On nonminimizing solutions of elliptic free boundary problems

TL;DR

This paper develops a variational approach to analyze solutions of elliptic free boundary problems that are not energy minimizers, revealing existence, regularity, and boundary properties in various dimensions.

Contribution

It introduces a framework for nonminimizing solutions, including mountain pass solutions, and proves regularity results for free boundaries in different dimensions.

Findings

Existence of mountain pass solutions for superlinear free boundary problems.

Full regularity of free boundaries in two dimensions.

Partial regularity results in higher dimensions.

Abstract

We present a variational framework for studying the existence and regularity of solutions to elliptic free boundary problems that do not necessarily minimize energy. As applications, we obtain mountain pass solutions of critical and subcritical superlinear free boundary problems, and establish full regularity of the free boundary in dimension $N = 2$ and partial regularity in higher dimensions.