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Zbl 0942.31002
Krutitskii, P.A.
The Dirichlet problem for the two-dimensional Laplace equation in a multiply connected domain with cuts. (English)
[J] Proc. Edinb. Math. Soc., II. Ser. 43, No.2, 325-341 (2000). ISSN 0013-0915; ISSN 1464-3839/e

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.
[P.A.Krutitskii (Moskva)]

MSC 2000:: *31A10 Integral representations of harmonic functions (two-dimensional)
31A05 Harmonic functions, etc. (two-dimensional)
35J05 Laplace equation, etc.
45E05 Integral equations with kernels of Cauchy type
30E25 Boundary value problems, complex analysis
31A25 Boundary value and inverse problems (two-dim.potential theory)

Keywords: Laplace equation; Dirichlet problem; cracked domain

Cited in: Zbl 1198.35075

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