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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]

In this paper, we combine the fractional ψ−$$ \psi - $$hyperholomorphic function theory with the fractional calculus with respect to another function. As a main result, a fractional Borel–Pompeiu type formula related to a fractional ψ−$$ \psi - $$Fueter operator with respect to a vector‐valued function is proved. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to combine the (q,q′)$$ \left(q,{q}^{\prime}\right) $$‐calculus in the quaternionic context, which is proposed via two kinds of (q,q′)$$ \left(q,{q}^{\prime}\right) $$‐operators, with the theory of slice regular functions. Specifically, we shall work in suitable subclasses of slice regular functions in which the (q,q′)$$ \left(q,{q}^{\prime}\right) $$‐operators can be related with the slice derivative. The paper presents some results such as an integral formula and series expansion. [ABSTRACT FROM AUTHOR]

The Moisil‐Teodorescu operator is considered to be a good analogue of the usual Cauchy–Riemann operator of complex analysis in the framework of quaternionic analysis and it is a square root of the scalar Laplace operator in ℝ3. In the present work, a general quaternionic structure is developed for the local fractional Moisil–Teodorescu operator in Cantor‐type cylindrical and spherical coordinate systems. Furthermore, in order to reveal the capacity and adaptability of the methods, we show two examples for the Helmholtz equation with local fractional derivatives on the Cantor sets by making use of the local fractional Moisil–Teodorescu operator. [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]

Euclidean Clifford analysis has become a well‐established theory of monogenic functions in higher‐dimensional Euclidean space with a variety of applications both inside and outside of mathematics. Noncommutativity of the geometric product in Clifford algebras leads to what are now known as inframonogenic functions, which are characterized by certain elliptic system associated to the orthogonal Dirac operator in ℝm. The main question we shall be concerned with is whether or not a higher‐order Lipschitz function on the boundary Γ of a Jordan domain Ω⊂ℝm can be decomposed into a sum of the two boundary values of a sectionally inframonogenic function with jump across Γ. To this end, a kind of Cauchy‐type integral and singular integral operator, very specific to the inframonogenic setting, are widely used. [ABSTRACT FROM AUTHOR]

In this paper, we consider functions satisfying the sandwich equation ∂x_f∂x_=0, where ∂x_ stands for the Dirac operator in Rm. Such functions are referred as inframonogenic and represent an extension of the monogenic functions, ie, null solutions of ∂x_. In particular, for odd m, we prove that a C2‐function is both inframonogenic and harmonic in Ω⊂Rm if and only if it can be represented in Ω as f=f1+f2+f3x_+x_f4,where f1 and f2 are, respectively, left and right monogenic functions in Ω, while f3 and f4 are two‐sided monogenic functions there. Finally, in deriving some applications of our results, we have made use of the deep connection between the class of inframonogenic vector fields and the universal solutions of the Lamé‐Navier system in R3. [ABSTRACT FROM AUTHOR]

Solutions of the sandwich equation ∂ x _ f ∂ x _ = 0, where ∂ x _ stands for the first‐order differential operator (called Dirac operator) in the Euclidean space R m, are known as inframonogenic functions. These functions generalize in a natural way the theory of kernels associated with ∂ x _, the nowadays well‐known monogenic functions, and can be viewed also as a refinement of the biharmonic ones. In this paper we deepen study the connections between inframonogenic functions and the solutions of the homogeneous Lamé‐Navier system in R 3. Our findings allow to shed some new light on the structure of the solutions of this fundamental system in 3‐dimensional elasticity theory. [ABSTRACT FROM AUTHOR]