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research
The three divergence free matrix fields problem
Authors
Mariapia Palombaro
Marcello Ponsiglione
Publication date
1 January 2003
Publisher
View
on
arXiv
Abstract
We prove that for any connected open set
Ω
⊂
R
n
\Omega\subset \R^n
Ω
⊂
R
n
and for any set of matrices
K
=
{
A
1
,
A
2
,
A
3
}
⊂
M
m
×
n
K=\{A_1,A_2,A_3\}\subset M^{m\times n}
K
=
{
A
1
​
,
A
2
​
,
A
3
​
}
⊂
M
m
×
n
, with
m
≥
n
m\ge n
m
≥
n
and rank
(
A
i
−
A
j
)
=
n
(A_i-A_j)=n
(
A
i
​
−
A
j
​
)
=
n
for
i
â‰
j
i\neq j
i
î€
=
j
, there is no non-constant solution
B
∈
L
∞
(
Ω
,
M
m
×
n
)
B\in L^{\infty}(\Omega,M^{m\times n})
B
∈
L
∞
(
Ω
,
M
m
×
n
)
, called exact solution, to the problem Div B=0 \quad \text{in} D'(\Omega,\R^m) \quad \text{and} \quad B(x)\in K \text{a.e. in} \Omega. In contrast, A. Garroni and V. Nesi \cite{GN} exhibited an example of set
K
K
K
for which the above problem admits the so-called approximate solutions. We give further examples of this type. We also prove non-existence of exact solutions when
K
K
K
is an arbitrary set of matrices satisfying a certain algebraic condition which is weaker than simultaneous diagonalizability.Comment: 15 pages, 1 figur
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Last time updated on 12/11/2016