INTERIOR INTERFACES WITH (OR WITHOUT) BOUNDARY INTERSECTION FOR AN ANISOTROPIC ALLEN-CAHN EQUATION

We consider the nonlinear problem of anisotropic Allen-Cahn equa-tion epsilon 2div(Va(y)u) +P(y)u(1 - u2) = 0 in ohm, Va(y)u center dot nu = 0 on partial differential ohm, where ohm is a bounded domain in R2 with smooth boundary, epsilon is a small positive parameter, nu denotes the unit outward normal of partial differential ohm, and P(y) is a uniformly positive smooth potential on ohm over bar . The operator Va(y)u is defined by Va(y)u = (a1(y)uy1 , a2(y)uy2) with a(y) = (a1(y), a2(y)), where a1(y) and a2(y) are two positive smooth functions on ohm over bar . Let Gamma be an interior curve intersecting orthogonally with partial differential ohm at exactly two points or a closed simple curve in ohm, and dividing ohm into two parts. Moreover, Gamma is a non-degenerate geodesic embedded in the Riemannian manifold R2 associated with metric P(y)(a2(y)dy1 (R) dy1 + a1(y)dy2 (R) dy2). By assuming some additional constraints on the functions a(y), P(y) and the curves Gamma, partial differential ohm, we prove that there exists a solution u epsilon with an interface such that: as epsilon -> 0, u epsilon approaches +1 in one part of ohm, while tends to -1 in the other part, except a small neighborhood of Gamma.