A general class of free boundary problems for fully nonlinear elliptic equations

TL;DR

This paper establishes optimal regularity results for solutions to a class of fully nonlinear free boundary problems involving elliptic equations, including regularity of solutions and free boundaries under certain conditions.

Contribution

It introduces a general class of free boundary problems for fully nonlinear elliptic equations and proves optimal regularity of solutions and free boundaries.

Findings

Solutions are locally $C^{1,1}$ inside the domain.

Regularity of the free boundary is established under additional conditions.

The results include optimal regularity for solutions in the specified class.

Abstract

In this paper we study the fully nonlinear free boundary problem $$ {{array}{ll} F(D^2u)=1 & \text{a.e. in}B_1 \cap \Omega |D^2 u| \leq K & \text{a.e. in}B_1\setminus\Omega, {array}. $$ where $K>0$, and $\Omega$ is an unknown open set. Our main result is the optimal regularity for solutions to this problem: namely, we prove that $W^{2,n}$ solutions are locally $C^{1,1}$ inside $B_1$. Under the extra condition that $\Omega \supset \{D u\neq 0 \}$, and a uniform thickness assumption on the coincidence set $\{D u = 0 \}$, we also show local regularity for the free boundary $\partial\Omega\cap B_1$.