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

Abstract

In this paper, we consider the fully nonlinear parabolic free boundary problem $$\begin{aligned} \left\{ \begin{array}{ll} F(D^2u) -\partial _{t} u=1 &{} \text {a.e. in }Q_1 \cap \Omega |D^2 u| + |\partial _{t} u| \le K &{} \text {a.e. in }Q_1{\setminus }\Omega , \end{array} \right. \end{aligned}$$ 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 Caffarelli and Huang (Duke Math J 118(1):1–17, 2003). 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 $$\Omega \supset \{ u \ne 0 \}$$ , and a uniform thickness assumption on the coincidence set $$\{ u = 0 \}$$ .