Let $g\in L^\infty(\Omega)$ be given, where $\Omega\subset \mathbb R^2$ is open bounded with smooth boundary.
Define, for $1<p\leq 2$, $$ u_p:=\operatorname{argmin}\left\{\int_\Omega|u-g|^pdx+\int_\Omega|\nabla u|^pdx,\,\,u\in W^{1,p}(\Omega)\right\} $$ and $$ u_1:=\operatorname{argmin}\left\{\int_\Omega|u-g|dx+|Du|(\Omega),\,\,u\in BV(\Omega)\right\} $$ where by $|Du|(\Omega)$ we denote the total variation of $u$ as a $BV$ function.
My question: could we expect, as $p\to 1$, $$ u_p\to u_1 $$ strongly in $L^1$?
Moreover, could we expect, as $p\to 1$, $$ \|\nabla u_p\|_{L^p}\to |Du_1|(\Omega) $$ too?