Home > Partial Differential Equations, shape optimization > New definition for the perimeter of a set.

New definition for the perimeter of a set.

As you can see in this post we can define the perimeter of a Lebesgue measurable set $A \subset D$ relative to an open set $D\subset \Bbb{R}^N$ (if $D=\Bbb{R}^N$ it is the usual perimeter) of a set by using the formula

$P_D(A)=\sup \left\{\int_D \chi_A {\rm div} \varphi dx | \varphi \in C_c^1(U;\Bbb{R}^N),\ |\varphi|\leq 1\right\}$

It is important that this definition would agree with the classical definition for $C^1$ sets, namely, if $\Omega$ is a bounded open set of class $C^1$ then $P_D(\Omega)=\displaystyle \int_{ D \cap \partial\Omega} d\sigma$, where $d\sigma$ represents the surface element on $\partial \Omega$.