A simple proof of the optimal power in Liouville theorems

2020/03/09 by Salvador Villegas, Villegas, Salvador
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2003.04400

openalex publication_date 2020/03/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider the equation div(φ² ∇ σ)=0 in ℝ^N, where φ>0. It is well-known that if there exists C>0 such that ∫_BR(φσ)² dx≤ CR² for every R≥ 1 then σ is necessarily constant. In this paper we prove that this result is not true if we replace R² by R^k for k>2 in any dimension N. This question is related to a conjecture by De Giorgi.

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