Approximation of a $W^{1,2}(Omega,mathbb{S}^2)$ function
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Suppose $Omega subset mathbb{R}^3$ is bounded domain with smooth boundary and $u in W^{1,2}(Omega,mathbb{S}^2)$ such that $|u|=1$ a.e. in $Omega$. It is known that there is a sequence $u_n in C^1(overline{Omega})$ such that $u_n to u$ in $W^{1,2}$ norm.
Is it possible to find such a sequence with $|u_n|=1$ for all $n$ in $Omega$ (or just $|u_n|=1$ on $partial Omega$)?
I found that the construction of such sequence is the use of mollifier but I can't figure out how to make $u_n$ satisfies $|u_n|=1$.
Also, I have tried to show that $u_n/|u_n|$ converges to $u$ in $W^{1,2}$ norm, but not success.
Thank you!
functional-analysis sobolev-spaces
asked Dec 13 '18 at 10:48
mnmn1993mnmn1993
402413
$endgroup$
add a comment |
$begingroup$
Suppose $Omega subset mathbb{R}^3$ is bounded domain with smooth boundary and $u in W^{1,2}(Omega,mathbb{S}^2)$ such that $|u|=1$ a.e. in $Omega$. It is known that there is a sequence $u_n in C^1(overline{Omega})$ such that $u_n to u$ in $W^{1,2}$ norm.
Is it possible to find such a sequence with $|u_n|=1$ for all $n$ in $Omega$ (or just $|u_n|=1$ on $partial Omega$)?
I found that the construction of such sequence is the use of mollifier but I can't figure out how to make $u_n$ satisfies $|u_n|=1$.
Also, I have tried to show that $u_n/|u_n|$ converges to $u$ in $W^{1,2}$ norm, but not success.
Thank you!
functional-analysis sobolev-spaces
asked Dec 13 '18 at 10:48
mnmn1993mnmn1993
402413
$endgroup$
1
$begingroup$
You can find some ideas in this paper: hal.archives-ouvertes.fr/hal-00747679/document See also: pitt.edu/~hajlasz/OriginalPublications/…
$endgroup$
– Siminore
Dec 13 '18 at 13:21
$begingroup$
Thank you so much! So I think my statement is true according to Theorem 2.4 in the second paper?
$endgroup$
– mnmn1993
Dec 13 '18 at 13:46
add a comment |
$begingroup$
Suppose $Omega subset mathbb{R}^3$ is bounded domain with smooth boundary and $u in W^{1,2}(Omega,mathbb{S}^2)$ such that $|u|=1$ a.e. in $Omega$. It is known that there is a sequence $u_n in C^1(overline{Omega})$ such that $u_n to u$ in $W^{1,2}$ norm.
Is it possible to find such a sequence with $|u_n|=1$ for all $n$ in $Omega$ (or just $|u_n|=1$ on $partial Omega$)?
I found that the construction of such sequence is the use of mollifier but I can't figure out how to make $u_n$ satisfies $|u_n|=1$.
Also, I have tried to show that $u_n/|u_n|$ converges to $u$ in $W^{1,2}$ norm, but not success.
Thank you!
functional-analysis sobolev-spaces
asked Dec 13 '18 at 10:48
mnmn1993mnmn1993
402413
$endgroup$
Suppose $Omega subset mathbb{R}^3$ is bounded domain with smooth boundary and $u in W^{1,2}(Omega,mathbb{S}^2)$ such that $|u|=1$ a.e. in $Omega$. It is known that there is a sequence $u_n in C^1(overline{Omega})$ such that $u_n to u$ in $W^{1,2}$ norm.
Is it possible to find such a sequence with $|u_n|=1$ for all $n$ in $Omega$ (or just $|u_n|=1$ on $partial Omega$)?
I found that the construction of such sequence is the use of mollifier but I can't figure out how to make $u_n$ satisfies $|u_n|=1$.
Also, I have tried to show that $u_n/|u_n|$ converges to $u$ in $W^{1,2}$ norm, but not success.
Thank you!
functional-analysis sobolev-spaces
functional-analysis sobolev-spaces
asked Dec 13 '18 at 10:48
mnmn1993mnmn1993
402413
asked Dec 13 '18 at 10:48
mnmn1993mnmn1993
402413
edited Dec 13 '18 at 13:15
mnmn1993
asked Dec 13 '18 at 10:48
mnmn1993mnmn1993
402413
asked Dec 13 '18 at 10:48
mnmn1993mnmn1993
402413
asked Dec 13 '18 at 10:48
mnmn1993mnmn1993
402413
402413
1
$begingroup$
You can find some ideas in this paper: hal.archives-ouvertes.fr/hal-00747679/document See also: pitt.edu/~hajlasz/OriginalPublications/…
$endgroup$
– Siminore
Dec 13 '18 at 13:21
$begingroup$
Thank you so much! So I think my statement is true according to Theorem 2.4 in the second paper?
$endgroup$
– mnmn1993
Dec 13 '18 at 13:46
add a comment |
1
$begingroup$
You can find some ideas in this paper: hal.archives-ouvertes.fr/hal-00747679/document See also: pitt.edu/~hajlasz/OriginalPublications/…
$endgroup$
– Siminore
Dec 13 '18 at 13:21
$begingroup$
Thank you so much! So I think my statement is true according to Theorem 2.4 in the second paper?
$endgroup$
– mnmn1993
Dec 13 '18 at 13:46
1
1
$begingroup$
You can find some ideas in this paper: hal.archives-ouvertes.fr/hal-00747679/document See also: pitt.edu/~hajlasz/OriginalPublications/…
$endgroup$
– Siminore
Dec 13 '18 at 13:21
$begingroup$
You can find some ideas in this paper: hal.archives-ouvertes.fr/hal-00747679/document See also: pitt.edu/~hajlasz/OriginalPublications/…
$endgroup$
– Siminore
Dec 13 '18 at 13:21
$begingroup$
Thank you so much! So I think my statement is true according to Theorem 2.4 in the second paper?
$endgroup$
– mnmn1993
Dec 13 '18 at 13:46
$begingroup$
Thank you so much! So I think my statement is true according to Theorem 2.4 in the second paper?
$endgroup$
– mnmn1993
Dec 13 '18 at 13:46
add a comment |
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$begingroup$
You can find some ideas in this paper: hal.archives-ouvertes.fr/hal-00747679/document See also: pitt.edu/~hajlasz/OriginalPublications/…
$endgroup$
– Siminore
Dec 13 '18 at 13:21
$begingroup$
Thank you so much! So I think my statement is true according to Theorem 2.4 in the second paper?
$endgroup$
– mnmn1993
Dec 13 '18 at 13:46