Let $\Omega$ be a bounded Lipschitz-domain with outward normal vector $\nu$ and let us take a look at the Sobolev-spaces $H^1:=H^1(\Omega)$, $H^1_0:=H^1_0(\Omega)$ and $H^1_{\nabla_0}:=H^1_{\nabla_0}(\Omega):=\{u\in H^1(\Omega)\ : \ \nabla u\cdot\nu=0\}$.
I know that (one type of possible characterizations) of the duals of $H^1:=H^1(\Omega)$ and $H^1_0:=H^1_0(\Omega)$ are given by
$$(H^1)^*=H^{-1}_0,\quad (H^1_0)^*=H^{-1}.$$
Now to my first question: How can we characterize the dual of $H^1_{\nabla_0}$ and how is this dual related to $H^{-1}_0$ and $H^{-1}$? I would, at least, expect $H^{-1}_0\subset(H^1_{\nabla_0})^*$ since $H^1_{\nabla_0}\subset H^1$.
My second question: Can anyone recommend a good book/paper/article/lecture notes etc. on that matter (not necessarily in regard to Sobolev spaces but to dual spaces in general)?