Free boundary regularity for semilinear variational problems with a topological constraint

TL;DR

This paper investigates the regularity of free boundaries in semilinear variational problems with topological constraints, establishing optimal Lipschitz regularity and analyzing singularities related to minimal capacity and spectral partition problems.

Contribution

It introduces new regularity results for free boundaries under topological constraints, including optimal Lipschitz regularity and analysis of singularities in related variational problems.

Findings

Minimizers exist for the studied free boundary problems.

Minimizers exhibit optimal Lipschitz regularity.

Free boundaries are analytic away from a codimension two singular set.

Abstract

We study a class of semilinear free boundary problems in which admissible functions $u$ have a topological constraint, or spanning condition, on their 1-level set. This constraint forces $\{u=1\}$, which is the free boundary, to behave like a surface with some special types of singularities attached to a fixed boundary frame, in the spirit of the Plateau problem \cite{HP16}. Two such free boundary problems are the minimization of capacity among surfaces sharing a common boundary and an Allen-Cahn formulation of the Plateau problem. We establish the existence of minimizers and study their regularity properties, obtaining the optimal Lipschitz regularity of minimizers and analytic regularity for the free boundaries away from a codimension two singular set. The singularity models for these problems are given by conical critical points of the minimal capacity problem, which are closely related to spectral optimal partition and segregation problems.