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

### Ido Bright

University of Washington, Seattle, USA### Monica Torres

Purdue University, West Lafayette, USA

## Abstract

Given two intersecting sets of finite perimeter, $E_{1}$ and $E_{2}$, with unit normals $ν_{1}$ and $ν_{2}$ respectively, we obtain a bound on the integral of $ν_{1}$ over the reduced boundary of $E_{1}$ inside $E_{2}$. This bound depends only on the perimeter of $E_{2}$. For any vector field $F:R_{n}→R_{n}$ with the property that $F∈L_{∞}$ 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.

## Cite this article

Ido Bright, Monica Torres, The integral of the normal and fluxes over sets of finite perimeter. Interfaces Free Bound. 17 (2015), no. 2, pp. 245–262

DOI 10.4171/IFB/341