Symmetry Lie Group of the Monge-Ampère Equation

C.Udrişte, N.Bîlă

Abstract We find the symmetry group of the Monge-Ampère equation and we describe

the geometric dynamics induced by an affine vector field.

Mathematics Subject Classification: 58G35, 35A30 Key-words: symmetry group, infinitesimal criterion of invariance, group-invariant solutions, geometric dynamics.

A symmetry group of a second order PDE is a group which transforms solutions of the equation to other solutions. We can determine explicitly special types of solutions which are themselves invariant under some subgroup of the full symmetry group. These ”group-invariant” solutions are found by solving a reduced system of partial differential equations. The theory of symmetry group for differential equations, partial differential equations and systems has manny applications to geometry, physics and mechanics.

In this paper we shall study the Monge-Ampère equation applying the general theory in the P.J.Olver book[6]. Also we present the geometric dynamics induced by an affine vector field on R^{n+1}. Ours results are original modulo the quoted papers.

The Monge-Ampère equation of two variables is closely related to the Minkovski problem and the Weyl embedding problem. For higher dimension, the Monge-Ampère equation is studied in connection to affine geometry because it is invariant under the special linear group. In this paper, the special linear group is a subgroup of the symmetry group of Monge-Ampère equation.

1 Symmetry Lie group of a PDE of order two

Let π :R^{n+1} →R^{n}, π(x, u) = x, x = (x^{1}, ..., x^{n}), be the projection map.

Let U ⊂R^{n+1} be an open set and U_{0} = π(U). Definition 1. A smooth map s : U_{0} → U, s(x) = (x, u(x)) is called local section of π (over U_{0}).

For the function u, we note

u_{i_{1}...i_{p}}(x) = \frac{∂^{p}u}{∂x^{i_{1}}...∂x^{i_{p}}} (x), x ∈ U_{0}, 1 ≤ i_{1} ≤ ... ≤ i_{p} ≤ n, p ≥ 1.

Balkan Journal of Geometry and Its Applications, Vol.3, No.2, 1998, pp. 121-134 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

Let us consider

J^{k}(U) = {(x^{i}, u, u_{i_{1}}, ..., u_{i_{1}...i_{k}})|(x^{i}, u) ∈ U},

J^{k}(U) ⊂R^{n+1}×R^{N_{1}} × ...× R^{N_{k}}, where N_{k} = \frac{(n+k−1)!}{k!(n−1)!}, k > 0, and the projection

π^{k} : J^{k}(U) → U_{0}, π^{k}(x, u, u_{i_{1}}, ..., u_{i_{1}...i_{k}}) = x.

Convention: for k = 0, J^{0}(U) = U and π^{0} = π. Definition 2. Let s : U_{0} → U be a local section of π (over U_{0}). The section j^{k}(s) of π^{k} over U_{0}, called the k-jet of s, is defined as follows:

j^{k}(s)(x) = (x, u, u_{i_{1}}, ..., u_{i_{1}...i_{k}}), x ∈ U_{0},

u_{i_{1}...i_{r}} = \frac{∂^{r}u}{∂x^{i_{1}}...∂x^{i_{r}}} (p), 0 ≤ r ≤ k.

The space J^{k}(U) is called the kth-order jet space.

Let Ω_{k}^{q}(U) be the vector space of q-forms on J^{k}(U) with exterior differential d. In particular Ω_{0}^{q}(U) = Ω^{q}(U) is the exterior algebra of q-forms on U and Ω_{k}^{0} = C^{∞}(J^{k}(U)) is the algebra of real-valued functions

f = f(x^{i}, u, u_{i_{1}}, ..., u_{i_{1}...i_{k}})

on J^{k}(U). A basis for Ω_{k}^{1}(U) (as a module over C^{∞}(J^{k}(U))) consists of the one forms dx^{i}, du, du_{i_{1}}, ..., du_{i_{1}...i_{k}}. For f ∈ C^{∞}(J^{k}(U)) we have

df = \frac{∂f}{∂x^{i}} dx^{i} + \frac{∂f}{∂u} du + \frac{∂f}{∂u_{i_{1}}} du_{i_{1}} + ... + ∑_{1≤i_{1}≤...≤i_{k}≤n} \frac{∂f}{∂u_{i_{1}...i_{k}}} du_{i_{1}...i_{k}}.

Definition 3. A form ω ∈ Ω_{k}^{q}(U) is called basic if it is of the type

ω = A_{j_{1}...j_{q}}dx^{j_{1}} ∧ ... ∧ dx^{j_{q}},

where A_{j_{1}...j_{q}} are smooth functions on J^{k}(U).

Denote the space of basic q-forms by B_{k}^{q}(U). Definition 4. The operator D : B_{k}^{q}(U) → B_{k+1}^{q+1}(U),

Df = (\frac{∂f}{∂x^{j}} + \frac{∂f}{∂u} u_{j} + \frac{∂f}{∂u_{i_{1}}} u_{i_{1}j} + ... + ∑_{1≤i_{1}≤...≤i_{k}≤n} \frac{∂f}{∂u_{i_{1}...i_{k}}} u_{i_{1}...i_{k}j}) dx^{j},

for f ∈ C^{∞}(J^{k}(U)), and

Dω = DA_{j_{1}...j_{q}} ∧ dx^{j_{1}} ∧ ... ∧ dx^{j_{q}},

for

ω = A_{j_{1}...j_{q}}dx^{j_{1}} ∧ ... ∧ dx^{j_{q}},

is called the total exterior derivative on B_{k}^{q}(U).

Convention: Df = D_{i}f dx^{i}, where D_{i}f is the total derivative of f with respect to x^{i}.

Let Ω_{2}^{n+1}(U) be the space of (n + 1)-forms, and θ ∈ Ω_{2}^{n+1}(U),

θ = F (x^{i}, u, u_{l}, u_{lk})du ∧ dx^{1} ∧ ... ∧ dx^{n}.

Definition 5. A solution of the equation

θ = 0

on U_{0} ⊂R^{n}, is a section s : U_{0} → U such that θ · j^{2}(s) = 0.

In other words, θ determines the PDE

(1)  F (x, u, u_{l}, u_{lk}) = 0,

a solution of which is a function u = u(x) such that

F (x, u(x), \frac{∂u}{∂x^{l}} (x), \frac{∂^{2}u}{∂x^{l}∂x^{k}} (x)) = 0, ∀x ∈ U_{0}.

We note u^{(2)} = (u, u_{l}, u_{lk}). Definition 6. The PDE (1) is called of maximal rank if the Jacobian matrix

J_{F} (x, u^{(2)}) = (F_{x^{i}}; F_{u}; F_{u_{i}}, F_{u_{ij}})

has rank 1 whenever F (x, u^{(2)}) = 0.

Then the subset

S = {(x, u^{(2)}) ∈ J^{(2)}(U)|F (x, u^{(2)}) = 0}

is a hypersurface. Definition 7. A symmetry group of PDE is a local group of transformations G acting on an open set U of the associated space of independent and dependent variables, with the property that whenever u = f(x) is a solution of the equation and whenever g · f is defined for g ∈ G, then v = g · f(x) is also a solution of the equation.

The computational procedure for finding the symmetry group uses the following infinitesimal criterion of invariance. Theorem 1. Let F (x, u^{(2)}) = 0 be a PDE of maximal rank defined on an open set U_{0}. If G is a local group of transformations acting on U and

(2)  pr^{(2)}X[F (x, u^{(2)})] = 0 whenever F (x, u^{(2)}) = 0,

for every infinitesimal generator X of G, then G is a symmetry group of the equation. Consider the vector field

X = ζ^{i}(x, u)\frac{∂}{∂x^{i}} + ϕ(x, u)\frac{∂}{∂u}

on U. The first prolongation of X is the vector field

pr^{(1)}X = X + Φ_{i} \frac{∂}{∂u_{i}} ,

where

Φ_{i} = D_{i}(ϕ − ζ^{k}u_{k}) + ζ^{k}u_{ik} = ϕ_{x^{i}} + u_{i}ϕ_{u} − u_{k} (ζ_{x^{i}}^{k} + u_{i}ζ_{u}^{k}) .

The second prolongation of X is the vector field

(3)  pr^{(2)}X = pr^{(1)}X + Φ_{ij} \frac{∂}{∂u_{ij}} ,

where

Φ_{ij} = D_{ij}(ϕ − ζ^{k}u_{k}) + ζ^{k}u_{ijk} =
= ϕ_{x^{i}x^{j}} + u_{i}ϕ_{x^{j}u} + u_{j}ϕ_{x^{i}u} + u_{i}u_{j}ϕ_{uu} + u_{ij}ϕ_{u}−
−u_{k} (ζ_{x^{i}x^{j}}^{k} + u_{i}ζ_{x^{j}u}^{k} + u_{j}ζ_{x^{i}u}^{k} + u_{i}u_{j}ζ_{uu}^{k} + u_{ij}ζ_{u}^{k}) −
−u_{ki} (ζ_{x^{j}}^{k} + u_{j}ζ_{u}^{k}) − u_{kj} (ζ_{x^{i}}^{k} + u_{i}ζ_{u}^{k}) .

Proposition 1. Let be a PDE of the maximal rank defined on U_{0}. The set of all infinitesimal symmetries of the equation forms a Lie algebra on U. Moreover, if this Lie algebra is finite-dimensional, the symmetry group of the equation is a local Lie group of transformations acting on U.

Algorithm for finding the symmetry group of PDE (1)

-one consider a vector field X on U and one writes the infinitesimal invariance condition (2);

-one eliminates any dependence between partial derivatives of the function u, determined by the PDE (1);

-one writes the condition (2) like a polynomial in the partial derivatives of u; -one equates with zero the coefficients of partial derivatives of u in (2), written

as a polynomial in the derivatives of the function u; it follows a PDE system with respect to the unknown functions ζ^{i}, ϕ, and this system defines the Lie symmetry group G of the given PDE.

Every s-parametric subgroup H of the group G determines a family of groupinvariant solutions. The problem of classification of group-invariant solutions reverts to the problem of classification of Lie subalgebras of Lie algebra g of the group G ([6],pp.186). For 1-dimensional algebras one considers a general element X, and we simplify this as much as possible using the adjoint transformations. Remark. We will compute the adjoint representation Ad G of the underlying Lie group G, by using the Lie series

(4) Ad(exp(εX)Y) = ∑_{n=0}^{∞} \frac{ε^{n}}{n!} (adX)^{n}(Y) = Y − ε[X, Y] + \frac{ε^{2}}{2} [X, [X, Y]] − ...

2 Symmetry Lie group of the Monge-Ampère equation

The general form of the Monge-Ampère equation is [16]

det(u_{ij} + v_{ij}) = H(x, u, u_{i}),

where v_{ij} are functions of x, u, u_{i}. This is a particular nonlinear second order PDE.

First, let us consider the special case

(5)  det(u_{ij}) = f(x, u),

where U_{0} is an open set of R^{2}, u ∈ C^{2}(U_{0}) and f is a nonconstant function. In the case of the equation (5) we have

F (x, u^{(2)}) = det(u_{ij}) − f(x, u)

and the Jacobian matrix is

J_{F} (x, u^{(2)}) = (−f_{x^{i}}; −f_{u}; 0; d_{u_{ij}}), d = det(u_{ij}).

Let A^{ij} be the cofactors of the elements of the determinant d. Because (u_{ij}) is a symmetric matrix, it follows

\frac{∂d}{∂u_{ij}} = \{ \begin{matrix}2A^{ij} & for & i ≠ j \\ A^{ii} & for & i = j. \end{matrix}

Then for F (x, u^{(2)}) = 0, rankJ_{F} = 1.

Let X be the vector field

X = ζ^{i}(x, u)\frac{∂}{∂x^{i}} + ϕ(x, u)\frac{∂}{∂u}

on U. In this case the condition (2) turns in

− ∑_{i=1}^{n} ζ^{i}f_{x^{i}} − ϕf_{u} + ∑_{i=1}^{n} Φ_{ii}A^{ii} + 2 ∑_{1≤i<j≤n} Φ_{ij}A^{ij} = 0.

Substituing the functions Φ_{i} and Φ_{ij} defined by the relation (3) and after eliminating any dependencies among the derivatives of the u’s caused by the equation (5) itself, we find

− ∑_{i=1}^{n} ζ^{i}f_{x^{i}} − ϕf_{u} + +nf (ϕ_{u} − \frac{2}{n} ∑_{i=1}^{n} ζ_{x^{i}}^{i}) +

∑_{i=1}^{n} A^{ii}ϕ_{x^{i}x^{i}} + 2 ∑_{i,j=1,i<j}^{n} A^{ij}ϕ_{x^{i}x^{j}}+

−(n + 2)f ∑_{i=1}^{n} u_{i}ζ_{u}^{i} + ∑_{i=1}^{n} A^{ii}u_{i} (2ϕ_{x^{i}u} − ζ_{x^{i}x^{i}}^{i}) −

− ∑_{i,k=1,k≠i}^{n} A^{ii}u_{k}ζ_{x^{i}x^{i}}^{k} + 2 ∑_{i,j=1,i<j}^{n} A^{ij}u_{j} (ϕ_{x^{i}u} − ζ_{x^{i}x^{j}}^{j}) +

+2 ∑_{i,j=1,i<j}^{n} A^{ij}u_{i} (ϕ_{x^{j}u} − ζ_{x^{i}x^{j}}^{i}) − 2 ∑_{i,j,k=1,i<j,k≠i,j}^{n} A^{ij}u_{k}ζ_{x^{i}x^{j}}^{k}+

+ ∑_{i=1}^{n} A^{ii}u_{i}^{2} (ϕ_{uu} − 2ζ_{x^{i}u}^{i}) − 2 ∑_{i,j=1,i≠j}^{n} A^{ii}u_{i}u_{j}ζ_{x^{i}u}^{j}+

+2 ∑_{i,j=1,i<j}^{n} A^{ij}u_{i}u_{j} (ϕ_{uu} − ζ_{x^{i}u}^{i} − ζ_{x^{j}u}^{j}) −

−2 ∑_{i,j=1,i<j}^{n} A^{ij}u_{j}^{2}ζ_{x^{i}u}^{j} − 2 ∑_{i,j=1,i<j}^{n} A^{ij}u_{i}^{2}ζ_{x^{j}u}^{i}−

−2 ∑_{i,j,k=1,i<j,k≠i,j}^{n} A^{ij}u_{k}u_{i}ζ_{x^{j}u}^{k} − ∑_{i=1}^{n} A^{ii}u_{i}^{3}ζ_{uu}^{i}−

− ∑_{i,k=1,k≠i}^{n} A^{ii}u_{i}^{2}u_{k}ζ_{uu}^{k} − 2 ∑_{i,j=1,i<j}^{n} A^{ij}u_{i}^{2}u_{j}ζ_{uu}^{i}−

−2 ∑_{i,j=1,i<j}^{n} A^{ij}u_{i}u_{j}^{2}ζ_{uu}^{j} − 2 ∑_{i,j,k=1,i<j,k≠i,j}^{n} A^{ij}u_{i}u_{j}u_{k}ζ_{uu}^{k} = 0.

Now we can equate the coefficients of the remaining unconstrained partial derivatives of u to zero. We obtain a large number of partial differential equations for the coefficient functions ζ^{i} and ϕ of the infinitesimal operator, namely

− ∑_{i=1}^{n} ζ^{i}f_{x^{i}} − ϕf_{u} + nf (ϕ_{u} − \frac{2}{n} ∑_{i=1}^{n} ζ_{x^{i}}^{i}) = 0,

ϕ_{x^{i}x^{i}} = 0, i = 1, ..., n
ϕ_{x^{i}x^{j}} = 0, 1 ≤ i < j ≤ n,
ζ_{u}^{i} = 0, i = 1, ..., n,
ϕ_{x^{i}u} − ζ_{x^{i}x^{j}}^{j} = 0, 1 ≤ i < j ≤ n,
ϕ_{x^{j}u} − ζ_{x^{i}x^{j}}^{i} = 0, 1 ≤ i < j ≤ n,

ζ_{x^{i}x^{j}}^{k} = 0, i, j, k = 1, ..., n, i < j, k ≠ i, j,

ϕ_{uu} − 2ζ_{x^{i}u}^{i} = 0, i = 1, ..., n,
ζ_{x^{i}u}^{j} = 0, 1 ≤ i < j ≤ n,
ϕ_{uu} − ζ_{x^{i}u}^{i} − ζ_{x^{j}u}^{j} = 0, 1 ≤ i < j ≤ n,
ζ_{x^{j}u}^{i} = 0, 1 ≤ i < j ≤ n,
ζ_{x^{j}u}^{k} = 0, j, k = 1, ..., n, k ≠ j.

called the defining equations for the symmetry group of the given equation. This PDEs system admits the solutions

ζ^{i} = x^{i} ∑_{j=1}^{n} a_{j}x^{j} + ∑_{j=1}^{n} b_{j}^{i}x^{j} + c^{i}, i = 1, ..., n

ϕ = u (∑_{j=1}^{n} a_{j}x^{j} + b) + ∑_{j=1}^{n} c_{j}x^{j} + c,

where a_{j}, b_{j}^{i}, b, c_{j}, c^{i}, c ∈R, with

− ∑_{i=1}^{n} ζ^{i}f_{x^{i}} − ϕf_{u} + nf (ϕ_{u} − \frac{2}{n} ∑_{i=1}^{n} ζ_{x^{i}}^{i}) = 0.

In the particular case

(6)  det(u_{ij}) = 1,

the preceding expressions are reduced to

ζ^{i} = ∑_{j=1}^{n} b_{j}^{i}x^{j} + c^{i}, i = 1, ..., n

ϕ = \frac{2u}{n} ∑_{j=1}^{n} b_{j}^{j} + ∑_{j=1}^{n} c_{j}x^{j} + c,

and it follows

X = ∑_{i,j=1}^{n} b_{i}^{j} (x^{i} \frac{∂}{∂x^{j}} + δ_{j}^{i} \frac{2u}{n} \frac{∂}{∂u}) +

+ ∑_{i=1}^{n} c_{i}x^{i} \frac{∂}{∂u} + ∑_{i=1}^{n} c^{i} \frac{∂}{∂x^{i}} + c\frac{∂}{∂u}.

Theorem 2. The Lie algebra of infinitesimal symmetries of the Monge-Ampère equation (6) is spanned by the vector fields

X_{ij} = x^{i} \frac{∂}{∂x^{j}} + δ_{j}^{i} \frac{2u}{n} \frac{∂}{∂u}, 1 ≤ i, j ≤ n,

(7)  Y_{i} = x^{i} \frac{∂}{∂u}, 1 ≤ i ≤ n,

Z_{i} = \frac{∂}{∂x^{i}} , 1 ≤ i ≤ n,

V = \frac{∂}{∂u}.

Let consider the case n = 2. The equation (6) turns in

(8)  u_{xx}u_{yy} − u_{xy}^{2} = 1.

The Lie algebra of infinitesimal symmetries of the Monge-Ampère equation (8) is spanned by nine vector fields

X_{1} = \frac{∂}{∂x}, X_{2} = \frac{∂}{∂y} , X_{3} = \frac{∂}{∂u},

(9)  X_{4} = y \frac{∂}{∂x}, X_{5} = x\frac{∂}{∂y} ,

X_{6} = x\frac{∂}{∂u}, X_{7} = y \frac{∂}{∂u}, X_{8} = x\frac{∂}{∂x} + u\frac{∂}{∂u},

X_{9} = y \frac{∂}{∂y} + u\frac{∂}{∂u},

where we note x^{1} = x, x^{2} = y, and Z_{1} = X_{1}, Z_{2} = X_{2}, V = X_{3}, X_{21} = X_{4}, X_{12} = X_{5}, Y_{1} = X_{6}, Y_{2} = X_{7}, X_{11} = X_{8}, X_{22} = X_{9}. Remark. It follows that each one-parameter subgroup G_{i} generated by X_{i} is a symmetry group and if u = f(x, y) is a solution of the Monge-Ampère equation (8), then every function

u^{(1)} = f(x − ε, y), u^{(2)} = f(x, y − ε),
u^{(3)} = f(x, y) + ε, u^{(4)} = f(x − εy, y),
u^{(5)} = f(x, y − εx), u^{(6)} = f(x, y) + εx,
u^{(7)} = f(x, y) + εy, u^{(8)} = e^{ε}f(xe^{−ε}, y),
u^{(9)} = e^{ε}f(x, ye^{−ε}),

is another solution for the equation, where ε is a real number.

We compute the adjoint representation Ad G of the underlying Lie group G, by using the Lie series (4) and we construct the table

\begin{matrix}Ad & X_{1} & X_{2} & X_{3} & X_{4} \\ X_{1} & X_{1} & X_{2} & X_{3} & X_{4} \\ X_{2} & X_{1} & X_{2} & X_{3} & X_{4} \\ X_{3} & X_{1} & X_{2} & X_{3} & X_{4} \\ X_{4} & X_{1} & X_{2} + εX_{1} & X_{3} & X_{4} \\ X_{5} & X_{1} + εX_{2} & X_{2} & X_{3} & X_{4} − ε(X_{8} − εX_{9}) − ε^{2}X_{5} \\ X_{6} & X_{1} + εX_{3} & X_{2} & X_{3} & X_{4} + εX_{7} \\ X_{7} & X_{1} & X_{2} + εX_{3} & X_{3} & X_{4} \\ X_{8} & e^{ε}X_{1} & X_{2} & e^{ε}X_{3} & e^{ε}X_{4} \\ X_{9} & X_{1} & e^{ε}X_{2} & e^{ε}X_{3} & e^{−ε}X_{4} \end{matrix}

\begin{matrix}Ad & X_{5} & X_{6} & X_{7} \\ X_{1} & X_{5} − εX_{2} & X_{6} − εX_{3} & X_{7} \\ X_{2} & X_{5} & X_{6} & X_{7} − εX_{3} \\ X_{3} & X_{5} & X_{6} & X_{7} \\ X_{4} & X_{5} − ε(X_{9} − X_{8}) − ε^{2}X_{4} & X_{6} − εX_{7} & X_{7} \\ X_{5} & X_{5} & X_{6} & X_{7} − εX_{6} \\ X_{6} & X_{5} & X_{6} & X_{7} \\ X_{7} & X_{5} + εX_{6} & X_{6} & X_{7} \\ X_{8} & e^{−ε}X_{5} & X_{6} & e^{ε}X_{7} \\ X_{9} & e^{ε}X_{5} & e^{ε}X_{6} & X_{7} \end{matrix}

\begin{matrix}Ad & X_{8} & X_{9} \\ X_{1} & X_{8} − εX_{1} & X_{9} \\ X_{2} & X_{8} & X_{9} − εX_{2} \\ X_{3} & X_{8} − εX_{3} & X_{9} − εX_{3} \\ X_{4} & X_{8} − εX_{4} & X_{9} + εX_{4} \\ X_{5} & X_{8} + εX_{5} & X_{9} − εX_{5} \\ X_{6} & X_{8} & X_{9} − εX_{6} \\ X_{7} & X_{8} − εX_{7} & X_{9} \\ X_{8} & X_{8} & X_{9} \\ X_{9} & X_{8} & X_{9} \end{matrix}

whith the (i, j)-th entry indicating Ad(exp(εX_{i}))X_{j}. For the equation (8) we find several optimal system of one-dimensional subalgebras spanned by

X_{2} + X_{4}, X_{8} + X_{9}, X_{8} − X_{9}, X_{3} + X_{8} − X_{9}.

In the case of

X_{2} + X_{4} = y \frac{∂}{∂x} + \frac{∂}{∂y} ,

we find the solution

u(x, y) = ±\frac{1}{3}(y^{2} − 2x + a)^{− \frac{3}{2}} + b, a, b ∈ R, y^{2} − 2x + a > 0.

In the case of

X_{8} + X_{9} = x\frac{∂}{∂x} + y \frac{∂}{∂y} + 2u\frac{∂}{∂u},

we get the solution

u(x, y) = cx^{2} + bxy + \frac{a}{2}y^{2}, a, b, c ∈ R, a^{2} + c^{2} > 0, 2ac − b^{2} = 1.

The function u is strictly convex.

Jörgens [4] showed that any convex solution u of the equation (8) defined on R^{2} is a convex quadratic polynomial.

In the case of

X_{8} − X_{9} = x\frac{∂}{∂x} − y \frac{∂}{∂y} ,

we find the following solution

u(x, y) = \sqrt{xy(a − xy)} − aarctg\sqrt{\frac{a − xy}{xy}} + b, a, b ∈ R, a > 0, 0 < xy < a.

For the case of

X_{3} + X_{8} − X_{9} = x\frac{∂}{∂x} − y \frac{∂}{∂y} + \frac{∂}{∂u},

we get the next solution u = u(x, y),

u = ln x − \frac{a}{2}arcsin (\frac{2xy + a}{\sqrt{a^{2} + 1}}) − \frac{1}{2} ln (1 − 2axy + \sqrt{1 − 4axy − 4x^{2}y^{2}}) +

+\frac{1}{2}\sqrt{1 − 4axy − 4x^{2}y^{2}} + b, x, y > 0, xy < \frac{−a + \sqrt{a^{2} + 1}}{2}, a > 0. b ∈ R.

The infinitesimal symmetries of the Monge-Ampère equation (6) can be classified as follows:

1) affine vector fields: X_{ij}, Y_{i}.Z_{i}, V;

2) Killing vector fields: Z_{i}, V;

3) solenoidal vector fields: Z_{i}, V, Y_{i}, X_{ij}(i ≠ j);

4) potential vector fields: Z_{i}, V, X_{ii};

5) torse forming vector fields: X_{ii} (concurrent), Y_{i}, X_{ij}(i ≠ j) (recurrent), Z_{i}, V (parallel).

Now we look for the converse of Theorem 2, for the case n = 2. Theorem 3. If the vector fields (9) are infinitesimal symmetries of the PDE

u_{xx} = H(x, y, u, u_{x}, u_{y}, u_{xy}, u_{yy}),

with maximal rank, then this PDE is reduced to the Monge-Ampère equation (8). Proof. We consider the chain of subalgebras

{X_{1}} ⊂ {X_{1}, X_{2}} ⊂ {X_{1}, X_{2}, X_{3}} ⊂ {X_{1}, X_{5}} ⊂ {X_{1}, X_{5}, X_{6}}
⊂ {X_{1}, X_{5}, X_{6}, X_{8}} ⊂ {X_{2}, X_{4}, X_{5}, X_{6}},

and we impose that the given PDE to be invariant with respect to each subalgebra ([1],pp.303).

1) \{X_{1}\} : X_{1} = \frac{∂}{∂x}, for which pr^{(2)}X_{1} = \frac{∂}{∂x}. The condition (2) becomes

pr^{(2)}X_{1}(F) = 0. It follows

F = u_{xx} − H_{1}(y, u, u_{x}, u_{y}, u_{xy}, u_{yy}).

2) \{X_{1}, X_{2}\} : X_{2} = \frac{∂}{∂y}, for which pr^{(2)}X_{2} = \frac{∂}{∂y}. The condition pr^{(2)}X_{2}(F) = 0

implies

F = u_{xx} − H_{2}(u, u_{x}, u_{y}, u_{xy}, u_{yy}).

3) \{X_{1}, X_{2}, X_{3}\} : X_{3} = \frac{∂}{∂u}, for which pr^{(2)}X_{2} = \frac{∂}{∂u}. The condition (2) is

equivalent to pr^{(2)}X_{3}(F) = 0. It follows

F = u_{xx} − H_{3}(u, u_{x}, u_{y}, u_{xy}, u_{yy}).

4) \{X_{1}, X_{5}\} : X_{5} = x\frac{∂}{∂y}, with pr^{(2)}X_{5} = x\frac{∂}{∂y} − u_{y} \frac{∂}{∂u_{x}} − u_{xy} \frac{∂}{∂u_{xx}} − u_{yy} \frac{∂}{∂u_{xy}} ,

for which pr^{(2)}X_{5}(F) = 0 implies

F = u_{xx} − \frac{u_{xy}^{2}}{u_{yy}} − \frac{1}{u_{yy}} H_{4}(u_{y}, u_{yy}, u_{y}u_{xy} − u_{x}u_{yy}).

5) \{X_{1}, X_{5}, X_{6}\} : X_{6} = x\frac{∂}{∂u} and pr^{(2)}X_{6} = x\frac{∂}{∂u} + \frac{∂}{∂u_{x}} for which pr^{(2)}X_{6}(F)=0

implies

F = u_{xx} − \frac{u_{xy}^{2}}{u_{yy}} − \frac{1}{u_{yy}} H_{5}(u_{y}, u_{yy}).

6) \{X_{1}, X_{5}, X_{6}, X_{8}\} : X_{8} = x\frac{∂}{∂x} + u\frac{∂}{∂u} and pr^{(2)}X_{8} = x\frac{∂}{∂x} + u\frac{∂}{∂u} + u_{x} \frac{∂}{∂u_{x}} +

u_{y} \frac{∂}{∂u_{y}} − u_{xx} \frac{∂}{∂u_{xx}} + u_{yy} \frac{∂}{∂u_{yy}} . The condition pr^{(2)}X_{8}(F) = 0 implies

F = u_{xx} − \frac{u_{xy}^{2}}{u_{yy}} − \frac{1}{u_{yy}} H_{6}(\frac{u_{yy}}{u_{y}}).

7) \{X_{2}, X_{4}, X_{5}, X_{6}\} : X_{4} = y \frac{∂}{∂x} and pr^{(2)}X_{4} = y \frac{∂}{∂x} − u_{x} \frac{∂}{∂u_{y}} − u_{xx} \frac{∂}{∂u_{xy}} −

−2u_{xy} \frac{∂}{∂u_{yy}} . The condition (2) becomes pr^{(2)}X_{4}(F) = 0. It follows

F = u_{xx} − \frac{u_{xy}^{2}}{u_{yy}} − \frac{k}{u_{yy}} , k ≠ 0.

Hence

u_{xx}u_{yy} − u_{xy}^{2} = k.

The condition of maximal rank requires k ≠ 0, and the changing u → ku shows that we can take k = 1.

For n > 2, in [9] A.V.Pogorelov showed that a generalized convex solution of the equation (6) is the function

(10)  u(x) = (∑_{i=2}^{n}(x^{i})^{2})^{1− \frac{1}{n}} · (h(x^{1}))^{\frac{2}{n} −1} ,

where h > 0 is a solution of the ODE y′′ +y^{n−1} = 0 (particular case of Emden-Fowler equation [5]).The solution of Pogorelov is invariant under the subgrup G_{1} generated by

X = ∑_{i=2}^{n} X_{ii} = ∑_{i=2}^{n} x^{i} \frac{∂}{∂x^{i}} + \frac{2(n − 1)}{n}u\frac{∂}{∂u}.

If g_{ε} ∈ G_{1}, g_{ε}(x^{1}, ..., x^{n}, u) = (x^{′1}, ..., x^{′n}, u′), ε ∈R, then x^{′1} = x^{1} + ε, x^{′2} = x^{2}e^{\frac{ε}{2}} , ..., x^{′n} = x^{n}e^{\frac{ε}{2}} , u′ = ue^{\frac{(n−1)ε}{n}} . If (10) is a solution it follows that u′ = f(x′, y′) is another solution of the equation. This implies that the solution (10) can be found by using the preceding algorithm for the vector field X.

3 Dynamics induced by an affine vector field

The generic element of the infinitesimal Lie algebra of symmetries of the Monge-Ampère equation is an affine vector field on R^{n+1}.

Let A = [a^{i}_{j}] be a quadratic matrix of order n + 1 and X(x) = Ax + b be the associated affine vector field on R^{n+1}. If f(x) = \frac{1}{2}∥Ax + b∥^{2} is the energy of X, then D_{Y} f = (AY, Ax + b), ∀Y ∈X (R^{n+1}) and hence zeros of the X (if they exist) are critical points of the energy f. Particularly, the relations

D_{X}X = A(Ax + b), D_{X}f = (A(Ax), Ax + b)

show that the set of the critical points of the energy f contains the field lines of X which are straight lines. The existence of a field line α which is a straight line of R^{n+1} imposes rankA ≤ n. If x_{0} ∈R^{n+1} is a critical points of the energy f and rankA = n + 1, then x_{0} is a zero of X.

If X = (X^{1}, ..., X^{n+1}), then the gradient of the energy f has the components

\frac{∂f}{∂x^{j}} = ∑_{i=1}^{n+1} \frac{∂X^{i}}{∂x^{j}} X^{i} = ∑_{i=1}^{n+1} a^{i}_{j}X^{i}.

The matrix of the Hessian d^{2}f of the energy f has the components

\frac{∂^{2}f}{∂x^{j}∂x^{k}} = ∑_{i=1}^{n+1} \frac{∂X^{i}}{∂x^{j}} \frac{∂X^{i}}{∂x^{k}} = ∑_{i=1}^{n+1} a^{i}_{j}a^{i}_{k}.

That in way d^{2}f ≥ 0 and hence the energy f is a convex function on R^{n+1}. This result implies the fact critical points of f coincide with global minimum points of f and hence with zeros of X. Also, the flow determined on R^{n+1} by grad f increases the volume excepting the case in which X is a parallel vector field.

The dynamics induced by a affine vector field is described by the second order differential system in the next [14] Theorem 4. Let X = Ax + b be an affine vector field on R^{n+1} and f be its energy. Any orbit of X is a trajectory of the (potential or nonpotential) dynamical system with n + 1 degrees of freedom

(11) \frac{d^{2}x^{i}}{dt^{2}} = \frac{∂f}{∂x_{i}} + ∑_{j=1}^{n+1}(a^{i}_{j} − a_{j}^{i})\frac{dx^{j}}{dt} , i = 1, ...n + 1, where a_{j}^{i} = δ^{ih}δ_{kj}a^{k}_{h}.

Theorem 5. 1) The trajectories of the dynamical system (11) are the extremal of the Lagrangian

L = \frac{1}{2}δ_{ij} \frac{dx^{i}}{dt} \frac{dx^{j}}{dt} − δ_{ij}X^{i}(x)\frac{dx^{j}}{dt} + f(x).

2) The dynamical system (11) is conservative, admitting the Hamiltonian

H = \frac{1}{2}δ_{ij} \frac{dx^{i}}{dt} \frac{dx^{j}}{dt} − f(x).

Theorem 6 (Lorentz-Udrişte world-force law). Every nonconstant trajectory of the dynamical system (11), which correspond to a constant value H of the Hamiltonian, is a reparametrized horizontal geodesic of the Riemann-Jacobi-Lagrange manifold

(R^{n+1} \ E, g_{ij} = (H + f)δ_{ij}, N^{i}_{j} = a^{i}_{j} − a_{j}^{i}),

where E is the set of zeros of the vector field X.

References

[1] G. Bluman, J.D.Cole -Similarity Methods for Differential Equations, Springer Verlag, New York, 1974.

[2] G. Caviglia -Symmetry transformations, isovectors and conservation laws, J.Math.Phys.27(4), April 1986, 972-978.

[3] B.K.Harrison, F.B.Estabrook -Geometric approach to invariance groups and solution of partial differential systems, J.Math.Phys.12(4), April, 1971, 653-666.

[4] K.Jörgens -Über die Lösungen der Differentialgleichung rt − s^{2} = 1, Math.Ann.127(1954), 130-134.

[5] I.T.Kiguradze, T.A.Chanturia -Asymptotic properties of solutions of nonautonomous ordinary differential equations, Mathematics and Its Applications, 89, Kluwer Academic Publishers, 1993.

[6] P.J.Olver -Applications of Lie Groups to Differential Equations, Springer Verlag Inc., 1986.

[7] P.J.Olver -Symmetry groups and group invariant solutions of partial differential equations, J.Diff.Geom. 14(1979), 497-542.

[8] D.Opris, I.Butulescu -Metode geometrice în studiul sistemelor de ecuaţii diferenţiale, Editura Mirton, Timişoara, 1997.

[9] A.V.Pogorelov, -On the regularity of generalized solutions of the equation

det (\frac{∂^{2}u}{∂x^{i}x^{j}} ) = Φ(x^{1}, x^{2}, ..., x^{n}) > 0, Dokl.Akad.Nauk SSSR 200(1971), 534-537.

[10] A.V.Pogorelov, -Monge-Ampère equation, Mockba Hayka, 1988.

[11] Shiu-Yen Cheng, Shing-Tung Yau -On the regularity of the Monge-Ampère equa-

tion det (\frac{∂^{2}u}{∂x^{i}x^{j}} ) = F (x, u), CPAM, 30(1977), 69-94.

[12] M.E.Taylor, -Partial differential equations III, Springer Verlag, New York, Inc., 1996.

[13] C.Udrişte -Linii de câmp, Editura Tehnică, 1988.

[14] C.Udrişte -Geometric Dynamics, Second Conference of Balkan Society of Geometers, Aristotle University of Thessaloniki, Greece, June, 23-26, 1998, to appear.

[15] G.Vrânceanu -Lecţii de geometrie diferenţială, Editura Didactică şi Pedagogică Bucureşti, 1976.

[16] S.T.Yau -Survey on partial differential equations in differential geometry, Seminar on Differential Geometry, Annals of Mathematics Studies, Princeton University Press, 102(1982), 3-72.

University ”Politehnica” of Bucharest Departament of Mathematics I Splaiul Independenţei 313 77206 Bucharest Romania email:udriste@mathem.pub.ro email:nbila@mathem.pub.ro