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

ESAIM: Control, Optimisation and Calculus of Variations (2008)

- Volume: 15, Issue: 3, page 555-568
- ISSN: 1292-8119

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topKrejčiřík, David. "Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions." ESAIM: Control, Optimisation and Calculus of Variations 15.3 (2008): 555-568. <http://eudml.org/doc/90926>.

@article{Krejčiřík2008,

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.
},

author = {Krejčiřík, David},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Laplacian in tubes;
Dirichlet and Neumann boundary conditions;
dimension reduction; norm-resolvent convergence;
binding effect of curvature; waveguides; Dirichlet and Neumann boundary conditions; dimension reduction; binding effect of curvature},

language = {eng},

month = {5},

number = {3},

pages = {555-568},

publisher = {EDP Sciences},

title = {Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions},

url = {http://eudml.org/doc/90926},

volume = {15},

year = {2008},

}

TY - JOUR

AU - Krejčiřík, David

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2008/5//

PB - EDP Sciences

VL - 15

IS - 3

SP - 555

EP - 568

AB -
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.

LA - eng

KW - Laplacian in tubes;
Dirichlet and Neumann boundary conditions;
dimension reduction; norm-resolvent convergence;
binding effect of curvature; waveguides; Dirichlet and Neumann boundary conditions; dimension reduction; binding effect of curvature

UR - http://eudml.org/doc/90926

ER -

## References

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