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The Resource Topics in harmonic analysis and partial differential equations : extension theorems and geometric maximum principles

Topics in harmonic analysis and partial differential equations : extension theorems and geometric maximum principles

Creator
Author
Summary
The present thesis consists of two main parts. In the first part, we prove that a function defined on a closed subset of a geometrically doubling quasi-metric space which satisfies a Hölder-type condition may be extended to the entire space with preservation of regularity. The proof proceeds along the lines of theoriginal work of Whitney in 1934 and yields a linear extension operator. A similar extension result is also proved in the absence of the geometrically doubling hypothesis, albeit the resulting extension procedure is nonlinear in this case. The results presented in this part are based upon work done in collaboration M. Mitrea. In the second part of the thesis we prove that an open, proper, nonempty subset of [R]n is a locally Lyapunov domain if and only if it satisfies a uniform hour-glass condition. The latter is a property of a purely geometrical nature, which amounts to the ability of threading the boundary, at any location, in between the two rounded components of a certain fixed region, whose shape resembles that of an ordinary hour-glass, suitably re-positioned. The limiting cases of the result are as follows: Lipschitz domains may be characterized by a uniform double cone condition, whereas domains of class [C]1,1 may be characterized by a uniform two-sided ball condition. Additionally, we discuss a sharp generalization of the Hopf-Oleinik boundary point principle for domains satisfying a one-sided, interior pseudo-ball condition, for semi-elliptic operators with singular drift. This, in turn, is used to obtain a sharp version of Hopf's Strong Maximum Principle for second-order, non-divergence form differential operators with singular drift. This part of my thesis originates from a recent paper in collaboration with D. Brigham, V. Maz'ya, M. Mitrea, and E. Ziad e
Language
eng
Work
Publication
Extent
1 online resource (vi, 174 pages)
Note
Advisor: Marius Mitrea
Instance
Label
Topics in harmonic analysis and partial differential equations : extension theorems and geometric maximum principles
Title
Topics in harmonic analysis and partial differential equations
Title remainder
extension theorems and geometric maximum principles
Creator
Author
Subject
Language
eng
Summary
The present thesis consists of two main parts. In the first part, we prove that a function defined on a closed subset of a geometrically doubling quasi-metric space which satisfies a Hölder-type condition may be extended to the entire space with preservation of regularity. The proof proceeds along the lines of theoriginal work of Whitney in 1934 and yields a linear extension operator. A similar extension result is also proved in the absence of the geometrically doubling hypothesis, albeit the resulting extension procedure is nonlinear in this case. The results presented in this part are based upon work done in collaboration M. Mitrea. In the second part of the thesis we prove that an open, proper, nonempty subset of [R]n is a locally Lyapunov domain if and only if it satisfies a uniform hour-glass condition. The latter is a property of a purely geometrical nature, which amounts to the ability of threading the boundary, at any location, in between the two rounded components of a certain fixed region, whose shape resembles that of an ordinary hour-glass, suitably re-positioned. The limiting cases of the result are as follows: Lipschitz domains may be characterized by a uniform double cone condition, whereas domains of class [C]1,1 may be characterized by a uniform two-sided ball condition. Additionally, we discuss a sharp generalization of the Hopf-Oleinik boundary point principle for domains satisfying a one-sided, interior pseudo-ball condition, for semi-elliptic operators with singular drift. This, in turn, is used to obtain a sharp version of Hopf's Strong Maximum Principle for second-order, non-divergence form differential operators with singular drift. This part of my thesis originates from a recent paper in collaboration with D. Brigham, V. Maz'ya, M. Mitrea, and E. Ziad e
Cataloging source
MUU
http://library.link/vocab/creatorName
Alvarado, Ryan
Degree
M.A.
Dissertation note
Thesis
Dissertation year
2011.
Government publication
government publication of a state province territory dependency etc
Granting institution
University of Missouri--Columbia,
Index
no index present
Literary form
non fiction
Nature of contents
dictionaries
Label
Topics in harmonic analysis and partial differential equations : extension theorems and geometric maximum principles
Instantiates
Publication
Contributor
Thesis advisor
Note
Advisor: Marius Mitrea
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier.
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent.
Control code
876610708
Extent
1 online resource (vi, 174 pages)
Form of item
online
Media category
computer
Media MARC source
rdamedia.
Media type code
  • c
Specific material designation
remote
System control number
(OCoLC)876610708
Label
Topics in harmonic analysis and partial differential equations : extension theorems and geometric maximum principles
Publication
Contributor
Thesis advisor
Note
Advisor: Marius Mitrea
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier.
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent.
Control code
876610708
Extent
1 online resource (vi, 174 pages)
Form of item
online
Media category
computer
Media MARC source
rdamedia.
Media type code
  • c
Specific material designation
remote
System control number
(OCoLC)876610708

Subject

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