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

Abstract

The boundary value problem of total reflection of time-harmonic electromagnetic waves is considered in an exterior domain Ω. Α Fredholm type alternative is shown to be valid under rather general assumptions on boundary regularity and regularity of the coefficients. The solution theory is developed in suitably weighted spaces. A cornerstone of the reasoning used to obtain the solution theory is a local compact imbedding property assumed to hold for Ω. A large class of domains is characterized featuring this property.

On Korn's first inequality for tangential or normal boundary conditions with explicit constants

“…Therefore, there exists , such that for all , holds. Because the embedding of into is compact even for bounded Lipschitz (or weaker) domains Ω, see , we also have the so‐called normal Maxwell estimate, that is, there exists , such that for all , there exists with …”

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…”

On the Maxwell constants in 3D

“…Another successful approach proving the Maxwell compactness property using a different technique from [2] has been shown in [3]. For the Maxwell compactness property in the case of full boundary conditions, we refer to [2,[4][5][6][7][8][9][10][11][12][13][14].…”