De Rham theorem gives the following application for distribution :
- Let $Q$ be an open subset of $\mathbb{R}^d$ and let $f$ a distribution in $D'(Q)$ satisfying :
$$ < f | \varphi>=0, \quad \forall \varphi \in D(Q), \ div \ \varphi=0 \quad \quad \quad (1)$$
Then there exists $p \in D'(\Omega)$ such as $f = \nabla p$.
- This theorem was furthered precised by Tartare who gaved the following :
if $f \in H^{-1}(Q)$ verifies $(1)$ and $Q$ is regular enough (connected and Lipschitz boundary) , there exists $p \in L^ 2(Q)$ such as $f = \nabla p$ and $p$ is unique up to an additive global constant.
- Later on, Chérif Amrouche and Vivette Girault gave an improved version for any Sobolev space :
if $f \in W^{-n,p}(Q)$ ($n \in \mathbb{N}^*$, $1 < p < \infty$) verifies $(1)$, then there exists $p \in W^{-n+1,p}(Q)$ such as $f=\nabla p$. If $Q$ is connected, $p$ is unique up to an additive constant.
Now, I wanted to know if there exists a version of this theorem where $p=1$. As an exemple, if I give myself $f \in L^1(Q)$ such as $(1)$ holds, I can found a primitive $p \in D'(Q)$ such as $f = \nabla p$. Is there any theorem that could give me some regularity about $p$ ?
Any advices and references are welcomed.