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 bounded domain in the plane having a locally Lipschitz boundary. For a C 2 boundary, we show 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.