Let $\Omega \subseteq \mathbb R^2$ be an open, connected, bounded domain with smooth boundary.
Let $f_n \in W^{1,2}(\Omega) $ be a bounded sequence in $W^{1,2}(\Omega) $ and suppose that $f_n|_U \to f|_U$ converges strongly in $W^{1,2}(U)$ for every $U \subset \subset \Omega$ compactly contained in $\Omega$.
Is it true that $f_n \to f$ in $W^{1,2}(\Omega)$?
I guess that I am asking if we can have "concentration at the boundary" in this setting.
No, and here is a counterexample :
Take $\Omega=(0,1)^2$ and $f_{n}(x,y)$ to be the piecewise linear function that satisfies $f_{n}(0,y)=\frac{1}{n}$ and $f_{n}(x,y)=0$ for $x\ge \frac{1}{n^2}$.
Then $||f_{n}||_{L^{2}(\Omega)}=\frac{1}{2n^3}$ and $||f_{n}'||_{L^{2}(\Omega)}=1$ , but $f_{n}'(x)=0$ for $x\ge \frac{1}{n}$ and subsequently $||f_{n}||_{H^{1}([\frac{1}{n},1)\times(0,1))} = 0$.
Thus $f_{n} \rightarrow 0$ in $H^{1}(U)$ for every $U \subset \subset \Omega$ because any $U$ will eventually be contained in $[\frac{1}{n},1)\times(0,1)$ for some sufficiently large $n$.
But since $||f_{n}'||_{L^{2}(\Omega)} = 1$ for all $n$, $f_{n}$ does not converge to $f$ in $H^{1}(\Omega)$.