One-Dimensional Symmetry for the Solutions of a Three-Dimensional Water Wave Problem

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 16. In this paper a crucial ingredient in the proof is given by an energy estimate for minimizers obtained via a comparison argument.