Juan Bory Reyes

On the Riemann Hilbert type problems in Clifford analysis

In this paper boundary value problems combining Jump — Riemann and Hilbert problems for monogenic functions in Ahlfors-David regular surfaces and in the upper half space respectively are investigated. The explicit formula of the solution is obtained.

Cauchy Transform and Rectifiability in Clifford Analysis

Let Γ be an n-dimensional rectifiable Ahlfors-David regular surface in Rn+1. Let u be a continuous R0,n-valued function on Γ, where R0,n is the Clifford algebra associated with Rn. Then we prove that the Cliffordian Cauchy transform

Hermitean Cauchy Integral Decomposition of Continuous Functions on Hypersurfaces

We consider Hölder continuous circulant matrix functions defined on the Ahlfors-David regular boundary of a domain in . The main goal is to study under which conditions such a function can be decomposed as , where the components are extendable to two-sided -monogenic functions in the interior and the exterior of , respectively. -monogenicity is 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. -monogenic functions then are the null solutions of a matrix Dirac operator, having these Hermitean Dirac operators as its entries; such functions have been crucial for the development of function theoretic results in the Hermitean Clifford context.

Boundary value problems for quaternionic monogenic functions on non-smooth sureaces

In this paper, analogous of the Compound Riemann-Hilbert boundary value problems are investigate for quaternionic monogenic functions. The solution (explicitly) of the problem is established over continuous surface, with little smoothness, which bounds a bounded domain of R3. In particular, smoothness property for high-dimensional Cauchy type integral are computed. We also use Zygmund type estimates to adapt existing one-variable complex results to ilustrate the Hölder-boundedness of the singular integral operator on 2-dimensional Ahlfors regular surfaces. At the end, uniqueness of solution for the Riemann boundary value problem have already built taking as a base the general Operator Theory.

The quaternionic riemann problem with a natural geometric condition on the boundary

We consider a generalization of the quaternionic Riemann problem whose formulation takes into account a reasonably natural geometric condition on the boundary and which may be solved under small constant perturbation via successive approximation. Smoothness properties of the Cauchy transform and the singular Cauchy integral operator are treated. Finally we discuss a reduction method to solve in closed form a class of singular integral equations which contains the characteristic equation as an application of our results.

Analytical and numerical solutions of the potential and electric field generated by different electrode arrays in a tumor tissue under electrotherapy

BackgroundElectrotherapy is a relatively well established and efficient method of tumor treatment. In this paper we focus on analytical and numerical calculations of the potential and electric field distributions inside a tumor tissue in a two-dimensional model (2D-model) generated by means of electrode arrays with shapes of different conic sections (ellipse, parabola and hyperbola).MethodsAnalytical calculations of the potential and electric field distributions based on 2D-models for different electrode arrays are performed by solving the Laplace equation, meanwhile the numerical solution is solved by means of finite element method in two dimensions.ResultsBoth analytical and numerical solutions reveal significant differences between the electric field distributions generated by electrode arrays with shapes of circle and different conic sections (elliptic, parabolic and hyperbolic). Electrode arrays with circular, elliptical and hyperbolic shapes have the advantage of concentrating the electric field lines in the tumor.ConclusionThe mathematical approach presented in this study provides a useful tool for the design of electrode arrays with different shapes of conic sections by means of the use of the unifying principle. At the same time, we verify the good correspondence between the analytical and numerical solutions for the potential and electric field distributions generated by the electrode array with different conic sections.

A Boundary Value Problem for Hermitian Monogenic Functions

We study the problem of finding a Hermitian monogenic function with a given jump on a given hypersurface in . Necessary and sufficient conditions for the solvability of this problem are obtained.

Cauchy Integral Decomposition of Multi-Vector Valued Functions on Hypersurfaces

Let Ω be a bounded open and connected subset of ℝm which has a C∞-boundary Σ and let Fk ∊ C∞(Σ) be a k-multi-vector valued function on Σ. Under which conditions can Fk be decomposed as Fk = Fk+ + Fk− where Fk+- are extendable to harmonic k-multi-vector fields in Ω± with Ω+ = Ω and

Generalized Moisil-Théodoresco Systems and Cauchy Integral Decompositions

\Omega \_ = \mathbb{R}^m \backslash \bar \Omega ?