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Upper bounds for a class of energies containing a non-local term

Published online by Cambridge University Press:  31 July 2009

Arkady Poliakovsky
Arkady Poliakovsky*
Affiliation:
Fachbereich Mathematik, Universität Duisburg-Essen, Lotharstrasse 65, 47057 Duisburg, Germany. arkady.poliakovsky@math.uzh.ch
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Abstract

In this paper we construct upper bounds for families of functionals of the form

$$ E_\varepsilon(\phi):=\int_\Omega\Big(\varepsilon |\nabla\phi|^2+\frac{1}{\varepsilon }W(\phi)\Big){\rm d}x+\frac{1}{\varepsilon }\int_{{\mathbb{R}}^N}|\nabla \bar H_{F(\phi)}|^2{\rm d}x $$

where Δ$\bar H_u$ = div {$\chi_\Omega$u}. Particular cases of such functionals arise in Micromagnetics. We also use our technique to construct upper bounds for functionals that appear in a variational formulation of the method of vanishing viscosity for conservation laws.

Keywords

Gamma-convergencemicromagneticsnon-local energy
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 4 , October 2010 , pp. 856 - 886
Copyright
© EDP Sciences, SMAI, 2009

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