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 ∫ ℝ 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).  
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  6. A. Poliakovsky, Upper bounds for singular perturbation problems involving gradient fields. J. Eur. Math. Soc.9 (2007) 1–43.  
  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.  
  8. A. Poliakovsky, A general technique to prove upper bounds for singular perturbation problems. J. Anal. Math.104 (2008) 247–290.  
  9. A. Poliakovsky, On a variational approach to the Method of Vanishing Viscosity for Conservation Laws. Adv. Math. Sci. Appl.18 (2008) 429–451.  
  10. T. Rivière and S. Serfaty, Limiting domain wall energy for a problem related to micromagnetics. Comm. Pure Appl. Math.54 (2001) 294–338.  
  11. 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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