Define a particular function

Question

Does anybody know how you can define a function $\eta \in C_c^1(B_R(0))$ such that $\eta = 1$ on $B_{\frac{R}{2}}(0)$ cause I need such a function in a particular proof, so I would really like to know if something like this is possible. (I don't know how you get the transitions smooth). Thank you very much in advance.

Answer 1

Consider the function:

$$ f(x) = 1-\exp\left(-\cot^2(\pi x)\right),\qquad f(0)=1,\;f\left(\frac{1}{2}\right)=f\left(-\frac{1}{2}\right)=0. $$ over $I=\left[-\frac{1}{2},\frac{1}{2}\right]$.

We have $f\in C^{\infty}(I)$ and all the derivatives of $f$ at $x=0$ vanish, hence the function defined by:

$$ g(x) = \left\{\begin{array}{rcl} 1 &\text{if}& -\frac{1}{2}\leq x\leq\frac{1}{2},\\f\left(x-\frac{1}{2}\right)&\text{if}&\frac{1}{2}\leq x\leq 1,\\f\left(x+\frac{1}{2}\right)&\text{if}&-1\leq x\leq -\frac{1}{2}\end{array}\right.\tag{2}$$ over $[-1,1]$ is a $C^{\infty}$ function that equals one over $\left[-\frac{1}{2},\frac{1}{2}\right]$.

By considering $g\!\left(\frac{\|x\|}{R}\right)\cdot\mathbb{1}_{\|x\|\leq R}\,$ you get your function.

See also the Wikipedia entry about mollifiers.