---
tag: A3413S0RTM
kind: definition
name: Monge–Ampère equation
msc: [35J96, 35J60]
defines: 14AKAC0S0T
uses: [2D3N1ZJWBW, 0693Y2METT, 07GR3TNW4H]
sources:
  - url: https://en.wikipedia.org/wiki/Monge%E2%80%93Amp%C3%A8re_equation
---

## 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|2D3N1ZJWBW]]
$$\det D^2 u(x) = f\big(x, u(x), \nabla u(x)\big), \qquad x \in \Omega,$$
for a twice differentiable function $u \colon \Omega \to \R$, where $D^2 u$ is the [[Hessian matrix|0693Y2METT]] of $u$.
The most studied case is $\det D^2 u = f(x)$ with $f > 0$, for an unknown [[convex function|07GR3TNW4H]] $u$.

## 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.