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Sibirsk. Mat. Zh., 2019, Volume 60, Number 1, Pages 201–213 (Mi smj3070)  

Reduction of vector boundary value problems on Riemann surfaces to one-dimensional problems

E. V. Semenkoab

a Novosibirsk State Technical University, Novosibirsk, Russia
b Novosibirsk State Pedagogical University, Novosibirsk, Russia

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.

DOI: https://doi.org/10.33048/smzh.2019.60.117

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English version:
Siberian Mathematical Journal, 2019, 60:1, 153–163

Bibliographic databases:

UDC: 517.53/55
MSC: 35R30
Received: 09.01.2018
Revised: 20.08.2018
Accepted:17.10.2018

Citation: E. V. Semenko, “Reduction of vector boundary value problems on Riemann surfaces to one-dimensional problems”, Sibirsk. Mat. Zh., 60:1 (2019), 201–213; Siberian Math. J., 60:1 (2019), 153–163

Citation in format AMSBIB
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\pages 201--213
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