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In this paper, we study existence of solutions to the scalar additive Jump problem and the Riemann boundary value problems in the context of vectorial Clifford analysis on domains with fractal boundaries. A reduction procedure is applied with great effectiveness to find the solution of the problems. [ABSTRACT FROM AUTHOR]

Let Ω⊂ℝm$$ \Omega \subset {\mathrm{\mathbb{R}}}^m $$ be a bounded regular domain, let ∂x_$$ {\partial}_{\underset{\_}{x}} $$ be the standard Dirac operator in ℝm$$ {\mathrm{\mathbb{R}}}^m $$, and let ℝ0,m$$ {\mathrm{\mathbb{R}}}_{0,m} $$ be the Clifford algebra constructed over the quadratic space ℝ0,m$$ {\mathrm{\mathbb{R}}}^{0,m} $$. For k∈{1,...,m}$$ k\in \left\{1,\dots, m\right\} $$ fixed, ℝ0,m(k)$$ {\mathrm{\mathbb{R}}}_{0,m}^{(k)} $$ denotes the space of k$$ k $$‐vectors in ℝ0,m$$ {\mathrm{\mathbb{R}}}_{0,m} $$. In the framework of Clifford analysis, we consider two boundary value problems for a second‐order elliptic system of partial differential equations of the form ∂x_Fk∂x_=fk$$ {\partial}_{\underset{\_}{x}}{F}_k{\partial}_{\underset{\_}{x}}={f}_k $$ in Ω$$ \Omega $$, where fk$$ {f}_k $$ is a smooth k$$ k $$‐vector valued function. The boundary conditions of the problems contain the inner and outer products of the k$$ k $$‐vector solution Fk$$ {F}_k $$ with both the Dirac operator and the normal vector to ∂Ω$$ \mathrm{\partial \Omega } $$, ensuring the well‐posedness for the problems. Investigation of the spectral properties of the sandwich operator ∂x_(.)∂x_$$ {\partial}_{\underset{\_}{x}}(.){\partial}_{\underset{\_}{x}} $$ is considered by using the Fredholm theory. Finally, it is shown that satisfactory problem‐solving properties, in general, fail when we replace the standard Dirac operator by those, obtained via unusual orthogonal bases of ℝm$$ {\mathrm{\mathbb{R}}}^m $$. [ABSTRACT FROM AUTHOR]

A theory of quaternion-valued functions, called hyperholomorphic, of two real variables has long been established. This theory is in the same relation to the two dimensional Helmholtz equation as the usual one-dimensional complex analysis is to the Laplace equation in R 2 . In this work we define a new Cauchy integral for domains with fractal boundary illustrating its applications and usage to study the jump and Dirichlet type boundary value problems in a fractal domain. [ABSTRACT FROM AUTHOR]

Abstract: Solvability conditions for some Dirichlet type boundary value problems in the framework of Hermitian Clifford analysis are established. [Copyright &y& Elsevier]

Abstract: The aim of this work is to further extend the notion of d-summability due to Harrison and Norton in the beginning of the 1990s. Explicit examples are given to illustrate how our notion can be applied to describe the geometry of a simply connected bounded open subset of in a more delicate manner than the latter one. Applications on the solvability conditions for the Riemann boundary value problems for analytic functions over closed curves merely required to be summable in the generalized sense are described. [Copyright &y& Elsevier]

Abstract: In this paper, we study a class of boundary value problems for a first order linear partial differential equation (all of whose solutions are harmonic functions), which is called the Cimmino system. With the help of the one-to-one correspondence between the theory of quaternion valued hyperholomorphic functions and that of Cimmino system’s solutions, necessary and sufficient conditions for the solvability of the non-homogeneous Cimmino system coupled by the boundary conditions are derived and its general solution is explicitly described. [Copyright &y& Elsevier]

Suppose that Ω is a bounded domain of ℝ with a fractal boundary Γ and let ℝ be the real Clifford algebra constructed over the quadratic space ℝ. Replacing the fractal dimensions of Γ with conditions of approximating character we will characterize the monogenicity of a ℝ-valued function F in the interior and exterior of Ω, in terms of its boundary value f = F|. Moreover, our geometric perspective allows for generalizations of certain two-sided monogenic extension results to a wide class of domains. [ABSTRACT FROM AUTHOR]

Abstract: As is the case for the Laplace operator, in Euclidean Clifford analysis also the Helmholtz operator can be factorized, more precisely by using perturbed Dirac operators. In this paper we consider the Helmholtz equation in a circulant matrix form in the context of Hermitian Clifford analysis. The aim is to introduce and study the corresponding inhomogeneous Hermitian Dirac operators, which will constitute a splitting of the traditional perturbed Dirac operators of the Euclidean Clifford analysis context. This will not only lead to special solutions of the Hermitian Helmholtz equation as such, but also to the study of boundary value problems of Riemann type for those solutions, which are, in fact, solutions of the Hermitian perturbed Dirac operators involved. [Copyright &y& Elsevier]

Abstract: In this paper we introduce an alternative way of defining the curvilinear Cauchy integral over non-rectifiable curves in the case of complex functions of one complex variable. Especially the jump behavior on the boundary is considered. As an application, solvability conditions of the Riemann boundary value problem are derived on very weak conditions to the boundary. Besides the complex case the consideration can be extended to the theory of Douglis algebra valued functions. [Copyright &y& Elsevier]

Abstract: We are interested in finding solvability conditions for the Riemann boundary value problems for hyperanalytic functions in a simply connected bounded open subset of the complex plane whose boundary is merely required to be a d-summable closed curve. [Copyright &y& Elsevier]

We consider Hölder continuous circulant (2 x 2) matrix functions Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. defined on the fractal boundary Γ of a domain Ω in ℝ2n. The main goal is to study under which conditions such a function Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. can be decomposed as Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed., where the components Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. are extendable to H-monogenic functions in the interior and the exterior of Ω, respectively. H-monogenicity are a concept from the framework of Hermitean Clifford analysis, a higher-dimensional function theory centered around the simultaneous null solutions of two first-order vector-valued differential operators, called Hermitean Dirac operators. H-monogenic functions then are the null solutions of a (2 x 2) matrix Dirac operator, having these Hermitean Dirac operators as its entries; such matrix functions play an important role in the function theoretic development of Hermitean Clifford analysis. In the present paper a matricial Hermitean Téodorescu transform is the key to solve the problem under consideration. The obtained results are then shown to include the ones where domains with an Ahlfors-David regular boundary were considered. [ABSTRACT FROM AUTHOR]

Monogenic extension theorem of complex Clifford algebras-valued functions over a bounded domain with fractal boundary is obtained. The paper is dealing with the class of Hölder continuous functions. Applications to holomorphic functions theory of several complex variables as well as to that of the so-called biregular functions will be deduced directly from the isotonic approach. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of a hypercomplex singular integral equation, which contains as special case the characteristic equation and other ones associated with it. The equation is solved in closed form by reduction to two Riemann boundary value problems for Douglis algebras-valued functions in generalized Holder spaces over a closed Jordan Carleson curve. [ABSTRACT FROM AUTHOR]

Abstract: This Note gives geometric conditions on a surface of so that the Hilbert transform on that surface in the framework of Clifford analysis defines a bounded operator in the Hölder continuous functions classes. This result provides a generalization of the well-known theorem of Plemelj and Privalov for curves in . To cite this article: R. Abreu Blaya et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009). [Copyright &y& Elsevier]

The purpose of this paper is to solve a kind of the Riemann–Hilbert boundary value problem for (φ , ψ) -harmonic functions, which are linked with the use of two orthogonal bases of the Euclidean space ℝ m . We approach this problem using the language of Clifford analysis for obtaining an explicit expression of the solution of the problem in a Jordan domain Ω ⊂ ℝ m with fractal boundary. Since our study is concerned with a second order differential operator, the boundary data are restricted to involve the higher order Lipschitz class Lip ⁡ (1 + α , Γ) . [ABSTRACT FROM AUTHOR]