ESAIM: Control, Optimisation and Calculus of Variations

Research Article

3D-2D Asymptotic Analysis for Micromagnetic Thin Films

Alicandro, Robertoa1 and Leone, Chiaraa2

a1 SISSA, Via Beirut 4, 34013 Trieste, Italy; alicandr@sissa.it.

a2 Dipartimento di Matematica, Università di Roma I, P.le A. Moro 2, 00185 Roma, Italy; leone@mat.uniroma1.it.

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.

(Received September 25 2000)

(Received April 2001)

(Online publication August 15 2002)

Key Words:

  • Γ-limit;
  • thin films;
  • micromagnetics;
  • relaxation of constrained functionals.

Mathematics Subject Classification:

  • 35E99;
  • 35M10;
  • 49J45;
  • 74K35
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