Spectrum of the Laplacian on a Domain Perturbed by Small Resonators

Kód výsledku v 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>

Nalezeny alternativní kódy

RIV/62690094:18470/23:50021108

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Spectrum of the Laplacian on a Domain Perturbed by Small Resonators

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Spectrum of the Laplacian on a Domain Perturbed by Small Resonators

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA22-18739S" target="_blank" >GA22-18739S: Asymptotická a spektrální analýza operátorů v matematické fyzice</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2023

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

SIAM Journal on Mathematical Analysis

ISSN

0036-1410

e-ISSN

1095-7154

Svazek periodika

55

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

36

Strana od-do

3677-3712

Kód UT WoS článku

001114782600011

EID výsledku v databázi Scopus

2-s2.0-85172657366