Monge–Ampère equation

Short description: Nonlinear second-order partial differential equation of special kind

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

Definition

In two dimension

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

$\begin{array}{rl}L[u]=& A\det ({\mathrm{\nabla }}^{2}u)+B\mathrm{\Delta }u+2C{u}_{xy}+(D-B)u_{yy}+E\\ =& A(u_{xx}{u}_{yy}-u_{xy}^2)+B{u}_{xx}+2C{u}_{xy}+D{u}_{yy}+E=0,\end{array}$

where $A$, $B$, $C$, $D$, and $E$ are functions depending on the first-order variables $x$, $y$, $u$, $u_x$, and $u_y$ only.

In general

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

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

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

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

Classification

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

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

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

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

where

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

The quadratic form

$Q_x(\xi )=a^{ij}(x){\xi }_i{\xi }_j,\xi \in {\mathbb{ℝ}}^n,$

is the principal symbol of the linearized operator at the point $x$.

The equation is said to be

• elliptic at $x$ if $Q_x(\xi )$ if all eigenvalues are of the same sign,
• hyperbolic at $x$ if $Q_x(\xi )$ takes both positive and negative values (the matrix is indefinite),
• parabolic at $x$ if $Q_x(\xi )$ 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 $f$ is a positive function and solutions satisfy the constraint of being uniformly convex. If $f$ is merely strictly convex, then the equation is degenerate-elliptic.({{{1}}}, {{{2}}})

Examples

The Monge–Ampère equation in its simplest form is

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

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

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

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

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 $f:\mathrm{\Omega }\to \mathbb{ℝ}$ defined over a domain $\mathrm{\Omega }\subset {\mathbb{ℝ}}^n$, its graph is a manifold of n dimensions. At any $x\in \mathrm{\Omega }$, the Gaussian curvature of the manifold at $f(x)$ is $\frac{\det D^{2}u}{(1+|Du{|}^2{)}^{(n+2)/2}}$. 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:

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

where $K$ 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}}}, {{{2}}})

For example, the rigidity of the 2-sphere manifests as the fact that if we require $n=3,K=1,u(0)=0,Du=0$, 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 ${\mathbb{ℝ}}^n$. That is, suppose $\mu ,\nu$ are distributions on ${\mathbb{ℝ}}^n$ with probability density functions ${\rho }_{\mu },{\rho }_{\nu }$. In this case, a map $T:\mathrm{supp}(\mu )\to \mathrm{supp}(\nu )$ is a transport map iff it satisfies$$\int h(T(x)){\rho }_{\mu }(x)dx=\int h(y){\rho }_{\nu }(y)dy$$for any integrable test function $h\in L^1(\mathrm{supp}(\nu ))$. The problem is to find the $T$ that minimizes the following quadratic cost function:$$\underset{T}{\min }\int \Vert x-T(x){\Vert }^{2}f(x)dx$$By a theorem of Brenier, the optimal transport map exists, and is the gradient of a convex function $\psi :\mathrm{supp}(\mu )\to {\mathbb{ℝ}}^n$, with $T=D\psi$. The convex function satisfies a Monge–Ampère equation:({{{1}}}, {{{2}}})({{{1}}}, {{{2}}})$$\{\begin{array}{l}\det (D^2\psi )=\frac{{\rho }_{\mu }}{{\rho }_{\nu }\circ D\psi }\\ D\psi (\partial \mathrm{supp}(\mu ))=\partial \mathrm{supp}(\nu )\end{array}$$The boundary condition simply states that the optimal transport maps the boundary of the source to the boundary of the target. Furthermore, the solution $\psi$ is almost everywhere unique.

The function $\psi$ 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.({{{1}}}, {{{2}}})

Rellich's theorem

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

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

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

If

$BD-C^2-AE>0,$

then the Dirichlet problem has at most two solutions.({{{1}}}, {{{2}}})

Ellipticity results

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

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

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

See also

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

Additional references

• Aubin, Thierry (1998). Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics. Berlin: Springer-Verlag. doi:10.1007/978-3-662-13006-3. ISBN 3-540-60752-8. 
• Courant, R.; Hilbert, D. (1962). Methods of mathematical physics. Volume II: Partial differential equations. New York–London: Interscience Publishers. doi:10.1002/9783527617234. ISBN 9780471504399. 
• Gilbarg, David; Trudinger, Neil S. (2001). Elliptic partial differential equations of second order. Classics in Mathematics (Reprint of the 1998 ed.). Berlin: Springer-Verlag. doi:10.1007/978-3-642-61798-0. ISBN 3-540-41160-7. 
• Spivak, Michael (1999). A comprehensive introduction to differential geometry: volume five (Third edition of 1975 original ed.). Publish or Perish, Inc.. ISBN 0-914098-74-8. 
• Villani, Cédric (2003). Topics in optimal transportation. Graduate Studies in Mathematics. 58. Providence, RI: American Mathematical Society. doi:10.1090/gsm/058. ISBN 0-8218-3312-X. 
• Villani, Cédric (2009). Optimal transport. Old and new. Grundlehren der mathematischen Wissenschaften. 338. Berlin: Springer-Verlag. doi:10.1007/978-3-540-71050-9. ISBN 978-3-540-71049-3. 
• Figalli, Alessio (2017). The Monge-Ampère equation and its applications. Zurich lectures in advanced mathematics. Zürich: European Mathematical Society. ISBN 978-3-03719-170-5. 
• Gutiérrez, Cristian E. (2016). The Monge-Ampère Equation. Progress in Nonlinear Differential Equations and Their Applications (2nd ed. 2016 ed.). Cham: Birkhäuser. ISBN 978-3-319-43372-1. 
• Lê, Nam Q. (2024). Analysis of Monge-Ampère equations. Graduate studies in mathematics. Providence, Rhode Island: American Mathematical Society. ISBN 978-1-4704-7420-1. 

External links

• Hazewinkel, Michiel, ed. (2001), "Monge–Ampère equation", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=M/m064620 
• Weisstein, Eric W.. "Monge-Ampère Differential Equation". http://mathworld.wolfram.com/Monge-AmpereDifferentialEquation.html.