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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@[email protected])
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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.

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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