Phase transition of an anisotropic Ginzburg-Landau equation

We study the effective geometric motions of an anisotropic Ginzburg-Landau equation with asmall parameter epsilon>0 which characterizes the width of the transition layer. For well-preparedinitial datum, we show that as epsilon tends to zero the solutions will develop a sharp interface limitwhich evolves under mean curvature flow. The bulk limits of the solutions correspond to avector fieldu(x,t)which is of unit length on one side of the interface, and is zero on the otherside. The proof combines the modulated energy method and weak convergence methods. Inparticular, by a (boundary) blow-up argument we show thatumust be tangent to the sharpinterface. Moreover, it solves a geometric evolution equation for the Oseen-Frank model inliquid crystals