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.
Categories: math.AP
