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arXiv · 2503.02029

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

Ovidiu Savin·Chilin Zhang
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Abstract

We establish Liouville theorems for global minimizers $u$ of the Allen-Cahn energy $$\int |\nabla u|^2 + W(u) \, dx,$$ which have subquadratic growth at infinity. In particular we extend the results of \cite{S1,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 \cite{ACKS}.

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Ovidiu Savin, Chilin Zhang. 2025-03-03. Minimizers of the Allen-Cahn energy with sub-quadratic growth. https://arxiv.org/abs/2503.02029

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