Riemann boundary value problems on the sphere in Clifford analysis
type: Item
instance of: scholarly article
title: Riemann boundary value problems on the sphere in Clifford analysis
zbMATH Open document ID: 1262.30056
author: Min Ku
author: Uwe Kähler
author: Daoshun Wang
published in: Advances in Applied Clifford Algebras
publication date: 2013-01-28T00:00:00Z
review text: Clifford analysis offers an elegant generalization of analytic functions from the complex plane to higher dimensional space. It focuses on the study of so-called monogenic functions, i.e. null solutions of the Dirac operator or the generalized Cauchy-Riemann operator \(\mathcal{D}\). Starting from the polynomial \(p(\lambda) = \lambda^k + a_1 \lambda^{k-1} + \ldots + a_k\) (\(a_i \in \mathbb{C}\), \( i=1,2,\ldots,k\)), a polynomially generalized Cauchy-Riemann operator \(p(\mathcal{D})\) can be defined: \[ p(\mathcal{D}) = \mathcal{D}^k +a_1 \mathcal{D}^{k-1} + \ldots + a_k, \] where \(a_k\) means \(a_k I\), \(I\) denoting the identity operator. The solutions to polynomially generalized Cauchy-Riemann equations \(p(\mathcal{D}) \phi = 0\) are called polynomially monogenic functions. In this paper a type of Riemann boundary value problems for polynomially monogenic functions over the sphere in \(\mathbb{R}^{n+1}\) is studied. To this end the Fischer type decomposition is established for null solutions of the differential operator \((\mathcal{D} - a(x))^k\), including the polynomial generalized Cauchy-Riemann operators \((\mathcal{D} - \lambda)^k\) and \(\mathcal{D} - \lambda\) as special cases, where \(a \in C^k(\Omega, \mathbb{C}^{n+1})\) satisfies \(\mathcal{D} \gamma(x) = a(x)\) with \(\gamma \in C^{k+1}(\Omega,\mathbb{C})\), \(\Omega \subset \mathbb{R}^{n+1}\) and \(\lambda \in \mathbb{C}\). Applying this Fischer type decomposition and the Clifford calculus for polynomial monogenic functions, explicit expressions of solutions to Riemann boundary value problems for polynomially monogenic functions over the sphere of \(\mathbb{R}^{n+1}\) are obtained. As special cases, the solutions to classical Riemann boundary value problems for polyanalytic functions and metaanalytic functions in the complex plane are derived, which are important not only for pure mathematical reasons, but also for concrete problems in physics and engineering, particularly when considered over t…
reviewed by: Nele De Schepper
Mathematics Subject Classification ID: 30G35
Mathematics Subject Classification ID: 30E25
zbMATH DE Number: 6131881
zbMATH Keywords: Clifford analysis
zbMATH Keywords: monogenic functions
zbMATH Keywords: generalized Cauchy-Riemann operator
zbMATH Keywords: Hölder continuous functions
Wikidata QID: Q58038893
MaRDI profile type: Publication
full work available at URL: s00006-011-0308-2
OpenAlex ID: W2069108625
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DOI: 10.1007/S00006-011-0308-2
description: scientific article; zbMATH DE number 6131881