Let $B_0(1)$ be the unit ball in $\mathbb R^n$, $n\geq2$. $h\in W_0^{1,2}(B_0(1))$. For $r\in (0,1)$, define a function $f_r(x):[0,1]\rightarrow \mathbb R$ by \begin{equation} f_r(x):= \begin{cases} 1,&\text{when} \ x\in(0,r],\\ \frac{1-x}{1-r},& \text{when} \ x\in(r,1]. \end{cases} \end{equation} Let $g_r(x):B_0(1)\rightarrow \mathbb R$ to be $g_r(x):=f_r(|x|)\cdot h(x)$.

Question: Can we prove that $\int_{B_0(1)\backslash B_0(r)}|\nabla g_r|^2dx\to0$, as $r\to1^-$?