Upper bounds for a class of energies containing a non-local term
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 4, p. 856-886

In this paper we construct upper bounds for families of functionals of the form ${E}_{\epsilon }\left(\phi \right):={\int }_{\Omega }\left({\epsilon |\nabla \phi |}^{2}+\frac{1}{\epsilon }W\left(\phi \right)\right)\mathrm{d}x+\frac{1}{\epsilon }{\int }_{{ℝ}^{N}}{|\nabla {\overline{H}}_{F\left(\phi \right)}|}^{2}\mathrm{d}x$ where Δ ${\overline{H}}_{u}$ = 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.

DOI : https://doi.org/10.1051/cocv/2009022
Classification:  35A15,  35J35,  82D40
Keywords: gamma-convergence, micromagnetics, non-local energy
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title = {Upper bounds for a class of energies containing a non-local term},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {16},
number = {4},
year = {2010},
pages = {856-886},
doi = {10.1051/cocv/2009022},
zbl = {pre05821901},
mrnumber = {2744154},
language = {en},
url = {http://www.numdam.org/item/COCV_2010__16_4_856_0}
}

Poliakovsky, Arkady. Upper bounds for a class of energies containing a non-local term. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 4, pp. 856-886. doi : 10.1051/cocv/2009022. http://www.numdam.org/item/COCV_2010__16_4_856_0/

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