TY - JOUR

T1 - On the fractional Korn inequality in bounded domains

T2 - Counterexamples to the case ps < 1

AU - Harutyunyan, Davit

AU - Mikayelyan, Hayk

N1 - Funding Information:
We thank the anonymous referee for useful comments and for pointing out some relevant literature that improved the presentation of the article. The work of Davit Harutyunyan is supported by the National Science Foundation (Grant No. DMS-1814361).
Publisher Copyright:
© 2023 the author(s), published by De Gruyter.

PY - 2023/2/27

Y1 - 2023/2/27

N2 - 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 for fractional W 0 s, p (ω)-regular domains ω. Also, in the case p s < 1, for any open bounded C 1 domain ω R n Ω Rn, 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 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 are interpolations of a constant affine rigid motion inside the domain away from the boundary and of the zero field close to the boundary.

AB - 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 for fractional W 0 s, p (ω)-regular domains ω. Also, in the case p s < 1, for any open bounded C 1 domain ω R n Ω Rn, 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 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 are interpolations of a constant affine rigid motion inside the domain away from the boundary and of the zero field close to the boundary.

KW - fractional Hardy inequality

KW - fractional Korn inequality

KW - peridynamics

UR - http://www.scopus.com/inward/record.url?scp=85149339593&partnerID=8YFLogxK

U2 - 10.1515/anona-2022-0283

DO - 10.1515/anona-2022-0283

M3 - Article

AN - SCOPUS:85149339593

SN - 2191-9496

VL - 12

JO - Advances in Nonlinear Analysis

JF - Advances in Nonlinear Analysis

IS - 1

M1 - 20220283

ER -