# 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

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topAlicandro, 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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- I. Fonseca and S. Müller, Quasi-convex integrands and lower semicontinuity in L1. SIAM J. Math. Anal.23 (1992) 1081-1098. Zbl0764.49012
- G. Gioia and R.D. James, Micromagnetics of very thin films. Proc. Roy. Soc. Lond. Ser. A453 (1997) 213-223.
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