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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.
Reviewer: P.A.Krutitskii (Moskva)

##### MSC:
 31A10 Integral representations of harmonic functions (two-dimensional) 31A05 Harmonic, subharmonic, superharmonic functions (two-dimensional) 35J05 Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation 45E05 Integral equations with kernels of Cauchy type 30E25 Boundary value problems, complex analysis 31A25 Boundary value and inverse problems (two-dimensional potential theory)
##### Keywords:
Laplace equation; Dirichlet problem; cracked domain
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