Homogenization of micromagnetics large bodies

Giovanni Pisante

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

  • Volume: 10, Issue: 2, page 295-314
  • ISSN: 1292-8119

Abstract

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A homogenization problem related to the micromagnetic energy functional is studied. In particular, the existence of the integral representation for the homogenized limit of a family of energies ε ( m ) = Ω φ x , x ε , m ( x ) d x - Ω h e ( x ) · m ( x ) d x + 1 2 3 | u ( x ) | 2 d x of a large ferromagnetic body is obtained.

How to cite

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Pisante, Giovanni. "Homogenization of micromagnetics large bodies." ESAIM: Control, Optimisation and Calculus of Variations 10.2 (2010): 295-314. <http://eudml.org/doc/90731>.

@article{Pisante2010,
abstract = { A homogenization problem related to the micromagnetic energy functional is studied. In particular, the existence of the integral representation for the homogenized limit of a family of energies $$ \mathcal\{E\}\_\{\varepsilon\}(m)=\int\_\{\Omega\} \phi\left(x,\frac\{x\}\{\varepsilon\},m(x)\right)\,\{\rm d\}x -\int\_\{\Omega\}h\_e(x)\cdot m(x)\,\{\rm d\}x+\frac\{1\}\{2\}\int\_\{\mathbb R^3\}|\nabla u(x)|^2\,\{\rm d\}x $$ of a large ferromagnetic body is obtained. },
author = {Pisante, Giovanni},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Micromagnetics; homogenization; Γ-convergence.; micromagnetics; integral representation; homgenized limit},
language = {eng},
month = {3},
number = {2},
pages = {295-314},
publisher = {EDP Sciences},
title = {Homogenization of micromagnetics large bodies},
url = {http://eudml.org/doc/90731},
volume = {10},
year = {2010},
}

TY - JOUR
AU - Pisante, Giovanni
TI - Homogenization of micromagnetics large bodies
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 10
IS - 2
SP - 295
EP - 314
AB - A homogenization problem related to the micromagnetic energy functional is studied. In particular, the existence of the integral representation for the homogenized limit of a family of energies $$ \mathcal{E}_{\varepsilon}(m)=\int_{\Omega} \phi\left(x,\frac{x}{\varepsilon},m(x)\right)\,{\rm d}x -\int_{\Omega}h_e(x)\cdot m(x)\,{\rm d}x+\frac{1}{2}\int_{\mathbb R^3}|\nabla u(x)|^2\,{\rm d}x $$ of a large ferromagnetic body is obtained.
LA - eng
KW - Micromagnetics; homogenization; Γ-convergence.; micromagnetics; integral representation; homgenized limit
UR - http://eudml.org/doc/90731
ER -

References

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  1. G. Anzellotti, S. Baldo and A. Visintin, Asymptotic behavior of the Landau-Lifshitz model of ferromagnetism. Appl. Math. Optim.23 (1991) 171-192.  Zbl0746.49032
  2. J.M. Ball, A. Taheri and M. Winter, Local minimizers in micromagnetics and related problems. Calc. Var. Partial Differ. Equ.14 (2002) 1-27.  Zbl1005.49002
  3. A. Braides and A. Defranceschi, Homogenization of multiple integrals. The Clarendon Press Oxford University Press, New York, Oxford Lecture Ser. Math. Appl.12 (1998).  Zbl0911.49010
  4. A. Braides, I. Fonseca and G. Leoni, A-quasiconvexity: relaxation and homogenization. ESAIM: COCV5 (2000) 539-577 (electronic).  Zbl0971.35010
  5. Jr. Brown and W. Fuller, Micromagnetics. Interscience Publishers, John Wiley & Sons, New York, London (1963).  
  6. B. Dacorogna, Direct methods in the calculus of variations. Appl. Math. Sci.78 (1989).  Zbl0703.49001
  7. B. Dacorogna and I. Fonseca, A-B quasiconvexity and implicit partial differential equations. Calc. Var. Partial Differ. Equ.14 (2002) 115-149.  Zbl1030.35022
  8. G. Dal Maso, An introduction to Γ -convergence. Birkhäuser Boston Inc., Boston, MA Prog. Nonlinear Differ. Equ. Appl.8 (1993).  Zbl0816.49001
  9. A. De Simone, Energy minimizers for large ferromagnetic bodies. Arch. Ration. Mech. Anal.125 (1993) 99-143.  Zbl0811.49030
  10. A. De Simone, Hysteresis and imperfection sensitivity in small ferromagnetic particles. Meccanica30 (1995) 591-603. Microstructure and phase transitions in solids (Udine, 1994).  
  11. A. De simone, R.V. Kohn, S. Müller and F. Otto, A reduced theory for thin-film micromagnetics. Commun. Pure Appl. Math.55 (2002) 1408-1460.  Zbl1027.82042
  12. A. De Simone, S. Müller, R.V. Kohn and F. Otto, A compactness result in the gradient theory of phase transitions. Proc. R. Soc. Edinb. Sect. A131 (2001) 833-844.  Zbl0986.49009
  13. I. Fonseca and G. Leoni, Relaxation results in micromagnetics. Ricerche Mat.49 (2000) (suppl.) 269-304. Contributions in honor of the memory of Ennio De Giorgi (Italian).  Zbl1072.49010
  14. R.D. James and D. Kinderlehrer, Frustration in ferromagnetic materials. Contin. Mech. Thermodyn.2 (1990) 215-239.  
  15. L.D. Landau and E.M. Lifshits, Teoreticheskaya fizika. Tome VIII. “Nauka”, Moscow, third edition (1992). Elektrodinamika sploshnykh sred. [Electrodynamics of continuous media], with a preface by Lifshits and L.P. Pitaevskiĭ, edited and with a preface by Pitaevskiĭ.  
  16. L. Tartar, On mathematical tools for studying partial differential equations of continuum physics: H-measures and Young measures. Plenum, New York, in Developments in partial differential equations and applications to mathematical physics (Ferrara, 1991), (1992) 201-217.  Zbl0897.35010
  17. L. Tartar, Beyond Young measures. Meccanica30 (1995) 505-526. Microstructure and phase transitions in solids (Udine, 1994).  Zbl0835.73062
  18. A. Visintin, On Landau-Lifshitz' equations for ferromagnetism. Japan J. Appl. Math.2 (1985) 69-84.  Zbl0613.35018

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