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 $Y\setminus X$?