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

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 $\partial\Omega\cap B_1$.