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.