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

We establish Liouville theorems for global minimizers $u$ of the Allen-Cahn energy $$\int \mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2+W(u)dx,$$ 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}.