Applications of the maximum term and the central index in the asymptotic growth analysis of entire solutions to higher dimensional polynomial Cauchy–Riemann equations

Abstract

In this article we deal with entire Clifford algebra valued solutions to polynomial Cauchy–Riemann equations in higher dimensional Euclidean spaces. We introduce generalizations of the maximum term and the central index within the context of this family of elliptic partial differential equations. These notions enable us to perform a basic study of the asymptotic growth behaviour of entire solutions to these systems. In this article we set up generalizations of some classical fundamental results of Wiman and Valiron's theory. Our results then enable us to get some insight on the structure of the solutions of a certain class of higher dimensional PDE.

Keywords:

• Polynomial Cauchy–Riemann equations
• Partial differential equations
• Asymptotic growth
• Maximum term
• Central index
• Valiron's inequalities
• Radial differential operators

AMS Subject Classification::

• 30G35

Acknowledgements

The first author gratefully acknowledges the financial support from BOF/GOA 01GA0405 of Ghent University. The second author gratefully acknowledges the partial support by the R&D unit Matemãtica e Aplicações (UIMA) of the University of Aveiro, sponsored through the Portuguese Foundation for Science and Technology (FCT), co-financed by the European Community fund FEDER, from the grant ProdepIII (2/5.3/2001) from FSE and from FCT (SFRH/BD/8330/2002).