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Siberian Mathematical Journal

, Volume 60, Issue 1, pp 153–163 | Cite as

Reduction of Vector Boundary Value Problems on Riemann Surfaces to One-Dimensional Problems

  • E. V. SemenkoEmail author
Article
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Abstract

This article lays foundations for the theory of vector conjugation boundary value problems on a compact Riemann surface of arbitrary positive genus. The main constructions of the classical theory of vector boundary value problems on the plane are carried over to Riemann surfaces: reduction of the problem to a system of integral equations on a contour, the concepts of companion and adjoint problems, as well as their connection with the original problem, the construction of a meromorphic matrix solution. We show that each vector conjugation boundary value problem reduces to a problem with a triangular coefficient matrix, which in fact reduces the problem to a succession of one-dimensional problems. This reduction to the well-understood one-dimensional problems opens up a path towards a complete construction of the general solution of vector boundary value problems on Riemann surfaces.

Keywords

Riemann surface vector conjugation boundary value problem companion problem adjoint problem holomorphic vector bundle 

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Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Novosibirsk State Technical University Novosibirsk State Pedagogical UniversityNovosibirskRussia

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