The Neumann problem for the equation $\Delta u - k^2u=0$ in the exterior of non-closed Lipschitz surfaces
Author:
P. A. Krutitskii
Journal:
Quart. Appl. Math. 72 (2014), 85-91
MSC (2010):
Primary 35J05, 35J25, 31A10, 31A25
DOI:
https://doi.org/10.1090/S0033-569X-2013-01319-4
Published electronically:
November 13, 2013
MathSciNet review:
3185133
Full-text PDF Free Access
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Abstract: We study the Neumann problem for the equation $\Delta u - k^2u=0$ in the exterior of non-closed Lipschitz surfaces in $R^3$. Theorems on existence and uniqueness of a weak solution of the problem are proved. The integral representation for a solution is obtained in the form of a double layer potential. The density in the potential is defined as a solution of the operator (integral) equation, which is uniquely solvable.
References
- William McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000. MR 1742312
- Gregory Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal. 59 (1984), no. 3, 572–611. MR 769382, DOI https://doi.org/10.1016/0022-1236%2884%2990066-1
- David S. Jerison and Carlos E. Kenig, The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 203–207. MR 598688, DOI https://doi.org/10.1090/S0273-0979-1981-14884-9
- Grisvard, P., Boundary value problems in nonsmooth domains. Pitman, London, 1985.
- I. K. Lifanov, L. N. Poltavskii, and G. M. Vainikko, Hypersingular integral equations and their applications, Differential and Integral Equations and Their Applications, vol. 4, Chapman & Hall/CRC, Boca Raton, FL, 2004. MR 2053793
- A. Ya. Povzner and I. V. Suharevskiĭ, Integral equations of the second kind in problems of diffraction by an infinitely thin screen, Soviet Physics. Dokl. 4 (1960), 798–801 (Russian). MR 0110445
- Berkhin, P. E., Problem of diffraction of a fine screen. Siberian Mathematical Journal, 1984, v. 25, 31–42.
- P. A. Krutitskii, The Neumann problem in a plane domain bounded by closed and open curves, Complex Variables Theory Appl. 38 (1999), no. 1, 1–20. MR 1685543, DOI https://doi.org/10.1080/17476939908815149
- P. A. Krutitskii, The $2$-D Neumann problem in an exterior domain bounded by closed and open curves, Math. Methods Appl. Sci. 20 (1997), no. 18, 1551–1562. MR 1486525, DOI https://doi.org/10.1002/%28SICI%291099-1476%28199712%2920%3A18%3C1551%3A%3AAID-MMA912%3E3.3.CO%3B2-H
- P. A. Krutitskii, The 2-D Neumann problem in a domain with cuts, Rend. Mat. Appl. (7) 19 (1999), no. 1, 65–88 (English, with English and Italian summaries). MR 1710125
- 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
References
- William McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000. MR 1742312 (2001a:35051)
- Gregory Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal. 59 (1984), no. 3, 572–611. MR 769382 (86e:35038), DOI https://doi.org/10.1016/0022-1236%2884%2990066-1
- David S. Jerison and Carlos E. Kenig, The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 203–207. MR 598688 (84a:35064), DOI https://doi.org/10.1090/S0273-0979-1981-14884-9
- Grisvard, P., Boundary value problems in nonsmooth domains. Pitman, London, 1985.
- I. K. Lifanov, L. N. Poltavskii, and G. M. Vainikko, Hypersingular integral equations and their applications, Differential and Integral Equations and Their Applications, vol. 4, Chapman & Hall/CRC, Boca Raton, FL, 2004. MR 2053793 (2005e:45001)
- A. Ya. Povzner and I. V. Suharevskiĭ, Integral equations of the second kind in problems of diffraction by an infinitely thin screen, Soviet Physics. Dokl. 4 (1960), 798–801 (Russian). MR 0110445 (22 \#1325)
- Berkhin, P. E., Problem of diffraction of a fine screen. Siberian Mathematical Journal, 1984, v. 25, 31–42.
- P. A. Krutitskii, The Neumann problem in a plane domain bounded by closed and open curves, Complex Variables Theory Appl. 38 (1999), no. 1, 1–20. MR 1685543 (2000a:35037)
- P. A. Krutitskii, The $2$-D Neumann problem in an exterior domain bounded by closed and open curves, Math. Methods Appl. Sci. 20 (1997), no. 18, 1551–1562. MR 1486525 (98m:35039), DOI https://doi.org/10.1002/%28SICI%291099-1476%28199712%2920%3A18%24%3C%241551%3A%3AAID-MMA912%24%3E%243.3.CO%3B2-H
- P. A. Krutitskii, The 2-D Neumann problem in a domain with cuts, Rend. Mat. Appl. (7) 19 (1999), no. 1, 65–88 (English, with English and Italian summaries). MR 1710125 (2000k:31004)
- 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 (2002g:35056), DOI https://doi.org/10.1006/jdeq.2000.3954
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Additional Information
P. A. Krutitskii
Affiliation:
KIAM, Miusskaya Sq. 4, Moscow, 125047, Russia
Email:
biem@mail.ru
Received by editor(s):
February 26, 2012
Published electronically:
November 13, 2013
Article copyright:
© Copyright 2013
Brown University
