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Harmonic reflection in quasicircles and well-posedness of a Riemann-Hilbert problem on quasidisks
Univ Manitoba, Dept Math, Canada..
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
2017 (English)In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 448, no 2, 864-884 p.Article in journal (Refereed) Published
Abstract [en]

A complex harmonic function of finite Dirichlet energy on a Jordan domain has boundary values in a certain conformally invariant sense, by a construction of H. Osborn. We call the set of such boundary values the Douglas-Osborn space. One may then attempt to solve the Dirichlet problem on the complement for these boundary values. This defines a reflection of harmonic functions. We show that quasicircles are precisely those Jordan curves for which this reflection is defined and bounded. We then use a limiting Cauchy integral along level curves of Green's function to show that the Plemelj-Sokhotski jump formula holds on quasicircles with boundary data in the Douglas-Osborn space. This enables us to prove the well-posedness of a Riemann-Hilbert problem with boundary data in the Douglas Osborn space on quasicircles.

Place, publisher, year, edition, pages
2017. Vol. 448, no 2, 864-884 p.
Keyword [en]
Riemann-Hilbert problem, Harmonic reflection, Limiting Cauchy integral, Quasidisks, Dirichlet space
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:uu:diva-329134DOI: 10.1016/j.jmaa.2016.11.041ISI: 000403630400008OAI: oai:DiVA.org:uu-329134DiVA: diva2:1148173
Funder
Wenner-Gren Foundations
Available from: 2017-10-10 Created: 2017-10-10 Last updated: 2017-10-10Bibliographically approved

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Staubach, Wolfgang

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