An overview of unconstrained free boundary problems

Abstract

In this paper we present a survey concerning unconstrained free boundary problems of type $$\{\begin{array}{ll}{F}_1(D^{2}u,\nabla u,u,x)=0& \text{in }{B}_1\cap \Omega ,\\ F_2(D^{2}u,\nabla u,u,x)=0& \text{in }{B}_1\setminus \Omega ,\\ u\in \mathbb{S}(B_1),& \end{array}$$ 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.

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