Estimates for second order derivatives of eigenvectors in thin anisotropic plates with variable thickness

S. A. Nazarov

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 33, Tome 308 (2004), pp. 161-181

Résumé

For second order derivatives of eigenvectors in a thin anisotropic heterogeneous plate $\Omega_h$, we derive estimates of the weighted $L_2$-norms with the majorants whose dependence on both, the plate thickness $h$ and the eigenvalue number, are expressed explicitly. These estimates keep the asymptotic sharpness along the whole spectrum while, inside its low-frequency range, the majorants remain bounded as $h\to+0$. The latter is rather unexpected fact because, for the first eigenfunction $u^1$ of the alike boundary value problem for a scalar second order differential operator with variable coefficients, the norm $\Vert\nabla_x^2u^0;L_2(\Omega_h)\Vert$ is of order $h^{-1}$ and grows as $h$ vanishes.

Bibliographie

[1] P. G. Ciarlet, S. Kesavan, “Two dimensional approximations of three dimensional eigenvalues in plate theory”, Comput. Methods Appl. Mech. Engrg., 26 (1980), 149–172 | MR

[2] S. A. Nazarov, “Obosnovanie asimptoticheskoi teorii tonkikh sterzhnei. Integralnye i potochechnye otsenki”, Problemy matem. analiza, 17, izd-vo SPbGU, SPb., 1997, 101–152 | MR

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

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

[5] S. A. Nazarov, B. A. Plamenevskii, Ellipticheskie zadachi v oblastyakh s kusochno gladkoi granitsei, Nauka, M., 1991

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

[7] S. A. Nazarov, “Ob asimptotike spektra zadachi teorii uprugosti dlya tonkoi plastiny”, Sibirsk. matem. zhurnal, 41:4 (2000), 895–912 | MR | Zbl

[8] S. A. Nazarov, “Estimating the convergence rate for eigenfrequencies of anisotropic plates with variable thickness”, C. R. Mecanique, 330 (2002), 603–607 | DOI | Zbl

[9] B. A. Shoikhet, “Ob asimptoticheski tochnykh uravneniyakh tonkikh plit slozhnoi struktury”, Prikladnaya matematika i mekhanika, 37:5 (1973), 913–924 | MR

[10] S. A. Nazarov, “Neravenstva Korna, asimptoticheski tochnye dlya tonkikh oblastei”, Vestnik SPbGU, Seriya 1, 2:8 (1992), 19–24 | MR

[11] S. A. Nazarov, “Asimptoticheskii analiz proizvolno anizotropnoi plastiny peremennoi tolschiny (pologoi obolochki)”, Matem. sbornik, 191:7 (2000), 129–159 | MR | Zbl

[12] M. S. Agranovich, M. I. Vishik, “Ellipticheskie zadachi s parametrom i parabolicheskie zadachi obschego vida”, Uspekhi matem. nauk, 19:3 (1999), 53–160

[13] V. G. Mazya, B. A. Plamenevskii, “Shauderovskie otsenki reshenii ellipticheskikh kraevykh zadach v oblastyakh s rebrami na granitse”, Trudy seminara S. L. Soboleva, 2, 1978, 69–102 | MR

[14] V. G. Mazya, B. A. Plamenevskii, “$L_p-$otsenki reshenii ellipticheskikh kraevykh zadach v oblastyakh s rebrami”, Trudy Moskovsk. matem. obschestva, 37, 1978, 49–93 | MR

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

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

[17] J. Nečas, Les méthodes in théorie des équations elliptiques, Masson-Academia, Paris–Prague, 1967

[18] 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

[19] S. Agmon, A. Douglis, L. Nirenberg, “Estimates near the boundary for solutions of elliptic differential equations satisfying general boundary conditions, 2”, Comm. Pure Appl. Math., 17 (1964), 35–92 | DOI | MR | Zbl

[20] V. L. Berdichevskii, “Vysokochastotnye dlinnovolnovye kolebaniya plastin”, Dokl AN SSSR, 236:6 (1977), 1319–1322 | MR

[21] I. S. Zorin, S. A. Nazarov, “Kraevoi effekt pri izgibe tonkoi trekhmernoi plastiny”, Prikladnaya matem. i mekhanika, 53:4 (1989), 642–650 | MR | Zbl

[22] M. Dauge, I. Djurdjevic, E. Faou, A. Rössle, “Eigenmode asymptotics in thin elastic plates”, J. Math. Pures Appl., 78:9 (1999), 925–964 | DOI | MR | Zbl

[23] S. A. Nazarov, “Asimptotika sobstvennykh chisel zadachi Dirikhle v tonkoi oblasti”, Izvestiya VUZov. Matem., 11 (1987), 54–64 | MR | Zbl

[24] I. V. Kamotskii, S. A. Nazarov, “O sobstvennykh funktsiyakh, lokalizovannykh okolo kromki tonkoi oblasti”, Problemy matem. analiza, 19, Nauchn. kniga, Novosibirsk, 1999, 105–148

[25] I. Roitberg, D. Vassiliev, T. Weidl, “Edge resonance in an elastic semi-strip”, Q. J. Mech. Appl. Math., 51:1 (1998), 1–13 | DOI | MR | Zbl

[26] S. A. Nazarov, “Otsenki vblizi rebra resheniya zadachi Neimana dlya ellipticheskoi sistemy”, Vestnik LGU, Seriya 1, 1:1 (1988), 37–42 | MR | Zbl

[27] S. A. Nazarov, B. A. Plamenevskii, “Zadacha Neimana dlya samosopryazhennykh ellipticheskikh sistem v oblasti s kusochno gladkoi granitsei”, Trudy leningradskogo matem. obschestva, 1, 1990, 174–211 | MR

[28] M. Dauge, Elliptic Boundary Value Problems on Corner Domains, Lect. Notes Math., 1341, 1988 | MR | Zbl

[29] S. A. Nazarov, “Vesovye funktsii i invariantnye integraly”, Vychislitelnaya mekhanika deformiruemogo tverdogo tela, 1 (1990), 17–31

[30] V. A. Kozlov, V. G. Mazya, “Spektralnye svoistva operatornykh puchkov, porozhdennykh kraevymi zadachami v konuse”, Funkts. analiz i ego prilozheniya, 22:2 (1988), 38–46 | MR | Zbl

[31] V. A. Kozlov, V. G. Maz'ya, J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations, Mathematical Surveys and Monographs, AMS, Providence, RI, 2001 | MR | Zbl

[32] A. A. Kulikov, S. A. Nazarov, M. A. Narbut, “Affinnye preobrazovaniya v ploskoi zadache teorii uprugosti”, Vestnik SPbGU, Ser. 1, 2:8 (2000), 91–95 | MR | Zbl

[33] M. L. Williams, “Stress singularities resulting from various boundary conditions in angular corners of plates in extensions”, J. Appl. Mech., 19 (1952), 526–528

[34] M. Costabel, M. Dauge, “Computation of corner singularities in linear elasticity”, Lect. Notes Pure and Appl. Maths., 167, 1994, 59–68 | MR

[35] P. J. Papadakis, I. Babuška, “A numerical procedure for the determination of certain quantities related to the stress intensity factors in two-dimensional elasticity”, Comput. Methods Appl. Mech. Engrg., 122 (1995), 69–92 | DOI | MR | Zbl