Consistency in fractional Korn's inequalities on Lipschitz domains

Abstract

We investigate fractional analogs of Korn's inequality on bounded Lipschitz domains in several contexts. Constrained and unconstrained inequalities are established; constraints include quotient spaces and spaces incorporating various classes of boundary conditions or volume constraints. We also expand the parameters of integrability for which the inequalities hold. Specifically, a fractional Korn's first unconstrained inequality holds for vector fields in the fractional Sobolev space $W^{s,p}(Ω;\mathbb{R}^n)$ for all $p \in [1,\infty]$ and all $s \in (0,1)$. In the cases $p = 1$ and $p = \infty$, the inequality has no local analog. We quantify this by investigating the consistency of the inequalities as the differentiability parameter $s \to 1$; the universal constant in the inequality remains stable for $p \in (1,\infty)$ but degenerates for $p \in \{1,\infty\}$. We introduce an explicit $L^p$-bounded projection operator onto infinitesimal rigid motions as a key tool to quantitatively strengthen the inequality.