A General Class of Free Boundary Problems for Fully Nonlinear Elliptic Equations

In this paper we study the fully nonlinear free boundary problem {F(D(2)u) = 1 almost everywhere in B-1 boolean AND Omega vertical bar D(2)u vertical bar <= K almost everywhhere in B-1\Omega, 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 and a uniform thickness assumption on the coincidence set {D u = 0}, we also show local regularity for the free boundary partial derivative Omega boolean AND B-1.