An asymptotic monotonicity formula for minimizers of elliptic systems of Allen-Cahn type and the Liouville property
Texas State University, Department of Mathematics
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 Δu = Wu(u), x ∈ ℝn, n ≥ 2, with W : ℝ, → ℝ, m ≥ 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.
Entire solutions, Monotonicity formula, Allen-Cahn equation, Liouville theorem, Multi-phase transitions
Sourdis, C. (2021). An asymptotic monotonicity formula for minimizers of elliptic systems of Allen-Cahn type and the Liouville property. <i>Electronic Journal of Differential Equations, 2021</i>(04), pp. 1-11.