A simple proof of the optimal power in Liouville theorems
Villegas Barranco, Salvador (Universidad de Granada. Departamento de Análisis Matemático)

Date:	2022
Abstract:	Consider the equation div(ϕ2∇σ) = 0 in RN , where ϕ > 0. It is well known [4, 2] that if there exists C > 0 such that R BR (ϕσ) 2 dx ≤ CR2 for every R ≥ 1, then σ is necessarily constant. In this paper we present a simple proof that this result is not true if we replace R2 with Rk for k > 2 in any dimension N. This question is related to a conjecture by De Giorgi.
Grants:	Ministerio de Ciencia e Innovación PGC2018-096422-B-I00
Note:	The author has been supported by Ministerio de Ciencia, Innovación y Universidades of Spain PGC2018-096422-B-I00 and by Junta de Andalucía A-FQM-187-UGR18 and P18-FR-667.
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió publicada
Subject:	Allen-cahn equation ; Liouville theorems ; Dirichlet and potential energies
Published in:	Publicacions matemàtiques, Vol. 66 Núm. 2 (2022) , p. 883-892 (Articles) , ISSN 2014-4350

Adreça original: https://raco.cat/index.php/PublicacionsMatematiques/article/view/402275
DOI: 10.5565/PUBLMAT6622212

10 p, 311.3 KB

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