On homogeneous polynomial solutions of generalized Moisil-Théodoresco systems in Euclidean space

Abstract

Let for s ∈ {0, 1, ..., m + 1} (m ≥ 2),  be the space of s-vectors in the Clifford algebra IR_0,m+1 constructed over the quadratic vector space IR^0,m+1 and let r, p, q, ∈ IN be such that 0 ≤ r ≤ m + 1, p < q and r + 2q ≤ m + 1. The associated linear system of first order partial differential equations derived from the equation ∂_xW = 0 where W is -valued and ∂_x is the Dirac operator in IR^m+1, is called a generalized Moisil-Théodoresco system of type (r, p, q) in IR^m+1. For k ∈ N, k ≥ 1, , denotes the space of -valued homogeneous polynomials W_k of degree k in IR^m+1 satisfying ∂_xW_k = 0. A characterization of W_k ∈ is given in terms of a harmonic potential H_k+1 belonging to a subclass of -valued solid harmonics of degree (k + 1) in IR^m+1. Furthermore, it is proved that each W_k ∈  admits a primitive W_k+1 ∈ . Special attention is paid to the lower dimensional cases IR³ and IR⁴. In particular, a method is developed for constructing bases for the spaces , r being even.