Monge–Ampère equation

Summary

The Monge–Ampère equation prescribes the determinant of the Hessian matrix of an unknown function: $\det D^2 u = f$. It is the basic example of a fully nonlinear PDE, since it is nonlinear in the highest derivatives. It is elliptic when $u$ is convex and $f > 0$. It appears in the problem of prescribing the Gauss curvature of a convex surface, in optimal transport and in complex geometry.