Riemann–Hilbert Problems on a Cut Plane

Abstract

We consider the Riemann–Hilbert-type boundary-value problems in the case of several unknown functions and obtain solvability conditions and index formulas for the linear conjugation problem, the Riemann–Hilbert problem, and the Riemann–Hilbert–Poincare problem on a plane cut along several regular arcs in various weighted functional classes. We also examine the general differential boundary-value problem for analytic functions and boundary-value problems with shifts and complex conjugation on a cut plane.