Document ESAIM: COCV
DOI: 10.1051/cocv:2008035
Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions
David KrejciríkDepartment of Theoretical Physics, Nuclear Physics Institute, Academy of Sciences, 250 68 Rez near Prague, Czech Republic; krejcirik@ujf.cas.cz
Received May 31, 2007. Revised December 10, 2007. Published online May 30, 2008.
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.
Mathematics Subject Classification. 35P15, 49R50, 58J50, 81Q15
Key words: Laplacian in tubes, Dirichlet and Neumann boundary conditions, dimension reduction, norm-resolvent convergence, binding effect of curvature, waveguides
© EDP Sciences, SMAI 2008