A remark on a compactness result in electromagnetic theory

Abstract

Communicated by R. LeisCalderon's extension theorem is a crucial tool in the proof of the compactness of the resolvent for the Maxwell operator, and whence this result is proved for domains with the strict cone property. However, the proof only requires an extension operator that extends WZ.2-functions compactly as W'.'-functions. It is shown that this can be achieved under weaker regularity conditions on the domain: the cone may be replaced by some cusp of an appropriate order. Bo(R):= { E E~( R ) :(curl H, E) = (H,curl E) for all H E~( Q ) } , 9(R):= { E E Y~( R ) : div EEL'(R)}, Qo(R):= {HEQ(R): (div H, u ) = -(H, Vu) for all U E W'*'(R)}, Maxwell's boundary value problem consists in finding EEB'(R), H E 9(Q) such that t For various function spaces we use the notation from [l]. The corresponding script letters are used if we deal with corresponding spaces of vector fields.

Time-Harmonic Maxwell Equations in the Exterior of Perfectly Conducting, Irregular Obstacles

“…there exists an open neighbourhood of ξ such that Ω Π U has only finitely many connected components which are either Lipschitz isomorphic to some cone C(M) with a tame base Μ C S 2 or to a domain having the "p-cusp-property" with ρ < 2 defined in [37].…”

“…In [37] the p-cusp property is supposed for the exterior of the domain. But this is equivalent to the p-cusp property of the domain itself.…”

“…The investigation of specific classes of boundaries having the (MCP) has in itself quite a long history, e.g. [9], [8], [16], [28], [27], [24], [20], [6], [37], [13], just to mark some of the contributions to characterize such boundary classes. The current paper provides in section 3 another contribution in this direction indeed enlarging the class of admissible boundaries substantially.…”

On Constants in Maxwell Inequalities for Bounded and Convex Domains

“…By the Maxwell compactness properties (see [5][6][7][12][13][14]), i.e., by the compactness of the two embeddings…”

A nonconforming finite element method for a two-dimensional curl–curl and grad-div problem

“…[23,41,42,49,51] and the discussion in Sect. 2 below), there exists a sequence of nonnegative numbers 0 ≤ λ γ,1 ≤ λ γ,2 ≤ · · · → ∞ such that the following eigenproblem has a nontrivial solution w ∈ H 0 (curl; ) ∩ H (div; ):…”