Mathematical Notes

, Volume 62, Issue 5, pp 629–641 | Cite as

Weighted korn inequalities in paraboloidal domains

  • S. A. Nazarov
Article

Abstract

A weighted Korn inequality in a domain Ω ⊂ ℝ n with paraboloidal exit II 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 II) 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\(C_0^\infty (\bar \Omega )^n\) 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 exponents∈(−∞, 1) in the “rate of expansion” of the paraboloid II.

Key words

Korn inequalities elasticity problem energy class boundary value problems in unbounded domains 

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References

  1. 1.
    S. A. Nazarov, “Asymptotically exact Korn inequalities for thin domains,”Vestnik St. Petersburg Univ. Mat. Mekh. Astronom. [Vestnik St. Petersburg Univ. Math.], No. 8, 19–24 (1992).Google Scholar
  2. 2.
    V. A. Kondrat'ev and O. A. Oleinik, “On the dependence of constants in the Korn inequality on a parameter characterizing the geometry of the domain,”Uspekhi Mat. Nauk [Russian Math. Surveys],44, No. 6, 157–158 (1989).MathSciNetGoogle Scholar
  3. 3.
    V. A. Kondratiev and O. A. Oleinik, “Korn's type inequalities for a class of unbounded domains and applications to boundary value problems in elasticity,” in:Elasticity. Mathematical Methods and Application (G. Eason and R. W. Ogden, editors), Horword, Chichester (1990), pp. 211–233.Google Scholar
  4. 4.
    V. A. Kondratiev and O. A. Oleinik, “Hardy's and Korn's type inequalities and their applications,”Rend. Mat. Appl. (7), 10, 641–666 (1990).MathSciNetGoogle Scholar
  5. 5.
    S. A. Nazarov and B. A. Plamenevsky,Elliptic Problems in Domains with Piecewise Smooth Boundaries, Walter de Gruyter, Berlin (1994).Google Scholar
  6. 6.
    G. Duvaut and J.-L. Lions,Les inéquations en méchanique et en physique, Dunod, Paris (1972).Google Scholar
  7. 7.
    V. A. Kondrat'ev and O. A. Oleinik, “Boundary value problems for an elasticity system in unbounded domains. Korn inequality,”Uspekhi Mat. Nauk [Russian Math. Surveys],43, No. 5, 55–98 (1988).MathSciNetGoogle Scholar
  8. 8.
    O. A. Oleinik, “Korn's type inequalities and applications to elasticity,” Convegno internazionale in memoria di Vito Volterra,Atti Accad. Naz. Lincei. Cl. Sci. Fis. Mat. Natur. Mem. (9) Mat. Appl.,92, 183–209 (1992).MATHMathSciNetGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • S. A. Nazarov
    • 1
  1. 1.Admiral S. O. Makarov State Naval AcademyRussia

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