Generalized spherical harmonics and exterior differentiation in weighted sobolev spaces

Abstract

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.

Generalized linear elasticity in exterior domains. I: Radiation problems

“…They consider compactly supported perturbations B and right-hand sides and reduce the problem to homogeneous Helmholtz equations. In the present case the results from [26] allow a reduction to Helmholtz equations, too, but these are inhomogeneous. So we need estimates for solutions of the Helmholtz equation for the case of a right-hand side which vanishes with a certain order.…”

“…Adding the two equalities in (27) yields (30). Notice (6) as well as the definitions of rot and div in the weak sense [26]. Then for any 3 CO ( ) the following holds:…”

“…On the one hand, the paper at hand demonstrates an application of [26]; On the other, it is the basis of a lowfrequency analysis for the generalized elastic equations which will be treated in the second part [27].…”

p -form spectra and Casimir energies on spherical tessellations

“…Another interesting discussion is provided by Weck and Witsch, [34] esp. §3 and appendix, using ordinary tensor analysis, who derive corresponding results and compute the various degeneracies.…”

Generalized linear elasticity in exterior domains. II: low-frequency asymptotics

“…I , f ! I as defined by (9), (10), (16), (17) will be called solutions of (2) of 'type' 1, 2 , 4 (in this order).…”