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

2017/10/03 by Cinti, Eleonora, Miraglio, Pietro, Valdinoci, Enrico
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1710.01137

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.

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