Riemann-Hilbert Boundary Value Problem with Piecewise Constant Transition Function
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Riemann-Hilbert Boundary Value Problem with Piecewise Constant Transition Function G. Giorgadze1
· G. Gulagashvili1
Received: 14 May 2020 / Revised: 28 September 2020 / Accepted: 23 October 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We prove that, for any Fuchsian system of differential equations on the Riemann sphere, there exists a rational matrix function whose partial indices coincide with the splitting type of the canonical vector bundle induced from the Fuchsian system. From this, we obtain solution of the Riemann-Hilbert boundary value problem for piecewise constant matrix function in terms of holomorphic sections of vector bundle and calculate the partial indices of the problem. Keywords Partial indices · Holomorphic vector bundle · Splitting type · Fuchsian system · Birkhoff factorization Mathematics Subject Classification (2010) 30E25 · 32L05
1 Introduction The Riemann-Hilbert boundary value problem, which is concerned with finding a piecewise holomorphic vector function Φ = (Φ + , Φ − ) on the extended complex plane C = CP 1 , such that Φ + is holomorphic in the interior of a given closed contour Γ, Φ − is holomorphic in the exterior of Γ, both are continuously extendable to Γ and satisfy the boundary condition Φ + = f Φ − , where f is a given transition matrix function on Γ , arises as an auxiliary problem in unfinished work of B. Riemann [15]. The main goal of Riemann’s research is the construction of the system of differential equations on the Riemann sphere CP 1 with simple poles at given points s1 , ..., sm ∈ CP 1 and given monodromy representation, i.e., representation of the fundamental group of the sphere punctured marked points in the complex linear group: π1 (CP 1 \ {s1 , ..., sm }, z0 ) → GL(n, C). Following the Riemannian scheme, J. Plemelj [14] proved the existence of the solution of the boundary value problem for a piecewise constant boundary matrix function. He reduced the problem to the
G. Giorgadze
[email protected] 1
Faculty of Exact and Natural Sciences, Tbilisi State University, 3 I.Chavchavadze ave., 0179 Tbilisi, Georgia
G. Giorgadze and G. Gulagashvili
case of continuous boundary matrix functions and solved the monodromy problem for the class of regular systems. In [14], Plemelj introduced a formula for the boundary values of analytic function, which today is known as the Sokhotski-Plemelj formula. For the investigation of the boundary value problem with H¨older continuous transition function, the representation of the solution of the problem by Cauchy-type integral and Sokhotski-Plemelj formula for the boundary values of the function are used. Due to this approach, the index (in scalar case) (see [6]) and partial indices (in matrix case) [13] were discovered. Index for scalar problem is a complete invariant; besides, in this case, the solution of the problem is expressed analytically by the exponent of Cauchy-type integral from the transition function [6]. The solution of the matrix boundary value problem in explicit form is
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