Volume 26, Issue 1

In the upper half-space H we study weak L q -solutions (1 < q < ∞) of the eigenvalue problem Δ v = λ div v in H , v | ∂ H = 0. It turns out that for all 1 < q < ∞ and all n ≥ 2 there exist only the eigenvalues λ 1 = 1 and λ 2 = 2 each of infinite multiplicity. We need no results for elliptic systems. Only the strong L q -Dirichlet and Neumann problem for the Laplacian and the weak L q -Dirichlet problem for Δ 2 is used. As a conclusion we get a direct decomposition of the Sobolev space under consideration. The weak L q -solvability of Stokes' system and of Lamé's system is a simple corollary. Then we study analogously for n = 3 the problem Δ u = λ rot rot u , u | ∂ H = 0. This is more complicated. Finally we prove existence of weak L q -solutions to Stokes' system under stress-free boundary conditions.

If G ⊂ R n is a bounded or an exterior domain with sufficiently smooth boundary then for the eigenvalue problem λ = 1 is an eigenvalue of infinite multiplicity and λ = 2 is an accumulation point of eigenvalues of finite multiplicity. For the L q -solutions corresponding to eigenvalues λ ∈ R \ {1,2}, one can prove the existence of higher (classical) derivatives. Furthermore they do not depend on q . As an application we obtain a relationship of Green's function to the reproducing kernel.

If G ⊂ R n ( n ≥ 2) is either a bounded or an exterior domain, weak L q -solutions of Δ v = ∇ p in G , v | ∂G = 0 are considered. Let Z q p := div v for p ∈ L q ( G ). Restricted to a certain closed subspace B (0) q ( G ) ⊂ L q ( G ) made up of harmonic functions, the operator ( Z q − ½ I )| B (0) q (G) is compact. On one hand this forms the basis for the investigation of the Cosserat spectrum [8]. On the other hand it admits an easy proof of a useful decomposition theorem for the Sobolev space Ĥ • 1,q ( G ) n . As a corollary existence and a priori estimates for Stokes' and Lamé's equation follow immediately. Finally we investigate the problem whether there exist eigensolutions v to the Cosserat eigenvalue λ = 1 such that div v is harmonic. This depends on topological properties of G .