On Cauchy estimates and growth orders of entire solutions of iterated Dirac and generalized Cauchy–Riemann equations

Abstract

AbstractIn this paper, we study the growth behaviour of entire Clifford algebra‐valued solutions to iterated Dirac and generalized Cauchy–Riemann equations in higher‐dimensional Euclidean space. Solutions to this type of systems of partial differential equations are often called k‐monogenic functions or, more generically, polymonogenic functions. In the case dealing with the Dirac operator, the function classes of polyharmonic functions are included as particular subcases. These are important for a number of concrete problems in physics and engineering, such as, for example, in the case of the biharmonic equation for elasticity problems of surfaces and for the description of the stream function in the Stokes flow regime with high viscosity.Furthermore, these equations in turn are closely related to the polywave equation, the poly‐heat equation and the poly‐Klein–Gordon equation.In the first part we develop sharp Cauchy‐type estimates for polymonogenic functions, for equations in the sense of Dirac as well as Cauchy–Riemann. Then we introduce generalizations of growth orders, of the maximum term and of the central index in this framework, which in turn then enable us to perform a quantitative asymptotic growth analysis of this function class. As concrete applications we develop some generalizations of some of Valiron's inequalities in this paper. Copyright © 2006 John Wiley & Sons, Ltd.