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

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 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) ⊂ 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.