In some paper I read, it says we understand the identity
$u(x) = v(x)\ \ \ \forall \ x \in K$
in the sense
$u-v \in W_{0,\text{ loc}}^{1,2}(\mathbb{R}^n\setminus K),$
where $K \subset \subset \mathbb{R}^n.$ Unfortunately, I don't know how to interpret this. How is this a statement about the relation of $u$ and $v$ on the set $K$? We only seem to have some statement on the boundary of $K$.
Thank you for any hints in advance.
This was too long for a comment so I had to post it as an answer, even though I am not 100% sure it is correct:
I am not familiar with this notation, but it intuitively makes sense to me. You have that $W_{0}^{1,2}(\mathbb R^n \setminus K)$ is the closure of all smooth functions with compact support in $\mathbb R^n \setminus K$ with respect to the $W^{1,2}$ norm. These functions are $0$ almost everywhere on $K$, which seems to be in line with what the definition wants to achieve. Now, $W_{0, loc}^{1,2}(\mathbb R^n \setminus K)$ would be the space of functions that are locally in $W_{0}^{1,2}(\mathbb R^n \setminus K)$. In other words, these are functions such that on any compact interval they can be approximated by a smooth function with compact support in $\mathbb R^n \setminus K$ with respect to the $W^{1,2}$ norm.