I am reading the book Mathematical Problems in Elasticity and Homogenization by Oleinik, Shamaev and Yosifan. You can find the passage here https://books.google.de/books?id=WbJFaB0zxVMC&printsec=frontcover&redir_esc=y#v=onepage&q&f=false on page 14/15 or have a look at the attached screenshots.
The author states lemma 2.2 and proves it. Here is a short summary: Let $\Omega \subseteq \mathbb{R}^n$ be a bounded Lipschitz domain, $\rho(x) = dist(x, \partial \Omega), \delta > 0, \Omega_\delta = \Omega \cap \{ x : \rho(x) > \delta\}$ and $v \in C^\infty(\Omega) \cap L^2(\Omega)$. The author says that the inequality $$ \int_{\Omega_\delta} (\rho(x) - \delta)^2 \vert \nabla v \vert^2 \ dx \le c_1 \left( \Vert v \Vert_{L^2(\Omega_\delta)} \Vert (\rho - \delta) \vert \nabla v \vert \Vert_{L^2(\Omega_\delta)} + \Vert v \Vert_{L^2(\Omega_\delta)} \Vert (\rho - \delta)^2 \Delta v \Vert_{L^2(\Omega_\delta)} \right) $$ implies the existence of a constant $c_2 >0$ which does not depend on $v$ such that $$ \Vert (\rho - \delta) \vert \nabla v \vert \Vert_{L^2(\Omega_\delta)} \le c_2 \left( \Vert v \Vert_{L^2(\Omega_\delta)} \Vert + \Vert (\rho - \delta)^2 \Delta v \Vert_{L^2(\Omega_\delta)} \right) . $$ I understand how we got to the first inequality but I do not understand how we got from there to the second inequality. When we divide the first inequality by $\Vert (\rho - \delta) \vert \nabla v \vert \Vert_{L^2(\Omega_\delta)}$ (we assume it is nonzero - otherwise the statement is trivial) on both sides we almost have the answer. We just have to show that there is a constant $c>0$ which does not depend on $v$ such that $$ \frac{\Vert v \Vert_{L^2(\Omega_\delta)}}{\Vert (\rho - \delta) \vert \nabla v \vert \Vert_{L^2(\Omega_\delta)}} \le c . $$ However this does not seem true. Of course the term is bounded since $v \in C^\infty(\Omega)$ but this bound depends on $v$. I tried introducing Friedrichs inequality but I have troubles since $v$ does not vanish on $\partial \Omega_\delta$. I have the feeling that I am missing an important detail.
Any help is appreciated. Thanks in advance
As a user in the comments pointed out all you have to do is apply the inequality $$ 2ab \le t a^2 + t^{-1} b^2 $$ for $a,b \in \mathbb{R}$ and $t > 0$ to both summands in the right side of the first inequality.