Boundary value problems in the theory of analytic functions in Hölder classes on Riemann surfaces

Abstract

This paper is based on the papers, written mainly during the last decade, on the investigation and solution of boundary value problems in the theory of analytic functions on finite oriented Riemann surfaces. In the introduction we give a short survey of the fundamental work on this topic, beginning with the classical results of Riemann and right up to the research of contemporary authors. The main content of this paper consists of the material presented in §§ 2-6. Here we find explicit expressions for analogues to the Cauchy kernel, we construct the general solution, and give a complete sketch of the solubility of Riemann's boundary value problem for a single unknown piecewise meromorphic function in the case of composite contours on a closed oriented Riemann surface. In this context we give a new version for the solution of Jacobi's inversion problem. In §§ 7 and 8 we consider the case of Riemann surfaces of algebraic functions, we investigate the hyperelliptic case in detail, and we give applications. § 9 is devoted to boundary value problems on Riemann surfaces with boundary. We present the ideas of the method of passage to the double and the method of pasting. In § 10 we give a survey of results on the Hilbert boundary value problem for multiply-connected domains and we mention some new results of the author. In § 11 we give a survey of the literature on our topic that is not treated in the main part of the paper.