The Neumann problem for the equation in the exterior of nonclosed Lipschitz surfaces
Author:
P. A. Krutitskii
Journal:
Quart. Appl. Math. 72 (2014), 8591
MSC (2010):
Primary 35J05, 35J25, 31A10, 31A25
Published electronically:
November 13, 2013
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Abstract: We study the Neumann problem for the equation in the exterior of nonclosed Lipschitz surfaces in . 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.
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 [1]
 William McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000. MR 1742312 (2001a:35051)
 [2]
 Gregory Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, J. Funct. Anal. 59 (1984), no. 3, 572611. MR 769382 (86e:35038), http://dx.doi.org/10.1016/00221236(84)900661
 [3]
 David S. Jerison and Carlos E. Kenig, The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 203207. MR 598688 (84a:35064), http://dx.doi.org/10.1090/S027309791981148849
 [4]
 Grisvard, P., Boundary value problems in nonsmooth domains. Pitman, London, 1985.
 [5]
 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)
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 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), 798801 (Russian). MR 0110445 (22 #1325)
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 Berkhin, P. E., Problem of diffraction of a fine screen. Siberian Mathematical Journal, 1984, v. 25, 3142.
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 P. A. Krutitskii, The Neumann problem in a plane domain bounded by closed and open curves, Complex Variables Theory Appl. 38 (1999), no. 1, 120. MR 1685543 (2000a:35037)
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 P. A. Krutitskii, The D Neumann problem in an exterior domain bounded by closed and open curves, Math. Methods Appl. Sci. 20 (1997), no. 18, 15511562. MR 1486525 (98m:35039), http://dx.doi.org/10.1002/(SICI)10991476(199712)20:181551::AIDMMA9123.3.CO;2H
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 P. A. Krutitskii, The 2D Neumann problem in a domain with cuts, Rend. Mat. Appl. (7) 19 (1999), no. 1, 6588 (English, with English and Italian summaries). MR 1710125 (2000k:31004)
 [11]
 P. A. Krutitskii, The Neumann problem in a 2D exterior domain with cuts and singularities at the tips, J. Differential Equations 176 (2001), no. 1, 269289. MR 1861190 (2002g:35056), http://dx.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
DOI:
http://dx.doi.org/10.1090/S0033569X2013013194
PII:
S 0033569X(2013)013194
Received by editor(s):
February 26, 2012
Published electronically:
November 13, 2013
Article copyright:
© Copyright 2013 Brown University
