Electron. J. Differential Equations, Vol. 2021 (2021), No. 04, pp. 1-11.

An asymptotic monotonicity formula for minimizers of elliptic systems of Allen-Cahn type and the Liouville property

Christos Sourdis

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.

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Christos Sourdis
Department of Mathematics
National and Kapodistrian University of Athens
Athens, Greece
email: sourdis@uoc.gr

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