The integral of the normal and fluxes over sets of finite perimeter (Q500884)

Summary: 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 \(\operatorname{div}F\) 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.