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.

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