Antisymmetry of solutions for some weighted elliptic problems
Abstract
This article concerns the antisymmetry, uniqueness, and monotonicity properties of solutions to some elliptic functionals involving weights and a double well potential. In the one-dimensional case, we introduce the continuous odd rearrangement of an increasing function and we show that it decreases the energy functional when the weights satisfy a certain convexity-type hypothesis. This leads to the antisymmetry or oddness of increasing solutions (and not only of minimizers). We also prove a uniqueness result (which leads to antisymmetry) where a convexity-type condition by Berestycki and Nirenberg on the weights is improved to a monotonicity condition. In addition, we provide with a large class of problems where antisymmetry does not hold. Finally, some rather partial extensions in higher dimensions are also given.
Keywords
Cite
@article{arxiv.1709.07656,
title = {Antisymmetry of solutions for some weighted elliptic problems},
author = {Xavier Cabre and Marcello Lucia and Manel Sanchon and Salvador Villegas},
journal= {arXiv preprint arXiv:1709.07656},
year = {2017}
}