A jump problem for β-analytic functions in domains with fractal boundaries

Abstract

Let γ be a simple closed Jordan curve in the complex plane ℂ, Ω⁺ and Ω⁻ the corresponding domains in ℂ, with 0∈Ω⁺ and ∞∈Ω⁻.

The classes of complex valued functions satisfying some boundary conditions as well as integral representations for them are considered.

Main goal of this paper is the study of the standard jump Riemann boundary value problem over a fractal curve γ

$$\varphi^{+}(t)-\varphi^{-}(t)=f(t),\quad{}t\in\gamma,$$

where φ ^±(t) are the boundary values of the β-analytic function φ at the point t, approaching the boundary from Ω⁺ and Ω⁻ respectively.