Boundary value problems for the Cimmino system via quaternionic analysis

Blaya, Ricardo Abreu; Reyes, Juan C.; Adán, Alí Guzmán; Schneider, Baruch

doi:10.1016/j.amc.2012.10.022

Cited by 7 publications

(4 citation statements)

References 4 publications

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“…There exists another conventional function theory to be considered. This is the case of the Cimmino system, which was introduced in 1941 by G. Cimmino [10] and recently researchers, following [35], have come to understand several aspects of the system, see [1,2,6,8,9,11,37,38]. The next step to our work would be to develop of Bergman spaces theory induced by the Cimmino system, whose solutions form a proper subset of the π 2 −hyperholomorphic functions class.…”

Section: Discussionmentioning

confidence: 99%

On hyperholomorphic Bergman type spaces in domains of $\mathbb C^2$

González-Cervantes¹,

Bory-Reyes²

2021

Preprint

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Quaternionic analysis is regarded as a broadly accepted branch of classical analysis referring to many different types of extensions of the Cauchy-Riemann equations to the quaternion skew field H.In this work we deals with a well-known (θ, u)−hyperholomorphic H−valued functions class related to elements of the kernel of the Helmholtz operator with a parameter u ∈ H, just in the same way as the usual quaternionic analysis is related to the set of the harmonic functions.Given a domain Ω ⊂ H ∼ = C 2 , we define and study a Bergman spaces theory for (θ, u)− hyperholomorphic quaternion-valued functions introduced as elements of the kernel of θ u D[f ] = θ D[f ] + uf with u ∈ H defined in C 1 (Ω, H), where

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Section: Discussionmentioning

confidence: 99%

On hyperholomorphic Bergman type spaces in domains of $\mathbb C^2$

González-Cervantes¹,

Bory-Reyes²

2021

Preprint

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“…Remark If

f : Ω \to H

has

u : Ω \to C

and

v : Ω \to C

as complex components then the ψ‐hyperholomorphicity of f can be interpreted as the following system of equations

^{ψ_{θ}} D [f] = 0 \Leftrightarrow \{\begin{matrix} right \partial_{{\overset{z}{true}}_{1}} u - e_{1} e^{e_{1} θ} \cdot \partial_{z_{2}} \overset{v}{true} = 0, \\ right e_{1} e^{- e_{1} θ} \cdot \partial_{{\overset{z}{true}}_{2}} u - \partial_{z_{1}} \overset{v}{true} = 0 . \end{matrix}

The system above may be interpreted as a generalization of that study by the distinguished Italian mathematician Gianfranco Cimmino (12 Marzo 1908–30 Mayo 1989), which can be recovered from if we take

θ = \frac{π}{2}

. In an in‐depth study of Cimmino system using quaternionic analysis has been presented. As the basic for the development of our theory, we need to point out that the class of all equivalent structural sets defines the same hyperholomorphicity, then we should in fact further restrict our analysis to consider only structural sets not to be equivalent from the same side of hyperholomorphicity we are assumed.…”

Section: Holomorphic Maps and Classes Of Hyperholomorphymentioning

confidence: 99%

“…The system above may be interpreted as a generalization of that study by the distinguished Italian mathematician Gianfranco Cimmino (12 Marzo 1908-30 Mayo 1989, which can be recovered from (3.2) if we take θ = π 2 . In [1] an in-depth study of Cimmino system using quaternionic analysis has been presented.…”

Section: Holomorphic Maps and Classes Of Hyperholomorphymentioning

confidence: 99%

On some structural sets and a quaternionic (φ,ψ)-hyperholomorphic function theory

et al. 2015

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Quaternionic analysis is regarded as a broadly accepted branch of classical analysis referring to many different types of extensions of the Cauchy‐Riemann equations to the quaternion skew field double-struckH. It relies heavily on results on functions defined on domains in R4(or R3) with values in double-struckH. This theory is centred around the concept of ψ‐hyperholomorphic functions related to a so‐called structural set ψ of H4(or H3) respectively. The main goal of this paper is to develop the nucleus of the (φ,ψ)‐hyperholomorphic function theory, i.e., simultaneous null solutions of two Cauchy‐Riemann operators associated to a pair φ,ψ of structural sets of double-struckH4. Following a matrix approach, a generalized Borel‐Pompeiu formula and the corresponding Plemelj‐Sokhotzki formulae are established.

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“…In the same paper the Authors address the problem whether conditions (3) are also su cient for the resolubility of the Dirichlet problem (2). More recently, Abreu Blaya et al [1] studied (1) by means of quaternionic analysis. In particular, they found some di↵erent necessary and su cient conditions involving some particular integral operators.…”

Section: Introductionmentioning

confidence: 99%