BERGMAN FUNCTION AND DIRICHLET PROBLEM IN CLIFFORD ANALYSIS

Abstract:

In this paper, we generalize the results of¹, on theory of hyperholomorphic analysis to the case for several variables, that is, Clifford analysis. These results are not only of theoretical importance but also parctical interest. In fact, we also generalize³, the Dirac operator to . When the base i ={i₁, …,i_n} is given, the Dirac operator D is determined, while ^ψD may consider as the rotation transform on the vector ψ = {ψ¹, … , ψⁿ}. Such a generalization is similar to Descartes coordinate system to the moving coordinate system, therefore, it is very convenient to build some basic formulas on Clifford analysis and to find some analytic representations of differential equations. From this idea, we obtain a very important representation of Bergman kernel function:

this is a key formula for proving Bergman kernel function ^ψB to be a projector of L_p(Ω,C(V_n,0)) onto _ψA^p(Ω,C(V_n,0)), where ^φ,ψC is a compact operator. Moreover, we build up some results on Clifford analysis, such as Stokes formula, Borel-Pompeiu formula and Cauchy formula, propreties of Bergman kernel function. Moreover as their application, we discuss the Dirichlet problem for related bi-analytic functions of several variables on the bounded simply-connected domain with smooth boundary, and then representations of solution are obtained.