Transition layer for the heterogeneous Allen–Cahn equation

Abstract

We consider the equation ɛ^{2}\mathrm{\Delta }u = \left(u−a(x)\right)\left(u^{2}−1\right)\:\text{in }\Omega ,\:\frac{\partial u}{\partial \nu } = 0\:\text{on }\partial \Omega ,\tag{1} where Ω is a smooth and bounded domain in \mathbb{R}^{n} , ν the outer unit normal to ∂Ω , and a a smooth function satisfying −1 < a(x) < 1 in \overline \Omega . We set K , \Omega _{ + } and \Omega _{−} to be respectively the zero-level set of a , \{a > 0\} and \{a < 0\} . Assuming \mathrm{∇}a \neq 0 on K and a \neq 0 on ∂Ω , we show that there exists a sequence ɛ_{j}\rightarrow 0 such that Eq. (1) has a solution u_{ɛ_{j}} which converges uniformly to ±1 on the compact sets of \Omega _{ \pm } as j\rightarrow + \infty . This result settles in general dimension a conjecture posed in [P. Fife, M.W. Greenlee, Interior transition layers of elliptic boundary value problem with a small parameter, Russian Math. Surveys 29 (4) (1974) 103–131], proved in [M. del Pino, M. Kowalczyk, J. Wei, Resonance and interior layers in an inhomogeneous phase transition model, SIAM J. Math. Anal. 38 (5) (2007) 1542–1564] only for n = 2 .