We consider the Allen–Cahn equation ε2∆u + u − u3 = 0 in Ω, ∂u ∂ν = 0 on ∂Ω, where Ω is a smooth and bounded domain in Rn such that the mean curvature is positive at each boundary point. We show that there exists a sequence εj → 0 such that the Allen–Cahn equation has a solution uεj with an interface which approaches the boundary as j → +∞.
Boundary Interface for the Allen-Cahn Equation
Malchiodi, Andrea;
2007
Abstract
We consider the Allen–Cahn equation ε2∆u + u − u3 = 0 in Ω, ∂u ∂ν = 0 on ∂Ω, where Ω is a smooth and bounded domain in Rn such that the mean curvature is positive at each boundary point. We show that there exists a sequence εj → 0 such that the Allen–Cahn equation has a solution uεj with an interface which approaches the boundary as j → +∞.File in questo prodotto:
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