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 SN?1 which replace the classical spherical harmonics. We characterize and classify homogeneous q-forms u which satisfy Δu = 0 on ?N??{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. |