Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-25T11:58:47.205Z Has data issue: false hasContentIssue false
  • English
  • Français

The integral representation for a solution of the 2-d Dirichlet problem with boundary data on closed and open curves

Part of: Fredholm integral equations

Published online by Cambridge University Press:  26 February 2010

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

Abstract

The integral representation for the solution of the 2-D Dirichlet problem for harmonic functions with boundary data on closed and open curves is obtained. The solution is expressed as a sum of potentials, the density of which obeys the uniquely solvable Fredholm integral equation of the second kind.

MSC classification

Secondary: 45B05: Fredholm integral equations
Type
Research Article
Information
Mathematika , Volume 47 , Issue 1-2 , December 2000 , pp. 339 - 354
Copyright
Copyright © University College London 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Gabov, S. A.. An angular potential and its applications. Math. U.S.S.R. Sbornik, 32 (1977), 423–436.CrossRefGoogle Scholar
2.Gakhov, F. D.. Boundary value problems. Pergamon Press, Oxford; Addison-Wesley, Reading, Mass., 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. Maths. Math. Phys., 34 (1994), 1073–1090.Google Scholar
5.Krutitskii, P. A.. Wave propagation in a 2-D external domain bounded by closed and open curves. Nonlinear Analysis, TMA, 32 (1998), 135–144.CrossRefGoogle Scholar
6.Krulitskii, P. A.. Fast stratified flow over several obstacles, including wings. IMA Journ. Appl. Math., 57 (1996), 243 256.Google Scholar
7.Krutitskii, P. A.. The Neumann problem for the 2-D Helmholtz equation in a domain bounded by closed and open curves. Internat. J. Math. & Math. Sci., 21 (1998), 209–216.CrossRefGoogle Scholar
8.Krutitskii, P. A.. The Dirichlet problem for the dissipative Helmholtz equation in a plane domain bounded by closed and open curves. Hiroshima Math. J., 28 (1998), 149–168.CrossRefGoogle Scholar
9.Muskhelishvili, N. I.. Singular integral equations. Noordhoff, Groningen, 1972. (3-d Russian edition: Nauka, Moscow, 1968).Google Scholar
10.Vladimirov, V. S.. Equations of Mathematical Physics. Marcel Dekker, N.Y., 1971.Google Scholar
11.Krutitskii, P. A.. The Neumann problem for the 2-D Helmholtz equation in a multiply connected domain with cuts. Zeitschrift Analysis Anwendungen, 16 (1997), 349–361.CrossRefGoogle Scholar
12.Helms, L. L.. Introduction to Potential Theory. Wiley-Interscience, N.Y., 1969.Google Scholar
13.Perron, O.. Eine neue Behandlung der Randwertaufgabe für Δu = 0. Math. Zeit., 18 (1923), 42–54.CrossRefGoogle Scholar
14.Kràl, J.. The Fredholm method in potential theory. Trans. Amer. Math. Soc., 125 (1966), 511–547.CrossRefGoogle Scholar
15.Kellogg, O. D.. Foundations of potential theory. Dover, N.Y., 1954.Google Scholar
16.Gunter, N. M.. La théorie du potentiel et ses applications aux problèmes fondamentaux de la physique mathematique. Gauthier-Villars, Paris, 1934.Google Scholar
17.Fabes, E., Jodeit, M. and Riviere, N.. Potential technique for boundary value problems on C 1 domains. Acta Math., 141 (1978), 165–186.CrossRefGoogle Scholar
18.Verchota, G.. Layer potentials and boundary value problems for Laplace equation in Lipshitz domains. J. Fund. Anal., 59 (1984), 572–611.CrossRefGoogle Scholar
19.Medková, D.. Solution of the Dirichlet problem for the Laplace equation. Math. Inst. Acad. Sci. Czech Republic. Preprint No. 128/1998. Prague, 1998.Google Scholar
20.Medková, D.. The boundary value problems for Laplace equation and domains with nonsmooth boundary. Archivum mathematicum, 34 (1998), 173–182.Google Scholar
21.Medková, D.. Solution of the Neumann problem for the Laplace equation. Czechoslovak Math. J., 48 (123), (1998), 763–784.CrossRefGoogle Scholar
22.Medková, D.. The third boundary value problem in potential theory for domains with a piecewise smooth boundary. Czechoslovak Math. J., 47 (122), (1997), 651–680.CrossRefGoogle Scholar
23.Nishimura, E.. Crack determination problems. Theor. Appl. Mech., 46 (1997), 39–57.Google Scholar
24.Nishimura, N. and Kobayashi, S.. A boundary integral equation method for an inverse problem related to crack detection. Int. J. Num. Meth. Eng., 32 (1991), 1371 1387.CrossRefGoogle Scholar
25.Nishimura, N. and Kobayashi, S.. Determination of cracks having arbitrary shapes with the boundary integral equation method. Eng. Anal. Boundary Elements, 15 (1995), 180–195.CrossRefGoogle Scholar