Antisymmetry of solutions for some weighted elliptic problems

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...

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Bibliographic Details
Published in:Communications in partial differential equations 2018-03, Vol.43 (3), p.506-547
Main Authors: Cabré, Xavier, Lucia, Marcello, Sanchón, Manel, Villegas, Salvador
Format: Article
Language:English
Subjects:
35 Partial differential equations
35B Qualitative properties of solutions
35J Partial differential equations of elliptic type
35Q Equations of mathematical physics and other areas of application
Antisymmetric solutions
Antisymmetry
bistable nonlinearity
Classificació AMS
continuous odd rearrangement
Convexity
Differential equations, Partial
Equacions diferencials parcials
Functionals
Matemàtiques i estadística
monotonicity
Uniqueness
weights
Àrees temàtiques de la UPC
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Summary: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.
ISSN:0360-5302
1532-4133
DOI:10.1080/03605302.2018.1446168