Existence and regularity of higher critical points for elliptic free boundary problems

TL;DR

This paper studies the existence and regularity of higher critical points in elliptic free boundary problems, introducing a new approach to handle nondifferentiable energy functionals related to plasma confinement.

Contribution

It develops a variational framework to find mountain pass type solutions for nondifferentiable problems and proves their regularity and free boundary smoothness.

Findings

Existence of nontrivial generalized solutions of mountain pass type.

Solutions are nondegenerate and satisfy free boundary conditions in viscosity sense.

Free boundary is smooth near points with measure-theoretic normal.

Abstract

Existence and regularity of minimizers in elliptic free boundary problems have been extensively studied in the literature. We initiate the corresponding study of higher critical points by considering a superlinear free boundary problem related to plasma confinement. The associated energy functional is nondifferentiable, and therefore standard variational methods cannot be used directly to prove the existence of critical points. Here we obtain a nontrivial generalized solution $u$ of mountain pass type as the limit of mountain pass points of a suitable sequence of $C^1$-functionals approximating the energy. We show that $u$ minimizes the energy on the associated Nehari manifold and use this fact to prove that it is nondegenerate. We use the nondegeneracy of $u$ to show that it satisfies the free boundary condition in the viscosity sense. Moreover, near any free boundary point that has a measure-theoretic normal, the free boundary is a smooth surface, and hence the free boundary condition holds in the classical sense.