Reduction of boundary value problems to fixed-point problems using real and complex fundamental solutions. (English) Zbl 1234.35070
Le Hung Son (ed.) et al., Algebraic structures in partial differential equations related to complex and Clifford analysis. Based on the selected lectures of the 17th international conference on finite and infinite dimensional complex analysis and applications, Ho Chi Minh City, Vietnam, August 3–7, 2009. Ho Chi Minh City: Ho Chi Minh City University of Education Press (ISBN 978-604-918-001-9/hbk). 85-105 (2010).
Summary: Using the Cauchy kernel of complex analysis and the Cauchy kernel of Clifford analysis, this survey will show how boundary value problems for (linear or nonlinear) first order systems can be reduced to fixed-point problems. Especially boundary value problems for generalized analytic functions in the plane and boundary value problems for generalized monogenic functions in higher-dimensional Euclidean spaces will be reduced to the corresponding boundary value problems for simplified linear systems.
Whereas in real analysis the fundamental solution depends on the coefficients of the differential equation, the explicitly given Cauchy kernels in the plane and in higher dimensions can be used for solving boundary value problems for a general class of systems of first order equations.
For the entire collection see [Zbl 1223.35010].
Whereas in real analysis the fundamental solution depends on the coefficients of the differential equation, the explicitly given Cauchy kernels in the plane and in higher dimensions can be used for solving boundary value problems for a general class of systems of first order equations.
For the entire collection see [Zbl 1223.35010].
MSC:
| 35F60 | Boundary value problems for systems of nonlinear first-order PDEs |
| 30G20 | Generalizations of Bers and Vekua type (pseudoanalytic, \(p\)-analytic, etc.) |
| 30G35 | Functions of hypercomplex variables and generalized variables |
| 47H10 | Fixed-point theorems |
| 35A08 | Fundamental solutions to PDEs |
| 35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |
