Abstract
AbstractWe 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 SN−1 which replace the classical spherical harmonics. We characterize and classify homogeneous q‐forms u which satisfy Δu = 0 on ℝN∖︁{0} 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.
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