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Generalized spherical harmonics and exterior differentiation in weighted sobolev spaces
Author(s) -
Weck Norbert,
Witsch Karl J.
Publication year - 1994
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670171303
Subject(s) - mathematics , sobolev space , spherical harmonics , spin weighted spherical harmonics , mathematical analysis , laplace operator , zonal spherical harmonics , differential form , vector spherical harmonics , differential operator , scalar (mathematics) , boundary value problem , directional derivative , pure mathematics , harmonics , geometry , physics , quantum mechanics , voltage
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.