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The 2-dimensional Dirichlet problem in an external domain with cuts. (English) Zbl 0907.35036
Earlier two-dimensional Dirichlet problems for the Laplace equation in a multiply connected domain bounded by closed curves and in the plane outside cuts were investigated separately, because different methods for their analysis were used. Here the author proposes a unified approach to these problems and considers an arbitrary multiply connected domain with cuts, provided that there are a finite number of sufficiently smooth boundary curves and cuts. Using the classical potential theory, he reduces the problem to a Fredholm equation of the second kinds, which is uniquely solvable. Additionally, the author investigates the solvability of the Dirichlet problem in internal domains with cuts, but here he applies the technique of conformal mappings. The results obtained can find applications by the modelling of thin wings, cracks or screens in physical problems.
Reviewer: O.Titow (Berlin)

MSC:
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
35J25Second order elliptic equations, boundary value problems
45E05Integral equations with kernels of Cauchy type
31A05Harmonic, subharmonic, superharmonic functions (two-dimensional)
31A25Boundary value and inverse problems (two-dimensional potential theory)
30E25Boundary value problems, complex analysis