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

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 Lp-norm for any p is an element of (1, infinity). However, if p = 1 or p = infinity the resulting inequality is false. It was previously shown that if one replaces the L-infinity- norm by the BMO-seminorm (Bounded Mean Oscillation) then one maintains Korn's inequality. (Recall that L-infinity(B) subset of BMO(B) subset of L-p(B) subset of L-1(B), 1 < p < infinity.) 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.