On a Riemann--Hilbert boundary value problem for (ϕ,ψ)-harmonic functions in ℝ m

Abstract

Abstract The purpose of this paper is to solve a kind of the Riemann–Hilbert boundary value problem for ( φ , ψ ) {(\varphi,\psi)} -harmonic functions, which are linked with the use of two orthogonal bases of the Euclidean space ℝ m {\mathbb{R}^{m}} . We approach this problem using the language of Clifford analysis for obtaining an explicit expression of the solution of the problem in a Jordan domain Ω ⊂ ℝ m {\Omega\subset\mathbb{R}^{m}} with fractal boundary. Since our study is concerned with a second order differential operator, the boundary data are restricted to involve the higher order Lipschitz class Lip ⁡ ( 1 + α , Γ ) {\operatorname{Lip}(1+\alpha,\Gamma)} .