Partial differential equations Definition

Monge–Ampère equation

Builds on Fully nonlinear PDE Hessian matrix Convex function on R^n

Definition

Let $n \ge 2$ , let $\Omega \subseteq \R^n$ be an open set, and let $f \colon \Omega \times \R \times \R^n \to \R$ be a given function. The Monge–Ampère equation is the fully nonlinear PDE $\det D^2 u(x) = f\big(x, u(x), \nabla u(x)\big), \qquad x \in \Omega,$ for a twice differentiable function $u \colon \Omega \to \R$ , where $D^2 u$ is the Hessian matrix of $u$ . The most studied case is $\det D^2 u = f(x)$ with $f > 0$ , for an unknown convex function $u$ .

Remarks

Example: $u(x) = \tfrac12 \abs{x}^2$ solves $\det D^2 u = 1$ on $\R^n$ , since $D^2 u$ is the identity matrix. Non-example: in $\R^2$ , $u(x) = \tfrac12 x_1^2$ does not solve $\det D^2 u = 1$ , since $D^2 u = \operatorname{diag}(1, 0)$ has determinant $0$ . The equation is elliptic at a solution $u$ when $D^2 u$ is positive definite, which is why one looks for convex solutions with $f > 0$ . For $n = 2$ it reads $u_{xx} u_{yy} - u_{xy}^2 = f$ . The graph of $u$ has Gauss curvature $K = \det D^2 u / (1 + \abs{\nabla u}^2)^{(n+2)/2}$ , so prescribing $K$ is a Monge–Ampère equation. For convex functions that are not twice differentiable, the equation is understood in the sense of Aleksandrov or in the viscosity sense.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`A3413S0RTM`
URI	`m2://A3413S0RTM`
Kind	definition
Writing	W0 stub
MSC	35J96, 35J60
Defines	Monge–Ampère equation
Uses	Fully nonlinear PDE, Hessian matrix, Convex function on R^n

Monge–Ampère equation

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here