Function with certain properties I want to find a function $\sigma \in C^{\infty}_0 (B_1 (0))$ with following properties:
$\kappa \in ]0,1[$
$$0\leq \sigma \leq1 ,$$
 $$\quad\sigma =1\quad \text{on} \quad B_{1- \kappa}(0) ,$$
$$|\nabla \sigma |\leq \frac{2}{\kappa}.$$
I think its easier to start with $n=1$ and then later transfer it to any $n$.
No clue how to find such a function. Maybe a convolution with a standard dirac squence $\eta_{\epsilon}$.
Anyone knows such a function and can help me?
 A: Let $n\in \mathbb N$. I assume, that for all $\eta >0$ we are able to construct $\phi_\eta \in C^\infty_0(\mathbb R^n)$ such that $\text{supp}\phi_\eta \subset B_{\eta}(0)$, and $\int\phi_\eta = 1$, and $\phi_\eta \geq 0$. I think this is also what you mean by Dirac sequence. Also, if you need a specific construction of such a function, you will find constructions all over this webpage.
Now define for $\eta < \kappa/2$
$$
h_{\kappa, \eta}(x) = \max\left\{\min\left\{1, \frac{1}{(\kappa-2\eta)} (1-\eta-|x|) \right\} , 0 \right\}.
$$
One readily verifies that $h_\kappa$ is equal to $1$ on $B_{1-(\kappa-\eta)}(0)$ and equal to $0$ on $B_{1-\eta}(0)^c$. By the reverse triangle inequality, it is easy to show, that $h_{\kappa, \eta}$ is Lipschitz continuous with Lipschitz constant $1/(\kappa-2\eta)$. 
Now, we set $\sigma = h_{\kappa, \eta} * \phi_\eta$, where $\eta = \kappa /4$ and $*$ denotes the convolution. By construction and elementary properties of the convolution, we have that $\sigma \in C^\infty$, supp$\sigma \in B_1(0)$, and $\sigma = 1$ on $B_{1-\kappa}(0)$, and $0 \leq \sigma \leq 1$. 
Finally, we need to address the question if $|\nabla \sigma| \leq 2/\kappa$. From the definition of the convolution, it is easy to see that $\sigma$ is Lipschitz continuous with Lipschitz constant $1/(\kappa-2\eta)$. By the Theorem of Rademacher, this implies that $|\nabla \sigma| \leq 1/(\kappa-2\eta) = 1/(\kappa/2) = 2/\kappa$. 
If anything is unclear, please do not hesitate to request clarification.
