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.
