An overview of unconstrained free boundary problems

TL;DR

This survey reviews unconstrained free boundary problems involving elliptic operators, focusing on solution regularity and highlighting open research questions in the field.

Contribution

It provides a unifying framework for understanding solution regularity in free boundary problems and discusses open problems for future research.

Findings

Unified approach to solution regularity

Identification of open problems in free boundary regularity

Discussion of elliptic operator conditions

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.