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 Mat. Zametki, 1997, Volume 62, Issue 5, Pages 751–765 (Mi mz1661)

Weighted Korn inequalities in paraboloidal domains

S. A. Nazarov

Abstract: A weighted Korn inequality in a domain $\Omega\subset\mathbb R^n$ with paraboloidal exit $\Pi$ to infinity is obtained. Asymptotic sharpness of the inequality is achieved by using different weight factors for the longitudinal (with respect to the axis of $\Pi$) and transversal displacement vector components and by making the weight factors of the derivatives depend on the direction of differentiation. The solvability of the elasticity problem in the energy class (the closure of $C_0^\infty(\overline\Omega)^n$ in the norm generated by the elastic energy functional) is studied; the dimensions of the kernel and the cokerned of the corresponding operator depend on the exponent $s\in(-\infty,1)$ in the “grate of expansion” of the paraboloid $\Pi$.

DOI: https://doi.org/10.4213/mzm1661

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English version:
Mathematical Notes, 1997, 62:5, 629–641

Bibliographic databases:

UDC: 517.946+539.3

Citation: S. A. Nazarov, “Weighted Korn inequalities in paraboloidal domains”, Mat. Zametki, 62:5 (1997), 751–765; Math. Notes, 62:5 (1997), 629–641

Citation in format AMSBIB
\Bibitem{Naz97} \by S.~A.~Nazarov \paper Weighted Korn inequalities in paraboloidal domains \jour Mat. Zametki \yr 1997 \vol 62 \issue 5 \pages 751--765 \mathnet{http://mi.mathnet.ru/mz1661} \crossref{https://doi.org/10.4213/mzm1661} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1627866} \zmath{https://zbmath.org/?q=an:0918.35028} \transl \jour Math. Notes \yr 1997 \vol 62 \issue 5 \pages 629--641 \crossref{https://doi.org/10.1007/BF02361301} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000075396200012} 

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Erratum

This publication is cited in the following articles:
1. Nazarov, SA, “Weighted Korn inequalities in paraboloidal domains (vol 62, pg 629, 1997)”, Mathematical Notes, 63:3–4 (1998), 565
2. 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
3. S. A. Nazarov, “Asymptotic analysis of an arbitrary anisotropic plate of variable thickness (sloping shell)”, Sb. Math., 191:7 (2000), 1075–1106
4. A. A. Kulikov, S. A. Nazarov, “Cracks in piezoelectric and electroconductive bodies”, J. Appl. Industr. Math., 1:2 (2007), 201–216
5. Nazarov, SA, “A criterion of the continuous spectrum for elasticity and other self-adjoint systems on sharp peak-shaped domains”, Comptes Rendus Mecanique, 335:12 (2007), 751
6. Nazarov, SA, “On eigenoscillations of a solid with a blunted pick”, Doklady Physics, 52:10 (2007), 560
7. S. A. Nazarov, “Korn inequalities for elastic junctions of massive bodies, thin plates, and rods”, Russian Math. Surveys, 63:1 (2008), 35–107
8. S. A. Nazarov, “The spectrum of the elasticity problem for a spiked body”, Siberian Math. J., 49:5 (2008), 874–893
9. S. A. Nazarov, “Concentration of trapped modes in problems of the linearized theory of water waves”, Sb. Math., 199:12 (2008), 1783–1807
10. Nazarov, SA, “The natural oscillations of an elastic body with a heavy rigid spike-shaped inclusion”, Pmm Journal of Applied Mathematics and Mechanics, 72:5 (2008), 561
11. S. A. Nazarov, “The Essential Spectrum of Boundary Value Problems for Systems of Differential Equations in a Bounded Domain with a Cusp”, Funct. Anal. Appl., 43:1 (2009), 44–54
12. Cardone, G, “A criterion for the existence of the essential spectrum for beak-shaped elastic bodies”, Journal de Mathematiques Pures et Appliquees, 92:6 (2009), 628
13. Cardone, G, ““Absorption” effect for elastic waves by the beak-shaped boundary irregularity”, Doklady Physics, 54:3 (2009), 146
14. Campbell A. Nazarov S.A. Sweers G.H., “Spectra of Two-Dimensional Models for Thin Plates with Sharp Edges”, SIAM J. Math. Anal., 42:6 (2010), 3020–3044
15. Nazarov S.A., Slutskij A.S., Taskinen J., “Korn Inequality For a Thin Rod With Rounded Ends”, Math. Meth. Appl. Sci., 37:16 (2014), 2463–2483
16. Neff P., Pauly D., Witsch K.-J., “Poincaré Meets Korn Via Maxwell: Extending Korn's First Inequality To Incompatible Tensor Fields”, J. Differ. Equ., 258:4 (2015), 1267–1302
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