If $\Gamma(f):=\frac a2|f'|^2$, are there $0\le\eta_k\in C_c^\infty$ with $\eta_k\uparrow1$ and $\Gamma(\eta_k)\le1k$?

Question

Let $$\Gamma(f):=\frac a2|f'|^2\;\;\;\text{for }f\in C_c^\infty(\mathbb R)$$ for some $a>0$. How can we show that there is a $(\eta_k)_{k\in\mathbb N}\subseteq C_c^\infty(\mathbb R)$ with $$0\le\eta_k\le\eta_{k+1}\;\;\;\text{for all }k\in\mathbb N,$$ $$\eta_k\xrightarrow{k\to\infty}1$$ and $$\Gamma(\eta_k)\le\frac1k\;\;\;\text{for all }k\in\mathbb N?$$

Answer 1

Yes. Let $\eta_k\colon \mathbb{R}\to [0,1]$ such that $\eta_k|_{[-k,k]}=1$, $\eta_k|_{\mathbb{R}\setminus[-2k,2k]}=0$ and $\eta_k$ is affine on $[-2k,-k]$ and on $[k,2k]$. Then the sequence $(\eta_k)$ is increasing to $1$ and $$ \frac {a}2|\eta_k(x)-\eta_k(y)|^2\leq \frac{a}{2k^2}. $$ Thus $\frac a 2|\eta_k'|^2\leq \frac 1 k$ for a $k$ sufficiently large. If you want if to be smooth, simply mollify. Similar constructions work in higher dimensions (and in fact on any complete Riemannian manifold without boundary).