Gradient ranges of bumps on the plane
Result description
We show that the gradient range of a $C^1$-smooth function $b$ on the plane is regularly closed (i.e., it is the closure of its interior), provided $b$ has non-empty bounded support and the gradient $grad b$ admits a modulus of continuity $omega = omega (t)$ that satisfies $omega (t)/sqrt{t} to 0$ as $t searrow 0$. Furthermore, under the same smoothness hypothesis, we show that the gradient range of a function $b fcolon Rn to R$ with non-empty bounded support has the topological dimension atleast two at points of a dense subset. The proof relies on a new Morse-Sard type result.
Keywords
The result's identifiers
Result code in IS VaVaI
Result on the web
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DOI - Digital Object Identifier
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Alternative languages
Result language
angličtina
Original language name
Gradient ranges of bumps on the plane
Original language description
We show that the gradient range of a $C^1$-smooth function $b$ on the plane is regularly closed (i.e., it is the closure of its interior), provided $b$ has non-empty bounded support and the gradient $grad b$ admits a modulus of continuity $omega = omega (t)$ that satisfies $omega (t)/sqrt{t} to 0$ as $t searrow 0$. Furthermore, under the same smoothness hypothesis, we show that the gradient range of a function $b fcolon Rn to R$ with non-empty bounded support has the topological dimension atleast two at points of a dense subset. The proof relies on a new Morse-Sard type result.
Czech name
Obory hodnot gradientů bumpů v rovině
Czech description
Obor hodnot gradientů $C^1$ hladké funkce $b$ v rovině je regulárně uzavřený (tj. je uzávěrem svého vnitřku) za předpokladu, že $b$ má neprázdný omezený nosič a gradient $nabla b$ má modul spojitosti $omega = omega(t)$ který splňuje $omega(t)/sqrt{t} to 0$ pro $tsearrow 0$. Za stejného předpokladu hladkosti má obor hodnot gradientu funkce $b colon Rn to R$ s neprázdným omezeným nosičem topologickou dimenzi alespoň 2 v hustě mnoha bodech. Je dokázána a použita nová věta typu Morse-Sard.
Classification
Type
Jx - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Result continuities
Project
GP201/02/D111: Real Analytic Methods in the Calculus of Variations
Continuities
Z - Vyzkumny zamer (s odkazem do CEZ)
Others
Publication year
2005
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Proceedings of the American Mathematical Society
ISSN
0002-9939
e-ISSN
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Volume of the periodical
133
Issue of the periodical within the volume
6
Country of publishing house
US - UNITED STATES
Number of pages
8
Pages from-to
1699-1706
UT code for WoS article
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EID of the result in the Scopus database
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Basic information
Result type
Jx - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP
BA - General mathematics
Year of implementation
2005