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Generalized spherical harmonics and exterior differentiation in weighted sobolev spaces

✍ Scribed by Norbert Weck; Karl J. Witsch

Publisher
John Wiley and Sons
Year
1994
Tongue
English
Weight
977 KB
Volume
17
Category
Article
ISSN
0170-4214

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✦ Synopsis

Abstract

We 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−1^ which replace the classical spherical harmonics. We characterize and classify homogeneous q‐forms u which satisfy Δ__u__ = 0 on ℝ^N^∖︁{0} and determine Fredholm properties, kernel and range of the exterior derivative d acting in weighted L^p^‐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.

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