Let
- $d\in\mathbb N$
- $\lambda$ denote the Lebesgue measure on $\mathbb R^d$
- $\Lambda\subseteq\mathbb R^d$ be bounded, convex, nonempty and open
- $p\in L^2(\Lambda)$ with $$\int_\Lambda p\:{\rm d}\lambda=0\tag1$$
I want to show that there is a unique $v\in H_0^1(\Lambda,\mathbb R^d)$ with $$\nabla\cdot v=p\tag2$$ and $$\left\|v\right\|_{H^1(\Lambda,\:\mathbb R^d)}\le C\left\|p\right\|_{L^2(\Lambda)}\tag3\;.$$
By Theorem 3.2.1.2 of Elliptic Problems in Nonsmooth Domains by Pierre Grisvard, there is a unique $u\in H_0^1(\Lambda)\cap H^2(\Lambda)$ with $$\Delta u=p\tag4\;.$$ So, $$v:=\nabla u$$ is a solution of $(2)$ in $H^1(\Lambda,\mathbb R^d)$.
Are we able to show $v\in H_0^1(\Lambda,\mathbb R^d)$ and $(3)$? If so, how?
If this is not possible without further assumptions on $\Lambda$ or $\partial\Lambda$: Where can I find a proof of the existence of such a $v$ (which is not necessarily the special $v$ constructed above) in the case where $\Lambda$ is a polyhedron?