Inequality involving perimeter and area - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T03:33:53Z http://mathoverflow.net/feeds/question/88507 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88507/inequality-involving-perimeter-and-area Inequality involving perimeter and area Beni Bogosel 2012-02-15T10:36:21Z 2012-04-11T21:01:08Z <p>I am studying an article: <em>The parametric problem of capillarity: the case of two and three fluids</em>, by U. Massari. In one of his proofs, he uses an inequality I can't manage to prove. It is like this:</p> <blockquote> <p>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 $\Omega$.</p> </blockquote> <p>where $\Omega_\varepsilon = \lbrace x \in \Omega : d(x,\partial \Omega) &lt;\varepsilon \rbrace$</p> <p>The previous inequality is stated without proof or reference in the article, but there is another similar inequality, with a reference to a proof:</p> <blockquote> <p>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 $n$.</p> </blockquote> <p>This inequality is proved in I. Tamanini: <em>Il problema della capillarita su domini non regolari</em>. 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 $\rho$. </p> <p>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. </p> http://mathoverflow.net/questions/88507/inequality-involving-perimeter-and-area/93559#93559 Answer by Beni Bogosel for Inequality involving perimeter and area Beni Bogosel 2012-04-09T10:16:57Z 2012-04-09T10:16:57Z <p>I have found an article which deals with this kind of inequalities. It is available in the following link: <a href="http://archive.numdam.org/ARCHIVE/RSMUP/RSMUP_1978__60_/RSMUP_1978__60__1_0/RSMUP_1978__60__1_0.pdf" rel="nofollow"> Funzioni BV e Tracce </a></p>