$L^p$-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^{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 \[ \|{ P }\|_{L^p(\Omega,\mathbb{R}^{3\times3})}\leq c\,\left(\|{\operatorname{dev} \operatorname{sym} P }\|_{L^p(\Omega,\mathbb{R}^{3\times3})} + \|{ \operatorname{dev} \operatorname{Curl} P }\|_{L^p(\Omega,\mathbb{R}^{3\times3})}\right) \] 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 $\partial\Omega$ and $\operatorname{dev} P := P -\frac13 \operatorname{tr}(P)\,\mathbb{1}_3$ denotes the deviatoric (trace-free) part of $P$. We also show the norm equivalence \[ \|{ P }\|_{L^p(\Omega,\mathbb{R}^{3\times3})}+\|{\operatorname{Curl} P }\|_{L^p(\Omega,\mathbb{R}^{3\times3})}\leq c\,\left(\|{\operatorname{dev} \operatorname{sym} P }\|_{L^p(\Omega,\mathbb{R}^{3\times3})} + \|{ \operatorname{dev}\operatorname{Curl} P }\|_{L^p(\Omega,\mathbb{R}^{3\times3})}\right) \] for tensor fields $P\in W^{1,\,p}_0(\operatorname{Curl}; \Omega,\mathbb{R}^{3\times3})$. These estimates also hold true for tensor fields with vanishing tangential trace only on a relatively open (non-empty) subset $\Gamma \subseteq \partial\Omega$ of the boundary.