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Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.12 no.2 Temuco  2010

http://dx.doi.org/10.4067/S0719-06462010000200010 

CUBO A Mathematical Journal Vol.12, N° 02, (145–167). June 2010

 

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


ABSTRACT

Let for s ∈ {0, 1, ...,m+ 1} (m ≥ 2) , IR(s)0,m+1 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 IR(r,p,q)0,m+1 = ∑qj=p ⊕IR(r+2j)0,m+1 -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,MT+(m+ 1; k; IR(r,p,q)0,m+1), denotes the space of IR(r,p,q)0,m+1-valued homogeneous polynomials Wc of degree k in IRm+1 satisfying ∂xWx = 0. A characterization of Wk∈ MT+(m + 1; k;IR(r,p,q)0,m+1) is given in terms of a harmonic potential Hk+1 belonging to a subclass of IR(r,p,q)0,m -valued solid harmonics of degree (k + 1) in IRm+1. Furthermore, it is proved that each Wk∈ MT+(m+ 1; k; IR(r,p,q)0,m+1) admits a primitive Wk+1 ∈ MT+(m+ 1; k + 1; IR(r,p,q)0,m+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 MT+(4; k; IR(r,p,q)0,4), r being even.

Key words and phrases: Clifford analysis; Moisil-Théodoresco systems; conjugate harmonic funtions; harmonic potentials; polynomial bases


RESUMEN

Para s ∈ {0, 1, ...,m+ 1} (m ≥ 2) , IR(s)0,m+1 el espacio de los s-vectors en el algebra de Clifford IR0,m+1 construida sobre el espacio de vectores cuadráticos IR0,m+1 sea r, p, q, ∈ IN tal que 0 ≤ r ≤ m + 1, p < q. El sistema lineal asociado de ecuaciones diferenciales parciales de primer orden derivado de la ecuaci´on ∂xW = 0 donde W es IR(r,p,q)0,m+1 = ∑qj=p ⊕IR(r+2j)0,m+1 1-valuada y ∂x es el operador de Dirac en IRm+1, es llamado un sistema de Moisil-Théodoresco generalizado de tipo (r, p, q) en IRm+1. Para k ∈ N, k ≥ 1,MT+(m+ 1; k; IR(r,p,q)0,m+1), denota el espacio de polinomios homogéneosWk IR(r,p,q)0,m+1- valuados de grado k en IRm+1. satisfaciendo ∂xWx = 0. Una caracterización de Wk∈ MT+(m+1; k; IR(r,p,q)0,m+1) es dada en términos de un potencial armónico Hk+1 perteneciendo a una subclase de armónicos consistentes IR(r,p,q)0,m -valuados de grado (k + 1) in IRm+1. Además es probado que todo Wk∈ MT+(m + 1; k; IR(r,p,q)0,m+1) admite una primitiva Wk+1 ∈ MT+(m + 1; k + 1; IR(r,p,q)0,m+1). Una especial atención es dada a los casos de dimensión baja IR3 y IR4. En particular, un metodo es desarrollado para construir bases para espaciosMT+(4; k; IR(r,p,q)0,4 ), r siendo par.

Mathematics Subject Classification (2000): 30G35


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Received: March 2009.

Revised: April 2009.