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 Zap. Nauchn. Sem. POMI, 2004, Volume 308, Pages 161–181 (Mi znsl833)

This article is cited in 4 scientific papers (total in 4 papers)

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

S. A. Nazarov

Institute of Problems of Mechanical Engineering, Russian Academy of Sciences

Abstract: 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.

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English version:
Journal of Mathematical Sciences (New York), 2006, 132:1, 91–102

Bibliographic databases:

UDC: 517.946
Received: 02.03.2004

Citation: S. A. Nazarov, “Estimates for second order derivatives of eigenvectors in thin anisotropic plates with variable thickness”, Mathematical problems in the theory of wave propagation. Part 33, Zap. Nauchn. Sem. POMI, 308, POMI, St. Petersburg, 2004, 161–181; J. Math. Sci. (N. Y.), 132:1 (2006), 91–102

Citation in format AMSBIB
\Bibitem{Naz04} \by S.~A.~Nazarov \paper Estimates for second order derivatives of eigenvectors in thin anisotropic plates with variable thickness \inbook Mathematical problems in the theory of wave propagation. Part~33 \serial Zap. Nauchn. Sem. POMI \yr 2004 \vol 308 \pages 161--181 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl833} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2092617} \zmath{https://zbmath.org/?q=an:1087.35090} \elib{http://elibrary.ru/item.asp?id=9128666} \transl \jour J. Math. Sci. (N. Y.) \yr 2006 \vol 132 \issue 1 \pages 91--102 \crossref{https://doi.org/10.1007/s10958-005-0478-3} \elib{http://elibrary.ru/item.asp?id=13511119} 

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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. S. A. Nazarov, “Korn's inequalities for junctions of elastic bodies with thin plates”, Siberian Math. J., 46:4 (2005), 695–706
2. S. A. Nazarov, “Korn inequalities for elastic junctions of massive bodies, thin plates, and rods”, Russian Math. Surveys, 63:1 (2008), 35–107
3. Nazarov S.A., Perez E., Taskinen J., “Localization effect for Dirichlet eigenfunctions in thin non-smooth domains”, Trans. Am. Math. Soc., 368:7 (2016), 4787–4829
4. Buttazzo G., Cardone G., Nazarov S.A., “Thin Elastic Plates Supported Over Small Areas. i: Korn'S Inequalities and Boundary Layers”, J. Convex Anal., 23:2 (2016), 347–386
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