Uwe Kähler

Elliptic boundary value problems of fluid dynamics over unbounded domains

In this paper we develop a Clifford operator calculus over unbounded domains whose complement contains a non-empty open set by using add-on terms to the Cauchy kernel. Using the knowledge about the Poisson equation allows us to prove a direct decomposition of the space L q (Ω), which will be applied to solve the linear Stokes problem in scales of W q k (Ω)-spaces over this kind of unbounded domains. This result will be used to investigate the Navier-Stokes equations by means of a Banach contraction principle. In the end, steady solutions of stream problems with free convection will be studied.

On Qp-spaces of quaternion-valued functions

We consider a definition of Qp-spaces for quaternion-valued functions of three real variables and study some of its basic properties.

Fischer Decomposition for Difference Dirac Operators

Abstract.We establish the basis of a discrete function theory starting with a Fischer decomposition for difference Dirac operators. Discrete versions of homogeneous polynomials, Euler and Gamma operators are obtained. As a consequence we obtain a Fischer decomposition for the discrete Laplacian.

Clifford analysis for finite reflection groups

In this article we establish the basis for a Clifford analysis over finite reflection groups. §Dedicated to Richard Delanghe on the occasion of his 65th birthday.

Difference potentials for the Navier–Stokes equations in unbounded domains

We develop a numerical method for the Navier–Stokes equations over unbounded domains. From the analytic methods used to show existence and uniqueness, we obtain their discrete counterparts which allows us to establish a problem-adapted numerical solver based on finite differences for functions with low regularity.

On a boundary value problem of the biharmonic equation

In this paper, we study a system of biharmonic equations coupled by the boundary conditions. These boundary conditions contain some combinations of the values, div, curl, and grad. Applications in mathematical physics are possible and the investigations will be done with the help of hypercomplex methods. It is also the aim of the paper to demonstrate the application of Clifford analytic methods to the solution of boundary value problems. The results on a special boundary value problem for the biharmonic equation will be used for the investigation of some first-order systems of partial differential equations. We study a theoretical problem connected with the ∂¯-problem and the solution of a Beltrami system by using a fixed-point iteration.

WEIGHTED LIPSCHITZ CONTINUITY AND HARMONIC BLOCH AND BESOV SPACES IN THE REAL UNIT BALL

The characterization by weighted Lipschitz continuity is given for the Bloch space on the unit ball of Rn. Similar results are obtained for little Bloch and Besov spaces.

A Quaternionic Beltrami-Type Equation and the Existence of Local Homeomorphic Solutions

The paper deals with a quaternionic Beltrami-type equation, which is a very natural generalization of the complex Beltrami equation to higher dimensions. Special attention is paid to the systematic use of the embedding of the set of quaternions H into C and the corresponding application of matrix singular integral operators. The proof of the existence of local homeomorphic solutions is based on a necessary and sufficient criterion, which relates the Jacobian determinant of a mapping from R into R to the quaternionic derivative of a monogenic function.

Fractional Clifford Analysis

In this paper we present the basic tools of a fractional function theory in higher dimensions by means of a fractional correspondence to the Weyl relations. A Fischer decomposition, Almansi decomposition, fractional Euler and Gamma operators, monogenic projection, and basic fractional homogeneous powers will be constructed.

Fueter's theorem and its generalizations in Dunkl–Clifford analysis

In this paper, we give a construction of Dunkl monogenic and Dunkl harmonic functions starting from holomorphic functions in the plane. This construction has the advantage of not needing Dunkls intertwining operator or Dunkl spherical harmonics. To this end we study Vekua-type systems and prove a version of Fueters theorem in the case of finite reflection groups. Important examples, such as a Dunkl monogenic Gaussian distribution or a Cauchy kernel, will be given at the end.