Transition layer for the heterogeneous Allen–Cahn equation

Abstract

We consider the equation(1)ɛ2Δu=(u−a(x))(u2−1)in Ω,∂u∂ν=0on ∂Ω, where Ω is a smooth and bounded domain in Rn, ν the outer unit normal to ∂Ω, and a a smooth function satisfying −1<a(x)<1 in Ω¯. We set K, Ω+ and Ω− to be respectively the zero-level set of a, {a>0} and {a<0}. Assuming ∇a≠0 on K and a≠0 on ∂Ω, we show that there exists a sequence ɛj→0 such that Eq. (1) has a solution uɛj which converges uniformly to ±1 on the compact sets of Ω± as j→+∞. 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.