Clustering of boundary interfaces for an inhomogeneous Allen-Cahn equation on a smooth bounded domain

摘要

We consider the inhomogeneous Allen-Cahn equation where Omega is a bounded domain in R2 with smooth boundary partial derivative Omega and V(x) is a positive smooth function, E>0 is a small parameter, nu denotes the unit outward normal of partial derivative Omega. For any fixed integer N >= 2, we will show the existence of a clustered solution uE with N-transition layers near partial derivative Omega with mutual distance O(E|lnE|), provided that the generalized mean curvature H of partial derivative Omega is positive and E stays away from a discrete set of values at which resonance occurs. Our result is an extension of those (with dimension two) by Malchiodi et al. (Pac. J. Math. 229(2):447-468, 2007) and Malchiodi et al. (J. Fixed Point Theory Appl. 1(2):305-336, 2007).

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