Sets of Finite Perimeter

We collect some basic definitions and facts about the set of finite perimeter.

If \(E\) is a set with smooth boundary, then we have the following Gauss-Green formula:

\[ \int_{E} \mathrm{div} \varphi d\mathcal{H}^n=\int_{ \partial E} \varphi \cdot \nu d\mathcal{H}^{n-1}, \]

for any \(\varphi \in C_c^1(\mathbb{R}^n,\mathbb{R}^n)\). In particular, if we require that \(|\varphi|\leq 1\), then the right-hand side can be bounded by the perimeter of \(E\) as

\[ \int_{ \partial E} \varphi\cdot \nu d\mathcal{H}^{n-1}\leq \mathcal{H}^{n-1}(\partial E). \]

and equality holds if and only if \(\varphi=\nu\) on \(\partial E\). This fact motivates the following definition of the perimeter of \(E\) as

\[ |\partial E|= \sup_{\varphi \in C_c^1(\mathbb{R}^n,\mathbb{R}^n), |\varphi|\leq 1} \int_{ \partial E} \varphi\cdot \nu d\mathcal{H}^{n-1}. \]

Note that the right-hand side of the equation above does not depend on the regularity of the boundary of \(E\). This motivates the following definition.

Definition 3.2.1

The perimeter of a set \(E\) in an open set \(U\) is defined as

\[ P(E,U)= \sup_{\varphi \in C_c^1(U,\mathbb{R}^n), |\varphi|\leq 1} \int_{ U\cap E} \mathrm{div} \varphi d\mathcal{H}^n. \]

We say that \(E\) has locally finite perimeter in \(U\) if \(P(E,W)\lt{}+\infty\) for any \(W \subset \subset U\). Such a set is also called a Caccioppoli set.

Suppose \(E\) has locally finite perimeter in \(U\). Then, we can consider the linear functional \(J_E\) on \(C_c^1(U,\mathbb{R}^n)\) defined by

\[ J_E(\varphi)= \int_{ U\cap E} \mathrm{div} \varphi d\mathcal{H}^n. \]

This functional is clearly linear. Since we have assumed that \(E\) has locally finite perimeter, we have that \(J_E\) is a bounded linear functional on \(C_c^1(W,\mathbb{R}^n)\) for any \(W \subset \subset U\). (Recall that an operator \(T\) is bounded if \(\sup_{|\varphi|\leq 1} |T(\varphi)|\lt{}+\infty\).) Now, we can apply the Riesz Representation Theorem to get a unique Radon measure \(\mu_E\) on \(U\), and a vector-valued function \(\nu_E\) on \(U\) such that \(|\nu_E(x)|=1\) for \(\mu_E\)-almost every \(x\) and

\[ J_E(\varphi)= \int_{ U} \varphi\cdot \nu_E d\mu_E, \]

for any \(\varphi \in C_c^1(U,\mathbb{R}^n)\).

The vector-valued measure \(\nu_E \mu_E\) is called the Gauss-Green measure of \(E\) in \(U\). We use \(\overrightarrow{\mu}_E\) to denote the vector-valued measure \(\nu_E \mu_E\). Then, the perimeter of \(E\) in \(U\) can also be written as

\[ P(E,U)= \mu_E(U). \]

We can understand \(\mu_E\) as a boundary measure of \(E\) in \(U\), and \(\nu_E\) as a boundary normal vector of \(E\) in \(U\), pointing outward.

Suppose \(E=[0,+\infty)^{2} \in \mathbb{R}^2\). Then, \(E\) is a set with locally finite perimeter. In particular,

\[ \mu_E=\mathcal{H}^1|_{[0,+\infty)\times \{0\}} + \mathcal{H}^1|_{\{0\}\times [0,+\infty)},\quad \nu_E=(0,-1)|_{[0,+\infty)\times \{0\}} + (1,0)|_{\{0\}\times [0,+\infty)}. \]

Recall that \(\mathrm{spt}\,\mu_E\) is the support of \(\mu_E\), which is defined as the set of points \(x\) such that \(\mu_E(B_r(x))\gt{}0\) for any \(r\gt{}0\).

Then, we have the following proposition:

Proposition 3.2.2

Suppose \(E\) is a set of locally finite perimeter in \(\mathbb{R}^n\). Then, \(\mathrm{spt}\,\mu_E\subset \partial E\), where \(\partial E\) is the topological boundary of \(E\).

Let

\[ E := (0,1)^2 \backslash \{ 0.5 \}\times [0,0.5]\subset \mathbb{R}^2. \]

Then \(\partial E\) contains both the outer boundary of the square and the slit:

\[ \partial E=\partial(0,1)^2\;\cup\;\bigl(\{0.5\}\times[0,0.5]\bigr). \]

On the other hand, removing a \(1\)-dimensional set does not change \(\chi_E\) in \(L^1\), so the perimeter measure is the same as for the open square:

\[ \mu_E=\mu_{(0,1)^2},\qquad \mathrm{spt}\,\mu_E=\partial(0,1)^2. \]

Hence

\[ \mathrm{spt}\,\mu_E \subsetneq \partial E. \]

To better illustrate the “true” boundary of a set of finite perimeter in the measure sense, we introduce the reduced boundary.

Definition 3.2.3

Given a set \(E\) of locally finite perimeter in \(\mathbb{R}^n\), the reduced boundary \(\partial^*E\) is the set of points \(x\in\mathbb{R}^n\) such that the limit

\[ \nu_E(x):=\lim_{r \to 0^+} \frac{\overrightarrow{\mu}_E(B_r(x))}{\mu_E(B_r(x))} \]

exists and belongs to \(\mathbb{S}^{n-1}\).

For the \(E=(0,1)^2 \backslash \{ 0.5 \}\times [0,0.5]\), we have

\[ \partial^* E = \partial(0,1)^2\backslash \{ (0,0), (1,0), (0,1), (1,1) \}. \]

In particular, we have

\[ \lim_{r \to 0^+} \frac{\overrightarrow{\mu}_E(B_r(0))}{\mu_E(B_r(0))} = (-\frac{1}{2},-\frac{1}{2})\notin \mathbb{S}^{n-1}. \]

So \((0,0)\notin \partial^* E\).

We have the following structure theorem for the reduced boundary.

Theorem 3.2.4

Suppose \(E\) is a set of locally finite perimeter in \(\mathbb{R}^n\). Then, the reduced boundary \(\partial^* E\) is a \((n-1)\)-countably rectifiable set in \(\mathbb{R}^n\), and we actually have

\[ \mu_E = \mathcal{H}^{n-1}|_{\partial^* E},\quad \nu_E = \text{ the outer normal vector field of } \partial^* E. \]

Note that since \(\partial^* E\) is rectifiable, so the approximate tangent space \(T_x \partial^* E\) is well-defined for \(\mu_E\)-almost every \(x \in \partial^* E\). Hence, we can choose the outer normal vector field perpendicular to \(T_x \partial^* E\) and point outward of \(E\).

Definition 3.2.5

Given Lebesgue measurable sets \(\{E_h\}_{h\in\mathbb{N}}\) and \(E\) in \(\mathbb{R}^n\), we say that \(E_h\) locally converges to \(E\), and write \(E_h \xrightarrow{\mathrm{loc}} E\), if

\[ \lim_{h \to \infty} \left| K \cap (E \Delta E_h) \right| = 0, \qquad \forall K \subset \mathbb{R}^n \text{ compact.} \]

This is equivalent to say that \(\chi_{E_h} \xrightarrow{} \chi_{E}\) in \(L^1_{loc}(\mathbb{R}^n)\).

Proposition 3.2.6

If \(\{E_h\}_{h\in\mathbb{N}}\) is a sequence of sets of locally finite perimeter in \(\mathbb{R}^n\), with

\[ E_h \xrightarrow{\mathrm{loc}} E, \qquad \limsup_{h \to \infty} P(E_h; K) \lt{} \infty, \]

for every compact set \(K\) in \(\mathbb{R}^n\), then \(E\) is of locally finite perimeter in \(\mathbb{R}^n\), \(\mu_{E_h} \rightharpoonup^* \mu_E\) and, for every open set \(A \subset \mathbb{R}^n\), we have

\[ P(E;A) \leq \liminf_{h \to \infty} P(E_h;A). \]

The inequality above can be strict.

Let \(B\) denote the closed unit disc \(B = \{x \in \mathbb{R}^2 : |x| \leq 1\}\). For each \(i \in \mathbb{N}\), let \(E_i = B \setminus \bigcup_{k=1}^{n_i} \bar{B}_{r_i}(x_{i,k})\), where \(\{x_{i,k}\}_{k=1}^{n_i}\) is a collection of centers inside \(B\), \(n_i\) increases as \(i \to \infty\), and \(r_i\to 0\) in such a way that \(n_i r_i\to c\gt{}0\) and

\[ \bigcup_{k=1}^{n_i} \bar{B}_{r_i}(x_{i,k}) \subset B, \]

with all the small discs disjoint and contained in \(B\).

For each \(E_i\), the perimeter is

\[ P(E_i) = P(B) + n_i P(\bar{B}_{r_i}) = 2\pi + n_i \cdot 2\pi r_i, \]

since each removed disc adds \(2\pi r_i\) to the perimeter.

As \(i \to \infty\), the number of holes \(n_i \to \infty\) and the radii \(r_i \to 0\), so the total removed area goes to \(0\). In the limit, the set \(E_i\) converges (locally in measure, or in \(L^1_{\mathrm{loc}}\)) to \(B\). However,

\[ \lim_{i \to \infty} P(E_i) = 2\pi+2\pi c, \]

because the total boundary length of the small holes remains visible before passing to the limit.

Thus, the perimeter functional is lower semicontinuous under local convergence, and it is possible that

\[ P(\lim_{i\to\infty} E_i) \lt{} \liminf_{i\to\infty} P(E_i), \]

because the limiting set does not retain the interior boundaries present in the approximating sequence.

Let \(D = [0,4] \times [0,2] \subset \mathbb{R}^2\). For each \(i\in\mathbb{N}\), construct \(E_i\) as the subset of \(D\) with lower and side boundaries of \(D\), but the top replaced with a zig-zag curve of \(n_i\) “teeth,” each of amplitude \(h_i\) (height above \(y=2\)), so that as \(i\to\infty\), \(n_i\to\infty\) and \(h_i\to 0\).

Each \(E_i\) is the set (shaded in the pictures above) contained below its zig-zag boundary and above \(y=0\). As \(i\to\infty\), the upper boundary of \(E_i\) converges to the straight line \(y=2\), and \(E_i\) converges (in measure) to \(D\).

The perimeter of \(E_i\) is

\[ P(E_i) = 2\cdot 2 + 4 + L_{\text{zigzag}}, \]

where \(L_{\text{zigzag}}\) is the total length of the zig-zag curve. If each tooth projects horizontally \(\delta x = 4/n_i\) and has vertical height \(h_i\), then

\[ L_{\text{zigzag}} = n_i \cdot 2\sqrt{ \left(\frac{\delta x}{2}\right)^2 + h_i^2 } = n_i \cdot 2 \sqrt{ (2/n_i)^2 + h_i^2 }. \]

If \(n_i h_i\to a\gt{}0\), then the zig-zag boundary length satisfies

\[ \lim_{i\to\infty} L_{\text{zigzag}} = 2\sqrt{4+a^2}\gt{}4. \]

Thus, in the limit, the region \(E_i\) converges to \(D\), but the perimeter keeps a positive excess:

\[ P(\lim_{i\to\infty} E_i) \lt{} \liminf_{i\to\infty} P(E_i). \]

This example again shows lower semicontinuity: the limiting domain loses the additional oscillating boundary length in the limit.

Theorem 3.2.7

If \(R\gt{}0\) and \((E_k)_{k\in\mathbb{N}}\) are sets of finite perimeter in \(\mathbb{R}^n\), with

\[ \begin{aligned} \sup_{k\in\mathbb{N}} P(E_k) &\lt{} \infty,\\ E_k &\subset B_R,\qquad \forall\, k\in\mathbb{N}, \end{aligned} \]

then one can find a set \(E\) of finite perimeter in \(\mathbb{R}^n\) and a subsequence of \((E_k)_{k\in\mathbb{N}}\), still denoted by \(E_k\), such that

\[ \chi_{E_{k}} \to \chi_E,\qquad \mu_{E_{k}} \stackrel{*}{\rightharpoonup} \mu_E,\qquad E \subset B_R. \]

Definition 3.2.8

Suppose \(A\subset \mathbb{R}^n\) is a bounded set and \(E_0\) is a set of finite perimeter in \(\mathbb{R}^n\). We say \(E_0\) is a perimeter minimizer in \(A\) if

\[ P(E_0;A)\leq P(E;A) \]

for every set of finite perimeter \(E\) such that \(E\setminus A=E_0\setminus A\).

In particular, we say \(E_0\) is a perimeter minimizer in \(\mathbb{R}^n\) if \(E_0\) is a perimeter minimizer for any bounded set \(A\).

Proposition 3.2.9

Suppose \(\{ E_k \}\) is a sequence of perimeter minimizers in \(\mathbb{R}^n\), and assume \(E_k\) converges to \(E\) in \(L^1_{\mathrm{loc}}(\mathbb{R}^n)\). Then \(E\) is also a perimeter minimizer in \(\mathbb{R}^n\).

Now, we are ready to state the general existence of minimizers in the following sense.

Proposition 3.2.10

Let \(A \subset \mathbb{R}^n\) be a bounded set and let \(E_0\) be a set of finite perimeter in \(\mathbb{R}^n\). Then there exists a set of finite perimeter \(E\) such that \(E \setminus A = E_0 \setminus A\) and

\[ P(E) \leq P(F) \]

for every \(F\) such that \(F \setminus A = E_0 \setminus A\).

For example, given a boundary curve \(\Gamma\), which lies on a boundary of a convex domain \(D\). We can construct a set of finite perimeter \(E\) such that \(\partial(\partial E \cap \Omega) = \Gamma\) and it is the one with the minimal perimeter.

The key here is actually the regularity of the boundary of the set of finite perimeter.

Sets of Finite Perimeter

Gaoming Wang

Assistant Professor