Let $u\in BV(I)$, $I=(0,1)$ be given. By Evans & Gariepy we know that there exists a sequence $u_n\in C^\infty(\bar I)$ such that $|u'_n|_{L^{1}(I)}\to |u|_{TV(I)}$ where we use $TV$ to denote the total variation.
My question: do I have $|u'_n|_{L^{1+1/n}(I)}\to |u|_{TV(I)}$ as well?
Thank you!