Branch Points for (Almost-)Minimizers of Two-Phase Free Boundary Problems

TL;DR

This paper investigates the existence and structure of branch points in two-phase free boundary problems, constructing minimizers with interior branch points and contrasting their structure in almost-minimizers.

Contribution

It introduces a family of minimizers with interior branch points and demonstrates that almost-minimizers can have very limited branch point structure, contrasting recent findings.

Findings

Constructed minimizers with interior branch points.

Showed almost-minimizers can have minimal branch point structure.

Contrasted with recent results on stationary solutions.

Abstract

We study the existence and structure of branch points in two-phase free boundary problems. More precisely, we construct a family of minimizers to an Alt- Caffarelli-Friedman type functional whose free boundaries contain branch points in the strict interior of the domain. We also give an example showing that branch points in the free boundary of almost-minimizers of the same functional can have very little structure. This last example stands in contrast with recent results of De Philippis- Spolaor-Velichkov on the structure of branch points in the free boundary of stationary solutions.