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Weak* convergence in $W^1,infty(Omega)$
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Let $Omega$ be a bounded domain in $Bbb R^m$ with smooth enough boundary so that $W^1,infty(Omega)=textLip(Omega)$.
Let $(u_n)$ be a sequence in $W^1,infty(Omega)$. What does it mean for the sequence to converge weak*-ly in $W^1,infty(Omega)$?
I don't know what the pre-dual of $W^1,infty(Omega)$ is. I know that for $p'=p/(p-1)$,
$$
W_0^k,p(Omega)^* = W^-k,p'(Omega)
$$
but I don't know if it makes sense to talk about $W_0^-1,1(Omega)$ or what it is, if it exists at all.
I would guess that $u_noverset*rightharpoonup u$ in $W^1,infty$means that
$$
int_Omega u_n f + nabla u_ncdot mathbf g dx to int_Omega u f + nabla ucdot mathbf g dx
$$
for all $fin L^1(Omega)$ and all $mathbf gin L^1(Omega;Bbb R^m)$. Even if this is true I still want to know the name of this space and how we know that its dual is indeed $W^1,infty(Omega)$.
functional-analysis pde sobolev-spaces weak-convergence
asked Jul 27 at 7:48
BigbearZzz
5,70311344
add a comment |Â
up vote
1
down vote
favorite
Let $Omega$ be a bounded domain in $Bbb R^m$ with smooth enough boundary so that $W^1,infty(Omega)=textLip(Omega)$.
Let $(u_n)$ be a sequence in $W^1,infty(Omega)$. What does it mean for the sequence to converge weak*-ly in $W^1,infty(Omega)$?
I don't know what the pre-dual of $W^1,infty(Omega)$ is. I know that for $p'=p/(p-1)$,
$$
W_0^k,p(Omega)^* = W^-k,p'(Omega)
$$
but I don't know if it makes sense to talk about $W_0^-1,1(Omega)$ or what it is, if it exists at all.
I would guess that $u_noverset*rightharpoonup u$ in $W^1,infty$means that
$$
int_Omega u_n f + nabla u_ncdot mathbf g dx to int_Omega u f + nabla ucdot mathbf g dx
$$
for all $fin L^1(Omega)$ and all $mathbf gin L^1(Omega;Bbb R^m)$. Even if this is true I still want to know the name of this space and how we know that its dual is indeed $W^1,infty(Omega)$.
functional-analysis pde sobolev-spaces weak-convergence
asked Jul 27 at 7:48
BigbearZzz
5,70311344
The preduals of Lip(X) are discussed in this paper
– user357151
Jul 27 at 16:19
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $Omega$ be a bounded domain in $Bbb R^m$ with smooth enough boundary so that $W^1,infty(Omega)=textLip(Omega)$.
Let $(u_n)$ be a sequence in $W^1,infty(Omega)$. What does it mean for the sequence to converge weak*-ly in $W^1,infty(Omega)$?
I don't know what the pre-dual of $W^1,infty(Omega)$ is. I know that for $p'=p/(p-1)$,
$$
W_0^k,p(Omega)^* = W^-k,p'(Omega)
$$
but I don't know if it makes sense to talk about $W_0^-1,1(Omega)$ or what it is, if it exists at all.
I would guess that $u_noverset*rightharpoonup u$ in $W^1,infty$means that
$$
int_Omega u_n f + nabla u_ncdot mathbf g dx to int_Omega u f + nabla ucdot mathbf g dx
$$
for all $fin L^1(Omega)$ and all $mathbf gin L^1(Omega;Bbb R^m)$. Even if this is true I still want to know the name of this space and how we know that its dual is indeed $W^1,infty(Omega)$.
functional-analysis pde sobolev-spaces weak-convergence
asked Jul 27 at 7:48
BigbearZzz
5,70311344
Let $Omega$ be a bounded domain in $Bbb R^m$ with smooth enough boundary so that $W^1,infty(Omega)=textLip(Omega)$.
Let $(u_n)$ be a sequence in $W^1,infty(Omega)$. What does it mean for the sequence to converge weak*-ly in $W^1,infty(Omega)$?
I don't know what the pre-dual of $W^1,infty(Omega)$ is. I know that for $p'=p/(p-1)$,
$$
W_0^k,p(Omega)^* = W^-k,p'(Omega)
$$
but I don't know if it makes sense to talk about $W_0^-1,1(Omega)$ or what it is, if it exists at all.
I would guess that $u_noverset*rightharpoonup u$ in $W^1,infty$means that
$$
int_Omega u_n f + nabla u_ncdot mathbf g dx to int_Omega u f + nabla ucdot mathbf g dx
$$
for all $fin L^1(Omega)$ and all $mathbf gin L^1(Omega;Bbb R^m)$. Even if this is true I still want to know the name of this space and how we know that its dual is indeed $W^1,infty(Omega)$.
functional-analysis pde sobolev-spaces weak-convergence
asked Jul 27 at 7:48
BigbearZzz
5,70311344
edited Jul 27 at 17:17
asked Jul 27 at 7:48
BigbearZzz
5,70311344
asked Jul 27 at 7:48
BigbearZzz
5,70311344
asked Jul 27 at 7:48
BigbearZzz
5,70311344
5,70311344
The preduals of Lip(X) are discussed in this paper
– user357151
Jul 27 at 16:19
add a comment |Â
The preduals of Lip(X) are discussed in this paper
– user357151
Jul 27 at 16:19
The preduals of Lip(X) are discussed in this paper
– user357151
Jul 27 at 16:19
The preduals of Lip(X) are discussed in this paper
– user357151
Jul 27 at 16:19
add a comment |Â
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The preduals of Lip(X) are discussed in this paper
– user357151
Jul 27 at 16:19