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Symmetry and monotonicity results for solutions of vectorial -Stokes systems
In this paper we shall study qualitative properties of a p p -Stokes type system, namely − Δ p u = − d i v ( | D u | p − 2 D u ) = f ( x , u ) in Ω , \begin{equation*} -{\boldsymbol \Delta }_p{\boldsymbol u}=-\operatorname {\mathbf {div}}(|D{\boldsymbol u}|^{p-2}D{\boldsymbol u}) = {\boldsymbol...
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Published in: | Transactions of the American Mathematical Society 2023-05, Vol.376 (5), p.3493-3514 |
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Main Authors: | , , |
Format: | Article |
Language: | English |
Citations: | Items that this one cites |
Online Access: | Get full text |
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Summary: | In this paper we shall study qualitative properties of a
p
p
-Stokes type system, namely
−
Δ
p
u
=
−
d
i
v
(
|
D
u
|
p
−
2
D
u
)
=
f
(
x
,
u
)
in
Ω
,
\begin{equation*} -{\boldsymbol \Delta }_p{\boldsymbol u}=-\operatorname {\mathbf {div}}(|D{\boldsymbol u}|^{p-2}D{\boldsymbol u}) = {\boldsymbol f}(x,{\boldsymbol u}) \text { in $\Omega $}, \end{equation*}
where
Δ
p
{\boldsymbol \Delta }_p
is the
p
p
-Laplacian vectorial operator. More precisely, under suitable assumptions on the domain
Ω
\Omega
and the function
f
\boldsymbol { f}
, it is deduced that system solutions are symmetric and monotone. Our main results are derived from a vectorial version of the weak and strong comparison principles, which enable to proceed with the moving-planes technique for systems. As far as we know, these are the first qualitative kind results involving vectorial operators. |
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ISSN: | 0002-9947 1088-6850 |
DOI: | 10.1090/tran/8867 |