On the eigenfunctions of the Stokes operator in a plane layer with a periodicity condition along it

In Rummler’s previous paper, formulas for the eigenfunctions of the Stokes operator were derived (in a rather concise form) in the case of a three-dimensional layer with a periodicity condition in orthogonal directions along the layer. In this paper, eigenfunctions and associated pressures are constructed and studied in a plane $n$-dimensional (specifically, two-dimensional) layer with a periodicity condition in orthogonal directions along the layer. A very simple and useful velocity representation in terms of the pressure gradient is used. As a result, the derivation of formulas is considerably simplified and reduced without applying cumbersome expressions and the eigenfunctions are expressed in terms of the associated pressures. Two-sided estimates are given, and the asymptotic behavior of nontrivial eigenvalues of the Stokes operator is analyzed.