Riemann-Hilbert problems for axially symmetric monogenic functions in $\mathbb{R}^{n+1}$

TL;DR

This paper develops a method to solve Riemann-Hilbert boundary value problems for axially symmetric monogenic functions in higher-dimensional Euclidean spaces by linking them to generalized analytic functions in the complex plane.

Contribution

It introduces a novel approach using the Vekua system to connect higher-dimensional RHBVPs with complex plane problems, enabling explicit solutions and solvability conditions.

Findings

Established a one-to-one correspondence between RHBVPs in axial domains and complex generalized analytic functions.

Derived solutions and solvability conditions for the boundary value problems.

Extended the method to null-solutions of generalized Dirac operators with a real parameter.

Abstract

We focus on the Clifford-algebra valued variable coefficients Riemann-Hilbert boundary value problems $\big{(}$for short RHBVPs$\big{)}$ for axially monogenic functions on Euclidean space $\mathbb{R}^{n+1},n\in \mathbb{N}$. With the help of Vekua system, we first make one-to-one correspondence between the RHBVPs considered in axial domains and the RHBVPs of generalized analytic function on complex plane. Subsequently, we use it to solve the former problems, by obtaining the solutions and solvable conditions of the latter problems, so that we naturally get solutions to the corresponding Schwarz problems. In addition, we also use the above method to extend the case to RHBVPs for axially null-solutions to $\big{(}\mathcal{D}-\alpha\big{)}\phi=0,\alpha\in\mathbb{R}$.