Best Possible Maximum Principles for Fully Nonlinear Elliptic Partial Differential Equations

Abstract

We investigate a class of equations including generalized Monge–Ampere equations as well as Weingarten equations and prove a maximum principle for suitable functions involving the solution and its gradient. Since the functions which enjoy the maximum principles are constant for special domains, we have a so called best possible maximum principle that can be used to find accurate estimates for the solution of the corresponding Dirichlet problem. For these equations we also give a variational form which may have its own interest.