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.

It is named after Gaspard Monge who introduced descriptive geometry and the first form of the partial differential equation in 1784, and after André-Marie Ampère who introduced the nonlinear partial differential equation in 1820 when studying the geometry of surfaces.

Luis Caffarelli earned the 2023 Abel Prize for his work on this equation.