Bernoulli free-boundary problems in strip-like domains and a property of permanent waves in water of finite depth

TL;DR

This paper investigates free boundary problems related to water waves of finite depth, establishing equivalences, regularity results, and conditions under which the free boundary forms a graph.

Contribution

It introduces a framework for analyzing weak solutions of free boundary problems in water wave models and proves the free boundary is generally a graph.

Findings

Free boundary problems are equivalent to fixed domain problems.

Solutions exhibit regularity properties.

Free boundary is often a graph of a function.

Abstract

We study weak solutions for a class of free boundary problems which includes as a special case the classical problem of traveling waves on water of finite depth. We show that such problems are equivalent to problems in fixed domains and study the regularity of their solutions. We also prove that in very general situations the free boundary is necessarily the graph of a function.