Abstract
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 language | English |
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Pages (from-to) | 1804-1835 |
Number of pages | 32 |
Journal | Journal of Geometric Analysis |
Volume | 30 |
Issue number | 2 |
Early online date | 17 Sep 2019 |
DOIs | |
Publication status | Published - 1 Apr 2020 |