# Asymptotic of the Discrete Volume-Preserving Fractional Mean Curvature Flow
via a Nonlocal Quantitative Alexandrov Theorem

Daniele De Gennaro  Dauphine University, Paris France  Andrea Kubin
Technische Universität München, Munich Germany  Anna Kubin  Politecnico di
Torino, Turin, Italy

###### Abstract

We characterize the long time behaviour of a discrete-in-time approximation of
the volume preserving fractional mean curvature flow. In particular, we prove
that the discrete flow starting from any bounded set of finite fractional
perimeter converges exponentially fast to a single ball. As an intermediate
result we establish a quantitative Alexandrov type estimate for normal
deformations of a ball. Finally, we provide existence for flat flows as limit
points of the discrete flow when the time discretization parameter tends to
zero.

###### Contents

  1. 1 Preliminaries
  2. 2 A fractional quantitative Alexandrov type estimate
  3. 3 The asymptotic of the discrete volume-preserving mean curvature flow
  4. A Existence of flat flows

## Introduction

We consider the geometric evolution of sets called the volume preserving
fractional mean curvature flow. The classical mean curvature flow is defined
as a flow of sets $(E_{t})_{0\leq t\leq T}$ in $\mathbb{R}^{N}$ following the
motion law

$v_{t}=-H_{E_{t}}\quad\text{on}\quad\partial E_{t},$

where $v_{t}$ denotes the component of the velocity relative to the outer
normal vector of $\partial E_{t}$ and $H_{E}$ is the mean curvature of the set
$E$. In order to include the volume constraint, one can consider the following
velocity

$v_{t}=\bar{H}_{E_{t}}-H_{E_{t}}\quad\text{on}\quad\partial E_{t}$

for all $t\in[0,T]$ , where $\bar{H}_{E_{t}}$ denotes the average of
$H_{E_{t}}$ over $\partial E_{t}$. The defined geometric evolution is called
volume preserving mean curvature flow, as one can observe that the volume of
the evolving sets is constant.

In the fractional setting, the velocity of the flow is given by the fractional
mean curvature, a geometric quantity introduced by Caffarelli, Roquejoffre and
Savin in [5] and defined as the first variation of the fractional perimeter
functional. The latter functional is defined on a measurable set
$E\subset\mathbb{R}^{N}$ simply as

$P^{s}(E)=\int_{E}\int_{E^{c}}\dfrac{1}{|x-y|^{N+s}}\,\textnormal{d}x\,\textnormal{d}y.$

It turns out that its first variation on any $C^{2}$ set $E$ is given by the
formula

$H^{s}_{E}(x):=\int_{\mathbb{R}^{N}}\dfrac{\chi_{E}(y)-\chi_{E^{c}}(y)}{|x-y|^{N+s}}\,\textnormal{d}y\qquad\forall
x\in\partial E.$

In both the previous formulae, the integrals are intended in the principal
value sense. In analogy with the classical case, the evolution law for the
volume preserving fractional mean curvature flow is given by

$v_{t}=\bar{H}^{s}_{E_{t}}-H^{s}_{E_{t}}\quad\text{on}\quad\partial E_{t},$
(0.1)

with the notations previously introduced.

Up to now, a satisfactory study of this type of evolution is still missing.
While the evolution without the volume constraint is well-understood (see e.g.
[10, 19]), the lack of a comparison principle in our case makes the study much
harder. Moreover, the generated flow may present singularities of different
kinds, as happens for the classical mean curvature flow: see [12] some some
explicit examples of pinch-like singularities. In [20] a short-time existence
is provided for the smooth flow (0.1), while existence of the smooth flow
starting from convex sets (under suitable assumptions) is proved in [11].

We will then follow the approach of the celebrated papers by Luckhaus and
Sturzenhecker [23] and Almgren, Taylor and Wang [2] consisting in building a
discrete-in-time approximation of the flow via a variational approach and then
sending the time discretization parameter to zero. Our approach follows
closely the work done by Mugnai, Seis and Spadaro in [25], where the
variational problem studied incorporates a volume penalization to take into
account the volume constraint. First of all we define a discrete-in-time
approximation of the flow that will be called the discrete flow. Given any
initial set $E_{0}$, with $|E_{0}|=m$, and a time-step $h>0$ we define
$E_{0}^{(h)}:=E_{0}$ and, iteratively, for $n>0$

$E_{n+1}^{(h)}\in\text{argmin}\left\\{P^{s}(F)+\dfrac{1}{h}\int_{F}\textnormal{sd}_{E_{n}^{(h)}}(x)\,\textnormal{d}x+\dfrac{1}{h^{\frac{s}{1+s}}}||F|-m|:F\subset\mathbb{R}^{N}\textnormal{
measurable}\right\\},$

where $\textnormal{sd}_{E_{n}^{(h)}}$ is the signed distance function from the
set $E_{n}^{(h)}$. We can define for every $t\geq 0$, the discrete flow by
$E^{(h)}(t):=E_{[t/h]}^{(h)}$. We will prove that such a flow is well-defined.
Any $L^{1}_{loc}$-limit point of this flow as the time-step $h$ converges to
zero will be called a flat flow. For the classical mean curvature flow, under
the hypothesis of convergence of the perimeters, this approach produces
global-in-time distributional solutions of the evolution law 0.1, as shown in
[25]. In the fractional case, we fall short of this result. Nonetheless, we
will prove in the Appendix the existence of flat flows, defined simply as
$L^{1}_{loc}-$limit points of the discrete flow as $h\to 0$, and address some
of its continuity properties. Moreover, under the additional hypothesis of
boundedness of the flow, we will prove that the flat flow is volume-
preserving.

In the recent years, the study of the long time behaviour of the volume
preserving mean curvature flow has attracted more and more attention. In the
classical case, after some preliminary studies [15, 18], in a recent paper
[24] the authors proved the asymptotic behaviour of the classical discrete
flow by showing its convergence to unions of equal balls. Then, they improved
their results in [21], proving uniform estimates with respects to the time
parameter $h$ in dimension $N=2$, thus obtaining the same result for the
classical flat flow. Also the situation in the periodic setting is quite
studied, with results for the discrete flow in [14] and for the flat flow in
dimensions $N=3,4$ in [26]. In the fractional setting some recent results have
been proved. For example, in [11] the authors prove that the smooth flow
starting from a convex set converges to a ball, up to translations possibly
depending on time and under the hypothesis of equiboundedness for the
fractional curvatures along the flow.

In this paper the long-time convergence analysis is developed in the
fractional setting. The main result of the paper is Theorem 0.1 below. It
provides a complete characterization of the long-time behaviour of the
discrete fractional mean curvature flow starting from any bounded set of
finite fractional perimeter, providing also an estimate on the convergence
speed. We will assume that the dimension $N$ is such that any
$\Lambda-$minimizer of the fractional perimeter is a smooth set. Namely, we
will assume that either:

  * •

$N=2$;

  * •

$N\leq 7$ and $s\in(s_{0},1)$, where $s_{0}$ is the constant of Proposition
3.1, item ii).

This is a technical hypothesis that could be dropped if we knew that the
evolving sets were smooth. In particular, it is essential to characterize the
possible long-time limit points for the discrete flow. We are then able to
prove the following result.

######  Theorem 0.1.

Let $m$, $M>0$ and let $E_{0}$ be an initial bounded set with
$P^{s}(E_{0})\leq M$, $|E_{0}|=m$. Then, for $h=h(s,M,m)>0$ small enough the
following holds: for any discrete flow $E_{n}^{(h)}$ starting from $E_{0}$,
there exists $\xi\in\mathbb{R}^{N}$ such that

$E_{n}^{(h)}-\xi\to B^{(m)}\quad\text{in}\quad C^{k}$

for all $k\in\mathbb{N}.$ Moreover, the convergence is exponentially fast.

We stress the difference between our result and the one holding in the classic
setting, where the limit points of the discrete flow are in general unions of
disjointed balls having the same radius. This is a peculiar feature of the
nonlocal perimeter considered, that penalizes non-connected components.

A crucial intermediate result consists in generalizing the Alexandrov-type
estimate [24, Theorem 1.3] and [14, Theorem 1.3] (see also [22]) to the
fractional setting. This result provides a stability inequality for
$C^{1}-$normal deformations of balls. We briefly give some definitions to
present some further details. Set $B=B_{1}(0)$ and let $f:\partial
B\to\mathbb{R}$ be a function with $\|f\|_{L^{\infty}(\partial B)}$ is
sufficiently small. The normal deformation $B_{f}$ of the set $B$ is defined
as

$\partial B_{f}:=\\{x(1+f(x))\ :\ x\in\partial B\\}.$

A normal deformation $B_{f}$ is said to be of class $C^{k}$ if $f\in
C^{k}(\partial E)$. The quantitative Alexandrov type estimate proved in [24]
is the following.

######  Theorem 0.2 (Theorem 1.1 in [24]).

There exist $\delta\in(0,1/2)$ and $C>0$ with the following property: for any
$f\in C^{1}(\partial B)\cap H^{2}(\partial B)$ such that
$\|f\|_{C^{1}(\partial B)}\leq\delta$, $|B_{f}|=\omega_{N}$ and
$\textnormal{bar}(B_{f})=\int_{B_{f}}x\,\textnormal{d}x=0$, we have

$\|f\|_{H^{1}(\partial B)}\leq C\|H_{B_{f}}-\bar{H}_{B_{f}}\|_{L^{2}(\partial
B)}.$

We are able to extend the previous result to the fractional setting. Namely,
we obtain the following.

######  Theorem 0.3.

There exist $\delta>0$ with the following property: for any $f\in
C^{1}(\partial B)$ such that $\|f\|_{C^{1}(\partial B)}\leq\delta$,
$|B_{f}|=\omega_{N}$ and
$\textnormal{bar}(B_{f})=\int_{B_{f}}x\,\textnormal{d}x=0$, then

  * i)

for any $s\in(0,1)$, there exists $C=C(N,s)>0$ such that

$\|f\|_{H^{\frac{1+s}{2}}(\partial B)^{2}}\leq
C\|H^{s}_{B_{f}}-\bar{H}^{s}_{B_{f}}\|_{L^{2}(\partial B)^{2}};$

  * ii)

there exist $s^{*}\in(0,1)$ and $C=C(N)>0$ such that, for any $s\in(s^{*},1)$,
it holds

$(1-s)\|f\|^{2}_{H^{\frac{1+s}{2}}(\partial B)}\leq
C\big{\|}(1-s)\left(H^{s}_{B_{f}}-\bar{H}^{s}_{B_{f}}\right)\big{\|}_{L^{2}(\partial
B)}^{2};$ (0.2)

  * iii)

if $f\in C^{2}(\partial B)$, as $s\to 1,$ estimate (0.2) tends to

$\|f\|_{H^{1}(\partial B)^{2}}\leq
C\|H_{B_{f}}-\bar{H}_{B_{f}}\|_{L^{2}(\partial B)^{2}}.$

In the previous results, we have set $\bar{H}^{s}_{B_{f}}:=\fint_{\partial
B}H^{s}_{B_{f}}(x+f(x)x)\,\textnormal{d}\mathcal{H}^{N-1}(x).$

In particular, we recover Theorem 0.2 as a corollary of our result. The proof
of the previous theorem follows closely the proof of the quantitative
Alexandrov type estimate obtained in the flat torus and contained in [14]. In
particular, the approach is based on some Taylor approximations of the factor
$\bar{H}^{s}_{B_{f}}-H^{s}_{B_{f}}(x)$ combined with the coercivity of the
second variation of the fractional perimeter, proved in [16].

After this work was completed, we were informed that a similar quantitative
Alexandrov type estimate, more precisely a result analogous to item i) in
Theorem 0.3, has been independently proved in [9]. In this paper the authors
use this result to prove both global existence of a flow of smooth sets
satisfying (0.1), starting from a smooth normal deformation of a ball, and its
asymptotic exponential convergence to the same ball.

### Acknowledgements

The authors warmly thank Professor Massimiliano Morini for insightful
discussions. D. De Gennaro and Anna Kubin wish to acknowledge the hospitality
of the Faculty of Mathematics of the Technical University of Munich, where
part of this research was carried out. Andrea Kubin and Anna Kubin are members
of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro
Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). D.
De Gennaro has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 94532. . Andrea Kubin was supported by the DFG Collaborative
Research Center TRR 109 “Discretization in Geometry and Dynamics”.

## Notation

We work in the Euclidian space $\mathbb{R}^{N}$, with $N\geq 1$. We denote
with $|\cdot|$ the standard Lebesgue measure in $\mathbb{R}^{N}$,
$\mathrm{M}(\mathbb{R}^{N})$ is the family of measurable set of
$\mathbb{R}^{N}$ and
$\mathrm{M}_{f}(\mathbb{R}^{N})\subset\mathrm{M}(\mathbb{R}^{N})$ is the
family of subsets of $\mathbb{R}^{N}$ having finite measure. We denote with
$E^{c}$ the complement of a set $E\subset\mathbb{R}^{N}$. We denote by
$\mathcal{H}^{N-1}$ the Hausdorff measure, and sometimes denote
$\,\textnormal{d}\mathcal{H}^{N-1}_{x}:=\,\textnormal{d}\mathcal{H}^{N-1}(x)$.
If $E$ is a set with smooth boundary the outer normal to $E$ at a point $x$ in
$\partial E$ is denoted by $\nu=\nu_{E}(x)$. We will denote $B=B(0,1)$. We
will denote the ball of radius $r$ and center $x$ both as $B(x,r)$ and
$B_{r}(x).$ Also, with $B^{(m)}$ we denote the ball centered in 0 and having
volume $|B^{(m)}|=m.$ Finally, we denote by $C(*,\cdots,*)$ a constant that
depends on $*,\cdots,*$; such a constant may change from line to line.

##  1 Preliminaries

Let $s\in(0,1)$ we define the $s$-fractional perimeter as the following
function

$P^{s}:\mathrm{M}_{f}(\mathbb{R}^{N})\rightarrow[0,+\infty],\quad
P^{s}(E)\coloneqq\int_{E}\int_{E^{c}}\frac{1}{|x-y|^{N+s}}\,\textnormal{d}x\,\textnormal{d}y=\frac{1}{2}[\chi_{E}]_{H^{\frac{s}{2}}}.$

More in general, for every $E,F\in\mathrm{M}_{f}(\mathbb{R}^{N})$ we set

$\mathcal{L}_{s}(E,F)\coloneqq\int_{E}\int_{F}\frac{1}{|x-y|^{N+s}}\,\textnormal{d}x\,\textnormal{d}y$

and, for any bounded set $\Omega$, we define the fractional perimeter of $E$
relative to $\Omega$ as

$P^{s}(E;\Omega)\coloneqq\mathcal{L}_{s}(E\cap\Omega,E^{c}\cap\Omega)+\mathcal{L}_{s}(E\cap\Omega,E^{c}\setminus\Omega)+\mathcal{L}_{s}(E\setminus\Omega,E^{c}\cap\Omega).$

Let $E\in\mathrm{M}_{f}(\mathbb{R}^{N})$ be a set of class $C^{2}$. Given a
vector field $X\in C^{1}_{c}(\mathbb{R}^{N};\mathbb{R}^{N})$, let

$\Phi:\mathbb{R}\times\mathbb{R}^{N}\to\mathbb{R}^{N},\quad\Phi(t,x)=x+tX(x).$

We recall that the first variation of the $s$-fractional perimeter of $E$ in
the direction of $X$ is given by

$\delta
P^{s}(E)[X]\coloneqq\frac{\,\textnormal{d}}{\,\textnormal{d}t}\Big{|}_{t=0}P^{s}(\Phi(t,E))=\int_{\partial
E}H^{s}_{E}(x)X(x)\cdot\nu_{E}(x)\,\textnormal{d}\mathcal{H}^{N-1}_{x},$

where $H^{s}_{E}(x)$ is the $s$-fractional mean curvature of $E$ evaluated at
$x\in\partial E$, that is

$H^{s}_{E}(x)\coloneqq\int_{\mathbb{R}^{N}}\frac{\chi_{E}(y)-\chi_{E^{c}}(y)}{|x-y|^{N+s}}\,\textnormal{d}y,$

where the integral has to be intended in the principal value sense. Applying
the divergence theorem, the fractional curvature can be written as

$H^{s}_{E}(p)=\int_{\partial
E}\dfrac{(x-p)\cdot\nu_{E}(x)}{|x-p|^{N+s}}\,\textnormal{d}\mathcal{H}^{N-1}(x)\quad\forall\,p\in\partial
E.$

We recall some useful results concerning sets of finite fractional perimeter.

######  Proposition 1.1 (Lower semi-continuity).

Let $\\{E_{n}\\}_{n\in\mathbb{N}}\subset\mathrm{M}_{f}(\mathbb{R}^{N})$ such
that $\chi_{E_{n}}\rightarrow\chi_{E}$ in $L^{1}_{loc}$, as
$n\rightarrow+\infty$, for some $E\in\mathrm{M}_{f}(\mathbb{R}^{N})$. Then,
for all $s\in(0,1)$, we have

$P^{s}(E)\leq\liminf_{n\rightarrow+\infty}P^{s}(E_{n}).$

######  Theorem 1.2 (Compactness).

If $R>0$ and $\\{E_{n}\\}_{n\in\mathbb{N}}\subset\mathrm{M}(\mathbb{R}^{N})$,
with

$E_{n}\subset B(0,R)\quad\forall
n\in\mathbb{N}\quad\textnormal{and}\quad\sup_{n\in\mathbb{N}}P^{s}(E_{n})<+\infty,$

then, up to a subsequence, ${E_{n}}\rightarrow E$ in $L^{1}(\mathbb{R}^{N})$,
where $E\subset B(0,R)$ and $P^{s}(E)<+\infty$.

######  Theorem 1.3.

(Relative isoperimetric inequality) Let $\Omega\subset\mathbb{R}^{N}$ be an
open bounded set with Lipschitz boundary and let $E\subset\mathbb{R}^{N}$ be a
measurable set. Then there exists a constant $C=C(s,N,\Omega)>0$ such that

$P^{s}(E,\Omega)\geq\mathcal{L}_{s}(E\cap\Omega,E^{c}\cap\Omega)\geq
C\min\left\\{|E\cap\Omega|^{\frac{N-s}{N}},\,|E\setminus\Omega|^{\frac{N-s}{N}}\right\\}.$

We recall the following pointwise convergence theorems. The first one concerns
the convergence of the fractional perimeter to the classical one: its proof
can be found in [6, Theorem 1].

######  Theorem 1.4.

Let $E$ be a bounded, $C^{1,\alpha}$ set for $\alpha\in(0,1)$. Then,

$\lim_{s\to 1}(1-s)P^{s}(E)=\omega_{N-1}P(E).$

The second one relates to the pointwise convergence of the fractional
curvatures. It was proved in a more general setting in [1, 6, 7].

######  Theorem 1.5.

Let $E$ be a bounded, $C^{2}$ set. Then,

$\lim_{s\to 1}(1-s)H^{s}_{E}=\omega_{N-1}H_{E}$

uniformly on $\partial E$.

Finally, we recall the pointwise convergence of the fractional Gagliardo
seminorms to the Sobolev one. The classical proof is contained in [3,
Corollary 2], see also [17, Proposition 3.7] for the same result in a more
general setting.

######  Theorem 1.6.

Assume $f\in H^{s}(\partial B)$. Then

$\lim_{s\to 1}(1-s)[f]_{H^{\frac{1+s}{2}}(\partial B)}^{2}=C\|\nabla
f\|_{L^{2}(\partial B)}^{2},$

where $C>0$ is a constant that depends only on $N$.

##  2 A fractional quantitative Alexandrov type estimate

In this section, we are going to prove the quantitative Alexandrov inequality
Theorem 0.3 in the nonlocal setting of the fractional perimeter. From now on
we set

$[f]_{\frac{1+s}{2}}^{2}\coloneqq[f]_{H^{\frac{1+s}{2}}(\partial
B)}^{2}=\int_{\partial B}\int_{\partial
B}\frac{|f(x)-f(y)|^{2}}{|x-y|^{N+s}}\,\textnormal{d}\mathcal{H}^{N-1}_{x}\,\textnormal{d}\mathcal{H}^{N-1}_{y}.$

We start by recalling representation formulas for the $s$-fractional perimeter
and its first variation on smooth sets.

######  Lemma 2.1.

The following equalities hold true:

  1. 1.

If $f\in C^{1}(\partial B)$ with $\|f\|_{\infty}$ sufficiently small, then

$\begin{split}P^{s}(B_{f})&=\frac{P^{s}(B)}{P(B)}\int_{\partial
B}(1+f)^{N-s}\,\textnormal{d}\mathcal{H}^{N-1}+\\\
&\quad+\frac{1}{2}\int_{\partial B}\int_{\partial
B}\int_{1+f(y)}^{1+f(x)}\int_{1+f(y)}^{1+f(x)}F_{|x-y|}(r,\rho)\,\textnormal{d}r\,\textnormal{d}\rho\,\textnormal{d}\mathcal{H}^{N-1}_{x}\,\textnormal{d}\mathcal{H}^{N-1}_{y},\end{split}$
(2.1)

where, for every $x,$ $y\in\partial B$, we have set

$F_{|x-y|}(r,\rho):=\frac{r^{N-1}\rho^{N-1}}{|rx-\rho y|^{N+s}},\quad\forall
r,\rho\in(0,+\infty).$

  2. 2.

If $f\in C^{1}(\partial B)$ with $\|f\|_{\infty}$ sufficiently small, then we
have

$\begin{split}&\delta
P^{s}(E_{f})[\psi]=(N-s)\frac{P^{s}(B)}{P(B)}\int_{\partial
B}(1+f)^{N-s-1}\psi\,\textnormal{d}\mathcal{H}^{N-1}\\\ &+\int_{\partial
B}\int_{\partial
B}\int_{f(y)}^{f(x)}\left(\psi(x)F_{|x-y|}(1+f(x),1+\rho)-\psi(y)F_{|x-y|}(1+f(y),1+\rho)\right).\end{split}$
(2.2)

###### Proof.

By explicit computations one can obtain equation (2.1), see for example the
calculations in the proof of [16, Theorem 2.1]. To prove (2.2), we take the
derivative

$\dfrac{\,\textnormal{d}}{\,\textnormal{d}t}\Big{|}_{t=0}P^{s}(B_{f+t\psi})$

in formula (2.1) and, recalling that

$\begin{split}\frac{\,\textnormal{d}}{\,\textnormal{d}t}\left[\int_{\alpha(t)}^{\beta(t)}\int_{\alpha(t)}^{\beta(t)}f(r,\rho)\,\textnormal{d}\rho\,\textnormal{d}r\right]&=\int_{\alpha(t)}^{\beta(t)}(f(\beta(t),\rho)\beta^{\prime}(t)-f(\alpha(t),\rho)\alpha^{\prime}(t))\,\textnormal{d}\rho\\\
&\quad+\int_{\alpha(t)}^{\beta(t)}(f(r,\beta(t))\beta^{\prime}(t)-f(r,\alpha(t))\alpha^{\prime}(t))\,\textnormal{d}r\end{split}$

for every function $\alpha,$ $\beta:\mathbb{R}\to\mathbb{R}$ of class $C^{1}$
and $f\in L^{1}_{loc}(\mathbb{R}\times\mathbb{R})$, we conclude

$\displaystyle\delta P^{s}(B_{f})[\psi]=$ $\displaystyle\int_{\partial
B}\int_{\partial
B}\int_{1+f(y)}^{1+f(x)}\left(\psi(x)F_{|x-y|}(1+f(x),\rho)-\psi(y)F_{|x-y|}(1+f(y),\rho)\right)\,\textnormal{d}\rho$
$\displaystyle+(N-s)\frac{P^{s}(B)}{P(B)}\int_{\partial
B}(1+f)^{N-s-1}\psi\,\textnormal{d}\mathcal{H}^{N-1}.$

A simple change of coordinates then yields the thesis. ∎

######  Lemma 2.2.

If $f\in C^{1}(\partial B)$ with $\|f\|_{\infty}$ sufficiently small, then we
have

$\begin{split}\delta P^{s}(B_{f})[1]&=(N-s)\frac{P^{s}(B)}{P(B)}\int_{\partial
B}\left(1+(N-s-1)f+O(f^{2}))\right)\,\textnormal{d}\mathcal{H}^{N-1}+O([f]_{\frac{1+s}{2}}^{2}),\end{split}$
(2.3) $\begin{split}\delta
P^{s}(B_{f})[f]&=(N-s)\frac{P^{s}(B)}{P(B)}\int_{\partial
B}\left(1+(N-s-1)f+O(f^{2}))\right)f\,\textnormal{d}\mathcal{H}^{N-1}\\\
\quad&+\int_{\partial B}\int_{\partial
B}\frac{(f(x)-f(y))^{2}}{|x-y|^{N+s}}+O([f]_{\frac{1+s}{2}}^{2})(\|f\|_{\infty}+\|f\|^{2}_{C^{1}}).\end{split}$
(2.4)

###### Proof.

First, we remark that, by expanding the first term in (2.2), we obtain

$\begin{split}\delta
P^{s}(B_{f})[\psi]&=(N-s)\frac{P^{s}(B)}{P(B)}\int_{\partial
B}(1+(N-s-1)f+O(f^{2}))\psi\,\textnormal{d}\mathcal{H}^{N-1}\\\
&+\int_{\partial B}\int_{\partial
B}\int_{f(y)}^{f(x)}\left(\psi(x)F_{|x-y|}(1+f(x),1+\rho)-\psi(y)F_{|x-y|}(1+f(y),1+\rho)\right).\end{split}$

Now, set $u=f(x)$ and $v=f(y)$, and define the auxiliary function

$g(1+u,1+v,\rho):=\psi(x)F_{|x-y|}(1+u,1+\rho)-\psi(y)F_{|x-y|}(1+v,1+\rho).$

We remark that

$\begin{split}F_{|x-y|}(1+u,1+\rho)&=\frac{(1+(N-1)u+O(u^{2}))(1+(N-1)\rho+O(\rho^{2}))}{|(1+u)x-(1+\rho)y|^{N+s}}\\\
&=\frac{(1+(N-1)u+O(u^{2}))(1+(N-1)\rho+O(\rho^{2}))}{|(u-\rho)^{2}+(1+u)(1+\rho)|x-y|^{2}|^{\frac{N+s}{2}}}\\\
&=\frac{1+(N-1)u+(N-1)\rho+O(u^{2}+\rho^{2})}{|x-y|^{N+s}((u-\rho)^{2}/|x-y|^{2}+u+\rho+u\rho+1)^{\frac{N+s}{2}}}.\end{split}$
(2.5)

Now, since $\|f\|_{C^{1}}\leq\delta$ and $\rho$ is between $f(y)$ and $f(x)$,
$|u-\rho|\leq\delta|x-y|$, we can then expand (2.5) and obtain

$\begin{split}&\frac{1+(N-1)u+(N-1)\rho+O(u^{2}+\rho^{2})}{|x-y|^{N+s}}\left(1-\frac{N+s}{2}\left(\frac{(u-\rho)^{2}}{|x-y|^{2}}+u+\rho+u\rho\right)\right)\\\
&=\frac{1+(N-1-\frac{N+s}{2})(u+\rho)}{|x-y|^{N+s}}-C\frac{(u-\rho)^{2}}{|x-y|^{N+s+2}}+\frac{O(u^{2}+\rho^{2})}{|x-y|^{N+s}},\end{split}$

from which we deduce

$\displaystyle g(1+u,1+v,\rho)$
$\displaystyle=\frac{\psi(x)-\psi(y)}{|x-y|^{N+s}}+C_{1}\left(\dfrac{\psi(x)(u+\rho)}{|x-y|^{N+s}}-\dfrac{\psi(y)(v+\rho)}{|x-y|^{N+s}}\right)$
$\displaystyle+C_{2}\left(-\frac{\psi(x)(u-\rho)^{2}}{|x-y|^{N+s+2}}+\frac{\psi(y)(v-\rho)^{2}}{|x-y|^{N+s+2}}\right)+\frac{O(u^{2}+v^{2}+\rho^{2})}{|x-y|^{N+s}},$

where the constants $C_{1,2}$ can also be non-positive. Finally, we obtain

$\begin{split}\int_{\partial B}&\int_{\partial
B}\int_{f(y)}^{f(x)}g(1+f(x),1+f(y),1+\rho)\\\ &\quad=\int_{\partial
B}\int_{\partial B}\frac{(\psi(x)-\psi(y))(f(x)-f(y))}{|x-y|^{N+s}}\\\
&\quad+C_{1}\int_{\partial B}\int_{\partial
B}\frac{(f(x)-f(y))(3f(x)/2+f(y))}{|x-y|^{N+s}}\psi(x)\\\
&\quad+C_{2}\int_{\partial B}\int_{\partial
B}\frac{(\psi(x)-\psi(y))(f(x)-f(y))^{3}}{|x-y|^{N+s+2}}\\\
&\quad+O\left(\int_{\partial B}\int_{\partial
B}\frac{(f(x)-f(y))}{|x-y|^{N+s}}f^{2}\right).\end{split}$

If $\psi=1$, using the fact that, by symmetry,

$\int_{\partial B}\int_{\partial
B}\frac{(f(x)-f(y))f(x)}{|x-y|^{N+s}}=\dfrac{1}{2}[f]_{\frac{1+s}{2}}^{2},$
(2.6)

we obtain

$\begin{split}&\int_{\partial B}\int_{\partial
B}\int_{f(y)}^{f(x)}g(1+f(x),1+f(y),1+\rho)=O([f]_{\frac{1+s}{2}}^{2}).\end{split}$

If $\psi=f$, using again (2.6), we get

$\begin{split}\int_{\partial B}\int_{\partial
B}\int_{f(y)}^{f(x)}g(1+f(x),1+f(y),1+\rho)&=\int_{\partial B}\int_{\partial
B}\frac{(f(x)-f(y))^{2}}{|x-y|^{N+s}}\\\
&\quad+O([f]_{\frac{1+s}{2}}^{2})(\|f\|_{\infty}+\|f\|^{2}_{C^{1}}).\end{split}$

∎

In order to prove Theorem 0.3, we need the following lemma, which states the
coercivity of the second variation of the fractional perimeter of a ball with
respect to normal deformations. Its proof is contained in [16, Theorem 8.1].
We start by defining

$\lambda_{1}^{s}:=s(N-s)\dfrac{P^{s}(B)}{P(B)}.$ (2.7)

######  Lemma 2.3.

There exists $\delta>0$ small such that, if $f\in C^{1}(\partial B)$ with
$\|f\|_{C^{1}(\partial B)}\leq\delta$, $|B_{f}|=\omega_{N}$ and
$\textnormal{bar}(B_{f})=0$, then we have

$\displaystyle\delta^{2}P^{s}(B)[f]$ $\displaystyle=\int_{\partial
B}\int_{\partial
B}\frac{(f(x)-f(y))^{2}}{|x-y|^{N+s}}\,\textnormal{d}\mathcal{H}^{N-1}_{x}\,\textnormal{d}\mathcal{H}^{N-1}_{y}-\lambda_{1}^{s}\int_{\partial
B}|f|^{2}\,\textnormal{d}\mathcal{H}^{N-1}$
$\displaystyle\geq\dfrac{1}{4}\Big{(}[f]_{\frac{1+s}{2}}^{2}+\lambda_{1}^{s}\|f\|_{L^{2}(\partial
B)}^{2}\Big{)}.$

We are now in position to prove Theorem 0.3.

###### Proof of Theorem 0.3.

We start by proving item i). Without loss of generality, we assume that
$\|H^{s}_{B_{f}}-\bar{H}^{s}_{B_{f}}\|_{L^{2}}\leq 1$. Let $\Phi:\partial
B\to\partial B_{f}\subset\mathbb{R}^{N}$ be the map defined by
$\Phi(x)=(1+~{}f(x))x,$ by direct computations one can prove that

$J\Phi(x)=(1+f(x))^{N-1}(1+(1+f(x))^{-2}|\nabla f(x)|^{1/2}.$

For every $\psi\in C^{1}(\partial B)$, let

$\overline{\psi}:\mathbb{R}^{N}\to\mathbb{R}^{N},\quad\bar{\psi}(x):=\frac{x}{|x|}\psi\left(\frac{x}{|x|}\right).$

Employing the area formula we get

$\begin{split}\delta P^{s}(B_{f})[\psi]&=\int_{\partial
B_{f}}H^{s}_{B_{f}}\nu_{B_{f}}\cdot\overline{\psi}\,\textnormal{d}\mathcal{H}^{N-1}\\\
&=\int_{\partial B}H^{s}_{B_{f}}(p)\nu_{B_{f}}(p)\cdot
x\psi(x)J\Phi(x)\,\textnormal{d}\mathcal{H}^{N-1}_{x}\\\ &=\int_{\partial
B}H^{s}_{B_{f}}(p)\,\psi(x)\,(1+f(x))^{N-1}\,\textnormal{d}\mathcal{H}^{N-1}_{x},\end{split}$

where we have set $p=(1+f(x))x$ (for more details see [24, Section 1] and [14,
Section 3]). Now, by a simple Taylor expansion we obtain

$\delta P^{s}(B_{f})[\psi]=\int_{\partial
B}H^{s}_{B_{f}}(p)\,\psi(x)\,(1+(N-1)f(x)+O(f^{2}))\,\textnormal{d}\mathcal{H}^{N-1}_{x}.$
(2.8)

We recall that

$H^{s}_{B}(x)=(N-s)\frac{P^{s}(B)}{P(B)}\quad\textnormal{for all }x\in\partial
B.$

If $\psi=1$, by combining formulas (2.8) and (2.3), we infer

$\begin{split}\int_{\partial
B}(H^{s}_{B_{f}}(p)-H^{s}_{B})(1+(N-1)f(x)+O(f^{2}))\,\textnormal{d}\mathcal{H}^{N-1}_{x}=\int_{\partial
B}O(f)\,\textnormal{d}\mathcal{H}^{N-1}+O([f]_{\frac{1+s}{2}}^{2})\end{split}$
(2.9)

and if $\psi=f$, by combining equations (2.8) and (2.4), we get

$\begin{split}&\int_{\partial B}\int_{\partial
B}\frac{(f(x)-f(y))^{2}}{|x-y|^{N+s}}\,\textnormal{d}\mathcal{H}^{N-1}_{x}\,\textnormal{d}\mathcal{H}^{N-1}_{y}-s(N-s)\frac{P^{s}(B)}{P(B)}\int_{\partial
B}f^{2}\,\textnormal{d}\mathcal{H}^{N-1}\\\ &=\int_{\partial
B}\left(H^{s}_{B_{f}}(p)-H^{s}_{B}\right)(1+(N-1)f(x)+O(f^{2}))f(x)\,\textnormal{d}\mathcal{H}^{N-1}_{x}\\\
&\quad+O([f]_{\frac{1+s}{2}}^{2})(\|f\|_{\infty}+\|f\|^{2}_{C^{1}}).\end{split}$
(2.10)

Using the same arguments of the proof of [14, Theorem 1.3] (see also [24,
Theorem 1.3]) we can conclude. Anyway for the interested reader we report a
sketch of the proof.

By (2.9), for $\delta$ sufficiently small, using Hölder’s inequality we obtain

$\displaystyle\left|\bar{H}^{s}_{B_{f}}-H^{s}_{B}\right|$
$\displaystyle\leq\left|-\fint_{\partial
B}(H^{s}_{B_{f}}-H^{s}_{B})((N-1)f+O(f^{2}))\,\textnormal{d}\mathcal{H}^{N-1}\right|$
$\displaystyle\quad+\int_{\partial
B}O(|f|)\,\textnormal{d}\mathcal{H}^{N-1}+O([f]^{2}_{\frac{1+s}{2}})$
$\displaystyle\leq\left|\fint_{\partial
B}(H^{s}_{B_{f}}-\bar{H}^{s}_{B_{f}})((N-1)f+O(f^{2}))\,\textnormal{d}\mathcal{H}^{N-1}\right|$
$\displaystyle\quad+\left|\fint_{\partial
B}(\bar{H}^{s}_{B_{f}}-H^{s}_{B})((N-1)f+O(f^{2}))\,\textnormal{d}\mathcal{H}^{N-1}\right|$
$\displaystyle\quad+\int_{\partial
B}O(|f|)\,\textnormal{d}\mathcal{H}^{N-1}+O([f]^{2}_{\frac{1+s}{2}})$
$\displaystyle\leq\delta\,\dfrac{N-1+C\delta}{P(B)^{1/2}}\|H^{s}_{B_{f}}-\bar{H}^{s}_{B_{f}}\|_{L^{2}}+\delta\left(N-1+C\delta\right)|\bar{H}^{s}_{B_{f}}-H^{s}_{B}|$
$\displaystyle\quad+\int_{\partial
B}O(|f|)\,\textnormal{d}\mathcal{H}^{N-1}+O([f]^{2}_{\frac{1+s}{2}}),$

with $C=C(N)$. For $\delta$ small enough, recalling that
$\|H_{B_{f}}-\bar{H}_{B_{f}}\|_{L^{2}}\leq 1$, the previous inequality implies

$\frac{1}{2}|\bar{H}^{s}_{B_{f}}-H^{s}_{B}|\leq
C\delta\|H^{s}_{B_{f}}-\bar{H}^{s}_{B_{f}}\|_{L^{2}}+\int_{\partial
B}O(|f|)\,\textnormal{d}\mathcal{H}^{N-1}+O([f]^{2}_{\frac{1+s}{2}})\leq
C\delta.$ (2.11)

By (2.10), using again Hölder’s inequality and by the previous remark, we get

$\displaystyle\int_{\partial B}\int_{\partial
B}\frac{(f(x)-f(y))^{2}}{|x-y|^{N+s}}\,\textnormal{d}\mathcal{H}^{N-1}_{x}\,\textnormal{d}\mathcal{H}^{N-1}_{y}-s(N-s)\frac{P^{s}(B)}{P(B)}\int_{\partial
B}f^{2}\,\textnormal{d}\mathcal{H}^{N-1}$ $\displaystyle=\int_{\partial
B}\left(H^{s}_{B_{f}}(p)-H^{s}_{B}\right)(1+(N-1)f+O(f^{2}))f\,\textnormal{d}\mathcal{H}^{N-1}$
$\displaystyle\quad+O([f]_{\frac{1+s}{2}}^{2})(\|f\|_{\infty}+\|f\|^{2}_{C^{1}})$
$\displaystyle=\int_{\partial
B}(H^{s}_{B_{f}}(p)-\bar{H}^{s}_{B_{f}})(1+(N-1)f+O(f^{2}))f\,\textnormal{d}\mathcal{H}^{N-1}$
$\displaystyle\quad+\int_{\partial
B}(\bar{H}^{s}_{B_{f}}-H^{s}_{B})(1+(N-1)f+O(f^{2}))f\,\textnormal{d}\mathcal{H}^{N-1}$
$\displaystyle\quad+O([f]_{\frac{1+s}{2}}^{2})(\|f\|_{\infty}+\|f\|^{2}_{C^{1}})$
$\displaystyle\leq
C\|H^{s}_{B_{f}}-\bar{H}^{s}_{B_{f}}\|_{L^{2}}\|f\|_{L^{2}}+|\bar{H}^{s}_{B_{f}}-H^{s}_{B}|\int_{\partial
B}(1+(N-1)f+O(f^{2}))f\,\textnormal{d}\mathcal{H}^{N-1}$
$\displaystyle\quad+O([f]_{\frac{1+s}{2}}^{2})(\|f\|_{\infty}+\|f\|^{2}_{C^{1}}).$
(2.12)

Since $|B_{f}|=\omega_{N}$, we have

$\Big{|}\int_{\partial
B}f\,\textnormal{d}\mathcal{H}^{N-1}\Big{|}=\int_{\partial
B}O(f^{2})\,\textnormal{d}\mathcal{H}^{N-1}.$ (2.13)

By (2.13) and (2.11), we obtain

$|\bar{H}^{s}_{B_{f}}-H^{s}_{B}|\int_{\partial
B}(f+O(f^{2}))\,\textnormal{d}\mathcal{H}^{N-1}\leq\delta\int_{\partial
B}O(f^{2}).$

Finally, by the above inequality, (2.13) again and by combining (2) with
(2.11) we deduce that, for any $\eta>0$, it holds

$\displaystyle\int_{\partial B}\int_{\partial
B}\frac{(f(x)-f(y))^{2}}{|x-y|^{N+s}}\,\textnormal{d}\mathcal{H}^{N-1}_{x}\,\textnormal{d}\mathcal{H}^{N-1}_{y}-s(N-s)\frac{P^{s}(B)}{P(B)}\int_{\partial
B}f^{2}\,\textnormal{d}\mathcal{H}^{N-1}$ $\displaystyle\quad\leq
C\|H^{s}_{B_{f}}-\bar{H}^{s}_{B_{f}}\|_{L^{2}}\|f\|_{L^{2}}+C\delta(\|f\|_{L^{2}}^{2}+[f]_{\frac{1+s}{2}}^{2})$
(2.14)
$\displaystyle\quad\leq\dfrac{1}{\eta}C^{2}\|H^{s}_{B_{f}}-\bar{H}^{s}_{B_{f}}\|_{L^{2}}^{2}+\eta\|f\|_{L^{2}}^{2}+C\delta(\|f\|_{L^{2}}^{2}+[f]_{\frac{1+s}{2}}^{2}).$
(2.15)

The conclusion then follows combining (2.15) with Lemma 2.3 and taking
$\delta$ and $\eta$ sufficiently small.

Item ii) then follows from item i), remarking that (2.14) can be read as

$\displaystyle(1-s)\left(\int_{\partial B}\int_{\partial
B}\frac{(f(x)-f(y))^{2}}{|x-y|^{N+s}}\,\textnormal{d}\mathcal{H}^{N-1}_{x}\,\textnormal{d}\mathcal{H}^{N-1}_{y}-s(N-s)\frac{P^{s}(B)}{P(B)}\int_{\partial
B}f^{2}\,\textnormal{d}\mathcal{H}^{N-1}\right)$ $\displaystyle\quad\leq
C\Big{\|}(1-s)\left(H^{s}_{B_{f}}-\bar{H}^{s}_{B_{f}}\right)\Big{\|}_{L^{2}}\|f\|_{L^{2}}+C\delta(\|f\|_{L^{2}}^{2}+[f]_{\frac{1+s}{2}}^{2}).$

Thus, we obtain

$\dfrac{1-s}{4}\left(\lambda_{1}^{s}\|f\|^{2}_{L^{2}}+[f]^{2}_{\frac{1+s}{2}}\right)\leq\dfrac{C^{2}}{\eta}\Big{\|}(1-s)\left(H^{s}_{B_{f}}-\bar{H}^{s}_{B_{f}}\right)\Big{\|}^{2}_{L^{2}}+\eta\|f\|^{2}_{L^{2}}+C\delta(\|f\|_{L^{2}}^{2}+[f]_{\frac{1+s}{2}}^{2}).$

Recalling (2.7) and the pointwise convergence Theorem 1.4, we conclude.
Finally, item iii) is a corollary of item ii) and Theorems 1.5 and 1.6,
recalling that Theorem 1.4 implies

$\lim_{s\to 1}(1-s)\lambda_{1}^{s}=(N-1)\omega_{N-1}.$

We just remark that, since the convergence is uniform,
$(1-s)\bar{H}^{s}_{B_{f}}\to\bar{H}_{B_{f}}.$ ∎

##  3 The asymptotic of the discrete volume-preserving mean curvature flow

In this section we start by introducing the incremental minimum problem which
defines the discrete-in-time approximation of the volume preserving fractional
mean curvature flow.

Let $E\neq\emptyset$ be a bounded, measurable subset of $\mathbb{R}^{N}$. In
the following we will always assume that $E$ coincides with its Lebesgue
representative. Fixed $h>0$, $m>0$, we consider the minimum problem

$\min\left\\{P^{s}(F)+\frac{1}{h}\int_{F}\textnormal{sd}_{E}(x)\,\textnormal{d}x+\frac{1}{h^{\frac{s}{s+1}}}||F|-m|:F\subset\mathbb{R}^{N}\right\\},$
(3.1)

where
$\textnormal{sd}_{E}(x):=\textnormal{dist}_{E}(x)-\textnormal{dist}_{E^{c}}(x)$
is the signed distance from the set $E$. Observe that the minimum problem
(3.1) is equivalent to the problem

$\min\left\\{P^{s}(F)+\frac{1}{h}\int_{F\triangle
E}\textnormal{dist}_{\partial
E}(x)\,\textnormal{d}x+\frac{1}{h^{\frac{s}{s+1}}}||F|-m|:F\subset\mathbb{R}^{N}\right\\}.$

We set
$\mathcal{F}_{h}(\cdot,\,E):\mathrm{M}_{f}(\mathbb{R}^{N})\rightarrow(-\infty,\,+\infty]$
the functional

$\mathcal{F}_{h}(F,\,E)=P^{s}(F)+\frac{1}{h}\int_{F}\textnormal{sd}_{E}(x)\,\textnormal{d}x+\frac{1}{h^{\frac{s}{s+1}}}||F|-m|.$

The following proposition recalls some properties of minimizers of problem
(3.1).

######  Proposition 3.1.

Let $M>0,h>0,s\in(0,1)$ and $m>0$. Let $E\subset\mathbb{R}^{N}$ be a bounded,
measurable set such that $P^{s}(E)\leq M$ and $|E|\leq M$. Then, there exists
a minimizer $F$ of (3.1), which is bounded. Moreover, the following properties
hold:

  * i)

There exists $\Lambda=\Lambda(h,N,s)>0$ such that $F$ is a $\Lambda$-minimizer
of the perimeter, namely

$P(F)\leq P(F^{\prime})+\Lambda\lvert F\triangle F^{\prime}\rvert$

for all measurable set $F^{\prime}\subset\mathbb{R}^{N}$ such that
$\textnormal{diam}(F\triangle F^{\prime})\leq 1$.

  * ii)

The boundary $\partial F$ is of class $C^{2,\alpha}$ for any $\alpha\in(0,s)$
outside of a closed set $\Sigma$ of Hausdorff dimension at most $N-3$.
Moreover, there exists $s_{0}\in(0,1)$ such that, if $s\in(s_{0},1)$, then
$\partial F$ is of class $C^{1,\alpha}$ for any $\alpha\in(0,1)$ outside a
closed set $\Sigma$ of Hausdorff dimension at most $N-8$.

  * iii)

There exist $c_{0}=c_{0}(N,s)>0$ and a radius $r_{0}=r_{0}(h,N,s)>0$ such that
for every $x\in\partial F\setminus\Sigma$ and $r\in(0,r_{0}]$ we have

$\lvert B_{r}(x)\cap F\rvert\geq c_{0}r^{N}\quad\text{and}\quad\lvert
B_{r}(x)\setminus F\rvert\geq c_{0}r^{N}.$

  * iv)

The following Euler-Lagrange equation holds: there exists
$\lambda\in\mathbb{R}$ such that for all $X\in
C^{1}_{c}(\mathbb{R}^{N},\mathbb{R}^{N})$ we have

$\int_{\partial
F}\dfrac{\textnormal{sd}_{E}}{h}X\cdot\nu_{F}\,\textnormal{d}\mathcal{H}^{N-1}+\int_{\partial
F}H^{s}_{F}X\cdot\nu_{F}\,\textnormal{d}\mathcal{H}^{N-1}=\lambda\int_{\partial
F}X\cdot\nu_{F}\,\textnormal{d}\mathcal{H}^{N-1},$ (3.2)

where $\lambda=\textnormal{sgn}(m-|F|)h^{-\frac{s}{1+s}}$ if $|F|\neq m$,
otherwise it is the Lagrange multiplier associated with the volume
penalization.

  * v)

There exist $k_{0}=k_{0}(h,N,s,M,m)\in\mathbb{N}$ and
$d_{0}=d_{0}(h,N,s,M,m)>0$ such that $F$ is made up of at most $k_{0}$
connected components having diameter larger than $d_{0}$.

###### Proof.

For the existence of minimizers of (3.1) see for example [8, Theorem 1.1]. The
$\Lambda-$minimality property is easily deduced, for instance we can choose
$\Lambda=2(h^{-1}+h^{-\frac{s}{1+s}})$. Concerning property $ii)$, it follows
from [8, Theorem 1.1] and [6, Theorem 5]. The density esimates can be found in
[5, Theorem 4.1]. The Euler-Lagrange equation in item iv) is easily deduced
performing variations of (3.1). The bound on the number of connected
components and on the diameter of the components follows from a covering
argument, as in [24, Proposition 2.3]. ∎

By induction we can now define the _discrete-in-time, volume preserving
fractional mean curvature flow_ $\\{E_{n}^{(h)}\\}_{n\in\mathbb{N}}$ and we
will refer to it as the discrete flow. Let $E_{0}\subset\mathbb{R}^{N}$ be a
measurable set such that $\lvert E_{0}\rvert=m$, we define $E_{1}^{(h)}$ as a
solution of (3.1) with $E_{0}$ instead of $E$. Assume that $E_{k}^{(h)}$ is
defined for $1\leq k\leq n-1$, we define $E_{n}^{(h)}$ as a solution of (3.1)
with $E$ replaced by $E_{n-1}^{(h)}$. We then recall the classical density
estimate (see [5, Theorem 4.1]).

######  Proposition 3.2.

There exists a constant $C=C(N,s)>0$ with the following property: given
$E\subset\mathbb{R}^{N}$, $R$, $\mu>0$ and $x_{0}\in\partial E$ such that

$P^{s}(E)\leq P^{s}(E\setminus B_{r}(x_{0}))+\mu|E\cap
B_{r}(x_{0})|\quad\forall 0<r<R,$

then

$Cr^{N}\leq|E\cap B_{r}(x_{0})|\quad\forall
0<r<\min\\{R,C_{s,N}\mu^{-1/s}\\}.$

######  Corollary 3.3.

Let $E\subset\mathbb{R}^{N}$ be a bounded set of finite fractional perimeter
and let $F$ be a minimizer of $\mathcal{F}_{h}(\cdot,E)$. Then, for every
$r\in(0,\gamma h^{1/1+s})$ and for every $x_{0}\in\partial^{*}F,$ it holds

$\min\\{|B_{r}(x_{0})\setminus F|,|F\cap B_{r}(x_{0})|\\}\geq cr^{N}$ (3.3)
$cr^{N-s}\leq P^{s}(F,B_{r}(x_{0}))\leq Cr^{N-s},$ (3.4)

where the constants only depend on $N$ and $s$.

###### Proof.

Since $F$ is a minimizer of $\mathcal{F}_{h}(\cdot,E)$, for any
$x_{0}\in\partial F$, it holds that
$\mathcal{F}_{h}(F,E)\leq\mathcal{F}_{h}(F\cup B_{r}(x_{0}),E)$, which implies

$\displaystyle P^{s}(F)$ $\displaystyle\leq P^{s}(F\cup
B_{r}(x_{0}))+\frac{1}{h}\int_{B_{r}(x_{0})\setminus
F}\textnormal{sd}_{E}\,\textnormal{d}x+\frac{1}{h^{s/1+s}}|B_{r}(x_{0})\setminus
F|$ $\displaystyle\leq P^{s}(F\cup
B_{r}(x_{0}))+\frac{C}{h^{s/1+s}}|B_{r}(x_{0})\setminus F|.$

Analogously, one can show that

$\displaystyle P^{s}(F)$ $\displaystyle\leq P^{s}(F\setminus
B_{r}(x_{0}))+\frac{C}{h^{s/1+s}}|F\cap B_{r}(x_{0})|$ (3.5)
$\displaystyle=\mathcal{L}_{s}(F\setminus B_{r}(x_{0}),F^{c}\setminus
B_{r}(x_{0}))+\mathcal{L}_{s}(F\setminus
B_{r}(x_{0}),B_{r}(x_{0}))+\frac{C}{h^{s/1+s}}|F\cap B_{r}(x_{0})|$

Therefore, by Proposition 3.2, we deduce

$\min\left\\{|F\cap B_{r}(x_{0})|,|B_{r}(x_{0})\setminus F|\right\\}\geq
cr^{N}\qquad\forall 0<r<\gamma h^{1/1+s}.$

The first inequality in (3.4) is now an immediate consequence of the relative
isoperimetric inequality. To prove the second inequality, by (3.5) we get

$\displaystyle P^{s}(F,B_{r}(x_{0}))$ $\displaystyle=\mathcal{L}_{s}(F\cap
B_{r}(x_{0}),F^{c})+\mathcal{L}_{s}(F\setminus B_{r}(x_{0}),F^{c}\cap
B_{r}(x_{0}))$ $\displaystyle=P^{s}(F)-\mathcal{L}_{s}(F\setminus
B_{r}(x_{0}),F^{c}\setminus B_{r}(x_{0}))$
$\displaystyle\leq\mathcal{L}_{s}(F\setminus
B_{r}(x_{0}),B_{r}(x_{0}))+\frac{C}{h^{s/1+s}}|B_{r}(x_{0})\setminus F|$
$\displaystyle\leq
P^{s}(B_{r}(x_{0}))+\frac{C\gamma^{s}}{r^{s}}\omega_{N}r^{N}\leq
C(N,s)r^{N-s},$

where we used that $r\leq\gamma h^{1/1+s}$. ∎

We employ the density estimates above to bound the distance function between
two consecutive sets of the discrete flow.

######  Proposition 3.4.

There exists a constant $\gamma=\gamma(N,s)>0$ with the following property.
Let $F\subset\mathbb{R}^{N}$ be a bounded set of finite fractional perimeter
and let $E$ be a minimizer of $\mathcal{F}_{h}(\cdot,F)$, then

$\sup_{E\triangle F}\textnormal{dist}_{\partial F}\leq\gamma h^{1/1+s}.$

###### Proof.

Let $\gamma=\max\\{3,2^{s+1/s}P^{s}(B)^{1/s}/C^{1/s}\\}$, where $C=C(N,s)$ is
the constant given by the Proposition 3.2. Let $c>\gamma$ and $x_{0}\in
E\triangle F$. Suppose by contradiction that $\textnormal{dist}_{\partial
F}(x_{0})>ch^{1/1+s}$. Since the other case is analogous, we assume $x_{0}\in
E\setminus F$. We then have

$\textnormal{sd}_{F}(x_{0})>ch^{1/1+s}$ (3.6)

and thus any ball $B_{r}(x_{0})$ of radius $r\leq ch^{1/1+s}/2$ is contained
in $F^{c}$. By the minimality of $E$, we have
$\mathcal{F}_{h}(E,F)\leq\mathcal{F}_{h}(E\setminus B_{r}(x_{0}),F)$,
therefore

$P^{s}(E)\leq P^{s}(E\setminus B_{r}(x_{0}))-\frac{1}{h}\int_{E\cap
B_{r}(x_{0})}\textnormal{sd}_{F}\,\textnormal{d}x+\frac{1}{h^{s/1+s}}|E\cap
B_{r}(x_{0})|.$

We use (3.6) and $r\leq ch^{1/1+s}/2$ to infer that

$-\frac{1}{h}\int_{E\cap
B_{r}(x_{0})}\textnormal{sd}_{F}\,\textnormal{d}x<-\frac{c}{2h^{s/1+s}}|E\cap
B_{r}(x_{0})|.$

Then we have

$P^{s}(E)\leq P^{s}(E\setminus
B_{r}(x_{0}))-\frac{1}{h^{s/1+s}}\left(\frac{c}{2}-1\right)|E\cap
B_{r}(x_{0})|.$ (3.7)

By assumption $c>3$ and we can apply the Proposition 3.2 with $\mu=0$ and
obtain

$Cr^{N}\leq|E\cap B_{r}(x_{0})|\qquad\forall 0<r<\frac{c}{2}h^{1/1+s}.$ (3.8)

On the other hand, from (3.7) we deduce, for every $0<r<ch^{1/1+s}/2$, that

$\frac{1}{h^{s/1+s}}\left(\frac{c}{2}-1\right)|E\cap B_{r}(x_{0})|\leq
P^{s}(E\setminus B_{r}(x_{0}))-P^{s}(E)\leq P^{s}(B_{r}^{c})=P^{s}(B)r^{N-s}$
(3.9)

(where the last inequality follows from the subadditivity of the perimeter on
$E$ and $B_{r}^{c}$). Combining (3.8) and (3.9), we get that

$Cr^{N}\leq|E\cap B_{r}(x_{0})|\leq
P^{s}(B)\left(\frac{c}{2}-1\right)^{-1}h^{s/1+s}r^{N-s}\leq
2P^{s}(B)h^{s/1+s}r^{N-s}$

for all $0<r<ch^{1/1+s}/2$, which gives the desired contradiction to the
choice of $c$ as soon as $r\to ch^{1/1+s}/2$. ∎

We now characterize the stationary sets $E$ for the discrete flow. We say that
$E$ is a stationary set for the discrete flow if it is a fixed set for the
functional (3.1), that is,

$E=E_{n}^{(h)}\quad\forall n\in\mathbb{N}.$

######  Remark 3.5.

From the monotonicity of the energy
$P_{s}(\cdot)+h^{-\frac{s}{1+s}}||\cdot|-m|$ along the discrete flow, one can
observe that, for $h=h(m,M)$ small, $|E_{n}^{(h)}|\in(m/2,3m/2)$ for all
$n\in\mathbb{N}$.

In the following, we will always assume that either:

  * •

$N=2$;

  * •

$N\leq 7$ and $s\in(s_{0},1)$, where $s_{0}$ is the constant of Proposition
3.1, item ii).

This hypothesis is essential for the proof of the following result.

######  Proposition 3.6.

Every stationary set $E$ for the discrete flow is a critical set of the
$s-$perimeter, that is, a single ball.

###### Proof.

It is an immediate consequence of the Euler-Lagrange equation (3.2). Since $E$
is a stationary point for the discrete flow, it satisfies

$\int_{\partial
E}H^{s}_{E}\,\textnormal{d}\mathcal{H}^{N-1}=\lambda\int_{\partial
E}X\cdot\nu_{E}\,\textnormal{d}\mathcal{H}^{N-1}$

for all $X\in C^{1}_{c}(\mathbb{R}^{N},\mathbb{R}^{N})$, i.e. $E$ is a
critical point for the $s-$perimeter. Employing [4, Theorem 1.1] and [13,
Theorem 1.1], we conclude that $E$ is a single ball having fractional mean
curvature $H^{s}_{E}=\lambda.$ ∎

Before proving the convergence of the flow up to translations, we recall the
following lemma from [24] that will be used in the proof of the next
proposition.

######  Lemma 3.7.

Let $\\{E_{n}^{(h)}\\}_{n\in\mathbb{N}}$ be a volume preserving discrete flow
starting from $E_{0}$ and let $E_{k_{n}}^{(h)}$ be a subsequence such that
$E_{k_{n}}^{(h)}+\tau_{n}\to F$ in $L^{1}$ for some set $F$ and a suitable
sequence $\\{\tau_{n}\\}_{n\in\mathbb{N}}\subset\mathbb{R}^{N}.$ Then
$\textnormal{dist}_{\partial
E_{k_{n}-1}^{(h)}}(\cdot+\tau_{n})\to\textnormal{dist}_{\partial F}$
uniformly.

The following result proves the convergence of the discrete flow to a union of
disjointed balls, all having the same radius. Moreover, we prove that the flow
eventually has fixed volume. A this point, we can not rule out that the flow
is converging to different balls (each at infinite distance from the others)
and that the translations introduced are different along different
subsequences. We will provide a sharper result in the final theorem.

######  Proposition 3.8.

Let $m$, $M>0$ and $E_{0}$ be an initial bounded set with $P^{s}(E_{0})\leq
M$, $|E_{0}|=m$. Then, for $h=h(s,M,m)$ small enough and for any discrete flow
$E_{n}^{(h)}$ starting from $E_{0}$ the following hold:

  * i)

for $n$ sufficiently large $|E_{n}^{(h)}|=m$;

  * ii)

there exists

$P_{\infty}^{s}=\lim_{n\to\infty}P^{s}(E_{n}^{(h)});$

  * iii)

$E_{n}^{(h)}$ is made of
$K=(P_{\infty}^{s}/\omega_{N}^{s})^{\frac{N}{s}}(\omega_{N}/m)^{\frac{N}{s}-1}$
distinct connected components $E_{n,i}^{(h)}$, and
$E_{n,i}^{(h)}-\textnormal{bar}(E_{n,i}^{(h)})$ converges in $C^{k}$, for
every $k\in\mathbb{N}$, to the ball centered at the origin and having mass
$m/K$.

###### Proof.

Let $\\{E_{k_{n}}^{(h)}\\}_{n\in\mathbb{N}}$ be any given subsequence of
$\\{E_{n}^{(h)}\\}_{n\in\mathbb{N}}$. By Proposition 3.1, each set
$E_{k_{n}}^{(h)}$ is made up of $l_{n}\leq k_{0}$ connected components having
diameter uniformly bounded by $d_{0}$. Therefore, there exist $l_{n}$ balls
$\\{B_{d_{0}}(\xi_{n}^{i})\\}$, each containing a different component of
$E_{k_{n}}^{(h)}$ and such that
$E_{k_{n}}^{(h)}\subset\cup_{i=1}^{l_{n}}B_{d_{0}}(\xi_{n}^{i})$. Up to
subsequences, we can assume that $l_{n}=\tilde{l}$, and for all $1\leq
i<j\leq\tilde{l}$ the following limits exist

$\limsup_{n\to\infty}|\xi_{n}^{i}-\xi_{n}^{j}|\eqqcolon
d^{i,j}\in[0,+\infty].$

Now we define the following equivalence classes: we say that $i\equiv j$ if
and only if $d^{i,j}<+\infty.$ Denote by $l\leq\tilde{l}$ the number of such
equivalence classes, let $j(i)$ be a representative for each class
$i\in\\{1,\ldots,l\\}$, and set $\sigma_{n}^{i}\coloneqq\xi^{j(i)}$ for
$i=1,\ldots,l.$ We have constructed a subsequence $E_{k_{n}}^{(h)}$ satisfying
$E_{k_{n}}^{(h)}\subset\cup_{i=1}^{l}B_{R}(\sigma_{n}^{i}),$ where
$R=d_{0}+\max\\{d^{i,j}:d^{i,j}<+\infty\\}+1,$ and for all $i\not\equiv j$ it
holds $|\sigma_{n}^{i}-\sigma_{n}^{j}|\to+\infty$ as $n\to+\infty$.

Now, fix $1\leq i\leq l$, and set

$F_{n}^{i}\coloneqq
E_{k_{n}}^{(h)}-\sigma_{n}^{i},\quad\tilde{F}_{n}^{i}\coloneqq(E_{k_{n}}^{(h)}-\sigma_{n}^{i})\cap
B_{R},\quad m_{n}^{i}\coloneqq|\tilde{F}_{n}^{i}|.$

Up to a subsequence, we have $m_{n}^{i}\to m^{i}>0$. Moreover, by Lemma 3.7
and by the compactness of sets of equi-bounded fractional perimeters, there
exist $\tilde{F}^{i}\Subset B_{R}$ such that, up to a subsequence,

$\tilde{F}_{n}^{i}\to\tilde{F}^{i}\textnormal{ in
}L^{1},\quad\textnormal{sd}_{E_{k_{n}-1}^{(h)}}(\cdot+\sigma_{n}^{i})\to\textnormal{sd}_{\tilde{F}^{i}}(\cdot)\textnormal{
locally uniformly.}$ (3.10)

Let $\tilde{G}^{i}$ be any bounded set with $|\tilde{G}^{i}|=m_{n}^{i}$ and
let
$\tilde{G}_{n}^{i}\coloneqq\left(\frac{m_{n}^{i}}{m^{i}}\right)^{\frac{1}{N}}\tilde{G}^{i}.$
We set now
$G_{n}^{i}\coloneqq(F_{n}^{i}\setminus\tilde{F}_{n}^{i})\cup\tilde{G}_{n}^{i}$
so that, for $n$ sufficiently large, $|F_{n}^{i}|=|G_{n}^{i}|$. By the
minimality of $E_{k_{n}}^{(h)}$ we have

$P^{s}(F_{n}^{i})+\frac{1}{h}\int_{F_{n}^{i}}\textnormal{sd}_{E_{k_{n}-1}^{(h)}}(x+\sigma_{n}^{i})\,\textnormal{d}x\leq
P^{s}(G_{n}^{i})+\frac{1}{h}\int_{G_{n}^{i}}\textnormal{sd}_{E_{k_{n}-1}^{(h)}}(x+\sigma_{n}^{i})\,\textnormal{d}x.$

For $n$ sufficiently large, we obtain

$\displaystyle
P^{s}(\tilde{F}_{n}^{i})+\int_{\tilde{F}_{n}^{i}}\int_{F_{n}^{i}\setminus\tilde{F}_{n}^{i}}\frac{1}{|x-y|^{N+s}}\,\textnormal{d}x\,\textnormal{d}y+\frac{1}{h}\int_{\tilde{F}_{n}^{i}}\textnormal{sd}_{E_{k_{n}-1}^{(h)}}(x+\sigma_{n}^{i})\,\textnormal{d}x$
$\displaystyle\leq
P^{s}(\tilde{G}_{n}^{i})+\int_{\tilde{G}_{n}^{i}}\int_{F_{n}^{i}\setminus\tilde{F}_{n}^{i}}\frac{1}{|x-y|^{N+s}}\,\textnormal{d}x\,\textnormal{d}y+\frac{1}{h}\int_{\tilde{G}_{n}^{i}}\textnormal{sd}_{E_{k_{n}-1}^{(h)}}(x+\sigma_{n}^{i})\,\textnormal{d}x.$

Passing to the limit as $n\to\infty$, using (3.10) and the uniform boundedness
of $\tilde{F}_{n}^{i}$ and $\tilde{G}_{n}^{i}$, we deduce that

$P^{s}(\tilde{F}^{i})+\frac{1}{h}\int_{\tilde{F}^{i}}\textnormal{sd}_{\tilde{F}^{i}}(x)\,\textnormal{d}x\leq
P^{s}(G^{i})+\frac{1}{h}\int_{G^{i}}\textnormal{sd}_{\tilde{F}^{i}}(x)\,\textnormal{d}x.$

This minimality property extends by density to all competitors $G^{i}$ with
finite perimeter and volume $m^{i}$, so that we deduce that $\tilde{F}^{i}$ is
a fixed point for the discrete scheme with prescribed volume $m^{i}$, and,
whence by Proposition 3.6, it is a ball. Moreover, since $\tilde{F}^{i}$ are
uniform $\Lambda-$minimizer by Proposition 3.1, we also deduce that
$\tilde{F}_{n}^{i}$ converge to $\tilde{F}^{i}$ in $C^{1,\alpha}$ for every
$\alpha\in(0,1).$ In particular, for $n$ large enough, $\tilde{F}_{n}^{i}$ has
only one connected component.

We have shown that, for $n$ large enough, $E_{k_{n}}^{(h)}$ is made up by a
fixed number $K$ of connected components $E_{k_{n}}^{(h),i}$, $i=1,\ldots,K$
and $E_{k_{n}}^{(h),i}-\textnormal{bar}(E_{k_{n}}^{(h),i})\to B_{R_{i}}$ where
$|B_{R_{i}}|=m_{i}$. Now, we show that all the radii $R_{i}$ are equal to $R$.
To this aim, we consider the Euler-Lagrange equation (3.2)

$\frac{1}{h}\textnormal{sd}_{E_{k_{n}-1}^{(h)}}+H^{s}_{E_{k_{n}}^{(h)}}=\lambda_{n}\quad\text{on
}\partial E_{k_{n}}^{(h)}.$

By Proposition 3.4, we deduce that

$|\lambda_{n}|\leq
h^{-1}\|\textnormal{sd}_{E_{k_{n}-1}^{(h)}}\|_{L^{\infty}(\partial
E_{k_{n}}^{(h)})}+\|H^{s}_{E_{k_{n}}^{(h)}}\|_{L^{\infty}(\partial
E_{k_{n}}^{(h)})}\leq c+\|H^{s}_{E_{k_{n}}^{(h)}}\|_{L^{\infty}(\partial
E_{k_{n}}^{(h)})}.$

To bound the right hand side, we use the $\Lambda-$minimality of
$E_{k_{n}}^{(h)}$ to obtain

$\|H^{s}_{E_{k_{n}}^{(h)}}\|_{L^{\infty}(\partial
E_{k_{n}}^{(h)})}\leq\Lambda.$

Therefore, by passing to a further subsequence, we can assume
$\lambda_{n}\to\lambda\in\mathbb{R}.$ Arguing as before, we can localize the
Euler-Lagrange equation to each single $F_{n}^{i}$ and obtain

$\frac{1}{h}\textnormal{sd}_{E_{k_{n}-1}^{(h)}}(x+\sigma_{n}^{i})+H^{s}_{F_{n}^{i}}(x)=\lambda_{n}\quad
x\in\partial F_{n}^{i}.$

We can pass to the limit as $n\to\infty$ thanks to Lemma 3.7 and the
continuity property of the fractional mean curvature (see e.g. [13, Lemma
2.1]). Thus, taking into account that $\tilde{F}^{i}$ is a fixed set for
(3.1), we deduce that

$H^{s}_{\tilde{F}^{i}}=\lambda\quad\text{on }\partial\tilde{F}^{i}.$

In particular, this shows that $R_{i}=c\lambda^{-s},$ for a suitable constant
$c$ depending only on $s$ and $N$.

In order to prove that, eventually, $|E_{n}^{(h)}|=m$, we proceed as follows.
Set $|B_{R_{i}}|=c_{1}\lambda^{-sN}$ and
$P^{s}(B_{R_{i}})=c_{2}\lambda^{-s(N-s)}.$ From Remark 3.5, we take $h=h(s,M)$
small enough such that

$|E_{k_{n}}^{(h)}|\in\left[\frac{m}{2},\frac{3m}{2}\right],\quad
P^{s}(E_{k_{n}}^{(h)})\leq P^{s}(E_{0})\leq M$

and, for $n$ large enough, this implies

$\sum_{i=1}^{K}m_{n}^{i}\in\left[\frac{m}{2},\frac{3m}{2}\right],\quad\sum_{i=1}^{K}P^{s}(\tilde{F}_{n}^{i})\leq
M.$

Passing to the limit as $n\to\infty$ we obtain

$Kc_{1}\lambda^{-sN}\in\left[\frac{m}{2},\frac{3m}{2}\right],\quad
Kc_{2}\lambda^{-s(N-s)}\leq M,$

which implies

$\lambda^{s^{2}}\leq\dfrac{2c_{1}\,M}{m\,c_{2}}.$ (3.11)

If we suppose that $|E_{k_{n}}^{(h)}|\neq m$ for infinitely many indexes, then
$\lambda=\textnormal{sgn}(m-|E_{k_{n}}^{(h)}|)h^{-\frac{s}{1+s}}$ which is a
contradiction to (3.11) if $h$ is sufficiently small. We have thus proved item
i). Since, for $n$ large enough, $|E_{n}^{(h)}|=m$, the sequence
$\\{P^{s}(E_{n}^{(h)})\\}_{n\in\mathbb{N}}$ is eventually non-increasing, from
which item ii) follows. Knowing the exact values of the volume and
$s-$perimeter of any limit point, we are able to compute $K$ and obtain the
convergence in $L^{1}$ of the whole sequence. Moreover, arguing as in [8] we
conclude the convergence in $C^{k}$ for every $k\in\mathbb{N}$ via a bootstrap
method. ∎

In order to prove the main theorem, we need to recall some results of [24].

######  Lemma 3.9.

Let $\eta>0$. There exists $\delta>0$ with the following property: if
$f_{1},f_{2}\in C^{1}(\partial B)$ with $\|f_{i}\|_{C^{1}(\partial
B)}\leq\delta$ and $|B_{f_{i}}|=|B|$ for $i=1,2$ we have

$\displaystyle C_{1}(1-\eta)\|f_{1}-f_{2}\|_{L^{2}(B)}^{2}\leq$
$\displaystyle\mathcal{D}(B_{f_{1}},B_{f_{2}})\leq
C_{1}(1+\eta)\|f_{1}-f_{2}\|_{L^{2}(B)}^{2}$
$\displaystyle\dfrac{1-\eta}{2}\int_{\partial
B_{f-1}}\textnormal{sd}^{2}_{B_{f_{2}}}\,\textnormal{d}\mathcal{H}^{N-1}\leq$
$\displaystyle\mathcal{D}(B_{f_{1}},B_{f_{2}})\leq\dfrac{1+\eta}{2}\int_{\partial
B_{f-1}}\textnormal{sd}^{2}_{B_{f_{2}}}\,\textnormal{d}\mathcal{H}^{N-1}$
$\displaystyle|\textnormal{bar}(B_{f_{1}})-\textnormal{bar}(B_{f_{2}})|^{2}\leq$
$\displaystyle
C_{2}\|f_{1}-f_{2}\|^{2}_{L^{2}(B)}\leq\frac{C_{2}}{C_{1}(1-\eta)}\mathcal{D}(B_{f_{1}},B_{f_{2}})$

for suitable constants $C_{1},\,C_{2}>0$.

The following lemma proves the crucial dissipation-dissipation inequality
(3.12). This result will play a central role in the proof of Theorem 0.1. Its
proof is based on the Alexandrov-type estimate contained in Theorem 0.3 and
will be omitted as it is the same of [24, Lemma 3.8].

######  Lemma 3.10.

Let $h>0$. There exist constants $C(h,m,s),$ $\delta>0$ with the following
property: given two normal deformations $B^{(m)}_{f_{1}},$ $B^{(m)}_{f_{2}}$
with $f_{i}\in C^{2}(\partial B^{(m)}),$ $\|f_{i}\|_{C^{1}(\partial
B^{(m)})}\leq\delta$, and such that $|B^{(m)}_{f_{2}}|=m,$
$\textnormal{bar}(B^{(m)}_{f_{2}})=0$ and

$H_{B^{(m)}_{f_{2}}}+\dfrac{\textnormal{sd}_{B^{(m)}_{f_{1}}}}{h}=\lambda\quad\text{on}\quad\partial
B^{(m)}_{f_{2}}$ (3.12)

for some $\lambda\in\mathbb{R}$, we have

$\mathcal{D}(B^{(m)},B^{(m)}_{f_{2}})\leq
C\mathcal{D}(B^{(m)}_{f_{2}},B^{(m)}_{f_{1}}).$

We are finally able to prove Theorem 0.1. We will follow closely the proofs of
[24, Theorem 3.3] and [14, Theorem 1.1]. The main difference is that we use
the fractional perimeter framework previously studied instead of the classical
one. We present a sketch of the proof.

###### Proof.

We start by proving the exponential decay of the dissipations, corresponding
to Step 1 in [24, Theorem 3.3].

From Proposition 3.8 we know that any limit point of the discrete flow is
given by the union of $K$ disjoint balls, all having volume $m/K$. We then use
two competitors to obtain a discrete Gronwall-type inequality. Firstly,
testing the minimality of $E_{n}^{(h)}$ with $E_{n-1}^{(h)}$ and summing from
$n$ to infinity, we obtain

$\sum_{k\geq n-1}\mathcal{D}(E_{k}^{(h)},E_{n-1}^{(h)})\leq
P(E_{n}^{(h)})-P_{\infty}^{s}=P(E_{n}^{(h)})-KP^{s}(B^{(m/K)}).$

On the other hand, recalling Proposition 3.8, the sets
$(E_{n}^{(h)})^{i}-\text{bar}((E_{n}^{(h)})^{i})=:(E_{n}^{(h)})^{i}-\xi_{n}^{i}$
are eventually $C^{1,\alpha}-$deformations of $B^{(m/K)}$, having volume
$|(E_{n}^{(h)})^{i}|=m_{n}^{i}$. We consider the admissible competitor for
$E_{n}^{(h)}$ given by

$\mathcal{B}_{n}=\bigcup_{i=1}^{K}\left(B^{(m_{n-1}^{i})}+\xi_{n-1}^{i}\right).$

Testing the minimality of $E_{n}^{(h)}$ against $\mathcal{B}_{n}$, one can
obtain

$P^{s}(E_{n}^{(h)})-P^{s}(\mathcal{B}_{n})\leq
C\mathcal{D}(E_{n-1}^{(h)},E_{n-2}^{(h)}).$

Recalling that, if a measurable set $F$ has $L$ disjointed connected
components $F^{i}$, $i=1,\dots,L$, then

$P^{s}(F)=\sum_{i=1}^{L}P^{s}(F^{i})-2\sum_{i<j}\int_{F^{i}}\int_{F^{j}}\frac{1}{|x-y|^{N+s}}\,\textnormal{d}x\,\textnormal{d}y,$

by concavity, we estimate

$P^{s}(\mathcal{B}_{n})\leq\sum_{i=1}^{K}P^{s}(B^{(m^{i}_{n-1})})\leq
KP^{s}(B^{(m/K)}).$

Thus, combining the previous two estimates, we obtain the discrete Gronwall-
type estimate

$\sum_{k\geq n-1}\mathcal{D}(E_{k}^{(h)},E_{n-1}^{(h)})\leq
C\mathcal{D}(E_{n-1}^{(h)},E_{n-2}^{(h)}).$

Finally, employing [24, Lemma 3.10] we conclude the exponential convergence of
the dissipations

$\mathcal{D}(E_{n}^{(h)},E_{n-1}^{(h)})\leq\left(1-\frac{1}{C+1}\right)^{\frac{n}{2}}(P^{s}(E_{0})-KP^{s}(B^{(m/K)})).$

From now on, one can follow directly the proof of [24, Theorem 3.3] to
conclude that the discrete flow $E_{n}^{(h)}$ is eventually contained in a
compact set and converges in $C^{k}$ to a union of $K$ disjoint balls. Now,
from Proposition 3.6 we deduce that the limit point is indeed a single ball,
having volume equal to $m$, thus reaching the conclusion of the proof. ∎

As a corollary of the previous result, we prove the following characterization
of the long-time behaviour of the discrete volume-preserving mean curvature
flow associated with the classical perimeter. The definition is analogous to
the discrete flow associated with the fractional perimeter, but in this case
we consider the following minimum problem (cp. the problem (3.1))

$\min\left\\{P(F)+\frac{1}{h}\int_{F}\textnormal{sd}_{E}(x)\,\textnormal{d}x+\frac{1}{\sqrt{h}}||F|-m|:F\subset\mathbb{R}^{N}\right\\}.$

A careful study of the previous results highlights that the classical
perimeter behaves essentially in the same manner. In particular, Proposition
3.8 still holds for $s=1$, replacing “$P^{1}$” with the classical perimeter.
Thus, we conclude the following.

######  Proposition 3.11.

Let $m$, $M>0$ and let $E_{0}$ be an initial set with $P(E_{0})\leq M$,
$|E_{0}|=m$. Then, for $h=h(M,m)$ small enough the following holds: for any
discrete flow $E_{n}^{(h)}$ starting from $E_{0}$, there exist
$x_{i}\in\mathbb{R}^{N}$, $i=1,\dots,K$, where
$K=N^{-N}\omega_{N}m^{1-N}P_{\infty}^{N}$ such that

$E_{n}^{(h)}\to\bigcup_{i=1}^{K}\left(B^{(m/K)}+x_{i}\right)\qquad\text{in
}C^{k}$

for all $k\in\mathbb{N}.$ Moreover, the convergence is exponentially fast.

The proof is essentially the same, the major difference being that the limit
point of the flow now are union of disjoint balls (see [24, Proposition 3.1]
for details) and not a single ball.

##  Appendix A Existence of flat flows

In order to prove the existence of flat flows, we follow the lines of [25,
Section 3]. In this appendix we drop the assumption on the dimension and work
in $\mathbb{R}^{N}$, $N\geq 2$. We start by remarking that the minimality of
the sets $E_{n}^{(h)}$ implies

$\displaystyle P^{s}(E_{t}^{(h)})+\frac{1}{h}\int_{E_{t}^{(h)}\triangle
E_{t-h}^{(h)}}\textnormal{dist}_{\partial
E_{t-h}^{(h)}}(x)\,\textnormal{d}x+\frac{1}{h^{s/1+s}}||E_{t}^{(h)}|-m|$
$\displaystyle\leq
P^{s}(E_{t-h}^{(h)})+\frac{1}{h^{s/1+s}}||E_{t-h}^{(h)}|-m|\qquad\forall
t\in[h,+\infty)$ (A.1)

and, by iterating (A) and using $|E_{0}|=m$, we get

$\displaystyle P^{s}(E_{t}^{(h)})\leq P^{s}(E_{0}),$ (A.2)
$\displaystyle\frac{1}{h^{s/1+s}}||E_{t}^{(h)}|-m|\leq P^{s}(E_{0}),$ (A.3)
$\displaystyle\int_{h}^{T}\int_{E_{t}^{(h)}\triangle
E_{t-h}^{(h)}}\frac{\textnormal{dist}_{\partial
E_{t-h}^{(h)}}(x)}{h}\,\textnormal{d}x\leq P^{s}(E_{0}),$

for all $t\geq 0$ and for every $T>h$. We are then able to bound the
$L^{1}-$distance between two consecutive sets of the discrete flow.

######  Proposition A.1.

Let $t>0$. Then

$|E_{t}^{(h)}\triangle E_{t-h}^{(h)}|\leq
C\left(l^{s}P^{s}(E_{t-h}^{(h)})+\frac{1}{l}\int_{E_{t}^{(h)}\triangle
E_{t-h}^{(h)}}\textnormal{dist}_{\partial
E_{t-h}^{(h)}}(x)\,\textnormal{d}x\right)\quad\forall l\leq\gamma h^{1/1+s}.$

###### Proof.

In order to estimate $|E_{t}^{(h)}\triangle E_{t-h}^{(h)}|$, we split it into
two parts:

$|E_{t}^{(h)}\triangle E_{t-h}^{(h)}|\leq|\\{x\in E_{t}^{(h)}\triangle
E_{t-h}^{(h)}:\textnormal{dist}_{\partial F}(x)\leq l\\}|+|\\{x\in
E_{t}^{(h)}\triangle E_{t-h}^{(h)}:\textnormal{dist}_{\partial
E_{t-h}^{(h)}}(x)\geq l\\}|.$

The second term is estimated by

$|\\{x\in E_{t}^{(h)}\triangle E_{t-h}^{(h)}:\textnormal{dist}_{\partial
E_{t-h}^{(h)}}(x)\geq l\\}|\leq\frac{1}{l}\int_{E_{t}^{(h)}\triangle
E_{t-h}^{(h)}}\textnormal{dist}_{\partial E_{t-h}^{(h)}}(x)\,\textnormal{d}x.$

To estimate the first term, we use a covering argument to find a collection of
disjoint balls $\\{B_{l}(x_{i})\\}_{i\in I}$ with $x_{i}\in\partial
E_{t-h}^{(h)}$ and $I\subset\mathbb{N}$ a finite set such that $\partial
E_{t-h}^{(h)}\subset\cup_{i\in I}B_{2l}(x_{i})$. Observe that, by (3.3), (3.4)
and the relative isoperimetric inequality, for every $i\in I$, we get

$\displaystyle|B_{3l}(x_{i})|$ $\displaystyle\leq C\min\\{|E_{t-h}^{(h)}\cap
B_{l}(x_{i})|,|B_{l}(x_{i})\setminus E_{t-h}^{(h)}|\\}$ $\displaystyle\leq
C\mathcal{L}_{s}(E_{t-h}^{(h)}\cap B_{l}(x_{i}),B_{l}(x_{i})\setminus
E_{t-h}^{(h)})^{N/N-s}$ $\displaystyle\leq
Cl^{s}\,\mathcal{L}_{s}(E_{t-h}^{(h)}\cap B_{l}(x_{i}),B_{l}(x_{i})\setminus
E_{t-h}^{(h)})$ $\displaystyle\leq Cl^{s}\,\mathcal{L}_{s}(E_{t-h}^{(h)}\cap
B_{l}(x_{i}),\mathbb{R}^{N}\setminus E_{t-h}^{(h)}).$

Since the set $\\{x\in E_{t}^{(h)}\triangle
E_{t-h}^{(h)}:\textnormal{dist}_{\partial E_{t-h}^{(h)}}(x)\leq l\\}$ is
covered by $\\{B_{3l}(x_{i})\\}_{i\in I}$, by summing over $i$ and by the
choice of the balls $\\{B_{l}(x_{i})\\}_{i\in I}$, we obtain

$\displaystyle|\\{x\in E_{t}^{(h)}\triangle
E_{t-h}^{(h)}:\textnormal{dist}_{\partial E_{t-h}^{(h)}}(x)\leq l\\}|$
$\displaystyle\leq\sum_{i\in I}|B_{3l}(x_{i})|$ $\displaystyle\leq
Cl^{s}\sum_{i\in I}\mathcal{L}_{s}(E_{t-h}^{(h)}\cap
B_{l}(x_{i}),\mathbb{R}^{N}\setminus E_{t-h}^{(h)})$ $\displaystyle\leq
Cl^{s}P^{s}(E_{t-h}^{(h)}).$

∎

######  Proposition A.2.

Let $h\leq 1$, then it holds

$|E_{t_{1}}^{(h)}\triangle E_{t_{2}}^{(h)}|\leq
C|t_{1}-t_{2}|^{s/s+1}\quad\forall 0\leq t_{1}\leq t_{2}<+\infty.$ (A.4)

###### Proof.

It is enough to consider the case $t_{2}-t_{1}\geq h$. Let $j\in\mathbb{N}$
and $k\in\mathbb{N}\setminus\\{0\\}$ be such that $t_{1}\in[jh,(j+1)h)$ and
$t_{2}\in[(j+k)h,(j+k+1)h).$ Then, we can use Proposition A.1 with $l=\gamma
h/|t_{1}-t_{2}|^{s/s+1}$ (note that $l\leq\gamma h^{1/s+1}$ by the assumption
$t_{2}-t_{1}\geq h$) and estimate in the following way

$\displaystyle|E_{t_{1}}^{(h)}\triangle E_{t_{2}}^{(h)}|$
$\displaystyle\leq\sum_{i=1}^{k}|E_{(j+i)h}^{(h)}\triangle
E_{(j+i-1)h}^{(h)}|$ $\displaystyle\leq
C\sum_{i=1}^{k}\frac{h^{s}}{|t_{1}-t_{2}|^{s^{2}/s+1}}P^{s}(E_{(j+i-1)h}^{(h)})$
$\displaystyle\
+C\sum_{i=1}^{k}\frac{|t_{1}-t_{2}|^{s/s+1}}{h}\int_{E_{(j+i)h}^{(h)}\triangle
E_{(j+i-1)h}}^{(h)}|\textnormal{sd}_{E_{(j+i-1)h}^{(h)}}|\,\textnormal{d}x.$

By using (A) we estimate the sum above by

$\displaystyle|E_{t_{1}}^{(h)}\triangle E_{t_{2}}^{(h)}|$ $\displaystyle\leq
C\sum_{i=1}^{k}\frac{h^{s}}{|t_{1}-t_{2}|^{s^{2}/s+1}}P(E_{0})$
$\displaystyle\
+C\sum_{i=1}^{k}|t_{1}-t_{2}|^{s/s+1}(P(E_{(j+i-1)h}^{(h)})-P(E_{(j+i)h}^{(h)}))$
$\displaystyle\
+C\sum_{i=1}^{k}\frac{|t_{1}-t_{2}|^{s/s+1}}{h^{s/s+1}}\left(||E_{(j+i-1)h}^{(h)}|-m|-||E_{(j+i)h}^{(h)}|-m|\right)$
$\displaystyle\leq
C\frac{kh^{s}}{|t_{1}-t_{2}|^{s^{2}/s+1}}P(E_{0})+C|t_{1}-t_{2}|^{s/s+1}(P(E_{t_{1}}^{(h)})-P(E_{t_{2}}^{(h)}))$
$\displaystyle\
+C\frac{|t_{1}-t_{2}|^{s/s+1}}{h^{s/s+1}}\left(||E_{t_{1}}^{(h)}|-m|-||E_{t_{2}}^{(h)}|-m|\right).$

Now, by (A.2) and (A.3), we get

$|E_{t_{1}}^{(h)}\triangle E_{t_{2}}^{(h)}|\leq
C|t_{1}-t_{2}|^{s/s+1}P(E_{0}),$

where we used that $kh^{s}\leq 2|t_{1}-t_{2}|h^{s-1}\leq 2|t_{1}-t_{2}|^{s}$
since $h\leq|t_{1}-t_{2}|$. ∎

We are now able to prove the existence of fractional flat flows, defined as
$L^{1}_{loc}-$limit points of the discrete flow previously introduced.

######  Proposition A.3.

Let $E_{0}$ be a bounded initial set of finite fractional perimeter and volume
$m$. For any $h>0$, let $\\{E_{t}^{(h)}\\}_{t\geq 0}$ be a discrete flow.
Then, there exist a family of sets $\\{E_{t}\\}_{t\geq 0}$ of finite
fractional perimeter and a subsequence $h_{k}\to 0$ such that, as
$k\to\infty$, it holds

$E^{(h_{k})}_{t}\to E_{t}\quad\text{in }L^{1}_{loc},\quad\text{for all
}t\in[0,+\infty).$

Moreover, the flow satisfies $\forall\,0\leq s\leq t$,

$\displaystyle|E_{t}\triangle E_{s}|$ $\displaystyle\leq
C|t-s|^{\frac{s}{1+s}},$ $\displaystyle P^{s}(E_{t})$ $\displaystyle\leq
P^{s}(E_{0}).$

###### Proof.

We follow [25, Theorem 2.2]. Considering $t\in\mathbb{Q}^{+}$, from (A.2) and
the compactness of sets of finite fractional perimeter, we find a subsequence
$h_{k}\to 0$ such that

$L^{1}_{loc}-\lim_{k\to\infty}E_{t}^{(h_{k})}=E_{t}\quad\forall
t\in\mathbb{Q}^{+}.$

By the triangular inequality, it’s easy to see that (A.4) passes to the limit.
Finally, a simple continuity argument implies that the whole sequence
$E^{(h_{k})}_{t}$ converges to the sets $E_{t}$ for all $t\in[0,+\infty)$. ∎

If we assume the following hypothesis:

  1. 1.

given $T>0$ and an initial bounded set $E_{0}$ of finite fractional perimeter,
there exists $R>0$ independent of $h$ such that $E_{t}^{(h)}\subset B_{R}$ for
all $h>0,t\in[0,T]$,

we are able to prove that the flat flow is indeed volume-preserving.

######  Corollary A.4.

Under hypothesis 1, let $E_{0}$ be a bounded initial set of finite fractional
perimeter and volume $m$. For any $h>0$, let $\\{E_{t}^{(h)}\\}_{t\geq 0}$ be
a discrete flow. Then, there exist a family of sets $\\{E_{t}\\}_{t\geq 0}$ of
finite fractional perimeter and a subsequence $h_{k}\to 0$ such that, as
$k\to\infty$, it holds

$E^{(h_{k})}_{t}\to E_{t}\quad\text{in }L^{1}\quad\text{for all
}t\in[0,+\infty).$

Moreover, the flow satisfies $\forall\,0\leq s\leq t$,

$\displaystyle|E_{t}|$ $\displaystyle=|E_{0}|$ $\displaystyle|E_{t}\triangle
E_{s}|$ $\displaystyle\leq C|t-s|^{\frac{s}{1+s}},$ $\displaystyle
P^{s}(E_{t})$ $\displaystyle\leq P^{s}(E_{0}).$

###### Proof.

The proof is analogous to the one of Proposition A.3. By uniform boundedness,
the limits are in $L^{1}$ instead of $L^{1}_{loc}$. Then, passing to the limit
$h\to 0$ in (A) we conclude that $|E_{t}|=m$ for all $t\geq 0$. ∎

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