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Pattern formation of the stationary Cahn-Hilliard model

Published online by Cambridge University Press:  14 November 2011

Hansjörg Kielhöfer
Hansjörg Kielhöfer
Affiliation:
Universität Augsburg, Institut für Mathematik, Universitätsstrasse 14, 86135 Augsburg, Germany
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Extract

We investigate critical points of the free energy Eε(u) of the Cahn–Hilliard model over the unit square under the constraint of a mean value ü. We show that for any fixed value ü in the so-called spinodal region and to any mode of an infinite class, there are critical points of Eε(u) having the characteristic symmetries of that mode provided ε > 0 is small enough. As ε tends to zero, these critical points have singular limits forming characteristic patterns for each mode. Furthermore, any singular limit is a stable critical point of E0(u)). Our method consists of a global bifurcation analysis of critical points of the energy Eε(u) where the bifurcation parameter is the mean value ü.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 6 , 1997 , pp. 1219 - 1243
Copyright
Copyright © Royal Society of Edinburgh 1997

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