Structure of non-rectifiable curves and solvability of the jump problem

Abstract

Introduction We consider the following boundary value problem for holomorphic functions. Let Γ be a closed Jordan curve on the complex plane C bounding finite domain D, and D− = C \D+. We seek a holomorphic in C Γ function Φ(z) such that Φ(∞) = 0, the boundary values limD+ z→t Φ(z) ≡ Φ(t) and limD− z→t Φ(z) ≡ Φ−(t) exist for any t ∈ Γ, and Φ(t) = G(t)Φ−(t) + g(t), t ∈ Γ. (1) This boundary value problem is called the Riemann problem. It is well known and has numerous applications in elasticity theory, hydro and aerodynamics and so on. If G(t) ≡ 1, then it turns to so called jump problem: