The distinguishing boundary for monogenic functions of Clifford analysis. (English) Zbl 1319.30041
Summary: Since all real-valued components of monogenic functions are solutions of the Laplace equation, a monogenic function is completely determined by the boundary values of its components. However, in order ro recover a monogenic function, one does not need the boundary data of all components on the whole boundary. We shall construct some (lowerdimensional) so-called distinguishing parts of the boundary from which all components of a monogenic function can be calculated.{
} This will be done by decomposing the Cauchy-Riemann system into completely integrable subsystems. Correspondingly the domain will be decomposed into (lower-dimensional) fibres. Then some of the components of the monogenic function turn out to be solutions of an initialvalue problem for a subsystem in the corresponding fibre.
MSC:
| 30G35 | Functions of hypercomplex variables and generalized variables |
Keywords:
calculation of monogenic functions from boundary data; decomposition of the Cauchy-Riemann system into completely integrable subsystems; decomposition of domains into fibresReferences:
| [1] | F. Brackx, R. Delanghe and F. Sommen, Clifford analysis. Pitman Research Notes in Mathematics no. 76, Boston 1982. · Zbl 0529.30001 |
| [2] | Doan Cong Dinh, Generalized Clifford analysis. Doctoral Thesis at Graz University of Technology, 2012. · Zbl 1295.35186 |
| [3] | Doan Cong Dinh, Dirichlet boundary value problem for monogenic functions in Clifford analysis. Complex Var. Elliptic Equ. (in print). · Zbl 1295.35186 |
| [4] | E. Escassut, W. Tutschke and C. C. Yang (eds.), Some topics on value distribution and differentiability in complex and p-adic analysis. Science Press Beijing 2008. · Zbl 1233.30005 |
| [5] | V. Kiryakova (ed.), Complex Analysis and Applications ’13. (Proc. Intern. Conf., Sofia, 2013) - Electronic Book (Full Length Papers), 345 pp. |
| [6] | Le Hung Son and W. Tutschke (eds.), Algebraic structures in partial differential equations related to complex and Clifford analysis. Ho Chi Minh City University of Education Press, Ho Chi Minh City, 2010. · Zbl 1223.35010 |
| [7] | Osgood, W. F., Lehrbuch der Funktionentheorie. Zweiter Band. New York, 1965. · Zbl 0184.29701 |
| [8] | E. N. Sattorov, Reconstruction of solutions to a generalized Moisil-Teodorescu system in a spatial domain from their values on a part of the boundary. Izv. Vyssh. Uchebn. Zaved. Math. 2011, No 1, 72-84. · Zbl 1231.35294 |
| [9] | W. Tutschke, Generalized analytic functions in higher dimensions. Georgian Math. J. 14 no. 3 (2007), pp. 581-595. · Zbl 1132.30351 |
| [10] | W. Tutschke, Reduction of boundary value problems to fixed-point problems using real and complex fundamental solutions. Contained in [6], pp. 85-105. · Zbl 1234.35070 |
| [11] | W. Tutschke, Boundary value problems for elliptic equations - real and complex methods in comparison. Contained in [5], pp. 322-345. · Zbl 1301.30045 |
| [12] | W. Tutschke and C. J. Vanegas, Clifford algebras depending on parameters and their applications to partial differential equations. Contained in [4], pp. 430-449. · Zbl 1232.30034 |
| [13] | W. Tutschke, A boundary value problem for monogenic functions in parameter-depending Clifford algebras. Complex Var. Elliptic Equ. 56 no. 1-4 (2011), pp. 113-118. · Zbl 1213.35182 |
| [14] | W Tutschke.: General algebraic structures of Clifford type and Cauchy-Pompeiu formulae for some piecewise constant structure relations. Adv. Appl. Clifford Algebr. 21(no. 4), pp. 829-838 (2011) · Zbl 1256.30047 · doi:10.1007/s00006-011-0282-8 |
| [15] | Z. Xu, J. Chen and W. Zhang, A harmonic conjugate of the Poisson kernel and a boundary value problem for monogenic functions in the unit ball of Rn (n ≥ 2). Simon Stevin, 64 (2) (1990), pp. 187-201. · Zbl 0722.31004 |
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