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

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 $\Delta u=\nabla W(u)$, $x\in {\mathbb{R}}^n$, $n\ge 2$, with $W:{\mathbb{R}}^m\to \mathbb{R}$, $m\ge 1$, 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.

References in corpus (3)

• The heteroclinic connection problem for general double-well potentials
• Optimal energy growth lower bounds for a class of solutions to the vectorial Allen-Cahn equation
• On the confinement of bounded entire solutions to a class of semilinear elliptic systems