## Abstract

Korn’s first inequality states that there exists a constant such that the L^{2}-norm of the infinitesimal displacement gradient is bounded above by this constant times the L^{2}-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 L^{2}-norm is replaced by the L^{p}-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) ⊂ L^{p}(B) ⊂ L^{1}(B) , 1 < p< ∞.) In this manuscript it is shown that Korn’s inequality is also maintained if one replaces the L^{1}-norm by the norm in the Hardy space H^{1}, 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