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Another thin-film limit of micromagnetics. (English) Zbl 1074.78012

The present paper deals with the study of the energy functional \[ E_h(m_h)=d^2h^{-2}| \log h| ^{-1}\int_{\Omega_h}| \nabla m_h| ^2+h^{-2}| \log h| ^{-1}\int_{\mathbb R^3}| \nabla u_h| ^2, \] where \(\Omega_h=\omega\times (0,h)\) (\(\omega\subset\mathbb R^2\)), \(| m_h| =1\) and \(u_h\) is a solution of the equation \(-\Delta u_h=\) \(\text{div} (m_h\chi (\Omega_h))\) in \(\mathbb R^3\). The main results of the paper establish the \(\Gamma\) convergence of the rescaled 3D micromagnetic problem to a suitable 2D problem. The proofs rely on the explicit form of the magnetostatic energy, expressed in terms of convolution operators, combined with standard estimates for convolution operators.

MSC:

78M35 Asymptotic analysis in optics and electromagnetic theory
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
58E30 Variational principles in infinite-dimensional spaces
78M30 Variational methods applied to problems in optics and electromagnetic theory
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