Riemann-Hilbert problems for monogenic functions in axially symmetric domains

Fuli He,Uwe Kähler,Frank Sommen,Swanhild Bernstein,Min Ku

doi:10.1186/s13661-016-0530-x

Abstract

We consider Riemann-Hilbert boundary value problems (for short RHBVPs) with variable coefficients for axially symmetric monogenic functions defined in axial symmetric domains. This is done by constructing a method to reduce the RHBVPs for axially symmetric monogenic functions defined in four-dimensional axial symmetric domains into the RHBVPs for analytic functions defined over the complex plane. Then we derive solutions to the corresponding Schwarz problem. Finally, we generalize the results obtained to null-solutions of $(\mathcal{D}-\alpha)\phi=0$ , $\alpha\in\mathbb{R}$ , where $\mathbb{R}$ denotes the field of real numbers.

Highlights

1 Introduction The classic theory of Riemann-Hilbert boundary value problems for analytic functions is closely connected with the theory of singular integral equations and has a wide range of applications in other fields, such as the theory of elasticity, quantum mechanics, statistical physics, the theory of orthogonal polynomials, and asymptotic analysis
Instead of trying to solve this problem directly, we are going to show that RHBVPs with variable coefficients for monogenic functions with axial symmetry can be solved in the context of quaternionic analysis
As a special case we present the solution to the Schwarz problem for monogenic functions with axial symmetry

Summary

Introduction

The classic theory of Riemann-Hilbert boundary value problems (for short RHBVPs) for analytic functions is closely connected with the theory of singular integral equations and has a wide range of applications in other fields, such as the theory of elasticity, quantum mechanics, statistical physics, the theory of orthogonal polynomials, and asymptotic analysis (cf. [ – ]). Even in the simplest case of quaternionic analysis, we have several major problems, such as the lack of a function which has the same properties as the logarithmic function of one complex variable These make it very difficult to solve RHBVPs with variable coefficients for monogenic functions in higher dimensions by directly employing classic methods from complex analysis. Instead of trying to solve this problem directly, we are going to show that RHBVPs with variable coefficients for monogenic functions with axial symmetry can be solved in the context of quaternionic analysis. Our idea is to transform a RHBVP with variable coefficients for monogenic functions with axial symmetry in R into a RHBVP for analytic functions over the complex plane, use the classic results from complex analysis, and transfer it back In this way we can obtain the solution in an explicit form.

Any element x

Proof Let the map ψ be given by

Using the Laplacian operator

Consider now the function