Search

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]

The paper provides integral representations for solutions to a certain first order partial differential equation natural arising in the factorization of the Lamé—Navier system with the help of Clifford analysis techniques. These representations look like in spirit to the Borel—Pompeiu and Cauchy integral formulas both in three and higher dimensional setting. [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]

In this paper we prove that a generalization of complex Π-operator in Clifford analysis, obtained by the use of two orthogonal bases of a Euclidean space, possesses several mapping and invertibility properties, as studied before for quaternion-valued functions as well as in the standard Clifford analysis setting. We improve and generalize most of those previous results in this direction and additionally other consequent results are presented. In particular, the expression of the jump of the generalized Π-operator across the boundary of the domain is obtained as well as an estimate for the norm of the Π-operator is given. At the end an application of the generalized Π-operator to the solution of Beltrami equations is studied where we give conditions for a solution to realize a local and global homeomorphism. [ABSTRACT FROM AUTHOR]

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]

Let Ω ⊂ ℝ be a Jordan domain with d-summable boundary Γ. The main gol of this paper is to estimate the Hölder norm of a fractal version of the Hilbert transform in the Clifford analysis context acting from Hölder spaces of Clifford algebra valued functions defined on Γ. The explicit expression for the upper bound of the norm provided here is given in terms of the Hölder exponents, the diameter of Γ and certain d-sum ( d > d) of the Whitney decomposition of Ω. The result obtained is applied to standard Hilbert transform for domains with left Ahlfors-David regular surface. [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]

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]

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]