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what is $W^{-1,2}$?
Announcing the arrival of Valued Associate #679: Cesar Manara
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I have been reading about the weak Poincare lemma in the book "Linear and Nonlinear Functional Analysis with Applications" by P.G. Ciarlet.
Let $Omega$ be a simply connected domain in $mathbb R^n$.
I was wondering what the notation $H^{-1}(Omega,mathbb R^n)$ means.
Earlier he defined $H^k:=W^{k,2}$, where $W^{k,p}$ denotes the Sobolev spaces.
The theorem goes as follows:
Weak Poincare Lemma:
Let there be given a vector field $hin H^{-1}(Omega,mathbb R^n)$ satisfying
$$mathbf{curl},,h=0quadtext{in},, H^{-2}(Omega)$$
Then there exists $pin L^2(Omega)$ such that
$$mathbf{grad},, p=hquadtext{in},, H^{-1}(Omega)$$
To me it seems to be defined as follows:
$fin H^{-1}(Omega,mathbb R^n)$ if there exists $tilde fin W^{1,2}(Omega,mathbb R^n)$ such that $Dtilde f=f$.
Is this correct?
functional-analysis notation sobolev-spaces
asked Mar 23 at 12:47
Pink PantherPink Panther
398114
$endgroup$
add a comment |
$begingroup$
I have been reading about the weak Poincare lemma in the book "Linear and Nonlinear Functional Analysis with Applications" by P.G. Ciarlet.
Let $Omega$ be a simply connected domain in $mathbb R^n$.
I was wondering what the notation $H^{-1}(Omega,mathbb R^n)$ means.
Earlier he defined $H^k:=W^{k,2}$, where $W^{k,p}$ denotes the Sobolev spaces.
The theorem goes as follows:
Weak Poincare Lemma:
Let there be given a vector field $hin H^{-1}(Omega,mathbb R^n)$ satisfying
$$mathbf{curl},,h=0quadtext{in},, H^{-2}(Omega)$$
Then there exists $pin L^2(Omega)$ such that
$$mathbf{grad},, p=hquadtext{in},, H^{-1}(Omega)$$
To me it seems to be defined as follows:
$fin H^{-1}(Omega,mathbb R^n)$ if there exists $tilde fin W^{1,2}(Omega,mathbb R^n)$ such that $Dtilde f=f$.
Is this correct?
functional-analysis notation sobolev-spaces
asked Mar 23 at 12:47
Pink PantherPink Panther
398114
$endgroup$
add a comment |
$begingroup$
I have been reading about the weak Poincare lemma in the book "Linear and Nonlinear Functional Analysis with Applications" by P.G. Ciarlet.
Let $Omega$ be a simply connected domain in $mathbb R^n$.
I was wondering what the notation $H^{-1}(Omega,mathbb R^n)$ means.
Earlier he defined $H^k:=W^{k,2}$, where $W^{k,p}$ denotes the Sobolev spaces.
The theorem goes as follows:
Weak Poincare Lemma:
Let there be given a vector field $hin H^{-1}(Omega,mathbb R^n)$ satisfying
$$mathbf{curl},,h=0quadtext{in},, H^{-2}(Omega)$$
Then there exists $pin L^2(Omega)$ such that
$$mathbf{grad},, p=hquadtext{in},, H^{-1}(Omega)$$
To me it seems to be defined as follows:
$fin H^{-1}(Omega,mathbb R^n)$ if there exists $tilde fin W^{1,2}(Omega,mathbb R^n)$ such that $Dtilde f=f$.
Is this correct?
functional-analysis notation sobolev-spaces
asked Mar 23 at 12:47
Pink PantherPink Panther
398114
$endgroup$
I have been reading about the weak Poincare lemma in the book "Linear and Nonlinear Functional Analysis with Applications" by P.G. Ciarlet.
Let $Omega$ be a simply connected domain in $mathbb R^n$.
I was wondering what the notation $H^{-1}(Omega,mathbb R^n)$ means.
Earlier he defined $H^k:=W^{k,2}$, where $W^{k,p}$ denotes the Sobolev spaces.
The theorem goes as follows:
Weak Poincare Lemma:
Let there be given a vector field $hin H^{-1}(Omega,mathbb R^n)$ satisfying
$$mathbf{curl},,h=0quadtext{in},, H^{-2}(Omega)$$
Then there exists $pin L^2(Omega)$ such that
$$mathbf{grad},, p=hquadtext{in},, H^{-1}(Omega)$$
To me it seems to be defined as follows:
$fin H^{-1}(Omega,mathbb R^n)$ if there exists $tilde fin W^{1,2}(Omega,mathbb R^n)$ such that $Dtilde f=f$.
Is this correct?
functional-analysis notation sobolev-spaces
functional-analysis notation sobolev-spaces
asked Mar 23 at 12:47
Pink PantherPink Panther
398114
asked Mar 23 at 12:47
Pink PantherPink Panther
398114
edited Mar 23 at 13:05
Pink Panther
asked Mar 23 at 12:47
Pink PantherPink Panther
398114
asked Mar 23 at 12:47
Pink PantherPink Panther
398114
asked Mar 23 at 12:47
Pink PantherPink Panther
398114
398114
add a comment |
add a comment |
1 Answer
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$H^{-1}(Omega )$ is the dual space of $H^1(Omega )$. In other word, $fin H^{-1}(Omega )$ if $$sup_{substack{|varphi |leq 1\ varphi in H^1(Omega )}}left|int_Omega fvarphi right|<infty $$
answered Mar 23 at 13:52
PierrePierre
19011
$endgroup$
add a comment |
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$begingroup$
$H^{-1}(Omega )$ is the dual space of $H^1(Omega )$. In other word, $fin H^{-1}(Omega )$ if $$sup_{substack{|varphi |leq 1\ varphi in H^1(Omega )}}left|int_Omega fvarphi right|<infty $$
answered Mar 23 at 13:52
PierrePierre
19011
$endgroup$
add a comment |
$begingroup$
$H^{-1}(Omega )$ is the dual space of $H^1(Omega )$. In other word, $fin H^{-1}(Omega )$ if $$sup_{substack{|varphi |leq 1\ varphi in H^1(Omega )}}left|int_Omega fvarphi right|<infty $$
answered Mar 23 at 13:52
PierrePierre
19011
$endgroup$
add a comment |
$begingroup$
$H^{-1}(Omega )$ is the dual space of $H^1(Omega )$. In other word, $fin H^{-1}(Omega )$ if $$sup_{substack{|varphi |leq 1\ varphi in H^1(Omega )}}left|int_Omega fvarphi right|<infty $$
answered Mar 23 at 13:52
PierrePierre
19011
$endgroup$
$H^{-1}(Omega )$ is the dual space of $H^1(Omega )$. In other word, $fin H^{-1}(Omega )$ if $$sup_{substack{|varphi |leq 1\ varphi in H^1(Omega )}}left|int_Omega fvarphi right|<infty $$
answered Mar 23 at 13:52
PierrePierre
19011
answered Mar 23 at 13:52
PierrePierre
19011
answered Mar 23 at 13:52
PierrePierre
19011
answered Mar 23 at 13:52
PierrePierre
19011
19011
add a comment |
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