Well-posedness of a Riemann–Hilbert problem on d-regular quasidisks

Abstract

Abstract. We prove the well-posedness of a Riemann-Hilbert problem on d-regular quasidisks, with boundary data in a class of Besov spaces.

Analysis on quasidisks; a unified approach through transmission and jump problems

“…The CNT boundary values of elements of D harm (Ω k ) are not continuous, so this technology was not available. The jump decomposition was shown to hold for a range of Besov spaces of boundary values by the authors, for d-regular quasidisks [57], which are not necessarily rectifiable. This result did not include the case of boundary values of the Dirichlet space, so it was also not available for use here.…”

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

“…The boundaries of quasidisks, called quasicircles, can be very rough curves with Hausdorff dimensions arbitrarily close to 2; examples include fractal-type curves such as snowflakes. The authors (together with D. Radnell) had previously studied Problem 1 in the case when the boundary curve is assumed to be a Weil-Petersson class quasicircle [15], and Schippers and Staubach extended the study of Problem 1 to d-regular quasicircles (which are a generalization of Ahlfors-regular quasicircles) where the boundary function was assumed to belong to a certain Besov space [18].…”

“…Here we consider Problem 1 when the domain is a general quasidisk and remove the assumption of d-regularity which was imposed in [18]. Moreover, as opposed to [18], we obtain the solution for an optimal class of boundary data and show that our solution yields yet another characterization of quasidisks. More precisely, to bypass the regularity issues of the boundary, we use a particular reflection of harmonic functions, and a limiting Cauchy integral.…”

“…Here we consider Problem 1 when the domain is a general quasidisk and remove the assumption of d-regularity which was imposed in [18]. Moreover, as opposed to [18], we obtain the solution for an optimal class of boundary data and show that our solution yields yet another characterization of quasidisks.…”