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

Abstract

SUMMARYIn this paper, we study the growth behaviour of entire Cli ord algebra-valued solutions to iterated Dirac and generalized Cauchy-Riemann equations in higher-dimensional Euclidean space. Solutions to this type of systems of partial di erential 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 ow 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 ÿrst 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.

Some Riemann boundary value problems in Clifford analysis

“…Combining the idea in [35] with the generalized Liouville theorem, we get explicit solutions for R m (m > 0) Riemann boundary value problems. This paper improves some results, which were obtained in [37] and generalizes the results of [11,12,35].…”

“…Thus, it is very important to deduce a generalized Liouville theorem in Clifford analysis, too. In [36,37], a generalized Liouville theorem for k-monogenic functions was obtained under some growth conditions. This is a very important result.…”

“…It is valuable to refine the conditions especially for Riemann boundary value problems. In this paper, by applying methods from [36,37], the Taylor expansion of k-regular functions in [38] as well as the integral representation formulas, we get a generalized Liouville theorem for harmonic functions and biharmonic functions under refined conditions. By using the integral representation formulas again and the Plemelj formula, a more generalized Liouville theorem is given.…”

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].…”

On the solutions of a class of iterated generalized Bers–Vekua equations in Clifford analysis

“…(sometimes called (k)-monogenic or polymonogenic functions) given in [5] (see also [6]) and further the classical Almansi theorem [7] for polyharmonic functions.…”