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

Dontová, E.; Dont, M.; Král, J.

Math. Bohem. 122, No. 3, 317-336 (1997).

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

Keywords:

boundary value problem; integral equations; Fredholm type equation