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

Abstract

Korn’s first inequality states that there exists a constant such that the ${\mathcal {L}}^{2}$ L 2 -norm of the infinitesimal displacement gradient is bounded above by this constant times the ${\mathcal {L}}^{2}$ 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 ${\mathcal {L}}^{2}$ L 2 -norm is replaced by the ${\mathcal {L}}^{p}$ L p -norm for any $p\in (1,\infty )$ p ∈ ( 1 , ∞ ) . However, if $p=1$ p = 1 or $p=\infty $ p = ∞ the resulting inequality is false. It was previously shown that if one replaces the ${\mathcal {L}}^{\infty}$ L ∞ -norm by the $\operatorname{BMO}$ BMO -seminorm (Bounded Mean Oscillation) then one maintains Korn’s inequality. (Recall that ${\mathcal {L}}^{\infty}({\mathcal {B}})\subset \operatorname{BMO}({\mathcal {B}}) \subset {\mathcal {L}}^{p}({\mathcal {B}})\subset {\mathcal {L}}^{1}({ \mathcal {B}})$ L ∞ ( B ) ⊂ BMO ( B ) ⊂ L p ( B ) ⊂ L 1 ( B ) , $1< p<\infty $ 1 < p < ∞ .) In this manuscript it is shown that Korn’s inequality is also maintained if one replaces the ${\mathcal {L}}^{1}$ L 1 -norm by the norm in the Hardy space ${\mathcal {H}}^{1}$ H 1 , the predual of $\operatorname{BMO}$ 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.