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

Abstract

We consider fully nonlinear obstacle-type problems of the form

where Ω is an open set and $K>0$. In particular, structural conditions on F are presented which ensure that $W^{2,n}(B_1)$ solutions achieve the optimal $C^{1,1}(B_{1/2})$ regularity when f is Hölder continuous. Moreover, if f is positive on ${\overline{B}}_1$, Lipschitz continuous, and $\{u\ne 0\}\subset \mathrm{\Omega }$, 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.

Highlights

• We show that $W^{2,n}$ solutions to our type of problems are $C^{1,1}$.

• The free boundary is shown to be $C^1$ under additional natural assumptions.

• The results above are extended to the parabolic setting.