Published Paper

Inserted: 18 dec 2019
Last Updated: 8 sep 2021

Journal: C.R., Math.
Volume: 339
Number: 6
Pages: 749-755
Year: 2021
Doi: 10.5802/crmath.216

Abstract:

For $n>2$ and $1<p<\infty$ we prove an $L^p$-version of the generalized Korn-type inequality for incompatible, $p$-integrable tensor fields $P:\Omega \to \mathbb{R}^{n\times n}$ having $p$-integrable generalized $\underline{\operatorname{Curl}}$ and generalized vanishing tangential trace $P\,\tau_l=0$ on $\partial \Omega$, denoting by $\{\tau_l\}_{l=1,\ldots, n-1}$ a moving tangent frame on $\partial\Omega$, more precisely we have: \[
P
_{L^p}\leq c\,(
\operatorname{sym} P
_{L^p} +
\underline{\operatorname{Curl}} P
_{L^p} ),\] where the generalized $\underline{\operatorname{Curl}}$ is given by $ (\underline{\operatorname{Curl}})_{ijk} :=\partial_i P_{kj}-\partial_j P_{ki}$ and $c=c(n,p,\Omega)>0$.

Keywords: Korn's inequality, Lions lemma, Nečas estimate, Poincaré's inequality