For
$1< p<\infty$
we prove an
$L^{p}$
-version of the generalized trace-free Korn inequality for incompatible tensor fields
$P$
in
$W^{1,p}_0(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$
. More precisely, let
$\Omega \subset \mathbb {R}^{3}$
be a bounded Lipschitz domain. Then there exists a constant
$c>0$
such that
\[ \lVert{ P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\leq c\,\left(\lVert{\operatorname{dev} \operatorname{sym} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})} + \lVert{ \operatorname{dev} \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\right) \]
holds for all tensor fields
$P\in W^{1,p}_0(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$
, i.e., for all
$P\in W^{1,p} (\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$
with vanishing tangential trace
$P\times \nu =0$
on
$\partial \Omega$
where
$\nu$
denotes the outward unit normal vector field to
$\partial \Omega$
and
$\operatorname {dev} P : = P -\frac 13 \operatorname {tr}(P) {\cdot } {\mathbb {1}}$
denotes the deviatoric (trace-free) part of
$P$
. We also show the norm equivalence
\begin{align*} &\lVert{ P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}+\lVert{ \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\\ &\quad\leq c\,\left(\lVert{P}\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})} + \lVert{ \operatorname{dev} \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\right) \end{align*}
for tensor fields
$P\in W^{1,p}(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$
. These estimates also hold true for tensor fields with vanishing tangential trace only on a relatively open (non-empty) subset
$\Gamma \subseteq \partial \Omega$
of the boundary.
Constant angle spacelike surface in de Sitter space S₁³
