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 $Y\setminus X$?
Best Answer
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 $H_0^1(\Omega)$.