I've encountered quite some papers in which it is simply assumed that
$\exists C>0 : \left(\displaystyle{\int\limits_{\Gamma}}((\nabla v)\cdot \hat{\bf{n}})^2d\Gamma\right)^{\dfrac{1}{2}}\leq C\left(\displaystyle{\int\limits_{\Omega}}(\nabla v)\cdot(\nabla v)d\Omega\right)^{\dfrac{1}{2}}\quad\forall v:\bar{\Omega}\rightarrow\mathbb{R},\quad\Gamma:=\partial\Omega,\bar{\Omega}:=\Omega\cup\Gamma$
instead of proven. The $\hat{\bf{n}}$ is unit vector normal to the boundary $\Gamma$. How do you prove this or where do you find a proof of this? Is there a name for such a type of inequality?
I think that the inequality is false. For $\Omega=D_1(0) \subset \mathbb{R}^2$ and $v(x)=|x|^n$, then $\nabla v(x)= n|x|^{n-2}x$, so the left term of the inequality gives $\sqrt{2 n^2 \pi} $, but the right one gives $C \sqrt{n \pi}$. I hope not to be wrong with integrals!