$L^p$-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with $p$-integrable exterior derivative

Abstract

For $n\ge 2$ and $1<p<\infty$ we prove an $L^p$-version of the generalized Korn-type inequality for incompatible, $p$-integrable tensor fields $P:\hat{I}\text{c◯}\to {\mathbb{R}}^{n\times n}$ having $p$-integrable generalized $\underline{\mathrm{Curl}}$ and generalized vanishing tangential trace $P\ddot{I}{„}_l=0$ on $\partial \hat{I}\text{c◯}$, denoting by $\{\ddot{I}{„}_l{\}}_{l=1,\dots ,n-1}$ a moving tangent frame on $\partial \hat{I}\text{c◯}$, more precisely we have: $$\Vert P{\Vert }_{L^p(\hat{I}\text{c◯},{\mathbb{R}}^{n\times n})}\le c(\Vert \mathrm{sym}P{\Vert }_{L^p(\hat{I}\text{c◯},{\mathbb{R}}^{n\times n})}+\Vert \underline{\mathrm{Curl}}P{\Vert }_{L^p(\hat{I}\text{c◯},(\mathfrak{so}(n){)}^n)}),$$ where the generalized $\underline{\mathrm{Curl}}$ is given by $(\underline{\mathrm{Curl}}{)}_{ijk}:={\partial }_i{P}_{kj}-{\partial }_j{P}_{ki}$ and $c=c(n,p,\hat{I}\text{c◯})>0$.