Monge–Ampère equation

Not to be confused with Monge equation.

In mathematics, a (real) Monge–Ampère equation is a nonlinear second-order partial differential equation of special kind. A second-order equation for the unknown function ${\displaystyle u}$ of two variables ${\displaystyle x}$, ${\displaystyle y}$ is of Monge–Ampère type if it is linear in the determinant of the Hessian matrix of ${\displaystyle u}$ and in the second-order partial derivatives of ${\displaystyle u}$. The independent variables (${\displaystyle x}$, ${\displaystyle y}$) vary over a given domain ${\displaystyle D}$ of ${\displaystyle \mathbb {R} ^{2}}$. The term also applies to analogous equations with ${\displaystyle n}$ independent variables. The most complete results so far have been obtained when the equation is elliptic.

Definition

In two dimension

Given two independent variables ${\displaystyle x}$ and ${\displaystyle y}$, and one dependent variable ${\displaystyle u}$, the general Monge–Ampère equation is of the form

${\displaystyle L[u]=A\det(\nabla ^{2}u)+B\Delta u+2Cu_{xy}+(D-B)u_{yy}+E=A(u_{xx}u_{yy}-u_{xy}^{2})+Bu_{xx}+2Cu_{xy}+Du_{yy}+E=0,}$

where ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$, ${\displaystyle D}$, and ${\displaystyle E}$ are functions depending on the first-order variables ${\displaystyle x}$, ${\displaystyle y}$, ${\displaystyle u}$, ${\displaystyle u_{x}}$, and ${\displaystyle u_{y}}$ only.

In general

Given a domain ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ and a real-valued function ${\displaystyle u\colon \Omega \to \mathbb {R} }$, a (real) Monge–Ampère equation is any fully nonlinear second-order equation that can be written in the form

${\displaystyle F(x,u,Du,\det D^{2}u)=0,}$

for some function ${\displaystyle F}$. More generally, ${\displaystyle \Omega }$ can be a Riemannian manifold, since ${\displaystyle Du,D^{2}u}$ are well-defined on a Riemannian manifold.

If ${\displaystyle F}$ also depends linearly on all principal minors of the Hessian matrix ${\displaystyle D^{2}u}$, then it is an equation of Monge–Ampère type.

Classification

Main article: Elliptic partial differential equation § Nonlinear and higher-order equations

As for other second-order fully nonlinear equations, the type of a Monge–Ampère equation is defined by the linearization of the operator at a sufficiently smooth solution. Of these, the most common is the elliptic case. When people say "Monge–Ampère equation" without adjective, they usually mean the elliptic case.

Let ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ be an open set, ${\displaystyle u\colon \Omega \to \mathbb {R} }$ be a ${\displaystyle C^{2}}$ function, and consider an operator of Monge–Ampère type

${\displaystyle L[u]=F(x,u,Du,D^{2}u),}$

where ${\displaystyle F}$ is smooth in all variables and depends on ${\displaystyle D^{2}u}$ only through its principal minors. The linearization of ${\displaystyle L}$ is of form

${\displaystyle \sum _{ij}a^{ij}(x)\,u_{ij}+{\text{lower order terms}},}$

where

${\displaystyle a^{ij}(x)={\frac {\partial F}{\partial u_{ij}}}(x,u(x),Du(x),D^{2}u(x)).}$

The quadratic form

${\displaystyle Q_{x}(\xi )=a^{ij}(x)\,\xi _{i}\xi _{j},\qquad \xi \in \mathbb {R} ^{n},}$

is the principal symbol of the linearized operator at the point ${\displaystyle x}$.

The equation is said to be

• elliptic at ${\displaystyle x}$ if ${\displaystyle Q_{x}(\xi )}$ if all eigenvalues are of the same sign,
• hyperbolic at ${\displaystyle x}$ if ${\displaystyle Q_{x}(\xi )}$ takes both positive and negative values (the matrix is indefinite),
• parabolic at ${\displaystyle x}$ if ${\displaystyle Q_{x}(\xi )}$ is degenerate (the matrix has vanishing determinant),
• degenerate elliptic if it is elliptic everywhere,
• elliptic if it is degenerate elliptic, and all eigenvalues are a bounded distance away from zero.

As follows from Jacobi's formula for the derivative of a determinant, this equation is elliptic if ${\displaystyle f}$ is a positive function and solutions satisfy the constraint of being uniformly convex. If ${\displaystyle f}$ is merely strictly convex, then the equation is degenerate-elliptic.^[1]

Examples

The Monge–Ampère equation in its simplest form is

${\displaystyle \det D^{2}u=f(x)}$

where ${\displaystyle f}$ is a given function on a domain ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$. This is a special case of equation (1) below with ${\displaystyle f(x,u,Du)=f(x)}$.

The classical Liouville theorem has an analogy here. If ${\displaystyle \det D^{2}u}$ is constant, and ${\displaystyle u}$ is defined on all of ${\displaystyle \mathbb {R} ^{n}}$, then ${\displaystyle u}$ is a quadratic function. This is the Jörgens–Calabi–Pogorelov theorem.^{[2]: Sec. 4.3}

If ${\displaystyle f(x)}$ is positive and uniformly convex, and ${\displaystyle u}$ is a solution to ${\displaystyle \det D^{2}u=f(x)}$, then its Legendre transform ${\displaystyle u^{*}}$ is a solution to ${\displaystyle \det D^{2}u^{*}=1/f(x)}$.

Geometry

Monge–Ampère equations arise naturally in several problems in Riemannian geometry, conformal geometry, affine geometry, and CR geometry.

Given a twice-differentiable real-valued function ${\displaystyle f:\Omega \to \mathbb {R} }$ defined over a domain ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$, its graph is a manifold of n dimensions. At any ${\displaystyle x\in \Omega }$, the Gaussian curvature of the manifold at ${\displaystyle f(x)}$ is ${\displaystyle {\frac {\det D^{2}u}{(1+|Du|^{2})^{(n+2)/2}}}}$. Thus, if we want to find a manifold whose Gaussian curvature is an arbitrary function we pick ourselves, then we need to solve the following Monge–Ampère equation:

${\displaystyle \det D^{2}u-K(x)(1+|Du|^{2})^{(n+2)/2}=0}$

where ${\displaystyle K}$ is the Gaussian curvature we want. Given such a function K, it is nontrivial to find a solution, if any. The problem of finding a solution is the Minkowski problem, or the prescribed Gaussian curvature problem.^[1]

For example, the rigidity of the 2-sphere manifests as the fact that if we require ${\displaystyle n=3,K=1,u(0)=0,Du=0}$, then there are just two unique solutions, which is the unit 2-sphere and its reflection.

The affine spheres can be characterized by a Monge–Ampère equation.

Optimal transport

Consider the problem of optimal transport with quadratic cost (this is also called the 2-Wasserstein metric problem) on ${\displaystyle \mathbb {R} ^{n}}$. That is, suppose ${\displaystyle \mu ,\nu }$ are distributions on ${\displaystyle \mathbb {R} ^{n}}$ with probability density functions ${\displaystyle \rho _{\mu },\rho _{\nu }}$. In this case, a map ${\displaystyle T:\operatorname {supp} (\mu )\to \operatorname {supp} (\nu )}$ is a transport map iff it satisfies$${\displaystyle \int h(T(x))\rho _{\mu }(x)dx=\int h(y)\rho _{\nu }(y)dy}$$for any integrable test function ${\displaystyle h\in L^{1}(\operatorname {supp} (\nu ))}$. The problem is to find the ${\displaystyle T}$ that minimizes the following quadratic cost function:$${\displaystyle \min _{T}\int \|x-T(x)\|^{2}f(x)dx}$$By a theorem of Brenier, the optimal transport map exists, and is the gradient of a convex function ${\displaystyle \psi$ :\operatorname {supp} (\mu )\to \mathbb {R} ^{n}} , with ${\displaystyle T=D\psi }$. The convex function satisfies a Monge–Ampère equation:^{[3]: 282 [4][5]}$${\displaystyle {\begin{cases}\det(D^{2}\psi )={\frac {\rho _{\mu }}{\rho _{\nu }\circ D\psi }}\\D\psi (\partial \operatorname {supp} (\mu ))=\partial \operatorname {supp} (\nu )\end{cases}}}$$The boundary condition simply states that the optimal transport maps the boundary of the source to the boundary of the target. Furthermore, the solution ${\displaystyle \psi }$ is almost everywhere unique.

The function ${\displaystyle \psi }$ is called the potential function of the problem in this case.

Conversely, some Monge–Ampère equations can be interpreted optimal transport. Weak-solutions of a Monge–Ampère equations obtained by optimal transport are often called Brenier solutions in the literature. Brenier solutions satisfy their corresponding Monge–Ampère equations almost everywhere.^[3]: 323

Rellich's theorem

Let ${\displaystyle \Omega }$ be a bounded domain in ${\displaystyle \mathbb {R} ^{3}}$, and suppose that on ${\displaystyle \Omega }$ the coefficients ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$, ${\displaystyle D}$, and ${\displaystyle E}$ are continuous functions of ${\displaystyle x}$ and ${\displaystyle y}$ only. Consider the Dirichlet problem to find ${\displaystyle u}$ so that

${\displaystyle L[u]=0,\quad {\text{on}}\ \Omega }$

${\displaystyle u|_{\partial \Omega }=g.}$

If

${\displaystyle BD-C^{2}-AE>0,}$

then the Dirichlet problem has at most two solutions.^[6]

Ellipticity results

Suppose now that ${\displaystyle \mathbf {x} }$ is a variable with values in a domain in ${\displaystyle \mathbb {R} ^{n}}$, and that ${\displaystyle f(\mathbf {x} ,u,Du)}$ is a positive function. Then the Monge–Ampère equation

${\displaystyle L[u]=\det D^{2}u-f(\mathbf {x} ,u,Du)=0\qquad \qquad (1)}$

is a nonlinear elliptic partial differential equation (in the sense that its linearization is elliptic), provided one confines attention to convex solutions.

Accordingly, the operator ${\displaystyle L}$ satisfies versions of the maximum principle, and in particular solutions to the Dirichlet problem are unique, provided they exist.^{[citation needed]}

See also

• List of nonlinear partial differential equations
• Complex Monge–Ampère equation