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.
Mots-clés : laplacian in tubes, Dirichlet and Neumann boundary conditions, dimension reduction, norm-resolvent convergence, binding effect of curvature, waveguides
@article{COCV_2009__15_3_555_0,
author = {Krej\v{c}i\v{r}{\'\i}k, David},
title = {Spectrum of the laplacian in a narrow curved strip with combined {Dirichlet} and {Neumann} boundary conditions},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {555--568},
publisher = {EDP-Sciences},
volume = {15},
number = {3},
year = {2009},
doi = {10.1051/cocv:2008035},
mrnumber = {2542572},
zbl = {1173.35618},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2008035/}
}
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 PY - 2009 SP - 555 EP - 568 VL - 15 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2008035/ DO - 10.1051/cocv:2008035 LA - en ID - COCV_2009__15_3_555_0 ER -
%0 Journal Article %A Krejčiřík, David %T Spectrum of the laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions %J ESAIM: Control, Optimisation and Calculus of Variations %D 2009 %P 555-568 %V 15 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2008035/ %R 10.1051/cocv:2008035 %G en %F COCV_2009__15_3_555_0
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, Tome 15 (2009) no. 3, pp. 555-568. doi : 10.1051/cocv:2008035. http://www.numdam.org/articles/10.1051/cocv:2008035/
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