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arXiv · 1604.08453

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

Eric Schippers·Wolfgang Staubach
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Abstract

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.

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Eric Schippers, Wolfgang Staubach. 2016-11-07. Harmonic reflection in quasicircles and well-posedness of a Riemann-Hilbert problem on quasidisks. https://arxiv.org/abs/1604.08453

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