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

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)dx+1 ε∫ ℝ N |∇H ¯ F(φ) | 2 dx
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.

DOI : 10.1051/cocv/2009022
Classification : 35A15, 35J35, 82D40
Mots clés : gamma-convergence, micromagnetics, non-local energy 
@article{COCV_2010__16_4_856_0,
     author = {Poliakovsky, Arkady},
     title = {Upper bounds for a class of energies containing a non-local term},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {856--886},
     publisher = {EDP-Sciences},
     volume = {16},
     number = {4},
     year = {2010},
     doi = {10.1051/cocv/2009022},
     mrnumber = {2744154},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2009022/}
}
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
PY  - 2010
SP  - 856
EP  - 886
VL  - 16
IS  - 4
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2009022/
DO  - 10.1051/cocv/2009022
LA  - en
ID  - COCV_2010__16_4_856_0
ER  - 
%0 Journal Article
%A Poliakovsky, Arkady
%T Upper bounds for a class of energies containing a non-local term
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2010
%P 856-886
%V 16
%N 4
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2009022/
%R 10.1051/cocv/2009022
%G en
%F 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, Tome 16 (2010) no. 4, pp. 856-886. doi : 10.1051/cocv/2009022. http://archive.numdam.org/articles/10.1051/cocv/2009022/

[1] F. Alouges, T. Rivière and S. Serfaty, Néel and cross-tie wall energies for planar micromagnetic configurations. ESAIM: COCV 8 (2002) 31-68. | Numdam | Zbl

[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). | Zbl

[3] A. Desimone, S. Müller, R.V. Kohn and F. Otto, Recent analytical developments in micromagnetics, in The Science of Hysteresis 2, G. Bertotti and I. Mayergoyz Eds., Elsevier Academic Press (2005) 269-381. | Zbl

[4] A. Hubert and R. Schäfer, Magnetic domains. Springer (1998).

[5] W. Jin and R.V. Kohn, Singular perturbation and the energy of folds. J. Nonlinear Sci. 10 (2000) 355-390. | Zbl

[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. | Numdam | Zbl

[8] A. Poliakovsky, A general technique to prove upper bounds for singular perturbation problems. J. Anal. Math. 104 (2008) 247-290. | Zbl

[9] A. Poliakovsky, On a variational approach to the Method of Vanishing Viscosity for Conservation Laws. Adv. Math. Sci. Appl. 18 (2008) 429-451. | Zbl

[10] T. Rivière and S. Serfaty, Limiting domain wall energy for a problem related to micromagnetics. Comm. Pure Appl. Math. 54 (2001) 294-338. | Zbl

[11] T. Rivière and S. Serfaty, Compactness, kinetic formulation and entropies for a problem related to mocromagnetics. Comm. Partial Differential Equations 28 (2003) 249-269. | Zbl

Cité par Sources :