Show that the function $u(x)=1 \in W^{m,\ 2}(\Omega)$, but not in $W_0^{m,\ 2}(\Omega)$ I have a problem:

For $\Omega$ be a domain in $\Bbb R^n$. Show that the function $u(x)=1 \in W^{m,\ 2}(\Omega)$, but not in $W_0^{m,\ 2}(\Omega)$, for all $m \ge 1$.

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Any help will be appreciated! Thanks!

HINT: any smooth function $u_n$ with compact support approximating $1$ well will be close to $1$ on most of $\Omega$ and close to $0$ elsewhere. This implies that there will always be a region where the approximating function has a big gradient. Try to derive a contradiction by showing that the norm of $1 - u_n$ stays bounded away frow $0$ thanks to the contribution from the gradient part of the norm.
