One-dimensional symmetry for the solutions of a three-dimensional water wave problem

TL;DR

This paper proves that solutions to a weighted water wave problem in three dimensions are one-dimensional, meaning they depend on only one spatial variable, extending previous results from two dimensions.

Contribution

It establishes a one-dimensional symmetry result for weighted water wave problems in three dimensions, using energy estimates and comparison arguments.

Findings

Minimizers depend on only one Euclidean variable.

Bounded monotone solutions are one-dimensional.

Energy estimates are crucial for the proof.

Abstract

We prove a one-dimensional symmetry result for a weighted Dirichlet-to-Neumann problem arising in a model for water waves in dimension 3. More precisely we prove that minimizers and bounded monotone solutions depend on only one Euclidean variable. The analogue of this result for the 2-dimensional case (and without weights) was established in an article by De La Llave and the third author. In this paper, a crucial ingredient in the proof is given by an energy estimate for minimizers obtained via a comparison argument.