Calculus of Variations and Geometric Measure Theory

E. Cinti - P. Miraglio - E. Valdinoci

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

created by cinti on 04 Oct 2017
modified on 27 Mar 2019


Accepted Paper

Inserted: 4 oct 2017
Last Updated: 27 mar 2019

Journal: J. Geom. Anal.
Year: 2017


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.