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An extension of the Bianchi-Egnell stability estimate to Bakry, Gentil, and Ledoux's generalization of the Sobolev inequality to continuous dimensions and applications

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Title
An extension of the Bianchi-Egnell stability estimate to Bakry, Gentil, and Ledoux's generalization of the Sobolev inequality to continuous dimensions and applications
Name (type = personal)
NamePart (type = family)
Seuffert
NamePart (type = given)
Francis
NamePart (type = date)
1986-
DisplayForm
Francis Seuffert
Role
RoleTerm (authority = RULIB)
author
Name (type = personal)
NamePart (type = family)
Carlen
NamePart (type = given)
Eric
DisplayForm
Eric Carlen
Affiliation
Advisory Committee
Role
RoleTerm (authority = RULIB)
chair
Name (type = corporate)
NamePart
Rutgers University
Role
RoleTerm (authority = RULIB)
degree grantor
Name (type = corporate)
NamePart
Graduate School - New Brunswick
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school
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Text
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theses
OriginInfo
DateCreated (qualifier = exact)
2016
DateOther (qualifier = exact); (type = degree)
2016-05
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2016
Place
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xx
Language
LanguageTerm (authority = ISO639-2b); (type = code)
eng
Abstract (type = abstract)
The main result of this dissertation is an extension of a stability estimate of the Sobolev Inequality established by Bianchi and Egnell in cite{BiEg}. Bianchi and Egnell's Stability Estimate answers the question raised by H. Brezis and E. H. Lieb in cite{BrLi}: ``Is there a natural way to bound $| abla varphi |_2^2 - C_N^2 | varphi |_frac{2N}{N-2}^2$ from below in terms of the `distance' of $varphi$ from the manifold of optimizers in the Sobolev Inequality?'' Establishing stability estimates - also known as quantitative versions of sharp inequalities - of other forms of the Sobolev Inequality, as well as other inequalities, is an active topic. See cite{CiFu}, cite{DoTo}, and cite{FiMa}, for stability estimates involving Sobolev inequalities and cite{CaFi}, cite{DoTo}, and cite{FuMa} for stability estimates on other inequalities. In this dissertation, we extend Bianchi and Egnell's Stability Estimate to a Sobolev Inequality for ``continuous dimensions.'' Bakry, Gentil, and Ledoux have recently proved a sharp extension of the Sobolev Inequality for functions on $mathbb{R}_+ imes mathbb{R}^n$, which can be considered as an extension to ``continuous dimensions.'' V. H. Nguyen determined all cases of equality. The dissertation extends the Bianchi-Egnell stability analysis for the Sobolev Inequality to this ``continuous dimensional'' generalization. The secondary result of this dissertation is a sketch of the proof of an extension of a stability estimate of a single case of a sharp Gagliardo-Nirenberg inequality to a whole family of Gagliardo-Nirenberg inequalities, whose sharp constants and extremals were calculated by Del Pino and Dolbeault in cite{DeDo}. The original stability estimate for the Gagliardo-Nirenberg inequality was stated and proved by E. Carlen and A. Figalli in cite{CaFi}. The proof for its extension to the entire class of sharp Gagliardo-Nirenberg inequalities of Del Pino and Dolbeault is a direct application of the extension of the Bianchi-Egnell Stability Estimate to Bakry, Gentil, and Ledoux's extension of the Sobolev Inequality to continuous dimensions.
Subject (authority = RUETD)
Topic
Mathematics
Subject (authority = ETD-LCSH)
Topic
Functional analysis
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Title
Rutgers University Electronic Theses and Dissertations
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ETD_7246
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electronic resource
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application/pdf
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text/xml
Extent
1 online resource (v, 95 p.)
Note (type = degree)
Ph.D.
Note (type = bibliography)
Includes bibliographical references
Note (type = statement of responsibility)
by Francis Seuffert
RelatedItem (type = host)
TitleInfo
Title
Graduate School - New Brunswick Electronic Theses and Dissertations
Identifier (type = local)
rucore19991600001
Location
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NjNbRU
Identifier (type = doi)
doi:10.7282/T3377BXB
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The author owns the copyright to this work.
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Name
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Seuffert
GivenName
Francis
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Permission or license
DateTime (encoding = w3cdtf); (qualifier = exact); (point = start)
2016-04-14 17:53:14
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Francis Seuffert
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Affiliation
Rutgers University. Graduate School - New Brunswick
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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2016-04-15T14:02:49
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2016-04-15T14:02:49
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