On a Riemann-Hilbert boundary value problem for $(\varphi,\psi)$-harmonic functions in $\mathbb{R}^m$

TL;DR

This paper addresses a Riemann-Hilbert boundary value problem for $(unction,unction)$-harmonic functions in Euclidean space, utilizing Clifford analysis to explicitly solve the problem in domains with fractal boundaries.

Contribution

It introduces a method to solve boundary value problems for $(unction,unction)$-harmonic functions using Clifford analysis, even with complex fractal boundary data.

Findings

Explicit solutions obtained for domains with fractal boundaries.

Boundary data involves higher order Lipschitz class functions.

Method extends classical boundary value problem techniques to complex geometries.

Abstract

The purpose of this paper is to solve a kind of Riemann-Hilbert boundary value problem for $(\varphi,\psi)$-harmonic functions, which are linked with the use of two orthogonal basis of the Euclidean space $\mathbb{R}^m$. We approach this problem using the language of Clifford analysis for obtaining the explicit expression of the solution of the problem in a Jordan domain $\Omega\subset\mathbb{R}^m$ with fractal boundary. One of the remarkable feature in this study is that the boundary data involves higher order Lipschitz class of functions.