Sobolev inequality cannot hold for all compactly supported smooth functions
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I am on a course on Sobolev Spaces and we had this as an exercise:
Let $1leq p<n$ and $q< p^*$, where $p^*=(pn)/(n-p)$. Show that
$||u||_{L^q(mathbb{R}^n)}leq C(q,p,n)||nabla u||_{L^p(mathbb{R}^n)}$
cannot hold for all $uin C_0^infty(mathbb{R}^n).$
We were given a hint that if we take some $uin C_0^infty(mathbb{R}^n)$ and then scale the variable inside, then we could show this. But I am at a loss, and don't know how to use the hint.
sobolev-spaces
asked Oct 17 at 13:00
Janne Nurminen
34
add a comment |
up vote
0
down vote
favorite
I am on a course on Sobolev Spaces and we had this as an exercise:
Let $1leq p<n$ and $q< p^*$, where $p^*=(pn)/(n-p)$. Show that
$||u||_{L^q(mathbb{R}^n)}leq C(q,p,n)||nabla u||_{L^p(mathbb{R}^n)}$
cannot hold for all $uin C_0^infty(mathbb{R}^n).$
We were given a hint that if we take some $uin C_0^infty(mathbb{R}^n)$ and then scale the variable inside, then we could show this. But I am at a loss, and don't know how to use the hint.
sobolev-spaces
asked Oct 17 at 13:00
Janne Nurminen
34
You mean that q is strictly smaller than $p^star$, right?
– Giuseppe Negro
Oct 31 at 9:26
Yes, I had a typo, thank you :)
– Janne Nurminen
Nov 19 at 16:11
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am on a course on Sobolev Spaces and we had this as an exercise:
Let $1leq p<n$ and $q< p^*$, where $p^*=(pn)/(n-p)$. Show that
$||u||_{L^q(mathbb{R}^n)}leq C(q,p,n)||nabla u||_{L^p(mathbb{R}^n)}$
cannot hold for all $uin C_0^infty(mathbb{R}^n).$
We were given a hint that if we take some $uin C_0^infty(mathbb{R}^n)$ and then scale the variable inside, then we could show this. But I am at a loss, and don't know how to use the hint.
sobolev-spaces
asked Oct 17 at 13:00
Janne Nurminen
34
I am on a course on Sobolev Spaces and we had this as an exercise:
Let $1leq p<n$ and $q< p^*$, where $p^*=(pn)/(n-p)$. Show that
$||u||_{L^q(mathbb{R}^n)}leq C(q,p,n)||nabla u||_{L^p(mathbb{R}^n)}$
cannot hold for all $uin C_0^infty(mathbb{R}^n).$
We were given a hint that if we take some $uin C_0^infty(mathbb{R}^n)$ and then scale the variable inside, then we could show this. But I am at a loss, and don't know how to use the hint.
sobolev-spaces
sobolev-spaces
asked Oct 17 at 13:00
Janne Nurminen
34
asked Oct 17 at 13:00
Janne Nurminen
34
edited Nov 19 at 16:12
asked Oct 17 at 13:00
Janne Nurminen
34
asked Oct 17 at 13:00
Janne Nurminen
34
asked Oct 17 at 13:00
Janne Nurminen
34
34
You mean that q is strictly smaller than $p^star$, right?
– Giuseppe Negro
Oct 31 at 9:26
Yes, I had a typo, thank you :)
– Janne Nurminen
Nov 19 at 16:11
add a comment |
You mean that q is strictly smaller than $p^star$, right?
– Giuseppe Negro
Oct 31 at 9:26
Yes, I had a typo, thank you :)
– Janne Nurminen
Nov 19 at 16:11
You mean that q is strictly smaller than $p^star$, right?
– Giuseppe Negro
Oct 31 at 9:26
You mean that q is strictly smaller than $p^star$, right?
– Giuseppe Negro
Oct 31 at 9:26
Yes, I had a typo, thank you :)
– Janne Nurminen
Nov 19 at 16:11
Yes, I had a typo, thank you :)
– Janne Nurminen
Nov 19 at 16:11
add a comment |
1 Answer
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Hint: For fixed $u in C_c^infty$, consider $u_t$ defined by $u_t(x) = u(tx)$ for $t>0$. Then express the norms of $u_t$ in the Sobolev inequality by scaled versions of the norms of $u$.
answered Oct 17 at 13:04
daw
24k1544
On the left I get the same norm? And on the right I get t times everything? I don't see where this leads?
– Janne Nurminen
Oct 17 at 16:29
Now let $tto 0$ or $tto infty$.
– daw
Oct 17 at 18:45
So now if $tto 0$ and the norm of u is positive (say 1) then the inequality doesn't work. But why do we assume that $q≤p^*$? I don't understand why the same conclusion would not surface with $p^*$.
– Janne Nurminen
Oct 18 at 5:24
The parameters do not play a role, as the right hand side is $|nabla u|_{L^p}$ and not $|u|_{W^{1,p}}$. The inequality is false on unbounded domains regardless of parameters $p,q,n$.
– daw
Oct 18 at 6:34
add a comment |
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1 Answer
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1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
up vote
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accepted
Hint: For fixed $u in C_c^infty$, consider $u_t$ defined by $u_t(x) = u(tx)$ for $t>0$. Then express the norms of $u_t$ in the Sobolev inequality by scaled versions of the norms of $u$.
answered Oct 17 at 13:04
daw
24k1544
On the left I get the same norm? And on the right I get t times everything? I don't see where this leads?
– Janne Nurminen
Oct 17 at 16:29
Now let $tto 0$ or $tto infty$.
– daw
Oct 17 at 18:45
So now if $tto 0$ and the norm of u is positive (say 1) then the inequality doesn't work. But why do we assume that $q≤p^*$? I don't understand why the same conclusion would not surface with $p^*$.
– Janne Nurminen
Oct 18 at 5:24
The parameters do not play a role, as the right hand side is $|nabla u|_{L^p}$ and not $|u|_{W^{1,p}}$. The inequality is false on unbounded domains regardless of parameters $p,q,n$.
– daw
Oct 18 at 6:34
add a comment |
up vote
1
down vote
accepted
Hint: For fixed $u in C_c^infty$, consider $u_t$ defined by $u_t(x) = u(tx)$ for $t>0$. Then express the norms of $u_t$ in the Sobolev inequality by scaled versions of the norms of $u$.
answered Oct 17 at 13:04
daw
24k1544
On the left I get the same norm? And on the right I get t times everything? I don't see where this leads?
– Janne Nurminen
Oct 17 at 16:29
Now let $tto 0$ or $tto infty$.
– daw
Oct 17 at 18:45
So now if $tto 0$ and the norm of u is positive (say 1) then the inequality doesn't work. But why do we assume that $q≤p^*$? I don't understand why the same conclusion would not surface with $p^*$.
– Janne Nurminen
Oct 18 at 5:24
The parameters do not play a role, as the right hand side is $|nabla u|_{L^p}$ and not $|u|_{W^{1,p}}$. The inequality is false on unbounded domains regardless of parameters $p,q,n$.
– daw
Oct 18 at 6:34
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Hint: For fixed $u in C_c^infty$, consider $u_t$ defined by $u_t(x) = u(tx)$ for $t>0$. Then express the norms of $u_t$ in the Sobolev inequality by scaled versions of the norms of $u$.
answered Oct 17 at 13:04
daw
24k1544
Hint: For fixed $u in C_c^infty$, consider $u_t$ defined by $u_t(x) = u(tx)$ for $t>0$. Then express the norms of $u_t$ in the Sobolev inequality by scaled versions of the norms of $u$.
answered Oct 17 at 13:04
daw
24k1544
answered Oct 17 at 13:04
daw
24k1544
answered Oct 17 at 13:04
daw
24k1544
answered Oct 17 at 13:04
daw
24k1544
24k1544
On the left I get the same norm? And on the right I get t times everything? I don't see where this leads?
– Janne Nurminen
Oct 17 at 16:29
Now let $tto 0$ or $tto infty$.
– daw
Oct 17 at 18:45
So now if $tto 0$ and the norm of u is positive (say 1) then the inequality doesn't work. But why do we assume that $q≤p^*$? I don't understand why the same conclusion would not surface with $p^*$.
– Janne Nurminen
Oct 18 at 5:24
The parameters do not play a role, as the right hand side is $|nabla u|_{L^p}$ and not $|u|_{W^{1,p}}$. The inequality is false on unbounded domains regardless of parameters $p,q,n$.
– daw
Oct 18 at 6:34
add a comment |
On the left I get the same norm? And on the right I get t times everything? I don't see where this leads?
– Janne Nurminen
Oct 17 at 16:29
Now let $tto 0$ or $tto infty$.
– daw
Oct 17 at 18:45
So now if $tto 0$ and the norm of u is positive (say 1) then the inequality doesn't work. But why do we assume that $q≤p^*$? I don't understand why the same conclusion would not surface with $p^*$.
– Janne Nurminen
Oct 18 at 5:24
The parameters do not play a role, as the right hand side is $|nabla u|_{L^p}$ and not $|u|_{W^{1,p}}$. The inequality is false on unbounded domains regardless of parameters $p,q,n$.
– daw
Oct 18 at 6:34
On the left I get the same norm? And on the right I get t times everything? I don't see where this leads?
– Janne Nurminen
Oct 17 at 16:29
On the left I get the same norm? And on the right I get t times everything? I don't see where this leads?
– Janne Nurminen
Oct 17 at 16:29
Now let $tto 0$ or $tto infty$.
– daw
Oct 17 at 18:45
Now let $tto 0$ or $tto infty$.
– daw
Oct 17 at 18:45
So now if $tto 0$ and the norm of u is positive (say 1) then the inequality doesn't work. But why do we assume that $q≤p^*$? I don't understand why the same conclusion would not surface with $p^*$.
– Janne Nurminen
Oct 18 at 5:24
So now if $tto 0$ and the norm of u is positive (say 1) then the inequality doesn't work. But why do we assume that $q≤p^*$? I don't understand why the same conclusion would not surface with $p^*$.
– Janne Nurminen
Oct 18 at 5:24
The parameters do not play a role, as the right hand side is $|nabla u|_{L^p}$ and not $|u|_{W^{1,p}}$. The inequality is false on unbounded domains regardless of parameters $p,q,n$.
– daw
Oct 18 at 6:34
The parameters do not play a role, as the right hand side is $|nabla u|_{L^p}$ and not $|u|_{W^{1,p}}$. The inequality is false on unbounded domains regardless of parameters $p,q,n$.
– daw
Oct 18 at 6:34
add a comment |
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You mean that q is strictly smaller than $p^star$, right?
– Giuseppe Negro
Oct 31 at 9:26
Yes, I had a typo, thank you :)
– Janne Nurminen
Nov 19 at 16:11