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Generalized Korn's inequality and conformal Killing vectors

  • Sergio Dain
Article

Abstract

Korn's inequality plays an important role in linear elasticity theory. This inequality bounds the norm of the derivatives of the displacement vector by the norm of the linearized strain tensor. The kernel of the linearized strain tensor are the infinitesimal rigid-body translations and rotations (Killing vectors). We generalize this inequality by replacing the linearized strain tensor by its trace free part. That is, we obtain a stronger inequality in which the kernel of the relevant operator are the conformal Killing vectors. The new inequality has applications in General Relativity.

Keywords

General Relativity System Theory Displacement Vector Strain Tensor Elasticity Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)Google Scholar
  2. 2.
    Bartnik, R., Isenberg, J.: The constraint equations. In: Chruściel, P.T., Friedrich, H., (eds.), The Einstein equations and large scale behavior of gravitational fields, pp. 1–38. Birhuser Verlag, Basel Boston Berlin. (2004) [gr-qc/0405092]Google Scholar
  3. 3.
    Chen, W., Jost, J.: A Riemannian version of Kornapos; inequality. Calc. Var. Partial Differential Equations 14(4), 517–530 (2002)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Ciarlet, P.G., Ciarlet, P.Jr.: Another approach to linearized elasticity and Kornapos; inequality. C. R. Math. Acad. Sci. Paris 339(4), 307–312 (2004)MathSciNetGoogle Scholar
  5. 5.
    Dain, S.: Trapped surfaces as boundaries for the constraint equations. Class. Quantum. Grav. 21(2), 555–573 (2004) [gr-qc/0308009]MathSciNetGoogle Scholar
  6. 6.
    Dain, S.: Corrigendum: Trapped surfaces as boundaries for the constraint equations. Class. Quantum. Grav. 22(4), 769 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Duvaut, G., Lions, J.-L.: Inequalities in mechanics and physics. Springer-Verlag, Berlin, (1976) Translated from the French by C.W. John, Grundlehren der Mathematischen Wissenschaften, p. 219Google Scholar
  8. 8.
    Horgan, C.O.: Kornapos; inequalities and their applications in continuum mechanics. SIAM Rev. 37(4), 491–511 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Lions, J.L., Magenes, E.: Non-homogeneous boundary value problems and applications., vol. 181 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York, Translated from the French by P. Kenneth (1972)Google Scholar
  10. 10.
    McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)Google Scholar
  11. 11.
    Nečas, J.: Les méthodes directes en théorie des équations elliptiques. Masson et Cie, Éditeurs, Paris (1967)Google Scholar
  12. 12.
    Nečas, J., Hlaváček, I.: Mathematical theory of elastic and elasto-plastic bodies: an introduction, vol. 3 of Studies in Applied Mechanics. Elsevier Scientific Publishing Co., Amsterdam (1980)Google Scholar
  13. 13.
    Taylor, M.E.: Partial Differential Equations. I, vol. 115 of Applied Mathematical Sciences. Springer-Verlag, New York (1996)Google Scholar
  14. 14.
    Tiero, A.: On Kornapos; inequality in the second case. J. Elasticity 54(3), 187–191 (1999)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Max-Planck-Institut für GravitationsphysikGolmGermany

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