On a boundary value problem for a class of generalized analytic functions. (English) Zbl 1262.30049
Almeida, Alexandre (ed.) et al., Advances in harmonic analysis and operator theory. The Stefan Samko anniversary volume on the occasion of his 70th birthday. Mainly based on the presentations at two conferences, Lisbon and Aveiro, Portugal, in June – July, 2011. Basel: Birkhäuser (ISBN 978-3-0348-0515-5/hbk; 978-3-0348-0516-2/ebook). Operator Theory: Advances and Applications 229, 91-99 (2013).
Summary: For solutions of the Bers-Vekua equation \(Dv:=v\bar{z}-c(z,\bar{z})\bar{v}=0\) defined in a domain \(\mathbb D\subset \mathbb C\) we consider Riemann-Hilbert type boundary conditions.
In the case of the existence of certain differential operators with which all the solutions of \(Dv = 0\) defined in \(\mathbb D\) can be generated from a function \(f\) holomorphic in \(\mathbb D\) the boundary value problem is reduced to a Goursat problem for \(f\) in essence. For certain classes of coefficients \(c\) and domains \(\mathbb D\) we show how this problem can be solved explicitly.
For the poly-pseudoanalytic functions obeying the differential equation \(D^nv=0\), \(n\in\mathbb N\), we investigate an appropriate boundary value problem and show the equivalence of this problem to \( n \) boundary value problems for generalized analytic functions.
For the entire collection see [Zbl 1258.00012].
In the case of the existence of certain differential operators with which all the solutions of \(Dv = 0\) defined in \(\mathbb D\) can be generated from a function \(f\) holomorphic in \(\mathbb D\) the boundary value problem is reduced to a Goursat problem for \(f\) in essence. For certain classes of coefficients \(c\) and domains \(\mathbb D\) we show how this problem can be solved explicitly.
For the poly-pseudoanalytic functions obeying the differential equation \(D^nv=0\), \(n\in\mathbb N\), we investigate an appropriate boundary value problem and show the equivalence of this problem to \( n \) boundary value problems for generalized analytic functions.
For the entire collection see [Zbl 1258.00012].
MSC:
| 30G20 | Generalizations of Bers and Vekua type (pseudoanalytic, \(p\)-analytic, etc.) |
| 35F15 | Boundary value problems for linear first-order PDEs |
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