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Korn’s Inequalities

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Divergence Operator and Related Inequalities

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Abstract

Introduced in the beginning of the past century [67, 69, 68], the Korn inequality \(\|\mathbf{D}\mathbf{u}\|_{L^{2}(\varOmega )^{n\times n}} \leq C\|\boldsymbol{\varepsilon }(\mathbf{u})\|_{L^{2}(\varOmega )^{n\times n}},\) has become a standard topic in the literature of continuum mechanics.

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Notes

  1. 1.

    Loosely speaking, the goal is to work in a proper subspace of H 1(Ω)n for which the condition \(\boldsymbol{\varepsilon }(\mathbf{u}) = 0\) implies u = 0.

  2. 2.

    “Real” rigid deformations or movements can be regarded as translations followed by rotations. Accordingly, they are associated to linear mappings defined by means of proper orthogonal matrices.

References

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Authors and Affiliations

  1. Department of Mathematics and IMAS, University of Buenos Aires and CONICET, Buenos Aires, Argentina

    Gabriel Acosta & Ricardo G. Durán

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  1. Gabriel Acosta
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  2. Ricardo G. Durán
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Acosta, G., Durán, R.G. (2017). Korn’s Inequalities. In: Divergence Operator and Related Inequalities. SpringerBriefs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6985-2_3

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