A simple proof of the optimal power in Liouville theorems

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Salvador Villegas Barranco
Universidad de Granada. Departamento de Análisis Matemático

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.

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Allen–Cahn equation, Liouville theorems, Dirichlet and potential energies

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Villegas Barranco, Salvador. «A simple proof of the optimal power in Liouville theorems». Publicacions Matemàtiques, 2022, vol.VOL 66, núm. 2, p. 883–892, https://raco.cat/index.php/PublicacionsMatematiques/article/view/402275.
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