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 $h\in H^{-1}(\Omega,\mathbb R^n)$ satisfying
$$\mathbf{curl}\,\,h=0\quad\text{in}\,\, H^{-2}(\Omega)$$
Then there exists $p\in L^2(\Omega)$ such that
$$\mathbf{grad}\,\, p=h\quad\text{in}\,\, H^{-1}(\Omega)$$
To me it seems to be defined as follows:
$f\in H^{-1}(\Omega,\mathbb R^n)$ if there exists $\tilde f\in W^{1,2}(\Omega,\mathbb R^n)$ such that $D\tilde f=f$.
Is this correct?
Best Answer
$H^{-1}(\Omega )$ is the dual space of $H^1(\Omega )$. In other word, $f\in H^{-1}(\Omega )$ if $$\sup_{\substack{\|\varphi \|\leq 1\\ \varphi \in H^1(\Omega )}}\left|\int_\Omega f\varphi \right|<\infty $$