Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Korn’s Inequality

  • Leonid P. Lebedev
  • Michael J. Cloud
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_217-1

Synonyms

Definitions

Korn’s inequality estimates the integral value of the squared displacement through the strain energy of an elastic body occupying volume V . This result was central to the development of linear elasticity theory.

A Version of Korn’s Inequality

The present article highlights the inequality
$$\displaystyle \begin{aligned} \int_V ( |\boldsymbol{u}|{}^2 + \| \nabla \boldsymbol{u} \|{}^2 ) \,dV \le c \int_V W({\boldsymbol{\varepsilon}}) \,dV \, , \end{aligned} $$
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References

  1. Bullen P (2015) Dictionary of inequalities, 2nd edn. Chapman & Hall/CRC, Boca RatonGoogle Scholar
  2. Ciarlet P (1988) Mathematical elasticity: vol. I, three-dimensional elasticity. North-Holland, AmsterdamGoogle Scholar
  3. Horgan C (1995) Korn’s inequalities and their applications in continuum mechanics. SIAM Rev 37(4):491–511MathSciNetCrossRefGoogle Scholar
  4. Lebedev L, Cloud M, Eremeyev V (2010) Tensor analysis with applications in mechanics. World Scientific, SingaporeCrossRefGoogle Scholar
  5. Lebedev L, Vorovich I, Cloud M (2012) Functional analysis in mechanics. Springer, New YorkzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2018

Authors and Affiliations

  1. 1.National University of ColombiaBogotaColombia
  2. 2.Department of Electrical and Computer EngineeringLawrence Technological UniversitySouthfieldUSA

Section editors and affiliations

  • Leonid P. Lebedev
    • 1
  • Michael J. Cloud
    • 2
  1. 1.Universidad Nacional de ColombiaBogotá D.C.Colombia
  2. 2.Department of Electrical and Computer EngineeringLawrence Technological UniversitySouthfieldUSA