Regularity of solutions to fully nonlinear elliptic and parabolic free boundary problems

We consider fully nonlinear obstacle-type problems of the form {F(D(2)u, x) = f (x) a.e. in B-1 boolean AND Omega, vertical bar D(2)u vertical bar <= K a.e. in B-1 \ Omega, where Omega is an open set and K > 0. In particular, structural conditions on F are presented which ensure that W-2,W-n (B-1) solutions achieve the optimal C-1,(1) (B-1/2) regularity when f is Holder continuous. Moreover, if f is positive on (B) over bar (1), Lipschitz continuous, and {u not equal 0} subset of Omega, we obtain interior C-1 regularity of the free boundary under a uniform thickness assumption on {u = 0}. Lastly, we extend these results to the parabolic setting. (C) 2015 Elsevier Masson SAS. All rights reserved.