3D-2D Asymptotic Analysis for Micromagnetic Thin Films

Roberto Alicandro; Chiara Leone

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

  • Volume: 6, page 489-498
  • ISSN: 1292-8119

Abstract

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Γ-convergence techniques and relaxation results of constrained energy functionals are used to identify the limiting energy as the thickness ε approaches zero of a ferromagnetic thin structure Ω ε = ω × ( - ε , ε ) , ω ⊂ ℝ 2 , whose energy is given by ℰ ε ( m ¯ ) = 1 ε ∫ Ω ε W ( m ¯ , ∇ m ¯ ) + 1 2 ∇ u ¯ · m ¯ d x subject to div ( - ∇ u ¯ + m ¯ χ Ω ε ) = 0 on ℝ 3 , and to the constraint | m ¯ | = 1 on Ω ε , where W is any continuous function satisfying p-growth assumptions with p> 1. Partial results are also obtained in the case p=1, under an additional assumption on W.

How to cite

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Alicandro, Roberto, and Leone, Chiara. "3D-2D Asymptotic Analysis for Micromagnetic Thin Films." ESAIM: Control, Optimisation and Calculus of Variations 6 (2010): 489-498. <http://eudml.org/doc/116575>.

@article{Alicandro2010,
abstract = {Γ-convergence techniques and relaxation results of constrained energy functionals are used to identify the limiting energy as the thickness ε approaches zero of a ferromagnetic thin structure $\Omega_\varepsilon=\omega\times(-\varepsilon,\varepsilon)$, $\omega\subset\mathbb R^2$, whose energy is given by $$ \{\cal E\}\_\{\varepsilon\}(\{\overline\{m\}\})=\frac\{1\}\{\varepsilon\} \int\_\{\Omega\_\{\varepsilon\}\}\left(W(\{\overline\{m\}\},\nabla\{\overline\{m\}\}) +\{\frac\{1\}\{2\}\}\nabla \{\overline\{u\}\}\cdot \{\overline\{m\}\}\right)\,\{\rm d\}x $$ subject to $$ \hbox\{div\}(-\nabla \{\overline\{u\}\}+\{\overline\{m\}\}\chi\_\{\Omega\_\varepsilon\})=0 \quad\hbox\{ on \}\mathbb R^3, $$ and to the constraint $$ |\overline\{m\}|=1 \hbox\{ on \}\Omega\_\varepsilon, $$ where W is any continuous function satisfying p-growth assumptions with p> 1. Partial results are also obtained in the case p=1, under an additional assumption on W. },
author = {Alicandro, Roberto, Leone, Chiara},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Γ-limit; thin films; micromagnetics; relaxation of constrained functionals.; Gamma-limit; relaxation of constrained functionals},
language = {eng},
month = {3},
pages = {489-498},
publisher = {EDP Sciences},
title = {3D-2D Asymptotic Analysis for Micromagnetic Thin Films},
url = {http://eudml.org/doc/116575},
volume = {6},
year = {2010},
}

TY - JOUR
AU - Alicandro, Roberto
AU - Leone, Chiara
TI - 3D-2D Asymptotic Analysis for Micromagnetic Thin Films
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 6
SP - 489
EP - 498
AB - Γ-convergence techniques and relaxation results of constrained energy functionals are used to identify the limiting energy as the thickness ε approaches zero of a ferromagnetic thin structure $\Omega_\varepsilon=\omega\times(-\varepsilon,\varepsilon)$, $\omega\subset\mathbb R^2$, whose energy is given by $$ {\cal E}_{\varepsilon}({\overline{m}})=\frac{1}{\varepsilon} \int_{\Omega_{\varepsilon}}\left(W({\overline{m}},\nabla{\overline{m}}) +{\frac{1}{2}}\nabla {\overline{u}}\cdot {\overline{m}}\right)\,{\rm d}x $$ subject to $$ \hbox{div}(-\nabla {\overline{u}}+{\overline{m}}\chi_{\Omega_\varepsilon})=0 \quad\hbox{ on }\mathbb R^3, $$ and to the constraint $$ |\overline{m}|=1 \hbox{ on }\Omega_\varepsilon, $$ where W is any continuous function satisfying p-growth assumptions with p> 1. Partial results are also obtained in the case p=1, under an additional assumption on W.
LA - eng
KW - Γ-limit; thin films; micromagnetics; relaxation of constrained functionals.; Gamma-limit; relaxation of constrained functionals
UR - http://eudml.org/doc/116575
ER -

References

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  1. A. Braides and A. Defranceschi, Homogenization of Multiple Integrals. Oxford University Press, Oxford (1998).  
  2. A. Braides and I. Fonseca, Brittle thin films, Preprint CNA-CMU. Pittsburgh (1999).  
  3. A. Braides, I. Fonseca and G. Francfort, 3D-2D asymptotic analysis for inhomogeneous thin films, Preprint CNA-CMU. Pittsburgh (1999).  
  4. W.F. Brown, Micromagnetics. John Wiley and Sons, New York (1963).  
  5. C. Castaing and M. Valadier, Convex analysis and measurable multifunctions. Springer-Verlag, New York, Lecture Notes in Math.580 (1977).  
  6. B. Dacorogna, Direct methods in Calculus of Variations. Springer-Verlag, Berlin (1989).  
  7. B. Dacorogna, I. Fonseca, J. Maly and K. Trivisa, Manifold constrained variational problems. Calc. Var.9 (1999) 185-206.  
  8. G. Dal Maso, An Introduction to Γ-convergence. Birkhäuser, Boston (1993).  
  9. I. Fonseca and G. Francfort, 3D-2D asymptotic analysis of an optimal design problem for thin films. J. Reine Angew. Math.505 (1998) 173-202.  
  10. I. Fonseca and G. Francfort, On the inadequacy of the scaling of linear elasticity for 3D-2D asymptotic in a nonlinear setting, Preprint CNA-CMU. Pittsburgh (1999).  
  11. I. Fonseca and S. Müller, Quasi-convex integrands and lower semicontinuity in L1. SIAM J. Math. Anal.23 (1992) 1081-1098.  
  12. G. Gioia and R.D. James, Micromagnetics of very thin films. Proc. Roy. Soc. Lond. Ser. A453 (1997) 213-223.  
  13. C.B. Morrey, Quasiconvexity and the semicontinuity of multiple integrals. Pacific J. Math.2 (1952) 25-53.  
  14. C.B. Morrey, Multiple integrals in the Calculus of Variations. Springer-Verlag, Berlin (1966).  

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