Let $\Omega$ be a domain in $\mathbb R^n$ (say, with smooth boundary for simplicity). I am looking for a reference for the fact that if $f \in H^{-1}$ (that is, the dual of $H^1 = W^{1,2}$) then, there is a weak solution $u \in H^1$ of $\Delta u = f$ in $\Omega$, $u = 0$ on $\partial \Omega$. In order words, the operator $\Delta:H^1_0 \to H^{-1}$ is an isomorphism.
I have found in many books the equivalent statement for $\Delta:H^1_0 \cap H^{k}\to H^{k-2}$ for $k \geq 2$, but not for $k=1$. Theorem 2.44 in these lecture notes gives what I want, but I would like to have a proper reference from a book/paper that I could cite. If you know of a reference dealing with the analogous question on compact manifolds, it would be even better.
Thanks in advance!