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

TL;DR

This paper establishes optimal regularity results for solutions to a broad class of fully nonlinear parabolic free boundary problems, including regularity of the free boundary under certain conditions.

Contribution

It introduces a new BMO-type estimate for parabolic problems and proves optimal regularity for solutions and free boundaries in this class.

Findings

Solutions are locally $C_x^{1,1} ext{ and } C_t^{0,1}$ inside $Q_1$.

New BMO-type estimate extends previous results to the parabolic setting.

Regularity of the free boundary is shown under additional geometric conditions.

Abstract

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 $\{u = 0 \}$,