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Solutions of Inhomogeneous Generalized Moisil–Teodorescu Systems in Euclidean Space

  • Juan Bory-Reyes
  • Marco Antonio Pérez-de la RosaEmail author
Article
  • 27 Downloads

Abstract

Let \(\mathbb R_{0, m+1}^{(s)}\) be the space of s-vectors (\(0\le s\le m+1\)) in the Clifford algebra \(\mathbb R_{0, m+1}\) constructed over the quadratic vector space \(\mathbb R^{0, m+1}\), let \(r, p, q\in \mathbb N\) with \(0\le r\le m+1\), \(0\le p\le q\) and \(r+2q\le m+1\) and let \(\mathbb R_{0, m+1}^{(r,p,q)}=\sum _{j=p}^q\bigoplus \mathbb R_{0, m+1}^{(r+2j)}\). Then a \(\mathbb R_{0, m+1}^{(r,p,q)}\)-valued smooth function F defined in an open subset \(\Omega \subset \mathbb R^{m+1}\) is said to satisfy the generalized Moisil–Teodorescu system of type (rpq) if \(\partial _x F=0\) in \(\Omega \), where \(\partial _x\) is the Dirac operator in \(\mathbb R^{m+1}\). To deal with the inhomogeneous generalized Moisil–Teodorescu systems \(\partial _x F=G\), with a \(\sum _{j=p}^{q} \bigoplus {\mathbb {R}}^{(r+2j-1)}_{0,m+1}\)-valued continuous function G as a right-hand side, we embed the systems in an appropriate Clifford analysis setting. Necessary and sufficient conditions for the solvability of inhomogeneous systems are provided and its general solution described.

Keywords

Clifford analysis Dirac operator Moisil–Teodorescu systems Conjugate harmonic pairs. 

Mathematics Subject Classification

Primary 30G359 Secondary 35F05 

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Juan Bory-Reyes
    • 1
  • Marco Antonio Pérez-de la Rosa
    • 2
    Email author
  1. 1.ESIME-Zacatenco, Instituto Politécnico NacionalMexico CityMexico
  2. 2.Department of Actuarial Sciences, Physics and MathematicsUniversidad de las Américas PueblaSan Andrés CholulaMexico

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