# Characterization of Sobolev spaces

Assume that for some smooth bounded open subset $\Omega$ in $\mathbb{R}^n$ and some $u\in H^{1,2}(\mathbb{R}^n)$ we know that $\chi_{\Omega} u\in H^{1,2}(\mathbb{R}^n)$. Is it then true that $u\in H^{1,2}_{0}(\Omega)$? Here $\chi_{\Omega}$ denotes the characteristic function which is one on $\Omega$ and zero elsewhere.

It is true that $$H_0^{1,2}(\Omega)= \bigl\{u\bigr|_\Omega\bigm| u\in H^{1,2}(\mathbb R^n),\, \operatorname{supp}u\subseteq\overline\Omega\bigr\}.$$ Indeed, the inclusion $\subseteq$ is obvious. To see the inclusion $\supseteq$, notice that, for $u\in H^{1,2}(\mathbb R^n)$, taking the trace $u\bigr|_{\partial\Omega}$ from either side of $\partial\Omega$ yields the same result, and if $\operatorname{supp}u\subseteq \overline\Omega$, then $\bigl(u\bigr|_\Omega\bigr)\bigr|_{\partial\Omega} = \bigl(u\bigr|_{\mathbb R^n\setminus\overline\Omega}\bigr)\bigr|_{\partial\Omega}=0$.