Plateau angle conditions for the vector-valued Allen-Cahn equation

TL;DR

This paper rigorously derives the Plateau angle conditions at triple junctions in three-dimensional diffused interfaces modeled by the vector-valued Allen-Cahn equation, using divergence theorem and stress tensor methods.

Contribution

It introduces a rigorous derivation of Plateau angle conditions for triple junctions from the vector-valued Allen-Cahn equation with a triple-well potential.

Findings

Derivation of Plateau angle conditions at triple junctions

Application of divergence theorem with stress tensor

Extension to three-dimensional diffused interfaces

Abstract

Under proper hypotheses, we rigorously derive the Plateau angle conditions at triple junctions of diffused interfaces in three dimensions, starting from the vector-valued Allen-Cahn equation with a triple-well potential. Our derivation is based on an application of the divergence theorem using the divergence-free form of the equation via an associated stress tensor.