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

TL;DR

This work extends the De Giorgi-type rigidity for the Allen-Cahn energy to global minimizers with subquadratic growth, proving that in dimensions $n\le7$ such minimizers are one-dimensional, and in the monotone graphical setting the conclusion holds up to $n\le8$. The authors connect blow-down limits to the ACKS Dirichlet/perimeter functional, establishing Gamma-convergence and compactness to a two-phase limit problem, and develop a suite of tools—density estimates, optimal growth bounds, asymptotic flatness, and improvement-of-flatness—to bootstrap 1D symmetry from large scales to small scales. These results extend the De Giorgi conjecture-type rigidity to unbounded growth and provide a framework for potential extensions to other tail potentials and to Alt-Phillips-type limits in the subquadratic regime. The approach combines barrier constructions, barrier continuity, and a robust blow-up/blow-down analysis to reveal that global minimizers with subquadratic growth exhibit highly rigid, one-dimensional behavior in low dimensions, with an extra dimension allowed in the graphical/monotone case.

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}.

Theorems & Definitions (51)

• Theorem 1.1: S1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4: ACF monotonicity formula
• Lemma 2.5
• proof
• Theorem 2.6: ACKS