Bistable Boundary Reactions in Two Dimensions

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

$Delta u = 0 quad {rm{in }}, Omega, qquad frac{partial u}{partial nu} = frac1varepsilon f(u) quad {rm{on }},partialOmega,$

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).