A Hermitian Cauchy formula on a domain with fractal boundary

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:369

Article

Abreu-Blaya, R.³  Bory-Reyes, J.²  Brackx, F.¹  De Schepper, H.¹  Sommen, F.¹ 

[1] Univ Ghent, Fac Engn, Clifford Res Grp, B-9000 Ghent, Belgium

[2] Univ Oriente, Dept Matemat, Santiago De Cuba 90500, Cuba

[3] Univ Holguin, Fac Informat & Matemat, Holguin 80100, Cuba

关键词: Hermitian Clifford analysis;    Cauchy integral;    Fractal geometry;   

DOI  :  10.1016/j.jmaa.2010.03.032

来源: Elsevier

【 摘 要 】

Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centered around the concept of monogenic functions, i.e. null solutions of a first order vector valued rotation invariant differential operator called Dirac operator, which factorizes the Laplacian: monogenic functions may thus also be seen as a generalization of holomorphic functions in the complex plane. Hermitian Clifford analysis offers yet a refinement of the Euclidean case: it focusses on the simultaneous null solutions, called Hermitian (or h-) monogenic functions, of two Hermitian Dirac operators which are invariant under the action of the unitary group. In Brackx et al. (2009) [8] a Clifford-Cauchy integral representation formula for h-monogenic functions has been established in the case of domains with smooth boundary, however the approach followed cannot be extended to the case where the boundary of the considered domain is fractal. At present, we investigate an alternative approach which will enable us to define in this case a Hermitian Cauchy integral over a fractal closed surface, leading to several types of integral representation formulae, including the Cauchy and Borel-Pompeiu representations. (C) 2010 Elsevier Inc. All rights reserved.

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