6533b837fe1ef96bd12a1e13

RESEARCH PRODUCT

On electric and magnetic problems for vector fields in anisotropic nonhomogeneous media

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r= 3~2, initiated by Saranen [ 131. In the above, n is the outward-drawn unit normal to the boundary and A denotes the exterior product. According to the simple models for static magnetic fields (resp. electric fields) which are governed by (0.1) (resp. (0.2)), we call (0.1) the magnetic type problem and (0.2) the electric type problem. Considering bounded smooth domains a c R3, we discussed in [ 131, by means of an appropriate Hilbert space method, the solvability and the representation of the solutions for both problems (0.1) and (0.2). Such a new approach was necessary to cover the general nonhomogeneous cases where v and E are matrix-valued functions. Here our aim is twofold. First, we will now include exterior domains in our consideration. This will be achieved by using certain weighted spaces which slightly restrict the behavior of the fields at infinity. To find the scalar potentials, we have to solve the Dirichlet and Neumann problems in exterior domains. Consequently, in order to define the vector potentials, we solve appropriate auxiliary second-order boundary problems for vector fields. The other aspect of this paper is to point out that the dimension of the space of solutions for (0.1) or (0.2) with homogeneous right side is 254 0022-247X/83/010254-22$03.00/0