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 do-main in the plane having a locally Lipschitz boundary. For a C2 bound-ary, we show that eigenvalues of the Stokes operator with Navier slip (friction) boundary conditions interpolate continuously between eigen-values of the Dirichlet Laplacian and of the classical Stokes operator. 1