Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions

Department of Theoretical Physics, Nuclear Physics Institute, Academy of Sciences, 250 68 Řež near Prague, Czech Republic; krejcirik@ujf.cas.cz

Received: 31 May 2007
Revised: 10 December 2007

Abstract

We consider the Laplacian in a domain squeezed between two parallel curves in the plane, subject to Dirichlet boundary conditions on one of the curves and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between the curves tends to zero. The asymptotics are uniform and local in the sense that the coefficients depend only on the extremal points where the ratio of the curvature radii of the Neumann boundary to the Dirichlet one is the biggest. We also show that the asymptotics can be obtained from a form of norm-resolvent convergence which takes into account the width-dependence of the domain of definition of the operators involved.