The sharp interface limit of the matrix-valued Allen-Cahn equation

TL;DR

This paper investigates the sharp interface limit of a matrix-valued Allen-Cahn equation with a specific potential, employing novel energy and convergence methods to analyze phase transitions and construct weak solutions.

Contribution

It introduces a new approach avoiding spectrum analysis, relaxes initial data assumptions, and constructs weak solutions for the limiting harmonic heat flow system.

Findings

Established the sharp interface limit for the matrix-valued Allen-Cahn equation.

Developed a method that bypasses spectrum analysis of linearized operators.

Constructed weak solutions to the harmonic heat flow system with boundary conditions.

Abstract

In this work, we study a matrix-valued Allen-Cahn equation with a Saint Venant-Kirchhoff potential $F(\mathbf{A})=\frac{1}{4}\|\mathbf{A}\mathbf{A}^\top-\mathbf{I}\|^2$. Our approach employs the modulated energy method together with weak convergence methods for nonlinear partial differential equations. This avoids the subtle spectrum analysis of the linearized operator at the so-called quasi-minimal orbits as well as the construction of asymptotic expansion. Moreover, it relaxes the assumption on the admissible initial data, which exhibits a phase transition along an initial interface. As a byproduct, we construct a weak solution to the limiting harmonic heat flow system with both minimal pair and Neumann-type boundary conditions across the interface.