Uniqueness of weak derivative

Question

In this result, I understand almost everything but I don't understand why we using
$\Omega' \Subset \Omega$ what is the major role of this why not we directly use $\Omega$.

thank you

Answer 1

You must use $\Omega' \Subset \Omega$ because $v, \tilde v$ are just assumed to be locally integrable $L^1$-functions (hence the notation $L^1_\text{loc}(\Omega)$), which means by definition that their restrictions to $\Omega'$ belong to $L^p(\Omega')$ for every compact subset $\Omega' \Subset \Omega$. $L^p_\text{loc}(\Omega)$ is strictly larger than $L^p(\Omega)$, e.g. $1 \in L^p_\text{loc}(\mathbb R^n) \setminus L^p(\mathbb R^n)$ or for bounded domains a function may explode near the boundary while remaining tame in the interior, e.g. $x \mapsto x^\gamma \in L^p_\text{loc}(0,1)$ for all $\gamma \in \mathbb R$ while for $\gamma < 0$ small (depending on $p$) $x\mapsto x^\gamma \not \in L^p(0,1)$.