THE RIEMANN - HILBERT PROBLEM FOR A MULTIDIMENSIONAL SYSTEM GENERALIZING THE CAUCHY-RIEMANN SYSTEM

Abstract

In this article, we construct a multidimensional — specifically, five-dimensional — analogue of the well-known Cauchy–Riemann system for analytic functions, using quaternions. The classical Cauchy–Riemann system plays a fundamental role in complex analysis and the theory of harmonic functions, and its higher-dimensional generalizations have attracted considerable attention in recent decades. Quaternions, with their non-commutative algebraic structure, provide a natural and effective framework for extending these ideas to five independent variables. We provide a representation of the solutions of the resulting system in terms of the derivatives of harmonic functions. This representation is of particular importance because it connects the solutions of the original first-order elliptic system with the well-studied theory of harmonic functions, thereby allowing classical methods to be applied. In addition, we study the corresponding Riemann–Hilbert boundary value problem and reduce it to the classical oblique derivative problem for harmonic functions. This reduction enables us to apply known results on the solvability and regularity of oblique derivative problems, and it clarifies the structure of the original boundary value problem for the five-dimensional system.