Hermitian clifford analysis

This paper gives an overview of some basic results on Hermitian Clifford analysis, a refinement of classical Clifford analysis dealing with functions in the kernel of two mutually adjoint Dirac operators invariant under the action of the unitary group. The set of these functions, called Hermitian monogenic, contains the set of holomorphic functions in several complex variables. The paper discusses, among other results, the Fischer decomposition, the Cauchy–Kovalevskaya extension problem, the axiomatic radial algebra, and also some algebraic analysis of the system associated with Hermitian monogenic functions. While the Cauchy–Kovalevskaya extension problem can be carried out for the Hermitian monogenic system, this system imposes severe constraints on the initial Cauchy data. There exists a subsystem of the Hermitian monogenic system in which these constraints can be avoided. This subsystem, called submonogenic system, will also be discussed in the paper

Integration over non-rectifiable curves and Riemann boundary value problems

In this paper we introduce an alternative way of defining the curvilinear Cauchy integral over non-rectifiable curves in the case of complex functions of one complex variable. Especially the jump behavior on the boundary is considered. As an application, solvability conditions of the Riemann boundary value problem are derived on very weak conditions to the boundary. Besides the complex case the consideration can be extended to the theory of Douglis algebra valued functions. © 2011 Elsevier Inc

Riemann-Hilbert problems for monogenic functions in axially symmetric domains

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 (D−α)ϕ=0, α∈R, where R denotes the field of real numbers

Riemann–Hilbert Problems for Monogenic Functions on Upper Half Ball of R^4

In this paper we are interested in finding solutions to Riemann– Hilbert boundary value problems, for short Riemann–Hilbert problems, with variable coefficients in the case of axially monogenic functions defined over the upper half unit ball centred at the origin in four-dimensional Euclidean space. Our main idea is to transfer Riemann– Hilbert problems for axially monogenic functions defined over the up- per half unit ball centred at the origin of four-dimensional Euclidean spaces into Riemann–Hilbert problems for analytic functions defined over the upper half unit disk of the complex plane. Furthermore, we extend our results to axially symmetric null-solutions of perturbed generalized Cauchy–Riemann equations