Upper bounds for a class of energies containing a non-local term
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 {$\...
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Veröffentlicht in: | ESAIM. Control, optimisation and calculus of variations optimisation and calculus of variations, 2010-10, Vol.16 (4), p.856-886 |
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Format: | Artikel |
Sprache: | eng |
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Online-Zugang: | Volltext |
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Zusammenfassung: | 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. |
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ISSN: | 1292-8119 1262-3377 |
DOI: | 10.1051/cocv/2009022 |