Bistable Boundary Reactions in Two Dimensions

Abstract

In a bounded domain ${\Omega \subset \mathbb R^2}$ with smooth boundary we consider the problem

$$\Delta u = 0 \quad {\rm{in }}\, \Omega, \qquad \frac{\partial u}{\partial \nu} = \frac1\varepsilon f(u) \quad {\rm{on }}\,\partial\Omega,$$

where ν is the unit normal exterior vector, ε > 0 is a small parameter and f is a bistable nonlinearity such as f(u) = sin(π 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 ∂Ω. We also construct an explicit solution when Ω is a disk and f(u) = sin(π u).