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

Abstract

We study the sharp-interface limit of a matrix-valued Allen--Cahn equation with the Saint Venant--Kirchhoff potential \[ F(\mathbf{A})=\frac14\|{\mathbf{A}\mathbf{A}^\top-\mathbf{I}}\|^2 . \] The zero set of this potential is the orthogonal group ${\mathbb{O}}_n={\mathbb{O}}_n^{+}\cup {\mathbb{O}}_n^{-}$, 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.

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