TY  - JOUR
A1  - Kähler, Uwe
T1  - On the solution of spatial generalizations of Beltrami equations
N2  - With the help of functional analytical methods complex analysis is a powerful tool in treating non-linear first-order partial differential equations in the plane. Some of the most important of these equations are the Beltrami equations. This is due to the fact that the theory of Beltrami systems is related to many problems of geometry and analysis, like non-linear subsonic two-dimensional hydrodynamics, problems of conformal and quasiconformal mappings of two-dimensional Riemannian manifolds, or complex analytic dynamics. The theory of Beltrami equations is strongly connected with the -operator. This singular integral operator is immediately recognized as two-dimensional Hilbert-transform, known also under the name of integral operator with Beurling kernel, acting as an isometry of L2(C) onto L2(C). In hypercomplex function theory the Beltrami equations have not yet this importance, but nevertheless, they are a basic condition for the transfer of complex methods and efforts for solving partial differential equations, especially of non-linear type, to the spatial case. Here we deal with hypercomplex Beltrami systems. For this we restrict ourselves to the quaternionic case, but without any loss of generality. We will show how a spatial generalization of the complex -operator can be used to solve systems of non-linear partial differential equations, in particular different types of spatial Beltrami systems. Also, the for practical purposes so important norm estimates will be derived. Some of our results are stronger as known results in the complex case, but they are applicable in the complex situation, too.
KW  - Beltrami-Gleichung
Y1  - 1997
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20111215-5018
ER  - 
TY  - JOUR
A1  - Cerejeiras, Paula
A1  - Kähler, Uwe
A1  - Legatiuk, Anastasiia
A1  - Legatiuk, Dmitrii
T1  - Discrete Hardy Spaces for Bounded Domains in Rn
JF  - Complex Analysis and Operator Theory
N2  - Discrete function theory in higher-dimensional setting has been in active development since many years. However, available results focus on studying discrete setting for such canonical domains as half-space, while the case of bounded domains generally remained unconsidered. Therefore, this paper presents the extension of the higher-dimensional function theory to the case of arbitrary bounded domains in Rn. On this way, discrete Stokes’ formula, discrete Borel–Pompeiu formula, as well as discrete Hardy spaces for general bounded domains are constructed. Finally, several discrete Hilbert problems are considered.
KW  - Dirac-Operator
KW  - Randwertproblem
KW  - Funktionentheorie
KW  - discrete Dirac operator
KW  - discrete monogenic functions
KW  - discrete boundary value problems
Y1  - 2020
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20210804-44746
UR  - https://link.springer.com/article/10.1007/s11785-020-01047-6
VL  - 2021
IS  - Volume 15, article 4
SP  - 1
EP  - 32
PB  - Springer
CY  - Heidelberg
ER  -