Mixing
Strong continuity of $langle Au,v rangle=int u^3 v dx$
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I am currently trying to figure out the following. If I consider the space $W^{1,p}_0$ is it possible to show that the operator given by
$$langle Au,v rangle=int u^3 v dx$$
is strongly (weak to strong) continuous for all $p$? We may assume we are on a bounded Lipschitz domain.
It is possible using the Sobolev embedding theorem but that will then restrict our choices of $p$ based on the dimension. I was told it is possible using the reverse dominated convergence theorem. Does anybody have any ideas?
functional-analysis analysis pde continuity operator-theory
edited Jan 20 at 2:30
Mattos
2,81721321
asked Jan 20 at 1:40
BennibenbenBennibenben
1248
$endgroup$
add a comment |
$begingroup$
I am currently trying to figure out the following. If I consider the space $W^{1,p}_0$ is it possible to show that the operator given by
$$langle Au,v rangle=int u^3 v dx$$
is strongly (weak to strong) continuous for all $p$? We may assume we are on a bounded Lipschitz domain.
It is possible using the Sobolev embedding theorem but that will then restrict our choices of $p$ based on the dimension. I was told it is possible using the reverse dominated convergence theorem. Does anybody have any ideas?
functional-analysis analysis pde continuity operator-theory
edited Jan 20 at 2:30
Mattos
2,81721321
asked Jan 20 at 1:40
BennibenbenBennibenben
1248
$endgroup$
$begingroup$
What is the "reverse dominated convergence theorem"? This operator is not defined if $p$ is too small (compared to the dimension).
$endgroup$
– gerw
Jan 21 at 7:10
add a comment |
$begingroup$
I am currently trying to figure out the following. If I consider the space $W^{1,p}_0$ is it possible to show that the operator given by
$$langle Au,v rangle=int u^3 v dx$$
is strongly (weak to strong) continuous for all $p$? We may assume we are on a bounded Lipschitz domain.
It is possible using the Sobolev embedding theorem but that will then restrict our choices of $p$ based on the dimension. I was told it is possible using the reverse dominated convergence theorem. Does anybody have any ideas?
functional-analysis analysis pde continuity operator-theory
edited Jan 20 at 2:30
Mattos
2,81721321
asked Jan 20 at 1:40
BennibenbenBennibenben
1248
$endgroup$
I am currently trying to figure out the following. If I consider the space $W^{1,p}_0$ is it possible to show that the operator given by
$$langle Au,v rangle=int u^3 v dx$$
is strongly (weak to strong) continuous for all $p$? We may assume we are on a bounded Lipschitz domain.
It is possible using the Sobolev embedding theorem but that will then restrict our choices of $p$ based on the dimension. I was told it is possible using the reverse dominated convergence theorem. Does anybody have any ideas?
functional-analysis analysis pde continuity operator-theory
functional-analysis analysis pde continuity operator-theory
edited Jan 20 at 2:30
Mattos
2,81721321
asked Jan 20 at 1:40
BennibenbenBennibenben
1248
edited Jan 20 at 2:30
Mattos
2,81721321
asked Jan 20 at 1:40
BennibenbenBennibenben
1248
edited Jan 20 at 2:30
Mattos
2,81721321
edited Jan 20 at 2:30
Mattos
2,81721321
edited Jan 20 at 2:30
Mattos
2,81721321
2,81721321
asked Jan 20 at 1:40
BennibenbenBennibenben
1248
asked Jan 20 at 1:40
BennibenbenBennibenben
1248
asked Jan 20 at 1:40
BennibenbenBennibenben
1248
1248
$begingroup$
What is the "reverse dominated convergence theorem"? This operator is not defined if $p$ is too small (compared to the dimension).
$endgroup$
– gerw
Jan 21 at 7:10
add a comment |
$begingroup$
What is the "reverse dominated convergence theorem"? This operator is not defined if $p$ is too small (compared to the dimension).
$endgroup$
– gerw
Jan 21 at 7:10
$begingroup$
What is the "reverse dominated convergence theorem"? This operator is not defined if $p$ is too small (compared to the dimension).
$endgroup$
– gerw
Jan 21 at 7:10
$begingroup$
What is the "reverse dominated convergence theorem"? This operator is not defined if $p$ is too small (compared to the dimension).
$endgroup$
– gerw
Jan 21 at 7:10
add a comment |
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$begingroup$
What is the "reverse dominated convergence theorem"? This operator is not defined if $p$ is too small (compared to the dimension).
$endgroup$
– gerw
Jan 21 at 7:10