A general class of free boundary problems for fully nonlinear parabolic equations

In this paper we consider the fully nonlinear parabolic free boundary problem $$\{\begin{array}{ll}F(D^{2}u)-{\mathrm{\partial }}_{t}u=1& \text{a.e. in}{Q}_1\cap \mathrm{\Omega }\\ \mathrm{\mid }{D}^{2}u\mathrm{\mid }+\mathrm{\mid }{\mathrm{\partial }}_{t}u\mathrm{\mid }\le K& \text{a.e. in}{Q}_1\setminus \mathrm{\Omega },\end{array}$$ where $K>0$ is a positive constant, and $\mathrm{\Omega }$ is an (unknown) open set. Our main result is the optimal regularity for solutions to this problem: namely, we prove that $W_x^{2,n}\cap W_t^{1,n}$ solutions are locally $C_x^{1,1}\cap C_t^{0,1}$ inside $Q_1$. A key starting point for this result is a new BMO-type estimate which extends to the parabolic setting the main result in \cite{CH}. Once optimal regularity for $u$ is obtained, we also show regularity for the free boundary $\mathrm{\partial }\mathrm{\Omega }\cap Q_1$ under the extra condition that $\Omega \supset {u \neq 0 }$, and a uniform thickness assumption on the coincidence set $\{u=0\}$,