Monge–Ampère equation

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

${\displaystyle L[u]=A\,{\text{det}}(\nabla ^{2}u)+B\Delta u+2Cu_{xy}+(D-B)u_{yy}+E=A(u_{xx}u_{yy}-u_{xy}^{2})+Bu_{xx}+2Cu_{xy}+Du_{yy}+E=0,}$

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

${\displaystyle L[u]=0,\quad {\text{on}}\ \Omega }$

${\displaystyle u|_{\partial \Omega }=g.}$

If

${\displaystyle BD-C^{2}-AE>0,}$

then the Dirichlet problem has at most two solutions.[5]

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

${\displaystyle L[u]=\det D^{2}u-f(\mathbf {x} ,u,Du)=0\qquad \qquad (1)}$

is a nonlinear elliptic partial differential equation (in the sense that its linearization is elliptic), provided one confines attention to convex solutions.

Accordingly, the operator L satisfies versions of the maximum principle, and in particular solutions to the Dirichlet problem are unique, provided they exist.

Monge–Ampère equations arise naturally in several problems in Riemannian geometry, conformal geometry, and CR geometry. One of the simplest of these applications is to the problem of prescribed Gauss curvature.[6] Suppose that a real-valued function K is specified on a domain Ω in Rⁿ, the problem of prescribed Gauss curvature seeks to identify a hypersurface of Rⁿ⁺¹ as a graph z = u(x) over x ∈ Ω so that at each point of the surface the Gauss curvature is given by K(x). The resulting partial differential equation is

${\displaystyle \det D^{2}u-K(\mathbf {x} )(1+|Du|^{2})^{(n+2)/2}=0.}$

The Monge–Ampère equations are related to the Monge–Kantorovich optimal mass transportation problem, when the "cost functional" therein is given by the Euclidean distance.[7]

• List of nonlinear partial differential equations
• Complex Monge–Ampère equation

• "Monge–Ampère equation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Monge-Ampère Differential Equation". MathWorld.