Advances in Applied Clifford Algebras

, Volume 19, Issue 1, pp 15–28 | Cite as

On the Structure of Solutions of the Moisil-Théodoresco System in Euclidean Space

  • Juan Bory Reyes
  • Richard Delanghe


Let \(\Omega \subset {\mathbb{R}}^{m+1}\) be open, let \(\partial_x\) be the Dirac operator in \({\mathbb{R}^{m+1}}\) and let \({\mathbb{R}_{0, m+1}}\) be the Clifford algebra constructed over the quadratic space \({\mathbb{R}}^{0, m+1}\). If for \(r \in \{0, 1, \ldots , m \}\) fixed, \(\mathbb{R}^{(r)}_{0, m+1}\) denotes the space of r-vectors in \(\mathbb{R}_{0, m+1}\), then an \((\mathbb{R}^{(r)}_ {0,m+1} \oplus {\mathbb{R}^{(r+2)}_{0,m+1}})\)-valued smooth function WW r  + W r+2 in Ω is said to satisfy the Moisil-Théodoresco system if \(\partial_{x}W = 0\, {\rm in}\,\Omega\). In terms of differential forms, this means that the corresponding \((\bigwedge^{r}(\Omega) \oplus \bigwedge^{r+2}(\Omega)) \)- valued smooth form w = w r  + w r+2 satisfies in Ω the system d * w r = 0, dw r  + d * w r+2 = 0; dw r+2 = 0.

Based on techniques and results concerning conjugate harmonic functions in the framework of Clifford analysis, a structure theorem is proved for the solutions of the Moisil-Théodoresco system.

Mathematics Subject Classification (2000).

30G35 58A10 


Monogenic functions self-conjugate differential forms conjugate harmonic pairs Moisil-Théodoresco system 


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Copyright information

© Birkhauser 2008

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de OrienteSantiago de CubaCuba
  2. 2.Department of Mathematical AnalysisGhent UniversityGhentBelgium

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