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The Dirichlet problem for the two-dimensional Laplace equation in a multiply connected domain with cuts. (English) Zbl 0942.31002
The Dirichlet problem for the two dimensional Laplace equation is studied in a multiply connected bounded region with cuts. The Dirichlet data is specified on the total boundary including sides of cuts. The existence of a classical solution is proved by potential theory. The integral representation for a solution is obtained in the form of potentials. The density in potentials obeys the uniquely solvable Fredholm equation of the second kind and index zero. Multiply connected interior region without cuts is a particular case of our problem. Uniqueness of the solution is proved.

MSC:
31A10Integral representations of harmonic functions (two-dimensional)
31A05Harmonic, subharmonic, superharmonic functions (two-dimensional)
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
45E05Integral equations with kernels of Cauchy type
30E25Boundary value problems, complex analysis
31A25Boundary value and inverse problems (two-dimensional potential theory)