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

Abstract.

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 W = W ^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.