Convergence Of The Solutions To The Allen-Cahn Type Equations With Neumann Boundary Condition

In this thesis, we study a singular limit problem on the following Allen-Cahn type equations with Neumann boundary condition where Ω(?)Rn is a bounded domain with smooth boundary,ε is a small positive parameter, v is the outer unit normal vector field on (?)Ω and W is a bi-stable potential with two equal wells at ±1, V(x)≥ a, a> 0 is a positive constant, and V ∈ C2(?), ▽V·v｜(?)Ω=0.Considering the above Allen-Cahn type equation with Neumann boundary con-ditions and general initial data of uniformly bounded energy, we prove that the limit of the time-parametrized family of energy measures is rectifiable and we derive a Brakke type inequality in the end. The main point of this paper is to generalize the result in [18] to more general singular limit problem under the appearance of the potential V.