Gradient flow of phase transitions with fixed contact angle

2024/11/27 by Kobe Marshall-Stevens, Marshall-Stevens, Kobe, Yoshihiro Tonegawa +4 · 1 citation
Computer Science · #35K20 #49Q20 #53E10 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2411.17979

openalex publication_date 2024/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the gradient flow of the Allen-Cahn equation with fixed boundary contact angle in Euclidean domains for initial data with bounded energy. Under general assumptions, we establish both interior and boundary convergence properties for the solutions and associated energy measures. Under various boundary non-concentration assumptions, we show that, for almost every time, the associated limiting varifolds satisfy generalised contact angle conditions and have bounded first variation, as well as deducing that the trace of the limit of the solutions coincides with the limit of their traces. Moreover, we derive an Ilmanen type monotonicity formula, for initial data with bounded energy, valid for the associated energy measures up to the boundary.

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• The vector-valued Allen-Cahn equation with potentials of high-dimensional double-wells under Robin boundary conditions

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