The Wiener criterion for fully nonlinear elliptic equations

TL;DR

This paper extends the Wiener criterion to fully nonlinear elliptic equations, providing a capacitary framework and potential-theoretic characterization of boundary regularity.

Contribution

It introduces a capacity concept for non-divergence form operators and establishes a Wiener criterion for boundary regularity in this context.

Findings

Defined a new capacity for nonlinear elliptic operators

Derived capacitary estimates for boundary points

Established Wiener criterion for boundary regularity

Abstract

We study the boundary continuity of solutions to fully nonlinear elliptic equations. We first define a capacity for operators in non-divergence form and derive several capacitary estimates. Secondly, we formulate the Wiener criterion, which characterizes a regular boundary point via potential theory. Our approach utilizes the asymptotic behavior of homogeneous solutions, together with Harnack inequality and the comparison principle.