20.9.2019 Cardioid-type domains and regularity of homeomorphic extensions

Extension problem is a classical problem in function theory. In his doctoral dissertation M.Sc Haiqing Xu studied existence and regularity of extension mappings. Start with a homeomorphism from the unit circle onto a Jordan curve, by the Schoenflies theorem there is a homeomorphic extension to the whole plane.
Published
20.9.2019

How good such extension can we find? 

We first care about extensions to the interior domain. If additionally the image domain is bounded convex, it was found that the extension can be harmonic diffeomorphic. Our contribution is on a special case of non-convex domains--internal chord-arc domains. We prove that a diffeomorphic extension can be found in this case. Moreover we study the relationship between the weighted Sobolev regularity of above extension and the regularity of the boundary homeomorphism. 

For extensions to the exterior domain, we study a special case: conformal boundary homeomorphisms. A nice extension can be found if the boundary of image domain is sufficiently regular, e.g. satisfies the three-point condition. Our work is on cardioid-type domains, which do not satisfy three-point condition. We show that a homeomorphic extension of finite distortion can be found in this case.
Furthermore we study the optimal regularity of such extension, in terms of the integrability degree of the derivatives and the distortion, and these for the inverse.   

Abstract:
Given a homeomorphism from the unit circle onto the boundary of an internal chord-arc domain, we study the existence and regularity of a diffeomorphic extension to the unit disk. If additionally the boundary homeomorphism is conformal and the image domain is cardioid-type, there is a homeomorphic extension of finite distortion to the whole plane. Moreover we achieve the optimal regularity of such extension.   

The dissertation is published in JYU dissertation series, number 126, 2019, Ä¢¹½Ö±²¥.  ISBN 978-951-39-7841-9 (pdf) ISSN 2489-9003
Link to publication: 

M.Sc. Haiqing Xu defends his doctoral dissertation in Mathematics "Cardioid-type domains and regularity of homeomorphic extensions" on Friday 20th of September at 12 o'clock in Mattilanniemi, lecture hall MaA  . Opponent is  Professor Xiao Zhong from University of Helsinki and Custos is Professor Pekka Koskela from Ä¢¹½Ö±²¥. The doctoral dissertation is held in English.

For further information:
M.Sc Haiqing Xu, haiqing.h.xu@student.jyu.fi
Comminications officer Tanja Heikkinen, tanja.s.heikkinen@jyu.fi, puh. 050 581 8351
The Faculty of Mathematics and Science
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