Abstract
AbstractWe characterise all linear maps $${\mathscr {A}}:\mathbb R^{n\times n}\rightarrow \mathbb R^{n\times n}$$
A
:
R
n
×
n
→
R
n
×
n
such that, for $$1\le p<n$$
1
≤
p
<
n
, $$\begin{aligned} \left\Vert P\right\Vert _{{\text {L}}^{p^{*}}(\mathbb R^{n})}\le c\,\Big (\left\Vert {\mathscr {A}}[P]\right\Vert _{{\text {L}}^{p^{*}}(\mathbb R^{n})}+\left\Vert {\text {Curl}}P\right\Vert _{{\text {L}}^{p}(\mathbb R^{n})} \Big ) \end{aligned}$$
P
L
p
∗
(
R
n
)
≤
c
(
A
[
P
]
L
p
∗
(
R
n
)
+
Curl
P
L
p
(
R
n
)
)
holds for all compactly supported $$P\in {\text {C}}_{c}^{\infty }(\mathbb R^{n};\mathbb R^{n\times n})$$
P
∈
C
c
∞
(
R
n
;
R
n
×
n
)
, where $${\text {Curl}}P$$
Curl
P
displays the matrix curl. Being applicable to incompatible, that is, non-gradient matrix fields as well, such inequalities generalise the usual Korn-type inequalities used e.g. in linear elasticity. Different from previous contributions, the results gathered in this paper are applicable to all dimensions and optimal. This particularly necessitates the distinction of different combinations between the ellipticities of $${\mathscr {A}}$$
A
, the integrability p and the underlying space dimensions n, especially requiring a finer analysis in the two-dimensional situation.
Funder
Deutsche Forschungsgemeinschaft
Hector Stiftung
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Analysis
Reference71 articles.
1. Adams, D.R., Hedberg, L.: Function Spaces and Potential Theory. Grundlehren der mathematischen Wissenschaften (A Series of Comprehensive Studies in Mathematics), vol. 314, Springer, Berlin-Heidelberg-New York, (1996). https://doi.org/10.1007/978-3-662-03282-4
2. Bauer, S., Neff, P., Pauly, D., Starke, G.: Dev-Div- and DevSym-DevCurl-inequalities for incompatible square tensor fields with mixed boundary conditions. ESAIM Control Optim. Calc. Var. 22(1), 112–133 (2016). https://doi.org/10.1051/cocv/2014068
3. Bourgain, J., Brezis, H.: New estimates for elliptic equations and Hodge type systems. J. Eur. Math. Soc. (JEMS) 9(2), 277–315 (2007). https://doi.org/10.4171/JEMS/80
4. Breit, D.: Existence Theory for Generalized Newtonian Fluids. Mathematics in Science and Engineering. Academic Press, Elsevier (2017).https://www.sciencedirect.com/book/9780128110447/existence-theory-for-generalized-newtonian-fluids
5. Breit, D., Cianchi, A., Diening, L.: Trace-free Korn inequalities in Orlicz spaces. SIAM J. Math. Anal. 49(4), 2496–2526 (2017). https://doi.org/10.1137/16M1073662
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