Basic exercise about Sobolev spaces let $\Omega  \subset R^n$  a open set .let $\varphi  \in H^{1}_{0} (\Omega)$ . Suppose that the  suport of $\varphi$ is compact. By definition , there exists a sequence of functions $  \varphi_i   , i \in N$   in $C_{0}^{\infty}(\Omega)$ converging to $\varphi$ in $H^{1}(\Omega)$ .
Consider an open set ${\Omega}^{'}$  satisfying (this set exists because the suport of $\varphi$ is compact) : 
$ \operatorname{supp}\varphi \subset {\Omega}^{'}  $
and $\overline{{\Omega}^{'}}  \subset \Omega$  
The affirmation " there exists $i_0 \in N$ such that   $i  \geq i_0 $ implies $ \operatorname{supp}\varphi_i \subset {\Omega}^{'}$  " is true       ?
My professor said that the affirmation is true . the hint of my professor is:
Suppose the affirmation is not true. since $\varphi_i   , i \in N$ converges to $\varphi$ in $H^{1}(\Omega)$, then $\varphi_i \rightarrow    \varphi$ a.e. the negation of your thesis contradicts this $"\varphi_i \rightarrow    \varphi$ a.e."
I dont know how to prove the affirmation..... someone can give me a hint ?
Thank you ( my english is terrible , sorry ) 
 A: I am thinking that the claim is false...
Let $\Omega=(-2,2)\subset\mathbb R$ and $\Omega'=(-\frac12,\frac12)$. Choose $\varphi=0\in H_0^1(\Omega)$ and
$$
  \varphi_n(x)
~=~
  \frac1n
  e^{-\frac{1}{x^2-1}}
  \chi_{(-1,1)}(x)
$$
where $\chi_A$ is the characteristic function of the set $A$, i.e.
$$
  \varphi_n(x)
~=~
  \begin{cases}
    \frac1ne^{-\frac{1}{x^2-1}} & \text{if }|x|< 1 \\[4pt]
    0 & \text{if }|x|\geq 1
  \end{cases}
$$
Now, clearly $\varphi_n\in C^\infty_c(\Omega)$ converges to $0$ in $L^2(\Omega)$. Besides,
$$
  \varphi_n'(x)
~=~
  -\frac1n
  \frac{2x}{(x^2-1)^2}
  e^{-\frac{1}{x^2-1}}
  \chi_{(-1,1)}(x)
$$
so $\varphi_n'$ too converges to $0=\varphi'$ in $L^2(\Omega)$. (To see this, just notice that you have convergence in $L^\infty$ too in both cases).
So, $\varphi_n\to0$ in $H^1(\Omega)$ but ${\sf supp}\varphi_n = [-1,1]$ for every $n$...
A: I think that the affirmation does not hold. Let's have some $\varphi  \in H^{1}_{0} (\Omega)$ and sequence of functions $\varphi_i$ converging to $\varphi$ in $H^1(\Omega)$. Assume that $\varphi_i(x),\varphi(x)\geq 0$ for all $x \in \Omega$
Than functions $$\psi_i(x) = \varphi_i(x)+\frac{e^{-||x||^2}}{i}$$
converges to $\varphi$ in $H^1$ and support of $\psi_i$ is $\Omega$
