Global second derivative estimates for the second boundary value problem of the prescribed affine mean curvature and Abreu's equations

TL;DR

This paper establishes global second derivative estimates for the second boundary value problem of prescribed affine mean curvature equations, extending previous results to cases where the affine mean curvature is only in L^p and including Abreu's equation.

Contribution

It extends existing second derivative estimates to broader conditions, covering cases with less regular affine mean curvature and including Abreu's equation.

Findings

Proved global second derivative estimates for prescribed affine mean curvature equations.

Extended previous results to cases with affine mean curvature in L^p.

Included analysis of Abreu's equation in the context of second boundary value problems.

Abstract

In this paper we prove the global second derivative estimates for the second boundary value problem of the prescribed affine mean curvature equation where the affine mean curvature is only assumed to be in $L^{p}$. Our result extends previous result by Trudinger and Wang in the case of globally bounded affine mean curvature and also covers Abreu's equation.