Energy estimates for minimizers to a class of elliptic systems of Allen-Cahn type and the Liouville property

TL;DR

This paper establishes energy growth estimates for bounded, globally minimizing solutions to certain elliptic systems of Allen-Cahn type, leading to Liouville theorems that classify solutions under specific conditions.

Contribution

It provides new energy growth estimates for minimizers of elliptic systems with potentials vanishing at a single point, extending Liouville theorems in this context.

Findings

Energy growth bounds for solutions

Liouville type classification results

Conditions under which solutions are trivial or constant

Abstract

We prove a theorem for the growth of the energy of bounded, globally minimizing solutions to a class of semilinear elliptic systems of the form $\Delta u=\nabla W(u)$, $x\in \mathbb{R}^n$, $n\geq 2$, with $W:\mathbb{R}^m\to \mathbb{R}$, $m\geq 1$, nonnegative and vanishing at exactly one point (at least in the closure of the image of the considered solution $u$). As an application, we can prove a Liouville type theorem under various assumptions.