Asymptotics of the spectrum of the mixed boundary value problem for the Laplace operator in a thin spindle-shaped domain

S. A. Nazarov ; J. Taskinen

Algebra i analiz, Tome 33 (2021) no. 2, pp. 136-196.

Bibliographie

[1] Ladyzhenskaya O. A., Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973

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

[3] Birman M. Sh., Skvortsov G. E., “O kvadratichnoi summiruemosti starshikh proizvodnykh resheniya zadachi Dirikhle v oblasti s kusochno gladkoi granitsei”, Izv. vuzov. Mat., 1962, no. 5, 11–21

[4] Kondratev V. A., “Kraevye zadachi dlya ellipticheskikh uravnenii v oblastyakh s konicheskimi ili uglovymi tochkami”, Tr. Moskovsk. mat. o-va, 16, 1967, 209–292 | Zbl

[5] Nazarov S. A., Plamenevsky B. A., Elliptic problems in domains with piecewise smooth boundaries, De Gruyter Expos. Math., 13, Walter de Gruyter, Berlin, 1994 | MR

[6] Mikhlin S. G., Variatsionnye metody v matematicheskoi fizike, Nauka, M., 1970 | MR

[7] Nazarov S. A., Slutskij A. S., Taskinen J., “Korn inequality for a thin rod with rounded ends”, Math. Methods Appl. Sci., 37:16 (2014), 2463–2483 | DOI | MR | Zbl

[8] Nazarov S. A., Slutskij A. S., Taskinen J., “Asymptotic analysis of an elastic rod with rounded ends”, Math. Methods Appl. Sci., 43:10 (2020), 6396–6415 | DOI | MR | Zbl

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

[10] Keldysh M. V., “O nekotorykh sluchayakh vyrozhdeniya uravnenii ellipticheskogo tipa na granitse oblasti”, Dokl. AN SSSR, 77:2 (1951), 181–183

[11] Oleinik O. A., “Ob uravneniyakh ellipticheskogo tipa, vyrozhdayuschikhsya na granitse oblasti”, Dokl. AN SSSR, 35:3 (1954), 885–887

[12] Vishik M. I., “Kraevye zadachi dlya ellipticheskikh uravnenii, vyrozhdayuschikhsya na granitse oblasti”, Mat. sb., 93:2 (1954), 513–568

[13] Polking J., Boggess A., Arnold D., Differential equations with boundary value problems, Second ed., Pearson Prentice Hall, New Jersey, 2006 | MR

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

[15] Babich V. M., Buldyrev V. S., “Iskusstvo asimptotiki”, Vestn. Leningr. un-ta. Ser. mat., mekh. i astronom., 1977, no. 3, 5–12 | Zbl

[16] Mazja W. G., Nasarow S. A., Plamenewski B. A., Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten, v. I, II, Math. Monogr., 83, Akademie-Verlag, Berlin, 1991 | MR

[17] Ilin A. M., Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989

[18] Kozlov V. A., Maz'ya V. G., Movchan A. B., Asymptotic analysis of fields in multi-structures, Oxford Math. Monogr., Clarendon Press, Oxford, 1999 | MR | Zbl

[19] Babich V. M., Kirpichnikova N. Ya., Metod pogranichnogo sloya v zadachakh difraktsii, Izd-vo LGU, L., 1974 | MR

[20] Van Daik M. D., Metody vozmuschenii v mekhanike zhidkostei, Mir, M., 1967

[21] Nazarov S. A., “Nesamosopryazhennye ellipticheskie zadachi s polinomialnym svoistvom v oblastyakh, imeyuschikh tsilindricheskie vykhody na beskonechnost”, Zap. nauch. semin. POMI, 249, 1997, 212–230 | Zbl

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

[23] Kozlov V. A., Maz'ya V. G., Rossmann J., Elliptic boundary value problems in domains with point singularities, Math. Surveys and Monogr., 52, Amer. Math. Soc., Providence, RI, 1997 | MR | Zbl

[24] Nazarov S. A., Plamenevskii B. A., “Zadacha Neimana dlya samosopryazhennykh ellipticheskikh sistem v oblasti s kusochno gladkoi granitsei”, Tr. Leningr. mat. o-va, 1 (1990), 174–211

[25] Arutyunyan N. Kh., Nazarov S. A., Shoikhet B. A., “Otsenki i asimptotika napryazhenno-deformirovannogo sostoyaniya trekhmernogo tela s treschinoi v teorii uprugosti i teorii polzuchesti”, Dokl. AN SSSR, 266:6 (1982), 1365–1369

[26] Nazarov S. A., Shoikhet B. A., “Koertsitivnye otsenki v vesovykh prostranstvakh reshenii zadach teorii uprugosti i polzuchesti v oblasti s dvumernoi treschinoi”, Izv. AN Arm. SSR, 36:4 (1983), 12–25 | MR | Zbl

[27] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, Mir, M., 1971

[28] Rofe-Beketov F. S., “Samosopryazhennye rasshireniya differentsialnykh operatorov v prostranstve vektor-funktsii”, Dokl. AN SSSR, 184:5 (1969), 1034–1037 | Zbl

[29] Berezin F. A., Faddeev L. D., “Zamechanie ob uravnenii Shredingera s singulyarnym potentsialom”, Dokl. AN SSSR, 137:5 (1961), 1011–1014 | Zbl

[30] Pavlov B. S., “Teoriya rasshirenii i yavno reshaemye modeli”, Uspekhi mat. nauk, 42:6 (1987), 99–132 | MR

[31] Nazarov S. A., “Samosopryazhennye rasshireniya operatora zadachi Dirikhle v vesovykh funktsionalnykh prostranstvakh”, Mat. sb., 137:2 (1988), 224–241

[32] Nazarov S. A., “Asimptoticheskie usloviya v tochkakh, samosopryazhennye rasshireniya operatorov i metod sraschivaemykh asimptoticheskikh razlozhenii”, Tr. Sankt-Peterburg. mat. o-va, 5 (1998), 112–183 | Zbl

[33] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972

[34] Rellich F., Perturbation theory of eigenvalue problems, Gordon and Breach Sci. Publ., New York, 1969 | MR | Zbl

[35] Ilin A. M., “Kraevaya zadacha dlya ellipticheskogo uravneniya vtorogo poryadka v oblasti s uzkoi schelyu. 1. Dvumernyi sluchai”, Mat. sb., 99:4 (1976), 514–537 | MR | Zbl

[36] Kamotski I. V., Maz'ya V. G., “On the linear water wave problem in the presence of of a critically submerged body”, SIAM J. Math. Anal., 44:6 (2012), 4222–4249 | DOI | MR | Zbl

[37] Mazya V. G., Nazarov S. A., Plamenevskii B. A., “Asimptoticheskie razlozheniya sobstvennykh chisel kraevykh zadach dlya operatora Laplasa v oblastyakh s malymi otverstiyami”, Izv. AN SSSR. Ser. mat., 48:2 (1984), 347–371 | MR

[38] Lamberti P. D., Lanza de Cristoforis M., “A real analyticity result for symmetric functions of the eigenvalues of a domain dependent Dirichlet problem for the Laplace operator”, J. Nonlinear Convex Anal., 5 (2004), 19–42 | MR | Zbl

[39] Lanza de Cristoforis M., “Asymptotic behavior of the solutions of the Dirichlet problem for the Laplace operator in a domain with a small hole. A functional analytic approach”, Analysis, 28 (2008), 63–93 | MR | Zbl

[40] Lanza de Cristoforis M., “Simple Neumann eigenvalues for the Laplace operator in a domain with a small hole. A functional analytic approach”, Rev. Mat. Complut., 25:2 (2012), 369–412 | DOI | MR | Zbl