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

Abstract

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 p s > 1 ps\gt 1 for fractional W 0 s , p ( Ω ) {W}_{0}^{s,p}\left(\Omega ) Sobolev fields in open and bounded C 1 {C}^{1} -regular domains Ω ⊂ R n \Omega \subset {{\mathbb{R}}}^{n} . Also, in the case p s < 1 ps\lt 1 , for any open bounded C 1 {C}^{1} domain Ω ⊂ R n \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 p s > 1 ps\gt 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 [A Fractional Korn-type inequality for smooth domains and a regularity estimate for nonlinear nonlocal systems of equations, Commun. Math. Sci. 20 (2022), no. 2, 405–423]. The counterexamples constructed in the case p s < 1 ps\lt 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.