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Zbl 0989.35009
Alicandro, Roberto; Leone, Chiara
3D-2D asymptotic analysis for micromagnetic thin films. (English)
[J] ESAIM, Control Optim. Calc. Var. 6, 489-498 (2001). ISSN 1292-8119; ISSN 1262-3377/e

Summary: $\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 $\Omega _\varepsilon =\omega \times (-\varepsilon ,\varepsilon)$, $\omega \subset \bbfR ^2$, whose energy is given by $$ {\Cal E}_{\varepsilon }({\overline m})={1\over \varepsilon }\int _{\Omega _{\varepsilon }}\left (W({\overline m},\nabla {\overline m}) +{1\over 2}\nabla {\overline u}\cdot {\overline m}\right)dx $$ subject to $$ \hbox {div}(-\nabla {\overline u} +{\overline m}\chi _{\Omega _\varepsilon })=0 \quad \hbox { on } \bbfR ^3, $$ 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 $W$.

MSC 2000:: *35A15 Variational methods (PDE)
49J45 Optimal control problems inv. semicontinuity and convergence
35M10 PDE of mixed type
74K35 Thin films

Keywords: Gamma-limit; relaxation of constrained functionals

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