On the mixed problem for harmonic functions in a 2-D exterior cracked domain with Neumann condition on cracks
Author:
P. A. Krutitskii
Journal:
Quart. Appl. Math. 65 (2007), 25-42
MSC (2000):
Primary 35J05, 35J25
DOI:
https://doi.org/10.1090/S0033-569X-07-01046-1
Published electronically:
January 2, 2007
MathSciNet review:
2313147
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The mixed Dirichlet-Neumann problem for the Laplace equation in an unbounded plane domain with cuts (cracks) is studied. The Dirichlet condition is given on closed curves making up the boundary of the domain, while the Neumann condition is specified on the cuts. The existence of a classical solution is proved by potential theory and a boundary integral equation method. The integral representation for a solution is obtained in the form of potentials. The density of the potentials satisfies a uniquely solvable Fredholm integral equation of the second kind and index zero. Singularities of the gradient of the solution at the tips of the cuts are investigated.
References
- Gabov S.A. An angular potential and its applications. Math. U.S.S.R. Sbornik, 1977, v.32, No.4, pp. 423-436.
- F. D. Gakhov, Boundary value problems, Pergamon Press, Oxford-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1966. Translation edited by I. N. Sneddon. MR 0198152
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
- P. A. Krutitskii, On the electric current from electrodes in a magnetized semiconductor film, IMA J. Appl. Math. 60 (1998), no. 3, 285–297. MR 1653863, DOI https://doi.org/10.1093/imamat/60.3.285
- P. A. Krutitskiĭ, The Dirichlet problem for the Helmholtz equation in the exterior of cuts in the plane, Zh. Vychisl. Mat. i Mat. Fiz. 34 (1994), no. 8-9, 1237–1258 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys. 34 (1994), no. 8-9, 1073–1090. MR 1300397
- Krutitskii P.A. The Neumann problem for the Helmholtz equation outside cuts in the plane. Comput. Math. Math. Phys., 1994, No.11, pp. 1421-1431.
- P. A. Krutitskii, The 2-dimensional Dirichlet problem in an external domain with cuts, Z. Anal. Anwendungen 17 (1998), no. 2, 361–378. MR 1632551, DOI https://doi.org/10.4171/ZAA/827
- P. A. Krutitskii, The Neumann problem in a 2-D exterior domain with cuts and singularities at the tips, J. Differential Equations 176 (2001), no. 1, 269–289. MR 1861190, DOI https://doi.org/10.1006/jdeq.2000.3954
- P. A. Krutitskii, Wave propagation in a $2$-D external domain bounded by closed and open curves, Nonlinear Anal. 32 (1998), no. 1, 135–144. MR 1491619, DOI https://doi.org/10.1016/S0362-546X%2897%2900470-7
- P. A. Krutitskii, Wave scattering in a 2-D exterior domain with cuts: the Neumann problem, ZAMM Z. Angew. Math. Mech. 80 (2000), no. 8, 535–546. MR 1775291, DOI https://doi.org/10.1002/1521-4001%28200008%2980%3A8%3C535%3A%3AAID-ZAMM535%3E3.0.CO%3B2-1
- Pavel A. Krutitskii, An explicit solution of the pseudo-hyperbolic initial-boundary value problem in a multiply connected region, Math. Methods Appl. Sci. 18 (1995), no. 11, 897–925. MR 1346665, DOI https://doi.org/10.1002/mma.1670181105
- P. A. Krutitskii, The oblique derivative problem for the Helmholtz equation and scattering tidal waves, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2001), no. 2011, 1735–1755. MR 1850934, DOI https://doi.org/10.1098/rspa.2001.0787
- Dagmar Medková, Solution of the Dirichlet problem for the Laplace equation, Appl. Math. 44 (1999), no. 2, 143–168. MR 1667634, DOI https://doi.org/10.1023/A%3A1022209421576
- I. K. Lifanov, Singular integral equations and discrete vortices, VSP, Utrecht, 1996. MR 1451377
- Gunter N.M. Potential theory and its application to the basic problems of mathematical physics. GITTL, Moscow, 1953.
- N. I. Muskhelishvili, Singulyarnye integral′nye uravneniya, Third, corrected and augmented edition, Izdat. “Nauka”, Moscow, 1968 (Russian). Granichnye zadachi teorii funktsiĭ i nekotorye ikh prilozheniya k matematicheskoĭ fizike. [Boundary value problems in the theory of function and some applications of them to mathematical physics]; With an appendix by B. Bojarski. MR 0355495
- V. S. Vladimirov, Uravneniya matematicheskoĭ fiziki, Izdat. “Nauka”, Moscow, 1971 (Russian). Second edition, revised and augmented. MR 0352661
- M. Durand, Layer potentials and boundary value problems for the Helmholtz equation in the complement of a thin obstacle, Math. Methods Appl. Sci. 5 (1983), no. 3, 389–421. MR 716663, DOI https://doi.org/10.1002/mma.1670050126
- V. V. Panasyuk, M. P. Savruk, and Z. T. Nazarchuk, Metod singulyarnykh integral′nykh uravneniĭ v dvumernykh zadachakh difraktsii, “Naukova Dumka”, Kiev, 1984 (Russian). MR 780185
- Yu. A. Tuchkin, Wave scattering by an unclosed cylindrical screen of arbitrary profile with the Neumann boundary condition, Dokl. Akad. Nauk SSSR 293 (1987), no. 2, 343–345 (Russian). MR 884044
References
- Gabov S.A. An angular potential and its applications. Math. U.S.S.R. Sbornik, 1977, v.32, No.4, pp. 423-436.
- Gakhov F.D. Boundary value problems. Pergamon Press, Oxford; Addison-Wesley, Reading, Mass., 1966. MR 0198152 (33:6311)
- Gilbarg D., Trudinger N.S. Elliptic partial differential equations of second order. Springer, Berlin, 1983. MR 0737190 (86c:35035)
- Krutitskii P.A. On the electric current from electrodes in a magnetized semiconductor film. IMA J. Appl. Math., 1998, v.60, pp. 285-297. MR 1653863 (99h:78026)
- Krutitskii P.A. The Dirichlet problem for the Helmholtz equation outside cuts in a plane. Comp. Math. Math. Phys., 1994, v.34, No.8/9, pp. 1073-1090. MR 1300397 (95f:35046)
- Krutitskii P.A. The Neumann problem for the Helmholtz equation outside cuts in the plane. Comput. Math. Math. Phys., 1994, No.11, pp. 1421-1431.
- Krutitskii P.A. The 2-Dimensional Dirichlet Problem in an External Domain with Cuts. Zeitscr. Analysis u. Anwend., 1998, v.17, No.2, pp. 361-378. MR 1632551 (99f:35036)
- Krutitskii P.A. The Neumann problem in a 2-D exterior domain with cuts and singularities at the tips. J. Differential Equations, 2001, v.176, No.1, pp. 269-289. MR 1861190 (2002g:35056)
- Krutitskii P.A. Wave propagation in a 2-D external domain bounded by closed and open curves. Nonlinear Analysis, Theory, Methods and Applications, 1998, v.32, No.1, pp. 135-144. MR 1491619 (98m:35040)
- Krutitskii P.A. Wave scattering in a 2-D exterior domain with cuts: The Neumann problem. ZAMM, 2000, v.80, No.8, pp. 535-546. MR 1775291 (2001d:35037)
- Krutitskii P.A. An explicit solution of the pseudo-hyperbolic initial boundary value problem in a multiply connected region. Mathematical Methods in the Applied Sciences, 1995, v.18, No.11, pp. 897-925. MR 1346665 (97d:76010a)
- Krutitskii P.A. The oblique derivative problem for the Helmholtz equation and scattering tidal waves. Proceedings of the Royal Society of London, ser. A, 2001, v.457, pp. 1735-1755. MR 1850934 (2002g:35051)
- Medkova D. Solution of the Dirichlet problem for the Laplace equation. Applications of Mathematics, 1999, v.44, pp. 143–168. MR 1667634 (2000a:35040)
- Lifanov I.K. Singular integral equations and discrete vortices. VSP, Utrecht, 1996. MR 1451377 (98g:65130)
- Gunter N.M. Potential theory and its application to the basic problems of mathematical physics. GITTL, Moscow, 1953.
- Muskhelishvili N.I. Singular integral equations, Nauka, Moscow, 1968 (3rd Russian edition). MR 0355495 (50:7969)
- Vladimirov V.S. Equations of Mathematical Physics, Marcel Dekker, N.Y., 1971. MR 0352661 (50:5148)
- Durand M. Layer potentials and boundary value problems for the Helmholtz equation in the complement of a thin obstacle. Math. Meth. Appl. Sci., 1983, v.5, pp. 389-421. MR 0716663 (84k:35040)
- Panasyuk, V.V., Savruk, M.P., Nazarchuk, Z.T. The method of singular integral equations in two-dimensional diffraction problems, Naukova Dumka, Kiev, 1984. (in Russian). MR 0780185 (86i:45009)
- Tuchkin Y. A., Scattering of waves by an unclosed cylindrical screen of arbitrary profile with Neumann boundary condition. Dokl. Akad. Nauk SSSR, 293 (1987), pp. 343-345. (in Russian). MR 0884044 (89d:78033)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2000):
35J05,
35J25
Retrieve articles in all journals
with MSC (2000):
35J05,
35J25
Additional Information
P. A. Krutitskii
Affiliation:
KIAM, Department 25, Miusskaya Sq. 4, Moscow 125047, Russia
Keywords:
Laplace equation,
Dirichlet–Neumann problem,
boundary integral equation method.
Received by editor(s):
October 27, 2005
Published electronically:
January 2, 2007
Article copyright:
© Copyright 2007
Brown University