3D-2D asymptotic analysis for micromagnetic thin films
ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 489-498

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 ¯dx
subject to
div(-∇u ¯+m ¯χ Ω ε )=0onℝ 3 ,
and to the constraint
|m ¯|=1onΩ 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.

Classification : 35E99, 35M10, 49J45, 74K35
Keywords: $\Gamma $-limit, thin films, micromagnetics, relaxation of constrained functionals 
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     author = {Alicandro, Roberto and Leone, Chiara},
     title = {3D-2D asymptotic analysis for micromagnetic thin films},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {489--498},
     year = {2001},
     publisher = {EDP Sciences},
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     zbl = {0989.35009},
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     url = {https://www.numdam.org/item/COCV_2001__6__489_0/}
}
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Alicandro, Roberto; Leone, Chiara. 3D-2D asymptotic analysis for micromagnetic thin films. ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 489-498. https://www.numdam.org/item/COCV_2001__6__489_0/

[1] A. Braides and A. Defranceschi, Homogenization of Multiple Integrals. Oxford University Press, Oxford (1998). | Zbl | MR

[2] A. Braides and I. Fonseca, Brittle thin films, Preprint CNA-CMU. Pittsburgh (1999). | Zbl | MR

[3] A. Braides, I. Fonseca and G. Francfort, 3D-2D asymptotic analysis for inhomogeneous thin films, Preprint CNA-CMU. Pittsburgh (1999). | Zbl | MR

[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). | Zbl | MR

[6] B. Dacorogna, Direct methods in Calculus of Variations. Springer-Verlag, Berlin (1989). | Zbl | MR

[7] B. Dacorogna, I. Fonseca, J. Malỳ and K. Trivisa, Manifold constrained variational problems. Calc. Var. 9 (1999) 185-206. | Zbl | MR

[8] G. Dal Maso, An Introduction to Γ-convergence. Birkhäuser, Boston (1993). | Zbl | MR

[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. | Zbl | MR

[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). | Zbl | MR

[11] I. Fonseca and S. Müller, Quasi-convex integrands and lower semicontinuity in L 1 . SIAM J. Math. Anal. 23 (1992) 1081-1098. | Zbl | MR

[12] G. Gioia and R.D. James, Micromagnetics of very thin films. Proc. Roy. Soc. Lond. Ser. A 453 (1997) 213-223.

[13] C.B. Morrey, Quasiconvexity and the semicontinuity of multiple integrals. Pacific J. Math. 2 (1952) 25-53. | Zbl | MR

[14] C.B. Morrey, Multiple integrals in the Calculus of Variations. Springer-Verlag, Berlin (1966). | Zbl | MR