A local compactness theorem for Maxwell's equations

Abstract

Abstract The paper gives a proof, valid for a large class of bounded domains, of the following compactness statements: Let G be a bounded domain, β be a tensor‐valued function on G satisfying certain restrictions, and let { n } be a sequence of vector‐valued functions on G where the L 2 ‐norms of { n }, {curl n }, and {div(β n )} are bounded, and where all n either satisfy x n = 0 or (β F n ) = 0 at the boundary ∂ G of G ( = normal to ∂ G ): then { n } has a L 2 ‐convergent subsequence. The first boundary condition is satisfied by electric fields, the second one by magnetic fields at a perfectly conducting boundary ∂ G if β is interpreted as electric dielectricity ϵ or as magnetic permeability μ, respectively. These compactness statements are essential for the application of abstract scattering theory to the boundary value problem for Maxwell's equations.