Transition layer for the heterogeneous Allen–Cahn equation

We consider the equationequation(1)ε2Δu=(u−a(x))(u2−1)in Ω,∂u∂ν=0on ∂Ω, where Ω   is a smooth and bounded domain in RnRn, ν the outer unit normal to ∂Ω, and a   a smooth function satisfying −10}{a>0} and {a<0}{a<0}. Assuming ∇a≠0∇a≠0 on K   and a≠0a≠0 on ∂Ω  , we show that there exists a sequence εj→0εj→0 such that Eq. (1) has a solution uεjuεj which converges uniformly to ±1 on the compact sets of Ω±Ω± as j→+∞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=2n=2.