Minimizers of the Allen-Cahn energy with sub-quadratic growth

2025/03/03 by Savin, Ovidiu, Zhang, Chilin
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.02029

Abstract

We establish Liouville theorems for global minimizers u of the Allen-Cahn energy ∫ |∇ u|² + W(u) dx, which have subquadratic growth at infinity. In particular we extend the results of \citeS1,S3 concerning the De Giorgi's conjecture to the setting of unbounded solutions. Part of the analysis relies on the regularity of minimizers for a Dirichlet/perimeter functional which was studied by Athanasopoulous-Caffarelli-Kenig-Salsa in \citeACKS.

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