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

TL;DR

This paper investigates harmonic reflection and boundary value problems on quasicircles, establishing conditions for well-posed Riemann-Hilbert problems using boundary data in the Douglas-Osborn space.

Contribution

It characterizes quasicircles via harmonic reflection boundedness and proves the Plemelj-Sokhotski formula and Riemann-Hilbert problem well-posedness for boundary data in the Douglas-Osborn space.

Findings

Quasicircles are exactly those Jordan curves with bounded harmonic reflection.

The Plemelj-Sokhotski jump formula holds on quasicircles for Douglas-Osborn boundary data.

Riemann-Hilbert problems are well-posed on quasicircles with this boundary data.

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.