Solutions of Inhomogeneous Generalized Moisil–Teodorescu Systems in Euclidean Space

Let R0,m+1(s) be the space of s-vectors (0 ≤ s≤ m+ 1) in the Clifford algebra R _{, m + 1} constructed over the quadratic vector space R ^{, m + 1} , let r, p, q∈ N with 0 ≤ r≤ m+ 1 , 0 ≤ p≤ q and r+ 2 q≤ m+ 1 and let R0,m+1(r,p,q)=∑j=pq⨁R0,m+1(r+2j). Then a R0,m+1(r,p,q)-valued smooth function F defined in an open subset Ω ⊂ R ^{m + 1} is said to satisfy the generalized Moisil–Teodorescu system of type (r, p, q) if ∂ _x F= 0 in Ω , where ∂ _x is the Dirac operator in R ^{m + 1} . To deal with the inhomogeneous generalized Moisil–Teodorescu systems ∂ _x F= G, with a ∑j=pq⨁R0,m+1(r+2j-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.