Hyperbolic solutions to Bernoulli's free boundary problem

TL;DR

This paper investigates hyperbolic solutions to Bernoulli's free boundary problem, introducing a new analytical approach to establish the existence of both hyperbolic and elliptic solutions within the same framework.

Contribution

It develops a novel implicit function theorem based on parabolic maximal regularity and analyzes spectral properties to prove existence of hyperbolic solutions.

Findings

Existence of foliated hyperbolic solutions

Existence of elliptic solutions in the same regularity class

New analytical methods for unstable solutions

Abstract

Bernoulli's free boundary problem is an overdetermined problem in which one seeks an annular domain such that the capacitary potential satisfies an extra boundary condition. There exist two different types of solutions called elliptic and hyperbolic solutions. Elliptic solutions are ``stable'' solutions and tractable by variational methods and maximum principles, while hyperbolic solutions are ``unstable'' solutions of which the qualitative behavior is less known. We introduce a new implicit function theorem based on the parabolic maximal regularity, which is applicable to problems with loss of derivatives. Clarifying the spectral structure of the corresponding linearized operator by harmonic analysis, we prove the existence of foliated hyperbolic solutions as well as elliptic solutions in the same regularity class.