AbstractFor $$n\ge 3$$
n
≥
3
and $$1<p<\infty $$
1
<
p
<
∞
, we prove an $$L^p$$
L
p
-version of the generalized trace-free Korn-type inequality for incompatible, p-integrable tensor fields $$P:\Omega \rightarrow \mathbb {R}^{n\times n}$$
P
:
Ω
→
R
n
×
n
having p-integrable generalized $${\text {Curl}}_{n}$$
Curl
n
and generalized vanishing tangential trace $$P\,\tau _l=0$$
P
τ
l
=
0
on $$\partial \Omega $$
∂
Ω
, denoting by $$\{\tau _l\}_{l=1,\ldots , n-1}$$
{
τ
l
}
l
=
1
,
…
,
n
-
1
a moving tangent frame on $$\partial \Omega $$
∂
Ω
. More precisely, there exists a constant $$c=c(n,p,\Omega )$$
c
=
c
(
n
,
p
,
Ω
)
such that $$\begin{aligned} \Vert P \Vert _{L^p(\Omega ,\mathbb {R}^{n\times n})}\le c\,\left( \Vert {\text {dev}}_n {\text {sym}}P \Vert _{L^p(\Omega ,\mathbb {R}^{n \times n})}+ \Vert {\text {Curl}}_{n} P \Vert _{L^p\left( \Omega ,\mathbb {R}^{n\times \frac{n(n-1)}{2}}\right) }\right) , \end{aligned}$$
‖
P
‖
L
p
(
Ω
,
R
n
×
n
)
≤
c
‖
dev
n
sym
P
‖
L
p
(
Ω
,
R
n
×
n
)
+
‖
Curl
n
P
‖
L
p
Ω
,
R
n
×
n
(
n
-
1
)
2
,
where the generalized $${\text {Curl}}_{n}$$
Curl
n
is given by $$({\text {Curl}}_{n} P)_{ijk} :=\partial _i P_{kj}-\partial _j P_{ki}$$
(
Curl
n
P
)
ijk
:
=
∂
i
P
kj
-
∂
j
P
ki
and "Equation missing" denotes the deviatoric (trace-free) part of the square matrix X. The improvement towards the three-dimensional case comes from a novel matrix representation of the generalized cross product.
L p -versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with p-integrable exterior derivative
