Bistable Boundary Reactions in Two Dimensions

In a bounded domain Ω ⊂ ℝ² with smooth boundary we consider the problem 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(πu) or f(u) = (1 - u²)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(πu).