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Solutions for the Cahn—Hilliard equation with many boundary spike layers

Published online by Cambridge University Press:  11 July 2007

Juncheng Wei
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong ([email protected])
Matthias Winter
Affiliation:
Mathematisches Institut A, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany ([email protected])

Abstract

In this paper we construct new classes of stationary solutions for the Cahn–Hilliard equation by a novel approach.

One of the results is as follows. Given a positive integer K and a (not necessarily non-degenerate) local minimum point of the mean curvature of the boundary, then there are boundary K-spike solutions whose peaks all approach this point. This implies that for any smooth and bounded domain there exist boundary K-spike solutions.

The central ingredient of our analysis is the novel derivation and exploitation of a reduction of the energy to finite dimensions (lemma 3.5), where the variables are closely related to the peak locations.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2001

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