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In the upper half-space  H  we study weak  L   q  -solutions (1 &lt;  q  &lt; ∞) of the eigenvalue problem Δ v  = λdiv  v  in  H ,  v | ∂ H   = 0. It turns out that for all 1 &lt;  q  &lt; ∞ 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.

The weak L q -Cosserat spectrum for the first boundary value problem in the half-space. Applications to Stokes' and Lamé's system