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

Abstract

We consider fully nonlinear obstacle-type problems of the form \begin{equation*} \begin{cases} F(D^{2}u,x)=f(x) & \text{a.e. in}B_{1}\cap\Omega,|D^{2}u|\le K & \text{a.e. in}B_{1}\backslash\Omega, \end{cases} \end{equation*} where $\Omega$ is an unknown 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 H\"older continuous. Moreover, if $f$ is positive on $\overline B_1$, Lipschitz continuous, and $\{u\neq 0\} \subset \Omega$, then we obtain local $C^1$ regularity of the free boundary under a uniform thickness assumption on $\{u=0\}$. Lastly, we extend these results to the parabolic setting.