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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  and
Matthias Winter
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])
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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
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 131 , Issue 1 , February 2001 , pp. 185 - 204
Copyright
Copyright © Royal Society of Edinburgh 2001

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