# Characterization of sets of finite local and non local perimeter via non
local heat equation

Andrea kubin  Zentrum Mathematik - M7, Technische Universitat München,
Boltzmannstrasse 3, 85748 Garching, Germany<EMAIL_ADDRESS>and  Domenico
Angelo La Manna  Università degli Studi di Napoli Federico II, Dipartimento di
Matematica ed Applicazioni R. Caccioppoli<EMAIL_ADDRESS>
###### Abstract.

In this paper we provide a characterization of sets of finite local and non
local perimeter via a $\Gamma-$convergence result. As an application we give a
short proof of the isoperimetric inequality, both in the local and in the non
local case.

###### Contents

  1. 1 Notation
  2. 2 Compactness
    1. 2.1 Proof of Theorem 2.1 when $s\in(0,\frac{1}{2})\cup(\frac{1}{2},1)$
    2. 2.2 Proof of Theorem 2.1 when $s=\frac{1}{2}$
  3. 3 $\Gamma$-convergence for $s\in(0,\frac{1}{2})$
    1. 3.1 Proof of Theorem 3.1(i).
    2. 3.2 Proof of Theorem 3.1(ii).
    3. 3.3 Characterization of sets of finite $2s$-fractional perimeter.
  4. 4 $\Gamma$-convergence for $s\in[\frac{1}{2},1)$.
    1. 4.1 Estimates on cubes
    2. 4.2 Proof of Theorem 4.1(i).
    3. 4.3 Proof of Theorem 4.1(ii).
    4. 4.4 Characterization of sets of finite perimeter.
  5. 5 Application: The local and the non local isoperimetric inequality

## Introduction

In this article we analyze the asymptotic behaviour of the energy

(0.1) $\mathcal{E}_{t}^{s}(E):=\int_{E}\int_{E^{c}}P^{s}(x-y,t)dydx$

as $t\rightarrow 0^{+}$ for all $s\in(0,1)$ and where $P^{s}(z,t)$ is the
fundamental solution of the fractional heat equation. In the spirit of [5, 22,
17, 8], our goal is to provide a characterization of sets of finite local and
nonlocal perimeter via the fractional heat semigroup. In [5] they prove a
characterization of the Sobolev space using as energy the $s$-Gagliardo
seminorms, they prove also pointwise convergence result as the parameter
$s\rightarrow 1$ (see also [10, 18, 19] for a $\Gamma$-convergence result).
The starting point of the investigation in [22] is the generalization of the
asymptotic of the heat semigroup shown in [21] to prove the isoperimetric
inequality via the heat semigroup. Their theorem provides a characterization
of sets of finite perimeter via the heat semigroup. Since the kernel
$\|P^{s}(\cdot,t)\|_{\L^{1}(\mathbb{R}^{n})}=1$ for all $t>0$ it is quite
immediate to understande that the functional in (0.1) is well-defined for all
measurable sets $E$ such that $E$ or $E^{c}$ has finite measure. A simple
computation shows that when $t\to 0$ then $\mathcal{E}_{t}^{s}(E)\to 0$ (see
the prof of Theorem 3.1) whenever $E$ is a set of finite measure. Our scope in
this article is to find the first non trivial order of the power series
expansion of such a functional under suitable assumpions.

Let us introduce the function

(0.2) $g_{s}(t)=\begin{cases}t\qquad&\text{if }s\in(0,\frac{1}{2})\\\ t\log
t\qquad&\text{if }s=\frac{1}{2}\\\ t^{\frac{1}{2s}}&\text{if
}s\in(\frac{1}{2},1).\end{cases}$

This function describes the leading order of $\mathcal{E}_{t}^{s}$ as $t$
approaches zero. Let us just synthetically state our (meta)theorem as follows.

######  Theorem 0.1.

Let $n\geq 2$ and $s\in(0,1)$ and define
$\mathbb{X}^{s}=\rm{BV}(\mathbb{R}^{n})$ if $s\geq\frac{1}{2}$ and
$\mathbb{X}^{s}=H^{2s}(\mathbb{R}^{n})$ if $s<\frac{1}{2}$. Then

$\Gamma-\lim_{t\to
0}\frac{\mathcal{E}_{s}^{t}(E)}{g_{s}(t)}=\begin{cases}\Gamma^{n,s}{\rm
P}_{\min\\{2s,1\\}}(E)\quad&\text{if }\;\chi_{E}\in\mathbb{X}^{s}\\\
+\infty&\text{otherwise.}\end{cases}$

In the above theorem ${\rm P}_{2s}(E)$ stands for the fractional perimeter,
defined in 1.5 and ${\rm P}_{1}(E)$ is the De Giorgi perimeter of the set $E$
defined in (1.1). Let us comment on the reasons to explore the topic under
investigation. Laux and Otto in [20] proved the convergence of the
trhesholding scheme towards a variational solution of the mean curvature flow
under suitable assumptions. It seems natural to extend such a result when one
consider the nonlocal heat equation. This $\Gamma-$convergence result is a
first step towards the understanding of the nonlocal thersholding scheme from
a variational point of view. We mention that in [7] Caffarelli and Souganidis,
studied the convergence of nonlocal threshold dynamics, showing that they
approximate viscosity solutions to the mean curvature and non local mean
curvature flow for $s\in[\frac{1}{2},1)$ and $s\in(0,\frac{1}{2})$
respectively.
One of the first difficulty in proving this theorem comes from the fact that
the first non trivial order in the expansion change with respect to the
parameter $s$ (see the definition of the function $g_{s}(t)$ in (0.2)) and
therefore the first thing to do is to find the correct space-time scaling
which ensures some basic properties, such as compactness. Once the function
$g_{s}$ is detected, we prove the compactness Theorem 2.1. When
$s\not=\frac{1}{2}$ the proof relies on first showing compactness in
$\mathrm{L}^{1}$ via the Frechet-Kolmogorov criterion and then we show that
the limit belongs to the correct class. When $s=\frac{1}{2}$ the logaritmic
behavior makes things slightly different and we prove our result in a
completely different way Inspired by [11]. After good compactness properties
are estabilished, we can prove our theorem and to do so we distinguish between
two cases as the proof is significantly different: in case $s<\frac{1}{2}$ the
proof does not need any geometric measure theory tool, as we just need to
carefully use Fourier analysis. The main reason for this fact is that the
fractional laplacian of a characteristic function, for $s<\frac{1}{2}$, is not
only merely a measure but it is a function. Thereofore, it is reasonable to
think that a strategy based on Fourier analysis provides a fast track to the
goal and it is actually the case.
For $s\geq\frac{1}{2}$ things change completely as the functional starts to
lose its nonlocal nature and there fore we need to borrow some tools from
georemetric measure theory. The strategy we decide to follow is the well
extabilished one used in [2] to study the $\Gamma-$convergence of the
fractional perimeter as the differentiation parameter $s$ goes to one (see
also [9], [12] for the Gaussian case). In our case, another difficulty comes
from the fact that the fractional heat kernel is not known when
$s>\frac{1}{2}$ and to deal with it we once again nedd to use Fourier
analysis. We stress that eventhough some decay properties are known (see
(1.11)), such a knowledge is not enough to prove $\Gamma-$convergence, as the
sole use of such estimates would provide different constant in the
$\Gamma-\liminf$ and $\Gamma-\limsup$. Note that the case $s>\frac{1}{2}$
could be recoverd from [1], but in our case we are able to provide the
explicit form of the limit functional. For this reason we believe that it is
worth to highlight it in here. In fact, in Section 4 we will provide the
argument to find the optimal constant by carefully computing the functional on
a halfspace and then we procede by proving our convergence result. We do not
enter into details of the proof of the equality $\Gamma^{n,s}=\Gamma_{n,s}$
(see (4.28) and (4.29) for the definition of these two contants) since the
proof follows without any change by a standard argument (gluing lemma and
calibration) as in [4]
To conclude, we use our result to prove the isoperimetric inequality, both in
the local and nonlocal framework. The proof relies on Hardy rearrangement
inequality (the approximating kernels are $\mathrm{L}^{1}$ normalized-
functions is the space variable) and our $\Gamma-$convergence result. Before
concluding this introduction, let us comment our result with a little
heuristic interpretation. By Bochner subordination formula, one way to obtain
the fractional laplacian of a function is

$(-\Delta)^{s}u=\int_{0}^{\infty}\frac{e^{t\Delta}u-u}{t^{1+s}}\,dt$

and therefore, if $s<\frac{1}{2}$ and $u=\chi_{E}$

${\rm
P}_{2s}(E)=\int_{\mathbb{R}^{n}}u\Delta^{s}u\,dx=\int_{0}^{\infty}\frac{1}{t^{1+s}}\int_{\mathbb{R}^{n}}u(x)(e^{t\Delta}u(x)-u(x))\,dx\,dt.$

Hence, the fractional perimeter of a set $E$ can be computed by taking the
evolution of the characteristic function of $E$ via the heat equation and then
computing the $s-$ derivative at time $0$ of the energy of the evolvomg
solution at time $t/2$. One of the results of this paper can be formally read
as

$\lim_{t\to
0^{+}}\frac{1}{t}\int_{\mathbb{R}^{n}}u(x)(e^{-t^{2s}(-\Delta)^{s}}u(x)-u(x))=\Gamma_{n,s}{\rm
P}(E)\,dx$

for $s>\frac{1}{2}$. In this sense, our result resamble the famous Bochner
subordination formula for the perimeter of a set: we found that the perimeter
of a set $E$ can be recovered by considering first the evolution of the
characteristic function of $E$ via the fractional heat equation evaluated at
time $t=t^{2s}$ and secondly computing the limit above, which is essentially
the derivative at time $0$ of the energy of the evolving solution at time
$\frac{1}{2}t^{2s}$. So, in a sense, we are saying that given a measurable set
$E$ smooth enough, if we first compute the evolution via local heat equation
and then take a non local time derivate of an appropriate energy we recover
the non local perimeter, while if we first compute the evolution of $\chi_{E}$
via non local heat equation and after we take the local time derivative of an
appropriate energy, we recover the local perimeter.

##  1\. Notation

We recall some notation and basic resultas of geometric measure theory from
[3]. We denote with $\mathrm{M}(\mathbb{R}^{n})$ the set of all Lebesgue
measurable subset of $\mathbb{R}^{n}$. For all $k\in\mathbb{N}$ we denote with
$\omega_{k}$ the Lebesgue measure of the unit ball of $\mathbb{R}^{k}$. For
every $E\in\mathrm{M}(\mathbb{R}^{n})$ we denote by ${\rm P}(E)$ the De Giorgi
perimeter of $E$ defined by

(1.1) ${\rm P}(E)=\sup\left\\{\int_{E}\mathrm{div}\phi(x)dx\colon\,\phi\in
C_{c}^{1}(\mathbb{R}^{n},\mathbb{R}^{n})\text{ and }\|\phi\|_{\infty}\leq
1\right\\}.$

For every $E\in\mathrm{M}(\mathbb{R}^{n})$ the set $\partial^{*}E$ identifies
the reduced boundary of $E$ and
$\nu_{E}:\partial^{*}E\rightarrow\mathbb{R}^{n}$ the outer normal vector
field. For a function $u\in\mathrm{L}^{2}(\mathbb{R}^{n})$ we denote the
Fourier transform of $u$ by $\mathcal{F}[u(\cdot)](\xi)$, i.e.

$\mathcal{F}[u(\cdot)](\xi)=\frac{1}{(2\pi)^{\frac{n}{2}}}\int_{\mathbb{R}^{n}}u(x)e^{-i\langle
x,\xi\rangle}\,dx$

and note that the Fourier transform is invertible and

$\mathcal{F}^{-1}[v(\cdot)](x)=\frac{1}{(2\pi)^{\frac{n}{2}}}\int_{\mathbb{R}^{n}}v(\xi)e^{-i\langle
x,\xi\rangle}\,d\xi$

For $s\in(0,1)$ and a smooth function $u:\mathbb{R}^{n}\to\mathbb{R}$ we
consider the operator

$(-\Delta)^{s}u(x)=C_{n,s}\int_{\mathbb{R}^{n}}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy$

where

(1.2)
$C_{n,s}=\left(\int_{\mathbb{R}^{n}}\frac{1-\cos(h_{1})}{|h|^{n+2s}}dh\right)^{-1}.$

For all $s\in(0,1)$ we define the $s$-fractional Gagliardo seminorm of a
function $u\colon\mathbb{R}^{n}\rightarrow\mathbb{R}$ as

(1.3)
$[u]_{H^{s}(\mathbb{R}^{n})}:=\left(\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{n+2s}}dxdy\right)^{\frac{1}{2}}$

and we denote with $\mathrm{H}^{s}(\mathbb{R}^{n})$ the space of the function
$u\in\mathrm{L}^{2}(\mathbb{R}^{n})$ such that
$[u]_{H^{s}(\mathbb{R}^{n})}<+\infty$. The fractional Gagliardo seminorm can
be written in Fourier as

(1.4)
$[u]_{H^{s}(\mathbb{R}^{n})}^{2}=2C_{n,s}^{-1}\int_{\mathbb{R}^{n}}|\xi|^{2s}|\mathcal{F}[u](\xi)|^{2}d\xi.$

If $s\in(0,\frac{1}{2})$ and $E\in\mathrm{M}(\mathbb{R}^{n})$ we define the
$2s$-fractional perimeter as

(1.5) ${\rm P}_{2s}(E):=\int_{E}\int_{E^{c}}\frac{1}{|x-y|^{n+2s}}dxdy$

and we observe that
$\mathrm{P}_{2s}(E)=\frac{1}{2}[\chi_{E}]_{H^{s}(\mathbb{R}^{n})}$. We also
recall that for all $u:\mathbb{R}^{n}\rightarrow\mathbb{R}$ smooth

$\mathcal{F}[(-\Delta)^{s}u(\cdot)](\xi)=|\xi|^{2s}\mathcal{F}[u(\cdot)](\xi).$

Given $u_{0}\in\mathrm{L}^{\infty}(\mathbb{R}^{n})$ let us consider the
solution to fractional heat equation

(1.6) $\begin{cases}\partial_{t}v+(-\Delta)^{s}v=0\\\
v(x,0)=u_{0}(x).\end{cases}$

We now recall some basic results about the fundamental solution of the heat
equation (see [15] for a detailed discussion and proofs). We define the
fundamental solution of the fraction heat equation the function $P^{s}(z,t)$
is such that

(1.7) $\begin{cases}\partial_{t}P^{s}(z,t)+(-\Delta)^{s}P^{s}(z,t)=0\quad
t>0,\,z\in\mathbb{R}^{n},\\\
P^{s}(\cdot,0)=\delta_{0}\end{cases}\quad\int_{\mathbb{R}^{n}}P^{s}(z,t)dz=1$

where $\delta_{0}$ is the Dirac delta measure in $\mathbb{R}^{n}$. It is well
known that for
$u_{0}\in\mathrm{H}^{s}(\mathbb{R}^{n})\cap\mathrm{L}^{\infty}(\mathbb{R}^{n})$
the solution of the problem (1.6) can be written as a convolution between the
fundamental solution and the initial data, i.e.

(1.8) $v(x,t)=\int_{\mathbb{R}^{n}}P^{s}(x-y,t)u_{0}(y)\,dy$

and $v\in C([0,+\infty),\mathrm{H}^{s}(\mathbb{R}^{n}))\cap
C^{1}([0,+\infty),\mathrm{L}^{2}(\mathbb{R}^{n}))\cap
C^{\infty}((0,+\infty),C^{\infty}(\mathbb{R}^{n}))$. Moreover, the function
$P^{s}(z,t)$ satisfies

(1.9)
$\mathcal{F}[P^{s}(\cdot,t)](\xi)=\frac{1}{(2\pi)^{\frac{n}{2}}}e^{-t|\xi|^{2s}}$

for $s\in(0,1)$. The function $P^{s}$ can not be explicitely computed unless
in the particular case $s=\frac{1}{2}$, for which we know

(1.10) $P^{\frac{1}{2}}(z,t)=\frac{t}{(|z|^{2}+t^{2})^{\frac{n+1}{2}}}.$

When $s\not=\frac{1}{2}$ it is well known that $P^{s}(z,t)$ satisfies the
decay estimantes

(1.11) $c(n,s)\left(\frac{t}{|z|^{n+2s}}\wedge t^{-\frac{n}{2s}}\right)\leq
P^{s}(z,t)\leq C(n,s)\left(\frac{t}{|z|^{n+2s}}\wedge
t^{-\frac{n}{2s}}\right)$

and the scaling property

(1.12) $P^{s}(z,t)=t^{-\frac{n}{2s}}P^{s}(zt^{-\frac{1}{2s}},1).$

##  2\. Compactness

In this section we study the compactness properties of the sequence
$\\{E_{i}\\}_{i\in\mathbb{N}}$ such that $E_{i}\subset U$ for all
$i\in\mathbb{N}$, with $U$ be open and bounded, and
$\frac{1}{g_{s}(t_{i})}\mathcal{E}^{s}_{t_{i}}(E_{i})\leq M$ for all
$i\in\mathbb{N}$, with $M\in\mathbb{R}$, and $t_{i}\rightarrow 0^{+}$ as
$i\rightarrow+\infty$, where

(2.1)
$\frac{1}{g_{s}(t)}\mathcal{E}^{s}_{t}(E)=\frac{1}{g_{s}(t)}\int_{E}\int_{E^{c}}P^{s}(x-y,t)dxdy\text{
for }s\in\big{(}0,1\big{)}$

and $P^{s}(x,t)$ is the solution of (1.7) and $g_{s}(t)$ is the function
defined in(0.2). This compactness properties suggests the candidate
$\Gamma$-limit: for $s\in(0,\frac{1}{2})$ such a candidate is the
$2s$-fractional perimeter defined in (1.5) while for $s\geq\frac{1}{2}$ is the
classical perimeter (1.1). The main theorem of this section is the following.

######  Theorem 2.1.

Let $E_{t_{i}}\subset U$ for all $i\in\mathbb{N}$ and $t_{i}\rightarrow 0^{+}$
where $U\subset\mathbb{R}^{n}$ open and bounded. Assume that

(2.2) $\frac{\mathcal{E}_{t_{i}}^{s}(\chi_{E_{t_{i}}})}{g_{s}(t_{i})}\leq M.$

There exists $E\subset U$ such that, up to a subsequence
$\chi_{E_{t_{i}}}\to\chi_{E}$ in $\mathrm{L}^{1}(\mathbb{R}^{n})$. Moreover,
if $s\in(0,\frac{1}{2})$ we have ${\rm P}_{2s}(E)<\infty$, while for
$s\in[\frac{1}{2},1)$ it holds ${\rm P}(E)<\infty$.

We divide the proof of the above theorem in two case
$s\in(0,\frac{1}{2})\cup(\frac{1}{2},1)$ and $s=\frac{1}{2}$, because in the
proof of the case $s=\frac{1}{2}$ we can’t apply the same argument of the
other cases.

###  2.1. Proof of Theorem 2.1 when $s\in(0,\frac{1}{2})\cup(\frac{1}{2},1)$

To prove compactness we need to use the following version of the Frechet-
Kolmogorov compactness criterion.

######  Lemma 2.2.

Let $X\subset\mathrm{L}^{p}(\mathbb{R}^{n})$ bounded . Assume that for all
$\Omega\subset\mathbb{R}^{n}$ open bounded

$\lim_{t\to 0}\sup_{t<|h|<2t}\sup_{u\in
X}\|\tau_{h}u-u\|_{\mathrm{L}^{p}(\Omega)}=0.$

Then $X$ is precompact in
$\mathrm{L}^{p}_{\operatorname{loc}}(\mathbb{R}^{n})$.

The next proposition is contained in [2][Proposition 5].

######  Proposition 2.3.

Let $u\in\mathrm{L}^{1}(\mathbb{R}^{n})$, $h\in\mathbb{R}^{n}$. Then for any
$z\in(0,|h|)$ we have

(2.3) $\|\tau_{h}u-u\|_{\mathrm{L}^{1}(\mathbb{R}^{n})}\leq
C(n)\frac{|h|}{z^{n+1}}\int_{B_{z}}\|\tau_{\xi}u-u\|_{\mathrm{L}^{1}(\mathbb{R}^{n})}\,d\xi.$

For all $s\in(0,\frac{1}{2})\cup(\frac{1}{2},1)$ and for all $t>0$ we define
the function $E_{s,t}:\mathrm{L}^{1}(\mathbb{R}^{n})\rightarrow[0,+\infty]$ as

(2.4)
$E_{s,t}(u):=t\int_{B^{c}_{t^{\frac{1}{2s}}}}\frac{\|\tau_{\xi}u-u\|_{\mathrm{L}^{1}(\mathbb{R}^{n})}}{|\xi|^{n+2s}}\,d\xi+t^{-\frac{n}{2s}}\int_{B_{t^{\frac{1}{2s}}}}\|\tau_{\xi}u-u\|_{\mathrm{L}^{1}(\mathbb{R}^{n})}\,d\xi.$

######  Lemma 2.4.

Let $C>0$, $s\in(0,\frac{1}{2})\cup(\frac{1}{2},1)$ and
$X_{C}=\bigcap_{t>0}\\{u\in\mathrm{L}^{1}(\mathbb{R}^{n}):E_{s,t}(u)\leq
Cg_{s}(t)\\}$. Then

(2.5)
$\sup_{2t^{\frac{1}{2s}}<|h|<4t^{\frac{1}{2s}}}\|\tau_{h}u-u\|_{\mathrm{L}^{1}(\mathbb{R}^{n})}\leq
Ct^{\min\\{1,(2s)^{-1}\\}}.$

###### Proof.

We will prove the result only for $s\in(0,\frac{1}{2})$, for $(\frac{1}{2},1)$
the proof it is analogous. Set $t_{1}:=t^{\frac{1}{2s}}$ and let
$h\in\mathbb{R}^{n}$ such that $2t_{1}<|h|<4t_{1}$. Multipling both sides of
(2.3) by $z^{-2s}$ and integrating with respect to $z$ in the interval
$[t_{1},|h|]$, we obtain that

(2.6)
$\|\tau_{h}u-u\|_{\mathrm{L}^{1}(\mathbb{R}^{n})}\frac{|h|^{1-2s}-t_{1}^{1-2s}}{1-2s}\leq
C(n)|h|\int_{t_{1}}^{|h|}\frac{1}{z^{n+1+2s}}\int_{B_{z}}\|\tau_{\xi}u-u\|_{\mathrm{L}^{1}(\mathbb{R}^{n})}\,d\xi\,dz.$

Setting $k(r)=\int_{\partial
B_{r}}\|\tau_{\xi}u-u\|_{\mathrm{L}^{1}(\mathbb{R}^{n})}d\mathcal{H}^{n-1}$
and using the integration in polar coordinate we have

(2.7)
$\begin{split}\int_{t_{1}}^{|h|}\frac{1}{z^{n+1+2s}}&\int_{B_{z}}\|\tau_{\xi}u-u\|_{\mathrm{L}^{1}(\mathbb{R}^{n})}\,dr\,dz=\int_{t_{1}}^{|h|}\frac{1}{z^{n+1+2s}}\int_{0}^{z}k(r)\,dr\,dz\\\
=&\int_{0}^{t_{1}}k(r)\int_{t_{1}}^{|h|}\frac{1}{z^{n+1+2s}}\,dz\,dr+\int_{t_{1}}^{|h|}k(r)\int_{r}^{|h|}\frac{1}{z^{n+1+2s}}\,dz\,dr\\\
=&\frac{1}{n+2s}\left(\frac{1}{t_{1}^{n+2s}}-\frac{1}{|h|^{n+2s}}\right)\int_{0}^{t_{1}}k(r)dr\\\
&+\frac{1}{n+2s}\int_{t_{1}}^{|h|}k(r)\left(\frac{1}{r^{n+2s}}-\frac{1}{|h|^{n+2s}}\right)dr\\\
\leq&\frac{1}{n+2s}\left(\frac{1}{t_{1}^{n+2s}}-\frac{1}{|h|^{n+2s}}\right)\int_{0}^{t_{1}}k(r)dr\\\
&+\frac{1}{n+2s}\int_{B_{|h|}\setminus
B_{t_{1}}}\frac{\|\tau_{\xi}u-u\|_{\mathrm{L}^{1}(\mathbb{R}^{n})}}{|\xi|^{n+2s}}\,d\xi.\end{split}$

Let us start by obseving that the requirement $u\in X_{C}$ implies that for
any $t_{1}>0$ it holds

(2.8)
$\int_{B^{c}_{t_{1}}}\frac{\|\tau_{\xi}u-u\|_{\mathrm{L}^{1}(\mathbb{R}^{n})}}{|\xi|^{n+2s}}\,d\xi\leq
C$

and

(2.9)
$\int_{B_{t_{1}}}\|\tau_{\xi}u-u\|_{\mathrm{L}^{1}(\mathbb{R}^{n})}\,d\xi\leq
Ct_{1}^{n+2s}.$

Observe also that since $u\in X_{C}$ the bounds in (2.8) and (2.9) hold.
Therefore, we have

(2.10)
$\int_{0}^{t_{1}}k(r)dr=\int_{B_{t_{1}}}\|\tau_{\xi}u-u\|_{\mathrm{L}^{1}(\mathbb{R}^{n})}\,d\xi\leq
Ct_{1}^{n+2s}$

and

(2.11) $\int_{B_{|h|}\setminus
B_{t_{1}}}\frac{\|\tau_{\xi}u-u\|_{\mathrm{L}^{1}(\mathbb{R}^{n})}}{|\xi|^{n+2s}}\,d\xi\leq
C.$

Thus using (2.10),(2.11) and (2.7) we get that (2.6) becomes

$\|\tau_{h}u-u\|_{\mathrm{L}^{1}(\mathbb{R}^{n})}\frac{|h|^{1-2s}-t_{1}^{1-2s}}{1-2s}\leq
C_{2}|h|+C_{3}|h|t_{1}^{n+2s}\left(\frac{1}{t_{1}^{n+2s}}-\frac{1}{|h|^{n+2s}}\right).$

Recalling that $h\in[2t_{1},4t_{1}]$ we obtain (2.5), which conclude the proof
$s\in(0,\frac{1}{2})$. Using the same argument we can obtain the proof also
for $s\in(\frac{1}{2},1)$, therefore we do not report it here. ∎

Now we are in position to prove Theorem 2.1 when
$s\in(0,\frac{1}{2})\cup(\frac{1}{2},1)$.

###### Proof.

Using Lemma 2.2 and Lemma 2.4 we find immediately that, up to subsequence,
$\chi_{E_{i}}$ converge to some function $u\in\mathrm{L}^{1}(\mathbb{R}^{n})$.
Moreover, we have $u=\chi_{E}$ for a measurable set such that $E\subset U$.
Finally, to show that $\mathrm{P}_{2s}(E)<\infty$ we just need to observe that
by using first monotone convergence Theorem and then Fatou Lemma we get

$\begin{split}{\rm P}_{2s}(E)&=\lim_{r\to
0}\int_{E^{c}}\int_{E\cap\\{y:|y-x|>r\\}}\frac{1}{|x-y|^{n+2s}}\,dxdy\\\
&\leq\lim_{r\to
0}\lim_{i\rightarrow+\infty}\int_{E_{i}^{c}}\int_{E_{i}\cap\\{y:|y-x|>r\\}}\frac{1}{|x-y|^{n+2s}}\,dxdy<C,\end{split}$

which conclude the case $s<\frac{1}{2}$. When $s>\frac{1}{2}$ we will show
that (2.5) implies that the limit $u=\chi_{E}$ is a
$\mathrm{BV}(\mathbb{R}^{n})$ function which means that $E$ has finite
perimeter. In, fact for any sequence $E_{i}$ as in the statement, arguing as
before we can assume that, up to a not relabeled subsequence, there exists
$E\subset\mathbb{R}^{n}$ such that $E_{i}\to E$ in
$\mathrm{L}^{1}(\mathbb{R}^{n})$. Let $\phi\in
C_{c}^{1}(\mathbb{R}^{n},\mathbb{R}^{n})$ with $\|\phi\|_{L^{\infty}}\leq 1$,
setting $r_{i}:=3t_{i}^{\frac{1}{2s}}$ it holds

$\begin{split}\int_{\mathbb{R}^{n}}u(x)\mathrm{div}\phi(x)\,dx&=\sum_{1\leq
j\leq
n}\lim_{i}\int_{\mathbb{R}^{n}}u_{i}(x)\frac{\phi_{j}(x+r_{i}e_{j})-\phi_{j}(x)}{r_{i}}\,dx\\\
&=\sum_{1\leq j\leq
n}\lim_{i}\int_{\mathbb{R}^{n}}\phi_{j}(x)\frac{u(x-r_{i}e_{j})-u(x)}{r_{i}}\,dx\\\
&\leq\sum_{1\leq j\leq
n}\frac{\|\tau_{r_{i}e_{j}}u-u\|_{L^{1}(\mathbb{R}^{n})}}{r_{i}}.\end{split}$

Therefore, recalling the definition of $g_{s}(t_{i})$ and $r_{i}$, using Lemma
2.4 we immediately get

$\int_{\mathbb{R}^{n}}u(x)\mathrm{div}\phi(x)\,dx\leq C.$

Taking the supremum over $\phi\in C_{c}^{1}(\mathbb{R}^{n},\mathbb{R}^{n})$
with $\|\phi\|_{L^{\infty}}\leq 1$ we readly get our result, namely
$u=\chi_{E}\in\mathrm{BV}(\mathbb{R}^{n})$. ∎

###  2.2. Proof of Theorem 2.1 when $s=\frac{1}{2}$

This subsection is devoted to the proof of Theorem 2.1 when $s=\frac{1}{2}$.
Ours proof it is inspired to the compactness result in [11]. To accomplish
this task we will need some preliminary results. We first recall the following
classical result

######  Theorem 2.5 (Compactness in $\mathrm{BV}$).

Let $\Omega\subset\mathbb{R}^{n}$ be an open set and let
$\\{u_{i}\\}_{i\in\mathbb{N}}\subset\mathrm{BV}_{loc}(\Omega)$ with

(2.12)
$\sup_{i\in\mathbb{N}}\left\\{\int_{A}|u_{i}(x)|dx+|Du_{i}|(A)\right\\}<+\infty\quad\forall
A\subset\\!\subset\Omega.$

Then, there exist a subsequence $\\{i_{k}\\}_{k\in\mathbb{N}}$ and a function
$u\in\mathrm{BV}_{loc}(\Omega)$ such that $u_{i_{k}}\rightarrow u$ in
$\mathrm{L}^{1}_{loc}(\Omega)$ as $k\rightarrow+\infty$.

Now we prove a non-local Poincaré-Wirtinger type inequality. In what follows
we set $Q:=[0,1)^{n}$.

######  Lemma 2.6.

Let $\xi\in\mathbb{R}^{n}$ and $u\in\mathrm{L}^{1}(lQ+\xi)$ where $0<t<l$.
Then we have

(2.13)
$\int_{lQ+\xi}\left|u(y)-\frac{1}{l^{n}}\int_{lQ+\xi}u(x)dx\right|dy\leq
lC(n)\int_{lQ+\xi}\int_{lQ+\xi}\frac{|u(x)-u(y)|}{(|x-y|^{2}+t^{2})^{\frac{n+1}{2}}}dxdy.$

###### Proof.

By translational invariance, it is enough to prove the claim only for $\xi=0$.
By assumption we have

(2.14)
$\begin{split}\int_{lQ}\left|u(y)-\frac{1}{l^{n}}\int_{lQ}u(x)dx\right|dy&\leq\frac{1}{l^{n}}\int_{lQ}\int_{lQ}\left|u(x)-u(y)\right|dxdy\\\
&=\frac{1}{l^{n}}\int_{lQ}\int_{lQ}\frac{|u(x)-u(y)|}{(|x-y|^{2}+t^{2})^{\frac{n+1}{2}}}(|x-y|^{2}+t^{2})^{\frac{n+1}{2}}dxdy\\\
&\leq
lC(n)\int_{lQ}\int_{lQ}\frac{|u(x)-u(y)|}{(|x-y|^{2}+t^{2})^{\frac{n+1}{2}}}dxdy\end{split}$

i.e., (2.13). ∎

######  Lemma 2.7.

Let $0<t<l$. For every $\xi\in\mathbb{R}^{n}$ and for every
$E\in\mathrm{M}(\mathbb{R}^{n})$, it holds

(2.15) $\frac{1}{l^{n}}|(lQ+\xi)\setminus E||(lQ+\xi)\cap
E|\leq\int_{lQ+\xi}\big{|}\chi_{E}(x)-\frac{1}{l^{n}}\int_{lQ+\xi}\chi_{E}(y)dy\big{|}dx.$

###### Proof.

We can assume without loss of generality that $\xi=0$. Then we have that

(2.16)
$\begin{split}&\int_{lQ}\big{|}\chi_{E}(x)-\frac{1}{l^{n}}\int_{lQ}\chi_{E}(y)dy\big{|}dx=\int_{lQ\cap
E}\big{|}1-\frac{|lQ\cap E|}{l^{n}}\big{|}dx+\int_{lQ\setminus E}\frac{|lQ\cap
E|}{l^{n}}dx\\\ &=\frac{1}{l^{n}}\big{(}\int_{lQ\cap E}|lQ\setminus
E|dx+\int_{lQ\setminus E}|lQ\cap E|dx\big{)}=\frac{2}{l^{n}}\int_{lQ\setminus
E}|lQ\cap E|dx\geq\frac{|lQ\cap E||lQ\setminus E|}{l^{n}}.\end{split}$

∎

The following result is a localized isoperimetric inequality for the non-local
energy in (2.1) for $s=\frac{1}{2}$.

######  Lemma 2.8.

Let $\Omega\subset\mathbb{R}^{n}$ be a open bounded with Lipschitz continuous
boundary and $|\Omega|=1$. For every $\eta\in(0,1)$ there exist $t_{0}>0$ and
$C=C(n,\Omega,\eta)>0$ such that

(2.17)
$\inf\left\\{\frac{1}{g_{\frac{1}{2}}(t)}\int_{\Omega}\int_{\Omega\setminus
A}P^{\frac{1}{2}}(x-y,t)dxdy\colon
A\subset\Omega,\,|A|\in(\eta,1-\eta),t\in(0,t_{0})\right\\}\geq C.$

Before proving Lemma 2.8, we state the following result which is a consequence
of [1, Theorem 1.4].

######  Lemma 2.9.

[16, Lemma 15] Let $\Omega\subset\mathbb{R}^{n}$ be open and bounded with
Lipschitz continuous boundary and $|\Omega|=1$. Let
$\\{\rho_{\varepsilon}\\}_{\varepsilon\in(0,1)}$ be the standard family of
Friedrichs mollifiers with support in $B(0,1)$. For every $\eta\in(0,1)$ there
exists $C=C(n,\Omega,\rho,\eta)>0$ such that

(2.18) $\inf\left\\{\frac{1}{\varepsilon}\int_{\Omega}\int_{\Omega\setminus
A}\rho_{\varepsilon}(|x-y|)dxdy\colon
A\subset\Omega,\,|A|\in(\eta,1-\eta),t\in(0,1)\right\\}\geq C.$

The proof of the above lemma it is shown in [16, Lemma 15] for $n=2$ and
$\Omega=\left(-\frac{1}{2},\frac{1}{2}\right)^{2}$, but the proof in every
dimension and for all type of $\Omega$ with Lipschitz continuous boundary is
fully analogous. We are now in position to prove Lemma 2.8.

###### Proof of Lemma 2.8.

Fix $\eta\in(0,1)$, $t\in(0,1)$ and let $I\in\mathbb{N}$ be such that
$2^{-I-1}\leq t\leq 2^{-I}$. We have that

(2.19) $\text{if }0\leq|z|\leq 2^{-i},\,i\in\mathbb{N}\text{ then
}\frac{1}{(|z|^{2}+t^{2})^{\frac{n+1}{2}}}\geq\frac{2^{(n+1)\min\\{i,I\\}}}{2^{\frac{n+1}{2}}}.$

Let $\rho$ and $\rho_{\varepsilon}$ (for $\varepsilon\in(0,1)$) be as in Lemma
(2.9). We claim that there exists $C(n,\rho)>0$ such that

(2.20)
$\sum_{i=1}^{I}2^{i}\rho_{2^{-i}}(z)\leq\frac{C(n,\rho)}{(|z|^{2}+t^{2})^{\frac{n+1}{2}}}\text{
for every }z\in\mathbb{R}^{n}.$

Indeed if $0\leq|z|\leq 2^{-I}$ then we obtain

(2.21)
$\begin{split}\sum_{i=0}^{I}2^{i}\rho_{2^{-i}}(z)&\leq\|\rho\|_{\infty}\sum_{i=0}^{I}\big{(}2^{n+1}\big{)}^{i}=\|\rho\|_{\infty}\sum_{i=0}^{I}\frac{\big{(}2^{n+1}\big{)}^{I}}{\big{(}2^{n+1}\big{)}^{I-i}}\\\
&\leq\|\rho\|_{\infty}\big{(}2^{n+1}\big{)}^{I}\sum_{i=0}^{+\infty}\frac{1}{\big{(}2^{n+1}\big{)}^{j}}=\big{(}2^{n+1}\big{)}^{I}C(\rho,n)\leq\frac{C(\rho,n)}{(|z|^{2}+t^{2})^{\frac{n+1}{2}}}\end{split}$

where in the last step we have used (2.19) for $i=I$. If
$2^{-\tilde{i}-1}\leq|z|\leq 2^{-\tilde{i}}$ for some
$\tilde{i}=0,1,\dots,I-1$, using that $\rho_{2^{-i}}(z)=0$ for every
$i=\tilde{i}+1,\dots,I$ we have

(2.22)
$\begin{split}&\sum_{i=0}^{I}2^{i}\rho_{2^{-i}}(z)=\sum_{i=0}^{\tilde{i}}2^{i}\rho_{2^{-i}}(z)\leq\|\rho\|_{\infty}\sum_{i=0}^{\tilde{i}}\big{(}2^{n+1}\big{)}^{i}\\\
&\leq\big{(}2^{n+1}\big{)}^{\tilde{i}}\|\rho\|_{\infty}\sum_{i=0}^{+\infty}\big{(}2^{n+1}\big{)}^{-i}=\big{(}2^{n+1}\big{)}^{\tilde{i}}C(\rho,n)\leq\frac{C(\rho,n)}{(|z|^{2}+t^{2})^{\frac{n+1}{2}}}\end{split}$

where the last we have used (2.19). If $|z|\geq 1$ we have that
$\rho_{2^{-i}}(z)=0$ for all $i\in\mathbb{N}$ and then

(2.23)
$\sum_{i=0}^{I}2^{i}\rho_{2^{-i}}(z)=0\leq\frac{C(\rho,n)}{(|z|^{2}+t^{2})^{\frac{n+1}{2}}}.$

Hence we conclude the claim. We prove that (2.20) implies (2.17). We observe
that

$\frac{|\log(t)|}{\log(2)}-1\leq I\leq\frac{|\log(t)|}{\log(2)}$

and hence

(2.24) $\sum_{i=0}^{I}1=I+1\geq\frac{|\log(t)|}{\log(2)}.$

Therefore by (2.20) and Lemma 2.9 with $\varepsilon$ replaced by $2^{-i}$ we
obtain

(2.25) $\begin{split}\int_{A}\int_{\Omega\setminus
A}\frac{1}{(|x-y|^{2}+t^{2})^{\frac{n+1}{2}}}dxdy&\geq
C(\Omega,\rho,n)\sum_{i=0}^{I}2^{i}\int_{A}\int_{\Omega\setminus
A}\rho_{2^{-i}}(x-y)dxdy\\\ &\geq
C(\Omega,\rho,n,\eta)\sum_{i=0}^{I}1=C(\Omega,\rho,n,\eta)|\log(t)|\end{split}$

and hence recalling the very definition of $g_{\frac{1}{2}}(t)$ and of
$z\rightarrow P^{\frac{1}{2}}(z,t)$ we obtain (2.17). ∎

We are now in position to prove Theorem 2.1 when $s=\frac{1}{2}$.

###### Proof.

We divide the proof into three steps.
Step 1. Let $\alpha\in(0,1)$ and set $l_{i}:=t_{i}^{\alpha}$ for every
$i\in\mathbb{N}$. Let $\\{Q_{h}^{i}\\}_{h\in\mathbb{N}}$ be a disjoint family
of cubes of sidelength $l_{i}$ such that
$\mathbb{R}^{n}=\cup_{h\in\mathbb{N}}Q_{h}^{i}$. Since $|E_{i}|\leq|U|$ there
exists $H(i)\in\mathbb{N}$ such that, up to permutation of indices,

(2.26) $\begin{split}&|Q_{h}^{i}\cap E_{i}|\geq\frac{l_{i}^{n}}{2}\text{ for
every }h=1,\dots,H(i),\\\ &|Q_{h}^{i}\setminus
E_{i}|>\frac{l_{i}^{n}}{2}\text{ for every }h>H(i).\end{split}$

For every $i\in\mathbb{N}$ we set

$\tilde{E}_{i}:=\bigcup_{h=1}^{H(i)}Q_{h}^{i}.$

We claim that there exists a constant $C(n)>0$ such that

(2.27) $|\tilde{E}_{i}\triangle E_{i}|\leq C(n)Ml_{i}|\log(t_{i})|\quad\text{
for every }i\in\mathbb{N},$

where $M$ is the constant in (2.2). Indeed we have

(2.28) $\begin{split}|E_{i}\triangle&\tilde{E}_{i}|=|\tilde{E}_{i}\setminus
E_{i}|+|E_{i}\setminus\tilde{E}_{i}|=\sum_{h=1}^{H(i)}|Q_{h}^{i}\setminus
E_{i}|+\sum_{h=H(i)+1}^{+\infty}|E_{i}\cap Q_{h}^{i}|\\\
&=2\sum_{h=1}^{H(i)}\frac{1}{l_{i}^{n}}|Q_{h}^{i}\setminus
E_{i}|\frac{l_{i}^{n}}{2}+2\sum_{h=H(i)+1}^{+\infty}\frac{1}{l_{i}^{n}}|E_{i}\cap
Q_{h}^{i}|\frac{l_{i}^{n}}{2}\leq
2\sum_{h=1}^{+\infty}\frac{1}{l_{i}^{n}}|Q_{h}^{i}\setminus
E_{i}||Q_{h}^{n}\cap E_{i}|\\\ &\leq
2\sum_{h=1}^{+\infty}\int_{Q_{h}^{i}}\big{|}\chi_{E_{i}}(x)-\frac{1}{l_{i}^{n}}\int_{Q_{h}^{i}}\chi_{E_{i}}(y)dy\big{|}dx\\\
&\leq C(n)\sum_{h=1}^{+\infty}l_{i}\int_{Q_{h}^{i}}\int_{Q_{h}^{i}\setminus
E_{i}}\frac{1}{(|x-y|^{2}+t_{i}^{2})^{\frac{n+1}{2}}}dxdy\leq
C(n)l_{i}|\log(t_{i})|\frac{\mathcal{E}_{t_{i}}(E_{i})}{g_{\frac{1}{2}}(t_{i})}\\\
&\leq C(n)Ml_{i}|\log(t_{i})|\end{split}$

where the second inequality follows by formula (2.15), the third inequality is
a consequence of (2.13), whereas the last one follows directly by (2.2).
Step 2. For every $i\in\mathbb{N}$ let $l_{i}$ and
$\tilde{E}_{i}:=\cup_{h=1}^{H(i)}Q_{h}^{i}$ be as in Step 1. We claim that
there exists a constant $C(\alpha,n)$ such that for $i$ large enough

(2.29) $\mathrm{P}(\tilde{E}_{i})\leq
C(\alpha,n)\frac{\mathcal{E}_{t_{i}}(E_{i})}{g_{\frac{1}{2}}(t_{i})}.$

We omit the dependence on $i$ by setting $t:=t_{i}$, $l:=l_{i}$, $E:=E_{i}$,
$Q_{h}:=Q_{h}^{i}$, $H:=H(i)$, $\tilde{E}:=\tilde{E}_{i}$. We define the
family $\mathcal{R}$ of rectangle $R=\tilde{Q}\cup\hat{Q}$ such that
$\tilde{Q}$ and $\hat{Q}$ are adjacent cubes (of the type $Q_{h}$) and
$\tilde{Q}\subset\tilde{E}$, $\hat{Q}\subset\tilde{E}^{c}$. We have that

(2.30) $\begin{split}&\mathrm{P}(\tilde{E})\leq C(n)l^{n-1}\\#\mathcal{R},\\\
&\frac{1}{g_{\frac{1}{2}}(t)}\sum_{R\in\mathcal{R}}\int_{R\cap
E}\int_{R\setminus
E}P^{\frac{1}{2}}(|x-y|,t)dydx\leq\frac{\mathcal{E}_{t}^{\frac{1}{2}}(E)}{g_{\frac{1}{2}}(t)}.\end{split}$

By Lemma 2.8, there exists $t_{0}>0$ such that for every rectangle $\bar{R}$
given by the union of two adjacent unitary cubes in $\mathbb{R}^{n}$

(2.31) $\inf\left\\{\frac{1}{g_{\frac{1}{2}}(t)}\int_{F}\int_{\bar{R}\setminus
F}P^{\frac{1}{2}}(|x-y|,t)dydx:0<t<t_{0},F\subset\bar{R},\frac{1}{2}\leq|F|\leq\frac{3}{2}\right\\}:=C(n)>0.$

For every $A\subset\mathbb{R}^{n}$ we set $A^{l}:=\frac{A}{l}$. By formula
(2.30) and (2.31) with $\bar{R}=R^{l}$ for every $R\in\mathcal{R}$, for $t$
small enough we have

(2.32)
$\begin{split}\frac{\mathcal{E}^{\frac{1}{2}}(E)}{g_{\frac{1}{2}}(t)}&\geq\frac{l^{2n}}{g_{\frac{1}{2}}(t)}\sum_{R\in\mathcal{R}}\int_{R^{l}\cap
E^{l}}\int_{R^{l}\setminus E^{l}}P^{\frac{1}{2}}(l|x-y|,t)dydx\\\
&=\frac{C(n,\alpha)l^{n-1}}{g_{\frac{1}{2}}(\frac{t}{l})}\sum_{R\in\mathcal{R}}\int_{R^{l}\cap
E^{l}}\int_{R^{l}\setminus E^{l}}P^{\frac{1}{2}}(|x-y|,\frac{t}{l})dxdy\\\
&\geq C(n,\alpha)l^{n-1}\\#\mathcal{R}\geq
C(n,\alpha)\mathrm{P}(\tilde{E}),\end{split}$

i.e., (2.29).
Step 3. Here we conclude the proof of the compactness result. Fix
$\alpha\in(0,1)$ so that, by (2.27)

(2.33) $|E_{i}\triangle\tilde{E}_{i}|\rightarrow 0\text{ as
}i\rightarrow+\infty.$

By Step 2. we have that $\\{\tilde{E}_{i}\\}_{i\in\mathbb{N}}$ satisfies the
assumption of Theorem 2.5. Therefore $\tilde{E}_{i}\rightarrow E$, up to
subsequence, in $\mathrm{L}^{1}(\mathbb{R}^{n})$ with $\mathrm{P}(E)<+\infty$.
Hence by (2.33) we have $E_{i}\rightarrow E$ in
$\mathrm{L}^{1}(\mathbb{R}^{n})$. ∎

##  3\. $\Gamma$-convergence for $s\in(0,\frac{1}{2})$

The main result of this section is the following.

######  Theorem 3.1.

Let $s\in(0,\frac{1}{2})$ and let $\\{t_{i}\\}_{i\in\mathbb{N}}\subset(0,1)$
such that $t_{i}\rightarrow 0^{+}$ as $i\rightarrow+\infty$. The following
$\Gamma$-convergence result holds true.

  * (i)

(Lower bound) Let $E\in\mathrm{M}(\mathbb{R}^{n})$ be a Lebesque measurable
set. For every $\\{E_{i}\\}_{i\in\mathbb{N}}\subset\mathrm{M}(\mathbb{R}^{n})$
with $\chi_{E_{i}}\to\chi_{E}$ strongly in $\mathrm{L}^{1}(\mathbb{R}^{n})$ it
holds

(3.1)
$C_{n,s}\mathrm{P}_{2s}(E)\leq\liminf_{i\rightarrow+\infty}\frac{\mathcal{E}_{t_{i}}(E_{i})}{g_{s}(t_{i})}\,.$

  * (ii)

(Upper bound) For every $E\in\mathrm{M}(\mathbb{R}^{n})$ there exists
$\\{E_{i}\\}_{i\in\mathbb{N}}\subset\mathrm{M}(\mathbb{R}^{n})$ such that
$\chi_{E_{i}}\to\chi_{E}$ strongly in $\mathrm{L}^{1}(\mathbb{R}^{n})$ and

(3.2) $C_{n,s}{\rm
P}_{2s}(E)\geq\limsup_{i\to+\infty}\frac{\mathcal{E}_{t_{i}}(E_{i})}{g_{s}(t_{i})}\,.$

Where $C_{n,s}$ is the constant defined in (1.2).

###  3.1. Proof of Theorem 3.1(i).

We remember that $g_{s}(t)=t$ for $s\in(0,\frac{1}{2})$. Let
$\\{E_{i}\\}_{i\in\mathbb{N}}\subset\mathrm{M}(\mathbb{R}^{n})$ such that
$\chi_{E_{i}}\rightarrow\chi_{E}$ in $\mathrm{L}^{1}(\mathbb{R}^{n})$ and
assume
$\liminf_{i\rightarrow+\infty}\frac{\mathcal{E}_{t_{i}}(E_{i})}{t_{i}}<+\infty$
(otherwise the formula (3.1) is trivial). We claim that

(3.3) $C_{n,s}{\rm
P}_{2s}(E)\leq\liminf_{i\rightarrow+\infty}\frac{\mathcal{E}_{t_{i}}(E_{i})}{t_{i}}.$

In fact

(3.4)
$\begin{split}\frac{\mathcal{E}_{t_{i}}(E_{i})}{t_{i}}&=\frac{1}{2}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{P^{s}(x-y,t_{i})}{t_{i}}|\chi_{E_{i}}(x)-\chi_{E_{i}}(y)|^{2}dxdy\\\
&=\frac{1}{2}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{P^{s}(h,t_{i})}{t_{i}}|\chi_{E_{i}}(x+h)-\chi_{E_{i}}(x)|^{2}dxdh\\\
&=\int_{\mathbb{R}^{n}}\frac{P^{s}(h,t_{i})}{t_{i}}\int_{\mathbb{R}^{n}}(1-\cos(\xi\cdot
h))|\hat{\chi_{E_{i}}}(\xi)|^{2}d\xi dh\end{split}$

where in the last step we have used the Plancherel theorem

$\int_{\mathbb{R}^{n}}|u(x+h)-u(x)|^{2}dx=\int_{\mathbb{R}^{n}}|1-e^{i\xi\cdot
h}|^{2}|\hat{u}(\xi)|^{2}d\xi=\int_{\mathbb{R}^{n}}2(1-\cos(\xi\cdot
h))|\hat{u}(\xi)|^{2}d\xi.$

We remember that $h\rightarrow P^{s}(h,t)$ is a Schwartz function and
$\mathcal{F}[P^{s}(\cdot,t)](\xi)=\frac{1}{(2\pi)^{\frac{n}{2}}}e^{-t|\xi|^{2s}}$.
We define

$\mathcal{I}(\xi):=\int_{\mathbb{R}^{N}}(1-\cos(\xi\cdot h))P^{s}(h,t)dh.$

We observe that the function $\mathcal{I}$ is rotationally invariant, that is

(3.5) $\mathcal{I}(\xi)=\mathcal{I}(|\xi|e_{1})$

where $e_{1}$ denotes the first direction vector in $\mathbb{R}^{N}$.
Therefore we have

(3.6) $\int_{\mathbb{R}^{N}}(1-\cos(\xi\cdot
h))P^{s}(h,t)dh=\int_{\mathbb{R}^{N}}(1-\cos(|\xi|h_{1}))P^{s}(h,t)dh.$

Now we remember that the Fourier transform of the function
$\alpha\rightarrow\cos(\alpha a)$ for $a\in\mathbb{R}$ is given by
$\mathcal{F}[\cos(a\cdot)](\beta)=\frac{\sqrt{\pi}}{\sqrt{2}}\left[\delta(\beta-a)+\delta(\beta+a)\right]$,
this Fourier transform should be read in the sense of the tempered
distributions (that is the dual space of the Schwartz function). Therefore
using the Plancherel theorem we have

(3.7)
$\begin{split}&\int_{\mathbb{R}^{n}}\frac{P^{s}(h,t)}{t}(1-\cos(|\xi|h_{1}))dh=\frac{1}{t}\left[1-\int_{\mathbb{R}^{n}}\cos(|\xi|h_{1})P^{s}(h,t)dh\right]\\\
&=\frac{1}{t}\left[1-\int_{\mathbb{R}^{n}}\frac{1}{2}\left(\delta_{0}(\eta_{1}-|\xi|)+\delta_{0}(\eta_{1}+|\xi|)\right)\delta_{0}(\eta_{2})\cdots\delta_{0}(\eta_{n})e^{-t|\eta|^{2s}}d\eta\right]\\\
&=\frac{1}{t}\left[1-e^{-t|\xi|^{2s}}\right]\end{split}$

where $\delta_{0}$ is the one dimensional Dirac delta with center in
$0\in\mathbb{R}$. By formulas (3.4), (3.6), (3.7) and for all $R>0$ we obtain

(3.8)
$\begin{split}&\liminf_{i\rightarrow+\infty}\frac{\mathcal{E}_{t_{i}}(E_{i})}{t_{i}}=\liminf_{i\rightarrow+\infty}\int_{\mathbb{R}^{n}}|\mathcal{F}[\chi_{E_{i}}](\xi)|^{2}\frac{1}{t}\left[1-e^{-t_{i}|\xi|^{2s}}\right]\\\
&\geq\liminf_{i\rightarrow+\infty}\int_{B(0,R)}|\mathcal{F}[\chi_{E_{i}}](\xi)|^{2}\frac{1}{t_{i}}\left[1-e^{-t_{i}|\xi|^{2s}}\right]=\int_{B(0,R)}|\mathcal{F}[\chi_{E}](\xi)|^{2}|\xi|^{2s}d\xi\end{split}$

where in the last step we have used that

$\lim_{t\rightarrow 0^{+}}\sup_{\xi\in
B(0,R)}\bigg{|}\frac{1}{t}\left[1-e^{-t|\xi|^{2s}}\right]-|\xi|^{2s}\bigg{|}=0.$

Now sending $R\rightarrow+\infty$ in (3.8) we obtain (3.3).

###  3.2. Proof of Theorem 3.1(ii).

Let $E\in\mathrm{M}(\mathbb{R}^{n})$ such that $\mathrm{P}_{2s}(E)<+\infty$
(otherwise the formula (3.2) is trivial) we prove that the limsup in (3.2) is
realized by the sequence $E_{i}=E$ for all $i\in\mathbb{N}$ and it is a limit.
We remember that $E$ has finite $2s$-fractional perimeter if
$\chi_{E}\in\mathrm{H}^{s}(\mathbb{R}^{n})$. Let
$v:\mathbb{R}^{n}\times[0,+\infty)\rightarrow\mathbb{R}$ be the solution to
the Cauchy problem in (1.6) with $u_{0}=\chi_{E}$ . By the definition of
$\mathcal{E}_{t}(E)$, see (0.1), and by formula (1.8) we have

$\begin{split}\mathcal{E}_{t}(u_{0})=\int_{\mathbb{R}^{n}}(1-u_{0}(x))v(x,t)dx&=\int_{\mathbb{R}^{n}}v(x,t)\,dx-\int_{\mathbb{R}^{n}}v(x,t)u_{0}(x)\,dx\\\
&=\int_{\mathbb{R}^{n}}u_{0}(x)\,dx-\int_{\mathbb{R}^{n}}v(x,t)u_{0}(x)\,dx\\\
&=\int_{\mathbb{R}^{n}}u_{0}(x)^{2}\,dx-\int_{\mathbb{R}^{n}}v(x,t)u_{0}(x)\,dx\end{split}$

where we used that

$\int_{\mathbb{R}^{n}}v(x,t)\,dx=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}P^{s}(x-y,t)u_{0}(y)\,dydx=\int_{\mathbb{R}^{n}}u_{0}(y)\,dy$

and the fact that $u_{0}(x)=\chi_{E}(x)\in\\{0,1\\}$. Now we observe that

$\begin{split}\int_{\mathbb{R}^{n}}u_{0}(x)^{2}\,dx-\int_{\mathbb{R}^{n}}v(x,t)u_{0}(x)\,dx=&-\int_{\mathbb{R}^{n}}u_{0}(x)\int_{0}^{t}\partial_{h}v(x,h)\,dh\,dx\\\
=&\int_{\mathbb{R}^{n}}\int_{0}^{t}u_{0}(x)(-\Delta)^{s}v(x,h)\,dh\,dx\end{split}$

where we have used that $v\in
C^{1}([0,+\infty),\mathrm{L}^{2}(\mathbb{R}^{n}))$. At this point we just need
to use that the Fourier transform is an isometry and
$\mathcal{F}[v(\cdot,t)](\xi)=\mathcal{F}[u_{0}(\cdot)](\xi)e^{-t|\xi|^{2s}}$
and therefore

$\int_{\mathbb{R}^{n}}u_{0}(x)(-\Delta)^{s}v(x,t)\,dx=\int_{\mathbb{R}^{n}}|\xi|^{2s}\frac{1}{(2\pi)^{\frac{n}{2}}}e^{-t|\xi|^{2s}}|\mathcal{F}[u_{0}(\cdot))](\xi)|^{2}\,d\xi.$

Since $u_{0}\in\mathrm{H}^{s}(\mathbb{R}^{n})$ we have

$\int_{\mathbb{R}^{n}}|\xi|^{2s}\frac{1}{(2\pi)^{\frac{n}{2}}}e^{-t|\xi|^{2s}}|\mathcal{F}[u_{0}(\cdot))](\xi)|^{2}\,d\xi\leq\int_{\mathbb{R}^{n}}|\xi|^{2s}|\mathcal{F}[u_{0}(\cdot))](\xi)|^{2}\,d\xi=C_{n,s}P_{2s}(E).$

Finally to conclude we just need to observe that

$\lim_{t\to
0^{+}}\frac{\mathcal{E}_{t}(u_{0})}{t}=\int_{\mathbb{R}^{n}}u_{0}(-\Delta)^{s}u_{0}=C_{n,s}{\rm
P}_{2s}(E).$

###  3.3. Characterization of sets of finite $2s$-fractional perimeter.

As a byproduct of our $\Gamma$\- convergence analysis, we prove that a set
$E\in\mathrm{M}(\mathbb{R}^{n})$ has finite $2s$-fractional perimeter if and
only if

$\limsup_{t\rightarrow 0^{+}}\frac{1}{t}\mathcal{E}_{t}^{s}(E)<+\infty.$

######  Theorem 3.2.

Let $E\subset\mathrm{M}(\mathbb{R}^{n})$. The following statement hold true.

(3.9) $\limsup_{t\rightarrow
0^{+}}\frac{1}{t}\mathcal{E}_{t}^{s}(E)<+\infty\iff{\rm P}_{2s}(E)<+\infty.$

###### Proof.

We notice that the implication $\Longrightarrow$ is a consequence of Theorem
2.1. The implication $\Longleftarrow$ is a consequence of Theorem 3.1(ii). ∎

##  4\. $\Gamma$-convergence for $s\in[\frac{1}{2},1)$.

In this section we threat the remaining cases. The main result of this section
is the following.

######  Theorem 4.1.

Let $s\in[\frac{1}{2},1)$ and let $\\{t_{i}\\}_{i\in\mathbb{N}}\subset(0,1)$
such that $t_{i}\rightarrow 0^{+}$ as $i\rightarrow+\infty$. The following
$\Gamma$-convergence result holds true.

  * (i)

(Lower bound) Let $E\in\mathrm{M}(\mathbb{R}^{n})$ be a Lebesque measurable
set. For every $\\{E_{i}\\}_{i\in\mathbb{N}}\subset\mathrm{M}(\mathbb{R}^{n})$
with $\chi_{E_{i}}\to\chi_{E}$ strongly in $\mathrm{L}^{1}(\mathbb{R}^{n})$ it
holds

(4.1)
$\Gamma^{n,s}\mathrm{P}(E)\leq\liminf_{i\rightarrow+\infty}\frac{\mathcal{E}_{t_{i}}(E_{i})}{g_{s}(t_{i})}\,.$

  * (ii)

(Upper bound) For every $E\in\mathrm{M}(\mathbb{R}^{n})$ there exists
$\\{E_{i}\\}_{i\in\mathbb{N}}\subset\mathrm{M}(\mathbb{R}^{n})$ such that
$\chi_{E_{i}}\to\chi_{E}$ strongly in $\mathrm{L}^{1}(\mathbb{R}^{n})$ and

(4.2) $\Gamma^{n,s}{\rm
P}(E)\geq\limsup_{i\to+\infty}\frac{\mathcal{E}_{t_{i}}(E_{i})}{g_{s}(t_{i})}\,.$

Where

(4.3)
$\Gamma^{n,s}:=\begin{cases}\frac{1}{2\pi}\Gamma(1-\frac{1}{2s})\,\qquad&s\in(\frac{1}{2},1),\\\
\frac{\omega_{n-2}}{2}\frac{\Gamma(\frac{n}{2})}{\Gamma(\frac{n+1}{2})}\qquad&s=\frac{1}{2}\end{cases}$

where $\Gamma(t):=\int_{0}^{+\infty}r^{t-1}e^{-r}dr$ is the Gamma function.

Note that the proof of the $\Gamma-$convergence in this case is going to be
completely different from the case $s\in(0,\frac{1}{2})$. We start computing
the limit as $t\to 0$ of our functional on hyperlanes.

###  4.1. Estimates on cubes

To begin, we consider the cae when $s=\frac{1}{2}$. This is the only instance
where we have a precise formula for the fundamental solution of the fractional
heat equation. In what follows we set
$Q^{\prime}_{\delta}=[-\delta,\delta]^{n-1}$, $Q^{+}_{\delta}=[0,\delta]\times
Q^{\prime}$ and $Q^{-}_{\delta}=[-\delta,0]\times Q^{\prime}$.

######  Lemma 4.2.

Let $\delta>0$. For $t$ small enough it holds

(4.4)
$\int_{Q_{\delta}^{-}}\int_{Q_{\delta}^{+}}\frac{t}{(|x-y|^{2}+t^{2})^{\frac{n+1}{2}}}\,dx\,dy\leq
c_{n}\mathcal{H}^{n-1}(Q^{\prime}_{\delta})t|\log t|+o(t|\log t|),$

where $c_{n}$ is given in (4.7)

###### Proof.

Using Fubini Tonelli Theorem we have

(4.5)
$\int_{Q_{\delta}^{-}}\int_{Q_{\delta}^{+}}\frac{t}{(|x-y|^{2}+t^{2})^{\frac{n+1}{2}}}\,dx\,dy\\\
=\int_{Q^{\prime}_{\delta}}\int_{0}^{\delta}\int_{-\delta}^{0}\int_{Q^{\prime}_{\delta}}\frac{t}{(|x^{\prime}-y^{\prime}|^{2}+(x_{n}-y_{n})^{2}+t^{2})^{\frac{n+1}{2}}}\,dy^{\prime}\,dx_{n}\,dy_{n}\,dx^{\prime}.$

We observe that

(4.6)
$\begin{split}\int_{0}^{\delta}\int_{-\delta}^{0}\int_{\mathbb{R}^{n-1}}P^{\frac{1}{2}}(x-y,t)\,dx_{n}dy_{n}dy^{\prime}=c_{n}t\int_{0}^{\delta}\int_{-\delta}^{0}\frac{1}{(x_{n}-y_{n})^{2}+t^{2}}\,dx_{n}dy_{n}\end{split}$

where we used the change of variable

$z=\frac{y^{\prime}-x^{\prime}}{\sqrt{(x_{n}-y_{n})^{2}+t^{2}}}$

and we have set

(4.7)
$c_{n}:=\int_{\mathbb{R}^{n-1}}\frac{1}{(1+|z|^{2})^{\frac{n+1}{2}}}\,dz.$

A straightforward computation gives

(4.8)
$\begin{split}t\int_{0}^{\delta}\int_{-\delta}^{0}\frac{1}{(x_{n}-y_{n})^{2}+t^{2}}\,dx_{n}dy_{n}&=2\delta(\arctan\frac{t}{\delta}-\arctan\frac{t}{2\delta})-t\log
t+t\log\left(\frac{\delta^{2}+t^{2}}{t^{2}+4\delta^{2}}\right)\\\
&:=g(t,\delta)+t|\log t|\end{split}$

with

(4.9) $c_{1}(\delta)t\leq g(t,\delta)\leq tc_{2}(\delta)$

where $c_{1}(\delta),c_{2}(\delta)\rightarrow 0$ as $\delta\rightarrow 0^{+}$.
Therefore, to conclude we just need to observe that by monoticity of the
integral and (4.8) we have

(4.10)
$\begin{split}\int_{Q_{\delta}^{-}}\int_{Q_{\delta}^{+}}&\frac{t}{(|x-y|^{2}+t^{2})^{\frac{n+1}{2}}}\,dx\,dy\\\
&\leq\int_{Q_{\delta}^{-}}\int_{\mathbb{R}^{n-1}}\frac{t}{(|x-y|^{2}+t^{2})^{\frac{n+1}{2}}}\,dx\,dy=c_{n}t|\log
t|+o(t\log t)\end{split}$

which is exactly (4.4). ∎

We now estimate the integral on cubes by above.

######  Lemma 4.3.

Let $c_{n}$ be the constant given in (4.7). For any $\delta>0$ and for $t$
small enough it holds true

$\int_{Q_{\delta}^{-}}\int_{Q_{\delta}^{+}}\frac{t}{(|x-y|^{2}+t^{2})^{\frac{n+1}{2}}}\,dx\,dy\geq
c_{n}\mathcal{H}^{n-1}(Q_{\delta})(t|\log t|+o(t\log t)).$

###### Proof.

To prove this lemma we just need to show that the error passing from a small
cube to the hyperplane $\mathbb{R}^{n-1}$ is negligible when $t$ is small. To
do that we start decomposing the integral

(4.11)
$\begin{split}&\int_{0}^{\delta}\int_{-\delta}^{0}\int_{Q^{\prime}_{\delta}}P^{\frac{1}{2}}(x-y,t)\,dx_{n}dy^{\prime}dy_{n}\\\
&=\int_{0}^{\delta}\int_{-\delta}^{0}\left(\int_{\mathbb{R}^{n-1}}P^{\frac{1}{2}}(x-y,t)\,dy^{\prime}-\int_{(Q^{\prime}_{\delta})^{c}}P^{\frac{1}{2}}(x-y,t)dy^{\prime}\right)\,dx_{n}dy_{n}\end{split}$

and observe that by a change of variable
$z=\frac{y^{\prime}-x^{\prime}}{\sqrt{(x_{n}-y_{n})^{2}+t^{2}}}$ we have

(4.12) $\int_{0}^{\delta}\int_{-\delta}^{0}\int_{\mathbb{R}^{n-1}\setminus
Q^{\prime}_{\delta}}P^{s}(x-y,t)\,dx_{n}dy_{n}dy^{\prime}\\\
=\frac{1}{t}\int_{0}^{\delta}\int_{-\delta}^{0}\int_{\mathbb{R}^{n-1}\setminus
Q^{\prime}_{\frac{\delta}{t}}(x^{\prime})}\frac{1}{\left(|z|^{2}+\frac{(x_{n}-y_{n})^{2}}{t^{2}}+1\right)^{\frac{n+1}{2}}}\,dz\,dx_{n}\,dy_{n}.$

Using the integration in polar coordinate we obtain

(4.13)
$\begin{split}\int_{0}^{\delta}\int_{-\delta}^{0}\int_{\mathbb{R}^{n-1}\setminus
Q^{\prime}_{\frac{\delta}{t}}(x^{\prime})}&\frac{\,dz\,dx_{n}\,dy_{n}}{\left(|z|^{2}+\frac{(x_{n}-y_{n})^{2}}{t^{2}}+1\right)^{\frac{n+1}{2}}}\leq\int_{0}^{\delta}\int_{-\delta}^{0}\int_{\mathbb{R}^{n-1}\setminus
Q^{\prime}_{\frac{\delta}{t}}(x^{\prime})}\frac{dz\,dx_{n}\,dy_{n}}{\left(|z|^{2}+1\right)^{\frac{n+1}{2}}}\,\\\
&\leq(n-1)\omega_{n-1}\int_{0}^{\delta}\int_{-\delta}^{0}\int_{\frac{\delta}{t}}^{\infty}\frac{\rho^{n-2}}{\left(\rho^{2}+1\right)^{\frac{n+1}{2}}}\,d\rho\,dx_{n}\,dy_{n}\\\
&\leq(n-1)\omega_{n-1}\delta^{2}\int_{\frac{\delta}{t^{2}}}^{\infty}\frac{\rho}{(\rho^{2}+1)^{2}}\,d\rho\\\
&=\frac{(n-1)\omega_{n-1}\delta^{2}}{\delta^{2}+t^{2}}\,t^{2}.\end{split}$

The conclusion now follows by combining (4.6), (4.8) and (4.13). ∎

We now procede performing the same computation for $s>\frac{1}{2}$. We stress
that in this instance we do not have the precise expression of the kernel, as
it is not known. However, the Fourier transform of the function $P^{s}(z,t)$
is explicit and can be used to perform the computations that we need. In what
follows we set $\mathbb{R}_{+}^{n}:=\mathbb{R}^{n-1}\times[0,+\infty)$ and
$R_{\delta}:=\\{(x^{\prime},x_{n}):x^{\prime}\in\mathbb{R}^{n-1},x_{n}\in(0,\delta)\\}$
for all $\delta>0$,

######  Lemma 4.4.

Let $\delta>0$, $s\in(\frac{1}{2},1)$. Then for $t$ small enough it holds true

(4.14)
$\int_{Q_{\delta}^{-}}\int_{\mathbb{R}_{+}^{n}}P^{s}(x-y,t)\,dx\,dy=\frac{1}{2\pi}\Gamma\left(1-\frac{1}{2s}\right)t^{\frac{1}{2s}}(\mathcal{H}^{n-1}(Q_{\delta})+o(1)).$

###### Proof.

We start by observing that

(4.15) $\int_{Q_{\delta}^{-}}\int_{\mathbb{R}_{+}^{n}\setminus
R_{\delta}}P^{s}(x-y,t)\,dx\,dy=o(1).$

We claim that

(4.16)
$\int_{\mathbb{R}^{n-1}}P^{s}((z^{\prime},z_{n}),t)\,dz^{\prime}=2\int_{0}^{\infty}e^{-t|r|^{2s}}\cos(z_{n}r)\,dr.$

For all $z_{n}\in\mathbb{R}$ we set
$v_{z_{n}}(z^{\prime},t):=P^{s}((z^{\prime},z_{n}),t)$. We have

(4.17)
$\begin{split}P^{s}(z,t)=\mathcal{F}^{-1}[\frac{1}{(2\pi)^{\frac{n}{2}}}e^{-t|\cdot|^{2s}}](z)&=\int_{\mathbb{R}^{n}}\frac{1}{(2\pi)^{n}}e^{-t|\xi|^{2s}}e^{i\langle
z,\xi\rangle}\,d\xi\\\
&=\int_{\mathbb{R}^{n-1}}\frac{1}{(2\pi)^{\frac{n-1}{2}}}e^{i\langle
z^{\prime},\xi^{\prime}\rangle}\int_{\mathbb{R}}\frac{1}{(2\pi)^{1+\frac{n-1}{2}}}e^{-t|\xi|^{2s}}e^{iz_{n}\xi_{n}}\,d\xi_{n}\,d\xi.\end{split}$

Since the Fourier transform is an isometry we have

$v_{z_{n}}(z^{\prime},t)=\mathcal{F}^{-1}[\mathcal{F}[v_{z_{n}}(\cdot,t)]](z^{\prime})=\frac{1}{(2\pi)^{\frac{n-1}{2}}}\int_{\mathbb{R}^{n-1}}e^{i\langle
z^{\prime},\xi^{\prime}\rangle}\mathcal{F}[v_{z_{n}}(\cdot,t)](\xi^{\prime})\,d\xi^{\prime},$

where with a little abuse of notations we denoted the Fourier transform in
$\mathbb{R}^{n-1}$ still by $\mathcal{F}[\cdot](\cdot)$. Therefore by
uniqueness of the Fourier transform we arrive at

$\begin{split}\mathcal{F}[v_{z_{n}}(\cdot,t)](\xi^{\prime})&=\frac{1}{(2\pi)^{1+\frac{n-1}{2}}}\int_{\mathbb{R}}e^{-t|\xi|^{2s}}e^{iz_{n}\xi_{n}}\,d\xi_{n}\\\
&=\frac{1}{(2\pi)^{1+\frac{n-1}{2}}}\int_{\mathbb{R}}e^{-t|\xi|^{2s}}\cos(z_{n}\xi_{n})\,d\xi_{n}\\\
&=\frac{1}{(2\pi)^{1+\frac{n-1}{2}}}\int_{\mathbb{R}}e^{-t(|\xi^{\prime}|^{2}+\xi_{n}^{2})^{s}}\cos(z_{n}\xi_{n})\,d\xi_{n}.\end{split}$

From the above formula and by

$\begin{split}\int_{\mathbb{R}^{n-1}}&P^{s}((z^{\prime},z_{n}),t)\,dz^{\prime}=\int_{\mathbb{R}^{n-1}}v_{z_{n}}(z^{\prime},t)\,dz^{\prime}\\\
&=(2\pi)^{\frac{n-1}{2}}\mathcal{F}[v_{z_{n}}(\cdot,t)](0)=\frac{1}{\pi}\int_{0}^{\infty}e^{-t|\xi_{n}|^{2s}}\cos(z_{n}\xi_{n})\,d\xi_{n},\end{split}$

we obtain (4.16).
Now using Fubini Theorem and (4.16) with $x_{n}-y_{n}$ instead of $z_{n}$ we
obtain that

(4.18)
$\begin{split}\int_{Q_{\delta}^{-}}\int_{R_{\delta}}P^{s}(x-y,t)\,dx\,dy=&\int_{Q_{\delta}}\int_{-\delta}^{0}\int_{\mathbb{R}^{n-1}}\int_{0}^{\delta}P^{s}((x^{\prime}-y^{\prime},x_{n}-y_{n}),t)\,dx^{\prime}\,dy^{\prime}\,dx_{n}\,dy_{n}\\\
=&\frac{1}{\pi}|Q_{\delta}|\int_{-\delta}^{0}\int_{0}^{\delta}\int_{0}^{\infty}e^{-tr^{2s}}\cos(r(x_{n}-y_{n})))\,dr\,dx_{n}\,dy_{n}.\end{split}$

Now we use basic trigonometry to have

$\int_{-\delta}^{0}\int_{0}^{\delta}\cos(r(x_{n}-y_{n})))\,dr\,dx_{n}\,dy_{n}=\frac{\sin^{2}(\delta
r)-(1-\cos(\delta r))^{2}}{r^{2}}=2\cos(\delta r)\frac{(1-\cos(\delta
r))}{r^{2}}$

Thus we have that (4.18) becomes

$\int_{Q_{\delta}^{-}}\int_{\mathbb{R}^{n}_{+}}P^{s}(x-y,t)\,dx\,dy=\frac{2}{\pi}|Q_{\delta}|\int_{0}^{\infty}e^{-tr^{2s}}\cos(\delta
r)\frac{(1-\cos(\delta r))}{r^{2}}\,dr$

Now we observe that

$\begin{split}\int_{0}^{\infty}e^{-tr^{2s}}\cos(\delta r)&\frac{(1-\cos(\delta
r))}{r^{2}}\,dr\\\ &=\int_{0}^{\infty}e^{-tr^{2s}}\frac{(\cos(\delta
r))-1}{r^{2}}\,dr+\int_{0}^{\infty}e^{-tr^{2s}}\frac{(1-\cos^{2}(\delta
r))}{r^{2}}\,dr\\\ &=\int_{0}^{\infty}e^{-tr^{2s}}\frac{(\cos(\delta
r))-1}{r^{2}}\,dr+\frac{1}{2}\int_{0}^{\infty}e^{-tr^{2s}}\frac{(1-\cos(2\delta
r)}{r^{2}}\,dr\\\ &=\int_{0}^{\infty}e^{-tr^{2s}}\frac{(\cos(\delta
r))-1}{r^{2}}\,dr+\int_{0}^{\infty}e^{\frac{-tr^{2s}}{2^{2s}}}\frac{(1-\cos(\delta
r)}{r^{2}}\,dr\\\ &=\int_{0}^{\infty}\frac{(1-\cos(\delta
r)}{r^{2}}(e^{\frac{-tr^{2s}}{2^{2s}}}-e^{-tr^{2s}})\,dr=t^{\frac{1}{2s}}h(t)\end{split}$

where

$h(t):=\int_{0}^{\infty}\frac{1-\cos\left(\frac{\delta
r}{t^{\frac{1}{2s}}}\right)}{r^{2}}(e^{\frac{-r^{2s}}{2^{2s}}}-e^{-r^{2s}})\,dr>0.$

It is quite easy to show that $h(t)$ is bounded and by applying Riemann-
Lebesgue lemma we finally have

$\lim_{t\to
0}h(t)=\int_{0}^{\infty}\frac{e^{\frac{-r^{2s}}{2^{2s}}}-e^{-r^{2s}}}{r^{2}}\,dr=\frac{1}{2}\int_{0}^{\infty}\frac{1-e^{-r^{2s}}}{r^{2}}\,dr=\frac{1}{4s}\int_{0}^{\infty}\frac{1-e^{-r}}{r^{1+\frac{1}{2}s}}\,dr=\frac{\Gamma(1-\frac{1}{2s})}{4}$

which readly gives (4.14). ∎

The next lemma is an obvious consequence of the previous one.

######  Lemma 4.5.

Let $Q_{\delta}^{+}.Q_{\delta}^{-}$ and $Q_{\delta}$ as above. For
$s\in(\frac{1}{2},1)$ and for $t$ small enough it holds true

(4.19)
$\int_{Q_{\delta}^{-}}\int_{Q_{\delta}^{+}}P^{s}(x-y,t)\,dx\,dy\leq\frac{\Gamma\left(1-\frac{1}{2s}\right)}{2\pi}t^{\frac{1}{2s}}(\mathcal{H}^{n-1}(Q_{\delta})+o(1)).$

Finally we prove the crucial estimate to prove the $\Gamma$ convergence
result.

######  Lemma 4.6.

Let $Q_{\delta}^{+}.Q_{\delta}^{-}$ and $Q_{\delta}$ as above. For
$s>\frac{1}{2}$ we have

(4.20)
$\int_{Q_{\delta}^{-}}\int_{Q_{\delta}^{+}}P^{s}(x-y,t)\,dx\,dy\geq\frac{\Gamma\left(1-\frac{1}{2s}\right)}{2\pi}t^{\frac{1}{2s}}(\mathcal{H}^{n-1}(Q_{\delta})+o(1)).$

###### Proof.

To prove this lemma we just need to show that the error passing from a small
cube to the hyperplane $\mathbb{R}^{n-1}$ is negligible when $t$ is small. To
do that we start decomposing the integral

(4.21)
$\begin{split}\int_{0}^{\delta}&\int_{-\delta}^{0}\int_{Q^{\prime}_{\delta}}P^{s}(x-y,t)\,dx_{n}dy\\\
&=\int_{0}^{\delta}\int_{-\delta}^{0}\int_{\mathbb{R}^{n-1}}P^{s}(x-y,t)\,dx_{n}dy-\int_{0}^{\delta}\int_{-\delta}^{0}\int_{\mathbb{R}^{n-1}\setminus
Q^{\prime}_{\delta}}P^{s}(x-y,t)\,dx_{n}dy\end{split}$

and observe that by (1.12) and using a change of variable we have

$\begin{split}\int_{0}^{\delta}\int_{-\delta}^{0}\int_{\mathbb{R}^{n-1}\setminus
Q^{\prime}_{\delta}}P(x-y,t)\,&dx_{n}dy=t^{-\frac{n}{2s}}\int_{0}^{\delta}\int_{-\delta}^{0}\int_{\mathbb{R}^{n-1}\setminus
Q^{\prime}_{\delta}}P^{s}\left(\frac{x-y}{t^{\frac{1}{2s}}},1\right)\,dx_{n}dy\\\
&=\frac{1}{t^{\frac{1}{2s}}}\int_{0}^{\delta}\int_{-\delta}^{0}\int_{\mathbb{R}^{n-1}\setminus
Q^{\prime}_{\frac{\delta}{t}}(x^{\prime})}P^{s}\left(z,\frac{y_{n}-x_{n}}{t^{\frac{1}{2s}}},1\right)\,dz\,dx_{n}\,dy_{n}.\end{split}$

Now we using (1.11) we have that

(4.22)
$\begin{split}t^{-\frac{1}{2s}}\int_{0}^{\delta}\int_{-\delta}^{0}\int_{\mathbb{R}^{n}\setminus
Q^{\prime}_{\frac{\delta}{t^{\frac{1}{2s}}}}(x^{\prime})}&P^{s}\left(z,\frac{y_{n}-x_{n}}{t^{\frac{1}{2s}}},1\right)\,dz\,dx_{n}\,dy_{n}\\\
&\leq
t^{-\frac{1}{2s}}\int_{0}^{\delta}\int_{-\delta}^{0}\int_{\mathbb{R}^{n}\setminus
B_{\frac{\delta}{t^{\frac{1}{2s}}}}(x^{\prime})}\frac{1}{|z|^{n+2s}}\,dz\,dx_{n}\,dy_{n}\\\
&\leq(n-1)\omega_{n-1}t^{-\frac{1}{2s}}\int_{0}^{\delta}\int_{-\delta}^{0}\int_{\frac{\delta}{t^{\frac{1}{2s}}}}^{\infty}\rho^{-2-2s}\,d\rho\,dx_{n}\,dy_{n}\\\
&=(n-1)\omega_{n-1}\delta^{1-2s}t=o(t^{\frac{1}{2s}})\end{split}$

The proof of (4.20) is now direct consequence of Lemma 4.4 and inequality
(4.22). ∎

Now we prove that $\Gamma^{n,\frac{1}{2}}$ defined in (4.3) is equal to the
constant $c_{n}$ defined in (4.7).

######  Lemma 4.7.

Let $a,b\in\mathbb{R}$ and $k\in\mathbb{N}$ such that $b>-k$ and $a>k+b$, then

(4.23)
$\int_{\mathbb{R}^{k}}\frac{|x|^{b}}{(1+|x|^{2})^{\frac{a}{2}}}dx=\frac{\omega_{k-1}}{2}\frac{\Gamma(\frac{a+k}{2})\Gamma(\frac{a-b-k}{2})}{\Gamma(\frac{a+b}{2})}.$

In particular

(4.24)
$c_{n}:=\int_{\mathbb{R}^{n-1}}\frac{1}{(1+|x|^{2})^{\frac{n+1}{2}}}dx=\frac{\omega_{n-2}}{2}\frac{\Gamma(\frac{n}{2})}{\Gamma(\frac{n+1}{2})}:=\Gamma^{n,\frac{1}{2}}.$

Before to prove Lemma (4.7) we recall the definition of Euler’s beta function

(4.25) $B:(0,+\infty)\times(0,+\infty)\rightarrow\mathbb{R},\quad
B(x,y):=2\int_{0}^{\frac{\pi}{2}}(\cos(\theta))^{2x-1}(\sin(\theta))^{2y-1}d\theta.$

The relation between the beta and the gamma function is the following

(4.26) $B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}.$

###### Proof of Lemma 4.7.

We observe that the function $h(x):=\frac{|x|^{b}}{(1+|x|^{2})^{\frac{a}{2}}}$
belong $\mathrm{L}^{1}(\mathbb{R}^{k})$. Integrating in polar coordinate and
by (4.26) we have that

(4.27)
$\begin{split}\int_{\mathbb{R}^{k}}&\frac{|x|^{b}}{(1+|x|^{2})^{\frac{a}{2}}}dx=\omega_{k-1}\int_{0}^{+\infty}\frac{\rho^{b+k-1}}{(1+\rho^{2})^{\frac{a}{2}}}d\rho\\\
&\overset{\rho=\tan(\eta)}{=}\omega_{k-1}\int_{0}^{\frac{\pi}{2}}\frac{(\tan(\eta))^{b+k-1}}{(1+\tan^{2}(\eta))^{\frac{a-2}{2}}}d\eta=\omega_{k-1}\int_{0}^{\frac{\pi}{2}}(\sin(\eta))^{b+k-1}(\cos(\eta))^{a-b-k-1}d\eta\\\
&=\frac{\omega_{k-1}}{2}B(\frac{b+k}{2},\frac{a-b-k}{2})=\frac{\omega_{k-1}}{2}\frac{\Gamma(\frac{a+k}{2})\Gamma(\frac{a-b-k}{2})}{\Gamma(\frac{a+b}{2})}\end{split}$

hence (4.23). Formula (4.24) follows by (4.23) with $b=0$ and $k=n-1$ and
$a=n+1$. ∎

In the next corollary we summirize all the previous result of this sections.

######  Corollary 4.8.

Let $g_{s}$ be the function defined in (0.2) and

(4.28)
$\Gamma^{n,s}=\begin{cases}(2\pi)^{n-2}\Gamma(1-\frac{1}{2s})\,\qquad&s\in(\frac{1}{2},1),\\\
\frac{\omega_{n-2}}{2}\frac{\Gamma(\frac{n}{2})}{\Gamma(\frac{n+1}{2})}\qquad&s=\frac{1}{2}.\end{cases}$

Then

$\lim_{t\to
0}\frac{1}{g_{s}(t)}\int_{Q^{-}_{\delta}}\int_{Q_{\delta}^{+}}P^{s}(x-y,t)\,dxdy=\Gamma^{n,s}\mathcal{H}^{n-1}(Q_{\delta}).$

We are now ready to prove our $\Gamma-$convergence theorem.

###  4.2. Proof of Theorem 4.1(i).

######  Proposition 4.9.

Let $s\in[\frac{1}{2},1)$, $Q=[-1,1]^{n}$, $g_{s}(t)$ defined in 4.8 and let
$E$ a set of finite perimeter. Set

(4.29) $\Gamma_{n,s}=\inf\left\\{\liminf_{t\to
0}\frac{1}{g_{s}(t)}\int_{E_{t}^{c}\cap Q}\int_{E_{t}\cap
Q}P^{s}(x-y,t)\,dxdy:\,E_{t}\to\mathbb{R}_{+}^{n}\text{ in
}\mathrm{L}^{1}(\mathbb{R}^{n})\right\\}.$

Then for any sequence $E_{i}\to E$ and $t_{i}\to 0$ we have

(4.30)
$\liminf_{i}\frac{1}{g_{s}(t_{i})}\int_{E_{i}}\int_{E_{i}^{c}}P^{s}(x-y,t)\,dxdy\geq\Gamma_{n,s}{\rm
P}(E)$

with the constant $\Gamma_{n,s}$ defined in (4.29).

###### Proof.

The result follows by following the strategy a well estabilished strategy
based on blow up, introduced in this context in[13]. Denote by ${\mathcal{C}}$
the family of all $n$-cubes in $\mathbb{R}^{1}{n}$

${\mathcal{C}}:=\left\\{R(x+rQ):\ x\in\mathbb{R}^{n},\,\,r>0,\,\,R\in
SO(n)\right\\}.$

Since the set $E$ is of finite perimeter $|\nabla\chi_{E}|$-a.e.
$x_{0}\in\mathbb{R}^{n}$ there exists $R_{x_{0}}\in SO(n)$ such that
$(E-x_{0})/r$ locally converge in measure to $R_{x_{0}}H$ as $r\to 0$ and

(4.31) $\lim_{r\to
0}\frac{|\nabla\chi_{E}|(x_{0}+rR_{x_{0}}Q)}{r^{n-1}}=1,\quad\text{for
$|\nabla\chi_{E}|$-a.e. }x_{0}.$

Now, given a cube $C\in{\mathcal{C}}$ contained in $\Omega$ we set

$\alpha_{i}(C):=\frac{1}{g(t_{i})}\int_{E_{i}^{c}\cap C}\int_{E_{i}\cap
C}P^{s}(x-y,t_{i})\,dxdy$

and

$\alpha(C):=\liminf_{i\to\infty}\alpha_{i}(C).$

We claim that, setting $C_{r}(x_{0}):=x_{0}+rR_{x_{0}}Q$, where $R_{x_{0}}$ is
as in (4.31), for $|\nabla\chi_{E}|$-a.e. $x_{0}$ we have

(4.32) $\Gamma_{n,s}\leq\liminf_{r\to
0}\frac{\alpha(C_{r}(x_{0}))}{r^{n-1}}\quad\text{for $|\nabla\chi_{E}|$-a.e.
}x\in\mathbb{R}{n}.$

Since from now on $x_{0}$ is fixed, we assume $R_{x_{0}}=I$, so that the limit
hyperplane is $H$ and the cubes $C_{r}(x_{0})$ are the standard ones
$x_{0}+rQ$. Let us choose a sequence $r_{k}\to 0$ such that

$\liminf_{r\to
0}\frac{\alpha(C_{r}(x_{0}))}{r^{n-1}}=\lim_{k\to\infty}\frac{\alpha(C_{r_{k}}(x_{0}))}{r_{k}^{n-1}}.$

For $k>0$ we can choose $i(k)$ so large that the following conditions hold:

$\begin{cases}&\alpha_{i_{k}}(C_{r_{k}}(x_{0}))\leq\alpha(C_{r_{k}}(x_{0}))+r_{k}^{n},\\\
&\int_{C_{r_{k}}(x)}|\chi_{E_{i_{k}}}-\chi_{E}|dx<\frac{|C_{r_{k}}|}{k}.\end{cases}$

Then we infer

(4.33)
$\begin{split}\frac{\alpha(C_{r_{k}}(x_{0}))}{r_{k}^{n-1}}&\geq\frac{\alpha_{i(k)}(C_{r_{k}}(x_{0}))}{r_{k}^{n-1}}-r_{k}\\\
&=\frac{1}{g(t_{i_{k}})r_{k}^{n-1}}\int_{E_{i_{k}}^{c}\cap
C_{r_{k}}(x_{0})}\int_{E_{i_{k}}\cap
C_{r_{k}}(x_{0})}P^{s}(x-y,t_{i_{k}})\,dxdy-r_{k}.\end{split}$

Now we distinguish the case $s=1/2$ and $s\in(1/2,1)$. In the former case the
above inequality reads as

(4.34) $\frac{\alpha(C_{r_{k}}(x_{0}))}{r_{k}^{n-1}}\geq\frac{1}{|\log
t_{i_{k}}|}\int_{(E_{i_{k}}-x_{0})/r_{k}\cap
C}\int_{(E_{i_{k}}-x_{0})/r_{k}\cap
C}\frac{1}{(|x-y|^{2}+\frac{t_{i_{k}}^{2}}{r_{k}^{2}})^{\frac{n+1}{2}}}\,dxdy$

Up to extract a further subsequecme we assume that $t_{i_{k}}\leq r_{k}^{k+1}$
and thereofore

(4.35)
$\log\frac{1}{t_{i_{k}}}=\log\frac{r_{k}}{t_{i_{k}}}+\frac{1}{k+1}\log\frac{1}{r^{k}_{k}}\leq\log\frac{r_{k}}{t_{i_{k}}}+\frac{1}{k}\log\frac{1}{r^{k}_{k}}\leq(1+\frac{1}{k})\log\frac{r_{k}}{t_{i_{k}}}$

Since $(E_{i_{k}}-x)/r_{k}\to H$ the above inequality implies joint with
(4.34) provides

$\liminf_{r\to 0}\frac{\alpha(C_{r}(x_{0}))}{r_{k}^{n-1}}\geq\Gamma_{n,s}$

where the constant $\Gamma_{n,s}$ is defined in (4.28)
Case two: $s>\frac{1}{2}$.
In this case we recall that $g(t)=t^{\frac{1}{2s}}$ and then using (1.12) we
get

$\begin{split}\frac{\alpha(C_{r_{k}}(x_{0}))}{r_{k}^{n-1}}&=\frac{t_{i_{k}}^{-\frac{n+1}{2s}}}{r_{k}^{n-1}}\int_{E_{i_{k}}^{c}\cap
C_{r_{k}}(x_{0})}\int_{E_{i_{k}}^{\cap}C_{r_{k}}(x_{0})}P^{s}((x-y)t^{-\frac{1}{2s}},1)\,dxdy-
r_{k}.\\\
&=(t_{i_{k}}^{-\frac{1}{2s}}r_{k})^{n+1}\int_{(E_{i_{k}}-x_{0})/r_{k}\cap
C}\int_{(E_{i_{k}}-x_{0})/r_{k}\cap
C)}P^{s}((x-y)r_{k}t_{i_{k}}^{-\frac{1}{2s}},1)\,dxdy-r_{k}\\\
&=(t_{i_{k}}r_{k}^{-2s})^{-\frac{1}{2s}}\int_{(E_{i_{k}}-x_{0})/r_{k}\cap
C}\int_{(E_{i_{k}}-x_{0})/r_{k}\cap
C)}P^{s}((x-y),t_{i_{k}}r_{k}^{-2s})\,dxdy-r_{k}\end{split}$

Therefore, chosing $i_{k}$ so larg that $t_{i_{k}}r_{k}^{-2s}\to 0$ we get

$\frac{\alpha(C_{r_{k}}(x_{0}))}{r_{k}^{n-1}}\geq\Gamma_{n,s}$

with $\Gamma_{n,s}$ defined in (4.29). ∎

We are now in position to prove Theorem 4.1(i).

###### Proof of Theorem 4.1(i).

To conclude the proof we need to show that $\Gamma_{n,s}=\Gamma^{n,s}$. The
proof of this fact is the same as in [2][Lemma 13]. We decide to not reproduce
it here in deteails but to just sketch it. First, via a gluing argument, which
is possibile since a coarea formula holds for our functional, one can show
that the infimum in the definition of $\Gamma_{n,s}$ can be take among sets
whch coincide with $H$ outside a smaller cube $Q_{\delta}$ and then, via a
nonlocal calibration argument, one also gets the minimality of the halfspace
(see [6, 23]), which readly provides $\Gamma^{n,s}=\Gamma_{n,s}$. ∎

###  4.3. Proof of Theorem 4.1(ii).

This subsection we prove Proof of Theorem 4.1(ii).

###### Proof of Theorem 4.1(ii).

We start by proving that for a polyhedron $\Pi$ it holds

(4.36) $\limsup_{t\to
0}\frac{1}{g_{s}(t)}\int_{\Pi}\int_{\Pi^{c}}P^{s}(x-y,t)\,dxdy\leq\Gamma^{n,s}{\rm
P}(\Pi).$

For any $\varepsilon>0$ there exists $\delta_{0}$ such that for any
$\delta\in(0,\delta_{0})$ there exists a collection of $N_{\delta}$ cubes of
volume $(2\delta)^{n}$ centered at $x_{i}\in\partial\Pi$ such that each cubes
$Q_{\delta}(x_{i})$ intersect one and only one face of $\partial\Pi$ and

  * •

if $\tilde{Q}_{\delta}(x_{i})$ denotes the dilation of $Q_{\delta}(x_{i})$ by
a factor $(1+\delta)$, then each cube $\tilde{Q}_{\delta}(x_{i})$ intersects
exactly one face $\Sigma$ of $\partial\Pi$ and each of its sides is either
parallel or orthogonal to $\Sigma$;

  * •

$|\mathcal{H}^{n-1}(\partial\Pi)-N_{\delta}(2\delta)^{n-1}|<\varepsilon$.

Moreover, since we are interested in the limit as $t\to 0$ we can assume that
$t<\delta$. For all $\delta>0$ we set

(4.37)
$(\partial\Pi)_{\delta}:=\left\\{x\in\mathbb{R}^{n}:d(x,\partial\Pi)<\delta\right\\},\quad(\partial\Pi)_{\delta}^{-}:=(\partial\Pi)_{\delta}\cap\Pi.$

For $x\in\Pi$ we define

$I(x,t)=\int_{\Pi^{c}}P^{s}(x-y,t)\,dy.$

We now have to distinguish among many cases.
Case 1: $x\in\Pi\setminus(\Pi)_{\delta}^{-}$.
In this case we note that (1.11) implies the existence of a constant $c_{n,s}$
such that

$P^{s}(x-y,t)\leq\frac{c_{n,s}t}{|x-y|^{2}+t^{\frac{1}{s}})^{\frac{n+2s}{2}}}$

and therefore

(4.38) $\begin{split}I_{s}(x,t)\leq
c_{n,s}\int_{\Pi^{c}}\frac{t}{(|x-y|^{2}+t^{\frac{1}{s}})^{\frac{n+2s}{2}}}&\leq
n\omega_{n}t\int_{\delta}^{\infty}\frac{\rho^{n-1}}{(\rho^{2}+t^{\frac{1}{s}})^{\frac{n+2s}{2}}}\,d\rho\\\
&\leq
n\omega_{n}t\int_{\delta}^{\infty}\frac{\rho}{(\rho^{2}+t^{\frac{1}{s}})^{1+s}}\,d\rho\\\
&=\frac{\delta t}{2s(\delta^{2}+t^{\frac{1}{s}})^{s}}.\end{split}$

Case two: $x\in\Pi_{\delta}^{-}\setminus\bigcup_{i}Q_{\delta}(x_{i})$.
In this case we write $\partial\Pi=\bigcup_{j=1}^{J}\Sigma_{j}$, and define

$(\partial\Pi)^{-}_{\delta,j}:=\\{x\in(\partial\Pi)^{-}_{\delta}:d(x,\Pi^{c})=d(x,\Sigma_{j})\\}.$

Clearly
$(\partial\Pi)^{-}_{\delta}=\bigcup_{j=1}^{J}(\partial\Pi)^{-}_{\delta,j}$ and

$(\partial\Pi)^{-}_{\delta,j}\subset\\{x+t\nu:x\in\Sigma_{\delta,j},\,t\in(0,\delta),\,\nu\text{
is the interior unit normal to }\Sigma_{\delta,j}\\},$

and $\Sigma_{\delta,j}$ is the set of points $x$ belonging to the same
hyperplane as $\Sigma_{j}$ and with $d(x,\Sigma_{j})\leq\delta$. Set
$d(x):=\rm{dist}(x,\Pi^{c})$ and observe that $B_{d(x)}(x)\subset\Pi$ and
therefore

$\begin{split}I(x,t)\leq\int_{B^{c}_{d(x)}(x)}P^{s}(x-y,t)\,dy&\leq
n\omega_{n}t\int_{d(x)}^{\infty}\frac{\rho^{n-1}}{(\rho^{2}+t^{\frac{1}{s}})^{\frac{n+2s}{2}}}\,d\rho\\\
&\leq\frac{n\omega_{n}}{2s}\frac{t}{(d(x)^{2}+t^{\frac{1}{s}})^{s}}.\end{split}$

Integrating the above inequality on the set
$(\partial\Pi)_{\delta}^{-}\setminus\bigcup_{i=1}^{N_{\delta}}Q_{i}^{\delta}$
gives

(4.39)
$\begin{split}\int_{(\partial\Pi)_{\delta}^{-}\setminus\bigcup_{i=1}^{N_{\delta}}Q_{\delta}(x_{i})}I(x,t)\,dx&\leq\frac{n\omega_{n}}{s}\sum_{j=1}^{J}\int_{(\partial\Pi)_{\delta,j}^{-}\setminus\bigcup_{i=1}^{N_{\delta}}Q_{\delta}(x_{i})}\frac{t}{(d(x)^{2}+t^{\frac{1}{s}})^{s}}\,dx\\\
&\leq\frac{n\omega_{n}}{s}\sum_{j=1}^{J}\int_{(\partial\Pi)_{\delta,j}^{-}\setminus\bigcup_{i=1}^{N_{\delta}}Q_{\delta}(x_{i})}\frac{t}{[{\rm
dist}(x,\Sigma_{\delta,j})]^{s}}\,dx\\\
&\leq\frac{n\omega_{n}t}{s}\sum_{j=1}^{J}\int_{(\Sigma_{\delta,j})\setminus\bigcup_{i=1}^{N_{\delta}}Q_{\delta}(x_{i})}\bigg{(}\int_{0}^{\delta}\frac{dr}{(r^{2}+t^{\frac{1}{s}})^{s}}\bigg{)}\,d\mathcal{H}^{n-1}\\\
&=\frac{n\omega_{n}t^{\frac{1}{2s}}}{s}\sum_{j=1}^{J}\int_{(\Sigma_{\delta,j})\setminus\bigcup_{i=1}^{N_{\delta}}Q_{\delta}(x_{i})}\bigg{(}\int_{0}^{\delta
t^{-\frac{1}{2s}}}\frac{dr}{(r^{2}+1)^{s}}\bigg{)}\,d\mathcal{H}^{n-1}.\end{split}$

Now we need to distinguish between the case $s=\frac{1}{2}$ and
$s>\frac{1}{2}$. In the former case we have

$\int_{0}^{\delta
t^{-1}}\frac{1}{r^{2}+1}\,dr=\log(\delta+\sqrt{\delta^{2}+t^{2}})-\log(t)\leq|\log
t|$

and therefore (4.39) becomes

$\int_{(\partial\Pi)_{\delta}^{-}\setminus\bigcup_{i=1}^{N_{\delta}}Q_{\delta}(x_{i})}I(x,t)\,dx\leq
2n\omega_{n}t|\log
t|\mathcal{H}^{n-1}\left(\bigg{(}\bigcup_{j=1}^{J}\Sigma_{\delta,j}\bigg{)}\setminus\bigcup_{i=1}^{N_{\delta}}Q_{\delta}(x_{i})\right),$

While $s>\frac{1}{2}$ from (4.39) we get

$\begin{split}\int_{(\partial\Pi)_{\delta}^{-}\setminus\bigcup_{i=1}^{N_{\delta}}Q_{i}^{\delta}}I(x,t)\,dx\leq\frac{n\omega_{n}Ct^{\frac{1}{2s}}}{s}\mathcal{H}^{n-1}\left(\bigg{(}\bigcup_{j=1}^{J}\Sigma_{\delta,j}\bigg{)}\setminus\bigcup_{i=1}^{N_{\delta}}Q_{i}^{\delta}\right).\end{split}$

with

$C=\int_{0}^{\infty}\frac{1}{(r^{2}+1)^{s}}\,dr.$

Therefore, in any case we found that for $s\geq\frac{1}{2}$ it holds

(4.40)
$\frac{1}{g_{s}(t)}\int_{(\partial\Pi)_{\delta}^{-}\setminus\bigcup_{i=1}^{N_{\delta}}Q_{\delta}(x_{i})}I(x,t)\leq
C^{2}_{n,s}\mathcal{H}^{n-1}\left(\bigg{(}\bigcup_{j=1}^{J}\Sigma_{\delta,j}\bigg{)}\setminus\bigcup_{i=1}^{N_{\delta}}Q_{\delta}(x_{i})\right)\leq
C^{2}_{n,s}\varepsilon.$

Case three: $x\in\Pi^{-}\cap\bigcup Q_{\delta}(x_{i})$.
In this case we write

$I(x,t)=\int_{\Pi^{c}\cap\\{y:|x-y|\geq\delta^{2}\\}}P^{s}(x-y,t)\,dy+\int_{\Pi^{c}\cap\\{y:|x-y|<\delta^{2}\\}}P^{s}(x-y,t)\,dy.$

For the first integral, arguing as in (4.38), we quickly have

$\int_{\Pi^{c}\cap\\{y:|x-y|\geq\delta^{2}\\}}P^{s}(x-y,t)\,dy\leq\frac{\delta^{2}t}{2s(\delta^{4}+t^{2})^{s}}.$

Therefore, we now estimate that for each $1\leq i\leq N_{\delta}$ it holds

(4.41)
$\begin{split}\int_{Q_{\delta}(x_{i})\cap\Pi}\int_{\Pi^{c}\cap\\{y:|x-y|<\delta^{2}\\}}P^{s}(x-y,t)&\,dy=\int_{Q_{\delta}(x_{i})\cap\Pi}\int_{Q_{\delta+\delta^{2}}(x_{i})\cap\Pi^{c}}P^{s}(x-y,t)\,dy\\\
&\leq\int_{Q_{\delta+\delta^{2}}(x_{i})\cap\Pi}\int_{Q_{\delta+\delta^{2}}(x_{i})\cap\Pi^{c}}P^{s}(x-y,t)\,dy\\\
&\leq\Gamma^{n,s}\left(g_{s}(t)+o(g_{s}(t))\right)\,|Q_{\delta+\delta^{2}}|.\end{split}$

Summing (4.41) for $1\leq i\leq N_{\delta}$ we get

(4.42)
$\frac{1}{g_{s}(t)}\int_{\bigcup_{i}Q_{\delta}(x_{i})}\int_{\Pi^{c}}P^{s}(x-y,t)\,dxdy\leq\Gamma^{n,s}N_{\delta}(\delta+\delta^{2})^{n-1}\leq\Gamma^{n,s}\mathcal{H}^{n-1}(\partial\Pi)+\varepsilon).$

Collecting (4.38),(4.39),(4.42),(4.41) and sendin $t\to 0$ fisrt and
$\varepsilon\to 0$ secondly we get

$\limsup_{t\to
0}\frac{1}{g_{s}(t)}\int_{\Pi}\int_{\Pi^{c}}P^{s}(x-y,t)\,dxdy\leq\Gamma^{n,s}{\rm
P}(\Pi).$

The argument for a generic set $E$ of finite perimeter follows by density. ∎

###  4.4. Characterization of sets of finite perimeter.

As a consequence of our $\Gamma$-convergence analysis, we prove that a set
$E\in\mathrm{M}(\mathbb{R}^{n})$ has finite perimeter if and only if for all
$s\in[\frac{1}{2},1)$

$\limsup_{t\to 0}\frac{\mathcal{E}^{s}_{t}(E)}{g^{s}(t)}<+\infty.$

######  Theorem 4.10.

Let $E\in\mathrm{M}(\mathbb{R}^{n})$. The following statement hold true.

  * (i)

If $\limsup_{t\to 0}\frac{\mathcal{E}^{s}_{t}(E)}{g_{s}(t)}<+\infty$ for some
$s\in[\frac{1}{2},1)$, then $E$ is a set of finite perimeter.

  * (ii)

If $E$ is a set of finite perimeter then

(4.43) $\limsup_{t\to
0}\frac{\mathcal{E}^{s}_{t}(E)}{g_{s}(t)}<+\infty\quad\text{for every
}s\in\big{[}\frac{1}{2},1\big{)}.$

###### Proof.

We notice that (i) is an immediate consequence of 2.1 taking $E_{i}=E$ for
every $i\in\mathbb{N}$. We prove (ii). By the very definition of
$\mathcal{E}^{s}_{t}(E)$ we have

(4.44)
$\begin{split}\frac{\mathcal{E}^{s}_{t}(E)}{g_{s}(t)}&=\frac{1}{g_{s}(t)}\int_{E}\int_{E^{c}}P^{s}(h,t)dhdx\\\
&=\frac{1}{2g_{s}(t)}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}|\chi_{E}(x+h)-\chi_{E}(x)|P^{s}(h,t)dhdx\\\
&\leq\frac{\mathrm{P}(E)}{2g_{s}(t)}\int_{\mathbb{R}^{n}}|h|P^{s}(h,t)dh.\end{split}$

Let $s\in(\frac{1}{2},1)$ by above formula and by (1.11) we obtain

(4.45) $\begin{split}\limsup_{t\to
0^{+}}\frac{\mathcal{E}^{s}_{t}(E)}{g_{s}(t)}&\leq\limsup_{t\to
0^{+}}\frac{\mathrm{P}(E)}{2}\bigg{[}\frac{1}{g_{s}(t)}\int_{B_{t^{\frac{1}{2s}}}}|h|t^{-\frac{n}{2s}}dh+\frac{1}{g_{s}(t)}\int_{B_{t^{\frac{1}{2s}}}^{c}}|h|\frac{t}{|h|^{n+2s}}dh\bigg{]}\\\
&=\frac{\mathrm{P}(E)}{2}\big{[}\frac{n\omega_{n}}{n-1}+\frac{n\omega_{n}}{2s-1}\big{]}\end{split}$

hence we have that (4.43) is true for all $s\in(\frac{1}{2},1)$. For
$s=\frac{1}{2}$ using the very definition of $P^{\frac{1}{2}}$, see (1.10), we
have

(4.46) $\begin{split}\limsup_{t\to
0^{+}}&\frac{\mathcal{E}^{\frac{1}{2}}_{t}(E)}{g_{s}(t)}=\frac{1}{g_{\frac{1}{2}}(t)}\int_{E}\int_{E^{c}}P^{\frac{1}{2}}(x-y,t)dydx\\\
&=\limsup_{t\to
0^{+}}\frac{1}{2g_{\frac{1}{2}}(t)}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}|\chi_{E}(x+h)-\chi_{E}(x)|P^{\frac{1}{2}}(h,t)dhdx\\\
&\leq\limsup_{t\to
0^{+}}\frac{\mathrm{P}(E)}{2t|\log(t)|}\bigg{[}\int_{B_{1}}\frac{t|h|}{(|h|^{2}+t^{2})^{\frac{n+1}{2}}}dh+\int_{B_{1}^{c}}\frac{t|h|}{(|h|^{2}+t^{2})^{\frac{n+1}{2}}}dh\bigg{]}\\\
&=\limsup_{t\to
0^{+}}\frac{\mathrm{P}(E)}{2|\log(t)|}\bigg{[}\int_{B_{\frac{1}{t}}}\frac{|\eta|}{(|\eta|^{2}+1)^{\frac{n+1}{2}}}d\eta+\int_{B_{\frac{1}{t}}^{c}}\frac{|\eta|}{(|\eta|^{2}+1)^{\frac{n+1}{2}}}d\eta\bigg{]}\\\
&=\limsup_{t\to
0^{+}}\frac{n\omega_{n}\mathrm{P}(E)}{2|\log(t)|}\bigg{[}\int_{0}^{\frac{1}{t}}\frac{\rho^{n}}{(\rho^{2}+1)^{\frac{n+1}{2}}}d\rho+\int_{\frac{1}{t}}^{+\infty}\frac{\rho^{n}}{(\rho^{2}+1)^{\frac{n+1}{2}}}d\rho\bigg{]}=C(n)\mathrm{P}(E)\end{split}$

where in the last step we have used the L’Hôpital’s rule. Hence by (4.46) we
obtain (4.43) for $s=\frac{1}{2}$. ∎

##  5\. Application: The local and the non local isoperimetric inequality

As a consequence of the $\Gamma$-convergence result we have a simple proof of
the isoperimetric inequality. Recall the fundamental properties of
$P^{s}(z,t)$ that $\int_{\mathbb{R}^{n}}P^{s}(z,t)\,dz=1$ and $P^{s}(|z|,t)$
is a decreasing function of the modulus of $z$. Given a measurable set $E$ of
finite measure and any $s\in(0,1)$ it is a straight forward application of
Riesz rearrangment inequality that

$\begin{split}\int_{E}\int_{E^{c}}P^{s}(x-y,t)\,dxdy&=|E|-\int_{E}\int_{E}P^{s}(x-y,t)\,dxdy\\\
&\geq|E^{*}|-\int_{E^{*}}\int_{E^{*}}P^{s}(x-y)\,dxdy\\\
&=\int_{E^{*}}\int_{(E^{*})^{c}}P^{s}(x-y,t)\,dxdy\end{split}$

where $E^{*}$ denotes the ball centered at the origin such that $|E^{*}|=|E|$.
Therefore if $\Pi\subset\mathbb{R}^{n}$ is a polyhedron, we have

$\frac{1}{g_{s}(t)}\left(\int_{\Pi}\int_{\Pi^{c}}P^{s}(x-y,t)\,dxdy-\int_{B}\int_{B^{c}}P^{s}(x-y,t)\,dxdy\right)\geq
0.$

with $|B|=|\Pi|$, Recalling (4.36), (4.30) and the fact
$\Gamma^{n,s}=\Gamma_{n,s}$, we take the $\limsup$ as $t\to 0$ in the above
inequality for $s\in[\frac{1}{2},1)$ to infer

${\rm P}(\Pi)\geq{\rm P}(B)=n\omega_{n}^{\frac{1}{n}}|\Pi|^{\frac{n-1}{n}}$

and by density we recover the isoperimetric inequality for all sets of finite
perimeter. For $s\in(0,\frac{1}{2})$ we even have pointwise convergence, hence
it is immediate to get that any set $E$ with finite $2s-$perimeter it holds

${\rm P}_{2s}(E)-{\rm P}_{2s}(B)\geq 0.$

which is the fractional isoperimetric inequality.

## Data availability and conflict of interest

On behalf of all authors, the corresponding author states that there is no
conflict of interest. No data was used for the research described in the
article.

## Aknowlegment

Andrea Kubin was supported by the DFG Collaborative Research Center TRR 109
“Discretization in Geometry and Dynamics”. D.A. La Manna is a member of GNAMPA
and has been partially supported by PRIN2022E9CF89.

## References

  * [1] G. Alberti, G. Bellettini. _A non-local anisotropic model for phase transitions: asymptotic behaviour of rescaled energies._ European Journal of Applied Mathematics. (1998),  9(3), 261-284.
  * [2] L. Ambrosio, G. De Philippis, L. Martinazzi. _Gamma-convergence of nonlocal perimeter functionals._ Manuscripta Math. (2011), 134 , 377-403.
  * [3] L. Ambrosio, N. Fusco, D. Pallara _Functions of Bounded Variation and FreeDiscontinuity Problems._ Oxford University Press, New York (2000).
  * [4] J. Berendsen and V. Pagliari,_On the asymptotic behaviour of nonlocal perimeters_ , ESAIM: COCV (2019) 25.
  * [5] J. Bourgain, H. Brezis, P. Mironescu,_Another look at Sobolev spaces._ ,In Optimal control and partial differential equations (2001) (439-455).
  * [6] X. Cabré,_Calibrations and null-Lagrangians for nonlocal perimeters and an application to the viscosity theory_ , Annali di Matematica Pura ed Applicata (2020) 199,1979–1995.
  * [7] L. Caffarelli and P.E. Souganidis,_Convergence of nonlocal threshold dynamics approximations to front propagation_ , Arch. Ration. Mech. Anal. 195 (2010), 1–23.
  * [8] A. Carbonaro, G. Mauceri,_A note on bounded variation and heat semigroup on Riemannian manifolds_ , Bull. Australian Math. Soc. (2007)76 (1), 155-161.
  * [9] A. Carbotti, S. Cito, D.A. La Manna, D. Pallara, _Gamma Convergence of Gaussian Fractional Perimeter_ , Adv. in Calc. Var.(2023), 16 (3) , 571-595.
  * [10] V. Crismale, L. De Luca, A. Kubin, A. Ninno, & M. Ponsiglione,  _The variational approach to s-fractional heat flows and the limit cases $s\rightarrow 0^{+}$ and $s\rightarrow 1^{-}$_, Journal of Functional Analysis.(2023), 284 (8), 109851.
  * [11] L. De Luca, A. Kubin, M. Ponsiglione, _The core-radius approach to supercritical fractional perimeters, curvatures and geometric flows_ , Nonlinear Analysis.(2022), 214 , 112585.
  * [12] A. De Rosa, D.A. La Manna, _A nonlocal approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties_ , Comm. Pure Appl. Anal., (2021),20(5): 2101-2116.
  * [13] I. Fonseca, S. Müller, _Quasi-convex integrands and lower semicontinuity in $L^{1}$_, SIAM J. Math. Anal. (1992) 23 , 1081-1098.
  * [14] F. Ferrari, M. Miranda, D. Pallara, A. Pinamonti, Y. Sire, _Fractional Laplacians, perimeters and heat semigroups in Carnot groups,_ Discrete Contin. Dyn. Syst. Ser. S (2018)11, 477-491
  * [15] N. Garofalo, _Fractional thoughts_ , New Developments in the Analysis of Nonlocal Operators (2017)
  * [16] A. Garroni, S. Müller,_A variational model for dislocations in the line tension limit_ , Archive for rational mechanics and analysis, (2006),181, 535-578.
  * [17] B. Güneysu, D. Pallara _Functions with bounded variation on a class of Riemannian manifolds with Ricci curvature unbounded from below_ Math. Ann. (2012) 365, 1307-1331.
  * [18] A. Kubin, M. Ponsiglione, , _Attractive Riesz potentials acting on hard spheres._ Nonlinearity, (2021)34(1), 350.
  * [19] A. Kubin, A. Tribuzio,  _Second order asymptotic of fractional Gagliardo seminorms as $s\nearrow 1$ and convergence of the associated gradient flows._ to appear.
  * [20] T. Laux, F. Otto, _Convergence of the thresholding scheme for multi-phase mean-curvature flow_ , Calc. Var. Par. Diff. Eq. (2016), 55 (5), 1–74.
  * [21] M. Ledoux, _Semigroup proofs of the isoperimetric inequality in Euclidean and Gauss space_. Bull. Sci. Math. (1994) 118, 485-510.
  * [22] M. Miranda (Jr), D. Pallara, F. Paronetto, M. Preunkert,_Heat Semigroup and Functions of Bounded Variation on Riemannian Manifolds_ , J. Reine Angew. Math. (2007), 613, 99-120.
  * [23] V. Pagliari,_Halfspaces minimise nonlocal perimeter: a proof via calibrations_ , Annali di Matematica Pura ed Applicata (2020)199, 1685–1696.