Matrix-valued Allen–Cahn equation and the Keller–Rubinstein–Sternberg problem
نویسندگان
چکیده
In this paper, we consider the sharp interface limit of a matrix-valued Allen–Cahn equation, which takes form: $$\begin{aligned} \partial _t{\textbf{A}}=\Delta {\textbf{A}}-\varepsilon ^{-2}({\textbf{A}}{\textbf{A}}^{\textrm{T}}{\textbf{A}}-{\textbf{A}})\quad \text {with}\quad {\textbf{A}}:\Omega \subset {\mathbb {R}}^m\rightarrow {R}}^{n\times n}. \end{aligned}$$ We show that system is two-phases flow system: evolves according to motion by mean curvature; in two bulk phase regions, solution obeys heat harmonic maps with values $$O^+(n)$$ and $$O^-(n)$$ (represent sets $$n\times n$$ orthogonal matrices determinant $$+1$$ $$-1$$ respectively); on interface, sides satisfy novel mixed boundary condition. The above result provides Keller–Rubinstein–Sternberg’s problem O(n) setting. Our proof relies key ingredients. First, order construct approximate solutions matched asymptotic expansions, as standard approach does not seem work, introduce notion quasi-minimal connecting orbits. They usual leading equations up some small higher terms. addition, linearized systems around these orbits needs be solvable good remainders. These flexibilities are needed for possible “degenerations” dimensional kernels operators functions due intriguing conditions at interface. second point establish spectral uniform lower bound estimate operator solutions. To end, additional decompositions reduce into coercive estimates several scalar singular product accomplished exploring special cancellation structures between eigenfunctions operators.
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ژورنال
عنوان ژورنال: Inventiones Mathematicae
سال: 2023
ISSN: ['0020-9910', '1432-1297']
DOI: https://doi.org/10.1007/s00222-023-01183-8