Spectrum of the Laplacian on a Domain Perturbed by Small Resonators
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F61389005%3A_____%2F23%3A00582305" target="_blank" >RIV/61389005:_____/23:00582305 - isvavai.cz</a>
Alternative codes found
RIV/62690094:18470/23:50021108
Result on the web
<a href="https://doi.org/10.1137/22M148207X" target="_blank" >https://doi.org/10.1137/22M148207X</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/22M148207X" target="_blank" >10.1137/22M148207X</a>
Alternative languages
Result language
angličtina
Original language name
Spectrum of the Laplacian on a Domain Perturbed by Small Resonators
Original language description
It is widely known that the spectrum of the Dirichlet Laplacian is stable under small perturbations of a domain, while in the case of the Neumann or mixed boundary conditions the spectrum may abruptly change. In this work we discuss an example of such a domain perturbation. Let Ω be a (not necessarily bounded) domain in ℝn. We perturb it to (Equation presented), where Sk, ε are closed surfaces with small suitably scaled holes (''windows'') through which the bounded domains enclosed by these surfaces (''resonators'') are connected to the outer domain. When ε goes to zero, the resonators shrink to points. We prove that in the limit ε → 0 the spectrum of the Laplacian on Ωε with the Neumann boundary conditions on Sk, ε and the Dirichlet boundary conditions on the outer boundary converges to the union of the spectrum of the Dirichlet Laplacian on Ω and the numbers γk, k = 1, ..., m, being equal to 1/4 times the limit of the ratio between the capacity of the kth window and the volume of the kth resonator. We obtain an estimate on the rate of this convergence with respect to the Hausdorff-type metrics. Also, an application of this result is presented: we construct an unbounded waveguide-like domain with inserted resonators such that the eigenvalues of the Laplacian on this domain lying below the essential spectrum threshold do coincide with the prescribed numbers.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
<a href="/en/project/GA22-18739S" target="_blank" >GA22-18739S: Asymptotic and spectral analysis of operators in mathematical physics</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2023
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
SIAM Journal on Mathematical Analysis
ISSN
0036-1410
e-ISSN
1095-7154
Volume of the periodical
55
Issue of the periodical within the volume
4
Country of publishing house
US - UNITED STATES
Number of pages
36
Pages from-to
3677-3712
UT code for WoS article
001114782600011
EID of the result in the Scopus database
2-s2.0-85172657366