# Free-boundary monotonicity for almost-minimizers of the relative perimeter

Gian Paolo Leonardi  Dipartimento di Matematica, via Sommarive 14, IT-38123
Povo - Trento (Italy<EMAIL_ADDRESS>and  Giacomo Vianello
Dipartimento di Matematica, via Sommarive 14, IT-38123 Povo - Trento (Italy)
<EMAIL_ADDRESS>
###### Abstract.

Let $E\subset\Omega$ be a local almost-minimizer of the relative perimeter in
the open set $\Omega\subset\mathbb{R}^{n}$. We prove a free-boundary
monotonicity inequality for $E$ at a point $x\in\partial\Omega$, under a
geometric property called “visibility”, that $\Omega$ is required to satisfy
in a neighborhood of $x$. Incidentally, the visibility property is satisfied
by a considerably large class of Lipschitz and possibly non-smooth domains.
Then, we prove the existence of the density of the relative perimeter of $E$
at $x$, as well as the fact that any blow-up of $E$ at $x$ is necessarily a
perimeter-minimizing cone within the tangent cone to $\Omega$ at $x$.

###### Key words and phrases:

relative perimeter, almost-minimizers, monotonicity, boundary regularity

###### 2020 Mathematics Subject Classification:

Primary: 49Q05. Secondary: 49Q10

This work has been supported by GNAMPA (INdAM) Project 2023: “Esistenza e
proprietà fini di forme ottime” and by PRIN Project 2022: “Geometric Measure
Theory: Structure of Singular Measures, Regularity Theory and Applications in
the Calculus of Variations”.

###### Contents

  1. 1 Introduction
    1. 1.1 Description of the main definitions and results
  2. 2 Preliminaries
    1. 2.1 A local extension result
  3. 3 Almost minimality
    1. 3.1 Some estimates for the minimality gap
    2. 3.2 Density estimates
  4. 4 The visibility property
    1. 4.1 Existence of the tangent cone
    2. 4.2 An off-centric visibility property
    3. 4.3 Foliation by off-centric spheres
    4. 4.4 Some examples
  5. 5 Free-boundary monotonicity
  6. 6 Blow-up limits of almost-minimizers are cones

##  1\. Introduction

Consider an open set $\Omega\subset\mathbb{R}^{n}$ with Lipschitz boundary,
and fix $x_{0}\in\partial\Omega$. The main goal of this work is to prove a
free-boundary monotonicity inequality for a local almost-minimizer
$E\subset\Omega$ of the relative perimeter at the boundary point $x_{0}$,
under a suitable geometric property of $\partial\Omega$ near $x_{0}$, that we
call “visibility”.

Monotonicity inequalities are key tools in the regularity theory for
minimizers and almost-minimizers of the area functional. In the prototypical
setting of $E$ being a local perimeter minimizer near $0\in\Omega$ (or even
just a critical point of the perimeter functional) and given $0<r_{1}<r_{2}$
such that the ball $B_{r_{2}}$ of radius $r_{2}$ and center $0$ is contained
in $\Omega$, it is known that

$\int_{(B_{r_{2}}\setminus B_{r_{1}})\cap\partial^{*}E}\frac{\left\langle
x,\nu_{E}\right\rangle^{2}}{|x|^{n+1}}d{\mathcal{H}}^{n-1}(x)\leq\frac{P(E;B_{r_{2}})}{r_{2}^{n-1}}-\frac{P(E;B_{r_{1}})}{r_{1}^{n-1}}\,,$
(1.1)

where $P(E;B_{r})$ is the perimeter of $E$ in $B_{r}$, ${\mathcal{H}}^{n-1}$
is the Hausdorff $(n-1)$-dimensional measure in $\mathbb{R}^{n}$,
$\partial^{*}E$ is the reduced boundary of $E$, and $\nu_{E}$ is the weak
interior normal defined for ${\mathcal{H}}^{n-1}$-almost every
$x\in\partial^{*}E$.

The first, fundamental consequences of an inequality like (1.1) are:

  * (i)

the monotonicity of the renormalized perimeter
$r\mapsto\frac{P(E;B_{r})}{r^{n-1}}$, which turns out to be a non-decreasing
function admitting a finite limit as $r\to 0$ (denoted as $\theta_{E}(0)$, the
perimeter density of $E$ at $0$);

  * (ii)

the fact that, if a perimeter minimizer has a constant renormalized perimeter,
then the left-hand side of (1.1) vanishes for all $0<r_{1}<r_{2}$ and
consequently $E$ coincides up to null sets with a cone with vertex at the
origin.

These two facts allow the application of the monotonicity formula in several
steps of the proof of the $C^{1,\alpha}$-regularity of the reduced boundary
$\partial^{*}E$, as well as in the analysis of the singular set $\partial
E\setminus\partial^{*}E$. After the pioneering work of De Giorgi [10] on the
partial regularity of local perimeter minimizers, the internal regularity
theory has been successfully extended to (rectifiable) sets of varying
topological type [26], area-minimizing integral currents [17, 4, 11, 12, 13],
and varifolds [3, 1]. In the codimension-$1$ case, in particular, the
dimension of the singular set is sharply estimated as $\leq n-8$, after the
work of Bombieri-De Giorgi-Giusti [7] combined with a dimension-reduction
argument due to Federer [29]. In these works, monotonicity formulas arise in
specific but substantially equivalent forms, through proofs based either on
comparison with cones or by testing the first variation of area with suitable
radially-symmetric vector fields.

A key feature of regularity theory is its stability under perturbations that,
at small scales, have a higher order of infinitesimality than area. This has
led to the extension of the regularity theory to wider classes of almost-
minimizers, including for instance minimizers of isoperimetric or prescribed
mean curvature problems (see, e.g., [25, 28, 8, 19, 30]).

Coming to the boundary properties of almost-minimizers, we record regularity
results proved for integral currents minimizing parametric elliptic integrals
and $k$-varifolds with mean curvature in $L^{p}$ for $p>k$, assuming
$C^{1,\alpha}$ regularity of their boundary [2, 14, 9]. In the free-boundary
case, regularity results have been proved for area-minimizing currents and
varifolds when the domain $\Omega$ is either of class $C^{2}$ [22, 21, 20, 23]
or, more recently, a local $C^{2}$-deformation of a wedge-type polyhedral cone
[15]. In these cases, monotonicity properties of the renormalized area are
shown by testing with locally constructed, almost-radial vector fields that
are smooth and tangent to $\partial\Omega$.

The background motivation of this paper is the study of the free-boundary
properties and regularity of perimeter almost-minimizers when the boundary of
the domain $\Omega$ is Lipschitz but not necessarily smooth. We aim to
consider $\Omega$ that might not be locally of class $C^{1,\alpha}$ or
coincide with a smooth deformation of a polyhedral wedge (as assumed in [15]).
Carrying over the full regularity program in such a general setting seems
particularly challenging. For this reason, in this paper, we focus on the
monotonicity property as a preliminary step towards the extension of the free-
boundary regularity results mentioned so far.

###  1.1. Description of the main definitions and results

Let $\Omega\subset\mathbb{R}^{n}$ be an open set with Lipschitz boundary and
$E\subset\Omega$ be a measurable set. In what follows, $E$ will be called a
local almost-minimizer of the relative perimeter in $\Omega$ if, for any
$x\in\overline{\Omega}$ there exists $r_{x}>0$ such that, for any $0<r<r_{x}$
and any measurable $F\subset\Omega$ with $F\Delta E\subset\subset B_{r}(x)$,
one has

$P(E;B_{r}(x)\cap\Omega)\leq P(F;B_{r}(x)\cap\Omega)+|F\Delta
E|^{\frac{n-1}{n}}\psi_{\Omega}(E;x,r)\,,$

for a suitable function $\psi_{\Omega}(E;x,r)$ such that $\lim_{r\to
0^{+}}\psi_{\Omega}(E;x,r)=0$. Of course, when the error term $|F\Delta
E|^{\frac{n-1}{n}}\psi_{\Omega}(E;x,r)$ vanishes, we have a local perimeter
minimizer. We conveniently introduce the following, classical notation. Given
a function $f\in BV_{\mathop{\mathrm{loc}}}(\Omega)$111Here we mean that $f\in
L^{1}_{\mathop{\mathrm{loc}}}(\Omega)$ and has bounded variation in
$\Omega\cap A$, for any bounded set $A\subset\mathbb{R}^{n}$., we define the
minimality gap of $f$ in $A$ as

$\Psi_{\Omega}(f;A)=|Df(\Omega\cap A)|-\inf\big{\\{}|Dg|(\Omega\cap
A)\,:\,g\in BV_{\mathop{\mathrm{loc}}}(\Omega),\
\mathop{\mathrm{spt}}(g-f)\subset\\!\subset A\big{\\}}$

and, when $f=\mathbf{1}_{E}$, we set
$\Psi_{\Omega}(E;A):=\Psi_{\Omega}(\mathbf{1}_{E};A)$. If $E$ is an almost-
minimizer and $x\in\overline{\Omega}$, it is immediate to check that
$\Psi_{\Omega}(E;B_{r}(x))\lesssim r^{n-1}\psi_{\Omega}(E;x,r)$. The two main
applications of Theorem 5.2 (our general monotonicity inequality) are
Corollary 5.3 and Theorem 6.1. For them, we will need to assume that the
minimality gap on balls centered at $0\in\partial\Omega$ decays to $0$
suitably fast as $r\to 0^{+}$, which is expressed through the following
summability property:

$\int_{0}^{R}\dfrac{\Psi_{\Omega}(E;B_{\rho})}{\rho^{n}}\,d\rho\simeq\int_{0}^{R}\dfrac{\psi_{\Omega}(E;0,r)}{\rho}\,d\rho<+\infty\,.$
(1.2)

Several notions of almost-minimality are available in the literature. Among
those where the minimality error is in additive form, like ours, we mention
the well-known $\Lambda$-minimality [5, 24], where the minimality error is
estimated by $\Lambda|F\Delta E|$ for some $\Lambda>0$, which is a special
case of our definition. We also remark that the definition of almost-
minimality given by Tamanini in [31]222in Tamanini’s definition, the term
$|F\Delta E|^{\frac{n-1}{n}}$ is replaced by $r^{n-1}$. is slightly more
general than ours, even though we take direct inspiration from it. The reason
why we are forced to consider a slightly stronger almost-minimality is that,
in the boundary case, it seems not possible to use the monotonicity inequality
to prove lower-density estimates for the perimeter and the volume of $E$ at
$x\in\partial\Omega$, as it happens for the interior case. On the other hand,
a local almost-minimizer in our sense turns out to fulfill volume and
perimeter density estimates directly (see Lemma 3.8). We point out that these
estimates are crucially used in the proof of Theorem 6.1.

The key assumption needed for the proof of our monotonicity formula is the
aforementioned visibility property, that we now briefly describe. Let
$\Omega\subset\mathbb{R}^{n}$ be an open set with Lipschitz boundary such that
$0\in\partial\Omega$ and in graphical form around $0$ with respect to the
variables $x^{\prime}=(x_{1},...,x_{n-1})$, so that $\Omega$ is locally the
epigraph of some Lipschitz function of $x^{\prime}$. We say that $\Omega$
satisfies the visibility property at $0$ if there exist $R>0$ and a function
$v\in C^{1}([0,R))$ such that:

  * (V1)

$v(0)=v^{\prime}(0)=0$ and $0\leq v^{\prime}\leq 2^{-1}$;

  * (V2)

The function

$\gamma_{v}(r):=r^{-1}\sup_{0<s\leq r}\sqrt{\frac{v(s)}{s}+v^{\prime}(s)}$

is summable on $(0,R)$;

  * (V3)

for all $0<r<R$, the segment joining the point $V_{r}=(0,\dots,0,-v(r))$ with
a point $x$ belonging to $\partial\Omega\cap B_{r}$ does not intersect
$\Omega$.

The visibility property allows us to construct a quasi-conical competitor for
a local almost-minimizer of the relative perimeter in $\Omega$, which is a key
step in the proof of our monotonicity inequality. It is worth observing that
this property guarantees the existence of the tangent cone to $\Omega$ at $0$
(Proposition 4.5). The visibility property at $0$ is satisfied for instance by
cones with vertex at $0$ (with the trivial choice $v\equiv 0$), by
$C^{1,\beta}$ open sets and also by convex sets suitably approximated by their
tangent cone at $0$ (see Section 4.4). Another consequence of the visibility
property is the existence of a foliation of a neighborhood of $0$ by spheres
with varying centers, that correspond to the level-sets of a function $\phi$
of class $C^{1}$, i.e., such that $\phi^{-1}(r)=\partial B_{r}(V_{r})$.
Moreover, by (V1) and (V2) one can show that $\phi$ is $C^{1}$-close to the
function $|x|$ (see Lemma 4.9).

In our main result, Theorem 5.2, we establish the following monotonicity
inequality for $f\in BV_{\mathop{\mathrm{loc}}}(\Omega)$, under the visibility
property:

$\displaystyle\left(\int_{\Omega\cap\mathcal{A}_{r_{1},r_{2}}}\phi^{1-n}\left|\left\langle\nu_{f},\nabla\phi\right\rangle\right|\,d|Df|\right)^{2}\leq
2\left(\int_{\Omega\cap\mathcal{A}_{r_{1},r_{2}}}\frac{|\nabla\phi(x)|}{\phi(x)^{n-1}}\,d|Df|\right)$
$\displaystyle\qquad\qquad\qquad\cdot\bigg{[}\mu_{f}(r_{2})-\mu_{f}(r_{1})+\int_{r_{1}}^{r_{2}}\dfrac{n-1}{\rho^{n}}\,\Psi_{\Omega}(f;B_{\rho}(V_{\rho}))d\rho+G(f;r_{1},r_{2})\bigg{]}\,.$
(1.3)

Here, $0<r_{1}<r_{2}$ are sufficiently small,
$\mathcal{A}_{r_{1},r_{2}}=B_{r_{2}}(V_{r_{2}})\setminus
B_{r_{1}}(V_{r_{1}})$, $\nu_{f}$ represents the Radon-Nikodym derivative of
$Df$ with respect to its total variation $|Df|$,

$\mu_{f}(r)=\frac{|Df|(\Omega\cap B_{r}(V_{r}))}{r^{n-1}}\,,$

and $G(f;r_{1},r_{2})$ is an error term whose precise expression is given in
(5.1), which depends on the properties of the boundary of $\Omega$, and goes
to $0$ when $r_{2}\to 0$ as soon as

$\mu_{f}(r)\leq C\,,\qquad\text{for all $r>0$ small enough and for some
$C>0$.}$ (1.4)

Inequality (1.1) has important consequences when $f=\mathbf{1}_{E}$ and $E$ is
a local almost-minimizer of the relative perimeter in $\Omega$ satisfying the
assumption (1.2). Indeed, in this case, the above-mentioned density estimates
guarantee that (1.4) holds. We then infer that the function

$\int_{0}^{r}\dfrac{n-1}{\rho^{n}}\,\Psi_{\Omega}(f;B_{\rho}(V_{\rho}))d\rho+G(f;0,r)$

is infinitesimal as $r\to 0^{+}$. Consequently, the density ratio $\mu_{f}(r)$
is almost-monotone, and hence it admits a finite limit $\theta_{E}(0)$ as
$r\to 0^{+}$ (see Corollary 5.3).

Finally, the left-hand side of (1.1) can be interpreted as an approximate
conical deviation term, since it is as small as $E$ is close to being a cone
with vertex at $0$ (at least when $\phi(x)=|x|$, i.e. if the visibility
property at $0$ is satisfied with $v\equiv 0$). This observation is exploited
in the proof of the last Theorem 6.1, where we show that any blow-up sequence
of an almost-minimizer of the relative perimeter in $\Omega$ admits a
subsequence that converge to a minimizing cone in the tangent cone to $\Omega$
at $0$.

##  2\. Preliminaries

Let $n\in\mathbb{N}$, $n\geq 2$. For $i=1,...,n$, we denote by $e_{i}$ the
$i$-th vector of the standard basis of $\mathbb{R}^{n}$. Given $x$,
$y\in\mathbb{R}^{n}$, their Euclidean product is $\langle
x,y\rangle:=\sum_{i=1}^{n}x_{i}y_{i}$, while the Euclidean norm of $x$ is
$|x|:=\sqrt{\langle x,x\rangle}$. When needed, we will write
$x=(x^{\prime},x_{n})$, where
$x^{\prime}=(x_{1},...,x_{n-1})\in\mathbb{R}^{n-1}$. We let $B_{r}(x)$ be the
open ball of radius $r>0$ centered at $x\in\mathbb{R}^{n}$, and we set
$B_{r}:=B_{r}(0)$. Similarly, we let $B^{\prime}_{r}(x^{\prime})$ be the
$(n-1)$-dimensional open ball with center $x^{\prime}$ and radius $r$. We
denote by either ${\mathcal{L}}^{n}(E)$ or $|E|$ the Lebesgue measure of
$E\subset\mathbb{R}^{n}$, and we set $\omega_{n}:={\mathcal{L}}^{n}(B_{1})$.
Given a measurable set $A\subset\mathbb{R}^{n}$, we set

$A^{(t)}:=\left\\{x\in\mathbb{R}^{n}\,:\,\lim_{r\rightarrow
0^{+}}\dfrac{|E\cap B_{r}(x)|}{\omega_{n}r^{n}}=t\right\\}\,,\qquad\text{for
all $0\leq t\leq 1$ \, .}$

Given two non-empty, bounded subsets $A$, $B\subset\mathbb{R}^{n}$, we denote
by $\mathop{\mathrm{dist}_{\mathcal{H}}}(A,B)$ the Hausdorff distance between
$A$ and $B$, that is

$\mathop{\mathrm{dist}_{\mathcal{H}}}(A,B)=\max\left\\{\sup_{x\in
A}\,\mathop{\mathrm{dist}}(x,B)\,,\,\sup_{y\in
B}\,\mathop{\mathrm{dist}}(y,A)\right\\}\,,$ (2.1)

where $\mathop{\mathrm{dist}}(x,Z):=\inf_{z\in Z}|x-z|$. Note that the
Hausdorff distance becomes a proper distance function when restricted to
compact sets (see [27]).

Given $\Omega\subset\mathbb{R}^{n}$ open and a vector-valued Radon measure
$\mu=(\mu_{1},...,\mu_{p})$ on $\Omega$, we denote by $|\mu|$ its total
variation. Let $u=(u_{1},...,u_{p}):\Omega\rightarrow\mathbb{R}^{p}$ be
summable with respect to $|\mu|$, then $u\cdot\mu$ denotes the Radon measure
defined by

$u\cdot\mu(E):=\int_{E}u\cdot d\mu=\sum_{q=1}^{p}\int_{E}u_{q}\,d\mu_{q}\,.$

It can be proved (see [6, Proposition 1.23]) that

$|u\cdot\mu|=|u|\cdot\mu\,.$ (2.2)

Given a Lipschitz map $g:\mathbb{R}^{n}\to\mathbb{R}^{p}$, we denote by
$\mathop{\mathrm{Lip}}(g)$ its Lipschitz constant, and we denote by
$\mathop{\mathrm{Lip}}(\mathbb{R}^{n})$ the space of Lipschitz real-valued
functions. We record for future reference the following, elementary fact.

######  Lemma 2.1.

Let $\\{f_{j}\\}_{j\in\mathbb{N}}\subset\mathop{\mathrm{Lip}}(\mathbb{R}^{n})$
be such that $\sup_{j\geq 1}\mathop{\mathrm{Lip}}(f_{j})<\infty$. Assume
further that $f_{j}(x)\rightarrow f(x)$ for all $x\in\mathbb{R}^{n}$. Then
$f_{j}\rightarrow f$ locally uniformly on $\mathbb{R}^{n}$.

Given $f\in L^{1}_{\mathop{\mathrm{loc}}}(\Omega)$, we define

$|Df|(\Omega):=\sup\left\\{\int_{\Omega}f(x)\mathop{\mathrm{div}}\phi(x)\,dx\,:\,\phi\in
C^{1}_{c}(\Omega;\mathbb{R}^{n})\,,\,||\phi||_{L^{\infty}(\Omega)}\leq
1\right\\}\,.$ (2.3)

We denote by $BV(\Omega)$ the space of functions $f\in L^{1}(\Omega)$ with the
property that $|Df|(\Omega)<\infty$. In other words, a function $f\in
L^{1}(\Omega)$ belongs to $BV(\Omega)$ if and only if its distributional
gradient $Df$ is represented by a $\mathbb{R}^{n}$-valued Radon measure with
finite total variation.

When $E$ is a measurable set, we define the perimeter of $E$ in $\Omega$ as

$P(E;\Omega):=|D\mathbf{1}_{E}|(\Omega)\,,$

where $\mathbf{1}_{E}$ is the characteristic function of $E$. We say that $E$
has _finite perimeter_ in $\Omega$ provided $P(E;\Omega)<+\infty$. In the
sequel, we will often use the following notation: given another open set
$A\subset\mathbb{R}^{n}$, we define

$P_{\Omega}(E;A):=P(E;A\cap\Omega)\,,$

with the short form $P_{\Omega}(E):=P_{\Omega}(E;A)$ whenever $A$ contains
$\Omega$. We call $P_{\Omega}$ the relative perimeter in $\Omega$, and
$P_{\Omega}(E;A)$ the relative perimeter of $E$ in $\Omega$ restricted to $A$.

For the specific purposes of this paper, it is convenient to define
$BV_{\mathop{\mathrm{loc}}}(\Omega)$ as the space of functions $f\in
L^{1}_{\mathop{\mathrm{loc}}}(\Omega)$ such that $f\in BV(\Omega\cap A)$ for
all open, bounded sets $A\subset\mathbb{R}^{n}$. We remark that $f\in
BV_{\mathop{\mathrm{loc}}}(\Omega)$ implies that $f$ has locally bounded
variation in $\Omega$, but the vice versa does not necessarily hold.

Let now $f\in BV_{\mathop{\mathrm{loc}}}(\Omega)$. As a consequence of [24,
Corollary 5.11] we can consider the density of $Df$ with respect to $|Df|$
defined by

$\nu_{f}(x)=\lim_{r\rightarrow
0^{+}}\dfrac{Df(B_{r}(x))}{|Df|(B_{r}(x))}\,,\qquad\text{for $|Df|$-a.e.
}x\in\mathbb{R}^{n}\,,$

and satisfying $Df=\nu_{f}\cdot|Df|$ and $|\nu_{f}(x)|=1$ for $|Df|$-a.e.
$x\in\Omega$. When $E$ has locally finite perimeter in $\Omega$, we set
$\nu_{E}=\nu_{\mathbf{1}_{E}}$.

Next we recall some well-known facts concerning the total variation
functional, $BV$ functions, and sets of locally finite perimeter. For further
details, the reader can consult, e.g., [24, 16, 18].

######  Theorem 2.2 (Lower-semicontinuity).

Let $f,f_{j}\in L^{1}_{\mathop{\mathrm{loc}}}(\Omega)$ for $j\in\mathbb{N}$ be
such that $f_{j}\to f$ in $L^{1}_{\mathop{\mathrm{loc}}}(\Omega)$. Then

$|Df|(\Omega)\leq\liminf_{j}|Df_{j}|(\Omega).$

BV functions can be approximated by smooth functions in a suitable sense.

######  Theorem 2.3 (Approximation).

Let $\Omega\subset\mathbb{R}^{n}$ be open and $f\in BV(\Omega)$. Then there
exists a sequence $\\{f_{j}\\}_{j\in\mathbb{N}}\in C^{\infty}(\Omega)\cap
BV(\Omega)$ such that

$||f_{j}-f||_{L^{1}(\Omega)}\rightarrow
0\,,\qquad|Df_{j}|(\Omega)\rightarrow|Df|(\Omega)\,.$ (2.4)

Moreover, provided (2.4) holds, we also have $Df_{j}\rightharpoonup^{\ast}Df$,
i.e., for all $\phi\in C^{\infty}_{c}(\mathbb{R}^{n};\mathbb{R}^{n})$,

$\int_{\mathbb{R}^{n}}\phi\cdot
dDf_{j}\rightarrow\int_{\mathbb{R}^{n}}\phi\cdot dDf\,,\qquad\text{as
}j\rightarrow\infty\,.$

######  Remark 2.4.

If property (2.4) holds, we say that the sequence $f_{j}$ _strictly converges_
to $f$ (see [6, Definition 3.14]). Thus Theorem 2.3 tells us in particular
that $C^{\infty}(\Omega)\cap BV(\Omega)$ is dense in $BV(\Omega)$ with respect
to the strict convergence. Moreover, the sequence
$\\{f_{j}\\}_{j\in\mathbb{N}}$ can be chosen in such a way that the following
extra property is satisfied (see [18, Remark 1.18])

$\lim_{\rho\rightarrow 0^{+}}\rho^{-N}\int_{\Omega\cap
B_{\rho}(x_{0})}|f_{j}-f|dx=0\,,\qquad\text{for all $N>0$,
$x_{0}\in\partial\Omega$, $j\in\mathbb{N}\,.$}$ (2.5)

######  Theorem 2.5 (Compactness).

Let $\Omega\subset\mathbb{R}^{n}$ be an open, bounded set with Lipschitz
boundary, let $C>0$ be a fixed constant, and for $j\in\mathbb{N}$ let
$f_{j}\in BV(\Omega)$ be such that

$\|f_{j}\|_{BV(\Omega)}:=\|f_{j}\|_{L^{1}(\Omega)}+|Df_{j}|(\Omega)\leq
C\,,\qquad\text{for every $j$.}$

Then there exists $f\in BV(\Omega)$ and a subsequence $f_{j_{k}}$ of $f_{j}$
such that

$\|f_{j_{k}}-f\|_{L^{1}(\Omega)}\rightarrow 0\,.$

######  Theorem 2.6 (Coarea for $BV$ functions).

Let $\Omega\subset\mathbb{R}^{n}$ be open and $f\in L^{1}(\Omega)$. For
$t\in\mathbb{R}$, let

$E_{t}=\\{x\in\Omega\,:\,f(x)>t\\}\,.$

Then the following statements hold:

  * (i)

if $f\in BV(\Omega)$, then $E_{t}$ has finite perimeter for a.e.
$t\in\mathbb{R}$, $t\mapsto P(E_{t};\Omega)$ is measurable, and

$|Df|(\Omega)=\int_{\mathbb{R}}P(E_{t};\Omega)\,dt\,;$

  * (ii)

conversely, if $f\in L^{1}(\Omega)$, $t\mapsto P(E_{t};\Omega)$, and

$\int_{\mathbb{R}}P(E_{t};\Omega)\,dt<+\infty\,,$

then $f\in BV(\Omega)$.

######  Theorem 2.7 (Coarea for Lipschitz functions).

If $\phi\in\mathop{\mathrm{Lip}}(\mathbb{R}^{n})$, $E\subset\mathbb{R}^{n}$ is
a Borel set, and $f:\mathbb{R}^{n}\to[0,\infty)$ is a Borel function, then

$\int_{E}f|\nabla\phi|=\int_{\mathbb{R}}\int_{E\cap\\{\phi=t\\}}f\,d{\mathcal{H}}^{n-1}\,dt\,,$

where ${\mathcal{H}}^{n-1}$ denotes the $(n-1)$-dimensional Hausdorff measure
in $\mathbb{R}^{n}$.

We will also need the following well-known result, which proof can be found in
[24].

######  Theorem 2.8 (Area Formula).

Let $M\subset\mathbb{R}^{n}$ be a locally ${\mathcal{H}}^{k}$-rectifiable set
and $f:\mathbb{R}^{n}\to\mathbb{R}^{m}$ be a Lipschitz continuous function,
with $1\leq k\leq m$. Then:

  * (i)

for ${\mathcal{H}}^{k}$-almost all $x\in M$, the restriction of $f$ to
$x+T_{x}M$, where $T_{x}M$ is the approximated tangent space to $M$ at $x$, is
differentiable at $x$ and we denote by $d^{M}f_{x}:T_{x}M\to\mathbb{R}^{m}$
its differential;

  * (ii)

the following identity holds

$\int_{\mathbb{R}^{m}}{\mathcal{H}}^{0}(M\cap\\{f=y\\})d{\mathcal{H}}^{k}(y)=\int_{M}J^{M}fd{\mathcal{H}}^{k}\,,$
(2.6)

where $J^{M}f(x):=\sqrt{\det(d^{M}f^{\ast}_{x}\circ d^{M}f_{x})}$ and
$d^{M}f^{\ast}_{x}$ denotes the adjoint of $d^{M}f_{x}$.

We continue by introducing the notion of trace of a $BV$ function on a
Lipschitz boundary.

######  Theorem 2.9.

Let $\Omega\subset\mathbb{R}^{n}$ be an open set with Lipschitz boundary and
denote by $\nu_{\Omega}$ the unit, inner normal vector defined
${\mathcal{H}}^{n-1}$-a.e. on $\partial\Omega$. Then there exists a unique
linear and bounded _inner trace operator_

${\mathrm{Tr}}^{+}(\cdot,\partial\Omega):BV(\Omega)\rightarrow
L^{1}(\partial\Omega,{\mathcal{H}}^{n-1})\,,$

such that, for all $f\in BV(\Omega)$,

$\lim_{r\rightarrow
0^{+}}\,\,-\hskip-10.80003pt\int_{B_{r}(x)\cap\Omega}|{\mathrm{Tr}}^{+}(f,\partial\Omega)(x)-f(y)|\,dy=0\,,\qquad\text{for
${\mathcal{H}}^{n-1}$-a.e. $x\in\Omega\,.$}$ (2.7)

Moreover, for all $f\in BV(\Omega)$ and $\phi\in
C^{1}_{c}(\mathbb{R}^{n},\mathbb{R}^{n})$, the following Gauss-Green formula
holds:

$\int_{\Omega}f\,\mathop{\mathrm{div}}\phi\,dx=-(\phi\cdot
Df)\Omega-\int_{\partial\Omega}\langle\phi,\nu_{\Omega}\rangle\,{\mathrm{Tr}}^{+}(f,\partial\Omega)\,d{\mathcal{H}}^{n-1}\,.$
(2.8)

The function ${\mathrm{Tr}}^{+}(f,\partial\Omega)$ is called the _inner trace_
of $f$ on $\partial\Omega$. Formula (2.7) ensures in particular that

${\mathrm{Tr}}^{+}(f,\partial\Omega)(x)=\lim_{r\rightarrow
0^{+}}\,\,-\hskip-10.80003pt\int_{B_{r}(x)\cap\Omega}f(y)\,dy\,,\qquad\text{for
${\mathcal{H}}^{n-1}$-a.e. $x\in\partial\Omega\,.$}$ (2.9)

######  Remark 2.10.

Under the assumptions of Theorem 2.9, the operator
${\mathrm{Tr}}^{+}(\cdot,\partial\Omega)$ is also continuous with respect to
the strict convergence in $\Omega$ (see [18, Theorem 2.11] or [6, Theorem
3.88]). Moreover, by (2.5) and (2.9), we have that the sequence
$\\{f_{j}\\}_{j\in\mathbb{N}}$ of Theorem 2.3 satisfies

$\displaystyle|{\mathrm{Tr}}^{+}(f_{j},\partial\Omega)(x)-{\mathrm{Tr}}^{+}(f,\partial\Omega)(x)|$
$\displaystyle=\left|\lim_{r\rightarrow
0^{+}}\,\,-\hskip-10.80003pt\int_{B_{r}(x)\cap\Omega}f_{j}(y)\,dy-\lim_{r\rightarrow
0^{+}}\,\,-\hskip-10.80003pt\int_{B_{r}(x)\cap\Omega}f(y)\,dy\right|$
$\displaystyle\leq\lim_{r\rightarrow
0^{+}}\,\,-\hskip-10.80003pt\int_{B_{r}(x)\cap\Omega}|f_{j}(y)-f(y)|\,dy=0$

for ${\mathcal{H}}^{n-1}$-a.e. $x\in\partial\Omega$. In other words,
${\mathrm{Tr}}^{+}(f_{j},\partial\Omega)={\mathrm{Tr}}^{+}(f,\partial\Omega)$
${\mathcal{H}}^{n-1}$-a.e. on $\partial\Omega$, for all $j$.

Let $\Omega$, $\Omega^{\prime}\subset\mathbb{R}^{n}$ open sets with Lipschitz
boundary, such that $\overline{\Omega}\subset\Omega^{\prime}$. For any $f\in
BV(\Omega^{\prime})$, we can also consider a trace of $f$ on $\partial\Omega$
computed with respect to $\Omega^{\prime}\setminus\overline{\Omega}$. We thus
set

${\mathrm{Tr}}^{-}(f,\partial\Omega):={\mathrm{Tr}}^{+}(f,\partial(\Omega^{\prime}\setminus\overline{\Omega}))\mathbin{\vrule
height=6.02773pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\,\partial\Omega\,.$

The function ${\mathrm{Tr}}^{-}(f,\partial\Omega)\in
L^{1}(\partial\Omega;{\mathcal{H}}^{n-1})$ is called the _outer trace_ of $f$
on $\partial\Omega$.

######  Lemma 2.11.

Let $\Omega$, $\Omega^{\prime}\subset\mathbb{R}^{n}$ be open sets. In
addition, let $\Omega$ be bounded and have Lipschitz boundary, and assume that
$\Omega\subset\subset\Omega^{\prime}$. For all $f\in BV(\Omega^{\prime})$, we
have

$|Df|(\partial\Omega)=\|{\mathrm{Tr}}^{+}(f,\partial\Omega)-{\mathrm{Tr}}^{-}(f,\partial\Omega)\|_{L^{1}(\partial\Omega;{\mathcal{H}}^{n-1})}\,.$
(2.10)

######  Lemma 2.12.

Fix an open set $A\subset\mathbb{R}^{n}$ and a Lipschitz function
$\phi:A\to\mathbb{R}$ of class $C^{1}$, such that $|\nabla\phi(x)|>0$ for all
$x\in A$. Set $A_{r}=\phi^{-1}(-\infty,r)$. Then for all $f\in BV(A)$, for
$\mathcal{L}^{1}$-a.e. $r\in\mathbb{R}$, and ${\mathcal{H}}^{n-1}$-a.e.
$x\in\partial A_{r}\cap A$, we have

$f(x)={\mathrm{Tr}}^{+}(f,\partial A_{r})(x)={\mathrm{Tr}}^{-}(f,\partial
A_{r})(x)\,.$ (2.11)

###### Proof.

We observe that, for a.e. $r\in\mathbb{R}$,

${\mathrm{Tr}}^{+}(f,\partial A_{r})={\mathrm{Tr}}^{-}(f,\partial
A_{r})\,,\qquad\text{${\mathcal{H}}^{n-1}$-a.e. on $\partial A_{r}\cap A$.}$
(2.12)

Indeed, for the proof of (2.12) we can combine (2.10) with $|Df|(\partial
A_{r}\cap A)=0$ for a.e. $r\in\mathbb{R}$, which in turn comes from the fact
that $|Df|$ is a finite measure and $\partial A_{r}\cap\partial A_{s}\cap
A=\emptyset$ whenever $r\neq s$. Let $x\in A$ be a Lebesgue point for $f$,
then $x\in\partial A_{r}$ if and only if $r=\phi(x)$. Thanks to the smoothness
of $\partial A_{r}\cap A$ (a consequence of the Implicit Function Theorem) we
have

$|A_{r}\cap B_{\rho}(x)|=\dfrac{1}{2}\,\rho^{n}+o(\rho^{n})\,,\qquad\text{as
}\rho\rightarrow 0^{+}\,,$

and consequently

$\displaystyle f(x)$ $\displaystyle=\lim_{\rho\rightarrow
0^{+}}\,\dfrac{1}{\omega_{n}\,\rho^{n}}\int_{B_{\rho}(x)}f(y)\,dy$
$\displaystyle=\lim_{\rho\rightarrow
0^{+}}\,\dfrac{1}{\omega_{n}\,\rho^{n}}\left(\int_{A_{r}\cap
B_{\rho}(x)}f(y)\,dy+\int_{B_{\rho}(x)\setminus A_{r}}f(y)\,dy\right)$
$\displaystyle=\lim_{\rho\rightarrow
0^{+}}\,\dfrac{1}{2}\,\dfrac{1}{|A_{r}\cap B_{\rho}(x)|}\int_{A_{r}\cap
B_{\rho}(x)}f(y)\,dy+\dfrac{1}{2}\,\dfrac{1}{|B_{\rho}(x)\setminus
A_{r}|}\int_{B_{\rho}(x)\setminus A_{r}}f(y)\,dy$
$\displaystyle=\dfrac{1}{2}\,{\mathrm{Tr}}^{+}(f,\partial
A_{r})(x)+\dfrac{1}{2}\,{\mathrm{Tr}}^{-}(f,\partial A_{r})(x)$
$\displaystyle={\mathrm{Tr}}^{\pm}(f,\partial A_{r})(x)\,.$

Since the set of Lebesgue points for $f$ coincides with $A$ up to a
$\mathcal{L}^{n}$-negligible set, by Theorem 2.7 we obtain that the first
equality in (2.11) is verified for $\mathcal{L}^{1}$-a.e. $r\in\mathbb{R}$ and
${\mathcal{H}}^{n-1}$-a.e. $x\in\partial A_{r}\cap A$, which together with
(2.12) concludes the proof. ∎

######  Proposition 2.13.

Under the assumptions of Lemma 2.12, we take $f,f_{j}\in BV(A)$ for
$j\in\mathbb{N}$, such that

$||f_{j}-f||_{L^{1}(A)}\longrightarrow
0\,,\qquad|Df_{j}|(A)\longrightarrow|Df|(A)\,.$

Then, for a.e. $0<r<1$, we have

$\displaystyle{\mathrm{Tr}}(f,\partial A_{r})$
$\displaystyle:={\mathrm{Tr}}^{+}(f,\partial
A_{r})={\mathrm{Tr}}^{-}(f,\partial A_{r})$ (2.13)
$\displaystyle{\mathrm{Tr}}(f_{j},\partial A_{r})$
$\displaystyle:={\mathrm{Tr}}^{+}(f_{j},\partial
A_{r})={\mathrm{Tr}}^{-}(f_{j},\partial A_{r})\,,\qquad\text{for all $j\geq
1\,,$}$ (2.14)

and

$|Df_{j}|(A_{r})\longrightarrow|Df|(A_{r})\,,\qquad||{\mathrm{Tr}}(f_{j},\partial
A_{r})-{\mathrm{Tr}}(f,\partial A_{r})||_{L^{1}(\partial
A_{r})}\longrightarrow 0\,,$ (2.15)

hence in particular $f_{j}$ strictly converges to $f$ on $A_{r}$.

###### Proof.

Thanks to Lemma 2.12, the two identities (2.13) and (2.14) hold for a.e.
$0<r<1$. In particular, for such $r$, we deduce that $|Df|(\partial A_{r})=0$,
hence $|Df|(\overline{A_{r}})=|Df|(A_{r})$. Moreover, by Theorem 2.2, we have

$\displaystyle\liminf_{j\rightarrow\infty}|Df_{j}|(A_{r})$
$\displaystyle\geq|Df|(A_{r})$
$\displaystyle=|Df|(\overline{A_{r}})=|Df|(A)-|Df|(A\setminus\overline{A_{r}})$
$\displaystyle=\lim_{j\rightarrow\infty}|Df_{j}|(A)-|Df|(A\setminus\overline{A_{r}})$
$\displaystyle\geq\limsup_{j\rightarrow\infty}|Df_{j}|(A)-|Df_{j}|(A\setminus\overline{A_{r}})$
$\displaystyle=\limsup_{j\rightarrow\infty}|Df_{j}|(\overline{A_{r}})$
$\displaystyle\geq\limsup_{j\rightarrow\infty}|Df_{j}|(A_{r})\,,$

which proves that

$|Df|(A_{r})=\lim_{j\rightarrow\infty}|Df_{j}|(A_{r})\,.$

Since $||f_{j}-f||_{L^{1}(A)}\rightarrow 0$, we have in particular
$||f_{j}-f||_{L^{1}(A_{r})}\rightarrow 0$, and thus $\\{f_{j}\\}_{j\geq 1}$
strictly converges to $f$ in $A_{r}$. Finally, (2.15) holds because, as
observed in Remark 2.10, the inner trace operator is continuous with respect
to the strict convergence. ∎

Let $E\subset\Omega$ be a set of locally finite perimeter in $\Omega$. We
define the _reduced boundary_ of $E$, and we denote it by $\partial^{\ast}E$,
as the set of those points $x\in\Omega$ such that
$|D\mathbf{1}_{E}(B_{r}(x))|>0$, for all $r>0$, and there exists a unit vector
$\nu_{E}(x)$ such that

$\nu_{E}(x)=\lim_{r\rightarrow
0^{+}}\dfrac{D\mathbf{1}_{E}(B_{r}(x))}{|D\mathbf{1}_{E}(B_{r}(x))|}\,.$

Clearly, we have that $\nu_{E}(x)=\nu_{\mathbf{1}_{E}}(x)$ for
$|D\mathbf{1}_{E}|$-a.e. $x\in\Omega$, therefore the perimeter measure
$|D\mathbf{1}_{E}|$ is concentrated on the reduced boundary $\partial^{*}E$.

######  Theorem 2.14 (De Giorgi-Federer’s Structure Theorem).

Let $E\subset\mathbb{R}^{n}$ be a set of locally finite perimeter. Then

$|D\mathbf{1}_{E}|={\mathcal{H}}^{n-1}\mathbin{\vrule
height=6.02773pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\,\partial^{\ast}E\,,\qquad
D\mathbf{1}_{E}=\nu_{E}\cdot{\mathcal{H}}^{n-1}\mathbin{\vrule
height=6.02773pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\,\partial^{\ast}E\,,$

and the following Gauss-Green Formula holds:

$\int_{E}\nabla\phi\,dx=\int_{\partial^{\ast}E}\phi\nu_{E}\,d{\mathcal{H}}^{n-1}\qquad\forall\,\phi\in
C^{1}_{c}(\mathbb{R}^{n})\,.$

Moreover $\partial^{\ast}E$ is countably $(n-1)$-rectifiable, i.e. there exist
countably many $C^{1}$ hypersurfaces $M_{j}\subset\mathbb{R}^{n}$ and a Borel
set $F$ with ${\mathcal{H}}^{n-1}(F)=0$, such that

$\partial^{\ast}E\subset F\cup\bigcup_{j\geq 1}M_{j}\,.$

Finally, $\partial^{*}E\subset E^{(1/2)}$ and
${\mathcal{H}}^{n-1}(\mathbb{R}^{n}\setminus(E^{(0)}\cup
E^{(1)}\cup\partial^{*}E))=0$.

###  2.1. A local extension result

From now on, we assume that $\Omega\subset\mathbb{R}^{n}$ is an open set with
Lipschitz boundary. We fix $\rho>0$ such that, up to an isometry, there exist
a Lipschitz function $\omega:B^{\prime}_{\rho}\rightarrow\mathbb{R}$ and a
constant $m>0$, with $\omega(0)=0$ and
$m>\|\omega\|_{L^{\infty}(B^{\prime}_{\rho})}$, satisfying the following
property: if we set $\mathcal{C}_{\rho,m}=B^{\prime}_{\rho}\times(-m,m)$, we
have

$\Omega\cap\mathcal{C}_{\rho,3m}=\\{x=(x^{\prime},x_{n})\in\mathbb{R}^{n}\,:\,x^{\prime}\in
B^{\prime}_{\rho}\,,\,\omega(x^{\prime})<x_{n}<3m\\}\,.$ (2.16)

We aim to prove that, under this assumption, any measurable set
$E\subset\Omega$ with $\mathbf{1}_{E}\in
BV_{\mathop{\mathrm{loc}}}(\Omega)$333We recall that in this paper $f\in
BV_{\mathop{\mathrm{loc}}}(\Omega)$ means $f\in BV(A)$ for all
$A\subset\Omega$ open and bounded. can be extended to a locally finite
perimeter set $\tilde{E}$ in $\Omega\cup\mathcal{C}_{\rho,m}$, in such a way
that $\tilde{E}\cap\Omega=E\cap\Omega$,
$P(\tilde{E};\partial\Omega\cap\mathcal{C}_{\rho,m})=0$, and
$P(\tilde{E};S(B))\leq C\,P(E;B)$, for all Borel sets
$B\subset\mathcal{C}_{\rho,m}\setminus\Omega$ and for some constant $C>0$
depending on the dimension $n$ and the function $\omega$. In what follows, we
will denote by $T_{x}E$ the approximate tangent space to $\partial^{\ast}E$ at
$x$444The approximate tangent space $T_{x}E$ is given by the orthogonal
complement of $\nu_{E}(x)$..

Set $\mathcal{C}_{\rho}=B^{\prime}_{\rho}\times\mathbb{R}$ and define the map
$S:\mathcal{C}_{\rho}\rightarrow\mathcal{C}_{\rho}$ as

$S(x):=(x^{\prime},2\omega(x^{\prime})-x_{n})\,.$ (2.17)

Note that $S$ satisfies $S^{2}(x)=x$ for all $x$. Moreover, elementary
computations show that

$\mathop{\mathrm{Lip}}(S)\leq\sqrt{3+6\mathop{\mathrm{Lip}}(\omega)^{2}}\,.$
(2.18)

Given $E\subset\Omega$ measurable with $\mathbf{1}_{E}\in
BV_{\mathop{\mathrm{loc}}}(\Omega)$, we define
$\tilde{E}\subset\Omega\cup\mathcal{C}_{\rho}$ as

$\tilde{E}=E\cup(S_{\rho}(E)\setminus\Omega)\,,$ (2.19)

where $S_{\rho}(E)=S(E\cap\mathcal{C}_{\rho})$. Clearly, we have
$\tilde{E}\cap\Omega=E\cap\Omega$. Further properties of $\tilde{E}$ are
stated in the next lemma.

######  Lemma 2.15.

Let $E\subset\Omega$ be a measurable set with $\mathbf{1}_{E}\in
BV_{\mathop{\mathrm{loc}}}(\Omega)$. Then, for almost all
$x\in\partial^{\ast}E$, $S$ restricted to $x+T_{x}E$ is differentiable at $x$,
and we denote by $d^{E}S_{x}:T_{x}E\rightarrow\mathbb{R}^{n}$ its
differential. Moreover, if $\tilde{E}$ is the set defined in (2.19), we have

$P(\tilde{E};\partial\Omega\cap\mathcal{C}_{\rho,m})=0$ (2.20)

and, for all Borel sets $B\subset S(\mathcal{C}_{\rho,m}\cap\Omega)$,

$P(\tilde{E};B)=P(S_{\rho}(E);B)=\int_{\partial^{\ast}E\cap
S(B)}J^{E}S(x)\,d{\mathcal{H}}^{n-1}(x)\leq C\,P(E;S(B))\,,$ (2.21)

where and $C=\mathop{\mathrm{Lip}}(S)^{n-1}$.

###### Proof.

Owing to Theorem 2.14, we know that $\partial^{\ast}E$ is countably
$(n-1)$-rectifiable. Then the fact that $S|_{T_{x}E}$ is differentiable at
${\mathcal{H}}^{n-1}$-a.e. $x\in\partial^{\ast}E$ follows immediately from
statement (i) of Theorem 2.8.

Now, we prove (2.20) in the following way. Thanks to Lemma 2.11 we only need
to check that, for ${\mathcal{H}}^{n-1}$-a.e.
$x\in\partial\Omega\cap\mathcal{C}_{\rho,m}$, we have

${\mathrm{Tr}}^{+}(\mathbf{1}_{\tilde{E}},\partial\Omega)(x)={\mathrm{Tr}}^{-}(\mathbf{1}_{\tilde{E}},\partial\Omega)(x)\,,$

that is,

${\mathrm{Tr}}^{+}(\mathbf{1}_{E},\partial\Omega)(x)={\mathrm{Tr}}^{+}(\mathbf{1}_{S_{\rho}(E)},\partial(\mathbb{R}^{n}\setminus\Omega))(x)\,.$
(2.22)

The proof of (2.22) goes as follows. We employ the characterization of the
trace as a limit of averages: for ${\mathcal{H}}^{n-1}$-a.e.
$x\in\partial\Omega$ we have

${\mathrm{Tr}}^{+}(\mathbf{1}_{E},\partial\Omega)(x)=\lim_{r\to
0^{+}}\frac{|E\cap B_{r}(x)\cap\Omega|}{|B_{r}(x)\cap\Omega|}$

and

${\mathrm{Tr}}^{+}(\mathbf{1}_{S_{\rho}(E)},\partial(\mathbb{R}^{n}\setminus\Omega))(x)=\lim_{r\to
0^{+}}\frac{|S_{\rho}(E)\cap
B_{r}(x)\setminus\Omega|}{|B_{r}(x)\setminus\Omega|}\,.$

Then we combine this characterization with a consequence of (2.7), i.e. that
the trace of a $BV$ characteristic function coincides with a characteristic
function ${\mathcal{H}}^{n-1}$-almost everywhere on $\partial\Omega$, to infer
that we only need to show the equivalence

${\mathrm{Tr}}^{+}(\mathbf{1}_{E},\partial\Omega)(x)=0\quad\Leftrightarrow\quad{\mathrm{Tr}}^{+}(\mathbf{1}_{S_{\rho}(E)},\partial(\mathbb{R}^{n}\setminus\Omega))(x)=0\,.$

One of the two required implications (the other can be discussed similarly) is

${\mathrm{Tr}}^{+}(\mathbf{1}_{E},\partial\Omega)(x)=0\quad\Rightarrow\quad{\mathrm{Tr}}^{+}(\mathbf{1}_{S_{\rho}(E)},\partial(\mathbb{R}^{n}\setminus\Omega))(x)=0\,.$

This implication can be restated as

$|E\cap B_{r}(x)\cap\Omega|=o(r^{n})\quad\Rightarrow\quad|S_{\rho}(E)\cap
B_{r}(x)\setminus\Omega|=o(r^{n})\qquad\text{as $r\to 0^{+}$.}$ (2.23)

Up to taking $r>0$ small enough, we have
$B_{r}(x)\subset\mathcal{C}_{\rho,m}$, hence setting
$L=\mathop{\mathrm{Lip}}(S)$ we get

$\displaystyle|S_{\rho}(E)\cap B_{r}(x)\setminus\Omega|$
$\displaystyle\leq|S\big{(}E\cap S(B_{r}(x))\cap\Omega\big{)}|$
$\displaystyle\leq L^{n}|E\cap B_{Lr}(x)\cap\Omega|$
$\displaystyle=L^{n}o(L^{n}r^{n})=o(r^{n})\qquad\text{as }r\to 0^{+}\,,$

which proves the implication (2.23) and concludes the proof of (2.20).

Finally, for the proof of (2.21), it is enough to show that

${\mathcal{H}}^{n-1}(\partial^{\ast}S(E)\Delta S(\partial^{\ast}E))=0\,.$
(2.24)

Indeed, if (2.24) holds, Theorem 2.14 ensures that

$P(S(E);B)={\mathcal{H}}^{n-1}(\partial^{\ast}S(E)\cap
B)={\mathcal{H}}^{n-1}(S(\partial^{\ast}E)\cap
B)={\mathcal{H}}^{n-1}(S(\partial^{\ast}E\cap S(B)))\,,$

thus (2.21) is an immediate consequence of the Area Formula for rectifiable
sets given in (2.6). Let us demonstrate (2.24). Again by Theorem 2.14, it
suffices to prove that

$S(E)^{(0)}=S(E^{(0)})\,,\qquad S(E)^{(1)}=S(E^{(1)})\,.$ (2.25)

Let us prove the first of the previous identities, as the proof of the other
one is obtained by observing that $(\mathbb{R}^{n}\setminus E)^{(0)}=E^{(1)}$.
Let $0<\delta<\rho$, and
$x\in\Omega\cap(B^{\prime}_{\rho-\delta}\times\mathbb{R})$. Set
$L=\mathop{\mathrm{Lip}}(S)$ as before, then by construction, for any
$0<r<\delta$, $S(B_{r}(S(x)))\subset B_{Lr}(x)$, and thus the Area Formula
(2.6) yields

$|S(E)\cap B_{r}(S(x))|=\int_{E\cap B_{r}(S(x))}JS\,dx\leq
L^{n}|B_{Lr}(x)|\,.$ (2.26)

Since $r$ is arbitrary and $S^{-1}=S$, it is easy to check that, thanks to
(2.26), if $x\in E^{(0)}$ then $S(x)\in S(E)^{(0)}$, which proves the
inclusion $S(E^{(0)})\subset S(E)^{(0)}$. The reverse inclusion is proved in a
completely analogous way. The proof of the lemma is then achieved thanks to
(2.18). ∎

##  3\. Almost minimality

Here we introduce the almost-minimality for the relative perimeter, which is
needed in the proofs of the free-boundary monotonicity results shown in
Section 5. As a consequence of our definition, we show some properties of the
minimality gap function, that will be needed later on, and (boundary) density
estimates for the volume and the perimeter of an almost-minimizer.

######  Definition 3.1 (Almost-minimality).

Given $\Omega\subset\mathbb{R}^{n}$ open and $E\subset\Omega$ measurable, we
say that $E$ is a _local almost-minimizer_ of $P_{\Omega}$ if, for any
$x\in\overline{\Omega}$ there exists $r_{x}>0$ such that, for any $0<r<r_{x}$
and any measurable $F\subset\Omega$ with $F\Delta E\subset\subset B_{r}(x)$,
one has

$P_{\Omega}(E;B_{r}(x))\leq P_{\Omega}(F;B_{r}(x))+|F\Delta
E|^{\frac{n-1}{n}}\psi_{\Omega}(E;x,r)\,,$ (3.1)

for a suitable function $\psi_{\Omega}(E;x,r)$ such that $\lim_{r\to
0^{+}}\psi_{\Omega}(E;x,r)=0$. If additionally

$\psi_{\Omega}(E;r):=\sup_{x\in\overline{\Omega}}\psi_{\Omega}(E;x,r)\rightarrow
0\qquad\text{as }r\to 0^{+}\,,$

then we say that $E$ is an _almost-minimizer_ of $P_{\Omega}$.

We note that if $\psi_{\Omega}(E;x,r)=0$, then $E$ is a minimizer of
$P_{\Omega}$ in $B_{r}(x)$. In particular, if $\psi_{\Omega}(E;x,r)=0$ for all
$x\in\overline{\Omega}$ and $r>0$, then $E$ is a (global) minimizer of
$P_{\Omega}$. We shall later discuss suitable summability properties of the
function $r\to\psi_{\Omega}((E;x,r)$, that are required in the proof of the
Monotonicity Formula.

It is worth recalling that, among the various notions of almost-minimality for
sets of locally finite perimeter that can be found in the literature, the one
expressed by (3.1) can be understood as a generalization of the well-known
$\Lambda$-minimality property [5, 24], however, it is slightly more
restrictive than the notion considered by Tamanini in [31] (however, we point
out that Tamanini’s work is focused on internal regularity theory). For more
completeness, we recall that a set $E$ is a _$\Lambda$ -minimizer_ of
$P_{\Omega}$ if, for any $x\in\overline{\Omega}$ there exists $r_{x}>0$ such
that, for any $0<r<r_{x}$ and any measurable subset $F$ of $\Omega$ with
$F\Delta E\subset\\!\subset B_{r}(x)$, one has

$P_{\Omega}(E;B_{r}(x))\leq P_{\Omega}(F;B_{r}(x))+\Lambda|F\Delta E|\,.$
(3.2)

It is then clear that a $\Lambda$-minimizer satisfies (3.1) with

$\psi_{\Omega}(E;x,r)=\Lambda\,r\,\omega_{n}^{1/n}\,.$

For better studying almost-minimizers, it is convenient to introduce the
notion of minimality gap.

######  Definition 3.2.

Let $\Omega$, $A\subset\mathbb{R}^{n}$ be open sets and $f\in
BV_{\mathop{\mathrm{loc}}}(\Omega)$. The _minimality gap_ of $f$ in $A$
relative to $\Omega$ is

$\Psi_{\Omega}(f;A)=|Df(\Omega\cap A)|-\inf\big{\\{}|Dg|(\Omega\cap
A)\,:\,g\in BV_{\mathop{\mathrm{loc}}}(\Omega),\
\mathop{\mathrm{spt}}(g-f)\subset\\!\subset A\big{\\}}\,.$

When $f=\mathbf{1}_{E}$, for some measurable subset $E\subset\mathbb{R}^{n}$,
we write $\Psi_{\Omega}(E;A)$ in place of $\Psi_{\Omega}(\mathbf{1}_{E};A)$.

###  3.1. Some estimates for the minimality gap

######  Lemma 3.3.

Let $\Omega$, $A\subset\mathbb{R}^{n}$ be open sets and $E\subset\Omega$ be
such that $\mathbf{1}_{E}\in BV_{\mathop{\mathrm{loc}}}(\Omega)$. Then

$\Psi_{\Omega}(E;A)=P_{\Omega}(E;A)-\inf\big{\\{}P_{\Omega}(F;A)\,:\,\mathbf{1}_{F}\in
BV_{\mathop{\mathrm{loc}}}(\Omega)\,,\,F\Delta E\subset\\!\subset
A\big{\\}}\,.$

###### Proof.

Let us define

$\displaystyle\mathcal{I}_{1}=\inf\\{|Dg|(\Omega\cap A)\,:\,g\in
BV_{\mathop{\mathrm{loc}}}(\Omega)\,,\,\mathop{\mathrm{spt}}(g-\mathbf{1}_{E})\subset\\!\subset
A\\}$
$\displaystyle\mathcal{I}_{2}=\inf\\{P_{\Omega}(F;A)\,:\,\mathbf{1}_{F}\in
BV_{\mathop{\mathrm{loc}}}(\Omega)\,,\,F\Delta E\subset\\!\subset A\\}\,.$

It suffices to show that $\mathcal{I}_{1}=\mathcal{I}_{2}$. For sure,
$\mathcal{I}_{1}\leq\mathcal{I}_{2}$ because we can take $g=\mathbf{1}_{F}$ in
the definition of $\mathcal{I}_{1}$. Fix now $\varepsilon>0$, and let $g\in
BV_{\mathop{\mathrm{loc}}}(\Omega)$ be such that
$\mathop{\mathrm{spt}}(g-\mathbf{1}_{E})\subset\subset A$ and

$|Dg|(\Omega\cap A)\leq\mathcal{I}_{1}+\varepsilon\,.$ (3.3)

For $t\in\mathbb{R}$, let us set

$G_{t}=\\{x\in\Omega\,:\,g(x)>t\\}\,.$

We observe that, for each $0<t<1$,
$G_{t}\setminus\mathop{\mathrm{spt}}(g-\mathbf{1}_{E})=E\setminus\mathop{\mathrm{spt}}(g-\mathbf{1}_{E})$,
and so

$G_{t}\Delta E\subset\mathop{\mathrm{spt}}(g-\mathbf{1}_{E})\subset\subset
A\,.$ (3.4)

We can now exploit Theorem 2.6 and (3.4) to infer that

$|Dg|(\Omega\cap
A)=\int_{\mathbb{R}}P_{\Omega}(G_{t};A)dt\geq\int_{0}^{1}P_{\Omega}(G_{t};A)dt\geq\mathcal{I}_{2}\,.$
(3.5)

By (3.3) and (3.5), we deduce that
$\mathcal{I}_{1}\leq\mathcal{I}_{2}\leq\mathcal{I}_{1}+\varepsilon$ and then,
since $\varepsilon>0$ is arbitrary, we get $\mathcal{I}_{1}=\mathcal{I}_{2}$
and complete the proof of the lemma. ∎

######  Remark 3.4.

The minimality gap of a local almost-minimizer $E$ satisfies the estimate

$\Psi_{\Omega}(E;B_{r}(x))\leq\omega_{n}^{1-\frac{1}{n}}r^{n-1}\psi_{\Omega}(E;x,r)$
(3.6)

for all $x\in\overline{\Omega}$, $0<r<r_{x}$, where $\psi_{\Omega}(E;x,r)$ is
the function appearing in (3.1). This means that
$\Psi_{\Omega}(E;B_{r}(x))=o(r^{n-1})$ as $r\to 0$. However, we anticipate
here that the monotonicity formula proved in Section 5 will require a slightly
stronger assumption on $\psi_{\Omega}(E;x,r)$, namely that
$r^{-1}\psi_{\Omega}(E;x,r)$ is summable on $(0,r_{x})$. We notice that this
kind of hypothesis is somehow well-known in the context of regularity theory
(see [31]).

######  Lemma 3.5.

Let $\Omega,A,A^{\prime}\subset\mathbb{R}^{n}$ be open sets and $f\in
BV_{\mathop{\mathrm{loc}}}(\Omega)$. If $A\subset\\!\subset A^{\prime}$, then

$\Psi_{\Omega}(f;A)\leq\Psi_{\Omega}(f;A^{\prime})\,.$

###### Proof.

Let us fix $\varepsilon>0$ and take $g\in BV_{\mathop{\mathrm{loc}}}(\Omega)$
such that

$\mathop{\mathrm{spt}}(g-f)\subset\\!\subset
A\qquad\text{and}\qquad|Df|(\Omega\cap A)-|Dg|(\Omega\cap
A)\geq\Psi_{\Omega}(f;A)-\varepsilon\,.$

Of course, $g$ satisfies also $\mathop{\mathrm{spt}}(g-f)\subset\\!\subset
A^{\prime}$, hence we get

$\displaystyle\Psi_{\Omega}(f;A^{\prime})$ $\displaystyle\geq|Df|(\Omega\cap
A^{\prime})-|Dg|(\Omega\cap A^{\prime})$ $\displaystyle=|Df|(\Omega\cap
A)+|Df|(\Omega\cap(A^{\prime}\setminus A))-|Dg|(\Omega\cap
A)-|Df|(\Omega\cap(A^{\prime}\setminus A))$ $\displaystyle=|Df|(\Omega\cap
A)-|Dg|(\Omega\cap A)$ $\displaystyle\geq\Psi_{\Omega}(f;A)-\varepsilon\,.$

Since $\varepsilon$ is arbitrary, we conclude the proof. ∎

We now prove two key results. The first one is the lower semicontinuity
property of the minimality gap for uniform sequences of local almost-
minimizers. An extra, technical difficulty arising in the proof, is that the
tangent cone to the domain may not locally contain the dilations of the domain
itself, since we are in the boundary case. This requires to suitably extending
a competitor from the tangent cone $\Omega_{0}$ to the rescaled domains of the
form $t^{-1}\Omega$, $t\to 0^{+}$ via Lemma 2.15. The second is an upper bound
for the difference between the minimality gaps of two
$BV_{\mathop{\mathrm{loc}}}$ functions.

In what follows, we will say that $\Omega_{j}\to\Omega$ locally in Hausdorff
distance if there exist $r_{0},m,L>0$, with $r_{0}<m$, and $L$-Lipschitz
functions $\omega_{j},\omega:B^{\prime}_{r_{0}}\to(-m,m)$ providing local
graphical representations of $\partial\Omega_{j},\ \partial\Omega$,
respectively, as in (2.16), such that $\omega_{j}\to\omega$ uniformly in
$B^{\prime}_{r_{0}}$.

######  Lemma 3.6.

For $j\in\mathbb{N}$ we let $\Omega_{j},\Omega\subset\mathbb{R}^{n}$ be open
sets with uniformly Lipschitz boundary, such that $0\in\partial\Omega$ and
$\Omega_{j}\to\Omega$ locally in Hausdorff distance. Let $E_{j},E$ be sets of
locally finite perimeter, such that $E_{j}\subset\Omega_{j},\ E\subset\Omega$,
and $E_{j}\to E$ in $L^{1}(B_{r_{0}})$. Finally, we assume that $E_{j}$
satisfies (3.1) for all $0<r<r_{0}$ and $x=0$, and that moreover we have

$\lim_{r\to 0^{+}}\,\sup_{j}\psi_{\Omega_{j}}(E_{j};0,r)=0\,.$

Then, $E$ satisfies (3.1) for all $0<r<r_{0}$, with
$\psi_{\Omega}(E;0,r)=\sup_{j}\psi_{\Omega_{j}}(E_{j};0,r)$, and

$\liminf_{j}\Psi_{\Omega_{j}}(E_{j};B_{r_{0}})\geq\Psi_{\Omega}(E;B_{r_{0}})\,.$
(3.7)

Moreover, if $\Psi_{\Omega_{j}}(E_{j};B_{r_{0}})\to 0$ as $j\to\infty$, then
for almost all $0<r<r_{0}$ we have

$\lim_{j\rightarrow\infty}P_{\Omega_{j}}(E_{j};B_{r})=P_{\Omega}(E;B_{r})\,,\qquad\text{for
almost all $r>0\,.$}$ (3.8)

###### Proof.

Let us fix $\varepsilon>0$. Let $F\subset\Omega$ be such that $F\Delta
E\subset\subset B_{r_{0}}$ and

$\Psi_{\Omega}(E;B_{r_{0}})\leq
P_{\Omega}(E;B_{r_{0}})-P_{\Omega}(F;B_{r_{0}})+\varepsilon\,.$ (3.9)

By Lemma 2.12, for all $j\geq 1$, for a.e. $0<r<r_{0}$ and
${\mathcal{H}}^{n-1}$-a.e. $x\in\partial B_{r}$, we have

$\mathbf{1}_{E_{j}}(x)={\mathrm{Tr}}^{\pm}(E_{j},\partial
B_{r})(x)\quad\text{and}\quad\mathbf{1}_{E}(x)={\mathrm{Tr}}^{\pm}(E,\partial
B_{r})(x)\,,$ (3.10)

where ${\mathrm{Tr}}^{\pm}(A,\partial
B_{r}):={\mathrm{Tr}}^{\pm}(\mathbf{1}_{A},\partial B_{r})$. By the
$L^{1}_{\mathop{\mathrm{loc}}}$-convergence of $E_{j}$ to $E$, we can choose
$r<r_{0}$ with the above property and the additional

$P_{\Omega}(E;B_{r})\geq P_{\Omega}(E;B_{r_{0}})-\varepsilon\,,$ (3.11)

then take $j_{\varepsilon}$ large enough, such that

$\displaystyle F\Delta E\subset\subset B_{r}\,,$ (3.12)
$\displaystyle\int_{\partial
B_{r}}|\mathbf{1}_{E_{j}}-\mathbf{1}_{E}|d{\mathcal{H}}^{n-1}<\varepsilon\qquad\text{for
$j\geq j_{\varepsilon}$}\,.$ (3.13)

Let us fix $\delta>0$, define

$U_{\delta}:=\big{\\{}x=(x^{\prime},x_{n})\in\mathcal{C}_{r,m}:\
\omega(x^{\prime})-\delta<x_{n}\leq\omega(x^{\prime})\big{\\}}\,,$

and assume $\delta$ so small that

${\mathcal{H}}^{n-1}(\partial B_{r}\cap U_{\delta})<\varepsilon$ (3.14)

and

$P_{\Omega}(E;B_{r})\leq P(E;B_{r}\cap(\Omega+\delta e_{n}))+\varepsilon\,.$
(3.15)

Owing to the uniform convergence of $\omega_{j}$ to $\omega$, we can select
$j_{\delta}\geq 1$ such that $\|\omega_{j}-\omega\|_{\infty}<\delta$ for all
$j\geq j_{\delta}$, then for those $j$ we define

$A_{j}:=\Omega_{j}\cap\Omega\cap B_{r}\,,\qquad
B_{j}:=(\Omega_{j}\setminus\Omega)\cap B_{r}\,,$ (3.16)

and observe that $B_{j}\subset U_{\delta}$. Now we set

$F_{j}:=(\tilde{F}\cap\Omega_{j}\cap
B_{r})\cup\big{(}E_{j}\cap(B_{r_{0}}\setminus B_{r})\big{)}\,,$ (3.17)

where $\tilde{F}=F\cup(S(F)\setminus\Omega)$ and $S$ is the symmetry through
$\partial\Omega$ defined in (2.17) (with $\rho=r_{0}$). Thanks to (3.12), we
have $F_{j}\subset\Omega_{j}$ and $F_{j}\Delta E_{j}\subset\subset B_{r_{0}}$,
which means that $F_{j}$ is a competitor for $E_{j}$ in the definition of
$\Psi_{\Omega_{j}}(E_{j},B_{r_{0}})$. Moreover, by (3.17), (3.13), and (3.14),
we have

$\displaystyle P_{\Omega_{j}}(F_{j};B_{r_{0}})$ $\displaystyle\leq
P_{\Omega_{j}}(F_{j};B_{r})+P_{\Omega_{j}}(E_{j};B_{r_{0}}\setminus\overline{B_{r}})$
$\displaystyle\qquad+\int_{\partial
B_{r}\cap\Omega_{j}\cap\Omega}|\mathbf{1}_{E_{j}}-\mathbf{1}_{E}|d{\mathcal{H}}^{n-1}+{\mathcal{H}}^{n-1}(\partial
B_{r}\cap U_{\delta})$ $\displaystyle\leq
P_{\Omega_{j}}(F_{j};B_{r})+P_{\Omega_{j}}(E_{j};B_{r_{0}}\setminus\overline{B_{r}})+2\varepsilon\,.$
(3.18)

Let us compute $P(F_{j};B_{r}\cap\Omega_{j})$. By Lemma 2.15,
$P(\tilde{F};\partial\Omega\cap\mathcal{C}_{r_{0},m})=0$, hence

$\displaystyle P_{\Omega_{j}}(F_{j};B_{r})$
$\displaystyle=P(F;A_{j})+P(S(F);B_{j})\,.$ (3.19)

Again by Lemma 2.15, up to possibly taking a smaller $\delta$, we have

$P(S(F);B_{j})=P(\tilde{F};B_{j})\leq
P(\tilde{F};U_{\delta})\leq\varepsilon\,.$ (3.20)

Putting together (3.18), (3.19), and (3.20), and taking into account (3.10),
we obtain

$\displaystyle\Psi_{\Omega_{j}}(E_{j};B_{r_{0}})$ $\displaystyle\geq
P_{\Omega_{j}}(E_{j};B_{r_{0}})-P_{\Omega_{j}}(F_{j};B_{r_{0}})$ (3.21)
$\displaystyle\geq P_{\Omega_{j}}(E_{j};B_{r})-P(F;A_{j})-3\varepsilon$
$\displaystyle\geq
P_{\Omega_{j}}(E_{j};B_{r})-P_{\Omega}(F;B_{r_{0}})-3\varepsilon\,.$

Now, since for all $j\geq j_{\varepsilon}$ we have
$\|\omega_{j}-\omega\|_{\infty}<\delta$, we infer that

$B_{r}\cap(\Omega+\delta e_{n})\subset B_{r}\cap\Omega_{j}\,.$

This inclusion combined with (3.21) and (3.15) gives

$\displaystyle\Psi_{\Omega_{j}}(E_{j};B_{r_{0}})$ $\displaystyle\geq
P(E_{j};B_{r}\cap(\Omega+\delta
e_{n}))-P(F;B_{r_{0}}\cap\Omega)-3\varepsilon\,,$ (3.22)

so that taking the liminf as $j\to\infty$ in (3.22), and using the lower
semicontinuity of the perimeter, (3.15), (3.11), and (3.9), we find

$\displaystyle\liminf_{j}\Psi_{\Omega_{j}}(E_{j};B_{r_{0}})$
$\displaystyle\geq\liminf_{j}P(E_{j};B_{r}\cap(\Omega+\delta
e_{n}))-P_{\Omega}(F;B_{r_{0}})-3\varepsilon$ $\displaystyle\geq
P(E;B_{r}\cap(\Omega+\delta e_{n}))-P_{\Omega}(F;B_{r_{0}})-3\varepsilon$
$\displaystyle\geq P_{\Omega}(E;B_{r})-P_{\Omega}(F;B_{r_{0}})-4\varepsilon$
$\displaystyle\geq
P_{\Omega}(E;B_{r_{0}})-P_{\Omega}(F;B_{r_{0}})-5\varepsilon$
$\displaystyle\geq\Psi_{\Omega}(E;B_{r_{0}})-6\varepsilon\,.$

Then, the arbitrary choice of $\varepsilon$ implies (3.7). Now, the fact that
$E$ satisfies (3.1) with $\psi_{\Omega}(E;0,r)$ as in the statement can be
proved with the same argument used to show (3.7), also taking into account
that $|E_{j}\Delta F_{j}|\to|E\Delta F|$ as $j\to\infty$. Finally, to prove
(3.8) we consider the sequence $F_{j}$ defined as before, but now with $F=E$.
Choosing $\varepsilon>0$ arbitrarily, for $j$ large enough we obtain as before

$\displaystyle\Psi_{\Omega_{j}}(E_{j};B_{r_{0}})$ $\displaystyle\geq
P_{\Omega_{j}}(E_{j};B_{r_{0}})-P_{\Omega_{j}}(F_{j};B_{r_{0}})$
$\displaystyle\geq
P_{\Omega_{j}}(E_{j};B_{r})-P_{\Omega}(E;B_{r})-3\varepsilon\,,$

which gives the desired conclusion. ∎

######  Lemma 3.7.

Let $\Omega,A\subset\mathbb{R}^{n}$ be open sets, with $\partial\Omega$
Lipschitz and $A$ bounded, of class $C^{2}$, and such that
${\mathcal{H}}^{n-1}(\partial A\cap\partial\Omega)=0$. Let $f,g\in
BV_{\mathop{\mathrm{loc}}}(\Omega)$, then

$\displaystyle|\Psi_{\Omega}(f,A)-\Psi_{\Omega}(g,A)|$
$\displaystyle\leq\Big{|}|Df|(\Omega\cap A)-|Dg|(\Omega\cap A)\Big{|}$ (3.23)
$\displaystyle\qquad+\|{\mathrm{Tr}}^{+}(f_{0},\partial
A)-{\mathrm{Tr}}^{+}(g_{0},\partial A)\|_{L^{1}(\partial A)}\,,$

where $f_{0},g_{0}$ denote the zero-extensions of, respectively, $f$ and $g$
on $A\setminus\Omega$.

###### Proof.

Given $\varepsilon>0$, there exists $h\in BV_{\mathop{\mathrm{loc}}}(\Omega)$
with $\mathop{\mathrm{spt}}(h-f)\subset\\!\subset A$, such that

$\displaystyle\Psi_{\Omega}(f,A)$
$\displaystyle\leq|Df|(A\cap\Omega)-|Dh|(A\cap\Omega)+\varepsilon$ (3.24)
$\displaystyle\leq\big{|}|Df|(A\cap\Omega)-|Dg|(A\cap\Omega)\big{|}+\Psi_{\Omega}(g,A)+|D\tilde{h}|(A\cap\Omega)-|Dh|(A\cap\Omega)+\varepsilon\,,$

where $\tilde{h}\in BV_{\mathop{\mathrm{loc}}}(\Omega)$ will be suitably
chosen, so that in particular
$\mathop{\mathrm{spt}}(\tilde{h}-g)\subset\\!\subset A$. For the definition of
$\tilde{h}$, we claim that it is possible to construct a sequence $A^{(k)}$ of
inner parallel sets of $A$ that converge to $A$, for which $|Df_{0}|(\partial
A^{(k)})=|Dg_{0}|(\partial A^{(k)})=0$ and, moreover,

$\lim_{k}\int_{\partial A^{(k)}}|{\mathrm{Tr}}^{+}(f_{0}-g_{0},\partial
A^{(k)})|\,d{\mathcal{H}}^{n-1}=\int_{\partial
A}|{\mathrm{Tr}}^{+}(f_{0}-g_{0},\partial A)|\,d{\mathcal{H}}^{n-1}\,.$ (3.25)

For the proof of (3.25) we argue as follows. Since $A$ is of class $C^{2}$,
there exists $\delta>0$ such that, for all $0<t<\delta$, the map
$\zeta_{t}(x)=x+t\nu_{A}(x)$ is a diffeomorphism of class $C^{1}$ between
$\partial A$ and the boundary $\partial A_{t}$ of the inner parallel set
$A_{t}=\\{x\in A:\ \mathop{\mathrm{dist}}(x,\partial A)>t\\}$. Now, we
consider two sequences $f_{0,j},g_{0,j}$ of smooth approximations of
$f_{0},g_{0}$ on $A$, with traces ${\mathrm{Tr}}^{+}(f_{0,j},\partial
A)={\mathrm{Tr}}^{+}(f_{0},\partial A)$ and
${\mathrm{Tr}}^{+}(g_{0,j},\partial A)={\mathrm{Tr}}^{+}(g_{0},\partial A)$,
respectively (see Remark 2.10). By inspecting the proof of Anzellotti-
Giaquinta’s approximation theorem, it is not restrictive to ask that the
sequences $f_{0,j},g_{0,j}$ also satisfy

$\displaystyle\int_{A\setminus
A_{1/k}}(|f_{0,j}-f_{0}|+|g_{0,j}-g_{0}|)\,dx\leq\frac{1}{k^{2}}\,,$ (3.26)

for all $j$ and for $k>\delta^{-1}$. We note that the tangential Jacobian of
$\zeta_{t}$ satisfies $J\zeta_{t}(x)=1+O(t)$, hence the area formula gives

$\int_{\partial
A_{t}}|f_{0,j}(y)-g_{0,j}(y)|\,d{\mathcal{H}}^{n-1}(y)=(1+O(t))\int_{\partial
A}|f_{0,j}(\zeta_{t}(x))-g_{0,j}(\zeta_{t}(x))|\,d{\mathcal{H}}^{n-1}(x)\,.$
(3.27)

As $t\to 0^{+}$ we have $\zeta_{t}(x)\to x$ uniformly. Therefore, by following
the same Cauchy-sequence argument as in the classical construction of the
trace (see, e.g., [16]), the compositions $f_{0,j}(\zeta_{t}(x))$ and
$g_{0,j}(\zeta_{t}(x))$ can be shown to converge in $L^{1}(\partial A)$ to
some limits $\hat{f}_{0,j}$ and $\hat{g}_{0,j}$, respectively. Hence (3.27)
implies

$\lim_{t\to 0^{+}}\int_{\partial
A_{t}}|f_{0,j}(y)-g_{0,j}(y)|\,d{\mathcal{H}}^{n-1}(y)=\int_{\partial
A}|\hat{f}_{0,j}(x)-\hat{g}_{0,j}(x)|\,d{\mathcal{H}}^{n-1}(x)\,.$ (3.28)

At the same time, if we choose a vector field $\xi\in
C^{1}(\mathbb{R}^{n};\mathbb{R}^{n})$ and set either $u_{j}=f_{0,j}$ or
$u_{j}=g_{0,j}$, by Gauss-Green Theorem we obtain

$\displaystyle\int_{A_{t}}(u_{j}\mathop{\mathrm{div}}\xi+\nabla
u_{j}\cdot\xi)\,dx$ $\displaystyle=-\int_{\partial
A_{t}}u_{j}\,\xi\cdot\nu_{A_{t}}\,d{\mathcal{H}}^{n-1}$
$\displaystyle=-(1+O(t))\int_{\partial
A}u_{j}(\zeta_{t}(x))\,\xi(\zeta_{t}(x))\cdot\nu_{A}(x)\,d{\mathcal{H}}^{n-1}(x)\,,$

hence taking the limit as $t\to 0^{+}$ gives

$\displaystyle\int_{A}(u_{j}\mathop{\mathrm{div}}\xi+\nabla
u_{j}\cdot\xi)\,dx$ $\displaystyle=-\int_{\partial
A}\hat{u}_{j}\,\xi\cdot\nu_{A}\,d{\mathcal{H}}^{n-1}\,,$

which means that $\hat{f}_{0,j}$ and $\hat{g}_{0,j}$ coincide, respectively,
with ${\mathrm{Tr}}^{+}(f_{0},\partial A)$ and
${\mathrm{Tr}}^{+}(g_{0},\partial A)$ up to ${\mathcal{H}}^{n-1}$-null sets
and for all $j$, by the uniqueness of the trace. We can thus rewrite (3.28) as

$\lim_{t\to 0^{+}}\int_{\partial
A_{t}}|f_{0,j}(y)-g_{0,j}(y)|\,d{\mathcal{H}}^{n-1}(y)=\int_{\partial
A}|{\mathrm{Tr}}^{+}(f_{0}-g_{0},\partial A)(x)|\,d{\mathcal{H}}^{n-1}(x)\,.$
(3.29)

To get (3.25) from (3.29), we must choose $A^{(k)}$ appropriately. To this
aim, we apply the coarea formula to the integral in (3.26) and average the
resulting inequality, deducing the existence of $0<t_{k}<1/k$ such that for
all $j$

$\int_{\partial A^{(k)}}(|f_{0,j}-{\mathrm{Tr}}(f_{0},\partial
A^{(k)})|+|g_{0,j}-{\mathrm{Tr}}(g_{0},\partial
A^{(k)})|)\,d{\mathcal{H}}^{n-1}\leq\frac{1}{k}\,,$ (3.30)

where we have set $A^{(k)}=A_{t_{k}}$. By the triangle inequality and (3.30)
we obtain

$\displaystyle\int_{\partial A^{(k)}}|{\mathrm{Tr}}^{+}(f_{0}-g_{0},\partial
A^{(k)})|\,d{\mathcal{H}}^{n-1}$ $\displaystyle\leq\int_{\partial
A^{(k)}}|f_{0,j}-g_{0,j}|\,d{\mathcal{H}}^{n-1}+\frac{1}{k}\,,$

which gives (3.25) at once from (3.29).

Now we observe that $|Df_{0}|(\partial A^{(k)})=|Dg_{0}|(\partial A^{(k)})=0$
because the inner and outer traces of $f_{0}$ and $g_{0}$ on $\partial
A^{(k)}$ coincide, hence we can define

$\tilde{h}=h\mathbf{1}_{A^{(k)}}+g\mathbf{1}_{A\setminus A^{(k)}}\,.$

Note that $\mathop{\mathrm{spt}}(\tilde{h}-g)\subset\\!\subset A$ and, if $k$
is large enough, $\mathop{\mathrm{spt}}(h-f)\subset\\!\subset A^{(k)}$, so
that

$\displaystyle|D\tilde{h}|(A\cap\Omega)$
$\displaystyle\leq|Dh|(A^{(k)}\cap\Omega)+|Dg_{0}|(A\setminus\overline{A^{(k)}})+\int_{\partial
A^{(k)}}|{\mathrm{Tr}}(f_{0}-g_{0},\partial
A^{(k)})|\,d{\mathcal{H}}^{n-1}\,.$

By choosing $k$ large enough we obtain $|Dg_{0}|(A\setminus
A^{(k)})<\varepsilon$ and thus

$|D\tilde{h}|(A\cap\Omega)-|Dh|(A\cap\Omega)\leq\int_{\partial
A}|{\mathrm{Tr}}^{+}(f_{0}-g_{0},\partial
A)|\,d{\mathcal{H}}^{n-1}+\varepsilon.$ (3.31)

By combining (3.24) and (3.31), we get

$\displaystyle\Psi_{\Omega}(f,A)$
$\displaystyle\leq\big{|}|Df|(A\cap\Omega)-|Dg|(A\cap\Omega)\big{|}+\Psi_{\Omega}(g,A)+\int_{\partial
A}|{\mathrm{Tr}}^{+}(f_{0}-g_{0},\partial
A)|\,d{\mathcal{H}}^{n-1}+2\varepsilon\,.$

Since $\varepsilon$ is arbitrary, we conclude

$\displaystyle\Psi_{\Omega}(f,A)-\Psi_{\Omega}(g,A)$
$\displaystyle\leq\big{|}|Df|(A\cap\Omega)-|Dg|(A\cap\Omega)\big{|}+\int_{\partial
A}|{\mathrm{Tr}}^{+}(f_{0}-g_{0},\partial A)|\,d{\mathcal{H}}^{n-1}$

and, by exchanging the role of $f$ and $g$ in the argument above, we obtain
(3.23). ∎

###  3.2. Density estimates

In this subsection we establish perimeter and volume density estimates for
almost-minimizers at a point either in $\Omega$ or on $\partial\Omega$.

######  Lemma 3.8.

Let $\Omega\subset\mathbb{R}^{n}$ be an open set with a Lipschitz boundary.
Let $E$ be an almost-minimizer in $\Omega$, and let $x\in\overline{\Omega}$.
Assume that $P_{\Omega}(E;B_{r}(x))>0$ for all $r>0$. Then, there exist
constants $C\geq 1,\ r_{0}>0$, depending on $\Omega$, $E$ and $x$, such that

$\displaystyle C^{-1}r^{n-1}\leq P_{\Omega}(E;B_{r}(x))\leq Cr^{n-1}$ (3.32)
$\displaystyle\min\big{(}|E\cap B_{r}(x)\cap\Omega|,\
|(B_{r}(x)\cap\Omega)\setminus E|\big{)}\geq C^{-1}r^{n}\,,$ (3.33)

for all $0<r<r_{0}$.

###### Proof.

Up to a translation, we assume that $x=0$. We start proving (3.33). Given
$r>0$ we set

$m(r):=|B_{r}\cap\Omega\cap E|\,,\quad\mu(r):=|B_{r}\cap\Omega\setminus E|\,.$

Both $m$ and $\mu$ are non-decreasing, thus differentiable for almost all
$r>0$. By [24, Example 13.3], for almost all $r>0$, we have

$m^{\prime}(r)={\mathcal{H}}^{n-1}(E\cap\partial
B_{r}\cap\Omega)\,,\quad\mu^{\prime}(r)={\mathcal{H}}^{n-1}(\partial
B_{r}\cap\Omega\setminus E)\,.$

Now, up to an isometry, and for $r_{0}$ sufficiently small and $L\geq 1$
sufficiently large, if we set
$Q_{L,r_{0}}=B^{\prime}_{r_{0}}\times(-Lr_{0},Lr_{0})$ and
$\Omega_{L,r_{0}}:=\Omega\cap Q_{L,r_{0}}$, we have that $\Omega_{L,r_{0}}$ is
connected and has a Lipschitz boundary, hence it supports a relative
isoperimetric inequality of the form

$P(F;\Omega_{L,r_{0}})\geq
C_{0}\min\\{|F\cap\Omega_{L,r_{0}}|,|\Omega_{L,r_{0}}\setminus
F|\\}^{\frac{n-1}{n}}\,.$ (3.34)

From (3.34) and the fact that $B_{r}\subset\\!\subset Q_{L,r_{0}}$ for all
$0<r<r_{0}$, we infer that

$m^{\prime}(r)+P_{\Omega}(E;B_{r})=P(E\cap B_{r};\Omega_{L,r_{0}})\geq
C_{0}\min\\{m(r),\mu(r)\\}^{\frac{n-1}{n}}\,.$ (3.35)

and

$\mu^{\prime}(r)+P_{\Omega}(E;B_{r})=P(B_{r}\setminus E;\Omega_{L,r_{0}})\geq
C_{0}\min\\{m(r),\mu(r)\\}^{\frac{n-1}{n}}$ (3.36)

for almost all $0<r<r_{0}$. Set $0<t<r$ and define the competitor

$F_{t}=\begin{cases}E\cup B_{t}\cap\Omega&\text{if }m(t)>\mu(t),\\\ E\setminus
B_{t}\cap\Omega&\text{otherwise.}\end{cases}$

We note that in the first case $F_{t}\Delta E=B_{t}\cap\Omega\setminus E$,
while in the second case $F_{t}\Delta E=B_{t}\cap\Omega\cap E$. In any case,
we have $F_{t}\Delta E\subset\subset B_{r}\cap\Omega$. Thus, by the almost-
minimality of $E$ in $\Omega$, and for almost all $0<t<r$, we infer that
either

$\displaystyle P_{\Omega}(E;B_{r})$ $\displaystyle\leq
P_{\Omega}(F_{t};B_{r})+\mu(t)^{\frac{n-1}{n}}\psi(r)$ (3.37)
$\displaystyle\leq
P_{\Omega}(E;B_{r}\setminus\overline{B_{t}})+{\mathcal{H}}^{n-1}(\partial
B_{t}\cap\Omega\setminus E)+\mu(r)^{\frac{n-1}{n}}\psi(r)\,,$

or

$\displaystyle P_{\Omega}(E;B_{r})$ $\displaystyle\leq
P_{\Omega}(F_{t};B_{r})+m(t)^{\frac{n-1}{n}}\psi(r)$ (3.38) $\displaystyle\leq
P_{\Omega}(E;B_{r}\setminus\overline{B_{t}})+{\mathcal{H}}^{n-1}(\partial
B_{t}\cap\Omega\cap E)+m(r)^{\frac{n-1}{n}}\psi(r)\,.$

where $\psi(r):=\psi_{\Omega}(E;0,r)$. Taking the limit as $t\nearrow r$ in
(3.37) and (3.38), and using (3.35), we deduce that, if $m(r)>\mu(r)$, then
for almost all $0<r<r_{0}$ we have

$2\mu^{\prime}(r)+\mu(r)^{\frac{n-1}{n}}\psi(r)\geq
C_{0}\mu(r)^{\frac{n-1}{n}}\,,$

while otherwise, by (3.36), we have

$2m^{\prime}(r)+m(r)^{\frac{n-1}{n}}\psi(r)\geq C_{0}m(r)^{\frac{n-1}{n}}\,.$

Therefore, calling $s(r):=\min\\{m(r),\mu(r)\\}$ and owing to the
infinitesimality of $\psi(r)$ as $r\to 0$, we obtain

$\dfrac{s^{\prime}(r)}{s(r)^{\frac{n-1}{n}}}\geq C\,,$

for $0<r<r_{0}$ and for some explicit constant $C>0$ depending on $C_{0}$.
Integrating this inequality on the interval $(0,r)$ we obtain (3.33). Then,
the first inequality in (3.32) follows from (3.33) and (3.35). Finally, the
second inequality in (3.32) follows from the observation that, taking the
limit as $t\nearrow r$ in (3.37) and possibly redefining $\overline{r}$ and
$C$, we have

$\displaystyle P_{\Omega}(E;B_{r})$ $\displaystyle\leq{\mathcal{H}}(\partial
B_{r}\cap\Omega\setminus E)+\mu(r)^{\frac{n-1}{n}}\psi(r)\leq Cr^{n-1}\,,$

for every $0<r<r_{0}$. ∎

##  4\. The visibility property

In this section, we introduce the visibility property and its main
consequences. In what follows, $\Omega\subset\mathbb{R}^{n}$ denotes an open
set with Lipschitz boundary such that $0\in\partial\Omega$ and
$\partial\Omega$ is a graph in a neighborhood of $0$, as in (2.16). For
notational convenience, we will only consider the visibility property at $0$,
but of course, we could equally define the property at a generic point of
$\partial\Omega$.

######  Definition 4.1.

We say that $\Omega$ satisfies the visibility property provided there exist
$T>0$ and a function $u\in C^{1}([0,T))$555By $u\in C^{1}([0,T))$ we mean that
$u\in C^{1}(0,T)$ and there exist finite the limits of $u(t)$ and
$u^{\prime}(t)$ as $t\to 0$. such that:

  * (V1)

$u(0)=u^{\prime}(0)=0$ and $0\leq u^{\prime}\leq 2^{-1}$;

  * (V2)

The function

$\gamma_{u}(t):=t^{-1}\sup_{0<s\leq t}\sqrt{\frac{u(s)}{s}+u^{\prime}(s)}$

is summable on $(0,T)$;

  * (V3)

for all $0<t<T$, the segment joining the point $U_{t}=-u(t)\,e_{n}$ with a
point $x$ belonging to $\partial\Omega\cap B_{t}$ does not intersect $\Omega$.

######  Remark 4.2.

We note that (V1) and (V2) imply that $u(t)=o(t)$ and

$t\gamma_{u}(t)\to 0\,,$ (4.1)

as $t\to 0$. Moreover, the summability of $\gamma_{u}(t)$ implies that of
$t^{-2}u(t)$, indeed $0\leq u(t)\leq t/2$ by (V1) and thus

$0\leq\frac{u(t)}{t^{2}}=t^{-1}\frac{u(t)}{t}\leq
t^{-1}\sqrt{\frac{u(t)}{t}}\leq\gamma_{u}(t)\,.$ (4.2)

In the following proposition, we rewrite the assumption (V3) in the form of a
property involving the functions $\omega(x^{\prime})$ and $u(t)$. This will be
particularly useful when checking the visibility property for relevant classes
of domains (see the examples at the end of the section).

######  Proposition 4.3.

$\Omega$ satisfies the property (V3) in Definition 4.1 if and only if, for any
$\nu\in\partial B^{\prime}_{1}$ and for all $0<t<T$, the slope $m_{t}(s)$ of
the line connecting $U_{t}$ with $(s,\omega(s\nu))$, that is given by

$m_{t}(s)=\dfrac{\omega(s\nu)+u(t)}{s}\,,$

is non-increasing as a function of $s$, for $s>0$ such that
$s^{2}+\omega(s\nu)^{2}<t^{2}$.

###### Proof.

Let us assume that (V3) holds, and set $\omega_{\nu}(s)=\omega(s\nu)$ for more
simplicity. By contradiction, let $s_{1}<s_{2}$ be such that
$s_{i}^{2}+\omega_{\nu}(s_{i})^{2}<t^{2}$, for $i=1,$ $2$, and
$m_{t}(s_{1})<m_{t}(s_{2})$. By definition of $m_{t}$, we have

$\dfrac{\omega_{\nu}(s_{1})+u(t)}{s_{1}}<\dfrac{\omega_{\nu}(s_{2})+u(t)}{s_{2}}\,,$

and so equivalently

$\omega_{\nu}(s_{1})<\dfrac{s_{1}}{s_{2}}\big{(}\omega_{\nu}(s_{2})+u(t)\big{)}-u(t)\,.$

This implies that the point

$x=\left(s_{1}\nu,\dfrac{s_{1}}{s_{2}}\big{(}\omega_{\nu}(s_{2})+u(t)\big{)}-u(t)\right)$

is internal to $\Omega$ and lies on the segment connecting
$(s_{2}\nu,\omega_{\nu}(s_{2}))$. This contradicts (V3).

Conversely, let us suppose that $m_{t}(s)$ is non-increasing in $s$, for $s>0$
such that $s^{2}+\omega_{\nu}(s)^{2}<t^{2}$. Set
$P(s)=(s\nu,\omega_{\nu}(s))$, then, arguing by contradiction, assume that
$s_{2}>0$ is such that $s_{2}^{2}+\omega_{\nu}(s_{2})^{2}<t^{2}$ and there
exists $\lambda\in(0,1)$ with the property

$(1-\lambda)U_{t}+\lambda P(s_{2})=(\lambda
s_{2},\lambda(u(t)+\omega_{\nu}(s_{2}))-u(t))\in\Omega\,.$

Then $\omega_{\nu}(\lambda s_{2})<\lambda(u(t)+\omega_{\nu}(s_{2}))-u(t)$. By
continuity, for all $\delta>0$ there exists $\lambda_{\delta}\in(\lambda,1)$
such that

$\lambda_{\delta}(u(t)+\omega_{\nu}(s_{2}))-u(t)-\delta<\omega_{\nu}(\lambda_{\delta}\,s_{2})<\lambda_{\delta}(u(t)+\omega_{\nu}(s_{2}))-u(t)\,.$
(4.3)

Since the segment $[U_{t},P(s_{2})]$ is compactly contained in $B_{t}$, by
(4.3), we can pick $\delta>0$ small enough and a correspondent
$\lambda_{\delta}$ such that

$P(\lambda_{\delta}s_{2})\in B_{t}\,.$ (4.4)

Let $s_{1}=\lambda_{\delta}s_{2}$. We observe that (4.4) and (4.3) imply

$m_{t}(s_{1})<m_{t}(s_{2})\,,\qquad
s_{1}^{2}+\omega_{\nu}(s_{1})^{2}<t^{2}\,,$

and this contradicts our hypothesis. This completes the proof of the
proposition. ∎

######  Corollary 4.4.

Assume that $\omega$ satisfies

$\langle
x^{\prime},\nabla\omega(x^{\prime})\rangle\leq\omega(x^{\prime})+u(|x^{\prime}|)\qquad\text{
for a.e. }x^{\prime}\in B^{\prime}_{\rho}\,,$ (4.5)

where $u:(0,T)\rightarrow\mathbb{R}$ is a non-decreasing function satisfying
properties (V1) and (V2). Then $\Omega$ satisfies the visibility property.

###### Proof.

Since $\omega$ is Lipschitz, the function $m_{t}$ defined in the statement of
Proposition 4.3 is a.e. differentiable, thus $m_{t}$ is non-increasing if and
only if $m_{t}^{\prime}(s)\leq 0$ at almost every $s$. We observe that

$m_{t}^{\prime}(s)=\dfrac{\omega_{\nu}^{\prime}(s)}{s}-\dfrac{\omega_{\nu}(s)+u(t)}{s^{2}}\,,$

thus $m_{t}^{\prime}\leq 0$ if and only if

$s\,\omega_{\nu}^{\prime}(s)\leq\omega_{\nu}(s)+u(t)\,.$ (4.6)

The hypothesis (4.5) implies that

$s\,\omega_{\nu}^{\prime}(s)\leq\omega_{\nu}(s)+u(s)\qquad\text{for almost all
$0<s<T\,.$}$ (4.7)

Hence if $s>0$ is such that $s^{2}+\omega_{\nu}(s)^{2}<t^{2}$, then $s<t$, and
by (4.7), since $u$ is non-decreasing, we obtain

$s\,\omega_{\nu}^{\prime}(s)\leq\omega_{\nu}(s)+u(t)\,,$

that is precisely (4.6). Consequently, (V3) is verified thanks to Proposition
4.3. ∎

###  4.1. Existence of the tangent cone

An important consequence of the visibility property is the existence of the
tangent cone to $\Omega$ at $0$.

######  Proposition 4.5.

Let $\Omega\subset\mathbb{R}^{n}$ satisfy the visibility property. Then there
exists the tangent cone to $\Omega$ at $0$, denoted by $\Omega_{0}$. More
precisely, if we set $\Omega_{s}:=s^{-1}\Omega$ for $s>0$, we obtain

$\lim_{s\rightarrow 0^{+}}\mathop{\mathrm{dist}_{\mathcal{H}}}(\Omega_{s}\cap
B_{R},\Omega_{0}\cap B_{R})=0\,,\qquad\text{for all $R>0$.}$ (4.8)

###### Proof.

Let us fix $\nu\in\partial B^{\prime}_{1}$, and let

$\omega_{\nu}(s)=\omega(s\nu)\,,\qquad s\geq 0\,,$

where $\omega$ is the function realizing (2.16). Let $s>0$ be a point where
$\omega_{\nu}$ is differentiable and let $t=\sqrt{s^{2}+\omega_{\nu}(s)^{2}}$.
By (V3) in Definition 4.1, we deduce that the slope of the line connecting
$(s,\omega_{\nu}(s))$ with $U_{t}$ needs to be bounded below by
$\omega_{\nu}^{\prime}(s)$, that is,

$\omega_{\nu}^{\prime}(s)\leq\dfrac{\omega_{\nu}(s)+u(t)}{s}\,.$

We set $L=\sqrt{1+\mathop{\mathrm{Lip}}(\omega)^{2}}$ and observe that $t\leq
Ls$. Since $u$ is non-decreasing by (V1), we infer

$\omega_{\nu}^{\prime}(s)\leq\dfrac{\omega_{\nu}(s)+u(Ls)}{s}\,,$

hence we get

$\left(\dfrac{\omega_{\nu}(s)}{s}\right)^{\prime}=\dfrac{\omega_{\nu}^{\prime}(s)}{s}-\dfrac{\omega_{\nu}(s)}{s^{2}}\leq\dfrac{u(Ls)}{s^{2}}\leq
L^{2}\dfrac{u(Ls)}{(Ls)^{2}}\,.$ (4.9)

We integrate (4.9) between $s_{1}<s_{2}$ thanks to (V2) (see Remark 4.2), and
obtain

$\dfrac{\omega_{\nu}(s_{2})}{s_{2}}-\dfrac{\omega_{\nu}(s_{1})}{s_{1}}\leq
L^{2}\int_{s_{1}}^{s_{2}}\dfrac{u(Ls)}{(Ls)^{2}}\,ds=L\int_{Ls_{1}}^{Ls_{2}}\dfrac{u(t)}{t^{2}}\,dt\,.$

Thus we conclude that the function

$s\mapsto\dfrac{\omega_{\nu}(s)}{s}-L\int_{0}^{Ls}\dfrac{u(t)}{t^{2}}\,dt$

is monotonically non-increasing in $s$ and bounded by
$\mathop{\mathrm{Lip}}(\omega)+L\int_{0}^{Ls}t^{-2}u(t)\,dt$, for
$0<s<L^{-1}T$. Therefore there exists

$D_{\nu}^{+}\omega(0):=\lim_{s\rightarrow
0^{+}}\dfrac{\omega_{\nu}(s)}{s}=\lim_{s\rightarrow
0^{+}}\dfrac{\omega_{\nu}(s)}{s}-L\int_{0}^{Ls}\dfrac{u(t)}{t^{2}}\,dt\in\mathbb{R}\,.$

Let us define

$\omega_{0}(x^{\prime})=\begin{cases}|x^{\prime}|D_{\frac{x^{\prime}}{|x^{\prime}|}}^{+}\omega(0)&\text{if
}x^{\prime}\neq 0,\\\ 0&\text{if }x^{\prime}=0\,.\end{cases}$

The function $\omega_{0}$ is $1$-homogeneous, therefore the set

$\Omega_{0}=\\{x\in\mathbb{R}^{n}\,:\,x_{n}>\omega_{0}(x^{\prime})\\}$

is a cone with vertex at $0$. Now, for all $s>0$, we set

$\omega_{s}(x^{\prime})=\begin{cases}\dfrac{\omega(sx^{\prime})}{s}&\text{if
}x^{\prime}\neq 0,\\\ 0&\text{if }x^{\prime}=0\,.\end{cases}$ (4.10)

It is immediate to observe that $\omega_{s}(0)=0$ and, setting
$t=s|x^{\prime}|$,

$\displaystyle\omega_{s}(x^{\prime})=\dfrac{\omega\left(t(x^{\prime}/|x^{\prime}|)\right)}{t}\,|x^{\prime}|\rightarrow|x^{\prime}|\,D^{+}_{\frac{x^{\prime}}{|x^{\prime}|}}\omega(0)=\omega_{0}(x^{\prime})\,,\qquad\text{as
$s\rightarrow 0^{+}\,.$}$

Since $\\{\omega_{s}\\}_{s>0}$ is a one-parameter family of locally equi-
bounded and equi-Lipschitz functions that pointwisely converge to $\omega_{0}$
as $s\rightarrow 0^{+}$, we can apply Lemma 2.1 to conclude that this
convergence is locally uniform. This easily implies the Hausdorff convergence
stated in (4.8). ∎

######  Remark 4.6.

We note that for the proof of Proposition 4.5, the hypothesis (V2) of
Definition 4.1 can be replaced by the weaker hypothesis of summability of
$t^{-2}u(t)$ on $(0,T)$. We also observe that, if $\Omega$ is convex, then the
existence of the tangent cone $\Omega_{0}$ is always granted, even though (V2)
may not be satisfied.

###  4.2. An off-centric visibility property

The next lemma shows that the assumption (V3) in Definition 4.1 can be
replaced by an equivalent assumption, where off-centric balls are taken
instead of balls centered at $0$. This off-centric visibility property will be
useful later on.

######  Lemma 4.7.

The following properties are equivalent:

  * (i)

$\Omega$ satisfies the visibility property;

  * (ii)

There exist $R>0$ and a function $v\in C^{1}(0,R)$ satisfying properties (V1),
(V2) of Definition 4.1, and

(V3’) for all $0<r<R$, any segment joining the point $V_{r}=-v(r)\,e_{n}$ with
a point $x$ belonging to $\partial\Omega\cap B_{r}(V_{r})$ does not intersect
$\Omega$.

###### Proof.

We prove that (i) implies (ii). Let $z(t)=t-u(t)$ where $u$ is as in
Definition 4.1. We can find $0<T^{\prime}<T$ such that $z(t)$ is an increasing
$C^{1}$ diffeomorphism of the interval $(0,T^{\prime})$ with the property

$\dfrac{1}{2}\,t\leq z(t)\leq t\,.$

Let $R=z(T^{\prime})$. Then we can consider the inverse $z^{-1}$ of $z$ in
$(0,R)$, that is an increasing diffeomorphism such that

$r\leq z^{-1}(r)\leq 2\,r\,.$ (4.11)

Setting $U_{t}=-u(t)e_{n}$, it follows that $B_{t-u(t)}(U_{t})\subset B_{t}$,
for all $0<t<T$, thus (V3) holds for all points $x\in\partial\Omega\cap
B_{t-u(t)}(U_{t})$. We then have that

$B_{r}(U_{z^{-1}(r)})\subset B_{z^{-1}(r)}\,,\qquad\text{for any $0<r<R$.}$

Any line segment joining $x\in\partial\Omega\cap B_{z^{-1}(r)}$ with
$U_{z^{-1}(r)}$ does not intersect $\Omega$, hence the same property holds for
any $x\in\partial\Omega\cap B_{r}(U_{z^{-1}(r)})$. Let

$v(r)=u(z^{-1}(r))\,,\qquad 0<r<R\,.$

It is clear that (V3’) holds. By (4.11), up to possibly reducing the value of
$R$, (V2) and $v^{\prime}\leq 2^{-1}$ follow. Since both $u$ and $z^{-1}$ are
non-decreasing, $v$ is non-decreasing, thus also (V1) is satisfied. A
completely analogous argument shows that (ii) implies (i), and the proof is
concluded. ∎

For all $0<r<R$, we define

${\mathfrak{C}}_{r}=\\{V_{r}+t(z-V_{r})\,:\,z\in\partial
B_{r}(V_{r})\cap\Omega\,,\,t>0\\}\,.$ (4.12)

The set ${\mathfrak{C}}_{r}$ is an open cone with vertex at $V_{r}$.

######  Lemma 4.8.

Assume that $\Omega$ satisfies the (off-centric) visibility property. Then,
for all $0<r<R$, the cone ${\mathfrak{C}}_{r}$ contains $\Omega\cap
B_{r}(V_{r})$.

###### Proof.

By contradiction, let $x\in(\Omega\cap
B_{r}(V_{r}))\setminus{\mathfrak{C}}_{r}$. For all $0<t\leq r$, let

$x_{t}=V_{r}+t\,\dfrac{x-V_{r}}{|x-V_{r}|}\,,$

and note that $x=x_{t_{0}}$ for a suitable $0<t_{0}<r$. It’s clear that
$x_{r}\notin\Omega$, otherwise $x_{t_{0}}$ would belong to
${\mathfrak{C}}_{r}$, for all $t>0$, which is against our assumption. We can
then select a value $s\in(t_{0},r]$ such that $x_{s}\in\partial\Omega$. This
leads to a contradiction with the visibility because the segment joining
$V_{r}$ and $x_{s}\in B_{r}(V_{r})\cap\partial\Omega$ contains
$x_{t_{0}}=x\in\Omega$. ∎

###  4.3. Foliation by off-centric spheres

Let us consider the family of off-centric balls $B_{r}(V_{r})$, with
$V_{r}=-v(r)e_{n}$, for $0<r<R$. By the Implicit Function Theorem we can
easily show the existence of a smooth function $\phi$, such that $\partial
B_{r}(V_{r})$ is the $r$-level set of $\phi$. This means that the punctured
ball $B_{R}(V_{R})\setminus\\{0\\}$ is foliated by the spheres $\partial
B_{s}(V_{s})=\phi^{-1}(s)$ for $0<s<R$.

######  Lemma 4.9.

There exists a function $\phi\in C^{1}(B_{R}(V_{R})\setminus\\{0\\})$ such
that $0\leq\phi<R$ and

$\partial B_{r}(V_{r})=\phi^{-1}(r)\,,\qquad\text{for any $0<r<R$.}$

In particular, for any $x\in B_{R}(V_{R})\setminus\\{0\\}$ we have

$\dfrac{\nabla\phi(x)}{|\nabla\phi(x)|}=\dfrac{x-V_{\phi(x)}}{|x-V_{\phi(x)}|}$
(4.13)

and

$\displaystyle\left|\nabla\phi(x)-\dfrac{x}{|x|}\right|\leq
4\,\sqrt{\dfrac{v(\phi(x))}{\phi(x)}+v^{\prime}(\phi(x))}\,.$ (4.14)

###### Proof.

If $v(r)$ is identically $0$, then there is nothing to prove because
$\phi(x)=|x|$ in this case. We then suppose $v\neq 0$. Let us start by proving
the existence of the function $\phi$. We observe that, for any $x\in
B_{R}(V_{R})$, there exists a unique $r=r_{x}\in[0,R)$ such that $x\in\partial
B_{r}(V_{r})$. Indeed, if $x=0$, we can take $r=0$. Otherwise, let
$F:(B_{R}(V_{R})\setminus\\{0\\})\times(0,R)\rightarrow\mathbb{R}$ be the
function defined by

$F(x,r)=|x-V_{r}|^{2}-r^{2}\,.$ (4.15)

It is immediate to observe that $F$ is continuous. Moreover,

$F(x,0)=|x|^{2}>0\quad\text{and}\quad F(x,R)<0\,,\qquad\text{for all }x\in
B_{R}(V_{R})\setminus\\{0\\}\,.$

Hence we can find $r\in(0,R)$ such that $F(x,r)=0$, i.e. such that
$x\in\partial B_{r}(V_{r})$. Let us show the uniqueness. Indeed, if $r$,
$r^{\prime}\in(0,R)$ have the property that

$x\in\partial B_{r}(V_{r})\cap\partial B_{r^{\prime}}(V_{r^{\prime}})\,,$

then we get

$|r-r^{\prime}|=||x-V_{r}|-|x-V_{r^{\prime}}||\leq|V_{r}-V_{r^{\prime}}|=|v(r)-v(r^{\prime})|\leq\dfrac{1}{2}\,|r-r^{\prime}|\,,$

thus we must have $r=r^{\prime}$. Now we can define $\phi(x)=r_{x}$. Let us
show that $\phi\in\mathcal{C}^{1}(B_{R}(R)\setminus\\{0\\})$. To do so, we
note that $\phi$ is implicitly defined by

$F(x,\phi(x))=0\,,$ (4.16)

where $F$ is the function defined in (4.15). Easy computations give

$\partial_{r}F(x,r)=-2r+2v^{\prime}(r)(x_{n}+v(r))\,.$

Therefore, if we assume $F(x,r)=0$ (that is, $x\in B_{r}(V_{r})$) we obtain

$\partial_{r}F(x,r)=-2r+2v^{\prime}(r)(x_{n}+v(r))\leq
2r+|x_{n}+v(r)|\leq-r\,,$

where the first inequality follows from the assumption $0\leq v^{\prime}\leq
2^{-1}$, while the second inequality from

$|x_{n}+v(r)|=|\langle x-V_{r},e_{n}\rangle|\leq|x-V_{r}|=r\,.$

By the Implicit Function Theorem we deduce that
$\phi\in\mathcal{C}^{1}(B_{R}(V_{R})\setminus\\{0\\})$. The identity (4.13) is
a consequence of the fact that, if $\phi(x)=r$, then the vector
$\nabla\phi(x)$ is orthogonal to the level set $\partial
B_{r}(V_{r})=\phi^{-1}(r)$ at $x$.

Let us now prove (4.14). We first observe that, if $\phi(x)=r>0$, then

$\nabla\phi(x)=-\dfrac{\partial_{x}F(x,r)}{\partial_{r}F(x,r)}=\dfrac{x+v(r)e_{n}}{r-v^{\prime}(r)(x_{n}+v(r))}$
(4.17)

Then (4.17) yields

$\displaystyle\left|\nabla\phi(x)-\dfrac{x}{|x|}\right|^{2}$
$\displaystyle=\dfrac{|x+ve_{n}|^{2}}{(r-v^{\prime}(x_{n}+v))^{2}}+1-2\,\left\langle\dfrac{x+ve_{n}}{r-v^{\prime}(x_{n}+v)},\dfrac{x}{|x|}\right\rangle$
$\displaystyle=\dfrac{|x+ve_{n}|^{2}}{(r-v^{\prime}(x_{n}+v))^{2}}+1-2\,\dfrac{|x+ve_{n}|^{2}-v(x_{n}+v)}{(r-v^{\prime}(x_{n}+v))|x|}$
$\displaystyle=\dfrac{|x|\,r^{2}+|x|(r-v^{\prime}(x_{n}+v))^{2}-2(r-v^{\prime}(x_{n}+v))[r^{2}-v(x_{n}+v)]}{|x|(r-v^{\prime}(x_{n}+v))^{2}}$
$\displaystyle=\dfrac{\frac{|x|}{r}+\frac{|x|}{r}\left(1-\frac{v^{\prime}(x_{n}+v)}{r}\right)^{2}-2\,\left(1-\frac{v^{\prime}(x_{n}+v)}{r}\right)\left(1-\frac{v}{r}\,\frac{x_{n}+v}{r}\right)}{\frac{|x|}{r}\left(1-\frac{v^{\prime}(x_{n}+v)}{r}\right)^{2}}$
$\displaystyle=\dfrac{1+\left(1-\frac{v^{\prime}(x_{n}+v)}{r}\right)^{2}-2\,\dfrac{r}{|x|}\left(1-\frac{v^{\prime}(x_{n}+v)}{r}\right)\left(1-\frac{v}{r}\,\frac{x_{n}+v}{r}\right)}{\left(1-\frac{v^{\prime}(x_{n}+v)}{r}\right)^{2}}\,,$
(4.18)

where the short forms $v=v(r)$ and $v^{\prime}=v^{\prime}(r)$ have been used.
Next we observe that $\big{|}|x|-r\big{|}=\big{|}x-|x-V_{r}|\big{|}$, hence

$r-v(r)\leq|x|\leq r+v(r)\,.$ (4.19)

Exploiting (4.19) in (4.3) and recalling that $v^{\prime}(r)\leq 1/2$ and
$|x_{n}+v(r)|\leq r$, we get

$\displaystyle\left|\nabla\phi(x)-\dfrac{x}{|x|}\right|^{2}$
$\displaystyle\leq\dfrac{2-2\,\left(1-\frac{v(r)}{r+v(r)}\right)(1-v^{\prime}(r))\left(1-\frac{v(r)}{r}\right)}{(1-v^{\prime}(r))^{2}}$
$\displaystyle\leq
8\left(1-\left(1-\frac{v(r)}{r}\right)^{2}(1-v^{\prime}(r))\right)$
$\displaystyle\leq 16\left(v^{\prime}(r)+\frac{v(r)}{r}\right)\,,$

and this concludes the proof. ∎

###  4.4. Some examples

We note that there are examples of sets with Lipschitz boundary that do not
satisfy the visibility property at $0$, like for instance the epigraph of the
function

$\omega(x)=\begin{cases}x^{2}\sin(|x|^{-1})&\text{if }x\neq 0\\\
0&\text{otherwise.}\end{cases}$

Indeed, one can easily check that at $x_{k}=\frac{1}{(2k+1)\pi}$ for
$k\in\mathbb{N}$ we have $\omega(x_{k})=0$ and $\omega^{\prime}(x_{k})=1$,
hence any visibility function $u$ for which (V3) holds must satisfy
$u(x_{k})\geq x_{k}$, which contradicts both (V1) and (V2) (see Remark 4.2).

In the following, we exhibit some examples of domains for which the visibility
holds. We recall that $\omega_{\nu}(s)=\omega(s\nu)$ for $s\geq 0$.

######  Example 4.10 (Lipschitz cones and outer star-shaped sets).

Let $\Omega$ be either a cone with respect to $0$, or such that its complement
$\mathbb{R}^{n}\setminus\Omega$ is locally star-shaped with respect to $0$. It
is immediate to check that $\Omega$ satisfies the visibility property with
visibility function $u(t)\equiv 0$.

######  Example 4.11 ($C^{1,\beta}$-sets).

Let $\Omega$ have $C^{1,\beta}$ boundary and assume $0\in\partial\Omega$. We
show that $\Omega$ satisfies the visibility property. Up to a rotation we can
assume that $\Omega$ admits a representation as in (2.16) with $\omega\in
C^{1,\beta}(B^{\prime}_{\rho})$. By Corollary 4.4, it is enough to show that
$\omega$ satisfies (4.5). Since $\nabla\omega$ is $\beta$-Hölder, we have

$\langle x^{\prime},\nabla\omega(x^{\prime})\rangle\leq\langle
x^{\prime},\nabla\omega(0)\rangle+u(|x^{\prime}|)\,.$ (4.20)

where $u(t)=C\,t^{1+\beta}$ for some $C>0$. Set
$\overline{\omega}(x^{\prime}):=\omega(x^{\prime})-\langle\nabla\omega(0),x^{\prime}\rangle$
and note that $\overline{\omega}$ is $C^{1,\beta}$, $\overline{\omega}(0)=0$,
$\nabla\overline{\omega}(0)=0$, and

$\displaystyle|\omega(x^{\prime})-\langle x^{\prime},\nabla\omega(0)\rangle|$
$\displaystyle\leq\max_{|y^{\prime}|\leq|x^{\prime}|}|\nabla\overline{\omega}(y^{\prime})||x^{\prime}|$
$\displaystyle\leq(|\nabla\overline{\omega}(0)|+C|x|^{\beta})|x^{\prime}|$
(4.21) $\displaystyle=C\,|x^{\prime}|^{1+\beta}\,.$

Putting together (4.20) and (4.11), we finally get

$\langle
x^{\prime},\nabla\omega(x^{\prime})\rangle\leq\omega(x^{\prime})+u(|x^{\prime}|),\,\qquad\text{
for a.e. }x^{\prime}\in B^{\prime}_{\rho}\,,$

that is precisely (4.5). Since trivially $u$ is non-decreasing and satisfies
(V2), by Corollary 4.4 we infer that $\Omega$ satisfies the visibility
property.

######  Example 4.12 (Convex sets satisfying (V2)).

Let $\Omega$ be a convex set with $0\in\partial\Omega$. For $s>0$, let
$\Omega_{s}:=s^{-1}\Omega$ and $\Omega_{0}:=\bigcup_{s>0}\Omega_{s}$. The set
$\Omega_{0}$ is the tangent cone to $\Omega$ at $0$. Let

$u(r):=\mathop{\mathrm{dist}_{\mathcal{H}}}(\Omega\cap B_{r},\Omega_{0}\cap
B_{r})\,,$ (4.22)

and assume that $u(r)$ satisfies (V2). We observe that $u$ is non-decreasing
in $r$. Let us prove that $\Omega$ satisfies the visibility property. As
before, we assume that $\Omega$ admits a graphical representation as in (2.16)
with the further property that $\omega:B^{\prime}_{\rho}\rightarrow\mathbb{R}$
is convex. Using the notations introduced in the proof of Proposition 4.5, by
the convexity of $\omega$, we can define

$\omega_{0}(x^{\prime})=\begin{cases}|x^{\prime}|D_{\frac{x^{\prime}}{|x^{\prime}|}}^{+}\omega(0)&\text{if
}x^{\prime}\neq 0,\\\ 0&\text{if }x^{\prime}=0\,,\end{cases}$

and deduce that

$\Omega_{0}=\\{x\in\mathbb{R}^{n}\,:\,x_{n}>\omega_{0}(x^{\prime})\\}\,.$

By the definition of $u$ given in (4.22), we have

$||\omega-\omega_{0}||_{L^{\infty}(B^{\prime}_{r})}\leq
Cu(r)\,,\qquad\text{for some $C>0\,.$}$ (4.23)

Owing to Corollary 4.4, the visibility property can be proved by showing that
(4.5) holds. Thanks to the convexity of $\omega$, for all $\nu\in\partial
B^{\prime}_{1}$, we have

$D^{+}_{\nu}\omega(0)\leq\omega^{\prime}_{\nu}\left(\frac{s}{2}^{+}\right):=\lim_{\sigma\rightarrow
0^{+}}\dfrac{\omega_{\nu}(s/2+\sigma)-\omega_{\nu}(s/2)}{\sigma}\,,\qquad\text{for
all $0\leq s<\rho\,.$}$ (4.24)

Moreover, for all $0<\sigma<s/2$, by (4.23) and the convexity of
$\omega_{\nu}$, we have

$\displaystyle\dfrac{\omega_{\nu}(s/2+\sigma)-\omega_{\nu}(s/2)}{\sigma}$
$\displaystyle\leq\dfrac{\omega_{\nu}(s)-\omega_{\nu}(s/2)}{s/2}$
$\displaystyle=2\,\dfrac{\omega_{\nu}(s)}{s}-\dfrac{\omega_{\nu}(s/2)}{s/2}$
$\displaystyle=2\,\dfrac{\omega_{\nu}(s)}{s}-2D^{+}_{\nu}\omega(0)+D^{+}_{\nu}\omega(0)-\dfrac{\omega_{\nu}(s/2)}{s/2}+D^{+}_{\nu}\omega(0)$
(4.25)
$\displaystyle=\dfrac{2}{s}\,\left(\omega_{\nu}(s)-\omega_{0}(s\nu)+\omega_{0}(s\nu/2)-\omega_{\nu}(s/2)\right)+D^{+}_{\nu}\omega(0)$
$\displaystyle\leq
D^{+}_{\nu}\omega(0)+\tilde{C}\,\dfrac{u(s)}{s}\,,\qquad\text{for some
$\tilde{C}>0\,.$}$

Putting together (4.24) and (4.12), we obtain

$\left|D^{+}_{\nu}\omega(0)-\omega^{\prime}_{\nu}\left(\frac{s}{2}^{+}\right)\right|\leq\tilde{C}\,\dfrac{u(s)}{s}\,.$

This suffices to conclude. In fact, if $x^{\prime}\in
B^{\prime}_{\rho}\setminus\\{0\\}$ is such that $\omega$ is differentiable at
$x^{\prime}$, setting $s:=|x^{\prime}|$, $\nu:=x^{\prime}/|x^{\prime}|$, for
some $C>0$, we achieve

$\displaystyle\langle x^{\prime},\nabla\omega(x^{\prime})\rangle$
$\displaystyle=\langle
s\nu,\nabla\omega(s\nu)\rangle=s\,\omega^{\prime}_{\nu}(s^{+})$
$\displaystyle\leq s\,D^{+}_{\nu}\omega(0)+C\,u(s)$
$\displaystyle\leq\omega(s\nu)+C\,u(s)=\omega(x^{\prime})+C\,u(|x^{\prime}|)\,.$

Since $\omega$ is convex, it is differentiable a.e. in $B^{\prime}_{\rho}$,
and so (4.5) is verified. Moreover, $u$ is non-decreasing and satisfies (V2)
by our assumption, thus we conclude.

######  Example 4.13.

For $i\in\mathbb{N}$, let $y_{i}=2^{-i}$ and $\tilde{y}_{i}=2^{-i}+4^{-i}$. We
observe that $y_{i+1}<\tilde{y}_{i+1}<y_{i}$. We define
$P_{i}=(y_{i},\tilde{y}_{i})$, $Q_{i}=(\tilde{y}_{i},\tilde{y}_{i})$, and
consider the polygonal curve formed by the segments $\overline{P_{i}Q_{i+1}}$,
$\overline{Q_{i+1}P_{i+1}}$, for all $i\geq 0$. It is immediate to observe
that this curve coincides with the graph of the function
$\omega:[0,1]\rightarrow\mathbb{R}$ defined by $\omega(0)=0$ and

$\omega(y)=\begin{cases}2^{-(i+1)}+4^{-(i+1)}&\text{if
}y\in(y_{i+1},\tilde{y}_{i+1}],\\\ a_{i}\,y-b_{i}&\text{if
}y\in(\tilde{y}_{i+1},y_{i}]\,,\end{cases}$ (4.26)

where

$a_{i}=\dfrac{1+3\cdot 2^{-(i+1)}}{1-2^{-(i+1)}}\,,\qquad
b_{i}=\dfrac{4^{-i}+2^{-(3i+1)}}{1-2^{-(i+1)}}\,.$

Figure 1. The graph of $\omega(y)$ "bounces" between the graphs of $y$ and
$y+y^{2}$.

Let $\Omega\subset\mathbb{R}^{2}$ be an open set such that

$\Omega\cap((-1,1)\times\mathbb{R})=\\{x=(y,z)\in\mathbb{R}^{2}\,:\,z>\omega(|y|)\\}\,.$

Let us show that $\Omega$ satisfies the visibility condition at $0$. Owing to
Corollary 4.4, it suffices to show the existence of a non-decreasing function
$u:(0,1)\rightarrow\mathbb{R}$ satisfying (V1), (V2) and such that

$y\,\omega^{\prime}(y)\leq\omega(y)+u(y)\,,\qquad\text{ for a.e.
}y\in(0,1)\,.$ (4.27)

By (4.26), for every $i\in\mathbb{N}$, we have

$y\,\omega^{\prime}(y)=\begin{cases}0&\text{if
}y\in(y_{i+1},\tilde{y}_{i+1}],\\\ a_{i}\,y&\text{if
}y\in(\tilde{y}_{i+1},y_{i}]\,.\end{cases}$

Thus, in order to verify (4.27), it suffices to choose a function $u=u(y)$
greater or equal than the function

$\overline{u}(y)=\begin{cases}0&\text{if }y\in(y_{i+1},\tilde{y}_{i+1}],\\\
b_{i}&\text{if }y\in(\tilde{y}_{i+1},y_{i}]\,.\end{cases}$

We look for $\alpha>0$ such that

$u_{\alpha}(y)=\alpha y^{2}\geq\overline{u}(y)\,.$ (4.28)

To obtain (4.28), it suffices to impose that

$u_{\alpha}(\tilde{y}_{i+1})=\alpha\,\tilde{y}_{i+1}^{2}\geq b_{i}\,.$

It is immediate to observe that

$\alpha\,\tilde{y}_{n+1}^{2}\geq\dfrac{\alpha}{4}\,2^{-2i}\,,\qquad b_{i}\leq
4\cdot 2^{-2i}\,,$

hence, if for instance we take $\alpha=16$, (4.28) holds. Since the function
$u(y)=16\,y^{2}$ fulfills (V1) and (V2), we conclude that $\Omega$ satisfies
the visibility condition at $0$.

##  5\. Free-boundary monotonicity

The present section is devoted to showing a free-boundary monotonicity
property for local almost-minimizers of $P_{\Omega}$ at a point
$x_{0}\in\partial\Omega$ satisfying the visibility property up to an isometry
(hence, from now on, we will directly assume $x_{0}=0$). Following the
notation of the previous section, given $0<r_{1}<r_{2}<R$, we recall that
$V_{r}=-v(r)e_{n}$ and define

$\mathcal{A}_{r_{1},r_{2}}:=B_{r_{2}}(V_{r_{2}})\setminus\overline{B_{r_{1}}(V_{r_{1}})}=\phi^{-1}(r_{1},r_{2})\,,$

where $\phi(x)$ is the function defined in Lemma 4.9. We also conveniently
introduce some further notation. Given $f\in
BV_{\mathop{\mathrm{loc}}}(\Omega)$ and $0<r_{1}<r_{2}$, we set

$\mu_{f}(r)=\dfrac{|Df|(\Omega\cap B_{r}(V_{r}))}{r^{n-1}}$

and

$\displaystyle G(f;r_{1},r_{2})$
$\displaystyle=\int_{r_{1}}^{r_{2}}\dfrac{n-1}{\rho^{n}}\,\int_{\Omega\cap
B_{\rho}(V_{\rho})}(|\nabla\phi|-1)\,d|Df|\,d\rho$ (5.1)
$\displaystyle\qquad+\dfrac{1}{r_{2}^{n-1}}\,\int_{\Omega\cap
B_{r_{2}}(V_{r_{2}})}(|\nabla\phi|-1)\,d|Df|-\dfrac{1}{r_{1}^{n-1}}\,\int_{\Omega\cap
B_{r_{1}}(V_{r_{1}})}(|\nabla\phi|-1)\,d|Df|\,.$

In the next proposition, we combine the visibility property with an upper
bound on $\mu_{f}(r)$ and obtain the finiteness of $\lim_{\rho\to
0}G(f;\rho,r)$.

######  Proposition 5.1.

Let $\Omega\subset\mathbb{R}^{n}$ satisfy the visibility property as in
Definition 4.1, and let $f\in BV_{\mathop{\mathrm{loc}}}(\Omega)$. Assume that
$\mu_{f}(r)\leq C$ for some constant $C>0$ and for all $r\in(0,R)$. Then for
$r\in(0,R)$ the limit

$\displaystyle G(f;r)$ $\displaystyle:=\lim_{\rho\to 0}G(f;\rho,r)$ (5.2)
$\displaystyle=\int_{0}^{r}\dfrac{n-1}{\rho^{n}}\,\int_{\Omega\cap
B_{\rho}(V_{\rho})}(|\nabla\phi|-1)\,d|Df|\,d\rho+\dfrac{1}{r^{n-1}}\,\int_{\Omega\cap
B_{r}(V_{r})}(|\nabla\phi|-1)\,d|Df|$

exists and is finite.

###### Proof.

By (4.14), for all $x\in B_{\rho}(V_{\rho})$ we have

$\displaystyle\left||\nabla\phi(x)|-1\right|$
$\displaystyle\leq\left|\nabla\phi(x)-\frac{x}{|x|}\right|\leq
4\sqrt{\frac{v(\phi(x))}{\phi(x)}+v^{\prime}(\phi(x))}\leq
4\sup_{0<r\leq\rho}\sqrt{\frac{v(r)}{r}+v^{\prime}(r)}=4\rho\gamma_{v}(\rho)\,,$

where $\gamma_{v}$ is the function defined in the visibility property (V2).
Then, using the upper bound on $\mu_{f}$, for $0<\rho<r<R$ we obtain

$\displaystyle\left|\rho^{1-n}\int_{\Omega\cap
B_{\rho}(V_{\rho})}(|\nabla\phi|-1)\,d|Df|\right|\leq
4C\rho\gamma_{v}(\rho)\,.$

Thanks to the summability of $\gamma_{v}$ (see property (V2) of the visibility
property) and to (4.1), from the last inequality we easily get the proof of
the proposition. ∎

Our monotonicity formula will then follow from the general inequality proved
in the next theorem.

######  Theorem 5.2 (Monotonicity inequality).

Let $\Omega\subset\mathbb{R}^{n}$ satisfy the visibility property, and let
$f\in BV_{\mathop{\mathrm{loc}}}(\Omega)$. Then, for $R>0$ small enough, for
almost every $0<r_{1}<r_{2}<R$, we have

$\displaystyle\left(\int_{\Omega\cap\mathcal{A}_{r_{1},r_{2}}}\phi^{1-n}\left|\left\langle\nu_{f},\nabla\phi\right\rangle\right|\,d|Df|\right)^{2}\leq
2\left(\int_{\Omega\cap\mathcal{A}_{r_{1},r_{2}}}\frac{|\nabla\phi(x)|}{\phi(x)^{n-1}}\,d|Df|\right)$
$\displaystyle\qquad\qquad\qquad\cdot\bigg{[}\mu_{f}(r_{2})-\mu_{f}(r_{1})+\int_{r_{1}}^{r_{2}}\dfrac{n-1}{\rho^{n}}\,\Psi_{\Omega}(f;B_{\rho}(V_{\rho}))d\rho+G(f;r_{1},r_{2})\bigg{]}\,,$
(5.3)

where $\nu_{f}$ is such that $Df=\nu_{f}|Df|$.

###### Proof.

We start by assuming $f\in BV(\Omega)\cap C^{1}(\Omega)$. For all $0<r<R$ and
$x\in{\mathfrak{C}}_{r}\cap B_{r}(V_{r})$, where ${\mathfrak{C}}_{r}$ is
defined in (4.12), we let

$Y_{r}(x)=V_{r}+r\,\dfrac{x-V_{r}}{|x-V_{r}|}\,.$

Standard computations yield

$\displaystyle DY_{r}(x)$
$\displaystyle=r\,\left[\dfrac{1}{|x-V_{r}|}D(x-V_{r})+(x-V_{r})\otimes\nabla|x-V_{r}|^{-1}\right]$
(5.4)
$\displaystyle=\dfrac{r}{|x-V_{r}|}\,\left[\mathrm{Id}-\dfrac{x-V_{r}}{|x-V_{r}|}\otimes\dfrac{x-V_{r}}{|x-V_{r}|}\right]\,.$

We define

$g_{r}(x)=f(Y_{r}(x))\qquad\text{for all $x\in{\mathfrak{C}}_{r}\,,$}$

then the “off-centric conical competitor” is

$f_{r}(x)=\begin{cases}g_{r}(x)&\text{if $x\in{\mathfrak{C}}_{r}\cap
B_{r}(V_{r})$}\\\ f(x)&\text{if $x\in\Omega\setminus
B_{r}(V_{r})\,.$}\end{cases}$ (5.5)

By definition, $f_{r}$ coincides with $f$ in $\Omega\setminus B_{r}(V_{r})$,
hence we infer

$\Psi_{\Omega}(f;B_{r}(V_{r}))\geq|Df|(\Omega\cap
B_{r}(V_{r}))-|Df_{r}|(\Omega\cap B_{r}(V_{r}))\,.$ (5.6)

Then, by (5.6), we deduce that

$\displaystyle|Df|(\Omega\cap B_{r}(V_{r}))-\Psi_{\Omega}(f;B_{r}(V_{r}))$
$\displaystyle\leq|Df_{r}|(\Omega\cap B_{r}(V_{r}))$
$\displaystyle\leq|Df_{r}|({\mathfrak{C}}_{r}\cap B_{r}(V_{r}))$ (5.7)
$\displaystyle=\int_{{\mathfrak{C}}_{r}\cap B_{r}(V_{r})}|\nabla
g_{r}(x)|dx\,.$

Let us now compute the gradient of $g_{r}$. By (5.4), setting

$\nu_{r}(x)=\dfrac{x-V_{r}}{|x-V_{r}|}\,,\qquad Y_{r}=Y_{r}(x)\,,$

we obtain

$\displaystyle\nabla g_{r}(x)=DY_{r}\cdot\nabla
f(Y_{r})=\dfrac{r}{|x-V_{r}|}\,\nabla f(Y_{r})^{\nu_{r}(x)^{\perp}}\,,$

where $\nabla f(Y_{r})^{\nu_{r}(x)^{\perp}}$ denotes the projection of $\nabla
f(Y_{r})$ onto the hyperplane

$\nu_{r}(x)^{\perp}:=\\{y\in\mathbb{R}^{n}\,:\,\langle\nu_{r}(x),y\rangle=0\\}\,.$

Going on with the computations, we obtain

$\displaystyle|\nabla g_{r}(x)|$
$\displaystyle=\dfrac{r}{|x-V_{r}|}\,\sqrt[]{|\nabla
f(Y_{r})|^{2}-\langle\nabla f(Y_{r}),\nu_{r}(x)\rangle^{2}}$
$\displaystyle=\dfrac{r}{|x-V_{r}|}\,\left|\nabla
f(Y_{r})\right|\,\sqrt[]{1-\dfrac{\left\langle\nabla
f(Y_{r}),\nu_{r}(x)\right\rangle^{2}}{|\nabla f(Y_{r})|^{2}}}\,.$

Consequently, we get

$\displaystyle\int_{{\mathfrak{C}}_{r}\cap B_{r}(V_{r})}|\nabla g_{r}|dx$
$\displaystyle=\int_{0}^{r}\int_{{\mathfrak{C}}_{r}\cap\partial
B_{\rho}(V_{r})}\dfrac{r}{\rho}\,\left|\nabla
f(Y_{r})\right|\,\sqrt{1-\dfrac{\left\langle\nabla
f(Y_{r}),\nu_{r}\right\rangle^{2}}{|\nabla
f(Y_{r})|^{2}}}\,d{\mathcal{H}}^{n-1}\,d\rho$
$\displaystyle=\int_{0}^{r}\left(\dfrac{\rho}{r}\right)^{n-2}d\rho\int_{{\mathfrak{C}}_{r}\cap\partial
B_{r}(V_{r})}\left|\nabla f\right|\,\sqrt{1-\dfrac{\left\langle\nabla
f,\nu_{r}\right\rangle^{2}}{|\nabla f|^{2}}}\,d{\mathcal{H}}^{n-1}$
$\displaystyle\leq\dfrac{r}{n-1}\,\Bigg{\\{}\int_{\Omega\cap\partial
B_{r}(V_{r})}|\nabla
f|d{\mathcal{H}}^{n-1}-\dfrac{1}{2}\,\int_{\Omega\cap\partial
B_{r}(V_{r})}\dfrac{\left\langle\nabla f,\nu_{r}\right\rangle^{2}}{|\nabla
f|}\,d{\mathcal{H}}^{n-1}\Bigg{\\}}\,.$ (5.8)

Combining (5) and (5), we get

$\displaystyle\dfrac{r}{2(n-1)}\,\int_{\Omega\cap\partial
B_{r}(V_{r})}\dfrac{\left\langle\nabla f,\nu_{r}\right\rangle^{2}}{|\nabla
f|}d{\mathcal{H}}^{n-1}$ (5.9)
$\displaystyle\qquad\leq\dfrac{r}{n-1}\,\int_{\Omega\cap\partial
B_{r}(V_{r})}|\nabla f|d{\mathcal{H}}^{n-1}-\int_{\Omega\cap
B_{r}(V_{r})}|\nabla f|dx+\Psi_{\Omega}(f;B_{r}(V_{r}))\,.$

Multiplying both sides of (5.9) by $(n-1)r^{-n}$ and observing that
$r=\phi(y)$ for any $y\in\partial B_{r}(V_{r})$, we get

$\displaystyle\dfrac{1}{2}\,\int_{\Omega\cap\partial B_{r}(V_{r})}$
$\displaystyle\left\langle\nabla
f,\dfrac{\nabla\phi}{|\nabla\phi|}\right\rangle^{2}\dfrac{1}{|\nabla
f|\,\phi^{n-1}}\,d{\mathcal{H}}^{n-1}$
$\displaystyle\leq\dfrac{1}{r^{n-1}}\,\int_{\Omega\cap\partial
B_{r}(V_{r})}|\nabla
f|\,d{\mathcal{H}}^{n-1}+\dfrac{n-1}{r^{n}}\,\int_{\Omega\cap
B_{r}(V_{r})}|\nabla f|dx+\dfrac{n-1}{r^{n}}\,\Psi_{\Omega}(f;B_{r}(V_{r}))$
$\displaystyle=\dfrac{d}{dr}\left(\dfrac{1}{r^{n-1}}\,\int_{\Omega\cap
B_{r}(V_{r})}|\nabla f||\nabla\phi|dx\right)$ (5.10)
$\displaystyle\qquad\qquad+\dfrac{n-1}{r^{n}}\,\int_{\Omega\cap
B_{r}(V_{r})}|\nabla
f|(|\nabla\phi|-1)dx+\dfrac{n-1}{r^{n}}\,\Psi_{\Omega}(f;B_{r}(V_{r}))\,.$

Let us now integrate (5) between $0<r_{1}<r_{2}<R$. We then achieve

$\displaystyle\dfrac{1}{2}\,\int_{\Omega\cap\mathcal{A}_{r_{1},r_{2}}}\left\langle\nabla
f(x),\dfrac{\nabla\phi(x)}{|\nabla\phi(x)|}\right\rangle^{2}\dfrac{|\nabla\phi(x)|}{|\nabla
f(x)|\,\phi(x)^{n-1}}\,dx$ $\displaystyle\qquad\leq\
\dfrac{1}{r_{2}^{n-1}}\,\int_{\Omega\cap B_{r_{2}}(V_{r_{2}})}|\nabla
f(x)||\nabla\phi(x)|dx-\dfrac{1}{r_{1}^{n-1}}\,\int_{\Omega\cap
B_{r_{1}}(V_{r_{1}})}|\nabla f(x)||\nabla\phi(x)|dx$ (5.11)
$\displaystyle\qquad\qquad\qquad+\int_{r_{1}}^{r_{2}}\dfrac{n-1}{r^{n}}\,\int_{\Omega\cap
B_{r}(V_{r})}|\nabla
f(x)|(|\nabla\phi(x)|-1)dx\,dr+\int_{r_{1}}^{r_{2}}\dfrac{n-1}{r^{n}}\,\Psi_{\Omega}(f;B_{r}(V_{r}))dr\,.$

By Hölder’s Inequality, we get

$\displaystyle\left(\int_{\Omega\cap\mathcal{A}_{r_{1},r_{2}}}\left|\left\langle\nabla
f(x),\nabla\phi(x)\right\rangle\right|\,\dfrac{dx}{\phi(x)^{n-1}}\right)^{2}$
(5.12)
$\displaystyle\qquad\leq\left(\int_{\Omega\cap\mathcal{A}_{r_{1},r_{2}}}\frac{|\nabla\phi(x)|}{\phi(x)^{n-1}}|\nabla
f(x)|dx\right)\cdot\left(\int_{\Omega\cap\mathcal{A}_{r_{1},r_{2}}}\left\langle\nabla
f(x),\dfrac{\nabla\phi(x)}{|\nabla\phi(x)|}\right\rangle^{2}\,\dfrac{|\nabla\phi(x)|\,dx}{|\nabla
f(x)|\phi(x)^{n-1}}\right)\,.$

Putting together (5) and (5.12), we obtain

$\displaystyle\left(\int_{\Omega\cap\mathcal{A}_{r_{1},r_{2}}}\left|\left\langle\nabla
f(x),\nabla\phi(x)\right\rangle\right|\,\dfrac{dx}{\phi(x)^{n-1}}\right)^{2}\leq
2\left(\int_{\Omega\cap\mathcal{A}_{r_{1},r_{2}}}\frac{|\nabla\phi(x)|}{\phi(x)^{n-1}}|\nabla
f(x)|dx\right)$
$\displaystyle\qquad\cdot\Bigg{(}\dfrac{1}{r_{2}^{n-1}}\,\int_{\Omega\cap
B_{r_{2}}(V_{r_{2}})}|\nabla f(x)|dx-\dfrac{1}{r_{1}^{n-1}}\,\int_{\Omega\cap
B_{r_{1}}(V_{r_{1}})}|\nabla f(x)|dx$ (5.13)
$\displaystyle\qquad\qquad+\int_{r_{1}}^{r_{2}}\dfrac{n-1}{\rho^{n}}\,\Psi_{\Omega}(f;B_{\rho}(V_{\rho}))d\rho+G(f;r_{1},r_{2})\Bigg{)}\,,$

where $G(f;r_{1},r_{2})$ is as in (5.1). This proves (5.2) for all $f\in
BV(\Omega)\cap C^{1}(\Omega)$.

Let now $f\in BV(\Omega)$. We can select a sequence $f_{j}\in BV(\Omega)\cap
C^{1}(\Omega)$ such that

$||f_{j}-f||_{L^{1}(\Omega)}\to
0\,,\qquad|Df_{j}|(\Omega)\to|Df|(\Omega)\,,\qquad
Df_{j}\mathop{\rightharpoonup}^{\ast}Df\,\,\,\,\,\text{in $\Omega\,.$}$ (5.14)

In particular, by the continuity of the trace with respect to the strict
convergence, we have

$||{\mathrm{Tr}}^{+}(f_{j},\partial\Omega)-{\mathrm{Tr}}^{+}(f,\partial\Omega)||_{L^{1}(\partial\Omega,{\mathcal{H}}^{n-1})}\to
0\,.$ (5.15)

Let us consider the extensions

$f_{0,j}(x)=\begin{cases}f_{j}(x)&\text{if $x\in\Omega$}\\\ 0&\text{if
$x\in\mathbb{R}^{n}\setminus\Omega\,,$}\end{cases}\qquad
f_{0}(x)=\begin{cases}f(x)&\text{if $x\in\Omega$}\\\ 0&\text{if
$x\in\mathbb{R}^{n}\setminus\Omega\,.$}\end{cases}$

We observe that, by (5.14), (5.15),

$||f_{0,j}-f_{0}||_{L^{1}(\mathbb{R}^{n})}\to
0\,,\qquad|Df_{0,j}|(\mathbb{R}^{n})\to|Df_{0}|(\mathbb{R}^{n})\,.$

By Proposition 2.13, for almost all $0<r<R$,

$|Df_{0,j}|(B_{r}(V_{r}))\to|Df_{0}|(B_{r}(V_{r}))\,,\qquad||{\mathrm{Tr}}^{+}(f_{0,j},\partial
B_{r}(V_{r}))-{\mathrm{Tr}}^{+}(f_{0},\partial B_{r}(V_{r}))||_{L^{1}(\partial
B_{r}(V_{r}))}\to 0\,,$ (5.16)

and in particular, owing to (5.15),

$|Df_{j}|(\Omega\cap B_{r}(V_{r}))=|Df_{0,j}|(\Omega\cap
B_{r}(V_{r}))\to|Df_{0}|(\Omega\cap B_{r}(V_{r}))=|Df|(\Omega\cap
B_{r}(V_{r}))\,.$ (5.17)

Now (5.16), (5.17) allow to apply Lemma 3.7, deducing that

$|\Psi_{\Omega}(f_{j};B_{r}(V_{r}))-\Psi_{\Omega}(f;B_{r}(V_{r}))|\to
0\,,\qquad\text{as $j\rightarrow\infty\,,$}$

for almost all $0<r<R$. This implies that

$\int_{r_{1}}^{r_{2}}\dfrac{n-1}{\rho^{n}}\,\Psi_{\Omega}(f_{j};B_{\rho}(V_{\rho}))d\rho\to\int_{r_{1}}^{r_{2}}\dfrac{n-1}{\rho^{n}}\,\Psi_{\Omega}(f;B_{\rho}(V_{\rho}))d\rho\,.$

Finally, to conclude that the RHS of (5) for $f=f_{j}$, passes to the limit as
$j\rightarrow\infty$, giving precisely the RHS of (5.2), it suffices to show
that the terms

$\int_{\Omega\cap\mathcal{A}_{r_{1},r_{2}}}\phi^{1-n}\,d|Df_{j}|\,,\quad\int_{\Omega\cap
B_{r_{1}}(V_{r_{1}})}(|\nabla\phi|-1)\,d|Df_{j}|\,,\quad\int_{\Omega\cap
B_{r_{2}}(V_{r_{2}})}(|\nabla\phi|-1)\,d|Df_{j}|\,,$

converge as $j\rightarrow\infty$ respectively to

$\int_{\Omega\cap\mathcal{A}_{r_{1},r_{2}}}\phi^{1-n}\,d|Df|\,,\quad\int_{\Omega\cap
B_{r_{1}}(V_{r_{1}})}(|\nabla\phi|-1)\,d|Df|\,,\quad\int_{\Omega\cap
B_{r_{2}}(V_{r_{2}})}(|\nabla\phi|-1)\,d|Df|\,.$

To see this, it suffices to construct a suitable partition of each domain, for
instance using portions of circular annuli whose boundaries are negligible for
$|Df|$ and $|Df_{j}|$ for all $j\geq 1$, to uniformly approximate each
integrand by simple functions (up to removing a small neighborhood of $0$ in
the case of the last two integrals). About the LHS of (5), we observe that
(5.14) implies

$Df_{j}\mathop{\rightharpoonup}^{\ast}Df\qquad\text{in
$\Omega\cap\mathcal{A}_{r_{1},r_{2}}\,.$}$

Now, for $f$ smooth, the LHS of (5.2) and the LHS of (5) coincide. Moreover,
we have

$\int_{\Omega\cap\mathcal{A}_{r_{1},r_{2}}}\phi(x)^{1-n}\left|\left\langle\nu_{f}(x),\nabla\phi(x)\right\rangle\right|\,d|Df|(x)=\left|\phi^{1-n}\nabla\phi\cdot
Df_{j}\right|(\Omega\cap\mathcal{A}_{r_{1},r_{2}})\,.$

In particular, (5.14) implies that

$\phi^{1-n}\nabla\phi\cdot Df_{j}\ \mathop{\rightharpoonup}^{\ast}\
\phi^{1-n}\nabla\phi\cdot Df\,,$

and well-known properties of the weak-star convergence of Radon measures (see
[24]) ensure that

$\left|\phi^{1-n}\nabla\phi\cdot
Df\right|(\Omega\cap\mathcal{A}_{r_{1},r_{2}})\leq\liminf_{j\rightarrow\infty}\left|\phi^{1-n}\nabla\phi\cdot
Df_{j}\right|(\Omega\cap\mathcal{A}_{r_{1},r_{2}})\,.$

This implies (5.2) and concludes the proof of the theorem. ∎

The next corollary, a first important consequence of Theorem 5.2, states the
monotonicity of a suitable function of the radius $r$, which is defined by
three terms: the renormalized perimeter $\mu_{E}(r)$, the integral of a
renormalized minimality gap $\Psi_{\Omega}(E;B_{r}(V_{r}))$, and the
visibility error $G(E,r)$. In particular, when $E$ is an almost-minimizer, the
infinitesimality of the second and third terms implies that $\mu_{E}(r)$ is
“almost-increasing”, hence that it admits a finite limit as $r\to 0$. This
limit represents the perimeter density of $E$ at $0$, see Remark 5.4 below.

######  Corollary 5.3 (Boundary monotonicity for almost-minimizers).

Let $\Omega$ be an open set satisfying the visibility property. Let
$E\subset\Omega$ be a local almost-minimizer, such that
$P_{\Omega}(E,B_{r})>0$ for all $r>0$ and

$\int_{0}^{R}\rho^{-n}\Psi_{\Omega}(E;B_{\rho}(V_{\rho}))\,d\rho<+\infty\,.$

Then, using the same notation introduced at the beginning of the section, we
can find $R^{\prime}>0$ such that the function

$r\mapsto\mu_{E}(r)+(n-1)\int_{0}^{r}\rho^{-n}\Psi_{\Omega}(E;B_{\rho}(V_{\rho}))\,d\rho+G(E;r)$

is non-decreasing on $(0,R^{\prime})$. Moreover, the two terms
$\int_{0}^{r}\rho^{-n}\Psi_{\Omega}(E;B_{\rho}(V_{\rho}))\,d\rho$ and $G(E;r)$
are infinitesimal as $r\to 0$, hence in particular $\mu_{E}(r)$ is “almost-
monotone” and the limit

$\theta_{E}(0):=\lim_{r\to 0^{+}}\mu_{E}(r)$

exists and is finite.

###### Proof of Corollary 5.3.

By Lemma 3.8 and the fact that $B_{r}(V_{r})\subset B_{r+v(r)}$, we can find
constants $C,R>0$ such that

$\dfrac{|Df|(\Omega\cap B_{r}(V_{r}))}{r^{n-1}}\leq\dfrac{|Df|(\Omega\cap
B_{r+v(r)})}{(r+v(r))^{n-1}}\left(1+\dfrac{v(r)}{r}\right)^{n-1}\leq
C\,,\qquad\text{for all $0<r<R$.}$

By combining Proposition 5.1 with (4.14) and the previous bound, up to
redefining the constants $C,R>0$, we obtain

$|G(E;r)|\leq
C\left(\int_{0}^{r}\gamma_{v}(\rho)\,d\rho+r\gamma_{v}(r)\right)\,,\qquad\text{for
all $0<r<R$.}$ (5.18)

Finally, the proof of the corollary follows directly from Theorem 5.2 and from
the observation that the RHS of (5.18) is infinitesimal as $r\to 0^{+}$. ∎

######  Remark 5.4.

It is easy to check that, under the assumptions of Corollary 5.3, one has

$\exists\,\lim_{r\to
0^{+}}\frac{P_{\Omega}(E;B_{r})}{r^{n-1}}=\theta_{E}(0)\,.$ (5.19)

Indeed, this is an immediate consequence of the inclusions

$B_{r-v(r)}(V_{r})\subset B_{r}\subset B_{r+v(r)}(V_{r})$

combined with $v(r)=o(r)$ as $r\to 0$.

##  6\. Blow-up limits of almost-minimizers are cones

We now apply Theorem 5.2 and prove that any blow-up limit of a local almost-
minimizer $E$ of $P_{\Omega}$ is a perimeter-minimizing cone.

######  Theorem 6.1.

Let $\Omega\subset\mathbb{R}^{n}$ be an open set satisfying the visibility
property. Let $E\subset\Omega$ be a local almost-minimizer in $\Omega$ such
that

$\int_{0}^{R}\dfrac{\Psi_{\Omega}(E;B_{r})}{r^{n}}dr<\infty\,.$ (6.1)

Fix a decreasing sequence $t_{j}\to 0$ and set $E_{t_{j}}=t_{j}^{-1}E$. Then,
up to subsequences, $E_{t_{j}}$ converges to $E_{0}\subset\Omega_{0}$ in
$L^{1}_{\mathop{\mathrm{loc}}}(\mathbb{R}^{n})$. Moreover, $E_{0}$ is a
nontrivial cone minimizing the relative perimeter in $\Omega_{0}$.

###### Proof.

Set $E_{j}=E_{t_{j}}$ and $\Omega_{j}=\Omega_{t_{j}}$ for more simplicity.
Then by the upper density estimate on the relative perimeter of $E$ (Lemma
3.8) coupled with analogous estimates satisfied by $\Omega$ Lipschitz, we can
find a constant $C>0$ such that, for every fixed $R>0$,

$\displaystyle P(E_{j};B_{R})$ $\displaystyle\leq
P(\Omega_{j};B_{R})+P_{\Omega_{j}}(E_{j};B_{R})$
$\displaystyle=t_{j}^{1-n}\big{(}P(\Omega;B_{Rt_{j}})+P_{\Omega}(E;B_{Rt_{j}})\big{)}$
$\displaystyle\leq CR^{n-1}\,.$

By the compactness property of sequences of sets with uniformly bounded
relative perimeter the ball $B_{R}$, we conclude that there exists a not
relabeled subsequence $E_{j}$ and a set $E_{0}$ of finite perimeter in
$B_{R}$, such that $E_{j}\to E_{0}$ in $L^{1}(B_{R})$ as $j\to\infty$. The
fact that $E_{0}\subset\Omega_{0}$ up to null sets is immediate, since
$E_{j}\subset\Omega_{j}$, for all $j$, and the sequence $\Omega_{j}$ converges
to the tangent cone $\Omega_{0}$ locally in Hausdorff distance (hence, in
$L^{1}_{\mathop{\mathrm{loc}}}(\mathbb{R}^{n})$) thanks to Proposition 4.5. Up
to a standard diagonal argument we can assume that the subsequence $E_{j}$
converges to $E_{0}$ in $L^{1}_{\mathop{\mathrm{loc}}}(\mathbb{R}^{n})$.
Moreover by the lower-density estimates on the volume of $E$ we also deduce
that $E_{0}$ can be neither the empty set, nor the whole $\Omega_{0}$ up to
null sets (that is, $E_{0}$ is nontrivial).

By the scaling properties of the perimeter, for any fixed $R>0$ we have

$\displaystyle\Psi_{\Omega_{t_{j}}}(E_{t_{j}};B_{R})=\dfrac{1}{t_{j}^{n-1}}\Psi_{\Omega}(E;B_{t_{j}R})\leq\omega_{n}^{1-\frac{1}{n}}\,R^{n-1}\psi_{\Omega}(E;0,t_{j}R)\longrightarrow
0\,,$

therefore we can apply Lemma 3.6 and deduce that

$\Psi_{\Omega_{0}}(E_{0};B_{R})\leq\liminf_{j}\Psi_{\Omega_{t_{j}}}(E_{t_{j}};B_{R})=0$

for all $R>0$ and, also owing to Corollary 5.3,

$\displaystyle P_{\Omega_{0}}(E_{0};B_{R})$
$\displaystyle=\lim_{j}P_{\Omega_{t_{j}}}(E_{t_{j}};B_{R})=\lim_{j}t_{j}^{1-n}P_{\Omega}(E;B_{t_{j}R}\cap\Omega)=R^{n-1}\theta_{E}(0)\,.$

Thus, $E_{0}$ is a minimizer for the relative perimeter in the cone
$\Omega_{0}$, such that

$\frac{P(E_{0};B_{R}\cap\Omega_{0})}{R^{n-1}}=\theta_{E}(0)\qquad\text{for all
$R>0$.}$

Now, the monotonicity inequality (5.2) written for $f=\mathbf{1}_{E_{0}}$ and
$\Omega=\Omega_{0}$ takes the form

$\displaystyle\Bigg{(}\int_{\Omega_{0}\cap(B_{r_{2}}\setminus B_{r_{1}})}$
$\displaystyle\dfrac{|\left\langle\nu_{E_{0}}(x),x\right\rangle|}{|x|^{n}}\,d|D\mathbf{1}_{E}|(x)\Bigg{)}^{2}$
$\displaystyle\leq\left(\int_{\Omega_{0}\cap(B_{r_{2}}\setminus
B_{r_{1}})}|x|^{1-n}\,d|D\mathbf{1}_{E_{0}}(x)|\right)\cdot\left(\dfrac{P_{\Omega_{0}}(E_{0};B_{r_{2}})}{r_{2}^{n-1}}-\dfrac{P_{\Omega_{0}}(E_{0};B_{r_{1}})}{r_{1}^{n-1}}\right)=0\,,$

for almost all $0<r_{1}<r_{2}$. The only possibility is then that
$\langle\nu_{E_{0}}(x),x\rangle=0$ at ${\mathcal{H}}^{n-1}$-a.e.
$x\in\partial^{\ast}E_{0}$. By [24, Proposition 28.8] we infer that $E_{0}$ is
a cone with vertex at the origin, up to negligible sets, and the proof is
concluded. ∎

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