## 00-22

#### The Eigenfunctions of the Stokes Operator in Special Domains III

Preprint series: 00-22, Preprints

The paper is published: Submitted to the ZAMM in August 2000

MSC:
34A30 Linear equations and systems
Abstract: We consider the eigenvalue problem of the Stokes operator in a bounded domain of ${\bf R}^{3}$ bounded by two concentrical cylinders with homogeneous Dirichlet boundary conditions on the curved part of the boundary and periodical conditions in along the cylinder axis (in $x_{1}$-direction). We deduce by separation the correspondent systems of ordinary differential equations and solve them explicitly looking for solenoidal vector fields fulfilling the boundary conditions. The investigation of possible cases yields either the explicit eigenfunctions and eigenvalues or equations for the determination of the eigenvalues and a general representation of the eigenfunctions. The completeness of the the calculated system of eigenfunctions in ${\bf S}$ can be proven analogous to the corresponding part in the habilitation of the second author.