| Title:
|
Spectrum of the Laplacian in narrow tubular neighbourhoods of hypersurfaces with combined Dirichlet and Neumann boundary conditions (English) |
| Author:
|
Krejčiřík, David |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
139 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
185-193 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Laplacian in tubes |
| Keyword:
|
Dirichlet boundary condition |
| Keyword:
|
Neumann boundary condition |
| Keyword:
|
eigenvalue asymptotics |
| Keyword:
|
dimension reduction |
| Keyword:
|
quantum waveguides |
| Keyword:
|
mean curvature |
| MSC:
|
35J05 |
| MSC:
|
35P15 |
| MSC:
|
49R05 |
| MSC:
|
58J50 |
| MSC:
|
81Q15 |
| idZBL:
|
Zbl 06362252 |
| idMR:
|
MR3238833 |
| DOI:
|
10.21136/MB.2014.143848 |
| . |
| Date available:
|
2014-07-14T08:13:09Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143848 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[6] Friedlander, L., Solomyak, M.: On the spectrum of the Dirichlet Laplacian in a narrow strip.Isr. J. Math. 170 (2009), 337-354. Zbl 1173.35090, MR 2506330, 10.1007/s11856-009-0032-y |
| Reference:
|
[7] Krejčiřík, D.: Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions.ESAIM, Control Optim. Calc. Var. 15 (2009), 555-568. Zbl 1173.35618, MR 2542572, 10.1051/cocv:2008035 |
| Reference:
|
[8] Krejčiřík, D., Raymond, N., Tušek, M.: The magnetic Laplacian in shrinking tubular neighbourhoods of hypersurfaces.J. Geom. Anal (to appear). |
| Reference:
|
[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. Zbl 1238.81100, MR 2885171 |
| Reference:
|
[10] Schatzman, M.: On the eigenvalues of the Laplace operator on a thin set with Neumann boundary conditions.Appl. Anal. 61 (1996), 293-306. Zbl 0865.35098, MR 1618236, 10.1080/00036819608840461 |
| . |
