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

This article is cited in 16 scientific papers (total in 16 papers)

Weighted Korn inequalities in paraboloidal domains

S. A. Nazarov

Admiral Makarov State Maritime Academy

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
Received: 20.04.1996

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
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\by S.~A.~Nazarov
\paper Weighted Korn inequalities in paraboloidal domains
\jour Mat. Zametki
\yr 1997
\vol 62
\issue 5
\pages 751--765
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1627866}
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\transl
\jour Math. Notes
\yr 1997
\vol 62
\issue 5
\pages 629--641
\crossref{https://doi.org/10.1007/BF02361301}
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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  mathnet  crossref  mathscinet  isi
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    3. S. A. Nazarov, “Asymptotic analysis of an arbitrary anisotropic plate of variable thickness (sloping shell)”, Sb. Math., 191:7 (2000), 1075–1106  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. A. A. Kulikov, S. A. Nazarov, “Cracks in piezoelectric and electroconductive bodies”, J. Appl. Industr. Math., 1:2 (2007), 201–216  mathnet  crossref  mathscinet  elib
    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  crossref  zmath  adsnasa  isi  elib  scopus  scopus
    6. Nazarov, SA, “On eigenoscillations of a solid with a blunted pick”, Doklady Physics, 52:10 (2007), 560  crossref  zmath  adsnasa  isi  elib  scopus  scopus
    7. S. A. Nazarov, “Korn inequalities for elastic junctions of massive bodies, thin plates, and rods”, Russian Math. Surveys, 63:1 (2008), 35–107  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    8. S. A. Nazarov, “The spectrum of the elasticity problem for a spiked body”, Siberian Math. J., 49:5 (2008), 874–893  mathnet  crossref  mathscinet  isi  elib  elib
    9. S. A. Nazarov, “Concentration of trapped modes in problems of the linearized theory of water waves”, Sb. Math., 199:12 (2008), 1783–1807  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    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  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi  scopus  scopus
    13. Cardone, G, ““Absorption” effect for elastic waves by the beak-shaped boundary irregularity”, Doklady Physics, 54:3 (2009), 146  crossref  zmath  adsnasa  isi  elib  scopus  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    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  crossref  mathscinet  zmath  isi  scopus  scopus
    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  crossref  mathscinet  zmath  isi  scopus  scopus
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