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

Published online by Cambridge University Press:  30 May 2008

David Krejčiřík
David Krejčiřík*
Affiliation:
Department of Theoretical Physics, Nuclear Physics Institute, Academy of Sciences, 250 68 Řež near Prague, Czech Republic; krejcirik@ujf.cas.cz
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Abstract

We consider the Laplacian in a domain squeezedbetween two parallel curves in the plane,subject to Dirichlet boundary conditions on one of the curvesand Neumann boundary conditions on the other.We derive two-term asymptotics for eigenvaluesin the limit when the distance between the curves tends to zero.The asymptotics are uniform and local in the sense thatthe coefficients depend only on the extremal points wherethe ratio of the curvature radii of the Neumann boundaryto the Dirichlet one is the biggest.We also show that the asymptotics can be obtainedfrom a form of norm-resolvent convergencewhich takes into account the width-dependenceof the domain of definition of the operators involved.

Keywords

Laplacian in tubesDirichlet and Neumann boundary conditionsdimension reductionnorm-resolvent convergencebinding effect of curvaturewaveguides
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 3 , July 2009 , pp. 555 - 568
Copyright
© EDP Sciences, SMAI, 2008

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