The Riemann boundary value problem on non-rectifiable curves and related questions

Abstract

We present a brief survey of results on the Riemann boundary value problem for non-rectifiable curves. The solution of the Riemann problem for piecewise smooth curves and arcs was one of the well-known achievements of the complex analysis in the first half of the twentieth century. But, for non-rectifiable contours, the problem was solved only during the latter three decades. In the present paper, we consider this solution and certain related questions such as various generalizations of the Cauchy integral over non-rectifiable curves. We describe the technique of investigations, recent results and open problems.

Keywords:

• Riemann boundary value problem
• non-rectifiable curve
• metric dimensions
• Cauchy integral

AMS Subject Classifications:

• Primary: 30E20
• 30E25

Acknowledgments

This work was supported by Russian Foundation for Basic Research, grants 12-01-00636-a, 13-01-00322-a, and 12-01-97015-r-povolzhie-a.