Monge–Ampère equation In mathematics , a 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 vary over a given domain D of R 2 . 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 .Description 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.Rellich's theorem Let Ω be a bounded domain in R 3 , 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.Ellipticity results Suppose now that x is a variable with values in a domain in R n , and that f is a positive function . Then the Monge–Ampère equation is a nonlinear elliptic partial differential equation , 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.Applications 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 . Suppose that a real-valued function K is specified on a domain Ω in R n , the problem of prescribed Gauss curvature seeks to identify a hypersurface of R n +1 as a graph z = u over x ∈ Ω so that at each point of the surface the Gauss curvature is given by K . The resulting partial differential equation is 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 .Additional references
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