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

Arkady Poliakovsky

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

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

Abstract

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In this paper we construct upper bounds for families of functionals of the form E ε ( φ ) : = Ω ε | φ | 2 + 1 ε W ( φ ) d x + 1 ε N | H ¯ F ( φ ) | 2 d x where Δ H ¯ u = div { χ Ω 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.

How to cite

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Poliakovsky, 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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  2. L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs. Oxford University Press, New York (2000).  Zbl0957.49001
  3. A. DeSimone, S. Müller, R.V. Kohn and F. Otto, Recent analytical developments in micromagnetics, in The Science of Hysteresis2, G. Bertotti and I. Mayergoyz Eds., Elsevier Academic Press (2005) 269–381.  Zbl1151.35426
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  7. A. Poliakovsky, Sharp upper bounds for a singular perturbation problem related to micromagnetics. Ann. Scuola Norm. Sup. Pisa Cl. Sci.6 (2007) 673–701.  Zbl1150.49006
  8. A. Poliakovsky, A general technique to prove upper bounds for singular perturbation problems. J. Anal. Math.104 (2008) 247–290.  Zbl1153.49018
  9. A. Poliakovsky, On a variational approach to the Method of Vanishing Viscosity for Conservation Laws. Adv. Math. Sci. Appl.18 (2008) 429–451.  Zbl1193.49004
  10. T. Rivière and S. Serfaty, Limiting domain wall energy for a problem related to micromagnetics. Comm. Pure Appl. Math.54 (2001) 294–338.  Zbl1031.35142
  11. T. Rivière and S. Serfaty, Compactness, kinetic formulation and entropies for a problem related to mocromagnetics. Comm. Partial Differential Equations28 (2003) 249–269.  Zbl1094.35125

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