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Complex Analysis and Operator Theory

, Volume 6, Issue 2, pp 407–424 | Cite as

Solutions to Polynomial Generalized Bers–Vekua Equations in Clifford Analysis

  • Min KuEmail author
  • Daoshun Wang
  • Lin Dong
Article

Abstract

In this paper, we mainly study polynomial generalized Vekua-type equation \({p(\mathcal{D})w=0}\) and polynomial generalized Bers–Vekua equation \({p(\mathcal{\underline{D}})w=0}\) defined in \({\Omega\subset\mathbb{R}^{n+1}}\) where \({\mathcal{D}}\) and \({\mathcal{\underline{D}}}\) mean generalized Vekua-type operator and generalized Bers–Vekua operator, respectively. Using Clifford algebra, we obtain the Fischer-type decomposition theorems for the solutions to these equations including \({\left(\mathcal{D}-\lambda\right)^{k}w=0,\left(\mathcal {\underline{D}}-\lambda\right)^{k}w=0\left(k\in\mathbb{N}\right)}\) with complex parameter λ as special cases, which derive the Almansi-type decomposition theorems for iterated generalized Bers–Vekua equation and polynomial generalized Cauchy–Riemann equation defined in \({\Omega\subset\mathbb{R}^{n+1}}\). Making use of the decomposition theorems we give the solutions to polynomial generalized Bers–Vekua equation defined in \({\Omega\subset\mathbb{R}^{n+1}}\) under some conditions. Furthermore we discuss inhomogeneous polynomial generalized Bers–Vekua equation \({p(\mathcal{\underline{D}})w=v}\) defined in \({\Omega\subset\mathbb{R}^{n+1}}\), and develop the structure of the solutions to inhomogeneous polynomial generalized Bers–Vekua equation \({p(\mathcal{\underline{D}})w=v}\) defined in \({\Omega\subset\mathbb{R}^{n+1}}\).

Keywords

Clifford analysis Polynomial generalized Bers–Vekua operator Monogenic functions 

Mathematics Subject Classification (2000)

30G35 32A25 35C10 

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of Computer Science and TechnologyTsinghua UniversityBeijingPeople’s Republic of China

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