Electronic Journal of Differential Equations, 
Vol. 2021 (2021), No. 04, pp. 1-11.

Title: An asymptotic monotonicity formula for minimizers of
       elliptic systems of Allen-Cahn type and the Liouville property
       
Authors: Christos Sourdis (National and Kapodistrian Univ. of Athens, Greece)

Abstract:
 We prove an asymptotic monotonicity formula for bounded, globally
 minimizing solutions (in the sense of Morse) to a class of semilinear elliptic systems of
 the form $\Delta u= W_u(u)$, $x\in \mathbb{R}^n$, $n\geq 2$, 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.

Submitted January 21, 2019. Published January 20, 2021.
Math Subject Classifications: 35J48, 35J20, 35J61.
Key Words: Entire solutions; monotonicity formula; Allen-Cahn equation;
           Liouville theorem; multi-phase transitions.