Non-rectifiable Riemann boundary value problem for bi-analytic functions

Abstract

We solve the Riemann boundary value problem for bi-analytic functions on a contour consisting of non-rectifiable closed Jordan curves. In the classical results on this problem, curves are piecewise-smooth. But the Riemann boundary value problem has a lot of applications in the theory of solid media and other fields, and some of these applications allow fractal (and, consequently, non-rectifiable) contours. In the present paper, we use the technique of integration over non-rectifiable curves, which was recently offered by the authors for the study of the Riemann problem on such contours.