Abstract
In a bounded domain \({\Omega \subset \mathbb R^2}\) 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(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).
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Communicated by P. Rabinowitz
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Dávila, J., del Pino, M. & Musso, M. Bistable Boundary Reactions in Two Dimensions. Arch Rational Mech Anal 200, 89–140 (2011). https://doi.org/10.1007/s00205-010-0337-3
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DOI: https://doi.org/10.1007/s00205-010-0337-3