# Nečas-Lions lemma revisited: An $L^p$-version of the generalized Korn inequality for incompatible tensor fields

created by lewintan on 17 Dec 2019
modified on 13 Oct 2021

[BibTeX]

Published Paper

Inserted: 17 dec 2019
Last Updated: 13 oct 2021

Journal: Mathematical Methods in the Applied Sciences
Year: 2021
Doi: 10.1002/mma.7498

ArXiv: 1912.08447 PDF

Abstract:

For $1<p<\infty$ we prove an $L^p$-version of the generalized Korn inequality for incompatible tensor fields $P$ in $W^{1,\,p}_0(\operatorname{Curl}; \Omega,\mathbb{R}^{3\times3})$. More precisely, let $\Omega\subset\mathbb{R}^3$ be a bounded Lipschitz domain. Then there exists a constant $c>0$ such that $| P|_{L^p}\leq c\,\left( |\operatorname{sym} P|_{L^p} + | {\operatorname{Curl}}P|_{L^p}\right)$ holds for all tensor fields $P\in W^{1,\,p}_0( {\operatorname{Curl}}; \Omega,\mathbb{R}^{3\times3})$, i.e., for all $P\in W^{1,\,p}( {\operatorname{Curl}}; \Omega,\mathbb{R}^{3\times3})$ with vanishing tangential trace $P\times \nu=0$ on $\partial\Omega$ where $\nu$ denotes the outward unit normal vector field to $\partial\Omega$.

For compatible $P= Du$ this recovers an $L^p$-version of the classical Korn's first inequality $|D u |_{L^p} \le c\, | {\operatorname{sym}} D u|_{L^p},$ with $D u \times \nu = 0$ on $\partial \Omega$ and for skew-symmetric $P=A\in\mathfrak{so}(3)$ an $L^p$-version of the Poincaré inequality $|A|_{L^p}\le c\, | {\operatorname{Curl}} A|_{L^p}$ with $A \times \nu = 0 \Leftrightarrow \ A=0$ on $\partial \Omega$.