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

TL;DR

This paper analyzes the vector-valued Allen-Cahn equation with high-dimensional double-well potentials under Robin boundary conditions, proving convergence to mean curvature flow with a fixed contact angle and deriving the sharp-interface limit.

Contribution

It extends previous analyses by establishing convergence and deriving the sharp-interface system for the vector-valued case with boundary effects.

Findings

Proves local-in-time convergence to mean curvature flow with contact angle.

Derives the limiting sharp-interface system with harmonic heat flows and minimal pair conditions.

Extends prior work to include boundary effects in the vector-valued setting.

Abstract

This work investigates the vector-valued Allen-Cahn equation with potentials of high-dimensional double-wells under Robin boundary conditions. We establish local-in-time convergence of solutions to mean curvature flow with a fixed contact angle $0<\alpha\leq 90^\circ$, for a broad class of boundary energy densities and well-prepared initial data. The limiting sharp-interface system is derived, comprising harmonic heat flows in the bulk and minimal pair conditions at phase boundaries. The analysis combines the relative entropy method with gradient flow calibrations and weak convergence techniques. These results extend prior works on the analysis of the vector-valued case without boundary effects (Comm. Pure Appl. Math., 78:1199-1247, 2025) and the scalar-valued case with boundary contact energy (Calc. Var. Partial Differ. Equ., 61:201, 2022).