construction of a smooth function using mollifiers

Question

let $r>0$ and $B(x_0, r) \subset R^n$ . My problem is construct a function $u \in C^{\infty}_{0}(B(x_0, 2r))$ using mollification satisfying

$$u = 1 \text{ on } \overline{B(x_0, r)} $$ and $$ |\nabla u| \leq \frac{2}{r}$$

maybe this help : using mollifiers I can construct a function $h \in C^\infty_0(B(x_0, 2r)) $ where $h(x) = 1$ if $x \in B(x_0, r)$ and $h(x) = 0$ near the boundary of $B(x_0, 2r) $

Someone can help me ?

Thanks in advance

Answer 1

Let $ x_0 =0 $. Let $ f(t)=e^{1+\frac{3}{t}} $ and $ g(t)= f(t-4) $. Note

$$ g(t)=1 \;\; 0 \leq t \leq 1 $$ $$ g(t)=0 \;\; t>4 $$ $$ |g'(t)| \leq \frac{1}{2} $$

Let $ \eta(x) = g(\frac{|x|^2}{r^2}) $. Then $ \eta \in C^{\infty}(R^n) $ and by a direct computation we have that

$$ \eta(x)= 1 \; \; on \; B(0,r) $$ $$ \eta(x) =0 \; \; on \; R^n- B(0,2r) $$ $$ |\nabla \eta(x) | \leq \frac{2}{r} $$