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

James P. Kelliher

Vol. 244 (2010), No. 1, 99–132

DOI: 10.2140/pjm.2010.244.99

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 C² 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.

Keywords

Laplacian, Stokes operator, eigenvalues, clamped buckling plate

Mathematical Subject Classification 2000

Primary: 35P99

Milestones

Received: 5 December 2008

Revised: 24 July 2009

Accepted: 9 September 2009

Published: 17 November 2009