Regularity theory for the Isaacs equation through approximation methods

TL;DR

This paper introduces an approximation approach to analyze the regularity of solutions to the Isaacs equation, a key non-convex fully nonlinear elliptic PDE, by relating it to Bellman equations.

Contribution

It develops a novel approximation method that links Isaacs equations to Bellman equations to establish regularity results in Sobolev and H"older spaces.

Findings

Regularity results in Sobolev spaces

Regularity results in H"older spaces

Implications for fully nonlinear elliptic equations

Abstract

In this paper, we propose an approximation method to study the regularity of solutions to the Isaacs equation. This class of problems plays a paramount role in the regularity theory for fully nonlinear elliptic equations. First, it is a model-problem of a non-convex operator. In addition, the usual mechanisms to access regularity of solutions fall short in addressing these equations. We approximate an Isaacs equation by a Bellman one, and make assumptions on the latter to recover information for the former. Our techniques produce results in Sobolev and H\"older spaces; we also examine a few consequences of our main findings.