The Riemann-Hilbert problem and the generalized Neumann kernel on unbounded multiply connected regions

TL;DR

This paper develops a Fredholm integral equation with the generalized Neumann kernel to solve the Riemann-Hilbert problem on unbounded multiply connected regions, providing a new approach for boundary value problems in complex analysis.

Contribution

It introduces a novel integral equation formulation for the Riemann-Hilbert problem on unbounded multiply connected regions, ensuring unique solvability.

Findings

Derivation of a Fredholm integral equation with the generalized Neumann kernel.

Establishment of unique solvability for the boundary integral equations.

Application to the modified Dirichlet problem on unbounded multiply connected regions.

Abstract

A Fredholm integral equation of the second kind with the generalized Neumann kernel associated with the Riemann-Hilbert problem on unbounded multiply connected regions will be derived and studied in this paper. The derived integral equation yields a uniquely solvable boundary integral equations for the modified Dirichlet problem on unbounded multiply connected regions.