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

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 16, Issue: 4, page 856-886
- ISSN: 1292-8119

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topPoliakovsky, Arkady. "Upper bounds for a class of energies containing a non-local term." ESAIM: Control, Optimisation and Calculus of Variations 16.4 (2010): 856-886. <http://eudml.org/doc/250729>.

@article{Poliakovsky2010,

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.
},

author = {Poliakovsky, Arkady},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Gamma-convergence; micromagnetics; non-local energy; gamma-convergence},

language = {eng},

month = {10},

number = {4},

pages = {856-886},

publisher = {EDP Sciences},

title = {Upper bounds for a class of energies containing a non-local term},

url = {http://eudml.org/doc/250729},

volume = {16},

year = {2010},

}

TY - JOUR

AU - Poliakovsky, Arkady

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/10//

PB - EDP Sciences

VL - 16

IS - 4

SP - 856

EP - 886

AB -
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.

LA - eng

KW - Gamma-convergence; micromagnetics; non-local energy; gamma-convergence

UR - http://eudml.org/doc/250729

ER -

## References

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- A. Poliakovsky, On a variational approach to the Method of Vanishing Viscosity for Conservation Laws. Adv. Math. Sci. Appl.18 (2008) 429–451.
- T. Rivière and S. Serfaty, Limiting domain wall energy for a problem related to micromagnetics. Comm. Pure Appl. Math.54 (2001) 294–338.
- T. Rivière and S. Serfaty, Compactness, kinetic formulation and entropies for a problem related to mocromagnetics. Comm. Partial Differential Equations28 (2003) 249–269.

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