Given an open bounded set $U\subseteq \mathbb{R}^n$ with $C^1$-boundary, we can define the trace function $T : H^1(U) \to L^2(\partial U)$ (Theorem 1, section 5.5. in Evan's PDE).
Similarly, I wish to extend the idea of a trace function to the open set $U=(0,1)^2 \subseteq \mathbb{R}^2$. Since the boundary is not $C^1$, I will fix $\epsilon >0$ small and define $$ T_{\epsilon} : C^1(U) \to L^2\left(\partial (\epsilon, 1-\epsilon)^2\right), \quad u \mapsto u\mid_{\partial (\epsilon, 1-\epsilon)^2} $$
Is there a (relatively elementary) way to show that $T_\epsilon$ extends to a function from $H^1(U) \to L^2\left(\partial (\epsilon, 1-\epsilon)^2\right)$?
If the extension can be defined, I would like the trace function to be the "limit" of $T_\epsilon$ as $\epsilon \to 0$. For this I would have to show that given $u \in H^1(U)$, there exists a function $v \in L^2(\partial U)$ such that $T_\epsilon (u) \to v$ as $\epsilon \to 0$.
Note here that I wish to take the limit in $L^2(\partial U)$. For this, I simply have to "scale" $T_\epsilon u$ to be a function on $\partial U$. To see what I mean here, consider for instance the side $\{0\} \times [0,1]$ of $\partial U$. Then on this side, I would write $$ \int_{{0}\times [0,1]}\lvert T_\epsilon u\left(\epsilon, \epsilon + (1-2\epsilon)y\right) - v(x, y) \rvert^2 $$ Doing as such for each side of $\partial U$, we obtain an integral of $T_\epsilon u -v$ over $\partial U$. If this integral tends to $0$ as $\epsilon \to 0$, then we say that $T_\epsilon u \to v$ in $L^2(\partial U)$.
But then the question is, will $T_\epsilon u$ always converge to some function in $L^2(\partial U)$. Is there a way to prove this is the case?
I read documents where this general approach was described for more general settings. To make sense of what is going on, I am trying to make sense of everything in this relatively simple space. Any input is appreciated!