Boundary value problems for elliptic equations – real and complex methods in comparison. (English) Zbl 1301.30045
Kiryakova, Virginia (ed.), Complex analysis and applications ’13. Proceedings of the international conference, CAA ’13, Sofia, Bulgaria, October 31 – November 2, 2013. Dedicated to the 100th anniversary of Academician Ljubomir Iliev. Sofia: Bulgarian Academy of Sciences, Institute of Mathematics and Informatics (ISBN 978-954-8986-38-0). 322-345 (2013).
Summary: Using fundamental solutions, boundary value problems for elliptic equations
and elliptic systems can be reduced to fixed-point problems for a suitably defined
operator. In order to solve the related fixed-point problems, we apply both the
contraction-mapping principle and the second version of the Schauder Fixed-point Theorem as well.
Since the right-hand sides are supposed to be only locally bounded or only
locally Lipschitz-continuous, the fixed-point theorems are applicable only in balls
of the underlying function space, not in the whole function space. We show also
how one can determine the best radius of the ball which leads to solvability
conditions which are as weak as possible.
Elliptic first order systems in higher dimensions can be reduced to operators whose definition contains a monogenic function. The real-valued components of a monogenic function are solutions of the Laplace equation. The paper
constructs also so-called distinguishing parts of the boundary from which the
real-valued components can completely be recovered.
For the entire collection see [Zbl 1298.00309].
For the entire collection see [Zbl 1298.00309].
MSC:
30G20 | Generalizations of Bers and Vekua type (pseudoanalytic, \(p\)-analytic, etc.) |
30G35 | Functions of hypercomplex variables and generalized variables |
35G30 | Boundary value problems for nonlinear higher-order PDEs |
35F60 | Boundary value problems for systems of nonlinear first-order PDEs |