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Interface regularity for Maxell and Stokes systems

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Kobayashi, T., Suzuki, T. and Watanabe, K. Osaka J. Math. 40 (2003), 925–943 INTERFACE REGULARITY FOR MAXELL AND STOKES SYSTEMS TAKAYUKI KOBAYASHI, TAKASHI SUZUKI and KAZUO WATANABE (Received March 1, 2002) 1. Introduction The purpose of the present paper is to study the interface regularity of three di- mensional Maxwell and Stokes systems. To our knowledge, not so much regards have been taken in this topic, but actually the solenoidal condition provides the regularity across interface to a specified component of the unknown vector field. Let ⊂ R3 be a bounded domain with Lipschitz boundary ∂ , and M ⊂ R3 be a 2 hypersurface cutting transversally. Then, it holds that M∩ 6= ∅ = + ∪ ( ∩M) ∪ − (disjoint union)(1) with the open subsets ± of . First, we take the Maxwell system in magnetostatics, (2) ∇× =∇ · = 0 } in ± where = ( 1( ) 2( ) 3( )) and = ( 1( ) 2( ) 3( )) stand for the three di- mensional vector fields, indicating the magnetic field and the total current density, re- spectively. Here and henceforth, ∇ = (∂1 ∂2 ∂3) denotes the gradient operator and × and · are the outer and the inner products in R3, so that ∇× and ∇· are the operations of the rotation and the divergence, respectively. In the context of magnetoencephalography, Suzuki, Watanabe, and Shimogawara [2] studied the case when the interface is given by the boundary ∂ of a smooth bounded domain ⊂ R3. Namely, from the properties of the layer potential, it showed that if is piecewise continuous on R3 \ ∂ and system (2) has a solution ∈ (R3)3 ∩ 1(R3 \ ∂ )3 for − = and + = R3 \ , then [∇( · )]+− = 0 on ∂ follows, regardless with the continuity of across ∂ . Here, denotes the outer unit 926 T. KOBAYASHI, T. SUZUKI AND K. WATANABE normal vector to ∂ , [ ]+− = + − −, and +(ξ) = lim→ξ ∈R3\ ( ) −(ξ) = lim→ξ ∈ ( ) for ξ ∈ ∂ . In this paper we study its local version, that is, the case where the bounded domain is given with the interface M∩ as in (1). To state the result, we take preliminaries on funct

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