University of Jyväskylä

Dissertation: 5.5.2017 Planar Sobolev extension domains (Zhang)

Start date: May 05, 2017 12:00 PM

End date: May 05, 2017 03:00 PM

Location: Mattilanniemi

Yi Zhang
Yi Zhang

M.Sc. Yi Zhang defends his doctoral dissertation in Mathematics ”Planar Sobolev extension domains”. Opponent Adjunct Professor Ritva Hurri-Syrjänen (University of Helsinki) and custos Professor Pekka Koskela (University of Jyväskylä). The doctoral dissertation is held in English.

In science, for example, one usually meets the following kind of problems: given data on some certain sets coming from observation, can one predict or recover the data at every point in the whole space with minimal error, or minimal energy? If so, what kind of sets are good for the prediction process?

In the mathematical words, we transfer this type of questions into the following question: given a function f on a subset of the whole space, is it possible to find a global function F in some certain functional spaces such that F coincides with f on the original set.

In the dissertation, we study the problem for the case where the whole space is the plane, and the set in question is priorly assumed to be open and simply connected. The functional spaces under consideration are Sobolev spaces; they are functions whose (pDirichlet) energy are finite. With the help of conformal geometry, we give geometric characterizations for those sets which admit every Sobolev function f defined in them has an extension to a global Sobolev function F. We expect that people working in partial differential equations and complex analysis would benefit from our results.

Background:

2009–2013 Math & Applied Math, Beihang University, Beijing, China, Bachelor of Science. 2013–2014 Math, University of Jyväskylä, Jyväskylä, Finland, Master of Science.

The dissertation is published in the series Department of Mathematics and Statistics, University of Jyväskylä / Report, number 159, 13 pp., ISBN: 978-951-39-7035-2. It is available at the University Library’s Publications Unit, tel. +358 (0)40 805 3825. Permanent link to the dissertation: http://urn.fi/URN:ISBN:978-951-39-7036-9

Abstract:

This doctoral thesis deals with geometric characterizations of bounded planar simply connected Sobolev extension domains. It consists of three papers. In the first and third papers we give full geometric characterizations of W1,p-extension domains for 1 < p < 2 and p = 1, respectively. The second paper establishes a density result for Sobolev functions on planar domains, necessary for the solution for the case p = 1. Combining with the known results, we obtain a full geometric characterization of W1,p-extension domains for every 1 ≤ p ≤∞.