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

Delanghe, Richard

CUBO, A Mathematical Journal, Tome 12 (2010), / Harvested from Cubo, A Mathematical Journal

Résumé

Let for s ∈ {0, 1, ..., m + 1} (m ≥ 2),  be the space of s-vectors in the Clifford algebra IR0,m+1 constructed over the quadratic vector space IR0,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 IRm+1, is called a generalized Moisil-Théodoresco system of type (r, p, q) in IRm+1. For k ∈ N, k ≥ 1, , denotes the space of -valued homogeneous polynomials Wk of degree k in IRm+1 satisfying ∂xWk = 0. A characterization of Wk ∈ is given in terms of a harmonic potential Hk+1 belonging to a subclass of -valued solid harmonics of degree (k + 1) in IRm+1. Furthermore, it is proved that each Wk ∈  admits a primitive Wk+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.

Détail

Publié le : 2010-06-01