Let $\Omega\subset\mathbb R^2$ be open bounded, smooth boundary. Given a sequence $(u_\epsilon)\subset BV(\Omega)$ such that $$ \|u_\epsilon-u_0\|_{L^2}^2+\epsilon |u_\epsilon|_{TV(\Omega)}\to 0 $$ as $\epsilon\to 0$, where $u_0\in BV(\Omega)$ as well, and by $|\cdot|_{TV(\Omega)}$ we denote the total variation of $u$.
My question: can we show that $u_\epsilon \to u_0$ weakly in $BV$?
My try: It is clear that $u_\epsilon\to u_0$ in $L^2$ and $\liminf_{\epsilon\to 0}|u_\epsilon|_{TV}\geq |u_0|_{TV}$. But I can not get an upper bounded of the total variation of $u_\epsilon$...
Sure, because there isn't one. The condition you stated allows the functions $u_\epsilon$ to have total variation of size $\epsilon^{-1/2}$; you can obtain such functions by adding small zigzags to $u_0$ (small in the $L^2$ norm, but large in the $BV$ norm). Then the sequence $u_\epsilon$ is unbounded in BV, hence is not weakly convergent.