Examples of Sobolev Spaces

Question

I would like to apologize in advance for this trivial question!

Does constant functions $u \equiv C$ and, in partucular, $u \equiv 0$ belong to $W_{0}^{1,2}(\Omega)$?

Update 2:

$\Omega \subset \mathbb{R}^n$ is bounded open set. I wonder whether $\exists \{u_i\}_{0}^{\infty} \subset C_{0}^{\infty},\mbox{ such that } u_i \rightarrow u \mbox{ in } W^{1,2}$?

Answer 1

You can characterize $W^{1,2}_0$ as "the 'compactly supported' $L^2$ functions that have $L^2$ weak derivatives." Since the weak derivatives of a constant function are all zero, hence $L^2$, the answer to your question is the answer to, "are constant functions compactly supported $L^2$?" This is never true unless the constant is zero.

Answer 2

Q1. Non-zero constant functions DO NOT BELONG to $W_0^{1,2}(\Omega)$.

Q2. The space $W_0^{1,2}(\Omega)$ is defined as the «completion» of $C_0^\infty(\Omega)$ with norm $$ \|u\|_{W^{1,2}_0(\Omega)}=\left(\int_\Omega\big(u^2(x)+\big|\nabla u(x)\big|^2\big)\,dx\right)^{1/2}. $$ Thus, indeed, every element of $W_0^{1,2}(\Omega)$ can be $\|\cdot\|_{W^{1,2}_0(\Omega)}-$approximated by functions in $C_0^\infty(\Omega)$, even if $\Omega$ is unbounded.