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

•We show that W2,nW2,n solutions to our type of problems are C1,1C1,1.•The free boundary is shown to be C1C1 under additional natural assumptions.•The results above are extended to the parabolic setting.

We consider fully nonlinear obstacle-type problems of the form{F(D2u,x)=f(x)a.e. in B1∩Ω,|D2u|≤Ka.e. in B1\Ω, where Ω is an open set and K>0K>0. In particular, structural conditions on F   are presented which ensure that W2,n(B1)W2,n(B1) solutions achieve the optimal C1,1(B1/2)C1,1(B1/2) regularity when f is Hölder continuous. Moreover, if f   is positive on B‾1, Lipschitz continuous, and {u≠0}⊂Ω{u≠0}⊂Ω, we obtain interior C1C1 regularity of the free boundary under a uniform thickness assumption on {u=0}{u=0}. Lastly, we extend these results to the parabolic setting.