Monge–Ampère 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.