## Reflection and a mixed boundary value problem concerning analytic functions.(English)Zbl 0903.30028

The authors study a mixed boundary value problem in complex function theory connected with hydrodynamics: Given a bounded domain $$D$$ with the boundary $$C$$ containing continuum $$A$$ with the boundary $$B$$ such that $$G=D\setminus A$$ is connected it is asked to find a holomorphic function $$f$$ on $$G$$ with prescribed continuous $$\text{Re } f=f_{C}$$ on $$C$$ and $$\text{Im } f=f_{B}+k$$ on $$B$$ ($$k$$ is a constant function). If $$B, C$$ are rectifiable Jordan curves satisfying some regularity conditions the solution $$f$$ can be expressed as the Cauchy type integral with continuous real densities. If $$D$$ admits reflection mapping the problem can be simplified as shown in the article. The solution is found with the help of Fredholm type equation under the “minimal” assumptions on $$B$$ and it is of interest also from the point of view of numerical methods.
Reviewer: J.Veselý (Praha)

### MSC:

 30E25 Boundary value problems in the complex plane 31A25 Boundary value and inverse problems for harmonic functions in two dimensions
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