Harmonic reflection in quasicircles and well-posedness of a Riemann-Hilbert problem on quasidisks

详细信息    查看全文

• 作者：Eric Schippers^a ; ^{eric_schippers@umanitoba.ca} ; Wolfgang Staubach^b ; ^{wulf@math.uu.se}
• 关键词：Riemann&ndash ; Hilbert problem ; Harmonic reflection ; Limiting Cauchy integral ; Quasidisks ; Dirichlet space
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 April 2017
• 年：2017
• 卷：448
• 期：2
• 页码：864-884
• 全文大小：502 K
• 卷排序：448

文摘

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.