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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.

Communicated by R. LeisWe generalize spherical harmonics expansions of scalar functions to expansions of alternating differential forms ('q-forms'). To this end we develop a calculus for the use of spherical co-ordinates for q-forms and determine the eigen-q-forms of the Beltrami-operator on S N --' which replace the classical spherical harmonics. We characterize and classify homogeneous q-forms u which satisfy Au = 0 on RN\{O} and determine Fredholm properties, kernel and range of the exterior derivative d acting in weighted Lp-spaces of q-forms (generalizing results of McOwen for the scalar Laplacian). These techniques and results are necessary prerequisites for the discussion of the low-frequency behaviour in exterior boundary value problems for systems occurring in electromagnetism and isotropic elasticity.

Communicated by R. LeisFor the theory of boundary value problems in linear elasticity, it is of crucial importance that the space of vector-valued L2-functions whose symmetrized Jacobians are square-integrable should be compactly embedded in Lz. For regions with the cone property this is usually achieved by combining Korn's inequalities and Rellich's selection theorem. We shall show that in a class of less regular regions Korn's second inequality fails whereas the desired compact embedding still holds true.

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