On the growth of the energy of entire solutions to the vector Allen-Cahn equation

2014/04/15 by Christos Sourdis, Sourdis, Christos
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Meromorphic and Entire Functions #Nonlinear Partial Differential Equations #Stochastic processes and statistical mechanics #math.AP

paper · pdf · doi:10.48550/arxiv.1404.3904

Improved the main result using the same method

openalex publication_date 2014/04/15 · arxiv created 2014/04/25 · arxiv updated 2014/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the energy over balls of entire, nonconstant, bounded solutions to the vector Allen-Cahn equation grows faster than (ln R)^k R^n-2, for any k>0, as the volume Rⁿ of the ball tends to infinity. This improves the growth rate of order R^n-2 that follows from the general weak monotonicity formula. Moreover, our estimate can be considered as an approximation to the corresponding rate of order R^n-1 that is known to hold in the scalar case.

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