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Is the composition of a Sobolev function and a smooth function Sobolev?
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Let $Omega subseteq mathbbR^n$ be an open bounded domain, and let $1<p<n$. Suppose that $f in W^1,p(Omega)$ is continuous*, and $g in C^infty(mathbbR)$.
Is it true that $g circ f in W^1,p_loc(Omega)$?
My guess was that the answer is positive, and that $partial_i (g circ f)(x)=g'(f(x)) partial_i f(x)$ but a naive calculation to prove it failed.
*Note that the continuity of $f$ does not follow from $f in W^1,p(Omega)$, since $p<n$; this is an additional assumption I am adding.
real-analysis sobolev-spaces regularity-theory-of-pdes weak-derivatives
asked Jul 30 at 9:06
Asaf Shachar
4,4823832
 |Â
show 4 more comments
up vote
1
down vote
favorite
Let $Omega subseteq mathbbR^n$ be an open bounded domain, and let $1<p<n$. Suppose that $f in W^1,p(Omega)$ is continuous*, and $g in C^infty(mathbbR)$.
Is it true that $g circ f in W^1,p_loc(Omega)$?
My guess was that the answer is positive, and that $partial_i (g circ f)(x)=g'(f(x)) partial_i f(x)$ but a naive calculation to prove it failed.
*Note that the continuity of $f$ does not follow from $f in W^1,p(Omega)$, since $p<n$; this is an additional assumption I am adding.
real-analysis sobolev-spaces regularity-theory-of-pdes weak-derivatives
asked Jul 30 at 9:06
Asaf Shachar
4,4823832
Isn't there some $|x|^θ$ that is in $W^1,p$, then using some $g(x) = |x|^phi$ should change the $p$
– Calvin Khor
Jul 30 at 9:27
the spike function only lives in local $W^1,p$ though
– Calvin Khor
Jul 30 at 9:33
the function i mentioned earlier, $|x|^-θ$ for $θ < (n-p)/p$
– Calvin Khor
Jul 30 at 9:36
You can use this result: math.stackexchange.com/questions/1110231/…
– Bob
Jul 30 at 11:11
@Bob Yes, but that assumes $g'$ is bounded and works also for $f$ that are not continuous.
– Calvin Khor
Jul 30 at 11:35
 |Â
show 4 more comments
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $Omega subseteq mathbbR^n$ be an open bounded domain, and let $1<p<n$. Suppose that $f in W^1,p(Omega)$ is continuous*, and $g in C^infty(mathbbR)$.
Is it true that $g circ f in W^1,p_loc(Omega)$?
My guess was that the answer is positive, and that $partial_i (g circ f)(x)=g'(f(x)) partial_i f(x)$ but a naive calculation to prove it failed.
*Note that the continuity of $f$ does not follow from $f in W^1,p(Omega)$, since $p<n$; this is an additional assumption I am adding.
real-analysis sobolev-spaces regularity-theory-of-pdes weak-derivatives
asked Jul 30 at 9:06
Asaf Shachar
4,4823832
Let $Omega subseteq mathbbR^n$ be an open bounded domain, and let $1<p<n$. Suppose that $f in W^1,p(Omega)$ is continuous*, and $g in C^infty(mathbbR)$.
Is it true that $g circ f in W^1,p_loc(Omega)$?
My guess was that the answer is positive, and that $partial_i (g circ f)(x)=g'(f(x)) partial_i f(x)$ but a naive calculation to prove it failed.
*Note that the continuity of $f$ does not follow from $f in W^1,p(Omega)$, since $p<n$; this is an additional assumption I am adding.
real-analysis sobolev-spaces regularity-theory-of-pdes weak-derivatives
asked Jul 30 at 9:06
Asaf Shachar
4,4823832
asked Jul 30 at 9:06
Asaf Shachar
4,4823832
asked Jul 30 at 9:06
Asaf Shachar
4,4823832
asked Jul 30 at 9:06
Asaf Shachar
4,4823832
4,4823832
Isn't there some $|x|^θ$ that is in $W^1,p$, then using some $g(x) = |x|^phi$ should change the $p$
– Calvin Khor
Jul 30 at 9:27
the spike function only lives in local $W^1,p$ though
– Calvin Khor
Jul 30 at 9:33
the function i mentioned earlier, $|x|^-θ$ for $θ < (n-p)/p$
– Calvin Khor
Jul 30 at 9:36
You can use this result: math.stackexchange.com/questions/1110231/…
– Bob
Jul 30 at 11:11
@Bob Yes, but that assumes $g'$ is bounded and works also for $f$ that are not continuous.
– Calvin Khor
Jul 30 at 11:35
 |Â
show 4 more comments
Isn't there some $|x|^θ$ that is in $W^1,p$, then using some $g(x) = |x|^phi$ should change the $p$
– Calvin Khor
Jul 30 at 9:27
the spike function only lives in local $W^1,p$ though
– Calvin Khor
Jul 30 at 9:33
the function i mentioned earlier, $|x|^-θ$ for $θ < (n-p)/p$
– Calvin Khor
Jul 30 at 9:36
You can use this result: math.stackexchange.com/questions/1110231/…
– Bob
Jul 30 at 11:11
@Bob Yes, but that assumes $g'$ is bounded and works also for $f$ that are not continuous.
– Calvin Khor
Jul 30 at 11:35
Isn't there some $|x|^θ$ that is in $W^1,p$, then using some $g(x) = |x|^phi$ should change the $p$
– Calvin Khor
Jul 30 at 9:27
Isn't there some $|x|^θ$ that is in $W^1,p$, then using some $g(x) = |x|^phi$ should change the $p$
– Calvin Khor
Jul 30 at 9:27
the spike function only lives in local $W^1,p$ though
– Calvin Khor
Jul 30 at 9:33
the spike function only lives in local $W^1,p$ though
– Calvin Khor
Jul 30 at 9:33
the function i mentioned earlier, $|x|^-θ$ for $θ < (n-p)/p$
– Calvin Khor
Jul 30 at 9:36
the function i mentioned earlier, $|x|^-θ$ for $θ < (n-p)/p$
– Calvin Khor
Jul 30 at 9:36
You can use this result: math.stackexchange.com/questions/1110231/…
– Bob
Jul 30 at 11:11
You can use this result: math.stackexchange.com/questions/1110231/…
– Bob
Jul 30 at 11:11
@Bob Yes, but that assumes $g'$ is bounded and works also for $f$ that are not continuous.
– Calvin Khor
Jul 30 at 11:35
@Bob Yes, but that assumes $g'$ is bounded and works also for $f$ that are not continuous.
– Calvin Khor
Jul 30 at 11:35
 |Â
show 4 more comments
1 Answer
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The answer is positive. Indeed, we have the following version of the chain rule in Sobolev spaces:
Assume $F : mathbbR to mathbbR$ is $C^1$, with $F'$ bounded. Suppose $U$ is bounded and $u in W^1,p(U)$ for some $1 le p < infty$. Then $$v :=F circ u in W^1,p(U) quad textand the weak derivatives satisfy quad v_x_i=F'(u)u_x_i.$$
As stated this theorem does not help us. However, inspecting its proof we see that it only uses the following fact:
There exist approximating functions $u_k in C^infty$ (i.e. $u_k to u$ in $W^1,p$) such that $F'$ is bounded in a ball containing $textImage(u_k),textImage(u)$ for all sufficiently large $k$.
In our case, $u$ is continuous hence bounded on compact subsets. Hence, there are approximations $u_k$ which are also uniformly bounded for sufficiently large $k$. (Think of the standard density proof, via convolution with mollifiers). Recall we are only looking for a local result.
From here essentially the same proof should work.
answered Aug 5 at 14:05
Asaf Shachar
4,4823832
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
The answer is positive. Indeed, we have the following version of the chain rule in Sobolev spaces:
Assume $F : mathbbR to mathbbR$ is $C^1$, with $F'$ bounded. Suppose $U$ is bounded and $u in W^1,p(U)$ for some $1 le p < infty$. Then $$v :=F circ u in W^1,p(U) quad textand the weak derivatives satisfy quad v_x_i=F'(u)u_x_i.$$
As stated this theorem does not help us. However, inspecting its proof we see that it only uses the following fact:
There exist approximating functions $u_k in C^infty$ (i.e. $u_k to u$ in $W^1,p$) such that $F'$ is bounded in a ball containing $textImage(u_k),textImage(u)$ for all sufficiently large $k$.
In our case, $u$ is continuous hence bounded on compact subsets. Hence, there are approximations $u_k$ which are also uniformly bounded for sufficiently large $k$. (Think of the standard density proof, via convolution with mollifiers). Recall we are only looking for a local result.
From here essentially the same proof should work.
answered Aug 5 at 14:05
Asaf Shachar
4,4823832
add a comment |Â
up vote
0
down vote
accepted
The answer is positive. Indeed, we have the following version of the chain rule in Sobolev spaces:
Assume $F : mathbbR to mathbbR$ is $C^1$, with $F'$ bounded. Suppose $U$ is bounded and $u in W^1,p(U)$ for some $1 le p < infty$. Then $$v :=F circ u in W^1,p(U) quad textand the weak derivatives satisfy quad v_x_i=F'(u)u_x_i.$$
As stated this theorem does not help us. However, inspecting its proof we see that it only uses the following fact:
There exist approximating functions $u_k in C^infty$ (i.e. $u_k to u$ in $W^1,p$) such that $F'$ is bounded in a ball containing $textImage(u_k),textImage(u)$ for all sufficiently large $k$.
In our case, $u$ is continuous hence bounded on compact subsets. Hence, there are approximations $u_k$ which are also uniformly bounded for sufficiently large $k$. (Think of the standard density proof, via convolution with mollifiers). Recall we are only looking for a local result.
From here essentially the same proof should work.
answered Aug 5 at 14:05
Asaf Shachar
4,4823832
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
The answer is positive. Indeed, we have the following version of the chain rule in Sobolev spaces:
Assume $F : mathbbR to mathbbR$ is $C^1$, with $F'$ bounded. Suppose $U$ is bounded and $u in W^1,p(U)$ for some $1 le p < infty$. Then $$v :=F circ u in W^1,p(U) quad textand the weak derivatives satisfy quad v_x_i=F'(u)u_x_i.$$
As stated this theorem does not help us. However, inspecting its proof we see that it only uses the following fact:
There exist approximating functions $u_k in C^infty$ (i.e. $u_k to u$ in $W^1,p$) such that $F'$ is bounded in a ball containing $textImage(u_k),textImage(u)$ for all sufficiently large $k$.
In our case, $u$ is continuous hence bounded on compact subsets. Hence, there are approximations $u_k$ which are also uniformly bounded for sufficiently large $k$. (Think of the standard density proof, via convolution with mollifiers). Recall we are only looking for a local result.
From here essentially the same proof should work.
answered Aug 5 at 14:05
Asaf Shachar
4,4823832
The answer is positive. Indeed, we have the following version of the chain rule in Sobolev spaces:
Assume $F : mathbbR to mathbbR$ is $C^1$, with $F'$ bounded. Suppose $U$ is bounded and $u in W^1,p(U)$ for some $1 le p < infty$. Then $$v :=F circ u in W^1,p(U) quad textand the weak derivatives satisfy quad v_x_i=F'(u)u_x_i.$$
As stated this theorem does not help us. However, inspecting its proof we see that it only uses the following fact:
There exist approximating functions $u_k in C^infty$ (i.e. $u_k to u$ in $W^1,p$) such that $F'$ is bounded in a ball containing $textImage(u_k),textImage(u)$ for all sufficiently large $k$.
In our case, $u$ is continuous hence bounded on compact subsets. Hence, there are approximations $u_k$ which are also uniformly bounded for sufficiently large $k$. (Think of the standard density proof, via convolution with mollifiers). Recall we are only looking for a local result.
From here essentially the same proof should work.
answered Aug 5 at 14:05
Asaf Shachar
4,4823832
answered Aug 5 at 14:05
Asaf Shachar
4,4823832
answered Aug 5 at 14:05
Asaf Shachar
4,4823832
answered Aug 5 at 14:05
Asaf Shachar
4,4823832
4,4823832
add a comment |Â
add a comment |Â
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Isn't there some $|x|^θ$ that is in $W^1,p$, then using some $g(x) = |x|^phi$ should change the $p$
– Calvin Khor
Jul 30 at 9:27
the spike function only lives in local $W^1,p$ though
– Calvin Khor
Jul 30 at 9:33
the function i mentioned earlier, $|x|^-θ$ for $θ < (n-p)/p$
– Calvin Khor
Jul 30 at 9:36
You can use this result: math.stackexchange.com/questions/1110231/…
– Bob
Jul 30 at 11:11
@Bob Yes, but that assumes $g'$ is bounded and works also for $f$ that are not continuous.
– Calvin Khor
Jul 30 at 11:35