Convergence of the Allen-Cahn equation with Neumann boundary conditions
  We study a singular limit problem of the Allen-Cahn equation with Neumann boundary conditions and general initial data of uniformly bounded energy. We prove that the time-parametrized family of limit energy measures is Brakke's mean curvature flow with a generalized right angle condition on the boundary. 
Masashi Mizuno, Yoshihiro Tonegawa
math.AP math.DG