Convergence Of Solutions Of The Weighted Allen-Cahn Equations On Non-convex Domains

To describe the limiting state and the motion of two phase boundaries,we study the singular limit problem of solutions to the following weighted parabolic Allen-Cahn type equations on bounded non-convex domains with smooth boundaries.(?)where ?(?)R~n is a bounded non-convex domain with smooth boundary,v is the outer unit normal vector field on(?)?,W indicates a bi-stable potential with two equal wells at ±1.We define the following energy measures(?)We may prove that varifolds Vt associated to the limit measures ?_t of energy measures ?_t~? are the generalized Brakke's mean curvature flow when the the initial data (?) satisfies some conditions.More precisely,we firstly obtain the convergence of energy measures ?t?.Furthermore,we prove that the limit measures (?) of energy measures ?t?are n-1-rectifiable Radon measures.At the end,we define a normalized varifold Vt,and obtain a Brakke's type inequality with the mean curvature vector h_b=(?).