Sharp interface limit of a matrix-valued Allen-Cahn equation

In this talk we consider the sharp interface limit of a matrix-valued Allen-Cahn equation. We show that the sharp interface limit system is a two-phases flow system: the interface evolves according to the motion by mean curvature; in the two bulk phase regions, the solution obeys the heat flow of harmonic maps with values in the sets of nˆn orthogonal matrices with determinant 1 and -1 respectively; on the interface, the phase matrices in two sides satisfy a novel mixed boundary condition. The above result provides a solution to the Keller-Rubinstein-Sternberg’s (conjecture) problem in the orthogonal matrix setting. This is a joint work with Prof. Fanghua Lin, Prof. Wei Wang and Prof. Zhifei Zhang.