Eigenvalues of the Stokes operator versus the Dirichlet Laplacian in the plane

Abstract

We show that the k-th eigenvalue of the Dirichlet Laplacian is strictly less than the k-th eigenvalue of the classical Stokes operator (equivalently, of the clamped buckling plate problem) for a ndary, we bounded domain in the plane having a locally Lipschitz boundary. For a C2 boushow that eigenvalues of the Stokes operator with Navier slip (friction) boundary conditions interpolate continuously between eigenvalues of the Dirichlet Laplacian and of the classical Stokes operator. © 2010 Pacific Journal of Mathematics.