I have a question regarding a decomposition of a vector field. So fix $ 1<p<\infty$ and let $ \Omega$ denote a smooth bounded domain in $ R^N$. Now let $ F $ denote a smooth vector field $F:\Omega \rightarrow R^N$ with $ \|F\|_{L^p}=1$. Now consider $\Delta \phi =div(F)$ in $\Omega$ with $ \phi=0$ on $ \partial \Omega$.
My question is: does there exist some $C_p$ (independent of $F$ and $ \phi$) such that $ \| \nabla \phi \|_{L^p} \le C_p$?
When $p=2$ this is fairly obvious. Can one prove it for another range of $p$ by some interpolation or ???
My real question here is given $ F \in L^p(\Omega, R^N)$ can we decompose $F= \nabla \phi +g$ where $ \phi$ is as above and $div(g)=0$ and where the projections define by $ F \mapsto \nabla \phi$ and $ F \mapsto g$ are continuous in $L^p$.
I was convinced at one point this was always true and I had easy proof...but now I not so sure. I have tried googling Helmholtz and Hodge decomposition but i don't appear to be getting what i want.
thanks.
