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 $$\{\begin{array}{ll}F(D^{2}u)-{\partial }_{t}u=1& \text{a.e. in}{Q}_1\cap \Omega \\ |D^{2}u|+|{\partial }_{t}u|\le K& \text{a.e. in}{Q}_1\setminus \Omega ,\end{array}$$ 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 $\Omega \supset \{u\ne 0\}$, and a uniform thickness assumption on the coincidence set $\{u=0\}$,

arXiv admin note: text overlap with arXiv:1212.5809

Cited by in corpus (5)

• Regularity of solutions to fully nonlinear elliptic and parabolic free boundary problems
• Boundary regularity and non-transversal intersection for the fully nonlinear obstacle problem
• Sharp Hessian estimates for fully nonlinear elliptic equations under relaxed convexity assumptions, oblique boundary conditions and applications
• Non-transversal intersection of free and fixed boundary for fully nonlinear elliptic operators in two dimensions
• Optimal Regularity of Solutions to No-Sign Obstacle-Type Problems for the Sub-Laplacian