Resonance and interior layers in an inhomogeneous phase transition model

Abstract

We consider the problem epsilon(2)Delta u + (u - a(x))(1 - u(2)) = 0 in Omega, partial derivative u/partial derivative v = 0 on partial derivative Omega, where Omega is a smooth and bounded domain in R-2, - 1 < a( x) < 1. Assume that G = {x is an element of Omega, a(x) = 0} is a closed, smooth curve contained in Omega in such a way that Omega = Omega(+) boolean OR Gamma boolean OR Omega- and. partial derivative a/partial derivative n > 0 on Gamma, where n is the outer normal to Omega(+). Fife and Greenlee [Russian Math. Surveys, 29 (1974), pp. 103-131] proved the existence of an interior transition layer solution u(epsilon) which approaches -1 in Omega_ and +1 in Omega(+), for all epsilon sufficiently small. A question open for many years has been whether an interior transition layer solution approaching 1 in Omega_ and -1 in Omega(+) exists. In this paper, we answer this question affirmatively when n = 2, provided that e is small and away from certain critical numbers. A main difficulty is a resonance phenomenon induced by a large number of small critical eigenvalues of the linearized operator.