3D-2D asymptotic analysis for micromagnetic thin films

Roberto Alicandro; Chiara Leone

ESAIM: Control, Optimisation and Calculus of Variations (2001)

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

Abstract

top
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

top

Alicandro, Roberto, and Leone, Chiara. "3D-2D asymptotic analysis for micromagnetic thin films." ESAIM: Control, Optimisation and Calculus of Variations 6 (2001): 489-498. <http://eudml.org/doc/90604>.

@article{Alicandro2001,
abstract = {$\Gamma $-convergence techniques and relaxation results of constrained energy functionals are used to identify the limiting energy as the thickness $\varepsilon $ approaches zero of a ferromagnetic thin structure $\Omega _\varepsilon =\omega \times (-\varepsilon ,\varepsilon )$, $\omega \subset \mathbb \{R\}^2$, whose energy is given by\[ \{\mathcal \{E\}\}\_\{\varepsilon \}(\{\overline\{m\}\})=\{1\over \varepsilon \}\int \_\{\Omega \_\{\varepsilon \}\}\left(W(\{\overline\{m\}\},\nabla \{\overline\{m\}\}) +\{1\over 2\}\nabla \{\overline\{u\}\}\cdot \{\overline\{m\}\}\right)\,\{\rm d\}x \]subject to\[ \text\{div\}(-\nabla \{\overline\{u\}\} +\{\overline\{m\}\}\chi \_\{\Omega \_\varepsilon \})=0 \quad \text\{on\} \mathbb \{R\}^3, \]and to the constraint\[ |\overline\{m\}|=1 \text\{on\} \Omega \_\varepsilon , \]where $W$ is any continuous function satisfying $p$-growth assumptions with $p&gt; 1$. 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 = {$\Gamma $-limit; thin films; micromagnetics; relaxation of constrained functionals; Gamma-limit},
language = {eng},
pages = {489-498},
publisher = {EDP-Sciences},
title = {3D-2D asymptotic analysis for micromagnetic thin films},
url = {http://eudml.org/doc/90604},
volume = {6},
year = {2001},
}

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
PY - 2001
PB - EDP-Sciences
VL - 6
SP - 489
EP - 498
AB - $\Gamma $-convergence techniques and relaxation results of constrained energy functionals are used to identify the limiting energy as the thickness $\varepsilon $ approaches zero of a ferromagnetic thin structure $\Omega _\varepsilon =\omega \times (-\varepsilon ,\varepsilon )$, $\omega \subset \mathbb {R}^2$, whose energy is given by\[ {\mathcal {E}}_{\varepsilon }({\overline{m}})={1\over \varepsilon }\int _{\Omega _{\varepsilon }}\left(W({\overline{m}},\nabla {\overline{m}}) +{1\over 2}\nabla {\overline{u}}\cdot {\overline{m}}\right)\,{\rm d}x \]subject to\[ \text{div}(-\nabla {\overline{u}} +{\overline{m}}\chi _{\Omega _\varepsilon })=0 \quad \text{on} \mathbb {R}^3, \]and to the constraint\[ |\overline{m}|=1 \text{on} \Omega _\varepsilon , \]where $W$ is any continuous function satisfying $p$-growth assumptions with $p&gt; 1$. Partial results are also obtained in the case $p=1$, under an additional assumption on $W$.
LA - eng
KW - $\Gamma $-limit; thin films; micromagnetics; relaxation of constrained functionals; Gamma-limit
UR - http://eudml.org/doc/90604
ER -

References

top
  1. [1] A. Braides and A. Defranceschi, Homogenization of Multiple Integrals. Oxford University Press, Oxford (1998). Zbl0911.49010MR1684713
  2. [2] A. Braides and I. Fonseca, Brittle thin films, Preprint CNA-CMU. Pittsburgh (1999). Zbl0999.49012MR1851742
  3. [3] A. Braides, I. Fonseca and G. Francfort, 3D-2D asymptotic analysis for inhomogeneous thin films, Preprint CNA-CMU. Pittsburgh (1999). Zbl0987.35020MR1836533
  4. [4] W.F. Brown, Micromagnetics. John Wiley and Sons, New York (1963). 
  5. [5] C. Castaing and M. Valadier, Convex analysis and measurable multifunctions. Springer-Verlag, New York, Lecture Notes in Math. 580 (1977). Zbl0346.46038MR467310
  6. [6] B. Dacorogna, Direct methods in Calculus of Variations. Springer-Verlag, Berlin (1989). Zbl0703.49001MR990890
  7. [7] B. Dacorogna, I. Fonseca, J. Malỳ and K. Trivisa, Manifold constrained variational problems. Calc. Var. 9 (1999) 185-206. Zbl0935.49006MR1725201
  8. [8] G. Dal Maso, An Introduction to Γ -convergence. Birkhäuser, Boston (1993). Zbl0816.49001MR1201152
  9. [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. Zbl0917.73052MR1662252
  10. [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). Zbl1029.35216MR1831435
  11. [11] I. Fonseca and S. Müller, Quasi-convex integrands and lower semicontinuity in L 1 . SIAM J. Math. Anal. 23 (1992) 1081-1098. Zbl0764.49012MR1177778
  12. [12] G. Gioia and R.D. James, Micromagnetics of very thin films. Proc. Roy. Soc. Lond. Ser. A 453 (1997) 213-223. 
  13. [13] C.B. Morrey, Quasiconvexity and the semicontinuity of multiple integrals. Pacific J. Math. 2 (1952) 25-53. Zbl0046.10803MR54865
  14. [14] C.B. Morrey, Multiple integrals in the Calculus of Variations. Springer-Verlag, Berlin (1966). Zbl0142.38701MR202511

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.