Spectrum of the Laplacian in narrow tubular neighbourhoods of hypersurfaces with combined Dirichlet and Neumann boundary conditions

David Krejčiřík

Mathematica Bohemica (2014)

  • Volume: 139, Issue: 2, page 185-193
  • ISSN: 0862-7959

Abstract

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We consider the Laplacian in a domain squeezed between two parallel hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet boundary conditions on one of the hypersurfaces and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between the hypersurfaces 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 area of the Neumann boundary to the Dirichlet one is locally the biggest.

How to cite

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Krejčiřík, David. "Spectrum of the Laplacian in narrow tubular neighbourhoods of hypersurfaces with combined Dirichlet and Neumann boundary conditions." Mathematica Bohemica 139.2 (2014): 185-193. <http://eudml.org/doc/261907>.

@article{Krejčiřík2014,
abstract = {We consider the Laplacian in a domain squeezed between two parallel hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet boundary conditions on one of the hypersurfaces and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between the hypersurfaces 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 area of the Neumann boundary to the Dirichlet one is locally the biggest.},
author = {Krejčiřík, David},
journal = {Mathematica Bohemica},
keywords = {Laplacian in tubes; Dirichlet boundary condition; Neumann boundary condition; eigenvalue asymptotics; dimension reduction; quantum waveguides; mean curvature; eigenvalue asymptotics; quantum waveguides; Laplacian in tubes; Dirichlet boundary condition; Neumann boundary condition; dimension reduction; mean curvature},
language = {eng},
number = {2},
pages = {185-193},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Spectrum of the Laplacian in narrow tubular neighbourhoods of hypersurfaces with combined Dirichlet and Neumann boundary conditions},
url = {http://eudml.org/doc/261907},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Krejčiřík, David
TI - Spectrum of the Laplacian in narrow tubular neighbourhoods of hypersurfaces with combined Dirichlet and Neumann boundary conditions
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 2
SP - 185
EP - 193
AB - We consider the Laplacian in a domain squeezed between two parallel hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet boundary conditions on one of the hypersurfaces and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between the hypersurfaces 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 area of the Neumann boundary to the Dirichlet one is locally the biggest.
LA - eng
KW - Laplacian in tubes; Dirichlet boundary condition; Neumann boundary condition; eigenvalue asymptotics; dimension reduction; quantum waveguides; mean curvature; eigenvalue asymptotics; quantum waveguides; Laplacian in tubes; Dirichlet boundary condition; Neumann boundary condition; dimension reduction; mean curvature
UR - http://eudml.org/doc/261907
ER -

References

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  5. Friedlander, L., Solomyak, M., On the spectrum of the Dirichlet Laplacian in a narrow infinite strip, Spectral Theory of Differential Operators American Mathematical Society, Translations Series 2, 225, Advances in the Mathematical Sciences 62 Providence, RI 103-116 (2008). (2008) Zbl1170.35487MR2509778
  6. Friedlander, L., Solomyak, M., 10.1007/s11856-009-0032-y, Isr. J. Math. 170 (2009), 337-354. (2009) Zbl1173.35090MR2506330DOI10.1007/s11856-009-0032-y
  7. Krejčiřík, D., 10.1051/cocv:2008035, ESAIM, Control Optim. Calc. Var. 15 (2009), 555-568. (2009) Zbl1173.35618MR2542572DOI10.1051/cocv:2008035
  8. Krejčiřík, D., Raymond, N., Tušek, M., The magnetic Laplacian in shrinking tubular neighbourhoods of hypersurfaces, J. Geom. Anal (to appear). 
  9. Lampart, J., Teufel, S., Wachsmuth, J., Effective Hamiltonians for thin Dirichlet tubes with varying cross-section, Mathematical Results in Quantum Physics Proceedings of the QMath11 Conference 2010, Czech Republic World Scientific, Hackensack (2011), 183-189. (2011) Zbl1238.81100MR2885171
  10. Schatzman, M., 10.1080/00036819608840461, Appl. Anal. 61 (1996), 293-306. (1996) Zbl0865.35098MR1618236DOI10.1080/00036819608840461

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