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

David Krejčiřík

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

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

Abstract

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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.

How to cite

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Krejčiřík, 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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