A Best Possible Maximum Principle and an Overdetermined Problem for a Generalized Monge-Ampère Equation

This paper investigates a P-function associated with solutions to boundary value problems of some generalized Monge-Ampere equations in bounded convex domains. It will be shown that P attains its maximum value either on the boundary or at a critical point of any convex solution. Furthermore, it turns out that such P-function is actually a constant when the underlying domain is a ball. Therefore, our results provide a best possible maximum principle in the sense of L. Payne. As an application, we will use these results to study an overdetermined boundary value problem. More specifically, we will show solvability of this overdetermined boundary value problem forces their P-function to be a constant.