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In this paper, we consider the sharp interface limit of a matrix-valuedAllen-Cahn equation, which takes the form: $$\partial_t A=\DeltaA-\varepsilon^{-2}( A A^{\mathrm{T}}A-A)~~\text{with}~~A:\Omega\subset\mathbb{R}^m\to\mathbb{R}^{n\times n}.$$ Weshow that the sharp interface system is a two-phases flow system: the interfaceevolves according to the motion by mean curvature; in the two bulk phaseregions, the solution obeys the heat flow of harmonic maps with values in$O^+(n)$ and $O^-(n)$ (represent the sets of $n\times n$ orthogonal matriceswith determinant $+1$ and $-1$ respectively); on the interface, the phasematrices in two sides satisfy a novel mixed boundary condition.  The above result provides a solution to the Keller-Rubinstein-Sternberg's(conjecture) problem in this $O(n)$ setting. Our proof relies on two keyingredients. First, in order to construct the approximate solution by matchedasymptotic expansions as the standard approach does not seem to work, weintroduce the notion of quasi-minimal connecting orbits. They satisfy the usualleading order equations up to some small higher order terms. In addition, thelinearized systems around these quasi-minimal orbits needs to be solvable up tosome good remainders. These flexibilities are needed for the possible"degenerations" or higher dimensional kernels for the linearized operators dueto intriguing boundary conditions at the sharp interface. The second key pointis to establish a spectral uniform lower bound estimate for the linearizedoperator around approximate solutions. To this end, we introduce additionaldecompositions to reduce the problem into the coercive estimates of severallinearized operators for scalar functions and some singular product estimateswhich is accomplished by exploring special cancellation structures betweeneigenfunctions of these linearized operators.

Matrix-valued Allen-Cahn equation and the Keller-Rubinstein-Sternberg problem