On Korn’s First Inequality in a Hardy-Sobolev Space

Daniel E. Spector, Scott J. Spector*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Pages (from-to)187-198
Number of pages12
JournalJournal of Elasticity
Issue number1-4
Publication statusPublished - 2023 Nov


  • Hardy-Sobolev spaces
  • Korn’s inequality
  • Linear elasticity
  • Riesz transforms

ASJC Scopus subject areas

  • General Materials Science
  • Mechanics of Materials
  • Mechanical Engineering


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