SUMMARYA class of linear material laws is considered, which covers a number of diverse initial boundary-value problems of classical mathematical physics. The claim that this class is indeed to a large extent sufficiently general is exemplified for a number of specific models from classical physics.

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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.

The paper extends well-posedness results of a previously explored class of time-shift invariant evolutionary problems to the case of non-autonomous media. The Hilbert space setting developed for the time-shift invariant case can be utilized to obtain an elementary approach to non-autonomous equations. The results cover a large class of evolutionary equations, where well-known strategies like evolution families may be difficult to use or fail to work. We exemplify the approach with an application to a Kelvin-Voigt-type model for visco-elastic solids.

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.