Second Order Conditions for Extrema of Functionals Defined on Regular Surfaces

L. Solanilla, A. Baquero and W. Naranjo

Dedicated to the Memory of Grigorios TSAGAS (1935-2003), President of Balkan Society of Geometers (1997-2003)

Abstract

Here we study second-order conditions for a functional of the form

∫_{V} J(x, u, ∇u)dV

which achieves extrema in a convenient Banach space. V denotes a type of compact subset of a smooth surface for which a Maximum Principle holds. The method consists of finding the second derivative of the functional and generalizing the notion of conjugate point to the boundary of V. The results include proofs of general Legendre’s and Jacobi’s necessary conditions as well as a complete set of sufficient conditions for the existence of maxima and minima. Also, we provide an interesting example showing the application of the theory to the case of the conformal prescribed Gaussian curvature deformation functional.

Mathematics Subject Classification: 59K10, 53A55 Key words: Legendre’s condition, Jacobi’s equation, maximum principle, conformal prescribed Gauss curvature equation

1 Introduction

For the moment, V is a compact subset of an oriented, simple, regular surface whose interior ıV is connected and whose boundary ∂V is simple, close and enough regular, including interior spheres at each of its points. Later on, we will impose on V an additional condition, namely, a hypothesis of the so-called “Maximum Principle when a supersolution exists”. C_{0}(V), C(V), C_{0}^{1}(V), C^{1}(V) and C_{0}^{2}(V), C^{2}(V) denote the usual Banach spaces of functions vanishing or not on ∂V furnished with their corresponding supremum norms. We assume some familiarity of the reader with the Classical Differential Geometry of Surfaces (as in [1], for example) and with the basics of Calculus of Variations, for instance, the following result, cf. [2].

∗ Balkan Journal of Geometry and Its Applications, Vol.8, No.2, 2003, pp. 97-104.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2003.

Lemma 1.1 Let U be an open subset of a Banach space B and F : U → R a twice differentiable functional at u ∈ U such that dF (u) = 0 and d^{2}F (u) ≠ 0. Then, F achieves a local minimum (maximum) at the critical point u if and only if there exists a c > 0 (c < 0) such that d^{2}F (u)(h) ≥ (≤) c∥h∥^{2}, for all h ∈ B, i. e., d^{2}F (u) is a definite positive (negative) quadratic functional.

In a previous work [3], it was proved the rightness of next theorem.

Theorem 1.2 (Euler-Lagrange) If a functional F : C_{0}^{1}(V) → R has the form

F (u) = ∫_{V} J(x, u, ∇u)dV, where J is enough differentiable, then F is differentiable

and its first derivative is given by

dF (u)h = ∫_{V} [∂_{u}J − div(∂_{∇u}J)]hdV.

Henceforth, at any local extremum u of F the Euler-Lagrange equation

∂_{u}J − div(∂_{∇u}J) = 0

holds in V.

The point of departure for what follows is to compute the second derivative of such a functional. Then, this derivative is used to establish necessary and sufficient conditions for the existence of maxima or minima. In the first place, we can generalize Legendre’s necessary condition from the Classical Calculus of Variations, cf. [2].

Theorem 1.3 (Legendre) Let F : C_{0}^{1}(V) → R, F (u) = ∫_{V} J(x, u, ∇u)dV, be an

enough differentiable functional. Then, its second derivative is given by

d^{2}F (u)(h) = ∫_{V} (∇h^{t}P ∇h + Qh^{2})dV,

where the tensor field P = \frac{1}{2}∂_{∇u∇u}^{2}J and the scalar field Q = \frac{1}{2}(∂_{u^{2}}^{2}J − div(∂_{u∇u}^{2}J)). Moreover, if F achieves a minimum (maximum) at a critical point u, then P is definite nonnegative (nonpositive).

Next, we need a general and convenient definition of conjugate point.

Definition 1.4 (Conjugate points to ∂V) Given a quadratic functional d^{2}F (u) :

C_{0}^{2}(V) → R of the form d^{2}F (u)(h) = ∫_{V} (∇h^{t}P ∇h + Qh^{2})dV, we say that ıV does

not have conjugate points to ∂V with respect to d^{2}F (u) if no one of the non-trivial solutions (if any), h ∈ C_{0}^{2}(V)\{h ≡ 0}, of Jacobi’s differential equation −div(P∇h)+ Qh = 0 vanishes in ıV.

This allows us to get a general version of other classical theorem on quadratic functionals.

Theorem 1.5 If ıV does not have conjugate points to ∂V with respect to d^{2}F (u)(h) =

∫_{V} (∇h^{t}P ∇h + Qh^{2})dV, where the tensor field P is definite positive (negative) in V,

then d^{2}F (u) is definite positive (negative).

Clearly, Jacobi’s equation is elliptic if and only if P is definite positive or negative and so, it is not surprising that the analytic ground for more elaborated conditions turns out to be the existence and uniqueness theory for elliptic partial differential equations as well as the Maximum Principles, cf. [4]. In particular, we are interested in those Principles which do not depend at all on the sign of the independent term in the elliptic operator.

Lemma 1.6 (Maximum principle for nonpositive functions) If L is an elliptic operator and u ∈ C^{2}(ıV) ∩ C(V) satisfies Lu = 0 in ıV and and u ≤ 0 in V, then either u < 0 in ıV, or u ≡ 0. Moreover, if ∂V satisfies an interior sphere condition at y and u ∈ C^{1}(ıV ∪ {y}) with u < u(y) = 0 in V, then ∂u/∂ν < 0 at y in any direction ν pointing into an interior sphere.

In order to obtain a general version of Jacobi’s condition, we need to assume the validity of the following Maximum Principle.

Lemma 1.7 (Maximum Principle if a positive supersolution exists) Suppose L is elliptic and there is a w ∈ C^{2}(V) with w > 0 and Lw ≥ 0 in V. If u ∈ C^{2}(ıV) ∩ C(V) satisfies Lu = 0 in ıV, then either there exists a constant r ∈ R such that u ≡ rw, or u/w does not attain a nonnegative maximum in ıV.

Accordingly, from now on we will demand our domain V to fulfil the supersolution hypothesis of Lemma 1.7. The existence of such a supersolution can be accomplished, for example, by demanding V to be narrow or, alternatively, by assuring the existence of a positive eigenfunction for Jacobi’s operator, cf. [5]. The notion of conjugate point together with this Maximum Principle lead to a deeper general necessary condition.

Theorem 1.8 (Jacobi) Assume u ∈ C_{0}^{2}(V) is a local minimum (maximum) point of a differentiable enough functional F : C_{0}^{2}(V) → R of the form

F (u) = ∫_{V} J(x, u, ∇u)dV, d^{2}F (u) ≠ 0

and the tensor field 2P = ∂_{∇u∇u}^{2}J is definite positive (negative) at this extremum. Then, ıV does not have conjugate points to ∂V with respect to d^{2}F (u).

Corollary 1.9 Let d^{2}F (u)(h) = ∫_{V} (∇h^{t}P ∇h + Qh^{2})dV be the second derivative of

the functional at some u ∈ C_{0}^{2}(V) and suppose P = P (x, u(x)) is definite positive (negative) in V. d^{2}F (u) is definite positive (negative) if and only if ıV does not have conjugate points to ∂V with respect to d^{2}F (u).

To conclude, we put together the conditions of Euler, Legendre and Jacobi to obtain a general theorem that gives a set of sufficient conditions for the existence of an extremum.

100 L. Solanilla, A. Baquero and W. Naranjo

Theorem 1.10 Let F : C_{0}^{2}(V) → R be a twice differentiable functional of the form

F (u) = ∫_{V} J(x, u, ∇u)dV.

Suppose also u ∈ C_{0}^{2}(V) fulfills the following conditions:

1. u is a solution of the Euler-Lagrange equation ∂_{u}J − div(∂_{∇u}J) = 0;

2. u makes the tensor field P (x, u(x)) definite positive (negative);

3. ıV does not contain conjugate points to ∂V with respect to d^{2}F (u). Then, F attains a local minimum (maximum) at u.

2 Proofs of theorems and corollaries

Proof. (Theorem 1.3.) An easy calculation shows that

d^{2}F (u)(h) = \frac{1}{2} ∫_{V} [\frac{∂^{2}J}{∂u^{2}} h^{2} + 2h⟨∂_{u∇u}^{2}J, ∇h⟩ + ∇h^{t}[∂_{∇u∇u}^{2}J]∇h]dV.

Next, usual integration by parts transforms the second term of this integral into

∫_{V} 2h⟨∂_{u∇u}^{2}J, ∇h⟩dV = − ∫_{V} h^{2}div(∂_{u∇u}^{2}J)dV.

Substituting this equation in the second derivative yields

d^{2}F (u)(h) = ∫_{V} (∇h^{t}P ∇h + Qh^{2})dV,

where P = \frac{1}{2}[∂_{∇u∇u}^{2}J] and Q = \frac{1}{2}(\frac{∂^{2}J}{∂u^{2}} − div(∂_{u∇u}^{2}J)).

Now, if P is negative at x_{0} ∈ V, the continuity of P implies that

v^{t}P (x)v < −β∥v∥^{2}, ∀v ∈ T_{x}(V),

for some β > 0 and for all x in a geodesic ball of radius α centered at x_{0}. Now, we consider the radially symmetric function h ∈ C_{0}^{1}(V) defined by

h(x) = \{ \begin{matrix}sin^{2} \frac{πr}{α} & if r ∈ [0, α] \\ 0 & otherwise, \end{matrix}

where r is the geodesic distance from x_{0} to x. In this way,

∫_{V} (∇h^{t}P ∇h + Qh^{2})dV < −βmax(\frac{π^{2}}{α^{2}} sin^{2}(2\frac{πr}{α}))πα^{2} + max|Q|πα^{2}
< −βπ^{3} + max|Q|πα^{2}.

Finally, making α small enough, we obtain that the second derivative is negative. This is an obstruction for u to be a minimum. Word by word, the proof for a maximum is accomplished. □ Proof. (Theorem 1.5.) With a non-trivial solution ĥ to Jacobi’s equation we construct the vector field w = −P ∇ĥ/ĥ in ıV. A straightforward computation shows this field satisfies “Riccati’s” equation

Q + divw = w^{t}P^{−1}w.

Hence, for any h ∈ C_{0}^{2}(V), we can complete a perfect square as follows.

∇h^{t}P ∇h + Qh^{2} + div(wh^{2}) = ∇h^{t}P ∇h + Qh^{2} + (divw)h^{2} + ⟨w, 2h∇h⟩
= ∇h^{t}P ∇h + 2h⟨∇h, w⟩ + (Q + divw)h^{2}
= (∇h + hP^{−1}w)^{t}P (∇h + hP^{−1}w).

The usual procedure of integration by parts together with the fact that h is null on ∂V yield

∫_{V} div(wh^{2})dV = 0.

In this way, the integral reduces to

d^{2}F (u)(h) = ∫_{V} (∇h^{t}P ∇h + Qh^{2})dV = ∫_{V} (∇h + hP^{−1}w)^{t}P (∇h + hP^{−1}w)dV.

Since P is definite positive, d^{2}F (u)(h) = 0 implies that ∇h+hP^{−1}w = 0. The unique solution to this equation in C_{0}^{2}(V) is h≡0. This shows the functional d^{2}F (u) is definite positive. □

Remark 2.1 For the rest of the proofs below, we make repeated use of the additional condition on the surface domain V and the Maximum Principles.

Proof. (Theorem 1.9.) If ıV has a conjugate point to ∂V, Jacobi’s problem

−div(P∇h) + Qh = 0, h ∈ C_{0}^{2}(V) \ {h ≡ 0},

possesses a solution h vanishing at some point in ıV. We claim this is impossible. For, we consider the family of functionals

d_{t}F (u)(h) = t ∫_{V} (∇h^{t}P ∇h + Qh^{2})dV + (1 − t) ∫_{V} ⟨∇h, ∇h⟩dV,

t ∈ [0, 1]. The members of this family are all definite positive after Lemma 1.1. Also, after rewriting

d_{t}F (u)(h) = ∫_{V} (∇h^{t}(tP + (1 − t)I)∇h + tQh^{2})dV,

we notice that the associated family of Jacobi’s elliptic equations is given by

−div((tP + (1 − t)I)∇h) + tQh = 0, t ∈ [0, 1].

From now on, we look at these equations as a unique equation with parameter t whose solutions are denoted by h(x, t), (x, t) ∈ V × [0, 1]. To t = 0, it corresponds

Dirichlet’s functional d_{0}F (u)(h) = ∫_{V} ⟨∇h, ∇h⟩dV whose Jacobi’s equation is just

but Laplace’s equation −div(∇h) = −Δh = 0. From the Weak Maximum Principle, it follows that ıV does not have conjugate points to ∂V with respect to d_{0}^{2}F (u). To t = 1, it corresponds the functional under consideration. Our claim on the absence of conjugate points to ∂V with respect to d_{1}^{2}F (u) relies on the fact that it cannot appear conjugate points as t varies smoothly from 0 to 1 (homotopy invariance).

Let h(x, 1) be the nontrivial solution with h(x_{0}, 1) = 0 at some point x_{0} ∈ ıV, conjugate to ∂V with respect to d_{1}^{2}F (u). We must also have ∇h(x_{0}, 1) ≠ 0. Indeed, the only choice in Lemma 1.7 is r = 0, which leads to the excluded possibility h(x, 1) ≡ 0. Therefore, h(x, 1)/w does not achieve a nonnegative maximum in ıV and since w > 0, so does h(x, t) and ∇h(x_{0}, 1) = 0 is not possible. This suggests to study a differentiable structure for the set

N = {(x, t) ∈ ıV × [0, 1] | h(x, t) = 0}.

After our assumption, N ≠ ∅. The Implicit Function Theorem implies then that N is a curve x(t) = (x_{1}(t), x_{2}(t)) near (x_{0}, 1). Since ∇h(x, t) ≠ 0 in N for the same reasons above, we consider its continuation in V × [0, 1] :

1. The curve cannot stop suddenly inside ıV × (0, 1) because this would contradict the smooth dependence of h(x, t) on t.

2. It cannot return to ıV × {1} as this would yield ∇h = 0 at some point and this never happens.

3. The curve cannot reach ∂V × [0, 1] as, in an interior sphere with h > 0, Lemma 1.3 forbids ∂h/∂ν = 0 in any interior direction ν.

4. From above, N cannot hit ıV × {0}.

To sum up, the existence of a conjugate point results in a contradiction. The proof for a maximum runs similarly. □ Proof. (Corollary 1.9.) It follows directly from Theorem 1.5. □ Proof. (Theorem 1.10.) It is an immediate consequence of Lemma 1.1 and Corollary 1.9. □

3 Application to the conformal prescribed Gauss curvature problem

The conformal Gaussian curvature deformation functional on a compact Riemannian surface V with connected interior, cf. [7], is defined to be

F (u) = ∫_{V} (\frac{1}{2}⟨∇u, ∇u⟩ − \frac{K}{2} e^{2u} + ku)dV.

The associated Euler-Lagrange equation is, therefore, given by

−div(∇u) − Ke^{2u} + k = −Δu − Ke^{2u} + k = 0.

A straight-forward computation yields

d^{2}F (u)(h) = ∫_{V} (∇h^{t}P ∇h + Qh^{2})dV,

where the tensor field P = \frac{1}{2}(∂_{∇u∇u}^{2}J) = I (constant and equal to the identity matrix, which is certainly positive definite) and

Q = \frac{1}{2}[∂_{uu}^{2}J − div(∂_{u∇u}^{2}J)] = −Ke^{2u}.

Now, if the prescribed curvature function K ≤ 0 in V, d^{2}F (u) is definite positive and any solution of the Euler-Lagrange equation (critical point) is a minimum.

The problem is not that easy if we allow K(x) > 0 at some x ∈ V. However, a simple sufficient condition can be stated. Let u be a critical point of the Gaussian conformal functional and suppose that

u_{max} < − log \sqrt{C_{p}|K|_{max}}

holds with the notations |K|_{max} = max {|K(x)| : x ∈ V} , u_{max} = max {u(x) : x ∈ V} and C_{p} is Poincaré’s constant (Lemma 3.1 below). Then, u is a minimum. Certainly,

| ∫_{V} Qh^{2}dV | ≤ C_{p}|K|_{max}e^{2u_{max}} ∫_{V} ⟨∇h, ∇h⟩dV

implies that

d^{2}F (u)(h) ≥ (1 − C_{p}|K|_{max}e^{2u_{max}}) ∫_{V} ⟨∇h, ∇h⟩dV ≥ 0.

We have used the following well-known result, cf. [6].

Lemma 3.1 (Poincaré’s Inequality) For any h ∈ C_{0}^{2}(V), we have

∫_{V} h^{2}dV ≤ C_{p} ∫_{V} ⟨∇h, ∇h⟩dV,

for some positive constant C_{p}.

It is perhaps more interesting to notice that the assumption for V of a positive eigenfunction with eigenvalue λ for Jacobi’s operator −Δh − Ke^{2u}h implies a Maximum Principle for Q = −K(x)e^{2u(x)} < λ, cf. [5].

4 Concluding remarks

Most (if not all) of the results presented above do not really need the functions to be identically zero on the boundary of the domain. It is important to point out that the given definition of conjugate point is appropriate because it applies to unordered sets as well to ordered sets. The reason for this is that it is based upon a boundary value problem instead of an initial value problem. Last but not least, we remark again that the generalization of Jacobi’s condition presented above depends on a domain satisfying a Maximum Principle and interior sphere conditions at each of its boundary points. These are indeed quite restrictive technical assumptions. Aknowledgements. This paper is partially supported by the research project 500 of the Comité Central de Investigaciones at the Universidad del Tolima, Colombia, South America.

References

[1] M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, 1976.

[2] I. Gelfand and S. Fomin, Calculus of Variations. Prentice-Hall, 1963.

[3] L. Solanilla, M. Rondón and C. Morales, Noether’s Theorem on Surfaces, to appear, Applicable Analysis, 2003.

[4] L.C. Evans, Partial Differential Equations, Berkeley Mathematics Lectures Notes, Berkeley, California, 1994.

[5] G. Sweeers, Maximum Principles, http://aw.twi.tudelft.nl/∼koelink /ellip-PDE.html, (Regulariteitstheorie voor Elliptische Differentiaalvergelijkingen).

[6] J. Jost, Compact Riemann Surfaces, Springer Verlag, 1991.

[7] Hulin and M. Troyanov, Prescribing curvature on open surfaces, Math. Ann. 293 (1992), 277-315.

Departamento de Matemáticas y Estadística Universidad del Tolima, Barrio Santa Helena Ibagué, Colombia, South America email : lsolanilla@yahoo.com