Higher critical points in a free boundary problem

TL;DR

This paper investigates higher critical points of a variational functional in a free boundary problem related to plasma confinement, establishing existence, regularity, and smoothness of solutions, especially in two dimensions.

Contribution

It introduces a novel approach to find nontrivial mountain pass critical points in non-smooth functionals and proves regularity results for the free boundary.

Findings

Existence of a nontrivial mountain pass critical point

Lipschitz continuity and nondegeneracy of solutions

Smoothness of the free boundary in 2D and partial regularity in higher dimensions

Abstract

We study higher critical points of the variational functional associated with a free boundary problem related to plasma confinement. Existence and regularity of minimizers in elliptic free boundary problems have already been studied extensively. But because the functionals are not smooth, standard variational methods cannot be used directly to prove the existence of higher critical points. Here we find a nontrivial critical point of mountain pass type and prove many of the same estimates known for minimizers, including Lipschitz continuity and nondegeneracy. We then show that the free boundary is smooth in dimension 2 and prove partial regularity in higher dimensions.