Comparing harmonic and inframonogenic functions in Clifford analysis
type: Item
instance of: scholarly article
title: Comparing harmonic and inframonogenic functions in Clifford analysis
zbMATH Open document ID: 1494.30087
author: Arsenio Moreno García
author: Tania Moreno-García
author: Ricardo Abreu-Blaya
published in: Mediterranean Journal of Mathematics
publication date: 2022-01-27T00:00:00Z
review text: Let \(\mathbb{R}_{0, m}\) be the real Clifford algebra generated by \(e_1, e_2, \dots, e_m\) over the field of real numbers \(\mathbb{R}\). Consider \(\Omega\) to be a domain of \(\mathbb{R}^m\), a \(\mathbb{R}_{0, m}\)-valued function \( f \in C^2(\Omega)\) is called inframonogenic function if \(\partial_{\underline{x}} f \partial_{\underline{x}} =0\), where \(\partial_{\underline{x}} = \sum_{k=1}^m \partial_{x_k} e_k\) is the Dirac operator. Introducing the operator \(\Psi : \mathbb{R}_{0, m} \to \mathbb{R}_{0, m} \) defined by \(\Psi(f) = \sum_{k=1}^m e_ k f e_k\), where \(f : \mathbb{R}^{ m}\to \mathbb{R}_{0, m}\), it is proved that, for the case \(m\) is odd, a function \(f\) is inframonogenic (harmonic) in \(\Omega \subset \mathbb{R}^m\) if and only if \(\Psi(f)\) is inframonogenic (harmonic) in \(\Omega\). Furthermore, the mean value property for inframonogenic functions is introduced, for which, in contrast to the case of harmonic functions, the average needs to involve on the bounday of the ball of center at the origin and radius \(r>0\) (\(\partial B(0, r)\)) the values of the function as well as those of its first-order partial derivatives. The maximum principle for this class of functions is also discussed. The uniqueness and existence of solution, for \(m\) odd, of the Dirichlet problem \[ \left\{ \begin{array}{lcll} \partial_{\underline{x}} u \partial_{\underline{x}}& =& 0, &\text{ in }\; B(0, r)\\ u &=& g,&\text{ in }\; \partial B(0, r) \end{array}\right. \] assuming \(g\) to be a polynomial, is presented. This solution depends on a harmonic polynomial \(\mathcal{H}\) and the inverse operator of \(\Psi\) applied to \(\partial_{\underline{x}} \mathcal{H} \partial_{\underline{x}}\).
Mathematics Subject Classification ID: 30G35
zbMATH DE Number: 7465524
zbMATH Keywords: harmonic functions
zbMATH Keywords: Clifford analysis
zbMATH Keywords: inframonogenic functions
reviewed by: Regina C. Almeida
MaRDI profile type: Publication
full work available at URL: s00009-021-01957-5
OpenAlex ID: W4206934725
cites work: A Cauchy integral formula for infrapolymonogenic functions in Clifford analysis
cites work: Integral Representations in Complex, Hypercomplex and Clifford Analysis
cites work: Q3310867
cites work: Q4115384
cites work: Q4115385
cites work: Q3414551
cites work: Quaternionic analysis and elliptic boundary value problems
cites work: Linear and complex analysis problem book 3. Part 2
cites work: Representation and uniqueness theorems for polyharmonic functions
cites work: A Cauchy integral formula for inframonogenic functions in Clifford analysis
cites work: Decomposition of inframonogenic functions with applications in elasticity theory
cites work: Inframonogenic functions and their applications in 3‐dimensional elasticity theory
cites work: A Cauchy-Kowalevski theorem for inframonogenic functions
cites work: Fischer decomposition by inframonogenic functions
cites work: Almansi‐type theorems in Clifford analysis
cites work: Real Bargmann spaces, Fischer decompositions, and sets of uniqueness for polyharmonic functions
cites work: Q3023771
DOI: 10.1007/S00009-021-01957-5
Recommended article: Q4996731
Recommended article: Q4501280
Recommended article: Deriving harmonic functions in Dunkl-Clifford analysis and applications
Recommended article: On \((\phi,\psi)\)-inframonogenic functions in Clifford analysis
Recommended article: Q4778298
Recommended article: Q4543263
Recommended article: Q4828301
Recommended article: Q4527634
Recommended article: A Cauchy integral formula for inframonogenic functions in Clifford analysis
Recommended article: Hilbert Transforms in Clifford Analysis
description: scientific article; zbMATH DE number 7465524