On the boundary value problems of electro- and magnetostatics

Abstract

SynopsisThe classically well-known relation between the number of linearly independent solutions of the electro- and magnetostatic boundary value problems (harmonic Dirichlet and Neumann vector fields) and topological characteristics (genus and number of boundaries) of the underlying domain in 3-dimensional euclidean space is investigated in the framework of Hilbert space theory. It can be shown that this connection is still valid for a large class of domains with not necessarily smooth boundaries (segment property). As an application the inhomogeneous boundary value problems of electro- and magnetostatics are discussed.

𝐿²-well-posedness of 3d div-curl boundary value problems

“…Its construction and properties are described by Cessenat in [9, Chapter 9, Section 1.3], Foias and Temam [12], Picard [16] and Saarinen [17] amongst other references. The dimension L of this space is a topological invariant of the region.…”

“…When ε is a general matrix obeying (E1), a basis of H εν0 (Ω) using transmission problems was described by Picard in [16] and he showed the dimension of the space is L. Also Saarinen in [17] gives an abstract proof, credited to K. J. Witsch, that under conditions similar to ours, the space H εν0 (Ω) has dimension L. Here we first describe an explicit construction of a basis for H εν0 (Ω) which is based on perturbation of the basis with ε = I 3 and does not require the solution of another transmission type problem.…”

“…Results on this problem have been described previously by Saranen [17] and [18], and by Picard [16]. They used Hilbert space methods to obtain certain existence theorems.…”

“…Earlier work on the normal and tangential boundary-value problems for these systems includes that by Picard [16] and Saranen [17] and [18]. Their results used the theory of closed linear operators on Hilbert spaces.…”

“…Various results on the finite-energy solvability of this problem have been described by Picard in [16] and Saranen in [17] and [18]. They used operator-theoretic methods on Hilbert spaces.…”