On the existence of non-flat profiles for a Bernoulli free boundary problem

TL;DR

This paper demonstrates the existence of non-flat solutions in Bernoulli free boundary problems with mixed boundary conditions by identifying them as global minimizers of the Alt-Caffarelli energy functional.

Contribution

It establishes the variational existence of non-flat profiles for a broad class of Bernoulli free boundary problems with mixed boundary conditions.

Findings

Non-flat solutions exist as global minimizers.

Solutions are found variationally via the Alt-Caffarelli energy functional.

Applicable to a large class of Bernoulli-type problems.

Abstract

In this paper we consider a large class of Bernoulli-type free boundary problems with mixed periodic-Dirichlet boundary conditions. We show that solutions with non-flat profile can be found variationally as global minimizers of the classical Alt-Caffarelli energy functional.