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Comptes Rendus Mathématique
Volume 335, n° 4
pages 393-398 (2002)
Doi : S1631-073X(02)02494-9
Received : 17 May 2002 ;  accepted : 1 July 2002
Discrete-to-continuum limit of magnetic forces
Large limites de forces magnétiques du discret au continu
 

Stefan Müller a , Anja Schlömerkemper b
a Max-Planck-Institute for Mathematics in the Sciences, Inselstraße 22-26, 04103 Leipzig, Germany 
b Mathematical Institute, University of Oxford, 24-26 St. Giles', Oxford OX1 3LB, United Kingdom 

Note presented by Pierre-Louis Lions

Abstract

We derive a formula for the forces within a magnetized body, starting from a discrete configuration of magnetic dipoles on a Bravais lattice. The resulting force consists of the usual (nonlocal) volume term and an additional local surface term, whose coefficients involve a singular sum over the lattice. The force thus obtained is different from the usual continuum expression, reflecting the different character of the lattice regularization of the underlying hypersingular integral. To cite this article: S. Müller, A. Schlömerkemper, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 393-398.

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Résumé

Dans cette Note, nous dérivons une formule pour les forces dans un corps magnétisé rigide, prenant comme point de départ une configuration de dipôles magnétiques dans un réseau de Bravais et considérant la limite lorsque le paramètre de réseau tend vers zéro. Le terme volumique correspond à la formule pour les forces utilisée dans les théories de milieux continus. Le terme local de surface dans notre formule est, cependant, différent de celui de la théorie du continu. Mathématiquement ceci s'explique par le fait que l'approximation du réseau équivaut à une régularisation différente d'une intégrale hyper-singulière. Pour citer cet article : S. Müller, A. Schlömerkemper, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 393-398.

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