Abstract
In this paper we deal with a special type of quaternionic Beltrami equation and discuss the existence of local and global homeomorphic solutions based on a necessary and sufficient condition, which relates the Jacobian determinant of a mapping from ℝ4 to ℝ4 with the hypercomplex derivative of a monogenic function.
Similar content being viewed by others
References
A — L. Ahlfors:Lectures on quasiconformal mappings. Van Nostrand, Princeton, 1966.
BDS — F. Brackx, R. Delanghe and F. Sommen:Clifford analysis. Pitman Research Notes76, 1982.
CGKM — P. Cerejeiras, K. Gürlebeck, U. Kähler, U. and H. Malonek: A quaternionic Beltrami type equation and the existence of local homeomorphic solutions.ZAA, 2001, to appear.
GK1 — K. Gürlebeck and U. Kähler: On a spatial generalization of the complex П-operator.ZAA 15 No. 2 (1996), 283–297.
GK2 — K. Gürlebeck and U. Kähler: On a boundary value problem of the biharmonic equation.Math. Meth. in the Applied Sciences 20 (1997), 867–883.
GM — K. Gürlebeck and H. Malonek: A Hypercomplex Derivative of Monogenic Functions in ℝn+1 and its Applications.Complex Variables 39 (1999), 199–228.
GS — K. Gürlebeck and W. Sprößig:Quaternionic and Clifford calculus for Engineers and Physicists. Chichester, John Wiley &. Sons, 1997.
K1 — U. Kähler: On the solutions of higher-dimensional Beltrami-equations.Digital Proc. of the IKM 97, Weimar, 1997.
K2 — U. Kähler:On quaternionic Beltrami equations. In: Clifford Algebras and their Applications in Mathematical Physics, Vol. II, ed. by J. Ryan and W. Sprößig, Birkhäuser, Basel, 2000, 3–16.
MS — I.M. Mitelman and M.V. Shapiro: Differentiation of the Martinelli-Bochner Integrals and the Notion of Hyperderivability.Math. Nachr.,172 (1995), 211–238.
SV1 — M.V. Shapiro and N.L. Vasilevski:On the Bergman kernel function in the Clifford analysis. In: Clifford Algebras and their Applications in Mathematical Physics, edited by F. Brackx, R. Delanghe, and H. Serras, Kluwer, Dordrecht, 1993, 183–192.
SV2 — M.V. Shapiro and N.L. Vasilevski: On the Bergman kernel function in hypercomplex analysis.Acta Appl. Math.,46 No. 1 (1997), 1–27.
Sh — B.И. Шевченко:О локальном ςомеоморфузме мрехмерноςо хросмрансмеа, осущесмеляемом решемвем мекоморои éллвнмвческоŭ свсмемы. Доклады Академии Наук CCCP5 (1962), 1035–1038.
Sp — W. Sprößig:Über eine mehrdimensionale Operatorrechnung über beschränkten Gebieten des ℝ n. Thesis, TH Karl-Marx-Stadt, 1979.
Sud — A. Sudbery: Quaternionic Analysis.Math. Proc. Cambr. Phil. Soc. 85 (1979), 199–225.
Y — A. Yanushauskas: On Multi-Dimensional Generalizations of the Cauchy-Riemann System.Complex Variables 26 (1994), 53–62.
Author information
Authors and Affiliations
Additional information
Dedicated to Loo-Keng Hua on the occasion of his 90th birthday
This paper was done while the first author was a recipient of a PRAXIS XXI-scholarship of the Fundação para a Ciência e a Tecnologia visiting the Universidade de Aveiro in Portugal.
Rights and permissions
About this article
Cite this article
Kähler, U., Martins, A.M. Quaternionic Beltrami-type equations and homeomorphic solutions. AACA 11 (Suppl 2), 177–182 (2001). https://doi.org/10.1007/BF03219130
Issue Date:
DOI: https://doi.org/10.1007/BF03219130