Fine structure of the two-phase Bernoulli free boundaries in 2D

TL;DR

This paper analyzes the structure of free boundaries in a 2D two-phase Bernoulli problem, revealing local finiteness of the branching set through geometric methods and linking it to capillary minimal surfaces.

Contribution

It introduces a Weierstrass representation formula to study free boundaries and connects the problem to capillary minimal surfaces and the linear thin two-membrane problem.

Findings

Branching set is locally finite in 2D Bernoulli problem.

Transformation to a geometric problem for capillary minimal surfaces.

Establishes a link between obstacle problem derivatives and the thin two-membrane problem.

Abstract

We prove that the branching set of a solution to a two-dimensional two-phase Bernoulli problem with constant coefficients is locally finite. We do this via a Weierstrass representation formula, which allows to transform the problem into a new geometric two-phase problem for capillary minimal surfaces. We also apply this method to the obstacle problem establishing a connection between the directional derivatives of solutions to the obstacle problem and the linear thin two-membrane problem.