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

Abstract

We study weak solutions for a class of free-boundary problems which includes as a special case the classical problem of travelling gravity 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. © 2008 The Royal Society of Edinburgh.