Structure of one-phase free boundaries in the plane

TL;DR

This paper analyzes the structure of one-phase free boundaries in the plane, showing that when two boundary components are close, the solution resembles a known double hairpin shape, linking free boundary problems to minimal surface theory.

Contribution

It establishes a local resemblance between free boundary solutions and the double hairpin minimal surface, extending understanding of free boundary configurations.

Findings

Solutions near close boundary components resemble the double hairpin shape.

Theorems are analogous to those characterizing minimal annuli.

Connections are made via Traizet's correspondence.

Abstract

We study classical solutions to the one-phase free boundary problem in which the free boundary consists of smooth curves and the components of the positive phase are simply-connected. We show that if two components of the free boundary are close, then the solution locally resembles an entire solution discovered by Hauswirth, H\'elein and Pacard, whose free boundary has the shape of a double hairpin. Our results are analogous to theorems of Colding and Minicozzi characterizing embedded minimal annuli, and a direct connection between our theorems and theirs can be made using a correspondence due to Traizet.