On the fractional Korn inequality in bounded domains: Counterexamples to the case $ps<1$

TL;DR

This paper establishes the validity of fractional Korn's first inequality for $ps>1$ in bounded domains and constructs counterexamples showing failure for $ps<1$, clarifying the inequality's domain of applicability.

Contribution

It proves fractional Korn's first inequality holds for $ps>1$ and provides counterexamples for $ps<1$, resolving an open problem in fractional Sobolev spaces.

Findings

Korn's inequality holds for $ps>1$ in bounded domains.

Counterexamples show failure of Korn's inequality for $ps<1$.

The proof combines compactness, Hardy inequality, and recent Korn results.

Abstract

The validity of Korn's first inequality in the fractional setting in bounded domains has been open. We resolve this problem by proving that in fact Korn's first inequality holds in the case $ps>1$ for fractional $W^{s,p}_0(\Omega)$ Sobolev fields in open and bounded $C^{1}$-regular domains $\Omega\subset \mathbb R^n$. Also, in the case $ps<1,$ for any open bounded $C^1$ domain $\Omega\subset \mathbb R^n$ we construct counterexamples to the inequality, i.e., Korn's first inequality fails to hold in bounded domains. The proof of the inequality in the case $ps>1$ follows a standard compactness approach adopted in the classical case, combined with a Hardy inequality, and a recently proven Korn second inequality by Mengesha and Scott [\textit{Commun. Math. Sci.,} Vol. 20, N0. 2, 405--423, 2022]. The counterexamples constructed in the case $ps<1$ are interpolations of a constant affine rigid motion inside the domain away from the boundary, and of the zero field close to the boundary.