An overview of unconstrained free boundary problems

Abstract

In this paper we present a survey concerning unconstrained free boundary problems of type $$ \left\{ \begin{array}{ll} F_1(D^2u,\nabla u,u,x)=0 & \text{in }B_1 \cap \Omega ,\\ F_2 (D^2 u,\nabla u,u,x)=0 & \text{in }B_1\setminus\Omega ,\\ u \in \mathbb{S}(B_1), \end{array} \right. $$ where $B_1$ is the unit ball, $\Omega$ is an unknown open set, $F_1, F_2$ are elliptic operators (admitting regular solutions), and $\mathbb{S}$ is a functions space to be specified in each case. Our main objective is to discuss a unifying approach to the optimal regularity of solutions to the above matching problems, and list several open problems in this direction.