A Liouville theorem for the Neumann problem of the Monge-Ampere equation

TL;DR

This paper establishes a Liouville theorem for the Neumann problem of the Monge-Ampère equation, showing that under certain conditions, solutions must be quadratic polynomials, extending known results to higher dimensions.

Contribution

It proves new Liouville-type results for the Neumann problem of Monge-Ampère equations in semi-space, including higher dimensions and specific boundary conditions.

Findings

Viscosity convex solutions in 2D are quadratic polynomials.

In dimensions n ≥ 3, solutions are quadratic if boundary value is zero.

Solutions restricted to certain subspaces are quadratic under boundedness conditions.

Abstract

In this paper, we study the Neumann problem of Monge-Amp\`ere equations in Semi-space. For two dimensional case, we prove that its viscosity convex solutions must be a quadratic polynomial. When the space dimension $n\geq 3$, we show that the conclusion still holds if either the boundary value is zero or the viscosity convex solutions restricted on some $n-2$ dimensional subspace is bounded from above by a quadratic function.