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

[摘要] We prove an asymptotic monotonicity formula for bounded, globallyminimizing solutions (in the sense of Morse) to a class of semilinear elliptic systems ofthe form,,,with,,nonnegative andvanishing at exactly one point (at least in the closure of theimage of the considered solution ). As an application, we canprove a Liouville type theorem under various assumptions.

[学科分类] 数学（综合）

[关键词] Entire solutions;monotonicity formula;Allen-Cahn equation;Liouville theorem;multi-phase transitions.