Geodesic
Asymptotics of eigenvalues of the elasticity theory problem with the Winkler–Steklov spectral conditions at small parts of the boundary
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 50, Tome 519 (2022), pp. 152-187 
S. A. Nazarov. Asymptotics of eigenvalues of the elasticity theory problem with the Winkler–Steklov spectral conditions at small parts of the boundary. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 50, Tome 519 (2022), pp. 152-187. http://geodesic.mathdoc.fr/item/ZNSL_2022_519_a6/
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     title = {Asymptotics of eigenvalues of the elasticity theory problem with the {Winkler{\textendash}Steklov} spectral conditions at small parts of the boundary},
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Asymptotics of eigenpairs of the elasticity theory system is constructed in a three-dimensional domain with the Winkler–Steklov spectral boundry conditions at several small parts (the contact blots) and the Neumann (traction-free) conditions at the remaining part of the boundary. The asymptotic structures are essentially dependent on the distribution of the blots and the elastic or springy type of the contact. Various examples are considered and open questions are formulated.

[1] S. G. Lekhnitskii, Teoriya uprugosti anizotropnogo tela, Nauka, M., 1977

[2] Yu. N. Rabotnov, Mekhanika deformiruemogo tverdogo tela, Nauka, M., 1988

[3] S. A. Nazarov, Asimptoticheskaya teoriya tonkikh plastin i sterzhnei. Ponizhenie razmernosti i integralnye otsenki, Nauchnaya kniga, Novosibirsk, 2002

[4] A. Bertram, Elasticity and placticity of large deformations, Springer-Verlag, Heidelberg, 2005 | MR

[5] O. A. Ladyzhenskaya, Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973 | MR

[6] G. Fikera, Teoremy suschestvovaniya v teorii uprugosti, Mir, M., 1974

[7] S. A. Nazarov, “Samosopryazhennye ellipticheskie kraevye zadachi. Polinomialnoe svoistvo i formalno polozhitelnye operatory”, Problemy matem. analiza, 16, izd-vo SPbGU, SPb, 1997, 167–192

[8] S. A. Nazarov, “Polinomialnoe svoistvo samosopryazhennykh ellipticheskikh kraevykh zadach i algebraicheskoe opisanie ikh atributov”, Uspekhi matem. nauk, 54:5 (1999), 77–142 | MR

[9] S. A. Nazarov, “Neravenstva Korna dlya uprugikh sochlenenii massivnykh tel, tonkikh plastin i sterzhnei”, Uspekhi matem. nauk, 63:1 (2008), 37–110 | MR

[10] V. A. Kondratev, O. A. Oleinik, “Kraevye zadachi dlya sistemy teorii uprugosti v neogranichennykh oblastyakh. Neravenstvo Korna”, Uspekhi matem. nauk, 43:5 (1988), 55–98 | MR

[11] M. Sh. Birman, M. Z. Solomyak, Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, izd-vo Leningr. un-ta, L., 1980

[12] S. A. Nazarov, “Asimptoticheskie razlozheniya sobstvennykh chisel zadachi Steklova v singulyarno vozmuschennykh oblastyakh”, Algebra i analiz, 26:2 (2014), 119–184

[13] S. A. Nazarov, “Interaction of concentrated masses in a harmonically oscillating spatial body with Neumann boundary conditions”, RAIRO Model. Math. Anal. Numer., 27:6 (1993), 777–799 | DOI | MR

[14] J. J. Stoker, Water Waves. The Mathematical Theory with Applications, Wiley, New-York, 1992 | MR

[15] N. Kuznetsov, V. Maz'ya, B. Vainberg, Linear Water Waves, Cambridge University, Cambridge, 2002 | MR

[16] V. A. Kondratev, “Kraevye zadachi dlya ellipticheskikh uravnenii v oblastyakh s konicheskimi ili uglovymi tochkami”, Trudy Moskovsk. matem. obschestva, 16, 1963, 219–292

[17] S. A. Nazarov, B. A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, Walter de Gruyter, Berlin–New York, 1994 | MR

[18] L. D. Landau, E. M. Lifshits, Teoreticheskaya fizika. Teoriya uprugosti, Nauka, M., 1982 | MR

[19] V. G. Mazya, B. A. Plamenevskii, “O koeffitsientakh v asimptotike reshenii ellipticheskikh kraevykh zadach v oblasti s konicheskimi tochkami”, Math. Nachr., 76 (1977), 29–60 | DOI | MR

[20] V. G. Mazya, B. A. Plamenevskii, “Otsenki v $L_p$ i v klassakh Geldera i printsip maksimuma Miranda-Agmona dlya reshenii ellipticheskikh kraevykh zadach v oblastyakh s osobymi tochkami na granitse”, Math. Nachr., 77 (1977), 25–82

[21] M. I. Vishik, L. A. Lyusternik, “Regulyarnoe vyrozhdenie i pogranichnyi sloi dlya lineinykh differentsialnykh uravnenii s malym parametrom”, Uspekhi matem. nauk, 12:5 (1957), 3–122

[22] W. G. Mazja, S. A. Nasarow, B. A. Plamenewski, Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten, v. 1, 2, Akademie-Verlag, Berlin, 1991 ; Maz'ya V., Nazarov S., Plamenevskij B., Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, v. 1, 2, Birkhäuser Verlag, Basel, 2000 | MR | MR

[23] V. Kiado Piat, S. A. Nazarov, “Smeshannye kraevye zadachi v singulyarno vozmuschennykh dvumernykh oblastyakh so spektralnym usloviem Steklova”, Problemy matem. analiza, 106, Novosibirsk, 2020, 91–124