Abstract
Korn’s first inequality states that there exists a constant such that the L2-norm of the infinitesimal displacement gradient is bounded above by this constant times the L2-norm of the infinitesimal strain, i.e., the symmetric part of the gradient, for all infinitesimal displacements that are equal to zero on the boundary of a body ℬ. This inequality is known to hold when the L2-norm is replaced by the Lp-norm for any p∈ (1 , ∞ ). However, if p= 1 or p= ∞ the resulting inequality is false. It was previously shown that if one replaces the L∞-norm by the BMO -seminorm (Bounded Mean Oscillation) then one maintains Korn’s inequality. (Recall that L∞(B) ⊂ BMO (B) ⊂ Lp(B) ⊂ L1(B) , 1 < p< ∞.) In this manuscript it is shown that Korn’s inequality is also maintained if one replaces the L1-norm by the norm in the Hardy space H1, the predual of BMO. One caveat: the results herein are only applicable to the pure-displacement problem with the displacement equal to zero on the entire boundary of ℬ.
Original language | English |
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Pages (from-to) | 187-198 |
Number of pages | 12 |
Journal | Journal of Elasticity |
Volume | 154 |
Issue number | 1-4 |
DOIs | |
Publication status | Published - 2023 Nov |
Keywords
- Hardy-Sobolev spaces
- Korn’s inequality
- Linear elasticity
- Riesz transforms
ASJC Scopus subject areas
- General Materials Science
- Mechanics of Materials
- Mechanical Engineering