Pointwise boundary estimates for fully nonlinear elliptic equations with nonzero Dirichlet boundary conditions

TL;DR

This paper develops pointwise boundary estimates for fully nonlinear elliptic equations with nonzero boundary conditions, enabling derivation of global regularity and extending previous Monge-Ampère results.

Contribution

It introduces new boundary estimate techniques for fully nonlinear elliptic equations with general boundary conditions, including nonzero cases, expanding existing theoretical frameworks.

Findings

Established pointwise boundary bounds for convex solutions

Derived global Hölder regularity from boundary estimates

Unified previous results on Monge-Ampère equations as special cases

Abstract

In this paper, we investigate boundary estimates for the Dirichlet problem for a class of fully nonlinear elliptic equations with general boundary conditions, including nonzero boundary conditions. Given specific structural conditions on the problem, we develop pointwise boundary upper and lower bound estimates for convex solutions based on the subsolution and supersolution method. The global H\"older regularity can be derived as a direct consequence of these pointwise boundary estimates. These results fundamentally hinge on careful descriptions of the convexity properties of both the domains and the functions involved. Moreover, previous results on Monge-Amp\`ere equations with nonzero Dirichlet boundary conditions can be regarded as a special case of our results.