SUMMARYWe consider parabolic Dirac operators which do not involve fractional derivatives and use them to show the solvability of the in-stationary Navier-Stokes equations over time-varying domains.

Abstract. In this paper we construct the main ingredients of a discrete function theory in higher dimensions by means of a new "skew" type of Weyl relations. We will show that this new type overcomes the difficulties of working with standard Weyl relations in the discrete case. A Fischer decomposition, Euler operator, monogenic projection, and basic homogeneous powers will be constructed.

We develop a constructive framework to define difference approximations of Dirac operators which factorize the discrete Laplacian. This resulting notion of discrete monogenic functions is compared with the notion of discrete holomorphic functions on quad-graphs. In the end Dirac operators on quad-graphs are constructed. (2000). Primary 30G35, 30G25; Secondary 05C78. Mathematics Subject Classification

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.

A modified Cauchy kernel is introduced over unbounded domains whose complement contains nonempty open sets. Basic results on Clifford analysis over bounded domains are now carried over to this more general context and to functions that are no longer assumed to be bounded. In particular Plemelj formulae are explicitly computed. Basic properties of the Cauchy transform over unbounded domains lying in a half space are investigated, and an orthogonal decomposition of the L 2 space for such a domain is set up. At the end a boundary value problem will be studied in the case of an unbounded domain without using weighted Sobolev spaces. ᮊ 1997 Academic Press