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Existence and uniqueness of radially symmetric stationary points within the gradient theory of phase transitions

Published online by Cambridge University Press:  26 September 2008

Barbara S. Niethammer
Barbara S. Niethammer
Affiliation:
Institut für Angewandte Mathematik, Sonderforschungsbereich 256, Wegelerstr. 6, D-53115 Bonn 1, Germany (email: igel@sfb256@iam.uni-bonn.de)
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Extract

We study radially symmetric stationary points of the functional

where u denotes the density of a fluid confined to a container Ω, W(u) is the course-grain free energy and ε accounts for surface energy. Under the further assumption of small energy, that is

for small ε, we prove existence of precisely two solutions for the corresponding Euler-Lagrange equation. Each of these solutions is monotone in the radial direction and converges as ε→0 to one of two possible radially symmetric single interface minimizers of E0. Our main tool is the method of matched asymptotic expansions from which we construct exact solutions.

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Type
Research Article
Information
European Journal of Applied Mathematics , Volume 6 , Issue 1 , February 1995 , pp. 45 - 67
Copyright
Copyright © Cambridge University Press 1995

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