2022/04/03 by Davit Harutyunyan, Harutyunyan, Davit, Hayk Mikayelyan +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2204.00999
openalex publication_date 2022/04/03 · openalex created_date 2022/04/26 · openalex updated_date 2026/07/28
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 Ws,p0(Ω) Sobolev fields in open and bounded C1-regular domains Ω⊂ \mathbb Rn. Also, in the case ps<1, for any open bounded C1 domain Ω⊂ \mathbb 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 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 [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.