Existence and regularity for mixtures of micromagnetic materials BY EMILIO ACERBI 1 ,I RENE FONSECA 2, * AND GIUSEPPE MINGIONE 1 1 Dipartimento di Matematica, Viale delle Scienze, 43100 Parma, Italy 2 Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA A new model for the energy of a mixture of micromagnetic materials is introduced within the context of functions with special bounded variation. Existence and regularity for the solution of an optimal design problem in micromagnetics are obtained. Keywords: micromagnetics; constraints; quasi-minimizer; decay lemma; deviation 1. Introduction The commonly adopted Weiss–Landau–Lifschitz model of micromagnetics applies to a single crystal of a magnetic material, and according to this theory the total energy associated with the magnetized crystal is given as a sum of several energy contributions, as described briefly in §2 (see Brown 1963; Landau & Lifschitz 1984; Visintin 1985; Anzellotti et al. 1991; Hubert & Scha¨fer 1998; Dacorogna & Fonseca 2000). When considering a body composed of several distinct magnetic materials, surface energy terms must be taken into account due to the interaction between grains with different magnetic properties, and this leads to the introduction in §3 of a new model for mixtures of magnetic materials, framed within the context of the space SBV of functions with special bounded variation (SBV). In this model, all material information is encapsulated in a function u, the composite magnetization, and the total magnetic energy associated with a body U 3R 3 composed of a finite number K of different magnetic materials has the form EðuÞ Z ð U ½aðjujÞjVuj 2 C fðuÞ Kf $mðuÞ K ~ h½u$mðuÞdx C ð J u gðu C ; u K ; nÞdH 2 ; where f is the external magnetic field, J u is the set of discontinuity points of u, and the composite magnetization must satisfy the pointwise constraint, juj 2f1; 2; .; K g a:e: in U: In §4, we apply this model to an optimal design problem, that of minimizing the total energy of the body U for a fixed external magnetic field f , given the K materials which U may be made of, and possibly under fixed volume fractions of Proc. R. Soc. A (2006) 462, 2225–2243 doi:10.1098/rspa.2006.1655 Published online 28 February 2006 * Author for correspondence (fonseca@andrew.cmu.edu). Received 15 November 2005 Accepted 3 January 2006 2225 q 2006 The Royal Society