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

Сибирский математический журнал

E. V. Semenko¹ 

[1] Novosibirsk State Technical University Novosibirsk State Pedagogical University

关键词: Riemann surface;    vector conjugation boundary value problem;    companion problem;    adjoint problem;    holomorphic vector bundle;   

DOI  :  10.1134/S0037446619010178

学科分类：数学（综合）

来源: Izdatel stvo Instituta Matematiki Rossiiskoi Akademii Nauk

【 摘 要 】

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.

【 授权许可】

CC BY