When sequence of compactly supported functions bounded in $L^2$?

Question

Assume $d>1.$ Let $f:\mathbb R^d \to \mathbb C$ be a smooth radial function supported on $\{x: \frac{1}{2}<|x|<1 \},$ $f(x)=1$ on $\frac{3}{4}-\epsilon <|x|< \frac{3}{4} + \epsilon $ for small $\epsilon >0,$ and $|f|\leq 1.$ We note that there exists $\Phi: \mathbb R \to \mathbb C$ such that $f(x)= \Phi (|x|)$ for $x\in \mathbb R^d.$

We define $$f_n(x):= \Phi(|x|-n), \ n\in \mathbb N.$$

We note that $f_n$ is supported on $A_n= \{x: n+\frac{1}{2}< |x|<1+n \},$ and Lebesgue measure of $m(A_n)= \frac{1}{2^d}.$ (Please correct me if I am wrong here.) (If this true, then $\nabla f_n \in L^2(\mathbb R^d)$ for each fixed $n$.)

Question: Can we say $\{ \|\nabla f_n \|_{L^2(\mathbb R^d)} \}_{n\in \mathbb N}$ is a bounded sequence in $\mathbb R$?

Answer 1

No, $\int_{\Bbb R^d}|\nabla f_n|^2$ grows like $n^{d-1}$. $$\begin{align} \int_{\Bbb R^d}|\nabla f_n|^2&=C_d\int_{n+1/2}^{n+1}|\Phi'(r-n)|^2r^{d-1}\,dr\\ &\ge C_d\,(n+1/2)^{d-1}\int_{1/2}^1|\Phi'(r)|^2r^{d-1}\,dr. \end{align}$$