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 u of two variables x,y is of Monge–Ampère type if it is linear in the determinant of the Hessian matrix of u and in the second-order partial derivatives of u. The independent variables (x,y) vary over a given domain D of R². The term also applies to analogous equations with n independent variables. The most complete results so far have been obtained when the equation is elliptic.

Monge–Ampère equations frequently arise in differential geometry, for example, in the Weyl and Minkowski problems in differential geometry of surfaces. They were first studied by Gaspard Monge in 1784 and later by André-Marie Ampère in 1820. Important results in the theory of Monge–Ampère equations have been obtained by Sergei Bernstein, Aleksei Pogorelov, Charles Fefferman, and Louis Nirenberg. More recently, Alessio Figalli and Luis Caffarelli were recognized for their work on the regularity of the Monge–Ampère equation, with the former winning the Fields Medal in 2018 and the latter the Abel Prize in 2023.

Given two independent variables x and y, and one dependent variable u, the general Monge–Ampère equation is of the form

where A, B, C, D, and E are functions depending on the first-order variables x, y, u, u_x, and u_y only.

Let Ω be a bounded domain in R³, and suppose that on Ω A, B, C, D, and E are continuous functions of x and y only. Consider the Dirichlet problem to find u so that

If

then the Dirichlet problem has at most two solutions.

Suppose now that x is a variable with values in a domain in Rⁿ, and that f(x,u,Du) is a positive function. Then the Monge–Ampère equation