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Dirichlet-Type Problems for the Iterated Dirac Operator on the Unit Ball in Clifford Analysis

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Advances in Hypercomplex Analysis

Part of the book series: Springer INdAM Series ((SINDAMS,volume 1))

Abstract

We study a class of Dirichlet-type problems for null solutions to iterated Dirac operators on the unit ball of R n with boundary data given by function in (1<p<+∞). Applying Almansi-type decomposition theorems for null solutions to iterated Dirac operators, our Dirichlet-type problems for null solutions to iterated Dirac operators is transferred to Dirichlet-type problems for monogenic functions or harmonic functions. By introducing shifted Euler operators and making use of Clifford-Cauchy transform, we get its unique solution and its integral representation.

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Acknowledgements

This work was supported by FEDER funds through COMPETE—Operational Programme Factors of Competitiveness (“Programa Operacional Factores de Competitividade”), by Portuguese funds through the Center for Research and Development in Mathematics and Applications (University of Aveiro) and the Portuguese Foundation for Science and Technology (“FCT—Fundação para a Ciência e a Tecnologia”), within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690 and by NNSF of China under Grant No. 60873249. The first author is the recipient of Postdoctoral Foundation from FCT (Portugal) under Grant No. SFRH/BPD/74581/2010 and from China under Grant No. 201003111.

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Authors and Affiliations

  1. Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193, Aveiro, Portugal

    Min Ku, Uwe Kähler & Paula Cerejeiras

Authors
  1. Min Ku
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  2. Uwe Kähler
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  3. Paula Cerejeiras
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Corresponding author

Correspondence to Uwe Kähler .

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Editors and Affiliations

  1. Department of Mathematics "U.Dini", University of Florence, viale Morgagni 67/A, Firenze, 50134, Italy

    Graziano Gentili

  2. Department of Mathematics, Politecnico di Milano, Via Bonardi 9, Milano, 20133, Italy

    Irene Sabadini

  3. Department of Mathematics, Instituto Politécnico Nacional, Escuela Sup. de Física y Matemáticas, Mexico D.F., Mexico

    Michael Shapiro

  4. Fac. Engineering, Dept. Mathematical Analysis, University of Gent, Building S22 Galglaan 2, Gent, B 9000, Belgium

    Franciscus Sommen

  5. Schmid College of Science and Technology, Chapman University, University Drive 1, Orange, 92866, California, USA

    Daniele C. Struppa

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Ku, M., Kähler, U., Cerejeiras, P. (2013). Dirichlet-Type Problems for the Iterated Dirac Operator on the Unit Ball in Clifford Analysis. In: Gentili, G., Sabadini, I., Shapiro, M., Sommen, F., Struppa, D. (eds) Advances in Hypercomplex Analysis. Springer INdAM Series, vol 1. Springer, Milano. https://doi.org/10.1007/978-88-470-2445-8_5

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