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

Abstract

For $n\ge 2$ and $1 we prove an $L^p$-version of the generalized Korn-type inequality for incompatible, $p$-integrable tensor fields $P:\Omega \rightarrow \mathbb{R}^{n\,\times \,n}$ having $p$-integrable generalized $\underline{\operatorname{Curl}}\,$ and generalized vanishing tangential trace $P\,\tau _l=0$ on $\partial \Omega $, denoting by $\lbrace \tau _l\rbrace _{l=1,\,\ldots ,\,n-1}$ a moving tangent frame on $\partial \Omega $, more precisely we have: \[ \left\Vert P \right\Vert _{L^p\left(\Omega ,\,\mathbb{R}^{n\,\times \,n}\right)}\le c\,\left(\left\Vert \operatorname{sym}P \right\Vert _{L^p\left(\Omega ,\,\mathbb{R}^{n \times n}\right)}+ \left\Vert \underline{\operatorname{Curl}}\,P \right\Vert _{L^p\left(\Omega ,\,\left(\mathfrak{so}(n)\right)^n\right)}\right), \] where the generalized $\underline{\operatorname{Curl}}\,$ is given by $(\underline{\operatorname{Curl}}\,P)_{ijk} :=\partial _i P_{kj}-\partial _j P_{ki}$ and $c=c(n,p,\Omega )>0$