Notation: $B_{1}$ is the unit closed ball in $\mathbb{R}^{n}$
$<.>$ is the canonical inner product of $\mathbb{R}^{n}$
Let $u \in H^{1}(B_{1})$, $\xi \in C^{1}_{0}(B_{1})$. Set $v=(u-k)^{+}$ for $k \geq 0$ and $\phi=v\xi^{2}$. My argument to give an estimation for gradient of $v\xi$ is correct? If not, how could I do?
$D(v\xi)=Dv\xi + vD\xi$
$Dv,D\xi \in \mathbb{R}^{n}$, then
$D(v\xi)| = v^{2}|D\xi|^2 + 2\xi v <Dv,D\xi> + \xi^{2}|Dv|^{2}$
From this equality we have:
$|D(v\xi)| \leq v^{2}|D\xi|^2 + 2|\xi||v|<Dv,D\xi> + \xi^{2}|Dv|^{2}$
Then $|D(v\xi)| \leq v^{2}|D\xi|^2 + 2|\xi||v||Dv|D\xi| + \xi^{2}|Dv|^{2}$
and finally by Cauchy inequality
$|D(v\xi)| \leq 2v^{2}|D\xi|^2 + 2\xi^{2}|Dv|^{2}$