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arXiv · 1709.07656

Antisymmetry of solutions for some weighted elliptic problems

Xavier Cabre·Marcello Lucia·Manel Sanchon·Salvador Villegas
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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.

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Xavier Cabre, Marcello Lucia, Manel Sanchon, Salvador Villegas. 2017-09-22. Antisymmetry of solutions for some weighted elliptic problems. https://arxiv.org/abs/1709.07656

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