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

2026/02/28 by Xingyu Wang
Mathematics · #math.AP

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arxiv created 2026/08/01 · arxiv updated 2026/08/04

Abstract

We study the sharp-interface limit of a matrix-valued Allen--Cahn equation with the Saint Venant--Kirchhoff potential F(A)=\frac14‖AA^\top-I‖² . The zero set of this potential is the orthogonal group \mathbbO_n=\mathbbO_n⁺∪\mathbbO_n^-, and the corresponding limiting problem combines mean-curvature motion of the interface with harmonic-map heat flow in the two bulk phases. The proof combines a modulated-energy argument with compactness estimates obtained from two skew-symmetric commutator formulations of the equation. The method avoids the spectral analysis of linearized operators around quasi-minimal connecting orbits and the construction of high-order matched asymptotic expansions. In particular, the limiting maps satisfy the minimal-pair condition on the moving interface and the weak transmission identities which, for smooth limits, are equivalent to the Neumann-type jump condition of the sharp-interface system.

Citations

• Existence of weak solutions for two-phase matrix-valued harmonic map flows
• Phase Transition of Parabolic Ginzburg--Landau Equation with Potentials of High-Dimensional Wells
• Phase transition of anisotropic Ginzburg--Landau equation
• Matrix-valued Allen-Cahn equation and the Keller-Rubinstein-Sternberg problem
• Convergence rates of the Allen-Cahn equation to mean curvature flow: A\n short proof based on relative entropies
• Interface dynamics for an Allen-Cahn-type equation governing a matrix-valued field
• Harmonic maps in connection of phase transitions with higher dimensional potential wells
• Dynamics of the Nematic-Isotropic sharp interface for the liquid crystal
• On the motion of a curve by its binormal curvature