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 Mat. Sb., 1992, Volume 183, Number 10, Pages 13–44 (Mi msb1078)

Elliptic problems with radiation conditions on edges of the boundary

S. A. Nazarov, B. A. Plamenevskii

Abstract: A study is made of formulations of elliptic boundary value problems connected with the addition of radiation conditions on edges of the piecewise smooth boundary $\partial G$ of a domain $G\subset\mathbb{R}^n$. Such formulations lead to Fredholm operators acting in suitable function spaces with weighted norms. The basic means of description is the generalized Green formula, which contains in addition to the usual boundary integrals also integrals over an edge $M$ of bilinear expressions formed by the coefficients of the asymptotics of the solutions near $M$. Thus, the edge and the $(n-1)$-dimensional smooth part of the boundary are on the same footing-both $M$ and $\partial G\setminus M$ are represented by their contributions to the generalized Green formula. This permits the construction of a theory of elliptic problems in which the generalized Green formula takes the role of the usual Green formula in the smooth situation.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1994, 77:1, 149–176

Bibliographic databases:

UDC: 517.9
MSC: Primary 35J55; Secondary 46E35, 26B20, 47A53

Citation: S. A. Nazarov, B. A. Plamenevskii, “Elliptic problems with radiation conditions on edges of the boundary”, Mat. Sb., 183:10 (1992), 13–44; Russian Acad. Sci. Sb. Math., 77:1 (1994), 149–176

Citation in format AMSBIB
\Bibitem{NazPla92} \by S.~A.~Nazarov, B.~A.~Plamenevskii \paper Elliptic problems with radiation conditions on edges of the~boundary \jour Mat. Sb. \yr 1992 \vol 183 \issue 10 \pages 13--44 \mathnet{http://mi.mathnet.ru/msb1078} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1202790} \zmath{https://zbmath.org/?q=an:0789.35048|0779.35036} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1994SbMat..77..149N} \transl \jour Russian Acad. Sci. Sb. Math. \yr 1994 \vol 77 \issue 1 \pages 149--176 \crossref{https://doi.org/10.1070/SM1994v077n01ABEH003434} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1994MZ10900010} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Nazarov S., “Asymptotic Solutions of a Variational Inequality with Small Obstacles”, Comptes Rendus Acad. Sci. Ser. I-Math., 318:11 (1994), 1059–1064
2. S. A. Nazarov, “The Operator of a Boundary Value Problem With Chaplygin–Zhukovskii–Kutta Type Conditions on an Edge of the Boundary Has the Fredholm Property”, Funct. Anal. Appl., 31:3 (1997), 183–192
3. S. A. Nazarov, “The polynomial property of self-adjoint elliptic boundary-value problems and an algebraic description of their attributes”, Russian Math. Surveys, 54:5 (1999), 947–1014
4. Nazarov S. Pileckas K., “On Steady Stokes and Navier–Stokes Problems with Zero Velocity at Infinity in a Three-Dimensional Exterior Domain”, J. Math. Kyoto Univ., 40:3 (2000), 475–492
5. S. A. Nazarov, “Elliptic Boundary Value Problems in Hybrid Domains”, Funct. Anal. Appl., 38:4 (2004), 283–297
6. S. A. Nazarov, G. H. Sweers, “Boundary value problems for the bi-harmonic equation and for the iterated Laplacian in a three-dimensional domain with an edge”, J. Math. Sci. (N. Y.), 143:2 (2007), 2936–2960
7. J. Appl. Industr. Math., 3:3 (2009), 377–390
8. S. A. Nazarov, “Asymptotics of trapped modes and eigenvalues below the continuous spectrum of a waveguide with a thin shielding obstacle”, St. Petersburg Math. J., 23:3 (2012), 571–601
9. S. A. Nazarov, “Asymptotics of eigen-oscillations of a massive elastic body with a thin baffle”, Izv. Math., 77:1 (2013), 87–142
10. S. A. Nazarov, “Asymptotics of the eigenvalues of boundary value problems for the Laplace operator in a three-dimensional domain with a thin closed tube”, Trans. Moscow Math. Soc., 76:1 (2015), 1–53
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