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

  • Richard Delanghe Department of Mathematical Analysis, Clifford Research Group, Ghent University, Galglaan 2, B-9000 Ghent, Belgium.
Keywords: Clifford analysis, Moisil-Théodoresco systems, conjugate harmonic funtions, harmonic potentials, polynomial bases

Abstract

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 ≤ rm + 1, p < q and r + 2qm + 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 kN, 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.

Published
2010-06-01
How to Cite
[1]
R. Delanghe, “On homogeneous polynomial solutions of generalized Moisil-Théodoresco systems in Euclidean space”, CUBO, A Mathematical Journal, vol. 12, no. 2, pp. 145–167, Jun. 2010.