Bistable Boundary Reactions in Two Dimensions

Abstract

In a bounded domain Omega subset of R(2) with smooth boundary we consider the problem Delta U =0 in Omega, du/dv = i/epsilon f(u) on d Omega

where nu is the unit normal exterior vector, epsilon > 0 is a small parameter and f is a bistable nonlinearity such as f(u) = sin(pi u) or f(u) = (1 - u (2))u. We construct solutions that develop multiple transitions from -1 to 1 and vice-versa along a connected component of the boundary a,I (c). We also construct an explicit solution when Omega is a disk and f(u) = sin(pi u).