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 ∂G∂G of a domain
G⊂Rn. 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 ∂G∖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.
Citation:
S. A. Nazarov, B. A. Plamenevskii, “Elliptic problems with radiation conditions on edges of the boundary”, Russian Acad. Sci. Sb. Math., 77:1 (1994), 149–176
\Bibitem{NazPla92}
\by S.~A.~Nazarov, B.~A.~Plamenevskii
\paper Elliptic problems with radiation conditions on edges of the~boundary
\jour Russian Acad. Sci. Sb. Math.
\yr 1994
\vol 77
\issue 1
\pages 149--176
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\crossref{https://doi.org/10.1070/SM1994v077n01ABEH003434}
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Linking options:
https://www.mathnet.ru/eng/sm1078
https://doi.org/10.1070/SM1994v077n01ABEH003434
https://www.mathnet.ru/eng/sm/v183/i10/p13
This publication is cited in the following 11 articles:
Renata Bunoiu, Giuseppe Cardone, Sergey A. Nazarov, “Scalar problems in junctions of rods and a plate”, ESAIM: M2AN, 52:2 (2018), 481
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
S. A. Nazarov, “Asymptotics of eigen-oscillations of a massive elastic body with a thin baffle”, Izv. Math., 77:1 (2013), 87–142
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
J. Appl. Industr. Math., 3:3 (2009), 377–390
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
S. A. Nazarov, “Elliptic Boundary Value Problems in Hybrid Domains”, Funct. Anal. Appl., 38:4 (2004), 283–297
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
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
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
Nazarov S., “Asymptotic Solutions of a Variational Inequality with Small Obstacles”, Comptes Rendus Acad. Sci. Ser. I-Math., 318:11 (1994), 1059–1064
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