Construction of a cut-off function

Question

So basically the cut-off function that I am looking for has to satisfy the following properties: $ \phi : \mathbb{R}^n \rightarrow \mathbb{R}$

$ \phi \in C^2 $

$ \phi = 1 $ when $ |x| \leq r $

$ \phi = 0$ when x =2r and x= -2r

$ \phi = 1 $ when x = r and x = -r

and zero when $|x| > 2r$.

Basically to look like this enter image description here

Is there such function? The function that I found is

$ 1 ,$$ |x| \leq r$

0, $|x| \geq 2r$

$exp(\frac{-r^2}{r^2 - ( |x| -r )^2})e$, $ r < |x| < 2r $

But its second derivative is not continuous. Thanks for your help in advance.

Answer 1

You may change that to $\exp\left(\dfrac{1}{9r^{4}}-\dfrac{1}{9r^{4}-(|x|^{2}-r^{2})^{2}}\right)$ for $r<|x|<2r$.