The spectrum of geodesic balls on spherically symmetric manifolds

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F62690094%3A18470%2F17%3A50013788" target="_blank" >RIV/62690094:18470/17:50013788 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.4310/CAG.2017.v25.n3.a1" target="_blank" >http://dx.doi.org/10.4310/CAG.2017.v25.n3.a1</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4310/CAG.2017.v25.n3.a1" target="_blank" >10.4310/CAG.2017.v25.n3.a1</a>

Jazyk výsledku

angličtina

Název v původním jazyce

The spectrum of geodesic balls on spherically symmetric manifolds

Popis výsledku v původním jazyce

We study the Dirichlet spectrum of the Laplace operator on geodesic balls centred at a pole of spherically symmetric manifolds. We first derive a Hadamard-type formula for the dependence of the first eigenvalue lambda(1) on the radius r of the ball, which allows us to obtain lower and upper bounds for.1 in specific cases. For the sphere and hyperbolic space, these bounds are asymptotically sharp as r approaches zero and we see that while in two dimensions lambda(1) is bounded from above by the first two terms in the asymptotics for small r, for dimensions four and higher the reverse inequality holds. In the general case we derive the asymptotic expansion of lambda(1) for small radius and determine the first three terms explicitly. For compact manifolds we carry out similar calculations as the radius of the geodesic ball approaches the diameter of the manifold. In the latter case we show that in even dimensions there will always exist logarithmic terms in these expansions.

Název v anglickém jazyce

The spectrum of geodesic balls on spherically symmetric manifolds

Popis výsledku anglicky

We study the Dirichlet spectrum of the Laplace operator on geodesic balls centred at a pole of spherically symmetric manifolds. We first derive a Hadamard-type formula for the dependence of the first eigenvalue lambda(1) on the radius r of the ball, which allows us to obtain lower and upper bounds for.1 in specific cases. For the sphere and hyperbolic space, these bounds are asymptotically sharp as r approaches zero and we see that while in two dimensions lambda(1) is bounded from above by the first two terms in the asymptotics for small r, for dimensions four and higher the reverse inequality holds. In the general case we derive the asymptotic expansion of lambda(1) for small radius and determine the first three terms explicitly. For compact manifolds we carry out similar calculations as the radius of the geodesic ball approaches the diameter of the manifold. In the latter case we show that in even dimensions there will always exist logarithmic terms in these expansions.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

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OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2017

Kód důvěrnosti údajů

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

Název periodika

COMMUNICATIONS IN ANALYSIS AND GEOMETRY

ISSN

1019-8385

e-ISSN

—

Svazek periodika

25

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

38

Strana od-do

507-544

Kód UT WoS článku

000410554400001

EID výsledku v databázi Scopus

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