This is cross-posted on MSE: <https://math.stackexchange.com/q/1596565/9464>


In the *Partial Differential Equations* by Evans (2nd edition p299), $H^{-1}(\Omega)$ denotes the dual space to $H^1\_0(\Omega)$ where $\Omega$ is an open subset of $\mathbb{R}^n$ and $H^1(\Omega)=W^{1,2}(\Omega)$, $H^1\_0(\Omega)=W^{1,2}\_0(\Omega)$:


$$
W^{1,2}\_0(\Omega)=\overline{C\_c^\infty(\Omega)}^{\|\cdot\|\_{W^{1,2}(\Omega)}}
$$


While in the *Navier Stokes Equations* by Constantin and Foias (p7), $H^{-1}(\Omega)$ denotes the dual space of $H^1(\Omega)$. 


Let $X$ be the (continuous) dual of $H^1(\Omega)$ and $Y$ the dual of $H^1\_0(\Omega)$. One has that $X\subset Y$. 


Could anybody give a quick example in $Y\setminus X$?



Difference between the dual space of $H^1(\Omega)$ and the dual of $H^1_0(\Omega)$
In general, $X$ does not embed into $Y$. Indeed, suppose that $\partial \Omega$ is sufficiently smooth. Then there is the trace map $\gamma\_0 \colon H^1(\Omega) \to H^{1/2}(\partial\Omega)$, $u\mapsto u\bigr|\_{\partial\Omega}$, which fits into a short exact sequence
$$
0 \longrightarrow H\_0^1(\Omega) \longrightarrow H^1(\Omega) \xrightarrow{ \ \gamma\_0 \ } H^{1/2}(\partial\Omega) \longrightarrow 0.
$$ 
By duality, one gets a *surjective* map $X\twoheadrightarrow Y$, $\Phi\mapsto \Phi\bigr|\_{H\_0^1(\Omega)}$, and an embedding $H^{-1/2}(\partial\Omega)= H^{1/2}(\partial\Omega)'\to X$, $\Psi \mapsto \langle\Psi,\gamma\_0 (\cdot)\rangle$. Functionals $\Psi\in H^{-1/2}(\partial\Omega)$ regarded as elements of $X$ are zero when restricted to $H\_0^1(\Omega)$.


