  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 $\partial\Omega\cap B_1$.
