Solutions of Inhomogeneous Generalized Moisil–Teodorescu Systems in Euclidean Space

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 (r, p, q) 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.