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The Dirichlet problem for the two-dimensional Laplace equation in a multiply connected domain with cuts

Published online by Cambridge University Press:  20 January 2009

P. A. Krutitskii
P. A. Krutitskii
Affiliation:
Department of Mathematics, Faculty of Physics, Moscow State University, Moscow 119899, Russia
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Abstract

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The Dirichlet problem for the Laplace equation in a connected-plane region with cuts is studied. The existence of a classical solution is proved by potential theory. The problem is reduced to a Fredholm equation of the second kind, which is uniquely solvable.

Keywords

Laplace equationDirichlet problemcracked adomain
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 43 , Issue 2 , June 2000 , pp. 325 - 341
Copyright
Copyright © Edinburgh Mathematical Society 2000

References

1. Gabov, S. A., An angular potential and its applications. Math. USSR Sbornik 32 (1977), 423–436.CrossRefGoogle Scholar
2. Gakhov, F. D., Boundary value problems (Pergamon, Oxford; Addison-Wesley, Reading, MA, 1966).CrossRefGoogle Scholar
3. Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of second order (Springer, Berlin, 1983).Google Scholar
4. Krutitskii, P. A., Dirichlet problem for the Helmholtz equation outside cuts in a plane. Comp. Math. Math. Phys. 34 (1994), 1073–1090.Google Scholar
5. Krutitskii, P. A., Wave propagation in a 2-D external domain with cuts. Applicable Analysis 62 (1996), 297–309.CrossRefGoogle Scholar
6. Krutitskii, P. A., Fast stratified flow over several obstacles, including wings. IMA J. Appl Math. 57 (1996), 243–256.CrossRefGoogle Scholar
7. Krutitskii, P. A., The Neumann problem for the 2-D Helmholtz equation in a multiply connected domain with cuts. Zeitschr. Analysis Anwendungen 16 (1997), 349–361.CrossRefGoogle Scholar
8. Krutitskii, P. A., The Dirichlet problem for the 2-D Helmholtz equation in a multiply connected domain with cuts. ZAMM 77 (1997), 883–890.CrossRefGoogle Scholar
9. Muskhelishvili, N. I., Singular integral equations (Noordhoff, Groningen, 1972) (3rd Russian edn: Nauka, Moscow, 1968).Google Scholar
10. Vladimirov, V. S., Equations of mathematical physics (Marcel Dekker, New York, 1971).Google Scholar