In some online notes regarding the equivalence of norms on Sobolev spaces, I came across the following point:
"In the case $W_0^{1,2}(\Omega)$, one does not need the regularity of the boundary."
I checked Evans to see if I could gain some more insight into this point, and found the following on page 280:
"In view of this estimate, on $W_0^{1,2}(\Omega)$ the norm $\|Du\|_{L_2(\Omega)}$ is equivalent to $\|u\|_{W^{1,2}(\Omega)}$, if $\Omega$ is bounded"
where the estimate in question is Poincare's inequality; $D$ here denotes the gradient operator. I also consider $\Omega$ to be open as well as bounded in $\mathbb R^n$.
My problems: (i) I don't understand the part from the first quote which reads "we do not need the regularity of the boundary", and (ii), how this relates to the fact that, on $W_0^{1,2}(\Omega)$ one doesn't need to consider the contribution from $\|u\|_{L_2(\Omega)}$ in the $W^{1,2}(\Omega)$-norm, whereas one does in $W_0^{1,2}(\Omega)$. How exactly does this simplification occur? Also (iii), can the norm $\|Du\|_{L_2(\Omega)}$ ever be regarded as equivalent to $\|u\|_{W^{1,2}(\Omega)}$ on $W^{1,2}(\Omega)$, or is it only for $W_0^{1,2}(\Omega)$?
Best Answer
(1) $W^{1,2}_0$ is defined as the completion of the space of smooth, compactly supported functions. So, every function in $W^{1,2}_0$ can be approximated by a smooth function. This is not true in general for $W^{1,2}$, where one needs regularity of the boundary of $\Omega$ to get this property. Approximation by smooth functions is essential, as embedding theorems etc are first proved for smooth functions.
(2) $W^{1,2}_0$ with $\|\cdot\|_{W^{1,2}}$ is a Banach space by construction. By Poincare inequality, the norm $u\mapsto \|Du\|_{L^2}$ is equivalent.
(3) This is of course not true for $W^{1,2}$: If $u$ is a non-zero constant function, then $Du=0$. And $u\mapsto \|Du\|_{L^2}$ is not a norm. This problem does not occur in $W^{1,2}_0$, as the only constant function in $W^{1,2}_0$ is the zero function.