Further results on the asymptotic growth of entire solutions of iterated Dirac equations in ℝn

Abstract

SUMMARYIn this paper, we establish some further results on the asymptotic growth behaviour of entire solutions to iterated Dirac equations in R n . Solutions to this type of systems of partial di erential equations are often called polymonogenic or k-monogenic. In the particular cases where k is even, one deals with polyharmonic functions. These are of central importance for a number of concrete problems arising in engineering and physics, such as for example in the case of the biharmonic equation for the description of the stream function in the Stokes ow regime with low Reynolds numbers and for elasticity problems in plates.The asymptotic study that we are going to perform within the context of these PDE departs from the Taylor series representation of their solutions. Generalizations of the maximum term and the central index serve as basic tools in our analysis. By applying these tools we then establish explicit asymptotic relations between the growth behaviour of polymonogenic functions, the growth behaviour of their iterated radial derivatives and that of functions obtained by applying iterations of the operator to them.

Basics on growth orders of polymonogenic functions

“…To achieve this we applied the classical Cauchy formula for monogenic functions from [10] on the monogenic component functions { f l } l=0,...,k−1 and the projection formulas (3) afterwards. To make this paper self-contained we recall from [12] that…”

“…In [12][13][14], we even managed to extend some of these techniques to the more general context of entire solutions of polynomial Cauchy-Riemann equations with arbitrary complex coefficients.…”

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

“…In our follow-up paper [18], these concepts are then applied to establish explicit asymptotic relations between the growth of solutions to these systems of partial di erential equations, that of their generalized iterated radial derivatives and that of function classes that arise from applying iterations of closely related linear di erential operators, as, e.g. the radial -operator from Reference [11], on these solutions.…”

On a Generalization of Valiron’s Inequality for k-hypermonogenic Functions on Upper Half-Space

“…The study of the Taylor coefficients of an entire monogenic solution was the basis for the asymptotic growth analysis of entire solutions to partial differential equations (PDEs) related to the generalized Euclidean Dirac and CauchyRiemann operator, see for example [3,4,5,8]. In the context of hypermonogenic functions defined on the upper half-space the study of the Taylor coefficients is replaced by studying their Fourier images, see [6].…”