Compact support of $W^{1,p}_0$ functions

Question

I know that the space $W^{1,p}_0(\Omega)$ is defined as the closure of $C^\infty_0(\Omega)$ in the $\|\cdot\|_{W^{1,p}}$ norm. Do functions in $W^{1,p}_0(\Omega)$ themselves have compact support in $\Omega$ (as the functions in $C^\infty_0(\Omega)$)?

Answer 1

A simple counterexample. Take $\Omega = B_1 \subseteq \mathbb{R}^n$ (unit ball) and a function $u \colon \mathbb{R}^n \to \mathbb{R}$ whose support is exactly $\overline{B_1}$. The $u$ doesn't lie in $C_0^\infty(\Omega)$ but each of the functions $u_\lambda(x) := u(\lambda x)$ for $\lambda > 1$ does, as $\operatorname{supp} u_\lambda = \overline{B_{1/\lambda}}$. And it's easy to convince yourself that $u_\lambda \to u$ in $W^{1,p}(\Omega)$ as $\lambda \to 1$.