This is in connection to this question. I understand the solution, but I want to ask something else regarding the extension of the function. The question is like this:
Suppose that $v$ is a positive real function with $v \in H^1(\Omega)$ and there is a ball $B$ such that $v$ has an extension in $w \in H^1(B)$. Is it true that the extension $w$ can be chosen to be also positive?
I searched a bit and didn't find a theorem about this matter.