Abstract
Hermitian Clifford analysis has emerged as a new and successful branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focuses on the simultaneous null solutions of two Hermitian Dirac operators. Using a circulant matrix approach, we will study the R 0 Riemann type problems in Hermitian Clifford analysis. We prove a mean value formula for the Hermitian monogenic function. We obtain a Liouville-type theorem and a maximum module for the function above. Applying the Plemelj formula, integral representation formulas, and a Liouville-type theorem, we prove that the R 0 Riemann type problems for Hermitian monogenic and Hermitian-2-monogenic functions are solvable. Explicit representation formulas of the solutions are also given.
Highlights
The classical Riemann boundary value problem (BVP for short) theory in the complex plane has been systematically developed, see [ ] and [ ]
The theory is centered around the concept of monogenic functions, see [ – ], etc
In [ ] and [ ], Riemann BVP for harmonic functions (i.e., -monogenic functions) and biharmonic functions were studied, the solutions are given in an explicit way
Summary
The classical Riemann boundary value problem (BVP for short) theory in the complex plane has been systematically developed, see [ ] and [ ]. In [ ], the Riemann BVP for (left) Helmholtz H-monogenic functions (i.e., null solutions of perturbed Hermitian Dirac operators in the framework of Hermitian Clifford analysis). If the perturbed value vanishes, D(KZ,Z†) is D(Z,Z†), the R– Riemann BVP for H-monogenic circulant ( × ) matrix functions was solved. In this paper, motivated by [ , , , , , ], we will consider R Riemann BVP for H- -monogenic circulant ( × ) matrix functions in Hermitian Clifford analysis.
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