$L^p$-trace-free generalized Korn inequalities for incompatible tensor fields in three space dimensions

Abstract

For $1<p<\infty$ we prove an $L^p$-version of the generalized trace-free Korn inequality for incompatible tensor fields $P$ in $W_0^{1,p}(\mathrm{Curl};\hat{I}\text{c◯},{\mathbb{R}}^{3\times 3})$. More precisely, let $\hat{I}\text{c◯}\subset {\mathbb{R}}^3$ be a bounded Lipschitz domain. Then there exists a constant $c>0$ such that \[ \|{ P }\|_{L^p(Ω,\mathbb{R}^{3\times3})}\leq c\,\left(\|{\operatorname{dev} \operatorname{sym} P }\|_{L^p(Ω,\mathbb{R}^{3\times3})} + \|{ \operatorname{dev} \operatorname{Curl} P }\|_{L^p(Ω,\mathbb{R}^{3\times3})}\right) \] holds for all tensor fields $P\in W_0^{1,p}(\mathrm{Curl};\hat{I}\text{c◯},{\mathbb{R}}^{3\times 3})$, i.e., for all $P\in W^{1,p}(\mathrm{Curl};\hat{I}\text{c◯},{\mathbb{R}}^{3\times 3})$ with vanishing tangential trace $P\times \hat{I}½=0$ on $\partial \hat{I}\text{c◯}$ where $\hat{I}½$ denotes the outward unit normal vector field to $\partial \hat{I}\text{c◯}$ and $\mathrm{dev}P:=P-\frac{1}{3}\mathrm{tr}(P)1_3$ denotes the deviatoric (trace-free) part of $P$. We also show the norm equivalence \[ \|{ P }\|_{L^p(Ω,\mathbb{R}^{3\times3})}+\|{\operatorname{Curl} P }\|_{L^p(Ω,\mathbb{R}^{3\times3})}\leq c\,\left(\|{\operatorname{dev} \operatorname{sym} P }\|_{L^p(Ω,\mathbb{R}^{3\times3})} + \|{ \operatorname{dev}\operatorname{Curl} P }\|_{L^p(Ω,\mathbb{R}^{3\times3})}\right) \] for tensor fields $P\in W_0^{1,p}(\mathrm{Curl};\hat{I}\text{c◯},{\mathbb{R}}^{3\times 3})$. These estimates also hold true for tensor fields with vanishing tangential trace only on a relatively open (non-empty) subset $\hat{I}“\subseteq \partial \hat{I}\text{c◯}$ of the boundary.