Caccioppoli set

Definition 1. Let ${\displaystyle \Omega }$ be an open subset of ${\displaystyle \mathbb {R} ^{n}}$ and let ${\displaystyle E}$ be a Borel set. The perimeter of ${\displaystyle E}$ in ${\displaystyle \Omega }$ is defined as follows

${\displaystyle P(E,\Omega )=V\left(\chi _{E},\Omega \right):=\sup \left\{\int _{\Omega }\chi _{E}(x)\mathrm {div} {\boldsymbol {\phi }}(x)\,\mathrm {d} x:{\boldsymbol {\phi }}\in C_{c}^{1}(\Omega ,\mathbb {R} ^{n}),\ \|{\boldsymbol {\phi }}\|_{L^{\infty }(\Omega )}\leq 1\right\}}$

where ${\displaystyle \chi _{E}}$ is the characteristic function of ${\displaystyle E}$. That is, the perimeter of ${\displaystyle E}$ in an open set ${\displaystyle \Omega }$ is defined to be the total variation of its characteristic function on that open set. If ${\displaystyle \Omega =\mathbb {R} ^{n}}$, then we write ${\displaystyle P(E)=P(E,\mathbb {R} ^{n})}$ for the (global) perimeter.

Definition 2. The Borel set ${\displaystyle E}$ is a Caccioppoli set if and only if it has finite perimeter in every bounded open subset ${\displaystyle \Omega }$ of ${\displaystyle \mathbb {R} ^{n}}$, i.e.

${\displaystyle P(E,\Omega )<+\infty }$ whenever ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ is open and bounded.

Therefore, a Caccioppoli set has a characteristic function whose total variation is locally bounded. From the theory of functions of bounded variation it is known that this implies the existence of a vector-valued Radon measure ${\displaystyle D\chi _{E}}$ such that

${\displaystyle \int _{\Omega }\chi _{E}(x)\mathrm {div} {\boldsymbol {\phi }}(x)\mathrm {d} x=\int _{E}\mathrm {div} {\boldsymbol {\phi }}(x)\,\mathrm {d} x=-\int _{\Omega }\langle {\boldsymbol {\phi }},D\chi _{E}(x)\rangle \qquad \forall {\boldsymbol {\phi }}\in C_{c}^{1}(\Omega ,\mathbb {R} ^{n})}$

As noted for the case of general functions of bounded variation, this vector measure ${\displaystyle D\chi _{E}}$ is the distributional or weak gradient of ${\displaystyle \chi _{E}}$. The total variation measure associated with ${\displaystyle D\chi _{E}}$ is denoted by ${\displaystyle |D\chi _{E}|}$, i.e. for every open set ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ we write ${\displaystyle |D\chi _{E}|(\Omega )}$ for ${\displaystyle P(E,\Omega )=V(\chi _{E},\Omega )}$.

In his papers (De Giorgi 1953) and (De Giorgi 1954), Ennio de Giorgi introduces the following smoothing operator, analogous to the Weierstrass transform in the one-dimensional case

${\displaystyle W_{\lambda }\chi _{E}(x)=\int _{\mathbb {R} ^{n}}g_{\lambda }(x-y)\chi _{E}(y)\mathrm {d} y=(\pi \lambda )^{-{\frac {n}{2}}}\int _{E}e^{-{\frac {(x-y)^{2}}{\lambda }}}\mathrm {d} y}$

As one can easily prove, ${\displaystyle W_{\lambda }\chi (x)}$ is a smooth function for all ${\displaystyle x\in \mathbb {R} ^{n}}$, such that

${\displaystyle \lim _{\lambda \to 0}W_{\lambda }\chi _{E}(x)=\chi _{E}(x)}$

also, its gradient is everywhere well defined, and so is its absolute value

${\displaystyle \nabla W_{\lambda }\chi _{E}(x)=\mathrm {grad} W_{\lambda }\chi _{E}(x)=DW_{\lambda }\chi _{E}(x)={\begin{pmatrix}{\frac {\partial W_{\lambda }\chi _{E}(x)}{\partial x_{1}}}\\\vdots \\{\frac {\partial W_{\lambda }\chi _{E}(x)}{\partial x_{n}}}\\\end{pmatrix}}\Longleftrightarrow \left|DW_{\lambda }\chi _{E}(x)\right|={\sqrt {\sum _{k=1}^{n}\left|{\frac {\partial W_{\lambda }\chi _{E}(x)}{\partial x_{k}}}\right|^{2}}}}$

Having defined this function, De Giorgi gives the following definition of perimeter:

Definition 3. Let ${\displaystyle \Omega }$ be an open subset of ${\displaystyle \mathbb {R} ^{n}}$ and let ${\displaystyle E}$ be a Borel set. The perimeter of ${\displaystyle E}$ in ${\displaystyle \Omega }$ is the value

${\displaystyle P(E,\Omega )=\lim _{\lambda \to 0}\int _{\Omega }|DW_{\lambda }\chi _{E}(x)|\mathrm {d} x}$

Actually De Giorgi considered the case ${\displaystyle \Omega =\mathbb {R} ^{n}}$: however, the extension to the general case is not difficult. It can be proved that the two definitions are exactly equivalent: for a proof see the already cited De Giorgi's papers or the book (Giusti 1984). Now having defined what a perimeter is, De Giorgi gives the same definition 2 of what a set of (locally) finite perimeter is.

It is natural to try to understand the relationship between the objects ${\displaystyle D\chi _{E}}$, ${\displaystyle |D\chi _{E}|}$, and the topological boundary ${\displaystyle \partial E}$. There is an elementary lemma that guarantees that the support (in the sense of distributions) of ${\displaystyle D\chi _{E}}$, and therefore also ${\displaystyle |D\chi _{E}|}$, is always contained in ${\displaystyle \partial E}$:

Lemma. The support of the vector-valued Radon measure ${\displaystyle D\chi _{E}}$ is a subset of the topological boundary ${\displaystyle \partial E}$ of ${\displaystyle E}$.

Proof. To see this choose ${\displaystyle x_{0}\notin \partial E}$: then ${\displaystyle x_{0}}$ belongs to the open set ${\displaystyle \mathbb {R} ^{n}\setminus \partial E}$ and this implies that it belongs to an open neighborhood ${\displaystyle A}$ contained in the interior of ${\displaystyle E}$ or in the interior of ${\displaystyle \mathbb {R} ^{n}\setminus E}$. Let ${\displaystyle \phi \in C_{c}^{1}(A;\mathbb {R} ^{n})}$. If ${\displaystyle A\subseteq (\mathbb {R} ^{n}\setminus E)^{\circ }=\mathbb {R} ^{n}\setminus E^{-}}$ where ${\displaystyle E^{-}}$ is the closure of ${\displaystyle E}$, then ${\displaystyle \chi _{E}(x)=0}$ for ${\displaystyle x\in A}$ and

${\displaystyle \int _{\Omega }\langle {\boldsymbol {\phi }},D\chi _{E}(x)\rangle =-\int _{A}\chi _{E}(x)\,\operatorname {div} {\boldsymbol {\phi }}(x)\,\mathrm {d} x=0}$

Likewise, if ${\displaystyle A\subseteq E^{\circ }}$ then ${\displaystyle \chi _{E}(x)=1}$ for ${\displaystyle x\in A}$ so

${\displaystyle \int _{\Omega }\langle {\boldsymbol {\phi }},D\chi _{E}(x)\rangle =-\int _{A}\operatorname {div} {\boldsymbol {\phi }}(x)\,\mathrm {d} x=0}$

With ${\displaystyle \phi \in C_{c}^{1}(A,\mathbb {R} ^{n})}$ arbitrary it follows that ${\displaystyle x_{0}}$ is outside the support of ${\displaystyle D\chi _{E}}$.

The topological boundary ${\displaystyle \partial E}$ turns out to be too crude for Caccioppoli sets because its Hausdorff measure overcompensates for the perimeter ${\displaystyle P(E)}$ defined above. Indeed, the Caccioppoli set

${\displaystyle E=\{(x,y):0\leq x,y\leq 1\}\cup \{(x,0):-1\leq x\leq 1\}\subset \mathbb {R} ^{2}}$

representing a square together with a line segment sticking out on the left has perimeter ${\displaystyle P(E)=4}$, i.e. the extraneous line segment is ignored, while its topological boundary

${\displaystyle \partial E=\{(x,0):-1\leq x\leq 1\}\cup \{(x,1):0\leq x\leq 1\}\cup \{(x,y):x\in \{0,1\},0\leq y\leq 1\}}$

has one-dimensional Hausdorff measure ${\displaystyle {\mathcal {H}}^{1}(\partial E)=5}$.

The "correct" boundary should therefore be a subset of ${\displaystyle \partial E}$. We define:

Definition 4. The reduced boundary of a Caccioppoli set ${\displaystyle E\subset \mathbb {R} ^{n}}$ is denoted by ${\displaystyle \partial ^{*}E}$ and is defined to be equal to be the collection of points ${\displaystyle x}$ at which the limit:

${\displaystyle \nu _{E}(x):=\lim _{\rho \downarrow 0}{\frac {D\chi _{E}(B_{\rho }(x))}{|D\chi _{E}|(B_{\rho }(x))}}\in \mathbb {R} ^{n}}$

exists and has length equal to one, i.e. ${\displaystyle |\nu _{E}(x)|=1}$.

One can remark that by the Radon-Nikodym Theorem the reduced boundary ${\displaystyle \partial ^{*}E}$ is necessarily contained in the support of ${\displaystyle D\chi _{E}}$, which in turn is contained in the topological boundary ${\displaystyle \partial E}$ as explained in the section above. That is:

${\displaystyle \partial ^{*}E\subseteq \operatorname {support} D\chi _{E}\subseteq \partial E}$

The inclusions above are not necessarily equalities as the previous example shows. In that example, ${\displaystyle \partial E}$ is the square with the segment sticking out, ${\displaystyle \operatorname {support} D\chi _{E}}$ is the square, and ${\displaystyle \partial ^{*}E}$ is the square without its four corners.

For convenience, in this section we treat only the case where ${\displaystyle \Omega =\mathbb {R} ^{n}}$, i.e. the set ${\displaystyle E}$ has (globally) finite perimeter. De Giorgi's theorem provides geometric intuition for the notion of reduced boundaries and confirms that it is the more natural definition for Caccioppoli sets by showing

${\displaystyle P(E)\left(=\int |D\chi _{E}|\right)={\mathcal {H}}^{n-1}(\partial ^{*}E)}$

i.e. that its Hausdorff measure equals the perimeter of the set. The statement of the theorem is quite long because it interrelates various geometric notions in one fell swoop.

Theorem. Suppose ${\displaystyle E\subset \mathbb {R} ^{n}}$ is a Caccioppoli set. Then at each point ${\displaystyle x}$ of the reduced boundary ${\displaystyle \partial ^{*}E}$ there exists a multiplicity one approximate tangent space ${\displaystyle T_{x}}$ of ${\displaystyle |D\chi _{E}|}$, i.e. a codimension-1 subspace ${\displaystyle T_{x}}$ of ${\displaystyle \mathbb {R} ^{n}}$ such that

${\displaystyle \lim _{\lambda \downarrow 0}\int _{\mathbb {R} ^{n}}f(\lambda ^{-1}(z-x))|D\chi _{E}|(z)=\int _{T_{x}}f(y)\,d{\mathcal {H}}^{n-1}(y)}$

for every continuous, compactly supported ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$. In fact the subspace ${\displaystyle T_{x}}$ is the orthogonal complement of the unit vector

${\displaystyle \nu _{E}(x)=\lim _{\rho \downarrow 0}{\frac {D\chi _{E}(B_{\rho }(x))}{|D\chi _{E}|(B_{\rho }(x))}}\in \mathbb {R} ^{n}}$

defined previously. This unit vector also satisfies

${\displaystyle \lim _{\lambda \downarrow 0}\left\{\lambda ^{-1}(z-x):z\in E\right\}\to \left\{y\in \mathbb {R} ^{n}:y\cdot \nu _{E}(x)>0\right\}}$

locally in ${\displaystyle L^{1}}$, so it is interpreted as an approximate inward pointing unit normal vector to the reduced boundary ${\displaystyle \partial ^{*}E}$. Finally, ${\displaystyle \partial ^{*}E}$ is (n-1)-rectifiable and the restriction of (n-1)-dimensional Hausdorff measure ${\displaystyle {\mathcal {H}}^{n-1}}$ to ${\displaystyle \partial ^{*}E}$ is ${\displaystyle |D\chi _{E}|}$, i.e.

${\displaystyle |D\chi _{E}|(A)={\mathcal {H}}^{n-1}(A\cap \partial ^{*}E)}$ for all Borel sets ${\displaystyle A\subset \mathbb {R} ^{n}}$.

In other words, up to ${\displaystyle {\mathcal {H}}^{n-1}}$-measure zero the reduced boundary ${\displaystyle \partial ^{*}E}$ is the smallest set on which ${\displaystyle D\chi _{E}}$ is supported.

From the definition of the vector Radon measure ${\displaystyle D\chi _{E}}$ and from the properties of the perimeter, the following formula holds true:

${\displaystyle \int _{E}\operatorname {div} {\boldsymbol {\phi }}(x)\,\mathrm {d} x=-\int _{\partial E}\langle {\boldsymbol {\phi }},D\chi _{E}(x)\rangle \qquad {\boldsymbol {\phi }}\in C_{c}^{1}(\Omega ,\mathbb {R} ^{n})}$

This is one version of the divergence theorem for domains with non smooth boundary. De Giorgi's theorem can be used to formulate the same identity in terms of the reduced boundary ${\displaystyle \partial ^{*}E}$ and the approximate inward pointing unit normal vector ${\displaystyle \nu _{E}}$. Precisely, the following equality holds

${\displaystyle \int _{E}\operatorname {div} {\boldsymbol {\phi }}(x)\,\mathrm {d} x=-\int _{\partial ^{*}E}{\boldsymbol {\phi }}(x)\cdot \nu _{E}(x)\,\mathrm {d} {\mathcal {H}}^{n-1}(x)\qquad {\boldsymbol {\phi }}\in C_{c}^{1}(\Omega ,\mathbb {R} ^{n})}$