I have a question concerning the excess of Caccioppoli sets: Given a Caccioppoli set $E\subset\mathbb{R}^{n}$, and let $A\in\mathbb{R}^{n}$ be open and bounded, we define according to Giusti (see http://www.springer.com/us/book/97808176315360) (Definition 5.1, page 63) the excess of $E$ in $A$ as
$$ \Psi(E,A):= |\boldsymbol{1}_{E}|_{TV(A)} - \inf\left\{|\boldsymbol{1}_{F}|_{TV(A)}: F\Delta E\Subset A \right\} \tag{*}\label{*} $$ with $|\cdot|_{TV}$ the total variation, $\Delta$ the symmetric difference and $X\Subset Y$ meaning that $\overline{X}$ compact subset of $Y$.
I have trouble in understanding this definition. For instance, pick $E=B_{1}(0)$ und $A=B_{2}(0)$, we have a set with a very smooth boundary :), however, when I compute $\Psi(B_{1}(0),B_{2}(0))$ I do not see what would prevent me from picking in the subtrahend in \eqref{*} $F=\emptyset$ resulting in $\Psi(B_{1}(0),B_{2}(0))>0$ when I was expecting that the excess of this set would be zero.
Obviously, I miss an important point or misunderstand the concept. Can you help?
Thanks in advance and best,
Alex
Edit: You can also find the definition in https://www.degruyter.com/view/j/crll.1982.issue-334/crll.1982.334.27/crll.1982.334.27.xml "Boundaries of Caccioppoli sets with Hölder continuous normal vector" by Italo Tamanini. In this publication, you can find it on the bottom of page 2 (Equation 1.1).
It's not. Nowhere is it claimed that excess is zero for smooth sets. It is claimed (in Remark 5.2) that excess is zero for minimal sets. $B_1(0)$ is not minimal, its boundary has positive curvature.
In general, if $E\Subset A$, then one can take $F$ to be the empty set, and therefore $\Psi(E, A) = |1_E|_{TV(A)}$ in this case.
A set with zero excess would be one obtained by cutting across $A$ by a minimal hypersurface. If $A=B_2(0)$, take any hyperplane $P$ crossing $A$ and let $E$ be either of two parts in which $P$ cuts $A$. Then $E$ is minimal, because a perturbation of its boundary within $A$ does not decrease the size of the boundary.
As a side remark, Giusti defines $$ \psi(f, A) = \int_A|Df| - \inf\left\{\int_A |Dg| : g\in BV(A), \operatorname{spt}(g-f)\subset A\right\} $$ I understand you are interested in the case when $f$ is a characteristic function of a set, but even then the definition does not require $g$ to be a characteristic function of a set.