Constructing steering-type solutions for higher order Cauchy-Riemann equations in ℝ^m+1

2025/12/18 by Santiesteban, Daniel Alfonso, Peña, Dixan Peña, Blaya, Ricardo Abreu
#30G30 #30G35 #35J15 #35J25 #35J47 #35J57 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2512.16594

Abstract

The multidimensional Cauchy-Riemann operator provides a framework for studying higher order partial differential equations in ℝ^m+1, whose solutions include polymonogenic and polyharmonic functions, among others. In this work, we aim to explicitly construct solutions to such systems, generated from families of complex valued functions which are closed under conjugation and under the action of the complex Cauchy-Riemann operator. Moreover, we prove that precisely some of these solutions also satisfy homogeneous linear differential equations involving the so-called hypercomplex derivative.

Citations

• Biaxial monogenic functions from Funk-Hecke's formula combined with Fueter's theorem
• The Fischer Decomposition for the H-action and Its Applications
• A Cauchy-Kowalevski theorem for inframonogenic functions
• Fischer decomposition by inframonogenic functions

Related

• Generalized Cauchy-Riemann equations and relevant PDE
• Hypercomplex method for solving linear PDEs with constant coefficients
• Solutions of Cauchy problem for multiple inhomogeneous wave equation
• The Cauchy problem for higher-order linear partial differential equation
• Hardy decomposition of higher order Lipschitz classes by polymonogenic functions