The integral of the normal and fluxes over sets of finite perimeter

Abstract

Given two intersecting sets of finite perimeter, $E_1$ and $E_2$, with unit normals ${\nu }_1$ and ${\nu }_2$ respectively, we obtain a bound on the integral of ${\nu }_1$ over the reduced boundary of $E_1$ inside $E_2$. This bound depends only on the perimeter of $E_2$. For any vector field $F:{\mathbb{R}}^n\to {\mathbb{R}}^n$ with the property that $F\in L^{\infty }$ and div$F$ is a (signed) Radon measure, we obtain bounds on the flux of $F$ over the portion of the reduced boundary of $E_1$ inside $E_2$. These results are then applied to study the limit of surfaces with perimeter growing to infinity.