expression (−y^{\frac{m}{2}} λ_{1} \frac{∂}{∂x} + \frac{∂}{∂y})(u_{x}−v_{1}) in a generalized sense. By virtue of (3.12) and the equality v_{1y} − v_{2x} = 0, for any function φ ∈ C_{0}^{∞}(D_{1}) we have

∫_{D_{1}} (u_{x} − v_{1})( − y^{\frac{m}{2}} λ_{1} \frac{∂}{∂x} + \frac{∂}{∂y} )φdxdy =

=−∫_{D_{1}} u\frac{∂}{∂x}( − y^{\frac{m}{2}} λ_{1} \frac{∂}{∂x} + \frac{∂}{∂y} )φdxdy+∫_{D_{1}} [( − y^{\frac{m}{2}} λ_{1} \frac{∂}{∂x} + \frac{∂}{∂y} )v_{1}]φdxdy =

= ∫_{D_{1}} [( − y^{\frac{m}{2}} λ_{1} \frac{∂}{∂x} + \frac{∂}{∂y} )u]\frac{∂}{∂x}φdxdy + ∫_{D_{1}} [( − y^{\frac{m}{2}} λ_{1} \frac{∂}{∂x} + \frac{∂}{∂y} )v_{1}]φdxdy =

= ∫_{D_{1}} ( − y^{\frac{m}{2}} λ_{1}v_{1} + v_{2})\frac{∂}{∂x}φdxdy + ∫_{D_{1}} [( − y^{\frac{m}{2}} λ_{1} \frac{∂}{∂x} + \frac{∂}{∂y} )v_{1}]φdxdy =

= − ∫_{D_{1}} [\frac{∂}{∂x}( − y^{\frac{m}{2}} λ_{1}v_{1} + v_{2})]φdxdy +

+ ∫_{D_{1}} [( − y^{\frac{m}{2}} λ_{1} \frac{∂}{∂x} + \frac{∂}{∂y} )v_{1}]φdxdy = ∫_{D_{1}} (v_{1y} − v_{2x})φdxdy = 0,

whence by Theorem 1.4.2 of [26, p. 19] we can conclude that the continuous function u_{x} − v_{1} is constant along the characteristics L_{1} : x + \frac{2λ_{1}}{m+2}y^{\frac{m+2}{2}} = const, and since (u_{x}−v_{1})|_{OP_{1}} = 0, we have u_{x}−v_{1} = 0 in \overline{D}_{1}. The remaining part of our discussion is similar.

As a result of the substitution v = K̃w of the unknown function, and owing to (3.15), instead of (3.11)–(3.14) we shall have

\begin{matrix}w_{y} + D̃_{0}w_{x} = B_{2}w + C_{2}u^{0} + F^{1}, \\ ( − y^{\frac{m}{2}} λ_{1} \frac{∂}{∂x} + \frac{∂}{∂y} )u = ( − y^{\frac{m}{2}} λ_{1}K̃_{1} + K̃_{2})w, \\ ( − y^{\frac{m}{2}} λ_{2n}K̃_{1} + K̃_{2})w( − \frac{2λ_{2n}}{m+2}y^{\frac{m+2}{2}} , y) = f_{1}^{(1)}(y), 0 ≤ y ≤ d_{1}, \\ ( − y^{\frac{m}{2}} λ_{1}K̃_{1} + K̃_{2})w( − \frac{2λ_{1}}{m+2}y^{\frac{m+2}{2}} , y) = f_{2}^{(1)}(y), 0 ≤ y ≤ d_{2}, \\ u( − \frac{2λ_{2n}}{m+2}y^{\frac{m+2}{2}} , y) = f_{1}(y), 0 ≤ y ≤ d_{1}, \end{matrix} \} (3.16)

where B_{2} = −K̃^{−1}K̃y − K̃^{−1}B_{0}K̃, C_{2} = −K̃^{−1}C_{0}, F^{1} = K̃^{−1}F^{0}, and K̃_{1} and K̃_{2} are the matrices of order n × 2n composed, respectively, of the first and the last n rows of the matrix K̃.

By (3.6)–(3.9) we have

\begin{matrix}K̃_{1} = y^{− \frac{m}{2}} K_{1}, K̃_{2} = K_{2}, \\ B_{2} = \frac{m}{2y} K_{1}^{0} × K_{1} − \frac{1}{y} B̃_{0} − B̃_{1}, \end{matrix}  (3.17)

−y^{\frac{m}{2}} λ_{1}K̃_{1} + K̃_{2} = −λ_{1}K_{1} + K_{2} =
= (0, (λ_{2} − λ_{1})ν_{2}, …, (λ_{2n} − λ_{1})ν_{2n}),
−y^{\frac{m}{2}} λ_{2n}K̃_{1} + K̃_{2} = −λ_{2n}K_{1} + K_{2} =
= ((λ_{1} − λ_{2n})ν_{1}, …, (λ_{2n−1} − λ_{2n})ν_{2n−1}, 0).

Taking into account (3.17), we rewrite the problem (3.16) in the form

w_{y} + D̃_{0}w_{x} = \frac{1}{y} (B_{3}w + yC_{2}u^{0}) + F^{1},  (3.18)
( − y^{\frac{m}{2}} λ_{1} \frac{∂}{∂x} + \frac{∂}{∂y} )u = ( − λ_{1}K_{1} + K_{2})w,  (3.19)

\begin{matrix}( − λ_{2n}K_{1} + K_{2})w( − \frac{2λ_{2n}}{m+2}y^{\frac{m+2}{2}} , y) = f_{1}^{(1)}(y), 0 ≤ y ≤ d_{1}, \\ ( − λ_{1}K_{1} + K_{2})w( − \frac{2λ_{1}}{m+2}y^{\frac{m+2}{2}} , y) = f_{2}^{(1)}(y), 0 ≤ y ≤ d_{2}, \end{matrix} \} (3.20)

u( − \frac{2λ_{2n}}{m + 2}y^{\frac{m+2}{2}} , y) = f_{1}(y), 0 ≤ y ≤ d_{1}.  (3.21)

Here B_{3} = \frac{m}{2}K_{1}^{0}K_{1} − B̃_{0} − yB̃_{1}, and by (3.10) we have

sup_{\overline{D}_{1}\O} ∥B_{3}∥ < ∞, sup_{\overline{D}_{1}\O} ∥B_{3x}∥ < ∞.  (3.22)

It follows from (3.6) that v_{1} = y^{− \frac{m}{2}} K_{1}w, v_{2} = K_{2}w, w = y^{\frac{m}{2}} K_{1}^{0}v_{1} + K_{2}^{0}v_{2}. Therefore (v_{1}, v_{2}) belongs to the class (3.15) if and only if w ∈ C_{α+ \frac{m}{2} , \frac{m}{2} +1,1}^{1}.

Let L_{i}(x_{0}, y_{0}) : x = z_{i}(x_{0}, y_{0}; t) = x_{0} + \frac{2λ_{i}}{m+2}y0^{\frac{m+2}{2}} − \frac{2λ_{i}}{m+2}t^{\frac{m+2}{2}} , y = t

be a parametrization of the characteristic curve L_{i}(x_{0}, y_{0}) passing through the point (x_{0}, y_{0}) ∈ \overline{D}_{1}, i = 1, …, 2n. Denote by ω_{i}(x, y) the ordinate of the point of intersection of the characteristic L_{i}(x, y) with the curve γ_{1} for 1 ≤ i ≤ n and with the curve γ_{2} for n < i ≤ 2n, (x, y) ∈ \overline{D}_{1}. It can be easily verified that

ω_{i}(x, y) =

= \begin{cases}[\frac{m+2}{2}(λ_{i} − λ_{2n})^{−1}(x + \frac{2λ_{i}}{m+2}y^{\frac{m+2}{2}} )]^{\frac{2}{m+2}} , & i = 1, …, n, \\ [\frac{m+2}{2}(λ_{i} − λ_{1})^{−1}(x + \frac{2λ_{i}}{m+2}y^{\frac{m+2}{2}} )]^{\frac{2}{m+2}} , & i = n + 1, …, 2n. \end{cases} (3.23)

Let D_{Q}, Q ∈ \overline{D}_{1}\O, be the curvilinear quadrangle with a vertex at O(0, 0), bounded by the characteristics γ_{1}, γ_{2}, L_{n}(Q) and L_{n+1}(Q). Since by the assumption λ_{n} > 0 and λ_{n+1} < 0, the domain D_{Q}, Q(x_{0}, y_{0}) ∈ \overline{D}_{1}\O, is located entirely in the half-plane y ≤ y_{0}. Therefore it follows from the construction of the function ω_{i}(x, y) that

0 ≤ ω_{i}(x, y) ≤ y, (x, y) ∈ \overline{D}_{1}, i = 1, …, 2n,  (3.24)

because the segment of the characteristic L_{i}(Q), issued from the point Q ∈ \overline{D}_{1}\O up to the intersection with the curve γ_{1} for 1 ≤ i ≤ n and with the curve γ_{2} for n < i ≤ 2n, is contained entirely in \overline{D}_{Q}.

By virtue of (3.23) we can easily see that

\begin{matrix}ω_{i}|_{OP_{1}} = \begin{cases}y, & i = 1, …, n, \\ τ_{i}y, & i = n + 1, …, 2n − 1, \\ 0, & i = 2n, \end{cases} \\ ω_{i}|_{OP_{2}} = \begin{cases}0, & i = 1, \\ τ_{i}y, & i = 2, …, n, \\ y, & i = n + 1, …, 2n. \end{cases} \end{matrix}  (3.25)

Here

τ_{i} = \begin{cases}[(λ_{i} − λ_{1})^{−1}(λ_{i} − λ_{2n})]^{\frac{2}{m+2}} , & i = n + 1, …, 2n − 1, \\ [(λ_{i} − λ_{2n})^{−1}(λ_{i} − λ_{1})]^{\frac{2}{m+2}} , & i = 2, …, n; \end{cases}

moreover, by (3.24) we have

τ_{1} = τ_{2} = 0, 0 < τ_{i} < 1, i = 2, …, 2n − 1.  (3.26)

Suppose

φ_{i}(y) = \begin{cases}w_{i}|_{OP_{1}} = w_{i}( − \frac{2λ_{2n}}{m+2}y^{\frac{m+2}{2}} , y), & 0 ≤ y ≤ d_{1}, i = 1, …, n, \\ w_{i}|_{OP_{2}} = w_{i}( − \frac{2λ_{1}}{m+2}y^{\frac{m+2}{2}} , y), & 0 ≤ y ≤ d_{2}, i = n + 1, …, 2n. \end{cases}

Since α > 0, it is obvious that φ_{i}(0) = w_{i}(0, 0) = 0, i = 1, …, 2n.

Integrating the i-th equation of the system (3.18) along the i-th characteristic L_{i}(x, y) from P (x, y) ∈ \overline{D}_{1} to the point of intersection of L_{i}(x, y) with the curve γ_{1} for i ≤ n and with the curve γ_{2} for i > n, while the equation (3.19) along the first characteristic, and taking into account (3.21), we obtain

w_{i}(x, y) = φ_{i}(ω_{i}(x, y)) +
+ ∫_{ω_{i}(x,y)}^{y} \frac{1}{t}( ∑_{j=1}^{2n} B_{3ij}w_{j} + ∑_{j=1}^{n} tC_{2ij}u_{j})(z_{i}(x, y; t), t)dt + F_{i}^{2}(x, y), (3.27)
i = 1, …, 2n,

u(x, y) = f_{1}(ω_{1}(x, y)) +
+ ∫_{ω_{1}(x,y)}^{y} (−λ_{1}K_{1} + K_{2})w(z_{1}(x, y; t), t)dt,  (3.28)

where F_{i}^{2}(x, y) = ∫_{ω_{i}(x,y)}^{y} F_{i}^{1}(z_{i}(x, y; t), t)dt.

We rewrite the system of equations (3.27) in the form of one equation

w(x, y) = φ̃(x, y) +
+ ∑_{i=1}^{2n} ∫_{ω_{i}(x,y)}^{y} \frac{1}{t}(B_{4i}w + C_{3i}u)(z_{i}(x, y; t), t)dt + F^{2}(x, y), (3.29)

where B_{4i} and C_{3i} are well-defined matrices of orders 2n × 2n and 2n × n, respectively, and φ̃(x, y) = (φ_{1}(ω_{1}(x, y)), …, φ_{2n}(ω_{2n}(x, y))).

Substituting the expression for the value w(x, y) from (3.29) into the boundary condition (3.20) and using the equalities (3.25), we get

\begin{matrix}G_{0}^{1}φ^{1}(y) + ∑_{i=n+1}^{2n−1} G_{i}^{1}φ^{2}(τ_{i}y) + T_{1}(w, u)(y) = f_{3}(y), \\ 0 ≤ y ≤ d_{1}, \\ G_{0}^{2}φ^{2}(y) + ∑_{j=2}^{n} G_{j}^{2}φ^{1}(τ_{j}y) + T_{2}(w, u)(y) = f_{4}(y), \\ 0 ≤ y ≤ d_{2}, \end{matrix}  (3.30)

where φ^{1}(y) = (φ_{1}(y), …, φ_{n}(y)), φ^{2}(y) = (φ_{n+1}(y), …, φ_{2n}(y)); G_{i}^{1}, G_{j}^{2} are well-defined constant n×n-matrices; T_{i}(w, u), i = 1, 2, are linear integral operators; f_{3} and f_{4} are given in terms of the known functions f_{1}, f_{2}, F.

Because of (3.17), (3.20) we can easily see that

G_{0}^{1} = ((λ_{1} − λ_{2n})ν_{1}, …, (λ_{n} − λ_{2n})ν_{n}),
G_{0}^{2} = ((λ_{n+1} − λ_{1})ν_{n+1}, …, (λ_{2n} − λ_{1})ν_{2n}).

Therefore, when the condition (3.3) is fulfilled, the matrices G_{0}^{1} and G_{0}^{2} are invertible, and we can rewrite equations (3.30) equivalently as

\begin{matrix}φ^{1}(y) − ∑_{i=n+1}^{2n−1} ∑_{j=2}^{n} G_{ij}^{1}φ^{1}(τ_{i}τ_{j}y) + T_{3}(w, u)(y) = f_{5}(y), \\ 0 ≤ y ≤ d_{1}, \\ φ^{2}(y) − ∑_{i=n+1}^{2n−1} ∑_{j=2}^{n} G_{ij}^{2}φ^{2}(τ_{i}τ_{j}y) + T_{4}(w, u)(y) = f_{6}(y), \\ 0 ≤ y ≤ d_{2}, \end{matrix}  (3.31)

where G_{ij}^{1} = (G_{0}^{1})^{−1}G_{i}^{1}(G_{0}^{2})^{−1}G_{j}^{2}, G_{ij}^{2} = (G_{0}^{2})^{−1}G_{j}^{2}(G_{0}^{1})^{−1}G_{i}^{1}, and T_{3} and T_{4} are linear integral operators.

It is easily seen that the operators T_{3} and T_{4} are given by

\begin{matrix}T_{3}(w, u)(y) = ∑_{i=n+1}^{2n} ∫_{τ_{i}y}^{y} \frac{1}{t}(B_{5i}w + C_{5i}u)(z_{i}(γ_{1}(y), y; t), t)dt + \\ + ∑_{i=n+1}^{2n} ∑_{j=1}^{n} ∫_{τ_{i}τ_{j}y}^{τ_{i}y} \frac{1}{t}(B_{6ij}w + C_{6ij}u)(z_{j}(γ_{2}(τ_{i}y), τ_{i}y; t), t)dt, \\ T_{4}(w, u)(y) = ∑_{j=1}^{n} ∫_{τ_{j}y}^{y} \frac{1}{t}(B_{7j}w + C_{7j}u)(z_{j}(γ_{2}(y), y; t), t)dt + \\ + ∑_{i=n+1}^{2n} ∑_{j=1}^{n} ∫_{τ_{i}τ_{j}y}^{τ_{j}y} \frac{1}{t}(B_{8ij}w + C_{8ij}u)(z_{i}(γ_{1}(τ_{j}y), τ_{j}y; t), t)dt. \end{matrix} (3.32)

Here x = γ_{i}(y) is the equation of the curve γ_{i}, i = 1, 2, and B_{5i}, C_{5i}, B_{6ij}, C_{6ij}, B_{7j}, C_{7j}, B_{8ij}, C_{8ij} are well-defined matrices.

By (3.23) and the requirements imposed on f_{1}, f_{2}, and F, one can easily verify that the values F^{2}, f_{5}, f_{6} from (3.29), (3.31) satisfy for α > 1 the following conditions:

F^{2} ∈ C_{α+ \frac{m}{2} , \frac{m}{2} +1,1}^{1}, f_{4+i} ∈ C^{1}(OP_{i}),

sup_{OP_{i}\O} ∥y^{−(α+ \frac{m}{2} )}f_{4+i}∥ < ∞, sup_{OP_{i}\O} ∥y^{−(α+ \frac{m}{2} −1)}f_{4+i}^{(1)}∥ < ∞, i = 1, 2.

Remark. Obviously, the problem (3.1), (3.2) in the class (3.4) is equivalent to the system of equations (3.28), (3.29), (3.31) with respect to unknown functions u, w, φ^{1} and φ^{2}, where

u ∈ C_{α+ \frac{m}{2} , \frac{m}{2} ,0}^{1}, w ∈ C_{α+ \frac{m}{2} , \frac{m}{2} +1,1}^{1},

φ^{i} ∈ C_{α+ \frac{m}{2} ,1}^{1} = \{φ^{i} ∈ C^{1}[0, d_{i}] : sup_{0<y≤d_{i}} ∥y^{−(α+ \frac{m}{2} )}φ^{i}∥ < ∞,

sup_{0<y≤d_{i}} ∥y^{−(α+ \frac{m}{2} −1)} \frac{d}{dy} φ^{i}∥ < ∞\}, i = 1, 2.

Indeed, w ∈ C_{α+ \frac{m}{2} , \frac{m}{2} +1,1}^{1} implies that v = (v_{1}, v_{2}) belongs to the class (3.15), and since u_{x} = v_{1} and u_{y} = v_{2}, the function u belonging to C_{α+ \frac{m}{2} , \frac{m}{2} ,0}^{1} will also belong to C_{α+ \frac{m}{2} +1, \frac{m}{2} +1,1}^{2}, i.e., to the class (3.4).

§ 4. Investigation of the System of Integro-Functional Equations (3.28), (3.29), (3.31) and the Proof of Theorem 3.1

Introduce into consideration the functional equations

(Λ_{p}φ^{p})(y) = φ^{p}(y) − ∑_{i=n+1}^{2n−1} ∑_{j=2}^{n} G_{ij}^{p}φ^{p}(τ_{i}τ_{j}y) = g_{p}(y),  (3.33)
0 ≤ y ≤ d_{p}, p = 1, 2,

where G_{ij}^{p}, τ_{i}, τ_{j} are defined in (3.31).

Assume h_{p}(ρ) = ∑_{i=n+1}^{2n−1} ∑_{j=2}^{n}(τ_{i}τ_{j})^{ρ}∥G_{ij}^{p}∥, p = 1, 2. By (3.3), (3.17), (3.20)

and (3.26) we have 0 < τ_{i}τ_{j} < 1, ∥G_{ij}^{p}∥ ≠ 0, i = n + 1, …, 2n − 1; j = 2, …, n; p = 1, 2. Therefore the functions h_{1}(ρ) and h_{2}(ρ) are continuous and strictly monotonically decreasing on (−∞, ∞); moreover, lim_{ρ→−∞} h_{i}(ρ) =

+∞ and lim_{ρ→+∞} h_{i}(ρ) = 0, i = 1, 2. Hence there exist the unique real num-

bers ρ_{1} and ρ_{2} such that h_{1}(ρ_{1}) = 1 and h_{2}(ρ_{2}) = 1. Let ρ_{0} = max(ρ_{1}, ρ_{2}). According to Lemma 2.2 of Chapter II, equations (3.33) are uniquely

solvable in the spaces C^{∘}_{α}[0, d_{p}], p = 1, 2, for α > ρ_{0}, and we have the estimates

∥(Λ_{p}^{−1}g_{p})(y)∥ = ∥φ^{p}(y)∥ ≤ ξ_{pα}y^{α}∥g_{p}∥_{C^{∘}_{α}[0,d_{p}]}, p = 1, 2, (3.34)

where ξ_{pα} = (1 − h_{p}(α))^{−1} > 0, lim_{\begin{matrix}α→+∞, \\ α>ρ_{0} \end{matrix}}ξ_{pα} = 1, p = 1, 2.

Equations (3.31) in terms of (3.33) take the form

\begin{matrix}(Λ_{1}φ^{1})(y) + T_{3}(w, u)(y) = f_{5}(y), 0 ≤ y ≤ d_{1}, \\ (Λ_{2}φ^{2})(y) + T_{4}(w, u)(y) = f_{6}(y), 0 ≤ y ≤ d_{2}. \end{matrix}  (3.35)

We shall solve the system of equations (3.28), (3.29), (3.35) with respect to unknown functions u, w, φ^{1}, φ^{2} by the method of successive approximations.

Assume

u_{0}(x, y) ≡ 0, w_{0}(x, y) ≡ 0, φ_{0}^{i}(y) ≡ 0, i = 1, 2,

u_{k}(x, y) = f_{1}(ω_{1}(x, y)) + ∫_{ω_{1}(x,y)}^{y} (−λ_{1}K_{1} + K_{2})w_{k−1}(z_{1}(x, y; t), t)dt,

w_{k}(x, y) = φ̃_{k}(x, y) +

+ ∑_{i=1}^{2n} ∫_{ω_{i}(x,y)}^{y} \frac{1}{t}(B_{4i}w_{k−1} + C_{3i}u_{k−1})(z_{i}(x, y; t), t)dt + F^{2}(x, y),

where φ̃_{k}(x, y) = (φ_{1,k}(ω_{1}(x, y)), …, φ_{2n,k}(ω_{2n}(x, y))), and the values φ_{k}^{1}(y) = (φ_{1,k}(y), …, φ_{n,k}(y)) and φ_{k}^{2}(y) = (φ_{n+1,k}(y), …, φ_{2n,k}(y)) are to be determined from the equations

(Λ_{1}φ_{k}^{1})(y) + T_{3}(w_{k−1}, u_{k−1})(y) = f_{5}(y),
(Λ_{2}φ_{k}^{2})(y) + T_{4}(w_{k−1}, u_{k−1})(y) = f_{6}(y).

Remark. By virtue of (3.22), the coefficients at the unknown functions u and w appearing in the equalities (3.29) and (3.30) along with their first derivatives with respect to x are bounded uniformly in the norm in \overline{D}_{1}\O.

Owing to the estimates (3.34), equality (3.32) and the above remark, we have the following

Lemma 3.1. There exists a real number α_{1} ≥ 1 depending only on the coefficients of the system (3.1) such that for α > α_{1} the estimates

∥u_{k+1}(x, y) − u_{k}(x, y)∥ ≤ M_{1α}y^{α+ \frac{m}{2}} q_{1α}^{k},
∥w_{k+1}(x, y) − w_{k}(x, y)∥ ≤ M_{1α}y^{α+ \frac{m}{2}} q_{1α}^{k},
∥φ_{k+1}^{i}(y) − φ_{k}^{i}(y)∥ ≤ M_{1α}y^{α+ \frac{m}{2}} q_{1α}^{k}, i = 1, 2,

are valid, where positive numbers M_{1α} and q_{1α} do not depend on k, q_{1α} as a function of α strictly monotonically decreases for α > α_{1}, and q_{1α} < 1,

lim_{α→+∞} q_{1α} = 0.

On the basis of Lemma 1 we prove

Lemma 3.2. There exists a positive number α_{2}, α_{2} ≥ α_{1}, depending only on the coefficients of the system (3.1) such that for α > α_{2} the estimates

∥\frac{∂}{∂x}u_{k+1}(x, y) − \frac{∂}{∂x}u_{k}(x, y)∥ ≤ M_{2α}y^{α}q_{2α}^{k},

∥\frac{∂}{∂y} u_{k+1}(x, y) − \frac{∂}{∂y} u_{k}(x, y)∥ ≤ M_{2α}y^{α+ \frac{m}{2}} q_{2α}^{k},

∥\frac{∂}{∂x}w_{k+1}(x, y) − \frac{∂}{∂x}w_{k}(x, y)∥ ≤ M_{2α}y^{α−1}q_{2α}^{k},

∥\frac{∂}{∂y} w_{k+1}(x, y) − \frac{∂}{∂y} w_{k}(x, y)∥ ≤ M_{2α}y^{α+ \frac{m}{2} −1}q_{2α}^{k},

∥\frac{∂}{∂y} φ_{k+1}^{i}(y) − \frac{∂}{∂y} φ_{k}^{i}(y)∥ ≤ M_{2α}y^{α+ \frac{m}{2} −1}q_{2α}^{k}, i = 1, 2,

are valid. Here positive numbers M_{2α} and q_{2α} do not depend on k, q_{2α} as a function of α strictly monotonically decreases for α > α_{2}, and q_{2α} < 1,

lim_{α→+∞} q_{2α} = 0.

The lemma below holds.

Lemma 3.3. The homogeneous system of equations corresponding to (3.28), (3.29), (3.31) has only the trivial solution in the class of functions

u, w ∈ C_{α+ \frac{m}{2} ,0,0}^{0}, φ^{i} ∈ C[0, d_{i}], sup_{0<y≤d_{i}} ∥y^{−(α+ \frac{m}{2} )}φ^{i}∥ < ∞, i = 1, 2,

where α > α_{2}.

From Lemmas 3.1–3.3 we have

Lemma 3.4. For α > α_{2} the system of equations (3.28), (3.29), (3.31) has the unique solution in the class of functions

u ∈ C_{α+ \frac{m}{2} , \frac{m}{2} ,0}^{1}, w ∈ C_{α+ \frac{m}{2} , \frac{m}{2} +1,1}^{1}, φ^{i} ∈ C_{α+ \frac{m}{2} ,1}^{1}, i = 1, 2.

From the remark at the end of §3 and Lemma 3.4 it follows that when the conditions (3.3) are fulfilled and α > α_{2}, the problem (3.1), (3.2) is uniquely solvable in the class (3.4); moreover, we can choose the number α_{2} to depend only on the coefficients A, B, C and a of the system (3.1). ■

§ 5. A Characteristic Problem for Hyperbolic Systems of Second Order with Characteristic Line of Parabolic Degeneration

Let us consider the system of the form

Au_{xx} + 2y^{\frac{m}{2}} Bu_{xy} + y^{m}Cu_{yy} + au_{x} + bu_{y} + cu = F,  (3.36)

where A, B, C, a, b, c are given real n × n-matrices, F is a given and u is an unknown n-dimensional real vector, 0 < m = const < 2, n > 1.

Below A, B, C are assumed to be constant matrices, det A ≠ 0, and the polynomial p_{0}(μ) = det(Aμ^{2} + 2Bμ + C) of degree 2n is assumed to have simple real roots μ_{1}, …, μ_{2n}. Under these assumptions the system (3.36) for y > 0 is strictly hyperbolic, and the line of parabolic degeneration y = 0 is a characteristic of the system (3.36). It is easily seen that the numbers y^{\frac{m}{2}} μ_{1}, …, y^{\frac{m}{2}} μ_{2n} are the roots of the characteristic polynomial p(y; μ) = det(Aμ^{2} + 2y^{\frac{m}{2}} Bμ + y^{m}C) of the system (3.36), while the curves defined by the equations

L_{i}(P) : μ_{i}x + \frac{2}{2 − m}y^{\frac{2−m}{2}} = μ_{i}x_{0} + \frac{2}{2 − m}y0^{\frac{2−m}{2}} , i = 1, …, 2n, y_{0} > 0,

and passing through P (x_{0}, y_{0}) are characteristics of the system (3.36).

Denote by D the domain lying in the half-plane y > 0 and bounded by the two adjoint characteristics

γ_{1} : μ_{i_{1}}x + \frac{2}{2 − m}y^{\frac{2−m}{2}} = 0, γ_{2} : μ_{i_{2}}x + \frac{2}{2 − m}y^{\frac{2−m}{2}} = 0,

μ_{i_{2}} < μ_{i_{1}} < 0,

of the system (3.36), coming out of the origin O(0, 0). Let us take arbitrarily on γ_{1} a point P_{1} different from zero and choose the numbering of the characteristic curves L_{i}(P_{1}), i = 1, …, 2n, coming out of P_{1} into the angle D, such that starting from L_{1}(P_{1}) they follow each other counter-clockwise. Let us fix on the curve γ_{2} a point P_{2} lying strictly between the two points of intersection of the characteristics L_{n}(P_{1}) and L_{n+1}(P_{1}) with the curve γ_{2}. Let D_{0} ⊂ D be the characterisitc quadrangle with a vertex at the point O, bounded by the characteristics γ_{1}, γ_{2}, L_{n}(P_{1}) and L_{n+1}(P_{2}).

Consider the characteristic problem formulated as follows [40]: find in the domain D_{0} a regular solution u(x, y) of the system (3.36) satisfying on the segments OP_{i} of characterisitcs γ_{i} the following conditions

u|_{OP_{i}} = f_{i}, i = 1, 2,  (3.37)

where f_{1}, f_{2} are given n-dimensional real vectors, f_{1}(0) = f_{2}(0).

Note that owing to the character of degeneration of the system (3.36) the condition m < 2 whose fulfilment is not needed when considering problem (3.1), (3.2), is of great importance. In contrast to the problem (3.1), (3.2) where a condition of Gellerstedt type is imposed on the lowest coefficient a at u_{x}, in considering the problem (3.36), (3.37) a condition of similar type is to be imposed on the coefficient b at u_{y}.

Below we assume that a, b, c, F ∈ C^{1}(\overline{D}_{0}), f_{i} ∈ C^{2}(OP_{i}), i = 1, 2, and moreover, in the domain D_{0}

sup_{\overline{D}_{0}\O} ∥y^{1−m}b∥ < ∞ for m > 1,

sup_{\overline{D}_{0}\O} ∥x^{−(α+ \frac{m}{2−m} −1)}F ∥ < ∞, sup_{\overline{D}_{0}\O} ∥x^{−(α−2)}F_{y}∥ < ∞, α = const > 0,

f_{i}(0) = 0, sup_{OP_{i}\O} ∥x^{−(α+ \frac{m}{2−m})}f_{i}^{(1)}∥ < ∞,

sup_{OP_{i}\O} ∥x^{−(α+ \frac{m}{2−m} −1)}f_{i}^{(2)}∥ < ∞, i = 1, 2.

Since the system (3.36) is strictly hyperbolic, we have dim Ker(Aμ_{i}^{2} + 2Bμ_{i} + C) = 1, i = 1, …, 2n. Let ν_{i} ∈ Ker(Aμ_{i}^{2} + 2Bμ_{i} + C), ∥ν_{i}∥ ≠ 0, i = 1, …, 2n.

Under the assumption that μ_{n} < μ_{2n} < μ_{n+1}, the following theorem is valid.

Theorem 3.2. If

rank {ν_{1}, …, ν_{n}} = rank {ν_{n+1}, …, ν_{2n}} = n,  (3.38)

then there exists a positive number α_{0} depending only on the coefficients A, B, C, b of the system (3.36) such that for all α > α_{0} the problem (3.36), (3.37) is uniquely solvable in the class of functions

\{u ∈ C^{2}(\overline{D}_{0}) : ∂^{i,j}u(0, 0) = 0,

sup_{\overline{D}_{0}\O} ∥x^{−(α+ \frac{2}{2−m} −i− \frac{2}{2−m} j)}∂^{i,j}u∥ < ∞, 0 ≤ i + j ≤ 2\},

∂^{i,j} = \frac{∂^{i+j}}{∂x^{i}∂y^{j}} .

As examples show, when either the condition (3.38) or the inequality α > α_{0} is violated, the homogeneous problem corresponding to (3.36), (3.37) may have an infinite number of linearly independent solutions.

The proof of Theorem 3.2 goes by the same scheme as that of Theorem 3.1. For details the reader may refer to [40].

CHAPTER IV

MULTIDIMENSIONAL ANALOGUES OF THE

GOURSAT AND DARBOUX PROBLEMS FOR LINEAR

DIFFERENTIAL EQUATIONS OF HYPERBOLIC TYPE

§ 1. Formulation of Multidimensional Analogues of the Goursat and Darboux Problems for the Wave Equation

In the space of variables x_{1}, x_{2}, t let us consider the wave equation

□u ≡ \frac{∂^{2}u}{∂t^{2}} − \frac{∂^{2}u}{∂x_{1}^{2}} − \frac{∂^{2}u}{∂x_{2}^{2}} = F,  (4.1)

where F is a given real function and u is an unknown real function.

Denote by D : k_{1}t < x_{2} < k_{2}t, 0 < t < t_{0}, −1 ≤ k_{i} = const ≤ 1, i = 1, 2, k_{1} < k_{2}, the domain lying in a half-space t > 0 bounded by the plane surfaces S_{i} : k_{i}t−x_{2} = 0, 0 ≤ t ≤ t_{0}, i = 1, 2, and by the plane t = t_{0}.

For equation (4.1) let us consider the boundary value problem formulated as follows [44, 45]: find in the domain D a solution u(x_{1}, x_{2}, t) of equation (4.1) satisfying the boundary conditions

u|_{S_{i}} = f_{i}, i = 1, 2,  (4.2)

where f_{i}, i = 1, 2, are given real functions on S_{i} with (f_{1} − f_{2})|_{S_{1}∩S_{2}} = 0.

Note that when |k_{i}| = 1, the surface S_{i} is a characteristic surface for the equation (4.1), while when |k_{i}| < 1, this surface is time-type. In the case where |k_{i}| = 1, i = 1, 2, the problem (4.1), (4.2) represents a multidimensional analogue of the formulated in the introduction Goursat problem for the equation of string oscillation. For |k_{1}| < 1 and |k_{2}| = 1 the problem (4.1), (4.2) represents a multidimensional analogue of the first Darboux problem and for |k_{i}| < 1, i = 1, 2, it represents a multidimensional analogue of the second Darboux problem.

For the equation (4.1) one can also consider the boundary value problem formulated as follows: find in the domain D a solution u(x_{1}, x_{2}, t) of equation (4.1) satisfying the boundary conditions

\frac{∂u}{∂n}|_{S_{1}} = f_{1},  (4.3)
u|_{S_{2}} = f_{2},  (4.4)

where f_{i}, i = 1, 2, are given real functions and \frac{∂}{∂n} is the derivative along the outer normal to S_{1}.

Below we shall prove existence and uniqueness theorems both for regular and for strong solutions of the problems (4.1), (4.2) and (4.1), (4.3), (4.4) in the class W_{2}^{1}.

Denote by C_{∗}^{∞}(\overline{D}) the space of functions of the class C^{∞}(\overline{D}), having bounded supports, i.e.

C_{∗}^{∞}(\overline{D}) = \{u ∈ C^{∞}(\overline{D}) : diam supp u < ∞\}.

The spaces C_{∗}^{∞}(S_{i}), i = 1, 2, are defined in a similar way.

Denote by W_{2}^{1}(D), W_{2}^{2}(D) and W_{2}^{1}(S_{i}), i = 1, 2, the well-known Sobolev spaces. Note that C_{∗}^{∞}(\overline{D}) is an everywhere dense subspace of the spaces W_{2}^{1}(D) and W_{2}^{2}(D), while C_{∗}^{∞}(S_{i}) is an everywhere dense subspace of the space W_{2}^{1}(S_{i}), i = 1, 2.

Definition. Let f_{i} ∈ W_{2}^{1}(S_{i}), i = 1, 2, F ∈ L_{2}(D). A function u ∈ W_{2}^{1}(D) is said to be a strong solution of the problem (4.1), (4.2) of the class W_{2}^{1} if there exists a sequence u_{n} ∈ C_{∗}^{∞}(\overline{D}) such that u_{n} → u, □u_{n} → F and u_{n}|_{S_{i}} → f_{i} i = 1, 2, in the spaces W_{2}^{1}(D), L_{2}(D) and W_{2}^{1}(S_{i}), i = 1, 2, respectively, i.e., for n → ∞

∥u_{n} − u∥_{W_{2}^{1}(D)} → 0, ∥□u_{n} − F ∥_{L_{2}(D)} → 0,
∥u_{n}|_{S_{i}} − f_{i}∥_{W_{2}^{1}(S_{i})} → 0, i = 1, 2.

Below we shall also introduce the notion of the strong solution of the problem (4.1), (4.3), (4.4) in the class W_{2}^{1}.

§ 2. A Priori Estimates for Solutions of the Problems (4.1), (4.2) and (4.1), (4.3), (4.4)

The following lemma holds.

Lemma 4.1. When −1 ≤ k_{1} < 0 and 0 < k_{2} ≤ 1, the estimate

∥u∥_{W_{2}^{1}(D)} ≤ C( ∑_{i=1}^{2} ∥f_{i}∥_{W_{2}^{1}(S_{i})} + ∥F ∥_{L_{2}(D)}),  (4.5)

is valid for any u ∈ W_{2}^{2}(D), where f_{i} = u|_{S_{i}}, i = 1, 2, F = □u, and a positive constant C does not depend on u.

Proof. Since the space C_{∗}^{∞}(D) (C_{∗}^{∞}(S_{i})) is a dense subspace of the spaces W_{2}^{1}(D) and W_{2}^{2}(D) (W_{2}^{1}(S_{i})), due to the known theorems of embedding of W_{2}^{2}(D) in W_{2}^{1}(D) and W_{2}^{2}(D) in W_{2}^{1}(S_{i}) it suffices to prove the validity of the estimate (4.5) for the functions u of the class C_{∗}^{∞}(\overline{D}).

Introduce the notation:

D_{τ} = {(x, t) ∈ D : t < τ}, D_{0τ} = ∂D_{τ} ∩ {t = τ}, 0 < τ ≤ t_{0},

S_{iτ} = ∂D_{τ} ∩ S_{i}, i = 1, 2,

S_{τ} = S_{1τ} ∪ S_{2τ} , α_{1} = cos({ \atop n, x_{1}}), α_{2} = cos({ \atop n, x_{2}}), α_{3} = cos(n̂, t).

Here n = (α_{1}, α_{2}, α_{3}) is the unit vector of the outer normal to ∂D_{τ}; moreover, as is easily seen,

n|_{S_{1τ}} = (0, \frac{−1}{\sqrt{1 + k_{1}^{2}}} , \frac{k_{1}}{\sqrt{1 + k_{1}^{2}}} ), n|_{S_{2τ}} = (0, \frac{1}{\sqrt{1 + k_{2}^{2}}} , \frac{−k_{2}}{\sqrt{1 + k_{2}^{2}}} ),

n|_{D_{0τ}} = (0, 0, 1).

Therefore for −1 ≤ k_{1} < 0 and 0 < k_{2} ≤ 1 we have

α_{3}|_{S_{iτ}} < 0, i = 1, 2, α_{3}^{−1}(α_{3}^{2} − α_{1}^{2} − α_{2}^{2})|_{S_{i}} > 0, i = 1, 2. (4.6)

Multiplying both parts of (4.1) by 2u_{t}, where u ∈ C_{∗}^{∞}(\overline{D}), F = □u, integrating the obtained expression over D_{τ}, and taking into account (4.6), we get

2 ∫_{D_{τ}} F u_{t}dxdt = ∫_{D_{τ}} (\frac{∂u_{t}^{2}}{∂t} + 2u_{x_{1}}u_{tx_{1}} + 2u_{x_{2}}u_{tx_{2}})dxdt −

−2 ∫_{S_{τ}} (u_{x_{1}}u_{t}α_{1} + u_{x_{2}}u_{t}α_{2})ds = ∫_{D_{0τ}} (u_{t}^{2} + u_{x_{1}}^{2} + u_{x_{2}}^{2})dx +

+ ∫_{S_{τ}} [(u_{t}^{2} + u_{x_{1}}^{2} + x_{x_{2}}^{2})α_{3} − 2(u_{x_{1}}u_{t}α_{1} + u_{x_{2}}u_{t}α_{2})]ds =

= ∫_{D_{0τ}} (u_{t}^{2} + u_{x_{1}}^{2} + u_{x_{2}}^{2})dx + ∫_{S_{τ}} α_{3}^{−1}[(α_{3}u_{x_{1}} − α_{1}u_{t})^{2} +

+(α_{3}u_{x_{2}} − α_{2}u_{t})^{2} + (α_{3}^{2} − α_{1}^{2} − α_{2}^{2})u_{t}^{2}]ds ≥

≥ ∫_{D_{0τ}} (u_{t}^{2} + u_{x_{1}}^{2} + u_{x_{2}}^{2})dx + ∫_{S_{τ}} α_{3}^{−1}[(α_{3}u_{x_{1}} − α_{1}u_{t})^{2} +

+(α_{3}u_{x_{2}} − α_{2}u_{t})^{2}]ds.  (4.7)

Putting

w(τ) = ∫_{D_{0τ}} (u_{t}^{2} + u_{x_{1}}^{2} + u_{x_{2}}^{2})dx, ũ_{i} = α_{3}u_{x_{i}} − α_{i}u_{t}, i = 1, 2,

C_{1} = max (\frac{\sqrt{1 + k_{1}^{2}}}{|k_{1}|} , \frac{\sqrt{1 + k_{2}^{2}}}{|k_{2}|} ),

from (4.5) we have

w(τ) ≤ C_{1} ∫_{S_{τ}} (u_{1}^{2} + u_{2}^{2})ds + ∫_{D_{τ}} (F^{2} + u_{t}^{2})dxdt ≤

≤ C_{1} ∫_{S_{τ}} (u_{1}^{2} + u_{2}^{2})ds + ∫_{0}^{τ} dξ ∫_{D_{0ξ}} u_{t}^{2}dx + ∫_{D_{τ}} F^{2}dxdt ≤

≤ C_{1} ∫_{S_{τ}} (u_{1}^{2} + u_{2}^{2})ds + ∫_{0}^{τ} w(ξ)dξ + ∫_{D_{τ}} F^{2}dxdt.  (4.8)

Let (x, τ_{x}) be the point of intersection of the surface S_{1} ∪ S_{2} and the straight line, parallel to the axis t and passing through (x, 0). We have

u(x, τ) = u(x, τ_{x}) + ∫_{τ_{x}}^{τ} u_{t}(x, t)dt,

which implies

∫_{D_{0τ}} u^{2}(x, τ)dx ≤

≤ 2 ∫_{D_{0τ}} u^{2}(x, τ_{x})dx + 2|τ − τ_{x}| ∫_{D_{0τ}} dx ∫_{τ_{x}}^{τ} u_{t}^{2}(x, t)dt =

= 2 ∫_{S_{τ}} α_{3}^{−1}u^{2}ds + 2|τ − τ_{x}| ∫_{D_{τ}} u_{t}^{2}dxdt ≤

≤ C_{2}( ∫_{S_{τ}} u^{2}ds + ∫_{D_{τ}} u_{t}^{2}dxdt),  (4.9)

where C_{2} = 2 max(C_{1}, t_{0}).

Introducing the notation

w_{0}(τ) = ∫_{D_{0τ}} (u^{2} + u_{t}^{2} + u_{x_{1}}^{2} + u_{x_{2}}^{2})dx

and adding inequalities (4.8) and (4.9), we obtain

w_{0}(τ) ≤ C_{2}[ ∫_{S_{τ}} (u^{2} + u_{1}^{2} + u_{2}^{2})ds + ∫_{0}^{τ} w_{0}(ξ)dξ + ∫_{D_{τ}} F^{2}dxdt],

from which by Gronwall’s lemma we find that

w_{0}(τ) ≤ C_{3}[ ∫_{S_{τ}} (u^{2} + u_{1}^{2} + u_{2}^{2})ds + ∫_{D_{τ}} F^{2}dxdt],  (4.10)

where C_{3} = const > 0.

We can easily see that the operator α_{3} \frac{∂}{∂x_{i}} −α_{i} \frac{∂}{∂t} is an interior differential operator on the surface S_{τ}. Therefore by virtue of (4.2) the inequality

∫_{S_{τ}} (u^{2} + u_{1}^{2} + u_{2}^{2})ds ≤ C_{4} ∑_{i=1}^{2} ∥f_{i}∥_{W_{2}^{1}(S_{iτ})}^{2}, C_{4} = const > 0, (4.11)

is valid.

It follows from (4.10) and (4.11) that

w_{0}(τ) ≤ C_{5}( ∑_{i=1}^{2} ∥f_{i}∥_{W_{2}^{1}(S_{iτ})}^{2} + ∥F ∥_{L_{2}(D_{τ})}^{2}), C_{5} = const > 0. (4.12)

Integrating both parts of (4.12) with respect to τ, we get (4.5). ■

Remark 1. It should be noted that the constant C in (4.5) tends to infinity for k_{1} → 0 or k_{2} → 0 and in the limiting case where k_{1} = 0 or k_{2} = 0, i.e., for S_{1} : x_{2} = 0, 0 ≤ t ≤ t_{0} or S_{2} : x_{2} = 0, 0 ≤ t ≤ t_{0}, this estimate becomes, generally speaking, invalid. At the same time, following the proof of Lemma 4.1, we can easily see that (4.5) is also valid for k_{1} = 0 or for k_{2} = 0 if f_{1} = u|_{S_{1}} = 0 or f_{2} = u|_{S_{2}} = 0.

Remark 2. Below along with (4.1) we consider the equation

Lu ≡ □u + au_{x_{1}} + bu_{x_{2}} + cu_{t} + du = F,  (4.13)

where the coefficients a, b, c and d are given bounded measurable functions in the domain D. Moreover, it will be shown that the solvability of the problem (4.13), (4.2) follows from the solvability of the problem (4.1), (4.2) and the fact that in specifically chosen equivalent norms of the spaces L_{2}(D), W_{2}^{1}(D), W_{2}^{1}(S_{i}), i = 1, 2, the lower terms in equation (4.13) cause arbitrarily small perturbations.

In the space W_{2}^{1}(D) we consider the following equivalent norm depending on a parameter γ

∥u∥_{D,1,γ}^{2} = ∫_{D} e^{−γt}(u^{2} + u_{t}^{2} + u_{x_{1}}^{2} + u_{x_{2}}^{2})dxdt, γ > 0.

In the same way we introduce the norms ∥F ∥_{D,0,γ}, ∥f_{i}∥_{S_{i},1,γ} in the spaces L_{2}(D), W_{2}^{1}(S_{i}), i = 1, 2.

Using the energetic estimate (4.12), we obtain an a priori estimate for u ∈ C_{∗}^{∞}(\overline{D}) with respect to the norms ∥·∥_{D,1,γ}, ∥·∥_{S_{i},1,γ}, ∥·∥_{D,0,γ}. Multiplying both parts of (4.12) by e^{−γτ} and integrating the obtained inequality with respect to τ from 0 to t_{0}, we get

∥u∥_{D,1,γ}^{2} = ∫_{0}^{t_{0}} e^{−γτ} w_{0}(τ)dτ ≤

≤ C_{5}( ∑_{i=1}^{2} ∫_{0}^{t_{0}} e^{−γτ} ∥f_{i}∥_{W_{2}^{1}(S_{iτ})}^{2}dτ + ∫_{0}^{t_{0}} e^{−γτ} ∥F ∥_{L_{2}(D_{τ})}^{2}dτ). (4.14)

We have

∫_{0}^{t_{0}} e^{−γτ} ∥F ∥_{L_{2}(D_{τ})}^{2}dτ = ∫_{0}^{t_{0}} e^{−γτ} [ ∫_{0}^{τ} ( ∫_{D_{0σ}} F^{2}dx)dσ]dτ =

= ∫_{0}^{t_{0}} [ ∫_{D_{0τ}} F^{2}dx ∫_{σ}^{t_{0}} e^{−γτ} dτ]dσ = \frac{1}{γ} ∫_{0}^{t_{0}} (e^{−γσ} − e^{−γt_{0}})[ ∫_{D_{0σ}} F^{2}dx]dσ ≤

≤ \frac{1}{γ} ∫_{0}^{t_{0}} e^{−γσ}[ ∫_{D_{0σ}} F^{2}dx]dσ = \frac{1}{γ} ∥F ∥_{D,0,γ}^{2},  (4.15)

where D_{0τ} = ∂D_{τ} ∩ {t = τ}, 0 < τ ≤ t_{0}. Analogously we obtain

∫_{0}^{t_{0}} e^{−γτ} ∥f_{i}∥_{W_{2}^{1}(S_{iτ})}^{2}dτ ≤ \frac{C_{6}}{γ} ∥f_{i}∥_{S_{i},1,γ}^{2}, i = 1, 2,  (4.16)

where C_{6} is a positive constant independent of f_{i} and γ.

Under conditions of Lemma 4.1 from (4.14)–(4.16) we obtain the following a priori estimate for u ∈ W_{2}^{2}(D)

∥u∥_{D,1,γ} ≤ \frac{C_{7}}{\sqrt{γ}} ( ∑_{i=1}^{2} ∥f_{i}∥_{S_{i},1,γ} + ∥F ∥_{D,0,γ}),  (4.17)

where C_{7} = const > 0 does not depend on u and γ.

Consider now the problem (4.1), (4.3), (4.4) in the case where k_{1} = 0, i.e., S_{1} : x_{2} = 0, 0 ≤ t ≤ t_{0}, 0 < k_{2} ≤ 1, and in the boundary condition (4.3) the function f_{1} = 0, that is,

\frac{∂u}{∂n}|_{S_{1}} = 0.  (4.18)

Lemma 4.2. For any u ∈ W_{2}^{2}(D) satisfying the homogeneous boundary condition (4.18), the estimate

∥u∥_{W_{2}^{1}(D)} ≤ C(∥f_{2}∥_{W_{2}^{1}(S_{1})} + ∥F ∥_{L_{2}(D)})  (4.19)

is valid, where f_{2} = u|_{S_{2}}, F = □u and a positive constant C does not depend on u.

Proof. Denote by D_{−} : −k_{2}t < x_{2} < 0, 0 < t < t_{0} the domain which is symmetric to D with respect to the plane x_{2} = 0 and by D_{0} : −k_{2}t < x_{2} < k_{2}t, 0 < t < t_{0} the domain which is the union of domains D and D_{−} together with a piece of a plane surface x_{2} = 0, 0 < t < t_{0}.

It can be easily verified that if we extend evenly the function u ∈ W_{2}^{2}(D) satisfying the homogeneous boundary condition (4.18) to the domain D_{−}, then the obtained function u_{0}

u_{0}(x_{1}, x_{2}, t) = \begin{cases}u(x_{1}, x_{2}, t), & x_{2} ≥ 0, \\ u(x_{1}, −x_{2}, t), & x_{2} < 0, \end{cases}  (4.20)

will belong to the class W_{2}^{2}(D_{0}). By (4.5) the function u_{0} ∈ W_{2}^{2}(D_{0}) satisfies the estimate

∥u_{0}∥_{W_{2}^{1}(D_{0})} ≤ C(∥f_{1}∥_{W_{2}^{1}(S_{2}^{−})} + ∥f_{2}∥_{W_{2}^{1}(S_{2})} + ∥F_{0}∥_{L_{2}(D_{0})}), (4.21)

where S_{2}^{−} : k_{2}t + x_{2} = 0, 0 ≤ t ≤ t_{0}, f_{1} = u|_{S_{2}^{−}}, f_{2} = u|_{S_{2}}, F_{0} = □u_{0}. Now it remains only to note that in (4.21)

∥u_{0}∥_{W_{2}^{1}(D_{0})} = \sqrt{2}∥u∥_{W_{2}^{1}(D)}, ∥f_{1}∥_{W_{2}^{1}(S_{2}^{−})} = ∥f_{2}∥_{W_{2}^{1}(S_{2})},
∥F_{0}∥_{L_{2}(D_{0})} = \sqrt{2}∥F ∥_{L_{2}(D)}

because of (4.20). ■

Remark 3. Arguments similar to those given in proving the estimate (4.17) enable us to prove that for any u ∈ W_{2}^{2}(D) satisfying the homogeneous boundary condition (4.18) the estimate

∥u∥_{D,1,γ} ≤ \frac{C}{\sqrt{γ}} (∥f_{2}∥_{S_{2,1,γ}} + ∥F ∥_{D,0,γ}),  (4.22)

is valid, where f_{2} = u|_{S_{2}}, F = □u, and C is a positive constant independent of u and γ.

Remark 4. It follows from (4.5) and (4.19) that when conditions of Lemmas 4.1 and 4.2 are fulfilled, the problems (4.1), (4.2) and (4.1), (4.3), (4.4), respectively, cannot have more than one strong solution of the class W_{2}^{1}.

We can also show that for the problem (4.1), (4.2) the uniqueness theorem is likewise valid for the weak solution of the class W_{2}^{1}.

Let k_{1} = 0 and k_{2} = 1, i.e., S_{1} : x_{2} = 0, 0 ≤ t ≤ t_{0}, while S_{2} : t − x_{2} = 0, 0 ≤ t ≤ t_{0} is a characteristic surface. Suppose S_{3} = ∂D ∩ {t = t_{0}}, V = {v ∈ W_{2}^{1}(D) : v|_{S_{1}∪S_{3}} = 0}.

Definition. Let f_{i} ∈ W_{2}^{1}(S_{i}), i = 1, 2, F ∈ L_{2}(D). A function u ∈ W_{2}^{1}(D) is called a weak solution of the problem (4.1), (4.2) of the class W_{2}^{1} if it satisfies both the boundary conditions (4.2) and the identity

∫_{D} (u_{t}v_{t} − u_{x_{1}}v_{x_{1}} − u_{x_{2}}v_{x_{2}})dxdt +

+ ∫_{S_{2}} \frac{∂f_{2}}{∂N} vds + ∫_{D} F vdxdt = 0  (4.23)

for any v ∈ V, where \frac{∂}{∂N} is a derivative with respect to a conormal to S_{2}, N is the unit conormal vector at the point (x, t) ∈ ∂D with the direction cosines

cos N̂x_{1} = cos n̂x_{1}, cos N̂x_{2} = cos n̂x_{2}, cos N̂t = − cos n̂t,

and n is the unit vector of the outward normal to ∂D. Since on the characteristic surface S_{2} the direction of the conormal N coincides with that of a bicharacteristic lying on S_{2}, the value \frac{∂f_{2}}{∂N} is determined correctly.

Lemma 4.3. For k_{1} = 0, k_{2} = 1 the problem (4.1), (4.2) cannot have more than one weak solution of the class W_{2}^{1}.

Proof. Let a function u ∈ W_{2}^{1}(D) satisfy the identity (4.23) with u|_{S_{i}} = f_{i} = 0, i = 1, 2, F = 0. In this identity we take as v the function

v(x_{1}, x_{2}, t) = \begin{cases}0 & for t ≥ τ, \\ ∫_{τ}^{t} u(x_{1}, x_{2}, σ)dσ & for |x_{2}| ≤ t ≤ τ, \end{cases}  (4.24)

where 0 < τ ≤ t_{0}.

Obviously, v ∈ V and

\begin{matrix}v_{t} = u, v_{x_{i}} = ∫_{τ}^{t} u_{x_{i}}(x_{1}, x_{2}, σ)dσ, i = 1, 2, \\ v_{tx_{i}} = u_{x_{i}}, v_{tt} = u_{t}. \end{matrix}  (4.25)

By virtue of (4.24) and (4.25) the identity (4.23) for f_{2} = 0, F = 0 will take the form

∫_{D_{τ}} (v_{tt}v_{t} − v_{tx_{1}}v_{x_{1}} − v_{tx_{2}}v_{x_{2}})dxdt = 0,

i.e.,

∫_{D_{τ}} \frac{∂}{∂t}(v_{t}^{2} − v_{x_{1}}^{2} − v_{x_{2}}^{2})dxdt = 0,  (4.26)

where D_{τ} = D ∩ {t < τ}.

Applying Gauss-Ostrogradsky’s formula to the left-hand side of (4.26), we obtain

∫_{∂D_{τ}} (v_{t}^{2} − v_{x_{1}}^{2} − v_{x_{2}}^{2}) cos n̂tds = 0.  (4.27)

Since ∂D_{τ} = S_{1τ} ∪ S_{2τ} ∪ S_{3τ}, for S_{iτ} = ∂D_{τ} ∩ S_{i}, i = 1, 2, S_{3τ} = ∂D_{τ} ∩ {t = τ} and

cos n̂t|_{S_{1τ}} = 0, cos n̂t|_{S_{2τ}} = −\frac{1}{\sqrt{2}}, cos n̂t|_{S_{3τ}} = 1,

u|_{S_{iτ}} = f_{i} = 0, i = 1, 2, v_{x_{i}} |_{S_{3τ}} = 0, i = 1, 2, v_{t} = u,

it follows from (4.27) that

∫_{S_{3τ}} u^{2}dx_{1}dx_{2} + \frac{1}{\sqrt{2}} ∫_{S_{2τ}} (v_{x_{1}}^{2} + v_{x_{2}}^{2})ds = 0.

Hence, u|_{S_{3τ}} = 0 for any τ from (0, t_{0}]. Therefore, u ≡ 0 in D. ■

It should be noted that Lemma 4.3 is also valid for k_{1} = −1, k_{2} = 1.

Remark 5. Since the strong solution of the problem (4.1), (4.2) of the class W_{2}^{1} is at the same time a weak solution of the class W_{2}^{1}, it follows from Lemma 4.3 that if the strong solution of that problem of the class W_{2}^{1} exists, then the same solution will be the unique weak solution of the class W_{2}^{1}.

§ 3. Domain of Dependence of Solutions of the Problems (4.1), (4.2) and (4.1), (4.3), (4.4)

Lemma 4.4. For a point P_{0}(x_{1}^{0}, x_{2}^{0}, t^{0}) ∈ D the domain of dependence of the solution u(x_{1}, x_{2}, t) of the problem (4.1), (4.2) of the class C^{2}(\overline{D}) or W_{2}^{2}(D) is contained inside the characteristic cone of the past

∂K_{P_{0}} : t = t^{0} − \sqrt{(x_{1} − x_{1}^{0})^{2} + (x_{2} − x_{2}^{0})^{2}}

with the vertex at P_{0}.

Proof. Suppose

Ω_{P_{0}} = D ∩ K_{P_{0}}, S_{iP_{0}} = S_{i} ∩ ∂Ω_{P_{0}}, i = 1, 2,

where K_{P_{0}} : t < t^{0} − \sqrt{(x_{1} − x_{1}^{0})^{2} + (x_{2} − x_{2}^{0})^{2}} is the interior of ∂K_{P_{0}}. To prove the above lemma it suffices to show that if

f_{i}|_{S_{iP_{0}}} ≡ u|_{S_{iP_{0}}} = 0, i = 1, 2, F |_{Ω_{P_{0}}} ≡ □u|_{Ω_{P_{0}}} = 0,  (4.28)

then u|_{Ω_{P_{0}}} = 0.

Consider first the case u ∈ C^{2}(\overline{D}). Denote by S_{3P_{0}} the remainder part of the boundary of Ω_{P_{0}}, i.e., S_{3P_{0}} = ∂Ω_{P_{0}}\(S_{1P_{0}} ∪ S_{2P_{0}}). According to our construction, the surface S_{3P_{0}} is a part of ∂K_{P_{0}}. Therefore

α_{3}|_{S_{3P_{0}}} = const > 0, (α_{3}^{2} − α_{1}^{2} − α_{2}^{2})|_{S_{3P_{0}}} = 0,  (4.29)

where n = (α_{1}, α_{2}, α_{3}) is the unit vector of outward normal to ∂Ω_{P_{0}}.

Multiplying both parts of (4.1) by 2u_{t} and integrating the obtained expression over Ω_{P_{0}}, taking into account (4.6), (4.28), (4.29) and the arguments we used in obtaining inequality (4.7), we get

0 = 2 ∫_{Ω_{P_{0}}} F u_{t}dxdt =

= ∫_{∂Ω_{P_{0}}} [(u_{t}^{2} + u_{x_{1}}^{2} + u_{x_{2}}^{2})α_{3} − 2(u_{x_{1}}u_{t}α_{1} + u_{x_{2}}u_{t}α_{2})]ds =

= ∫_{∂Ω_{P_{0}}} α_{3}^{−1}[(α_{3}u_{x_{1}} − α_{1}u_{t})^{2} + (α_{3}u_{x_{2}} − α_{2}u_{t})^{2} +

+(α_{3}^{2} − α_{1}^{2} − α_{2}^{2})u_{t}^{2}]ds ≥
≥ ∫_{S_{3P_{0}}} α_{3}^{−1}[(α_{3}u_{x_{1}} − α_{1}u_{t})^{2} + (α_{3}u_{x_{2}} − α_{2}u_{t})^{2}]ds.  (4.30)

When deducing inequality (4.30), we have used the fact that the operator α_{3} \frac{∂}{∂x_{i}} − α_{i} \frac{∂}{∂t} is an inner differential operator on the surface ∂Ω_{P_{0}} and, in particular, by virtue of (4.28) the equalities

(α_{3} \frac{∂u}{∂x_{i}} − α_{i} \frac{∂u}{∂t})|_{S_{1P_{0}} ∪S_{2P_{0}}} = 0, i = 1, 2,

hold on S_{1P_{0}} ∪ S_{2P_{0}}.

Since α_{3} > 0 on S_{3P_{0}}, inequality (4.30) implies

(α_{3}u_{x_{i}} − α_{i}u_{t})|_{S_{3P_{0}}} = 0, i = 1, 2.  (4.31)

Taking into account that u ∈ C^{2}(\overline{D}) and the inner differential operators α_{3} \frac{∂}{∂x_{i}} − α_{i} \frac{∂}{∂t}, i = 1, 2, are linearly independent on the two-dimensional connected surface S_{3P_{0}}, (4.31) immediately yields

u|_{S_{3P_{0}}} ≡ const .  (4.32)

But because of (4.28)

u|_{S_{3P_{0}} ∩(S_{1P_{0}} ∪S_{2P_{0}} )} = 0,

from which due to (4.32) we conclude that

u|_{S_{3P_{0}}} ≡ 0.  (4.33)

In particular, (4.33) implies u(P_{0}) = 0.

If now we take an arbitrary point Q ∈ Ω_{P_{0}}, then (4.28) implies the validity of the same equalities after substitution of the point P_{0} by Q. Therefore, repeating the above arguments for the domain Ω_{Q}, we obtain u(Q) = 0. Hence, in the case u ∈ C^{2}(\overline{D}) we have u|_{Ω_{P_{0}}} = 0.

Let now u ∈ W_{2}^{2}(D) and equalities (4.28) be valid. It can be easily verified that for any point Q ∈ Ω_{P_{0}} the inequality (4.30) is also valid after substitution of the point P_{0} by Q, that is

∫_{S_{3Q}} α_{3}^{−1}[(α_{3}u_{x_{1}} − α_{1}u_{t})^{2} + (α_{3}u_{x_{2}} − α_{2}u_{t})^{2}]ds ≤ 0.

whence, due to the fact that α_{3}|_{S_{3Q}} = const > 0, we get

∫_{S_{3Q}} [(α_{3}u_{x_{1}} − α_{1}u_{t})^{2} + (α_{3}u_{x_{2}} − α_{2}u_{t})^{2}]ds = 0.  (4.34)

Denote by Γ_{Q} a piecewise smooth curve which at the same time is the boundary of a two-dimensional connected surface S_{3Q}. Obviously,

Γ_{Q} = S_{3Q} ∪ (S_{1Q} ∪ S_{2Q}).  (4.35)

Using the fact that on S_{3Q} inner differential operators α_{3} \frac{∂}{∂x_{i}} −α_{i} \frac{∂}{∂t}, i = 1, 2, are independent, it is not difficult to obtain for any v ∈ W_{2}^{1}(S_{3Q}) the following estimate

∫_{S_{3Q}} v^{2}ds ≤ C( ∫_{Γ_{Q}} v^{2}ds +

+ ∫_{S_{3Q}} [(α_{3}v_{x_{1}} − α_{1}v_{t})^{2} + (α_{3}v_{x_{2}} − α_{2}v_{t})^{2}]ds),  (4.36)

where C = const > 0 does not depend on v, and the trace v|_{Γ_{Q}} ∈ L_{2}(Γ_{Q}) is correctly determined in virtue of the corresponding embedding theorem.

Since u ∈ W_{2}^{2}(D), the traces u|_{S_{3Q}} ∈ W_{2}^{1}(S_{3Q}) and u|_{Γ_{Q}} ∈ L_{2}(Γ_{Q}) are correctly determined in virtue of the embedding theorems. Therefore, due to (4.25) and (4.35) we have

u|_{Γ_{Q}} = 0.  (4.37)

From (4.34), (4.36) and (4.37) we obtain

∫_{S_{3Q}} u^{2}ds ≤

≤ C( ∫_{Γ_{Q}} u^{2}ds + ∫_{S_{3Q}} [(α_{3}u_{x_{1}} − α_{1}u_{t})^{2} + (α_{3}u_{x_{2}} − α_{2}u_{t})^{2}]ds) = 0,

from which it immediately follows that

∫_{S_{3Q}} u^{2}ds = 0, u|_{S_{3Q}} = 0, ∀Q ∈ Ω_{P_{0}}.  (4.38)

Since u ∈ W_{2}^{2}(D), in virtue of (4.38) and applying Fubini’s theorem we can conclude that

u|_{Ω_{P_{0}}} = 0. ■

Remark 1. The assertion of Lemma 4.4 is also valid for the problem (4.1), (4.3), (4.4). Moreover, the above arguments should be modified only on the part S_{1P_{0}} of the boundary Ω_{P_{0}}. In this case for k_{1} = 0 and due to (4.18) we have

∫_{S_{1P_{0}}} [(u_{t}^{2} + u_{x_{1}}^{2} + u_{x_{2}}^{2})α_{3} − 2(u_{x_{1}}u_{t}α_{1} + u_{x_{2}}u_{t}α_{2})]ds = 0.

Remark 2. It follows from Lemma 4.4 that the wave process described by the problem (4.1), (4.2) or (4.1), (4.3), (4.4) propagates with a finite speed. Therefore, if u ∈ C^{∞}(\overline{D}) is a solution of the problem (4.1), (4.2) or (4.1), (4.3), (4.4) for f_{i} ∈ C_{∗}^{∞}(S_{i}), i = 1, 2, F ∈ C_{∗}^{∞}(\overline{D}), then u ∈ C_{∗}^{∞}(\overline{D}).

§ 4. Solvability of Multidimensional Analogues of the Goursat and the First Darboux Problems

In this section we intend to concern ourselves with the question of solvability of the problem (4.1), (4.2) in the case where

k_{1} = −1, k_{2} = 1,  (4.39)

that is, a multidimensional analogue of the Goursat problem, and in the case where

−1 < k_{1} < 0, k_{2} = 1,  (4.40)

that is, a multidimensional analogue of the first Darboux problem.

First we shall prove the existence of regular solutions of these problems of the class C_{∗}^{∞}(\overline{D}) and then the existence of strong solutions of the class W_{2}^{1}.

Below we shall get an integral representation of regular solutions of the problem (4.1), (4.2) by using the method suggested in [6].

Let us denote by D_{εδ} the part of the domain D which is bounded by the surfaces S_{1}, S_{2}, a circular cone K_{ε} : r^{2} = (t − t^{0})^{2}(1 − ε) with the vertex at (x^{0}, t^{0}) ∈ D and by a cylinder H_{δ} : r^{2} = δ^{2}, where r^{2} = (x_{1} − x_{1}^{0})^{2} + (x_{2} − x_{2}^{0})^{2}, and ε and δ are sufficiently small positive numbers.

For any two twice continuously differentiable functions u and v we have the obvious identity

u□v − v□u = ∑_{i=1}^{2} \frac{∂}{∂x_{i}} (v \frac{∂u}{∂x_{i}} − u\frac{∂v}{∂x_{i}} ) − \frac{∂}{∂t}(v \frac{∂u}{∂t} − u\frac{∂v}{∂t} ). (4.41)

Integrating (4.41) over the domain D_{εδ}, where u ∈ C^{1}(\overline{D}) ∩ C^{2}(D) is a regular solution of (4.1), and

v = E(r, t, t^{0}) = \frac{1}{2π} log \frac{t − t^{0} − \sqrt{(t − t^{0})^{2} − r^{2}}}{r} ,

we shall have

∫_{∂D_{εδ}} [E(r, t, t^{0})\frac{∂u}{∂N} − \frac{∂E(r, t, t^{0})}{∂N}u]ds +

+ ∫_{D_{εδ}} F E(r, t, t^{0})dxdt = 0,  (4.42)

where N is the unit conormal vector at the point (x, t) = (x_{1}, x_{2}, t) ∈ ∂D_{εδ} with direction cosines

cos N̂x_{1} = cos n̂x_{1}, cos N̂x_{2} = cos n̂x_{2}, cos N̂t = − cos n̂t,

and n is the unit vector of the outer normal to ∂D_{εδ}.

Passing in equality (4.42) to the limit for ε → 0, δ → 0, we obtain

∫_{x_{2}^{0}}^{t^{0}} u(x_{1}^{0}, x_{2}^{0}, t)dt =

= ∫_{S_{1}^{∗}∪S_{2}^{∗}} [\frac{∂E(r, t, t^{0})}{∂N}u − E(r, t, t^{0})\frac{∂u}{∂N} ]ds − ∫_{D^{∗}} F E(r, t, t^{0})dxdt,

where D^{∗} is the domain D_{εδ} for ε = δ = 0, and S_{i}^{∗} = S_{i} ∩ ∂D^{∗}, i = 1, 2. By differentiation we find that

u(x_{1}^{0}, x_{2}^{0}, t^{0}) = \frac{d}{dt^{0}} [ ∫_{S_{1}^{∗}∪S_{2}^{∗}} [\frac{∂E(r, t, t^{0})}{∂N}u − E(r, t, t^{0})\frac{∂u}{∂N} ]ds −

− ∫_{D^{∗}} F E(r, t, t^{0})dxdt].  (4.43)

Remark 1. Since in the case (4.39) the direction of the conormal N on the characteristic surface S_{i}^{∗} coincides with that of a bicharacteristic lying on S_{i}^{∗}, i = 1, 2, we can, alongside with the value u|_{S_{i}^{∗}} = f_{i}, calculate \frac{∂u}{∂N} over the surface S_{i}^{∗}. Therefore in the case (4.39) equality (4.43) gives the integral representation of a regular solution of the multidimensional analogue (4.1), (4.2) of the Goursat problem.

Remark 2. In the case (4.40) the surface S_{1}^{∗} is not characteristic. Therefore to obtain an integral representation of a regular solution of the multidimensional analogue (4.1), (4.2) of the first Darboux problem one should eliminate the value \frac{∂u}{∂N} |_{S_{1}^{∗}} in the right-hand side of the representation (4.43).

In the case (4.40) without loss of generality we can assume that for the domain D the value k_{1} = 0, i.e., D : 0 < x_{2} < t, 0 < t < t_{0}, since the case where k_{1} ≠ 0 is reduced to the case k_{1} = 0 by a suitable Lorentz transform under which the wave operator □ is invariant. Let us introduce the point P′(x_{1}^{0}, −x_{2}^{0}, t^{0}) which is symmetric to P (x_{1}^{0}, x_{2}^{0}, t^{0}) with respect to the plane x_{2} = 0. For this aim we denote by D_{ε} a part of the domain D bounded by the cone K_{ε}^{0} : (x_{1} − x_{1}^{0})^{2} + (x_{2} + x_{2}^{0})^{2} = (t − t^{0})^{2}(1 − ε) with the vertex at P′ and the boundary ∂D. Obviously, ∂D_{ε} ∩ S_{1} ⊂ S_{1}^{∗} and ∂D_{0} ∩ S_{1} = S_{1}^{∗}. Assume ∂D_{0} ∩ S_{2} = S̃_{2}, r̃ = \sqrt{(x_{1} − x_{1}^{0})^{2} + (x_{2} + x_{2}^{0})^{2}}. Integrating now (4.41) over D_{ε}, where u ∈ C^{1}(\overline{D}) ∩ C^{2}(D) is a regular solution of (4.1), and

v = E(r̃, t, t^{0}) = \frac{1}{2π} log \frac{t − t^{0} − \sqrt{(t − t^{0})^{2} − r^{2}}}{r̃} ,

and taking into account that the function E(r̃, t, t^{0}) has no singularities in the domain D_{0}, we obtain, after passing to the limit for ε → 0, the equality

\frac{d}{dt^{0}} [ ∫_{S_{1}^{∗}∪S_{2}^{∗}} [\frac{∂E(r̃, t, t^{0})}{∂N}u − E(r̃, t, t^{0})\frac{∂u}{∂N} ]ds −

− ∫_{D_{0}} F E(r̃, t, t^{0})dxdt] = 0.  (4.44)

Since r = r̃ for x_{2} = 0, we have E(r̃, t, t^{0}) = E(r, t, t^{0}) on S_{1}^{∗}. Therefore eliminating \frac{∂u}{∂N} |_{S_{1}^{∗}} from (4.43) and (4.44), we finally obtain the integral representation of a regular solution of a multidimensional analogue of the first Darboux problem (4.1), (4.2) for k_{1} = 0, k_{2} = 1

u(x_{1}^{0}, x_{2}^{0}, t^{0}) = \frac{d}{dt^{0}} [ ∫_{S_{1}^{∗}} [\frac{∂E(r, t, t^{0})}{∂N} − \frac{∂E(r̃, t, t^{0})}{∂N}]uds +

+ ∫_{S_{2}^{∗}} [\frac{∂E(r, t, t^{0})}{∂N} − E(r, t, t^{0})\frac{∂u}{∂N} ]ds −

− ∫ ; _{S_{2}} [\frac{∂E(r̃, t, t^{0})}{∂N}u − E(r̃, t, t^{0})\frac{∂u}{∂N} ]ds +

+ ∫_{D_{0}} F E(r̃, t, t^{0})dxdt − ∫_{D^{∗}} F E(r, t, t^{0})dxdt].  (4.45)

Remark 3. According to the above remarks, the formulas (4.43) and (4.45) determine uniquely regular solutions of multidimensional analogues of the Goursat and the first Darboux problems, respectively. Moreover, using the arguments of paper [25], we can show that for any F ∈ C_{∗}^{∞}(\overline{D}), f_{i} ∈ C_{∗}^{∞}(S_{i}), i = 1, 2, these solutions belong to the class C_{∗}^{∞}(\overline{D}).

Below, using a somewhat different method, we shall show that for any F ∈ C_{∗}^{∞}(\overline{D}), f_{i} ∈ C_{∗}^{∞}(S_{i}), i = 1, 2, the solution of the multidimensional analogue of the Goursat problem (4.1), (4.2) will belong to the class C_{∗}^{∞}(\overline{D}) in the case (4.39). This method consists in reducing the spatial-type problem (4.1), (4.2) to the plane Goursat problem with a parameter. For the solution of the problem the necessary estimates depending on the parameter will be obtained.

If u is a solution of the problem (4.1), (4.2) of the class C_{∗}^{∞}(\overline{D}) in the case (4.39), then after the Fourier transform with respect to the variable x_{1} equation (4.1) and the boundary conditions (4.2) take the form

\frac{∂^{2}v}{∂t^{2}} − \frac{∂^{2}v}{∂x_{2}^{2}} + λ^{2}v = Φ,  (4.46)
v|_{l_{i}} = g_{i}, i = 1, 2,  (4.47)

where

v(λ, x_{2}, t) = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} u(x_{1}, x_{2}, t)e^{−ix_{1}λ}dx_{1}

is the Fourier transform of the function u(x_{1}, x_{2}, t) and Φ, g_{1}, g_{2} are the Fourier transforms respectively of the functions F, f_{1}, f_{2} with respect to the variable x_{1}. Here l_{1} : t − x_{2} = 0, 0 ≤ t ≤ t_{0}, l_{2} : t + x_{2} = 0, 0 ≤ t ≤ t_{0} are the segments of beams lying in the plane of variables x_{2}, t and coming out of the origin O(0, 0).

Thus, after the Fourier transform with respect to x_{1}, the spatial-type problem (4.1), (4.2) is reduced to the plane Goursat problem (4.46), (4.47) with a parameter λ in the domain D_{0} : −t < x_{2} < t, 0 < t < t_{0} of the plane of variables x_{2}, t.

Remark 4. If u(x_{1}, x_{2}, t) is a solution of the problem (4.1), (4.2) of the class C_{∗}^{∞}(\overline{D}), then v(λ, x_{2}, t) will be a solution of the problem (4.46), (4.47) of the class C^{∞}(\overline{D}_{0}) which at the same time, according to the Paley-Wiener theorem, is an entire analytic function with respect to λ, satisfying the following growth condition: for any integer N ≥ 0 there exists a constant K_{N} such that [26, 73]

|v(λ, x_{2}^{0}, t^{0})| ≤ K_{N}(1 + |λ|^{2})^{−N}e^{d| Im λ|},  (4.48)

where

d = d(x_{2}^{0}, t^{0}) = max_{(x_{1},x_{2}^{0},t^{0})∈supp u} |x_{1}|;

moreover, as the constant K_{N} we can take the value [73]

K_{N} = K_{N}(x_{2}^{0}, t^{0}) = \frac{1}{\sqrt{2π}} ∫_{|x_{1}|<d} |(1 − \frac{∂^{2}}{∂x_{1}^{2}} )^{N}u(x_{1}, x_{2}^{0}, t^{0})|dx_{1}.

According to the same theorem, if v(λ, x_{2}, t) belongs to the class C^{∞}(\overline{D}_{0}) with respect to the variables x_{2}, t for fixed λ, and with respect to λ it is an entire analytic function satisfying the estimates (4.48) for some d = const > 0, then the function u(x_{1}, x_{2}, t), being the inverse Fourier transform of the function v(λ, x_{2}, t), belongs to the class C_{∗}^{∞}(\overline{D}).

According to our assumptions, the estimates similar to (4.48) are valid for the functions Φ, g_{1}, g_{2} which belong respectively to the classes C^{∞}(\overline{D}_{0}), C^{∞}(l_{1}), C^{∞}(l_{2}) and are entire analytic functions with respect to λ.

In new variables

ξ = \frac{1}{2}(t + x_{2}), η = \frac{1}{2}(t − x_{2}),  (4.49)

retaining the same notations for the functions v, Φ, g_{i} the problem (4.46), (4.47) will take the form

\frac{∂^{2}v}{∂ξ∂η} + λ^{2}v = Φ,  (4.50)
v|_{γ_{i}} = g_{i}, i = 1, 2.  (4.51)

Here a solution v = v(λ, ξ, η) of equation (4.50) is considered in the domain Ω_{0} of the plane of variables ξ, η which is the image of the domain Ω_{0} under the linear transform (4.49), γ_{i} being the image of l_{i} under the same transform. Obviously, the domain Ω_{0} is the triangle OP_{1}P_{2} with vertices O(0, 0), P_{1}(t_{0}, 0), P_{2}(0, t_{0}), and γ_{1} : η = 0, 0 ≤ ξ ≤ t_{0} and γ_{2} : ξ = 0, 0 ≤ η ≤ t_{0} are the sides OP_{1} and OP_{2} of the triangle.

As is well known, under the assumptions with respect to the functions Φ, g_{i} the problem (4.50), (4.51) has a unique solution v of the class C^{∞}(\overline{Ω}_{0}) which can be represented in the form [6]

v(λ, ξ, η) = R(ξ, 0; ξ, η)g_{1}(λ, ξ) + R(0, η; ξ, η)g_{2}(λ, η) −

− R(0, 0; ξ, η)g_{1}(λ, 0) − ∫_{0}^{ξ} \frac{∂R(σ, 0; ξ, η)}{∂σ}g_{1}(λ, σ)dσ −

− ∫_{0}^{η} \frac{∂R(0, τ; ξ, η)}{∂τ}g_{2}(λ, τ)dτ +

+ ∫_{0}^{ξ} dσ ∫_{0}^{η} R(σ, τ; ξ, η)Φ(λ, σ, τ)dτ,  (4.52)

where g_{1}(λ, ξ) = v(λ, ξ, 0), 0 ≤ ξ ≤ t_{0}, g_{2}(λ, η) = v(λ, 0, η), 0 ≤ η ≤ t_{0}, are the Goursat data for v, and R(ξ_{1}, η_{1}; ξ, η) is the Riemann function for the equation (4.50).

As is known, the Riemann function R(ξ_{1}, η_{1}; ξ, η) for the equation (4.50) can be expressed by the Bessel function of zero order as [17]

R(ξ_{1}, η_{1}; ξ, η) = J_{0}(2λ\sqrt{(ξ − ξ_{1})(η − η_{1})}).  (4.53)

Remark 5. Since the Bessel function J_{0}(z) of a complex argument z is an entire analytic function, the formula (4.52) in virtue of (4.53) gives a solution of (4.50) satisfying the Goursat data

\begin{matrix}v(λ, ξ, 0) = g_{1}(ξ), 0 ≤ ξ ≤ t_{0}, \\ v(λ, 0, η) = g_{2}(η), 0 ≤ η ≤ t_{0}. \end{matrix}  (4.54)

The solution is the entire analytic function with respect to the complex parameter λ.

From the well-known representation of the Bessel function [63]

J_{0}(z) = \frac{1}{2π} ∫_{−π}^{π} exp(iz sin Θ)dΘ  (4.55)

we can easily get that

J′_{0}(z) = −\frac{z}{2π} ∫_{−π}^{π} cos^{2} Θ exp(iz sin Θ)dΘ,

whence

\frac{dJ_{0}(2λ\sqrt{νx})}{dx} = −\frac{λ^{2}ν}{2π} ∫_{−π}^{π} cos^{2} Θ exp(i2λ\sqrt{νx} sin Θ)dΘ.  (4.56)

Now (4.53), (4.55) and (4.56) yield the following equalities and estimates

R(ξ, 0; ξ, η) = R(0, η; ξ, η) = 1,
|R(0, 0; ξ, η)| ≤ exp (2\sqrt{ξη}| Im λ|) ≤ exp (2t_{0}| Im λ|),

|\frac{∂R(σ, 0; ξ, η)}{∂σ}| ≤ 2|λ|^{2}η exp (2\sqrt{ξη}| Im λ|) ≤ 2|λ|^{2}t_{0} exp (2t_{0}| Im λ|),

|\frac{∂R(0, τ; ξ, η)}{∂τ}| ≤ 2|λ|^{2}ξ exp (2\sqrt{ξη}| Im λ|) ≤ 2|λ|^{2}t_{0} exp (2t_{0}| Im λ|),

|R(σ, τ; ξ, η)| ≤ exp (2\sqrt{ξη}| Im λ|) ≤ exp (2t_{0}| Im λ|).

From this, assuming without restriction of generality that for the functions Φ, g_{1}, g_{2} the estimates (4.48) are, owing to our assumptions, valid with respect to λ with the same constants K_{N} and d, we obtain for a solution v(λ, ξ, η) of the problem (4.50) representable in the form (4.52), the following estimates

|v(λ, ξ, η)| ≤ |g_{1}(λ, ξ)| + |g_{2}(λ, η)| + |g_{1}(λ, 0)| exp (2t_{0}| Im λ|) +

+2|λ|^{2}t_{0} exp (2t_{0}| Im λ|) ∫_{0}^{ξ} |g_{1}(λ, σ)|dσ +

+2|λ|^{2}t_{0} exp (2t_{0}| Im λ|) ∫_{0}^{η} |g_{2}(λ, τ)|dτ +

+ exp (2t_{0}| Im λ|) ∫_{0}^{ξ} dσ ∫_{0}^{η} |Φ(λ, σ, τ)|dτ ≤

≤ 2K_{N}(1 + |λ|^{2})^{−N} exp (d| Im λ|) +

+ exp (2t_{0}| Im λ|)K_{N}(1 + |λ|^{2})^{−N} exp (d| Im λ|) +
+2|λ|^{2}t_{0} exp (2t_{0}| Im λ|)ξK_{N}(1 + |λ|^{2})^{−N} exp (d| Im λ|) +
+2|λ|^{2}t_{0} exp (2t_{0}| Im λ|)ηK_{N}(1 + |λ|^{2})^{−N} exp (d| Im λ|) +
+ exp (2t_{0}| Im λ|)ξηK_{N}(1 + |λ|^{2})^{−N} exp (d| Im λ|) ≤
≤ K̃_{N−1}(1 + |λ|^{2})^{N−1} exp (d̃| Im λ|).  (4.57)

Here

K̃_{N−1} = (3 + 5t_{0}^{2})K_{N}, d̃= 2t_{0} + d,

d = max_{(x_{1},x_{2},t)∈I} |x_{1}|, I = supp F ∪ supp f_{1} ∪ supp f_{2},

K_{N} = \frac{1}{2π} ∫_{|x_{1}|<d} max_{0≤i≤2} max_{(x_{2}^{0},t)∈\overline{D}_{0}} |φ_{i}(x_{1}, x_{2}^{0}, t^{0})|dx_{1},

φ_{0} = (1 − \frac{∂^{2}}{∂x_{1}^{2}} )^{N}F, φ_{i} = (1 − \frac{∂^{2}}{∂x_{1}^{2}} )^{N}f_{i}, i = 1, 2.

Owing to (4.57) and the Paley-Wiener theorem, the function v(λ, ξ, η), after returning to the initial variables x_{2}, t will, by the formulas (4.49), be the Fourier transform of a function u(x_{1}, x_{2}, t) of the class C_{∗}^{∞}(\overline{D}). Moreover, due to (4.50) and (4.51) the function u(x_{1}, x_{2}, t) ∈ C_{∗}^{∞}(\overline{D}) will be the unique solution of the problem (4.1), (4.2) of the above-mentioned class. ■

Now, using the fact that the problem (4.1), (4.2) is solvable in the class C_{∗}^{∞}(\overline{D}), we shall prove the existence of a strong solution of the class W_{2}^{1} of that problem.

It is well-known that the spaces C_{∗}^{∞}(\overline{D}), C_{∗}^{∞}(S_{i}), i = 1, 2, are everywhere dense in the spaces L_{2}(D), W_{2}^{1}(S_{i}), i = 1, 2, respectively. Therefore there exist sequences F_{n} ∈ C_{∗}^{∞}(\overline{D}) and f_{in} ∈ C_{∗}^{∞}(S_{i}), i = 1, 2, such that

lim_{n→∞} ∥F − F_{n}∥_{L_{2}(D)} = lim_{n→∞} ∥f_{i} − f_{in}∥_{W_{2}^{1}(S_{i})} = 0, i = 1, 2. (4.58)

Moreover, since by the condition (f_{1} − f_{2})|_{S_{1}∩S_{2}} = 0, one can take the sequences f_{1n} and f_{2n} such that (f_{1n} − f_{2n})|_{S_{1}∩S_{2}} = 0, n = 1, 2, . . ..

As it was shown above, under the conditions (4.39) or (4.40) there exists a sequence u_{n} ∈ C_{∗}^{∞}(\overline{D}) of solutions of the problem (4.1), (4.2) for F = F_{n}, f_{i} = f_{in}, i = 1, 2.

By virtue of (4.5) we have

∥u_{n} − u_{m}∥_{W_{2}^{1}(D)} ≤
≤ C( ∑_{i=1}^{2} ∥f_{in} − f_{im}∥_{W_{2}^{1}(S_{i})} + ∥F_{n} − F_{m}∥_{L_{2}(D)}).  (4.59)

It follows from (4.58) and (4.59) that the sequence of functions u_{n} is fundamental in the space W_{2}^{1}(D). Therefore, due to the completeness of the space W_{2}^{1}(D) there exists a function u ∈ W_{2}^{1}(D) such that u_{n} → u, □u_{n} → F and u_{n}|_{S_{i}} → f_{i}, i = 1, 2, in W_{2}^{1}(D), L_{2}(D) and W_{2}^{1}(S_{i}), i = 1, 2, respectively, for n → ∞. Consequently, the function u is the strong solution of the problem (4.1), (4.2) of the class W_{2}^{1}. The uniqueness of the strong solution of the problem (4.1), (4.2) of the class W_{2}^{1} follows from inequality (4.5).

Thus the following theorem is valid.

Theorem 4.1. Let the condition (4.39) or (4.40) be fulfilled. Then for any f_{i} ∈ W_{2}^{1}(S_{i}), i = 1, 2, F ∈ L_{2}(D) there exists a unique strong solution u of the problem (4.1), (4.2) of the class W_{2}^{1} for which the estimate (4.5) is valid.

Consider now the question of solvability of multi-dimensional analogues of the Goursat and the first Darboux problem for the hyperbolic equation (4.13) with the wave operator □ in the principal part. To prove the solvability of the problem (4.13), (4.2) under the conditions (4.39) or (4.40), we shall use the solvability of the problem (4.1), (4.2) and the a priori estimate (4.17) in specifically chosen norms of spaces L_{2}(D), W_{2}^{1}(D), W_{2}^{1}(S_{i}), i = 1, 2, from which it follows that the lowest terms in the equation (4.13) give arbitrarily small perturbations.

Consider the space

V_{0} = L_{2}(D) × W_{2}^{1}(S_{1}) × W_{2}^{1}(S_{2}).

To the problem (4.13), (4.2) there corresponds the unbounded operator

T : W_{2}^{1}(D) → V_{0}

with the domain of definition Ω_{T} = C_{∗}^{∞}(\overline{D}) ⊂ W_{2}^{1}(D), acting by the formula

T u = (Lu, u|_{S_{1}}, u|_{S_{2}}), u ∈ Ω_{T} .

It can be easily proved that the operator T admits a closure \overline{T}. In fact, let u_{n} ∈ Ω_{T}, u_{n} → 0 in W_{2}^{1}(D) and let T u_{n} → (F, f_{1}, f_{2}) in V_{0}. First we shall show that F = 0. For φ ∈ C_{0}^{∞}(D) we have

(Lu_{n}, φ) = (u_{n}, □φ) + (Ku_{n}, φ),  (4.60)

where Ku = au_{x_{1}} +bu_{x_{2}} +cu_{t}+du. Since in W_{2}^{1}(D), u_{n} → 0, from (4.60) we have that (Lu_{n}, φ) → 0. On the other hand, by the assumption, Lu_{n} → F in L_{2}(D). Therefore (F, φ) = 0 for any φ ∈ C_{0}^{∞}(D), and hence, F = 0. The equalities f_{1} = f_{2} = 0 follow from the facts that u_{n} → 0 in W_{2}^{1}(D), and the contraction operator u → (u|_{S_{1}}, u|_{S_{2}}) acts boundedly from W_{2}^{1}(D) to L_{2}(S_{1}) × L_{2}(S_{2}). ■

To the problem (4.1), (4.2) there corresponds an unbounded operator T_{0} : W_{2}^{1}(D) → V_{0} obtained from the operator T for a = b = c = d = 0. As it was shown above, the operator T_{0} also admits a closure \overline{T}_{0}. Obviously, the operator K_{0} : W_{2}^{1}(D) → V_{0} acting by the formula K_{0}u = (Ku, 0, 0) is bounded, and

T = T_{0} + K_{0}.  (4.61)

Note that the domains of definition Ω\overline{T} and Ω_{\overline{T}_{0}} of the closed operators \overline{T} and \overline{T}_{0} coincide by virtue of (4.61) and the fact that K_{0} is bounded.

It is easily seen that from the existence of the bounded operator \overline{T}^{−1} right inverse to \overline{T}, defined on the whole space V_{0} follow the existence and uniqueness of the strong solution of the problem (4.13), (4.2) of the class W_{2}^{1}, as well as the estimate (4.5) for this solution.

The fact that under the conditions (4.39) or (4.40) the operator \overline{T}_{0} has its bounded right inverse \overline{T} _{0}^{−1} : V_{0} → W_{2}^{1}(D), follows from the Theorem 4.1 and the estimate (4.5) which, as it is shown above, can be rewritten in equivalent norms in terms of (4.17). It is easy to see that the operator

K_{0}\overline{T}^{−1} : V_{0} → V_{0}

is bounded, and in virtue of (4.17) its norm admits the estimate

∥K_{0}\overline{T} _{0}^{−1}∥ ≤ \frac{C_{7}C_{8}}{\sqrt{γ}} ,  (4.62)

where C_{8} is a positive constant depending only on the coefficients a, b, c and d of equation (4.13).

By virtue of (4.62), the operator (I + K_{0}\overline{T} _{0}^{−1}) : V_{0} → V_{0} has a bounded inverse (I + K_{0}\overline{T} _{0}^{−1})^{−1} for sufficiently large γ, where I is the unit operator. Now it remains for us only to note that the operator

\overline{T} _{0}^{−1}(I + K_{0}\overline{T} _{0}^{−1})^{−1}

is a bounded operator right inverse to \overline{T} and defined on the whole space V_{0}. Thus the following theorem is proved.

Theorem 4.2. Let the condition (4.39) or (4.40) be fulfilled. Then for any f_{i} ∈ W_{2}^{1}(S_{i}), i = 1, 2, and F ∈ L_{2}(D) there exists a unique strong solution u of the problem (4.13), (4.2) of the class W_{2}^{1} for which the estimate (4.5) is valid.

§ 5. Solvability of a Multidimensional Analogue of the Second Darboux Problem

Discussion of this paragraph will be concerned with the question of solvability of the problem (4.1), (2) in the case

−1 < k_{1} < 0, 0 < k_{2} < 1,  (4.63)

that is, with a multidimensional analogue of the second Darboux problem.

Unlike the cases (4.39) and (4.40) considered in the previous section, the fact that for (4.63) none of the surfaces S_{1} and S_{2} is characteristic, means that for regular solutions of the problem (4.1), (4.2) there is no integral representation. To a certain extent this circumstance makes investigation of this problem difficult. Below we shall prove the existence of regular and strong solutions of the problem (4.1), (4.2) of the class W_{2}^{1} in the case (4.63) by reducing the problem to a mixed type problem for a hyperbolic equation of second order in a cylinder.

To this end we shall need the following

Lemma 4.5. Let G be a bounded subdomain of D with a piecewise smooth boundary, bounded from above by the plane t = t_{0} and at the sides by the planes S_{1}, S_{2}, as well as by piecewise smooth time-type surfaces S_{3}, S_{4} on which the following inequalities are valid:

α_{3}|_{S_{3}} < 0, α_{3}|_{S_{4}} < 0,  (4.64)

where n = (α_{1}, α_{2}, α_{3}) is the unit vector of the outer normal to ∂G; moreover, S_{3} ∩ S_{4} = ∅. Let K_{P_{0}}^{+} : t > t^{0} + \sqrt{(x_{1} − x_{1}^{0})^{2} + (x_{2} − x_{2}^{0})^{2}} be the domain bounded by the characteristic cone of the future with the vertex at P_{0}(x_{1}^{0}, x_{2}^{0}, t^{0}). Let u_{0} ∈ C^{∞}(\overline{G}) and g_{i} = u_{0}|_{∂G∩S_{i}}, i = 1, 2, F_{0} = □u_{0}, X = supp g_{1} ∪ supp g_{2} ∪ supp F_{0}, Y = ∪_{P_{0}∈X}K_{P_{0}}^{+}. Denote by S_{3}^{ε}, S_{4}^{ε} the ε-neighbourhoods of surfaces S_{3}, S_{4}, where ε is a fixed sufficiently small positive number. Then, if

u_{0}|_{S_{3}∪S_{4}} = 0,  (4.65)
Y ∩ (S_{3}^{ε} ∪ S_{4}^{ε}) = ∅,  (4.66)

then the function

u(P) = \begin{cases}u_{0}(P), & P ∈ G, \\ 0, & P ∈ D\G \end{cases}

is a solution of the problem (4.1), (4.2) of the class C_{∗}^{∞}(\overline{D}) with

f_{i}(P) = \begin{cases}g_{i}(P), & P ∈ ∂G ∩ S_{i}, \\ 0, & P ∈ S_{i}\(∂G ∩ S_{i}), \end{cases} i = 1, 2,

F (P) = \begin{cases}F_{0}(P), & P ∈ G, \\ 0, & P ∈ D\G. \end{cases}

Proof. To prove the lemma it suffices to show that the function u_{0} ∈ C^{∞}(\overline{G}) vanishes on the set G ∩ (S_{3}^{ε} ∪ S_{4}^{ε}).

Let P_{0} ∈ G ∩ (S_{3}^{ε} ∪ S_{4}^{ε}) be an arbitrary point of this set. We shall show that u_{0}(P_{0}) = 0.

The use will be made of the notation of Lemma 4.4 and of §3:

Ω_{P_{0}} = G ∩ K_{P_{0}}, S_{iP_{0}} = S_{i} ∩ ∂Ω_{P_{0}}, i = 1, 2, 3, 4,
S_{5P_{0}} = ∂K_{P_{0}} ∩ ∂Ω_{P_{0}}.

Obviously, ∂Ω_{P_{0}} = ∪_{i=1}^{5}S_{iP_{0}}.

According to the assumptions of Lemma 4.5, we have

\begin{matrix}α_{3}|_{S_{iP_{0}}} < 0, i = 1, 2, 3, 4, \\ α_{3}^{−1}(α_{3}^{2} − α_{1}^{2} − α_{2}^{2})|_{S_{iP_{0}}} > 0, i = 1, 2, 3, 4, \end{matrix}  (4.67)
α_{3}|_{S_{5P_{0}}} > 0, (α_{3}^{2} − α_{1}^{2} − α_{2}^{2})|_{S_{5P_{0}}} = 0,  (4.68)

where n = (α_{1}, α_{2}, α_{3}) is the unit vector of the outer normal to ∂Ω_{P_{0}}.

On account of (4.65) and (4.66) and the fact that P_{0} ∈ G ∩ (S_{3}^{ε} ∪ S_{4}^{ε}), we have

u_{0}|_{S_{iP_{0}}} = 0, i = 1, 2, 3, 4, □u_{0}|_{Ω_{P_{0}}} = F_{0}|_{Ω_{P_{0}}} = 0.  (4.69)

Multiplying both parts of the equation □u_{0} = F_{0} by 2\frac{∂u_{0}}{∂t}, integrating the obtained expression over Ω_{P_{0}}, taking into account (4.67)–(4.69) and the arguments used when obtaining inequalities (4.7) and (4.30), we get

0 = 2 ∫_{Ω_{P_{0}}} F_{0} \frac{∂u_{0}}{∂t} dxdt =

= ∫_{∂Ω_{P_{0}}} α_{3}^{−1}[(α_{3} \frac{∂u_{0}}{∂t} − α_{1} \frac{∂u_{0}}{∂t} )^{2} + (α_{3} \frac{∂u_{0}}{∂x_{2}} − α_{2} \frac{∂u_{0}}{∂t} )^{2} +

+(α_{3}^{2} − α_{1}^{2} − α_{2}^{2})(\frac{∂u_{0}}{∂t} )^{2}]ds ≥

≥ ∫_{S_{5P_{0}}} α_{3}^{−1}[(α_{3} \frac{∂u_{0}}{∂x_{1}} − α_{1} \frac{∂u_{0}}{∂t} )^{2} + (α_{3} \frac{∂u_{0}}{∂x_{2}} − α_{2} \frac{∂u_{0}}{∂t} )^{2}]ds,

whence on account of α_{3}|_{S_{5P_{0}}} > 0, we find

(α_{3} \frac{∂u_{0}}{∂x_{i}} − α_{i} \frac{∂u_{0}}{∂t} )|_{S_{5P_{0}}} = 0, i = 1, 2.

The remaining reasonings repeat word by word the proof of Lemma 4.4. Consequently, u(P_{0}) = 0 and Lemma 4.5 is proved completely. ■

Remark 1. It is easy to see that Lemma 4.5 remains also valid in the case when conditions (4.64) are violated on a set ω ⊂ S_{3} ∪ S_{4} of zero twodimensional measure, i.e., α_{3}|_{ω} = 0. In particular, if ω = ∪^{m}_{i=1}γ_{i} is a union of a

finite number of smooth curves γ_{i} ⊂ S_{3} ∪S_{4} and α_{3}|_{ω} = 0, α_{3}|_{(S_{3}∪S_{4})\ω} < 0, then Lemma 4.5 remains correct.

We shall need this circumstance below in proving Theorem 4.3.

Remark 2. It should be also noted that Lemmas 4.4 and 4.5, in fact, suggest us a way of constructing the solution of the problem (4.1), (4.2) in the case (4.63) which is given below and consists in reduction of the initial problem (4.1), (4.2) to a mixed-type problem for a second order hyperbolic equation in a cylinder.

Below the functions f_{1} and f_{2} in the boundary conditions (4.2) are assumed to vanish on the straight line Γ = S_{1} ∩ S_{2}, i.e.,

f_{i}|_{Γ} = 0, i = 1, 2.  (4.70)

The set of functions of the class W_{2}^{1}(S_{i}) satisfying (4.70) is denoted by

W^{∘} _{2}^{1}(S_{i}, Γ), that is,

W^{∘} _{2}^{1}(S_{i}, Γ) = {f ∈ W_{2}^{1}(S_{i}) : f|_{Γ} = 0}, i = 1, 2.

We have the following

Theorem 4.3. Let the condition (4.63) be fulfilled. Then for any f_{i} ∈

W^{∘} _{2}^{1}(S_{i}, Γ), i = 1, 2, and F ∈ L_{2}(D) there exists a unique strong solution u of the problem (4.1), (4.2) of the class W_{2}^{1} for which the estimate (4.5) is valid.

Proof. Denote by S_{i}^{0} : k_{i}t − x_{2} = 0, 0 ≤ t < +∞, i = 1, 2, the half-plane containing the carrier S_{i} in the boundary conditions (4.2) and by D_{0} the dihedral angle contained between the half-planes S_{1}^{0} and S_{2}^{0}. It is well-

known that the function f_{i} ∈ W^{∘} _{2}^{1}(S_{i}, Γ) can be extended to the half-plane

S_{i}^{0} as a function f̃_{i} of the class W^{∘} _{2}^{1}(S_{i}), i.e., (f_{i} − f̃_{i})|_{S_{i}} = 0, f̃_{i} ∈ W^{∘} _{2}^{1}(S_{i}^{0}), i = 1, 2. Assume

F̃(P) = \begin{cases}F (P), & P ∈ D, \\ 0, & P ∈ D_{0}\D. \end{cases}

Obviously, F̃ ∈ L_{2}(D_{0}).

If C_{0}^{∞}(D_{0}), C_{0}^{∞}(S_{i}^{0}), i = 1, 2, are the spaces of finite infinitely differentiable functions, then, as we know, they are everywhere dense respectively

in L_{2}(D_{0}), W^{∘} _{2}^{1}(S_{i}^{0}), i = 1, 2. Therefore there exist sequences F_{n} ∈ C_{0}^{∞}(D_{0}) and f_{in} ∈ C_{0}^{∞}(S_{i}^{0}), i = 1, 2, such that

lim_{n→∞} ∥F̃ − F_{n}∥_{L_{2}(D_{0})} = lim_{n→∞} ∥f̃_{i} − f_{in}∥_{W_{2}^{1}(S_{i}^{0})} = 0, i = 1, 2. (4.71)

In the plane of variables x_{2}, t let us introduce the polar coordinates r, φ taking the axis t as the polar axis. We count the polar angle φ from the polar axis assuming it to be positive clockwise. Denote by φ_{i} the size of a bihedral angle contained between the half-planes S_{i}^{0} and x_{2} = 0, 0 ≤ t < +∞, i = 1, 2. Since the half-planes S_{i}^{0} are of time-type (−1 < k_{1} < 0, 0 < k_{2} < 1), we have 0 < φ_{i} < \frac{π}{4} , i = 1, 2.

In passing from the rectangular coordinates x_{1}, x_{2}, t to the system of coordinates x_{1}, τ = log r, φ, the bihedral angle D_{0} transforms to an infinite layer

H = { − ∞ < x_{1} < ∞, −∞ < τ < ∞, −φ_{1} < φ < φ_{2}},

and the equation (4.1), written in terms of the former notation for the functions u and F, will take the form

e^{−2τ} L(τ, φ, ∂)u = F,  (4.72)

where ∂ = (\frac{∂}{∂x_{1}} , \frac{∂}{∂τ} , \frac{∂}{∂φ}), L(τ, φ, ∂) is a second order differential operator of hyperbolic type with respect to τ with infinitely differentiable coefficients depending on τ and φ.

In the plane x_{1}, φ let us consider a convex domain Ω of the class C^{∞}, bounded by the segments of straight lines l_{1} : φ = −φ_{1}, l_{2} : φ = φ_{2} and by the curves γ_{1} : x_{1} = g(φ), −φ_{1} ≤ φ ≤ φ_{2}, γ_{2} : x_{2} = −g(φ), −φ_{1} ≤ φ ≤ φ_{2}. Here g(φ) ∈ C^{∞}(−φ_{1}, φ_{2}) ∩ C[−φ_{1}, φ_{2}], g(φ) > 0 for −φ_{1} ≤ φ ≤ φ_{2}, g^{(1)}(φ) > 0 for −φ_{1} < φ < 0, g^{(1)}(0) = 0, g^{(1)}(φ) < 0 for 0 < φ < φ_{2} and g^{(2)}(φ) < 0 for −φ_{1} < φ < φ_{2}; moreover,

min (g(−φ_{1}), g(φ_{2})) > 1 + t_{0} + d,  (4.73)

where d = max(d_{1}, d_{2}, d_{3}),

d_{i} = sup_{(x_{1},x_{2},t)∈supp f_{i}}|x_{1}|, i = 1, 2,

d_{3} = sup_{(x_{1},x_{2},t)∈supp F}|x_{1}|.

Denote by H_{0} ⊂ H a cylindrical domain Ω × (−∞, ∞) of the class C^{∞}, where (−∞, ∞) is the τ-axis, and denote by ∂H_{0} its lateral surface ∂Ω × (−∞, ∞). Upon the inverse transform (x_{1}, τ, φ) → (x_{1}, x_{2}, t), the cylindrical domain H_{0} transforms to an unbounded domain G_{0} ⊂ D_{0} bounded by surfaces S̃_{i} = S_{i}^{0} ∩ ∂G_{0}, i = 1, 2, S̃_{3} and S̃_{4}.

Below we shall show that the surfaces S̃_{3} and S̃_{4} are of time-type on which the following conditions

α_{3}|_{(_{S_{3}}∪_{S_{4}})\ω} < 0, α_{3}|_{ω} = 0,  (4.74)

are fulfilled, where ω is the union of two smooth curves ω_{1} and ω_{2} lying on S̃_{3} ∪ S̃_{4}.

Indeed, it can be easily seen that S̃_{1} and S̃_{2} are the images of cylindrical surfaces S′_{1} = l_{1} ×(−∞, ∞) ⊂ ∂H_{0} and S′_{2} = l_{2} ×(−∞, ∞) ⊂ ∂H_{0}, while S̃_{3} and S̃_{4} are the images of the surfaces S_{3}^{0} = γ_{1} × (−∞, ∞) ⊂ ∂H_{0} and S_{4}^{0} = γ_{2} × (−∞, ∞) ⊂ ∂H_{0} when the inverse transform (x_{1}, τ, φ) → (x_{1}, x_{2}, t) is applied. Dividing the surface S_{3}^{0} into two parts S_{3}^{0} = S_{3+}^{0} ∪ S_{3−}^{0}, where

S_{3+}^{0} = γ_{1+} × (−∞, ∞), S_{3−}^{0} = γ_{1−} × (−∞, ∞),
γ_{1+} : x_{1} = g(φ), 0 < φ < φ_{2}, γ_{1−} : x_{1} = g(φ), −φ_{1} < φ < 0,

we can see that the image S̃_{3+} ⊂ S̃_{3} of S_{3+}^{0} admits upon the inverse transform (x_{1}, τ, φ) → (x_{1}, x_{2}, t) the following parametric representation

S̃_{3+} : x_{1} = g(φ), x_{2} = σ sin φ,
t = σ cos φ, 0 < φ < φ_{2}, 0 < σ < +∞,

from which for the unit vector n = (α_{1}, α_{2}, α_{3}) of the outer normal to ∂G_{0} we obtain the following on the part S̃_{3+}

n|_{S_{3+}} = (\frac{σ}{\sqrt{σ^{2} + g^{′2}(φ)}}, \frac{−g′(φ) cos φ}{\sqrt{σ^{2} + g^{′2}(φ)}}, \frac{g′(φ) sin φ}{\sqrt{σ^{2} + g^{′2}(φ)}}). (4.75)

Taking into account the structure of the domain Ω, we can conclude from (4.75) that S̃_{3+} is a time-type surface on which α_{3}|_{S_{3+}} < 0. Assertion

similar to this one is proved for the remaining parts S̃_{3−}, S̃_{4+} and S̃_{4−} of the surfaces S̃_{3} and S̃_{4}. To prove finally the validity of (4.74), it suffices to note that on the curves

ω_{1} = ∂S̃_{3+} ∩ ∂S̃_{3−}, ω_{2} = ∂S̃_{4+} ∩ ∂S̃_{4−},

which are the images of the straight lines ω̃_{1} : x_{1} = g(0), φ = 0, −∞ < τ < ∞ and ω̃_{2} : x_{1} = −g(0), φ = π, −∞ < τ < ∞, the third component α_{3} of the unit vector of the normal n vanishes.

Let us determine on the boundary ∂G_{0} of the domain G_{0} the function ν_{n} of the class C^{∞} as follows

ν_{n}|_{S_{i}} = f_{in}, i = 1, 2, ν_{n}|_{S_{3}} = ν_{n}|_{S_{4}} = 0, n = 1, 2, ….

The fact that ν_{n} ∈ C_{0}^{∞}(∂G_{0}) follows from the structure of the domain G_{0} and inequality (4.73), as well as from the smoothness and location of supports of the functions f_{in} ∈ C_{0}^{∞}(S_{i}^{0}), i = 1, 2.

When passing to the variables x_{1}, τ, φ the functions ν_{n} and F_{n} will transform to some functions for which we retain the same notation. Obviously,

ν_{n} ∈ C_{0}^{∞}(∂H_{0}), F_{n} ∈ C_{0}^{∞}(H_{0}).  (4.76)

For hyperbolic equation (4.72) with F = F_{n} let us consider in the cylinder H_{0} the following mixed-type problem with ”zero Cauchy data” for τ = −∞:

e^{−2τ} L_{1}(τ, φ, ∂)v = F_{n},  (4.77)
v|_{∂H_{0}} = ν_{n}.  (4.78)

Taking into account (4.76), the mixed problem (4.77), (4.78), due to the results of [4], [74], has a unique solution v = v_{n} of the class C^{∞}(\overline{H}_{0}) which turns into identical zero for τ < −M, where M = const is a sufficiently large positive number.

Returning to the initial variables x_{1}, x_{2}, t and retaining former notation for the functions v_{n} and F_{n}, we get that:

1) the function u_{n}^{0} = v_{n}|_{∂G_{0}∩D} belongs to the class C^{∞}(\overline{G_{0} ∩ D}) and satisfies the equation

□u_{n}^{0} = F_{n};

2) u_{n}^{0} on the lateral part ∪_{i=1}^{4}S̃_{i}^{0} of the boundary domain G_{0} ∩D satisfies the conditions

u_{n}|_{_{S_{3}^{0}}∪_{S_{4}^{0}}} = 0, u_{n}|_{S_{i}} = f_{in}, i = 1, 2,

where, as is easily seen, the surface S̃_{i}^{0} is a part of S_{i} for i = 1, 2 and is a part of S̃_{i} for i = 3, 4 appearing in conditions (4.74).

Therefore, on account of (4.73), (4.74) as well as of Lemma 4.5 and Remark 1, the function

u_{n}(P) = \begin{cases}u_{n}^{0}(P), & P ∈ G_{0}, \\ 0, & P ∈ D\G_{0} \end{cases}

belongs to the class C_{∗}^{∞}(\overline{D}) and is a solution of the problem (4.1), (4.2) for f_{i} = f_{in}, i = 1, 2, and F = F_{n}.

By virtue of (4.5) we have

∥u_{n} − u_{m}∥_{W_{2}^{1}(D)} ≤
≤ C( ∑_{i=1}^{2} ∥f_{in} − f_{im}∥_{W_{2}^{1}(S_{i})} + ∥F_{n} − F_{m}∥_{L_{2}(D)}).  (4.79)

From (4.71) and (4.79) it follows that the sequence of the functions u_{n} is fundamental in the space W_{2}^{1}(D). Therefore, since the space W_{2}^{1}(D) is complete, there exists a function u ∈ W_{2}^{1}(D) such that u_{n} → u, □u_{n} → F and u_{n}|_{S_{i}} → f_{i}, i = 1, 2, in the spaces W_{2}^{1}(D), L_{2}(D) and W_{2}^{1}(S_{i}), i = 1, 2, respectively, as n → ∞. Consequently, u is a strong solution of problem (4.1), (4.2) of the class W_{2}^{1}. The uniqueness of the strong solution of the problem (4.1), (4.2) of the class W_{2}^{1} follows from the inequality (4.5). Thus Theorem 4.3 is proved completely. ■

Repeating word by word the same arguments connected with equivalent norms which led us to Theorem 4.2, we get that the following theorem is valid.

Theorem 4.4. Let the condition (4.63) be fulfilled. Then for any f_{i} ∈

W^{∘} _{2}^{1}(S_{i}, Γ), i = 1, 2, and F ∈ L_{2}(D) there exists a unique strong solution u of the problem (4.13), (4.2) of the class W_{2}^{1} for which estimate (4.5) is valid.

§ 6. Solvability of the Problem (4.1), (4.3), (4.4)

Consider the problem (4.1), (4.3), (4.4) in the case where

k_{1} = 0, k_{2} = 1,  (4.80)

that is, S_{1} : x_{2} = 0, 0 ≤ t ≤ t_{0} is a time-type surface, S_{2} : t − x_{2} = 0, 0 ≤ t ≤ t_{0} is a characteristic surface, and let in the boundary condition (4.3) the function f_{1} = 0, that is,

\frac{∂u}{∂n}|_{S_{1}} = 0.  (4.81)

We have the following

Lemma 4.6. Let the condition (4.80) be fulfilled. Then for any f_{2} ∈ C_{∗}^{∞}(S_{2}) and F ∈ C_{∗}^{∞}(\overline{D}) satisfying

\frac{∂^{k}F}{∂n^{k}} |_{S_{1}} = 0, k = 1, 3, 5, …,  (4.82)

the problem (4.1), (4.81), (4.4) is uniquely solvable in the class C_{∗}^{∞}(\overline{D}).

Proof. Denote by D_{−} : −t < x_{2} < 0, 0 < t < t_{0} the domain which is symmetric to the domain D : 0 < x_{2} < t, 0 < t < t_{0}, with respect to the plane x_{2} = 0 and by D_{0} : −t < x_{2} < t, 0 < t < t_{0} the domain being the union of the domains D and D_{−} and the piece of the plane surface x_{2} = 0, 0 < t < t_{0}.

If we extend evenly the function F ∈ C_{∗}^{∞}(\overline{D}) to the domain D_{−}, then because of (4.82) the function F_{0} obtained in the domain D_{0},

F_{0}(x_{1}, x_{2}, t) = \begin{cases}F (x_{1}, x_{2}, t), & x_{2} ≥ 0, \\ F (x_{1}, −x_{2}, t), & x_{2} < 0, \end{cases}

will belong to the class C_{∗}^{∞}(\overline{D}_{0}). Denote by f_{1}^{−} the function defined on S_{1}^{−} : t + x_{2} = 0, 0 ≤ t ≤ t_{0} by

f_{1}^{−}|_{S_{1}^{−}} = f_{1}^{−}(x_{1}, x_{2}, −x_{2}) = f_{2}(x_{1}, −x_{2}, −x_{2}) = f_{2}|_{S_{2}}.  (4.83)

Obviously, f_{1}^{−} ∈ C_{∗}^{∞}(S_{1}^{−}).

In the domain D_{0} let us now consider the problem of determination of a solution u_{0}(x_{1}, x_{2}, t) of the equation

□u_{0} = F_{0}  (4.84)

belonging to the class C_{∗}^{∞}(\overline{D}_{0}) and satisfying the boundary conditions

u_{0}|_{S_{1}^{−}} = f_{1}^{−}, u_{0}|_{S_{2}} = f_{2}.  (4.85)

It is shown in §4 of the present chapter that a multidimensional analogue of the Goursat problem (4.84), (4.85) for F_{0} ∈ C_{∗}^{∞}(\overline{D}_{0}), f_{1}^{−} ∈ C_{∗}^{∞}(S_{1}^{−}), f_{2} ∈ C_{∗}^{∞}(S_{2}) has a unique solution u_{0} of the class C_{∗}^{∞}(\overline{D}_{0}). Let us show now that the restriction of this function to the domain D, i.e., u = u_{0}|_{D}, is a solution of the problem (4.1), (4.81), (4.4) of the class C_{∗}^{∞}(\overline{D}). To this end it suffices to show that the function u_{0}(x_{1}, x_{2}, t) is even with respect to the variable x_{2}. Because the function F_{0} is even with respect to the variable x_{2}, and the functions f_{1} and f_{2} are connected by equality (4.83), we can easily verify that the function ũ(x_{1}, x_{2}, t) = u_{0}(x_{1}, −x_{2}, t) is also a solution of the same problem (4.84), (4.85) of the class C_{∗}^{∞}(\overline{D}_{0}). But due to a priori estimate (4.5), the problem (4.84), (4.85) cannot have more than one solution of the above-mentioned class. Therefore, ũ(x_{1}, x_{2}, t) ≡ u_{0}(x_{1}, x_{2}, t), i.e., the solution u_{0}(x_{1}, x_{2}, t) of equation (4.84) is an even function with respect to x_{2}. This implies \frac{∂u_{0}}{∂n} |_{x_{2}=0} = 0, i.e., the boundary condition (4.81) is fulfilled for u = u_{0}|_{D}. Thus, the function u = u_{0}|_{D} ∈ C_{∗}^{∞}(\overline{D}) is a solution of the problem (4.1), (4.81), (4.4). The uniqueness of this solution of the problem (4.1), (4.81), (4.4) follows from a priori estimate (4.19). ■

Definition. Let f_{2} ∈ W_{2}^{1}(S_{2}), F ∈ L_{2}(D). The function u ∈ W_{2}^{1}(D) is said to be a strong solution of the problem (4.1), (4.81), (4.4) of the class W_{2}^{1} if there exists a sequence u_{n} ∈ C_{∗}^{∞}(\overline{D}) such that \frac{∂u_{n}}{∂n} |_{S_{1}} = 0, u_{n} → u, □u_{n} → F and u_{n}|_{S_{2}} → f_{2} in the spaces W_{2}^{1}(D), L_{2}(D) and W_{2}^{1}(S_{2}), respectively.

The following theorem holds.

Theorem 4.5. Let the condition (4.80) be fulfilled. Then for any f_{2} ∈ W_{2}^{1}(S_{2}) and F ∈ L_{2}(D) there exists a unique strong solution u of the problem (4.1), (4.81), (4.4) of the class W_{2}^{1} for which the estimate (4.19) is valid.

Proof. It is known that the space C_{0}^{∞}(D) ⊂ C_{∗}^{∞}(\overline{D}) of infinitely differentiable finite functions in the domain D is everywhere dense in L_{2}(D), while the space C_{∗}^{∞}(S_{2}) is everywhere dense in W_{2}^{1}(S_{2}). Therefore there exist the sequences F_{n} ∈ C_{0}^{∞}(D) and f_{2n} ∈ C_{∗}^{∞}(S_{2}) such that

lim_{n→∞} ∥F − F_{n}∥_{L_{2}(D)} = lim_{n→∞} ∥f_{2} − f_{2n}∥_{W_{2}^{1}(S_{2})} = 0.  (4.86)

Since the functions F_{n} ∈ C_{0}^{∞}(D) satisfy the conditions (4.82), according to Lemma 4.6 there exists a sequence u_{n} ∈ C_{∗}^{∞}(\overline{D}) of solutions of the problem (4.1), (4.81), (4.4) with F = F_{n}, f_{2} = f_{2n}.

On account of (4.19) we have

∥u_{n} − u_{m}∥_{W_{2}^{1}(D)} ≤
≤ C(∥f_{2n} − f_{2m}∥_{W_{2}^{1}(S_{2})} + ∥F_{n} − F_{m}∥_{L_{2}(D)}).  (4.87)

It follows from (4.86), (4.87) that the sequence of functions u_{n} is fundamental in the space W_{2}^{1}(D). Therefore, since the space W_{2}^{1}(D) is complete, there exists a function u ∈ W_{2}^{1}(D) such that u_{n} → u, □u_{n} → F and u_{n}|_{S_{2}} → f_{2} respectively in the spaces W_{2}^{1}(D), L_{2}(D) and W_{2}^{1}(S_{2}) as n → ∞. Consequently, u is a strong solution of the problem (4.1), (4.81), (4.4) of the class W_{2}^{1}. The uniqueness of this solution follows from (4.19). ■

Using equivalent norms depending on a parameter and arguing as while proving Theorem 4.2 of §4, we can prove

Theorem 4.6. Let the condition (4.80) be fulfilled. Then for any f_{2} ∈ W_{2}^{1}(S_{2}), and F ∈ L_{2}(D) there exists a unique strong solution u of the problem (4.13), (4.81), (4.4) of the class W_{2}^{1} for which the estimate (4.19) is valid.

§ 7. One Multidimensional Analogue of the Second Darboux Problem for Hyperbolic Equations of Higher Orders

Consider in the space R^{n}, n > 2, a strictly hyperbolic equation of the type

p(x, ∂)u(x) = f(x),  (4.88)

where ∂ = (∂_{1}, …, ∂_{n}), ∂_{j} = \frac{∂}{∂x_{j}}, p(x, ξ) is a real polynomial of order 2m, m > 1, with respect to ξ = (ξ_{1}, …, ξ_{n}), f is a given function, and u is an unknown real function. It is assumed that in (4.88) the coefficients at higher derivatives are constant and the other coefficients are finite and infinitely differentiable in R^{n}.

Let D be a conic domain in R^{n}, i.e., D together with a point x ∈ D contains the entire beam tx, 0 < t < ∞. Denote by Γ the cone ∂D. D is assumed to be homeomorphic to the conic domain x_{1}^{2} + ⋯ + x_{n−1}^{2} − x_{n}^{2} < 0, x_{n} > 0, and Γ′ = Γ\O is assumed to be a connected (n − 1)-dimensional manifold of the class C^{∞}, where O is the vertex of Γ.

Consider the boundary value problem [43]: find in the domain D a solution u(x) of the equation (4.88) satisfying the boundary conditions

\frac{∂^{i}u}{∂ν^{i}} |_{Γ′} = g_{i}, i = 0, …, m − 1,  (4.89)

where ν = ν(x) is the outer normal to Γ′ at the point x ∈ Γ′, and g_{i}, i = 0, …, m − 1, are given real functions.

In this section we investigate the question whether the problem (4.88), (4.89) can be correctly formulated in special weighted spaces W_{α}^{k}(D) when the cone Γ is assumed not to be characteristic and to have a quite definite orientation.

Denote by p_{0}(ξ) the characteristic polynomial of (4.88), i.e., the higher homogeneous part of the polynomial p(x, ξ). The strict hyperbolicity of the equation (4.88) implies the existence of a vector ζ ∈ R^{n} such that the straight line ξ = λζ + η, where η ∈ R^{n} is an arbitrarily chosen vector not parallel to ζ and λ is a real parameter, intersects the cone of normals K : p_{0}(ξ) = 0 of the equation (4.88) at 2m real different points. In other words, the equation p_{0}(λζ + η) = 0 has 2m real different roots with respect to λ. The vector ζ is called a spatial-type normal. As is known, the set of all spatial-type normals form two connected centrally symmetric convex conic domains whose boundaries K_{1} and K_{2m} give the internal cavity of the cone of normals K [17]. The surface S ⊂ R^{n} is called characteristic at a point x ∈ S if the normal to S at x belongs to K.

Let the vector ζ be a spatial-type normal and the vector η ≠ 0 vary in the plane orthogonal to ζ. Then the roots of the characteristic polynomial p_{0}(λζ + η) with respect to λ can be renumbered so that λ_{2m}(η) < λ_{2m−1}(η) < ⋯ < λ_{1}(η). It is obvious that the vectors λ_{i}(η)ζ + η cover the cavities K_{i} of K, when the η varies on the plane orthogonal to ζ. Since λ_{m−j}(η) = −λ_{m+j+1}(−η), 0 ≤ j ≤ m − 1, the cones K_{m−j} and K_{m+j+1} are centrally symmetric with respect to the point (0, …, 0). It is well-known that the straight beams whose orthogonal planes are tangential planes to one of the cavities K_{i} at a point different from the vertex, are bicharacteristics of equation (4.88).

Assume that there exists a plane π_{0} such that π_{0} ∩ K_{m} = {(0, …, 0)}. This means that the cones K_{1}, …, K_{m} are located on one side of π_{0} and the cones K_{m+1}, …, K_{2m} on the other. Put K_{i}^{∗} = ∩_{η∈K_{i}}{ξ ∈ R^{n} : ξ · η < 0}, where ξ · η is the scalar product of the vectors ξ and η. Since π_{0} ∩ K_{m} = {(0, …, 0)}, K_{i}^{∗} is a conic domain and

K_{m}^{∗} ⊂ K_{m−1}^{∗} ⊂ … ⊂ K_{1}^{∗}, K_{m+1}^{∗} ⊂ K_{m+2}^{∗} ⊂ … ⊂ K_{2m}^{∗}.

It is easy to verify that ∂(K_{i}^{∗}) is a convex cone whose generatrices are bicharacteristics; note that in this case none of the bicharacteristics of equation (4.88) comes from the point (0, …, 0) into the cone ∂(K_{m}^{∗}) or ∂(K_{m+1}^{∗}) [17].

Let us consider

Condition 1. The surface Γ′ is characteristic at none of its points and each generatrix of the cone Γ has the direction of a spatial-type normal; moreover, Γ ⊂ K_{m}^{∗} ∪ O or Γ ⊂ K_{m+1}^{∗} ∪ O.

Denote by W_{α}^{k}(D), k ≥ 2m, −∞ < α < ∞, the function space with the norm [48]

∥u∥_{W_{α}^{k}(D)}^{2} = ∑_{i=0}^{k} ∫_{D} r^{−2α−2(k−i)}|\frac{∂^{i}u}{∂x^{i}} |^{2}dx,

where

r = (x_{1}^{2} + ⋯ + x_{n}^{2})^{\frac{1}{2}} , \frac{∂^{i}u}{∂x^{i}} = \frac{∂^{i}u}{∂x^{i_{1}} … ∂x_{n}^{i_{n}}} , i = i_{1} + ⋯ + i_{n}.

The space W_{α}^{k}(Γ) is defined in a similar manner.

Consider the space

V = W_{α−1}^{k+1−2m}(D) × ∏_{i=0}^{m−1} W_{α− \frac{1}{2}}^{k−i} (Γ).

Assume that to the problem (4.88), (4.89) there corresponds an unbounded operator

T : W_{α}^{k}(D) → V

with the domain of definition Ω_{T} = W_{α−1}^{k+1}(D) ⊂ W_{α}^{k}(D), acting as

T u = (p(x, ∂)u, u|_{Γ′}, …, \frac{∂^{i}u}{∂ν^{i}} |_{Γ′}, …, \frac{∂^{m−1}u}{∂ν^{m−1}} |_{Γ′}), u ∈ Ω_{T} .

It is obvious that the operator T admits the closure \overline{T}.

The function u is called a strong solution of the problem (4.88), (4.89) of the class W_{α}^{k}(D) if u ∈ Ω\overline{T}, \overline{T} u = (f, g_{0}, …, g_{m−1}) ∈ V, which is equivalent to the existence of a sequence u_{i} ∈ Ω_{T} = W_{α−1}^{k+1}(D) such that u_{i} → u in W_{α}^{k}(D) and

(p(x, ∂)u_{i}, u_{i}|_{Γ′}, …, \frac{∂^{m−1}u_{i}}{∂ν^{m−1}} |_{Γ′}) → (f, g_{0}, …, g_{m−1})

in the space V. Below, by a solution of the problem (4.88), (4.89) of the class W_{α}^{k}(D) will be meant a strong solution of this problem in the sense indicated above.

We shall prove

Theorem 4.7. Let condition 1 be fulfilled. Then there exists a real number α_{0} = α_{0}(k) > 0 such that for α ≥ α_{0} problem (4.88), (4.89) is uniquely solvable in the class W_{α}^{k}(D) for any f ∈ W_{α−1}^{k+1−2m}(D), g_{i} ∈ W_{α− \frac{1}{2}}^{k−i} (Γ), i = 0, …, m − 1, and for the solution u we have the estimate

∥u∥_{W_{α}^{k}(D)} ≤ c( ∑_{i=0}^{m−1} ∥g_{i}∥_{W_{α− \frac{1}{2}}^{k−i} (Γ)} + ∥f∥_{W_{α−1}^{k+1−2m}(D)}),  (4.90)

where c is a positive constant not depending on f, g_{i}, i = 0, …, m − 1.

First we shall show that Condition 1 implies the following conditions: Take any point P ∈ Γ′ and choose a Cartesian system of coordinates x_{1}^{0}, …, x_{n}^{0} having the vertex at P and such that the x_{n}^{0}-axis is directed along the generatrix of Γ passing through P, while the x_{n−1}^{0}-axis is directed along the inner normal to Γ at that point.

Condition 2. The surface Γ′ is characteristic at none of its point. Each generatrix of the cone Γ has the direction of a spatial-type normal, and exactly m characteristic planes of the equation (4.88) pass through the (n− 2)-dimensional plane x_{n}^{0} = x_{n−1}^{0} = 0 connected with an arbitrary point P ∈ Γ′ into the angle x_{n}^{0} > 0, x_{n−1}^{0} > 0.

Denote by p̃_{0}(ξ) the characteristic polynomial of (4.88) written in terms of the coordinate system x_{1}^{0}, …, x_{n}^{0} connected with an arbitrarily chosen point P ∈ Γ′.

Condition 3. The surface Γ′ is characteristic at none of its point. Each generatrix of Γ has the direction of a spatial-type normal and for Re s > 0 the number of roots λ_{j}(ξ_{1}, …, ξ_{n−2}, s) of the polynomial p̃_{0}(iξ_{1}, …, iξ_{n−2}, λ, s) with Re λ_{j} < 0, taking into account their multiplicities, is equal to m, i = \sqrt{−1}.

When condition 3 is fulfilled, the polynomial p̃_{0}(iξ_{1}, …, iξ_{n−2}, λ, s) can be written as Δ_{−}(λ)Δ_{+}(λ), where for Re s > 0 the roots of the polynomials Δ_{−}(λ) and Δ_{+}(λ) lie, respectively, to the left and to the right of the imaginary axis, while the coefficients are continuous for s, Re s ≥ 0, (ξ_{1}, …, ξ_{n−2}) ∈ R^{n−2}, ξ_{1}^{2} + ⋯ + ξ_{n−2}^{2} + |s|^{2} = 1 [4]. On the left side of the boundary conditions (4.89), to the differential operator b_{j}(x, ∂), 0 ≤ j ≤ m − 1, written in terms of the coordinate system x_{1}^{0}, …, x_{n}^{0} connected with P ∈ Γ′, there corresponds the characteristic polynomial b_{j}(ξ) = ξ_{n−1}^{j}. Therefore, since the degree of Δ_{−}(λ) is equal to m, the following condition will be fulfilled.

Condition 4. For any point P ∈ Γ′ and any s, Re s ≥ 0, and (ξ_{1}, …, ξ_{n−2}) ∈ R^{n−2}, such that ξ_{1}^{2}+⋯+ξ_{n−2}^{2}+|s|^{2} = 1, the polynomials b_{j}(iξ_{1}, …, iξ_{n−2}, λ, s) = λ^{j}, j = 0, …, m − 1, are linearly independent, as polynomials of λ, modulo Δ_{−}(λ).

We shall now show that condition 1 implies condition 2, while the latter implies condition 3. Let us consider the case Γ ⊂ K_{m+1}^{∗} ∪ O. The second case Γ ⊂ K_{m}^{∗} ∪ O is treated analogously. Let P ∈ Γ′ and x_{1}^{0}, …, x_{n}^{0} be the coordinate system connected with this point. Since the generatrix γ of Γ passing through P is a spatial-type normal, the plane x_{n}^{0} = 0 passing through P is a spatial-type plane.

Denote by K_{j}ˆ the boundary of the convex shell of K_{j} and by K_{j}^{⊥} the set which is the union of all bicharacteristics corresponding to K_{j} and coming out of O along the outer normal to K_{j}, 1 ≤ j ≤ 2m. Obviously, (K_{j}ˆ )^{∗} = K_{j}^{∗}, ∂(K_{j}^{∗}) = (K_{j}ˆ )^{⊥}.

Let us now show that the plane π_{1}, parallel to the plane x_{n}^{0} = 0 and passing through the point (0, …, 0), is the plane of support to the cone K_{m} ˆ at the point (0, …, 0). Indeed, it is evident that the plane N · ξ = 0, N ∈ R^{n}\(0, …, 0), ξ ∈ R^{n}, is the plane of support to K_{m} ˆ at the point (0, …, 0) if and only if the normal vector N to this plane taken with the sign + or − belongs to the closure of the conic domain (K_{m} ˆ )^{∗} = K_{m}^{∗}. Now it remains for us to note that the conic domains K_{m}^{∗} and K_{m+1}^{∗} are centrally symmetric with respect to (0, …, 0), and the generatrix Γ passing through P is perpendicular to π_{1} and, by the condition, belongs to K_{m+1}^{∗} ∪ O.

Since x_{n}^{0} = 0 is a spatial-type plane, the two-dimensional plane σ : x_{1}^{0} = ⋯ = x_{n−2}^{0} = 0 which passes through the generatrix γ directed along the spatial-type normal, intersects the cone of normals K_{P} of the equation (4.88) with the vertex at the point P by 2m different real straight lines [17]. The planes orthogonal to these straight lines and passing through the (n − 2)- dimensional plane x_{n}^{0} = x_{n−1}^{0} = 0, give all 2m characteristic planes passing through this plane.

The straight lines x_{n}^{0} = 0 and x_{n−1}^{0} = 0 divide the two-dimensional plane σ into four right angles σ_{1} : x_{n−1}^{0} > 0, x_{n}^{0} > 0; σ_{2} : x_{n−1}^{0} < 0, x_{n}^{0} > 0; σ_{3} : x_{n−1}^{0} < 0, x_{n}^{0} < 0; σ_{4} : x_{n−1}^{0} > 0, x_{n}^{0} < 0. It is easily seen that exactly m characteristic planes of equation (4.88) pass through the (n − 2)- dimensional plane x_{n}^{0} = x_{n−1}^{0} = 0 into the angle x_{n}^{0} > 0, x_{n−1}^{0} > 0, if and only if exactly m straight lines from the intersection of K_{P} with the twodimensional plane σ pass into the angle σ_{4}. The latter fact really occurs, since: 1) the plane x_{n}^{0} = 0 is the plane of support to K_{m} ˆ and therefore to all K_{1}, …, K_{2m}; 2) the planes x_{n}^{0} = 0, x_{n−1}^{0} = 0 are not characteristic because the generatrices of Γ have a spatial-type direction and Γ is not characteristic at the point P.

Now it will be shown that condition 2 implies condition 3. By virtue of Condition 2 the plane x_{n−1}^{0} = 0 is not characteristic and therefore the polynomial p̃_{0}(iξ_{1}, …, iξ_{n−2}, λ, s) for λ has exactly 2m roots. In this case, if Re s > 0, the number of roots λ_{j}(ξ_{1}, …, ξ_{n−2}, s), with the multiplicity of the polynomial p̃_{0}(iξ_{1}, …, iξ_{n−2}, λ, s) taken into account, will be equal to m provided that Re λ_{j} < 0.

Indeed, recalling that equation (4.88) is hyperbolic, for Re s > 0 the equation p̃_{0}(iξ_{1}, …, iξ_{n−2}, λ, s) = 0 has no purely imaginary roots with respect to λ. Since the roots λ_{j} are the continuous functions of s, we can determine the number of roots λ_{j} with Re λ_{j} < 0 by passing to the limit as Re s → +∞.

Since the equality

p̃_{0}(iξ_{1}, …, iξ_{n−2}, λ, s) = s^{2m}p̃_{0}(i\frac{ξ_{1}}{s} , …, i\frac{ξ_{n−2}}{s} , \frac{λ}{s}, 1)

holds, it is clear that the ratios \frac{λ_{j}}{s}, where λ_{j} are the roots of the equation p̃_{0}(iξ_{1}, …, iξ_{n−2}, λ, s) = 0, tend to the roots μ_{j} of the equation p̃_{0}(0, …, 0, μ, 1) = 0 as Re s → +∞. The latter roots are real and different because equation (4.88) is hyperbolic. If s is taken positive and sufficiently large, then for μ_{j} ≠ 0 we have λ_{j} = sμ_{j} + o(s). But μ_{j} ≠ 0, since the plane x_{n}^{0} = 0 is not characteristic. Therefore the number of roots λ_{j} with Re λ_{j} < 0 coincides with the number of roots μ_{j} with μ_{j} < 0. Since the characteristic planes of equation (4.88) passing through the (n − 2)-dimensional plane x_{n}^{0} = x_{n−1}^{0} = 0, are determined by the equalities μ_{j}x_{n−1}^{0}+x_{n}^{0} = 0, j = 1, …, 2m, condition 2 implies that the number of roots λ_{j} with Re λ_{j} < 0 is equal to m.

We give another equivalent description of the space W_{α}^{k}(D). On the unit sphere S^{n−1} : x_{1}^{2} + ⋯ + x_{n}^{2} = 1 let us choose a coordinate system (ω_{1}, …, ω_{n−1}) such that in the domain D the transformation

I : τ = log r, ω_{j} = ω_{j}(x_{1}, …, x_{n}), j = 1, …, n − 1,

is one-to-one, nondegenerate and infinitely differentiable. Since the cone Γ = ∂D is strictly convex at the point O(0, …, 0), such coordinates evidently exist. Under the above transformation the domain D turns to an infinite cylinder G bounded by an infinitely differentiable surface ∂G = I(Γ′).

Introduce the functional space H_{γ}^{k}(G), −∞ < γ < ∞, with the norm

∥v∥_{H_{γ}^{k}(G)}^{2} = ∑_{i_{1}+j=0}^{k} ∫_{G} e^{−2γτ} |\frac{∂^{i_{1}+j}v}{∂τ^{i_{1}}∂ω^{j}} |^{2}dωdτ,

where

\frac{∂^{i_{1}+j}v}{∂τ^{i_{1}}∂ω^{j}} = \frac{∂^{i_{1}+j}v}{∂τ^{i_{1}}∂ω_{1}^{j_{1}} … ∂ω_{n−1}^{j_{n−1}}} , j = j_{1} + ⋯ + j_{n−1}.

As it is shown in [48], the function u(x) ∈ W_{α}^{k}(D) if and only if ũ = u(I^{−1}(τ, ω)) ∈ H_{(α+k)− \frac{n}{2}}^{k} (G), and the estimates

c_{1}∥ũ∥_{H_{(α+k)− \frac{n}{2}}^{k} (G)} ≤ ∥u∥_{W_{α}^{k}(D)} ≤ c_{2}∥ũ∥_{H_{(α+k)− \frac{n}{2}}^{k} (G)}

hold, where I^{−1} is the transformation inverse to I and the positive constants c_{1} and c_{2} do not depend on u.

It is easy to see that the condition v ∈ H_{γ}^{k}(G) is equivalent to the condition e^{−γτ} v ∈ W^{k}(G), where W^{k}(G) is the Sobolev space. Denote by H_{γ}^{k}(∂G) the set of all ψ such that e^{−γτ} ψ ∈ W^{k}(∂G), and by W_{α− \frac{1}{2}}^{k} (Γ) the

set of all φ for which φ̃ = φ(I^{−1}(τ, ω)) ∈ H_{(α+k)− \frac{n}{2}}^{k} (∂G). Assume that

∥φ∥_{W_{α− \frac{1}{2}}^{k} (Γ)} = ∥φ̃∥_{H_{(α+k)− \frac{n}{2}}^{k} (∂G)}.

Spaces W_{α}^{k}(D) possess the following simple properties:

1) if u ∈ W_{α}^{k}(D), then \frac{∂^{i}u}{∂x^{i}} ∈ W_{α}^{k−i}(D), 0 ≤ i ≤ k;

2) W_{α−1}^{k+1}(D) ⊂ W_{α}^{k}(D); 3) if u ∈ W_{α−1}^{k}(D), then by the well-known embedding theorems we have

u|_{Γ} ∈ W_{α− \frac{1}{2}}^{k} (Γ), \frac{∂^{i}u}{∂ν^{i}} |_{Γ′} ∈ W_{α− \frac{1}{2}}^{k−i} (Γ), i = 1, …, m − 1;

4) f = p(x, ∂)u ∈ W_{α−1}^{k+1−2m}(D) if u ∈ W_{α−1}^{k+1}(D).

In what follows we will need in spaces W_{α}^{k}(D), W_{α− \frac{1}{2}}^{k} (Γ) other norms depending on the parameter γ = (α + k) − \frac{n}{2} and equivalent to the original norms.

Put

R_{ω,τ}^{n} = { − ∞ < τ < ∞, −∞ < ω_{i} < ∞, i = 1, …, n − 1},
R_{ω,τ,+}^{n} = {(ω, τ) ∈ R_{ω,τ}^{n} : ω_{n−1} > 0}, ω′ = (ω_{1}, …, ω_{n−2}),
R_{ω′,τ}^{n−1} = { − ∞ < τ < ∞, −∞ < ω_{i} < ∞, i = 1, …, n − 2}.

Denote by ṽ(ξ_{1}, …, ξ_{n−2}, ξ_{n−1}, ξ_{n}−iγ) the Fourier transform of the function e^{−γτ} v(ω, τ), i.e.,

ṽ(ξ_{1}, …, ξ_{n−1}, ξ_{n} − iγ) = (2π)^{− \frac{n}{2}} ∫ v(ω, τ)e^{−iωξ′−iτξ_{n}−γτ} dωdτ,
i = \sqrt{−1}, ξ′ = (ξ_{1}, …, ξ_{n−1}),

and by v̂(ξ_{1}, …, ξ_{n−2}, ω_{n−1}, ξ_{n} − iγ) the partial Fourier transform of the function e^{−γτ} v(ω, τ) with respect to ω′, τ .

In the above-considered spaces H_{γ}^{k}(R_{ω,τ}^{n}) and H_{γ}^{k}(R_{ω,τ,+}^{n}) we can introduce the following equivalent norms:

|∥v|∥_{R^{n},k,γ}^{2} = ∫_{R^{n}} (γ^{2} + |ξ|^{2})^{k}|ṽ(ξ_{1}, …, ξ_{n−1}, ξ_{n} − iγ)|^{2}dξ,

|∥v|∥_{R_{+}^{n},k,γ}^{2} = ∫_{0}^{∞} ∫_{R^{n−1}} ∑_{j=0}^{k} (γ^{2} + |ξ′|^{2})^{k−j} ×

× |\frac{∂^{j}}{∂ω_{n−1}^{j}} v̂(ξ_{1}, …, ξ_{n−2}, ω_{n−1}, ξ_{n} − iγ)|^{2}dξ′dω_{n−1}.

Let φ_{1}, …, φ_{N} be the partitioning of unity in G′ = G ∩ {τ = 0}, where

G = I(D), i.e., ∑_{j=1}^{N} φ_{j}(ω) ≡ 1 in G′, φ_{j} ∈ C^{∞}(\overline{G}′), the supports of the func-

tions φ_{1}, …, φ_{N−1} lie in boundary half-neighborhoods, while the support of the function φ_{N} lies inside G′. Then for γ = (α + k) − \frac{n}{2} the equalities

\begin{matrix}|∥u|∥_{G,k,γ}^{2} = ∑_{j=1}^{N−1} |∥φ_{j}u|∥_{R_{+}^{n},k,γ}^{2} + |∥φ_{N} u|∥_{R^{n},k,γ}^{2}, \\ |∥u|∥_{∂G,k,γ}^{2} = ∑_{j=1}^{N−1} |∥φ_{j}u|∥_{R_{ω′,τ,k,γ}^{n−1}}^{2} \end{matrix}  (4.91)

define equivalent norms in the spaces W_{α}^{k}(D) and W_{α− \frac{1}{2}}^{k} (Γ), where the norms on the right side of these equalities are taken in terms of local coordinates [4].

Assume first that the equation (4.88) contains only higher terms, i.e., p(x, ξ) ≡ p_{0}(ξ). Equation (4.88) and the boundary conditions (4.89) written in terms of the coordinates ω, τ will take the form

e^{−2mτ} A(ω, ∂)u = f,
e^{−iτ} B_{i}(ω, ∂)u|_{∂G} = g_{i}, i = 0, …, m − 1,

i.e.,

A(ω, ∂)u = f̃,  (4.92)
B_{i}(ω, ∂)u|_{∂G} = g̃_{i}, i = 0, …, m − 1,  (4.93)

where A(ω, ∂) and B_{i}(ω, ∂) are, respectively, differential operators of orders 2m and i with infinitely differentiable coefficients depending only on ω, while f̃ = e^{2mτ} f and g̃_{i} = e^{iτ} g_{i}, i = 0, 1, …, m − 1.

Thus under the transformation I : D → G, the unbounded operator T of the problem (4.88), (4.89) transforms to the unbounded operator

T̃ : H_{γ}^{k}(G) → H_{γ}^{k+1−2m}(G) × ∏_{i=0}^{m−1} H_{γ}^{k−i}(∂G)

with the domain of definition H_{γ}^{k+1}(G), acting as

T̃u = (A(ω, ∂)u, B_{0}(ω, ∂)u|_{∂G}, …, B_{m−1}(ω, ∂)u|_{∂G}),

where γ = (α + k) − \frac{n}{2}. Note that written in terms of the coordinates ω, τ, the functions f(ω, τ) ∈ H_{γ−2m}^{k+1−2m}(G), g_{i}(ω, τ) ∈ H_{γ−i}^{k−i}(∂G), i = 0, …, m−1, if f(x) ∈ W_{α−1}^{k+1−2m}(D), g_{i}(x) ∈ W_{α− \frac{1}{2}}^{k−i} (Γ), i = 0, 1, …, m − 1. Therefore

the functions f̃ = e^{2mτ} f ∈ H_{γ}^{k+1−2m}(G), g̃_{i} = e^{iτ} g_{i} ∈ H_{γ}^{k−i}(∂G), i = 0, …, m − 1.

Since by Condition 1 each generatrix of the cone Γ has the direction of a spatial-type normal, due to the convexity of K_{m} each beam coming out of the vertex O into the conic domain D also has the direction of a spatial-type normal. Therefore equation (4.92) is strictly hyperbolic with respect to the τ-axis. It was shown above that the fulfillment of Condition 1 implies that of Condition 4. Therefore, according to the results of [4], for γ ≥ γ_{0}, where γ_{0} is a sufficiently large positive number, the operator \overline{T̃} has a bounded right inverse operator \overline{T̃}^{−1}. Thus for any f̃ ∈ H_{γ}^{k+1−2m}(G), g̃_{i} ∈ H_{γ}^{k−i}(∂G), i = 0, 1, …, m − 1, γ ≥ γ_{0} the problem (4.92), (4.93) is uniquely solvable in the class H_{γ}^{k}(G) and for the solution u we have the estimate

|∥u|∥_{G,k,γ}^{2} ≤ c( ∑_{i=0}^{m−1} |∥g̃_{i}|∥_{∂G,k−i,γ} + \frac{1}{γ} |∥f̃|∥_{G,k+1−2m,γ}) (4.94)

with a positive constant c not depending on γ, f̃ and g̃_{i}, i = 0, 1, …, m − 1. Hence it immediately follows that the theorem and the estimate (4.90) are valid in the case p(x, ξ) ≡ p_{0}(ξ).

Remark. Estimate (4.94) with the coefficient \frac{1}{γ} at |∥f̃|∥_{G,k+1−2m,γ}, obtained in the appropriately chosen norms (4.91), enables one to prove Theorem 4.7 also when equation (4.88) contains lower terms, since the latter give arbitrarily small perturbations for sufficiently large γ.

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CONTENTS

S. Kharibegashvili.

Goursat and Darboux TypeProblems for Linear Hyperbolic Partial Differential Equations and Systems . . . . . . . . . . . . . . . . . . . . . 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Chapter I. Boundary Value Problems for A Hyperbolic Equation of Second Order

§1. Statement of the Problem and Its Reduction to the

Functional Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 §2. The Case Where the Curves γ_{1} and γ_{2} do not Have a Common

Tangent Line at the Point O(0, 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 §3. The Case Where the Curves γ_{1} and γ_{2} Have a Common

Tangent Line at O(0, 0), But the Directions of Differentiation in the Boundary Conditions (1.4) do not Coincide at This Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20

§4. The Case Where Tangents to the Curves γ_{1} and γ_{2} as Well as the Directions of Differentiation in the Boundary Conditions (1.4) Coincide at the Point O(0, 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

§5. Influence of the Lower Terms on the Correctness of Statement of the Problem (1.1), (1.2) in the Case Where Conditions (1.13) are Violated on the Whole Curve γ_{1} or γ_{2} . . . . . . . . . . . . . . . . . . . . .25

§6. The Case Where the Conditions (1.13) are Violated at One Point O(0, 0) Only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Chapter II. Boundary Value Problems for Linear Second Order Normally Hyperbolic Systems

§1. Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36 §2. Requirements Imposed on the Curves γ_{1}, γ_{2} and on the

Characteristics of the System (2.1). Determination of Numbers m_{1} and m_{2}. Construction of Domains D_{1} and D_{p} . . . . . . . . . . . . . 37

§3. Reduction of the Problem (2.1)–(2.3) to the System of

Integro-Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 §4. Investigation of the System of Integro-Functional

Equations (2.18), (2.19), (2.22) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 §5. The Case of Hyperbolic Systems with Constant Coefficients … 62

Chapter III. Characteristic Problems for Linear Hyperbolic Systems of Second Order with Parabolic Degeneration

§1. Characteristic Problem for Hyperbolic System of Second Order

with Non-Characteristic Line of Parabolic Degeneration . . . . . . . 70 §2. Some Structural Properties of the Hyperbolic System (3.1) …. 72 §3. Reduction of the Problem (3.1), (3.2) to a System of

Integro-Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 §4. Investigation of the System of Integro-Functional Equations

(3.28), (3.29), (3.31) and the Proof of Theorem 3.1 . . . . . . . . . . . . 79 §5. A characteristic Problem for Hyperbolic System of Second Order

with Characteristic Line of Parabolic Degeneration . . . . . . . . . . . . 81 Chapter IV. Multidimensional Analogues of the Goursat and

Darboux Problems for Linear Differential Equations of Hyperbolic Type

§1. Formulation of Multidimensional Analogues of the Goursat

and Darboux Problems for the Wave Equation . . . . . . . . . . . . . . . . 84 §2. A Priori Estimates for Solutions of the Problems (4.1), (4.2),

and (4.1), (4.3), (4.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 §3. Domain of Dependence of Solutions of the Problems (4.1), (4.2),

and (4.1), (4.3), (4.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 §4. Solvability of Multidimensional Analogues of the Goursat and

the First Darboux Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 §5. Solvability of a Multidimensional Analogue of the Second

Darboux Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 §6. Solvability of the Problem (4.1), (4.3), (4.4) . . . . . . . . . . . . . . . . . . 110 §7. One Multidimensional Analogue of the Second Darboux Problem

for Hyperbolic Equations of Higher Orders . . . . . . . . . . . . . . . . . . . 112 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121