Structure of Caccioppoli sets

Question

In $\mathbb{R}^N$, I think of a Caccioppoli set as being a set with the following description:

(1) some points are (Lebesgue) density 0, essentially the exterior

(2) some are density 1, essentially the interior

(3) some are density 1/2, essentially the "smooth" part of the boundary

(4) some other points

Of course the three intuitions I gave are not precise, but I am looking for confirmation on my understanding of the 4th set of points, which I understand to consist essentially of corners. Is that a correct intuition, or can more complicated behavior arise?

Answer 1

"Consists essentially of corners" is too vague, especially considering how hairy the sets of finite perimeter can be, but your intuition is correct in the following sense: the set of points of type (4) has zero $(n-1)$-dimensional measure. This makes it negligible not only as a subset of $\mathbb{R}^n$ but also as a subset of the boundary of our set.

For a set $E\subset \mathbb{R}^n$ introduce the following notation:

• $E^{(s)}$ is the set of all points of $\mathbb{R}^n$ where $E$ has Lebesgue density equal to $s$;
• $\partial^*E$ is the reduced boundary of $E$, i.e., the set where one can define a normal vector in measure-theoretical sense.
• $\partial^c E$ is the essential boundary of $E$, defined as $\mathbb{R}^n\setminus (E^{(0)}\cup E^{(1)})$.

In your notation, $\partial^c E$ is the union of sets (3) and (4), and the main point is that (4) is much smaller than (3). Specifically, $$ \partial^*E \subset E^{(1/2)} \subset \partial^c E $$ and $$ \mathcal H^{n-1}(\partial^c E \setminus \partial^*E) = 0 $$ Source: Theorem 16.2 in Sets of Finite Perimeter and Geometric Variational Problems by Francesco Maggi, page 184. The theorem is due to Federer. Its message is that $\partial^*E$, $E^{(1/2)}$, and $\partial^c E$ do about the same job in terms of defining where the boundary of $E$ is.

For an example of a point that does not really look like a "corner", take $E$ to be the union of disjoint balls $B(a_k, r_k)$ where $a_k\to 0$ and $r_k\to 0$ under conditions that $$\sum r_k^{n-1} < \infty,\qquad \lim_{R\to 0}\frac{1}{R^n}\sum_{r_k < R} r_k^{n} \to c \in (0,1)$$ Then $0$ is probably going to be of type (4), but it does not look like a corner.