The spectrum of geodesic balls on spherically symmetric manifolds
The result's identifiers
Result code in 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>
Result on the web
<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>
Alternative languages
Result language
angličtina
Original language name
The spectrum of geodesic balls on spherically symmetric manifolds
Original language description
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.
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
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2017
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
COMMUNICATIONS IN ANALYSIS AND GEOMETRY
ISSN
1019-8385
e-ISSN
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Volume of the periodical
25
Issue of the periodical within the volume
3
Country of publishing house
US - UNITED STATES
Number of pages
38
Pages from-to
507-544
UT code for WoS article
000410554400001
EID of the result in the Scopus database
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