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Capacity estimates and Poincaré inequalities for the weighted bow-tie
Linköping University, Department of Mathematics. Linköping University, Faculty of Science & Engineering.
2017 (English)Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
Abstract [en]

We give a short introduction to various concepts related to the theory of p-harmonic functions on Rn, and some modern generalizations of these concepts to general metric spaces. The article Björn-Björn-Lehrbäck [6] serves as the starting point of our discussion. In [6], among other things, estimates of the variational capacity for thin annuli in metric spaces are given under the assumptions of a Poincaré inequality and an annular decay property. Most of the parameters in the various results of the article are proven to be sharp by counterexamples at the end of the article. The main result of this thesis is the verification of the sharpness of a parameter.

At the center of our discussion will be a concrete metric subspace of weighted Rn, namely the so-called weighted bow-tie, where the weight function is assumed to be radial. A similar space was used in [6] to verify the sharpness of several parameters. We show that under the assumption that the variational p-capacity is nonzero for any ball centered at the origin, the p-Poincaré inequality holds in Rif and only if it holds on the corresponding bow-tie

Finally, we consider a concrete weight function, show that it is a Muckenhoupt A1 weight, and use this to construct a counterexample establishing the sharpness of the parameter in the above mentioned result from [6].

Place, publisher, year, edition, pages
2017. , p. 84
Keywords [en]
Bow-tie, Capacity, Metric space, Muckenhoupt weight, Poincaré inequality, Upper gradient, Weight function
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:liu:diva-138111ISRN: LiTH-MAT-EX--2017/07--SEOAI: oai:DiVA.org:liu-138111DiVA, id: diva2:1107147
Subject / course
Applied Mathematics
Supervisors
Examiners
Available from: 2017-06-09 Created: 2017-06-09 Last updated: 2017-06-09Bibliographically approved

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CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf