I am studying an article: *The parametric problem of capillarity: the case of two and three fluids*, by U. Massari. In one of his proofs, he uses an inequality I can't manage to prove. It is like this:



> 
> Let $\Omega \subset \Bbb{R}^n$ be an open, bounded set, with Lipschitz boundary with constant $L$. The following inequality holds:
>  $$(1) \ \ \int\_{\partial \Omega} \chi\_E d\mathcal{H}^{n-1} \leq \sqrt{1+L^2}\int\_{\Omega\_\varepsilon}| \nabla \chi\_E|+c \int\_{\Omega\_\varepsilon} \chi\_E dV $$
>  where $E$ is a measurable set of finite perimeter (i.e. $\chi\_E \in BV(\Omega)$), and the integral on $\partial \Omega$ is in fact the integral of the trace of $\chi\_E$ on the boundary of $\Omega$.
> 
> 
> 


where $\Omega\_\varepsilon = \lbrace x \in \Omega : d(x,\partial \Omega) <\varepsilon \rbrace$


The previous inequality is stated without proof or reference in the article, but there is another similar inequality, with a reference to a proof:



> 
> Let $\Omega \subset \Bbb{R}^n$ be an open, bounded set, with the property that there exists $\rho>0$ such that for every $x \in \Omega$ there is a ball $B\_\rho$ of radius $\rho$ (not necessarily centered in $x$) with $x \in B\_\rho \subset \Omega$. The following inequality holds:
>  $$ (2) \ \ \int\_{\partial \Omega} \chi\_E d\mathcal{H}^{n-1} \leq \int\_{\Omega\_\varepsilon}| \nabla \chi\_E|+c \int\_{\Omega\_\varepsilon} \chi\_E dV $$
>  where $E$ is a measurable set of finite perimeter (i.e. $\chi\_E \in BV(\Omega)$), and the integral on $\partial \Omega$ is in fact the integral of the trace of $\chi\_E$ on the boundary of $\Omega$. The constant $c$ depends on $\varepsilon, \rho, \Omega$ and $n$.
> 
> 
> 


This inequality is proved in I. Tamanini: *Il problema della capillarita su domini non regolari*. The hypothesys with the interior spheres of radius $\rho$ is used heavily in the proof. First $\Omega$ is written as a countable union of balls of radius $\rho$. 


The question is: do you know an article which proves the inequality $(1)$? If not, it is possible to deduce $(1)$ using $(2)$? Thank you. 



Inequality involving perimeter and area
I have found an article which deals with this kind of inequalities. It is available in the following link:  [Funzioni BV e Tracce](http://archive.numdam.org/ARCHIVE/RSMUP/RSMUP_1978__60_/RSMUP_1978__60__1_0/RSMUP_1978__60__1_0.pdf) 


