In this paper we consider the fully nonlinear parabolic free boundary problem $$ \left\{ \begin{ array}{ ll} F(D^2u) -\partial_t u=1 & \text{ a.e. in}Q_1 \cap \Omega\\ |D^2 u| + |\partial_t u| \leq K & \text{ a.e. in}Q_1\setminus\Omega, \end{ array} \right. $$ where $K>0$ is a positive constant, and $\Omega$ 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 $\partial\Omega\cap Q_1$ under the extra condition that $\Omega \supset \{ u \neq 0 \}$, and a uniform thickness assumption on the coincidence set $\{ u = 0 \}$,
Categories: math.AP