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

Eleonora Cinti, Pietro Miraglio, Enrico Valdinoci

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


We prove a one-dimensional symmetry result for a weighted Dirichlet-to-Neumann problem arising in a model for water waves in dimensions 2 and 3. More precisely, we prove that stable solutions in dimension 2 and minimizers and monotone solutions in dimension 3 depend on only one Euclidean variable. Monotone solutions in the 2-dimensional case without weights were studied in de la Llave and Valdinoci (Math Res Lett 16(5):909–918, 2009). In our paper, a crucial ingredient in the proof is given by an energy estimate for minimizers obtained via a comparison argument.

Original languageEnglish
Pages (from-to)1804-1835
Number of pages32
JournalJournal of Geometric Analysis
Issue number2
Early online date17 Sep 2019
Publication statusPublished - 1 Apr 2020


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