# Existence and symmetry of periodic nonlocal-CMC
surfaces via variational methods

Xavier Cabré  X. Cabré 1,2,3
1 ICREA, Pg. Lluis Companys 23, 08010 Barcelona, Spain
2 Universitat Politècnica de Catalunya, Departament de Matemàtiques and
IMTech, Diagonal 647, 08028 Barcelona, Spain
3 Centre de Recerca Matemàtica, Edifici C, Campus Bellaterra, 08193
Bellaterra, Spain<EMAIL_ADDRESS>,  Gyula Csató  G. Csató 1,2
1 Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran
Via 585, 08007 Barcelona, Spain
2 Centre de Recerca Matemàtica, Edifici C, Campus Bellaterra, 08193
Bellaterra, Spain<EMAIL_ADDRESS>and  Albert Mas  A. Mas 1,2
1 Departament de Matemàtiques, Universitat Politècnica de Catalunya, Campus
Diagonal Besòs, Edifici A (EEBE), Av. Eduard Maristany 16, 08019 Barcelona,
Spain
2 Centre de Recerca Matemàtica, Edifici C, Campus Bellaterra, 08193
Bellaterra, Spain<EMAIL_ADDRESS>
(Date: September 3, 2024)

###### Abstract.

This paper provides the first variational proof of the existence of periodic
nonlocal-CMC surfaces. These are nonlocal analogues of the classical Delaunay
cylinders. More precisely, we show the existence of a set in
${\mathbb{R}}^{n}$ which is periodic in one direction, has a prescribed (but
arbitrary) volume within a slab orthogonal to that direction, has constant
nonlocal mean curvature, and minimizes an appropriate periodic version of the
fractional perimeter functional under the volume constraint. We show, in
addition, that the set is cylindrically symmetric and, more significantly,
that it is even as well as nonincreasing on half its period. This monotonicity
property solves an open problem and an obstruction which arose in an earlier
attempt, by other authors, to show the existence of minimizers.

###### Key words and phrases:

Constant nonlocal mean curvature, periodic surfaces, Delaunay cylinders

###### 2010 Mathematics Subject Classification:

47G20,35J60,35B10,49Q05

The three authors are supported by the Spanish grants MTM2017-84214-C2-1-P,
PID2021-123903NB-I00, and RED2018-102650-T funded by
MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”. The
second and third authors are also supported by the Spanish grant
MTM2017-83499-P. The second author is in addition supported by the Spanish
grant PID2021-125021NA-I00. This work is supported by the Spanish State
Research Agency, through the Severo Ochoa and María de Maeztu Program for
Centers and Units of Excellence in R&D (CEX2020-001084-M)

##  1\. Introduction

For a general set $E\subset{\mathbb{R}}^{n}$, the nonlocal (or $s$-fractional,
$0<s<1$) mean curvature of $\partial E$ (or of $E$) at a boundary point
$x\in\partial E$ is defined by

$H_{s}[E](x):=PV\int_{{\mathbb{R}}^{n}}dy\,\frac{\chi_{E^{c}}(y)-\chi_{E}(y)}{|x-y|^{n+s}}=\lim_{\epsilon\downarrow
0}\int_{\\{|x-y|>\epsilon\\}´}dy\,\frac{\chi_{E^{c}}(y)-\chi_{E}(y)}{|x-y|^{n+s}}.$
(1.1)

Here $PV$ denotes the principal value sense, $\chi_{E}$ is the characteristic
function of the set $E,$ and $E^{c}$ is the complement in ${\mathbb{R}}^{n}$
of the set $E$. The nonlocal mean curvature arises when computing the first
variation of the fractional perimeter, an extension of the classical perimeter
that was introduced, in its localized or Dirichlet version, in the pioneering
article of Caffarelli, Roquejoffre, and Savin [13]. Their work has allowed for
a natural extension of the theory of minimal surfaces to the nonlocal
framework.

The current paper concerns the nonlocal analogue of the classical result of
Delaunay [17] on the existence of periodic surfaces of revolution with
constant mean curvature (CMC). We provide the first variational proof of the
existence of periodic surfaces of revolution having constant nonlocal mean
curvature and a prescribed (but arbitrary) volume within one period.
Throughout the paper, surfaces with constant nonlocal mean curvature will be
called nonlocal-CMC surfaces. Before our work, and up to our knowledge, the
only papers which had established the existence of some periodic nonlocal-CMC
surfaces are [10, 11, 12], by Cabré, Fall, Solà-Morales, and Weth. These works
did not use variational methods, but perturbative techniques (the implicit
function theorem, essentially). Thus, they only obtained nonlocal-CMC surfaces
which are very close to certain explicit configurations, as explained next.

In arbitrary dimension, [11] showed the existence of a one parameter family of
periodic nonlocal-CMC surfaces bifurcating from a straight cylinder. The paper
extended a previous analogous result in ${\mathbb{R}}^{2}$ from [10].111Such
nonlocal-CMC curves in the plane have no analogue, obviously, for the
classical mean curvature. Later, also by perturbation techniques, [12] proved
the existence of a periodic array of disjoint near-balls, the full array
having constant nonlocal mean curvature. They bifurcate from infinity,
starting from a single round ball of radius $1$ —in particular, the near-balls
have all the same shape (which is very close to a unit ball) and are very
distant from each other. Thus, after rescaling them to have their consecutive
centers at distance $2\pi$, they become of very small size (and, hence, of
small volume within a period). Instead, the $2\pi$-periodic near-cylinders
from [11] have a certain positive, bounded below, width. Prior to the current
paper, no result connecting both configurations, through other periodic
nonlocal-CMC surfaces, was available.222Recall that, in the local case, the
classical Delaunay surfaces vary continuously from a cylinder to an infinite
compound of tangent spheres.

This paper establishes for the first time the existence of a periodic
nonlocal-CMC surface in ${\mathbb{R}}^{n}$ for every given volume within a
slab. By a variational method we show the existence of a minimizer of the
functional

${\mathcal{P}}_{s}[E]:=\int_{E\cap\\{-\pi<x_{1}<\pi\\}}\\!dx\int_{{\mathbb{R}}^{n}\setminus
E}dy\,\frac{1}{|x-y|^{n+s}}$ (1.2)

(introduced for the first time in [16]), under a volume constraint within the
slab $\\{x\in{\mathbb{R}}^{n}:\,-\pi<x_{1}<\pi\\}$, among subsets $E$ of
${\mathbb{R}}^{n}=\\{x=(x_{1},x^{\prime})\in{\mathbb{R}}\times{\mathbb{R}}^{n-1}\\}$
which are $2\pi$-periodic in the coordinate $x_{1}$. Notice that the
functional ${\mathcal{P}}_{s}$ differs from the fractional $s$-perimeter of
$E$ relative to $\\{x\in{\mathbb{R}}^{n}:\,-\pi<x_{1}<\pi\\}$, a quantity
which is given by adding an additional term to ${\mathcal{P}}_{s}$, as
follows:

${\mathcal{P}}_{s}[E]+\int_{({\mathbb{R}}^{n}\setminus
E)\cap\\{-\pi<x_{1}<\pi\\}}\\!dx\int_{E\cap\\{|y_{1}|>\pi\\}}dy\,\frac{1}{|x-y|^{n+s}}.$
(1.3)

Still, and somehow surprisingly, we establish that the first variation of
these two functionals —the periodic first variation for (1.2) and the
Dirichlet first variation in the slab for (1.3)— lead both to the nonlocal
mean curvature.333The Dirichlet first variation of (1.3) was derived in the
seminal paper [13] by Caffarelli, Roquejoffre, and Savin. Uncovering this fact
originated the current article.

The following is our first main result. Apart from the existence of a
minimizer, it also establishes symmetry and monotonicity properties of any
minimizer.

######  Theorem 1.1.

Given $n\geq 2$, $s\in(0,1)$, and $\mu>0$, there exists a minimizer
$E\subset{\mathbb{R}}^{n}$ of ${\mathcal{P}}_{s}$ among all measurable sets
$F\subset{\mathbb{R}}^{n}$ which are $2\pi$-periodic in the $x_{1}$-variable
and satisfy

$|F\cap\\{-\pi<x_{1}<\pi\\}|=\mu.$ (1.4)

In addition, up to sets of measure zero:

  * $(i)$

Up to a translation in the $x^{\prime}$-variable, every minimizer $E$ is of
the form

$E=\\{x=(x_{1},x^{\prime})\in{\mathbb{R}}\times{\mathbb{R}}^{n-1}:\,|x^{\prime}|<u(x_{1})\\}$
(1.5)

for some $2\pi$-periodic nonnegative function $u$ whose power $u^{n-1}$
belongs to $W^{s,1}(-\pi,\pi)$. In particular, we have $u\in
W^{\frac{s}{n-1},\,n-1}(-\pi,\pi)$.

  * $(ii)$

Up to a translation in the $x_{1}$-direction, $u$ is even, as well as
nonincreasing in $(0,\pi)$.

  * $(iii)$

For every minimizer $E$, there exists a constant $\lambda\in{\mathbb{R}}$ such
that if $x\in\partial E$, $u$ is $C^{1,\alpha}$ in a neighborhood of $x_{1}$
for some $\alpha>s$, and $u(x_{1})>0$, then $H_{s}[E](x)=\lambda$.

  * $(iv)$

For $\mu$ small enough depending only on $n$ (and hence uniformly as
$s\uparrow 1$ and as $s\downarrow 0$), every minimizer is not a straight
circular cylinder, i.e., the function $u$ is nonconstant.

The symmetry and monotonicity properties of points $(i)$ and $(ii)$ will be
discussed in the next Subsection 1.1. Of particular importance is point
$(ii)$, since it relies on a deep and not so well-known result on
rearrangements in spheres ${\mathbb{S}}^{m}$. However, we emphasize that our
proof of existence of a minimizer only uses simple tools and, in particular,
it is independent of this rearrangement result.

The work [16] was the first attempt to show the existence of periodic
nonlocal-CMC surfaces with a minimization technique. This paper introduced,
for the first time, the functional ${\mathcal{P}}_{s}$ in (1.2) for periodic
sets. However, it did not relate its Euler-Lagrange equation to the nonlocal
mean curvature.444At that time, we knew that the first variation of
${\mathcal{P}}_{s}$ for periodic sets is the nonlocal mean curvature, since we
were working on the analogous functional for periodic solutions to fractional
semilinear equations (work of us still to appear). When [16] became available,
we communicated this fact to its authors, we pointed it out at the beginning
of page 35 in the printed version of [11], and was also mentioned in the
Master’s Thesis of M. Alvinyà [2] directed within our group. At the same time,
[16] did not prove the existence of a periodic constrained minimizer of
${\mathcal{P}}_{s}$ —it only obtained a minimizer within the class of sets
which are even in the $x_{1}$-variable and nonincreasing in $(0,\pi)$. Note
that this restricted minimization class (made of nonincreasing functions)
prevents from checking the Euler-Lagrange equation at a minimizer. The result
on rearrangements in spheres ${\mathbb{S}}^{m}$, mentioned above in connection
with point $(ii)$ of our theorem, is the additional tool that would have
completed the proof in [16]. Instead, as explained later, [16] proved a much
stronger result than our statement $(iv)$ (on small volumes) in Theorem 1.1.

The work [16] made apparent, thus, that showing the existence of a periodic
constrained minimizer of ${\mathcal{P}}_{s}$ was a delicate issue. In this
article we succeed to find a proof which does not rely, in contrast with the
approach in [16], on the nonincreasing property of the functions $u$ in a
minimizing sequence. To accomplish this, the main difficulty was finding a
suitable, more treatable expression for the periodic nonlocal perimeter. This
expression is given in Lemma 2.2, a result that also contains the important
lower bound (2.5). With these tools at hand, the existence of minimizer will
follow readily from the compact embedding of $W^{s,1}(-\pi,\pi)$ into
$L^{1}(-\pi,\pi)$.

In the course of this work, we first completed our existence proof in the case
of the plane, $n=2$. This was a simpler task, since it relied on an expression
for the fractional perimeter which is similar to some for the nonlocal mean
curvature contained in our 2D results from [10]. Since the expression for
$n=2$ is more explicit than the one in Lemma 2.2, we will present it in
Appendix A.

Our proof of point $(iv)$ in Theorem 1.1 —stating that minimizers for small
volumes are not straight cylinders, uniformly in $s$— is simple. It is based
on an energy-comparison argument and does not require the knowledge of
minimizers being nonincreasing in $(0,\pi)$. Instead, Theorem 3 of the above
mentioned paper [16] proved a much stronger, quantitative version of this
result. Using the monotonicity of the minimizer, it established that for small
volume constraints a minimizer must be close, in a measure theoretical sense,
to a periodic array of round balls. Notice here that the function $u$ defining
the minimizer could be identically zero in a subinterval $(a,\pi)$ of
$(0,\pi)$ and that this allows for the minimizer to be a periodic array of
near-balls. At the same time, the near-balls found in [12] are very close to
round balls since they were found through the implicit function theorem. Thus,
the result of [16] strongly suggests that for small volumes the minimizers
should agree with the near-balls constructed in [12]. However, this is still
an open problem.

Regarding point $(iii)$ in Theorem 1.1, we recall that the nonlocal mean
curvature is well defined at points $x\in\partial E$ where $\partial E$ is of
class $C^{1,\alpha}$, for some $\alpha>s$, in a neighborhood of $x$. However,
a complete $C^{1,\alpha}$ regularity theory for volume constrained minimizers
is not yet available.555In the nonperiodic case, the best result (up to our
knowledge) on the regularity of volume constrained minimizers of a certain
fractional perimeter functional is stated in [23] (which treats sets of
revolution in a half-space), claiming their smoothness except for a countable
set of points in the rotational axis. Notice that the Euler-Lagrange equation
of the functional in [23] is closely related, but not equal to, the nonlocal
mean curvature.

The following subsection addresses the symmetry and monotonicity properties of
minimizers.

###  1.1. Symmetry and monotonicity properties of minimizers

Theorem 1.1 contains two results on the symmetry and monotonicity properties
of minimizing sets: their cylindrical symmetry, (1.5), and the fact that $u$
is an even function which is nonincreasing in $(0,\pi).$ These properties will
follow from two rearrangement inequalities.

The first one concerns the cylindrical rearrangement, defined as follows. For
a measurable set $E\subset{\mathbb{R}}^{n}$, consider its sections
$E_{x_{1}}=\\{x^{\prime}\in{\mathbb{R}}^{n-1}:\,(x_{1},x^{\prime})\in E\\}$,
where $x_{1}\in{\mathbb{R}}$. Define $E^{*\\!\operatorname{cyl}}$ as the set
whose sections $(E^{*\\!\operatorname{cyl}})_{x_{1}}$ for every
$x_{1}\in{\mathbb{R}}$ are concentric open balls of ${\mathbb{R}}^{n-1}$
centered at $0$ and of the same $(n-1)$-dimensional Lebesgue measure as
$E_{x_{1}}$. The fact that the cylindrical rearrangement does not increase the
functional ${\mathcal{P}}_{s}$ follows from the Riesz rearrangement
inequality, and it was already used in [16, Proposition 13]. It will lead to
the existence of a cylindrical symmetric minimizer. A stronger fact is that
every minimizer is cylindrically symmetric, as stated in Theorem 1.1. This
will follow from a strict version of the Riesz rearrangement inequality which
analyzes the case of equality; see Lemma 3.1.

The second rearrangement that we need to consider is the symmetric decreasing
periodic rearrangement, which is defined in the following way. First, for a
measurable set $A\subset{\mathbb{R}}^{n}$ consider

$A^{*}:=\text{Steiner symmetrization of $A$ with respect to }\\{x_{1}=0\\}.$

This means that $A^{*}$ is symmetric with respect to the hyperplane
$\\{x_{1}=0\\}$ and that $A^{*}\cap\\{x^{\prime}=c\\}$ is an open interval of
the same $1$-dimensional Lebesgue measure as $A\cap\\{x^{\prime}=c\\}$ for
every $c\in{\mathbb{R}}^{n-1}.$ Now, given a set $E\subset{\mathbb{R}}^{n}$
which is $2\pi$-periodic in the $x_{1}$-variable we define its symmetric
decreasing periodic rearrangement by

$E^{*\\!\operatorname{per}}:=\text{$2\pi$-periodic extension of
}\left(E\cap\\{-\pi<x_{1}<\pi\\}\right)^{*},$

where the periodic extension is meant in the coordinate $x_{1}$. It is called
“symmetric decreasing” since the characteristic function
$\chi_{E^{*\\!\operatorname{per}}}$ is nonincreasing in $x_{1}$ within half of
the period, $(0,\pi)$, for each given $x^{\prime}\in{\mathbb{R}}^{n-1}$. Note
also that if $E$ is cylindrically symmetric, i.e., as in (1.5), then

$E^{*\\!\operatorname{per}}=\\{x\in{\mathbb{R}}^{n}\,:\,|x^{\prime}|<u^{*\\!\operatorname{per}}(x_{1})\\},$
(1.6)

where $u^{*\\!\operatorname{per}}$ is the $2\pi$-periodic function which is
even, nonincreasing in $(0,\pi)$, and equimeasurable with $u$ (see Section 3
for the precise definition).

Our second main result establishes that the periodic fractional perimeter
${\mathcal{P}}_{s}$ in (1.2) does not increase under symmetric decreasing
periodic rearrangement. This was left as an open problem in the last paragraph
of [16]. Our result also characterizes the case of equality.

######  Theorem 1.2.

Let $E\subset{\mathbb{R}}^{n}$ be a measurable set which is $2\pi$-periodic in
the coordinate $x_{1}$ and satisfies ${\mathcal{P}}_{s}[E]<+\infty.$

Then, ${\mathcal{P}}_{s}[E^{*\\!\operatorname{per}}]\leq{\mathcal{P}}_{s}[E].$
Moreover, if
${\mathcal{P}}_{s}[E^{*\\!\operatorname{per}}]={\mathcal{P}}_{s}[E]$ then, up
to a set of measure zero, $E=E^{*\\!\operatorname{per}}+ce_{1}$ for some
$c\in{\mathbb{R}}$, where $e_{1}:=(1,0,\ldots,0)\in{\mathbb{R}}^{n}$.

A key ingredient in the proof of Theorem 1.2 is a Riesz rearrangement
inequality on the circle ${\mathbb{S}}^{1}$, a result which is not as well-
known as the classical Riesz rearrangement inequality on ${\mathbb{R}}^{n}$.
Such periodic rearrangement inequality is originally due to Baernstein and
Taylor [7] and to Friedberg and Luttinger [19], who proved independently
different versions of the inequality around the same time; see more details in
the comments following Theorem 3.2. The case of equality in the Riesz
rearrangement inequality on the circle (and more generally on the sphere
$\mathbb{S}^{n}$) was first treated by Burchard and Hajaiej in [8].

###  1.2. On the shape of minimizers: stability of straight cylinders

Let us first recall what is known about the shape of minimizers in the local
case, that is, for the classical isoperimetric problem relative to a
slab.666In the local case, the isoperimetric problem in a slab is commonly
formulated without a periodicity condition. To relate this setting with the
periodic one, one considers an even reflection of the slab minimizer. In this
way, a period corresponds to twice the width of the slab. It turns out that,
for $n\leq 8$, the minimizers are either spheres or straight cylinders,
depending on whether the volume constraint is smaller or greater,
respectively, than a critical volume; see [4, 27] or [20, Theorem 6] for
$n=3$, and [25] for $n\leq 8$. In particular, the classical Delaunay unduloids
are not minimizers for $n\leq 8$. Furthermore, they are known to be unstable,
as proven independently in [4, 27] for $n=3$, and in [25] for $3\leq n\leq 8.$
In fact, when $n\leq 8$ the infinite array of balls and the cylinders are the
only stable periodic solutions to the constant mean curvature equation; see
[25, Corollary 5.4].

When $n\geq 10$ it has been proven in [25, Proposition 3.4] that there exist
unduloids which are minimizers under certain volume constraints (namely for
volumes which are close to that of a ball that has the diameter of one
period). For large volumes, however, cylinders are the only minimizers in all
dimensions, as shown in [26, Theorem 1.1]. For $n=9$, [21, Section 5] provides
numerical evidence that there exist stable unduloids in this dimension (as it
is the case for $n\geq 10$).

A simpler fact is to characterize the stability of straight cylinders, still
in the local case. A cylinder
$\\{(x_{1},x^{\prime})\in{\mathbb{R}}^{n}:\,|x^{\prime}|<R\\}$ is stable if
and only if

$R^{2}\geq n-2,$ (1.7)

as can be verified by a simple calculation (see [4] in the case $n=3$ and use
the Euler-Lagrange equation [21, Equation (4)] for general $n$).

In view of these facts for the local case, and at least in low dimensions, one
may expect the minimizers of Theorem 1.1 to be the near-balls found in [12]
for small volumes and straight cylinders for large volumes. To sustain this
statement for small volumes (which is still an open problem), recall first
that the function $u$ defining the minimizer in (1.5) could be identically
zero in a subinterval $(a,\pi)$ of $(0,\pi)$ —allowing the minimizer, in this
way, to be a periodic array of near-balls. Recall also that [16, Theorem 3]
proved that, for small volumes, a minimizer must be close (in a measure
theoretical sense) to a periodic array of round balls. On the other hand, the
periodic array of near-spheres found in [12] are obtained as small
perturbations of a single round sphere, and hence remains as a stable
configuration.

With regard to large volumes, in [9] we will establish the following result on
the stability of straight cylinders. Its conclusion supports the conjecture
that the only minimizers for large volumes should be straight cylinders. By
stability we mean that the second variation of ${\mathcal{P}}_{s}[E_{t}]$ is
nonnegative for all $2\pi$-periodic volume preserving variations
$\\{E_{t}\\}_{t\in(-\epsilon,\epsilon)}$ of the set $E$.777The proof of
Theorem 1.3 in [9] will rely mainly on the techniques developed in [10, 11].
In particular, it will involve rather lengthy technical computations. This is
the reason why we present the proof of this third theorem in a separate paper.

######  Theorem 1.3 ([9]).

Let $n\geq 2$ and $s\in(0,1)$.

Then, there exists a radius $R_{s}>0$ (depending only on $n$ and $s$) such
that the set $E=\\{(x_{1},x^{\prime})\in{\mathbb{R}}^{n}:\,|x^{\prime}|<R\\}$
is stable (for ${\mathcal{P}}_{s}$ and among sets which are $2\pi$-periodic in
the $x_{1}$ direction) if and only if $R\geq R_{s}$. Moreover,

$\lim_{s\uparrow 1}R^{2}_{s}=n-2.$

Observe that the last statement fits with (1.7), also in the case $n=2$ —a
dimension for which all above questions are trivial in the local case, but not
in the nonlocal case. The proof of Theorem 1.3 in [9] uses tools from [10, 11]
mainly, but also the symmetric decreasing periodic rearrangement of the
present paper. Indeed, we will use the fact that, thanks to Theorem 1.2, to
show stability we will only need to consider even variations of $E.$ In this
way, we will be able to use the basis $\cos(k\cdot)$, $k\in\mathbb{Z},$ of
even eigenfunctions to the linearized operator of ${\mathcal{P}}_{s}$ at a
straight cylinder $E.$

###  1.3. Plan of the paper

  * —

In Section 2 we prove Theorem 1.1, even though we will borrow the results of
Section 3 only to prove the symmetry and monotonicity properties of minimizers
claimed in the equality (1.5) of $(i)$ and in $(ii)$.

  * —

Section 3 deals with the behaviour of ${\mathcal{P}}_{s}$ under cylindrical
rearrangement and under symmetric decreasing periodic rearrangement. In
particular, here we prove Theorem 1.2.

  * —

In Appendix A we present a more explicit expression for ${\mathcal{P}}_{s}$
when $n=2$. This gives an alternative proof of the existence part in Theorem
1.1 for $n=2$.

  * —

In Appendix B we provide a simple proof of a well-known monotonicity property
of the fundamental solution of the one-dimensional heat equation under
periodic boundary conditions. The result is crucially used in the proof of
Theorem 1.2.

##  2\. Existence of periodic minimizers

In this section we establish Theorem 1.1. Before stating its key lemma, we
first prove an elementary one which will be used several times in the paper.

######  Lemma 2.1.

Let $m$ and $l$ be two positive integers, $u=(u_{1},\ldots,u_{m})$ and
$v=(v_{1},\ldots,v_{l})$ be $2\pi$-periodic functions in ${\mathbb{R}}$ with
values in ${\mathbb{R}}^{m}$ and ${\mathbb{R}}^{l}$, respectively. Suppose
that
$\Phi:{\mathbb{R}}^{m}\times{\mathbb{R}}^{l}\times[0,+\infty)\to{\mathbb{R}}$
is such that the function $(x,y)\mapsto\Phi(u(x),v(y),|x-y|)$ belongs to
$L^{1}((-\pi,\pi)\times{\mathbb{R}}).$

Then, this last function belongs also to $L^{1}({\mathbb{R}}\times(-\pi,\pi))$
and

$\int_{-\pi}^{\pi}\\!dx\int_{{\mathbb{R}}}\\!dy\;\Phi\big{(}u(x),v(y),|x-y|\big{)}=\int_{{\mathbb{R}}}\\!dx\int_{-\pi}^{\pi}\\!dy\;\Phi\big{(}u(x),v(y),|x-y|\big{)}.$
(2.1)

###### Proof.

In the integral on the left-hand side of (2.1), both for $|\Phi|$ and for
$\Phi$, we can write the domain of integration ${\mathbb{R}}$ as the union
over all $\big{(}(2k-1)\pi,(2k+1)\pi\big{)},$ where $k\in{\mathbb{Z}}$. Thus
the left-hand side is the countable sum of integrals over the domains
$(x,y)\in(-\pi,\pi)\times\big{(}(2k-1)\pi,(2k+1)\pi\big{)}$. For each
$k\in{\mathbb{Z}}$ we carry out the change of variables $x=\overline{x}+2k\pi$
and $y=\overline{y}+2k\pi,$ where
$(\overline{x},\overline{y})\in\big{(}(-2k-1)\pi,(-2k+1)\pi\big{)}\times(-\pi,\pi).$
Now, the periodicity of $u$ and $v$, and the identity
$|x-y|=|\overline{x}-\overline{y}|$, yield the claim of the lemma. ∎

The following lemma will be a key ingredient in the proof of existence of
minimizer.

######  Lemma 2.2.

Let $n\geq 2$ and $E\subset{\mathbb{R}}^{n}$ be a set of the form

$E=\\{x\in{\mathbb{R}}^{n}:\,|x^{\prime}|<u(x_{1})\\}$ (2.2)

with $u:{\mathbb{R}}\to{\mathbb{R}}$ measurable, nonnegative, and
$2\pi$-periodic in $x_{1}$.

Then,

${\mathcal{P}}_{s}(u):={\mathcal{P}}_{s}[E]=\int_{-\pi}^{\pi}\\!dx_{1}\int_{-\infty}^{+\infty}\\!dy_{1}\,{|x_{1}-y_{1}|^{n-2-s}}\,\phi\Big{(}\frac{u(x_{1})}{|x_{1}-y_{1}|},\frac{u(y_{1})}{|x_{1}-y_{1}|}\Big{)},$
(2.3)

where, for nonnegative numbers $p$ and $q$,

$\phi(p,q):=\int_{\\{w^{\prime}\in{\mathbb{R}}^{n-1}:\,|w^{\prime}|<p\\}}\\!dw^{\prime}\int_{\\{z^{\prime}\in{\mathbb{R}}^{n-1}:\,|z^{\prime}|>q\\}}\\!dz^{\prime}\,(1+|w^{\prime}-z^{\prime}|^{2})^{-\frac{n+s}{2}}.$
(2.4)

Moreover, we have

$\phi(p,q)\geq c_{0}(p^{n-1}-q^{n-1})-2c_{0}\qquad\text{if }0\leq q\leq p,$
(2.5)

for some constant $c_{0}>0$ depending only on $n$.

###### Proof.

From (1.2), we have

$\begin{split}{\mathcal{P}}_{s}[E]&=\int_{-\pi}^{\pi}\\!dx_{1}\int_{-\infty}^{+\infty}\\!dy_{1}\int_{\\{x^{\prime}\in{\mathbb{R}}^{n-1}:\,|x^{\prime}|<u(x_{1})\\}}\\!dx^{\prime}\\\
&\hskip
98.0pt\int_{\\{y^{\prime}\in{\mathbb{R}}^{n-1}:\,|y^{\prime}|>u(y_{1})\\}}\\!dy^{\prime}\,(|x_{1}-y_{1}|^{2}+|x^{\prime}-y^{\prime}|^{2})^{-\frac{n+s}{2}}\\\
&=\int_{-\pi}^{\pi}\\!dx_{1}\int_{-\infty}^{+\infty}\\!dy_{1}\,|x_{1}-y_{1}|^{-n-s}\int_{\\{x^{\prime}\in{\mathbb{R}}^{n-1}:\,|x^{\prime}|<u(x_{1})\\}}\\!dx^{\prime}\\\
&\hskip
98.0pt\int_{\\{y^{\prime}\in{\mathbb{R}}^{n-1}:\,|y^{\prime}|>u(y_{1})\\}}\\!dy^{\prime}\,\Big{(}1+\Big{|}\frac{x^{\prime}}{|x_{1}-y_{1}|}-\frac{y^{\prime}}{|x_{1}-y_{1}|}\Big{|}^{2}\Big{)}^{-\frac{n+s}{2}}\\\
&=\int_{-\pi}^{\pi}\\!dx_{1}\int_{-\infty}^{+\infty}\\!dy_{1}\,|x_{1}-y_{1}|^{n-2-s}\int_{\big{\\{}w^{\prime}\in{\mathbb{R}}^{n-1}:\,|w^{\prime}|<\frac{u(x_{1})}{|x_{1}-y_{1}|}\big{\\}}}\\!dw^{\prime}\\\
&\hskip
98.0pt\int_{\big{\\{}z^{\prime}\in{\mathbb{R}}^{n-1}:\,|z^{\prime}|>\frac{u(y_{1})}{|x_{1}-y_{1}|}\big{\\}}}\\!dz^{\prime}\,(1+|w^{\prime}-z^{\prime}|^{2})^{-\frac{n+s}{2}},\end{split}$
(2.6)

which gives (2.3)-(2.4).

Next, to prove (2.5) for $0\leq q\leq p$, we see that

$\begin{split}\phi(p,q)&=\int_{\\{w^{\prime}\in{\mathbb{R}}^{n-1}:\,|w^{\prime}|<p\\}}\\!dw^{\prime}\int_{q}^{+\infty}\\!dr\,r^{n-2}\int_{{\mathbb{S}}^{n-2}}\\!d\mathcal{H}^{n-2}(\sigma)\,(1+|w^{\prime}-r\sigma|^{2})^{-\frac{n+s}{2}}\\\
&=\int_{q}^{+\infty}\\!dr\,r^{n-2}\int_{{\mathbb{S}}^{n-2}}\\!d\mathcal{H}^{n-2}(\sigma)\,\int_{\\{w^{\prime}\in{\mathbb{R}}^{n-1}:\,|w^{\prime}|<p\\}}\\!dw^{\prime}\,(1+|w^{\prime}-r\sigma|^{2})^{-\frac{n+s}{2}}\\\
&=|{\mathbb{S}}^{n-2}|\int_{q}^{+\infty}\\!dr\,r^{n-2}\int_{\\{w^{\prime}\in{\mathbb{R}}^{n-1}:\,|w^{\prime}|<p\\}}\\!dw^{\prime}\,(1+|w^{\prime}-re_{1}^{\prime}|^{2})^{-\frac{n+s}{2}}\\\
&\geq|{\mathbb{S}}^{n-2}|\int_{q}^{p}\\!dr\,r^{n-2}\int_{\\{w^{\prime}\in{\mathbb{R}}^{n-1}:\,|w^{\prime}|<p\\}}\\!dw^{\prime}\,(1+|w^{\prime}-re_{1}^{\prime}|^{2})^{-\frac{n+s}{2}},\end{split}$
(2.7)

where $e_{1}^{\prime}=(1,0,\ldots,0)\in{\mathbb{R}}^{n-1}$.

We now claim that, for some constant $c>0$ depending only on $n$,888Here and
throughout the paper, $B_{r}^{k}(y)$ denotes the open ball in
${\mathbb{R}}^{k}$ of radius $r$ centered at $y\in{\mathbb{R}}^{k}$. We also
use the notation $B_{r}^{k}$ when $y=0$, and $B_{r}$ if in addition $k=n$.

$|B^{n-1}_{1}(re_{1}^{\prime})\cap B^{n-1}_{p}|\geq c\qquad\text{if }1\leq
r\leq p.$ (2.8)

This is simple. We have $B^{n-1}_{p}\supset B^{n-1}_{r}\supset
B^{n-1}_{1}((r-1)e_{1}^{\prime}).$ Hence,

$|B^{n-1}_{1}(re_{1}^{\prime})\cap
B^{n-1}_{p}|\geq|B^{n-1}_{1}(re_{1}^{\prime})\cap
B^{n-1}_{1}((r-1)e_{1}^{\prime})|=|B^{n-1}_{1}\cap
B^{n-1}_{1}(-e_{1}^{\prime})|=:c.$ (2.9)

Therefore, from (2.7) and (2.8) we deduce that, for $1\leq q\leq p$,

$\begin{split}\phi(p,q)&\geq|{\mathbb{S}}^{n-2}|\int_{q}^{p}\\!dr\,r^{n-2}\int_{B^{n-1}_{1}(re_{1}^{\prime})\cap
B^{n-1}_{p}}\\!dw^{\prime}\,(1+|w^{\prime}-re_{1}^{\prime}|^{2})^{-\frac{n+s}{2}}\\\
&\geq
2^{-\frac{n+s}{2}}|{\mathbb{S}}^{n-2}|\int_{q}^{p}\\!dr\,r^{n-2}|B^{n-1}_{1}(re_{1}^{\prime})\cap
B^{n-1}_{p}|\\\ &\geq
2^{-\frac{n+1}{2}}|{\mathbb{S}}^{n-2}|c\int_{q}^{p}\\!dr\,r^{n-2}=:2c_{0}(p^{n-1}-q^{n-1}).\end{split}$
(2.10)

We conclude that (2.5) holds if $1\leq q\leq p$.

Next, if $q<1$ and $p^{n-1}\geq 2$, we see that $\phi(p,q)\geq\phi(p,1)\geq
2c_{0}(p^{n-1}-1)$ by definition of $\phi$ and the previous bound. But
$p^{n-1}-1\geq(p^{n-1}-q^{n-1})/2$ since we assumed $p^{n-1}\geq 2$. Thus, in
this case (2.5) also holds.

It only remains to consider the case $0\leq q<1$ and $p^{n-1}<2$. But then we
have

$c_{0}(p^{n-1}-q^{n-1})-2c_{0}\leq c_{0}p^{n-1}-2c_{0}<0\leq\phi(p,q),$ (2.11)

which finishes the proof. ∎

We now establish Theorem 1.1. The proof of existence of a minimizer relies
only on the two previous basic lemmas and on the fact (already used in [16])
that ${\mathcal{P}}_{s}$ does not increase under cylindrical rearrangement (as
stated in Lemma 3.1). Instead, the proof of the eveness and nonincreasing
property of any minimizer uses our new result Theorem 1.2 on the periodic
rearrangement.

###### Proof of Theorem 1.1.

We split the proof into four parts.

Part 1 $($Existence of minimizer$)$. Here we prove the existence of a
minimizer. Note first that ${\mathcal{P}}_{s}\geq 0$ and that
${\mathcal{P}}_{s}\not\equiv+\infty$ —this last fact will be easily seen in
Part 4 of the proof when we check that ${\mathcal{P}}_{s}$ is finite when
computed on a periodic array of balls.

Given $\mu>0,$ we take a minimizing sequence of sets with volume $\mu$ within
the slab $\\{-\pi<x_{1}<\pi\\}$. In view of Lemma 3.1, the sets may be assumed
to be of the form

$E_{k}=\\{x\in{\mathbb{R}}^{n}:\,|x^{\prime}|<u_{k}(x_{1})\\}$ (2.12)

for some measurable, nonnegative, and $2\pi$-periodic functions $u_{k}$.

By Lemma 2.2 we have that, for the constant $c_{0}$ in (2.5) and some positive
constant $C_{n,s}$ depending only on $n$ and $s$,

$\begin{split}\int_{-\pi}^{\pi}\\!dx_{1}&\int_{-\pi}^{\pi}\\!dy_{1}\,{|x_{1}-y_{1}|^{n-2-s}}\Big{\\{}c_{0}+\phi\Big{(}\frac{u_{k}(x_{1})}{|x_{1}-y_{1}|},\frac{u_{k}(y_{1})}{|x_{1}-y_{1}|}\Big{)}\Big{\\}}\\\
&=C_{n,s}\,c_{0}+\int_{-\pi}^{\pi}\\!dx_{1}\int_{-\pi}^{\pi}\\!dy_{1}\,{|x_{1}-y_{1}|^{n-2-s}}\phi\Big{(}\frac{u_{k}(x_{1})}{|x_{1}-y_{1}|},\frac{u_{k}(y_{1})}{|x_{1}-y_{1}|}\Big{)}\\\
&\leq C_{n,s}\,c_{0}+{\mathcal{P}}_{s}[E_{k}]\leq C\end{split}$ (2.13)

since $\phi\geq 0$, where $C$ is a constant independent of $k$. From this,
symmetrizing in $x_{1}$ and $y_{1}$, using (2.5) and, again, $\phi\geq 0$, we
see that

$\begin{split}C&\geq\int_{-\pi}^{\pi}\\!dx_{1}\int_{-\pi}^{\pi}\\!dy_{1}\,{|x_{1}-y_{1}|^{n-2-s}}\Big{\\{}c_{0}+\phi\Big{(}\frac{u_{k}(x_{1})}{|x_{1}-y_{1}|},\frac{u_{k}(y_{1})}{|x_{1}-y_{1}|}\Big{)}\Big{\\}}\\\
&=\frac{1}{2}\int_{-\pi}^{\pi}\\!dx_{1}\int_{-\pi}^{\pi}\\!dy_{1}\,{|x_{1}-y_{1}|^{n-2-s}}\Big{\\{}c_{0}+\phi\Big{(}\frac{u_{k}(x_{1})}{|x_{1}-y_{1}|},\frac{u_{k}(y_{1})}{|x_{1}-y_{1}|}\Big{)}\Big{\\}}\\\
&\hskip
80.0pt+\frac{1}{2}\int_{-\pi}^{\pi}\\!dx_{1}\int_{-\pi}^{\pi}\\!dy_{1}\,{|x_{1}-y_{1}|^{n-2-s}}\Big{\\{}c_{0}+\phi\Big{(}\frac{u_{k}(y_{1})}{|x_{1}-y_{1}|},\frac{u_{k}(x_{1})}{|x_{1}-y_{1}|}\Big{)}\Big{\\}}\\\
&\geq\frac{1}{2}\int_{-\pi}^{\pi}\\!dx_{1}\int_{-\pi}^{\pi}\\!dy_{1}\,{|x_{1}-y_{1}|^{n-2-s}}\\\
&\hskip
80.0pt\times\Big{\\{}2c_{0}+\phi\Big{(}\frac{\max\\{u_{k}(x_{1}),u_{k}(y_{1})\\}}{|x_{1}-y_{1}|},\frac{\min\\{u_{k}(x_{1}),u_{k}(y_{1})\\}}{|x_{1}-y_{1}|}\Big{)}\Big{\\}}\\\
&\geq\frac{c_{0}}{2}\int_{-\pi}^{\pi}\\!dx_{1}\int_{-\pi}^{\pi}\\!dy_{1}\,{|x_{1}-y_{1}|^{n-2-s}}\frac{|u_{k}(x_{1})^{n-1}-u_{k}(y_{1})^{n-1}|}{|x_{1}-y_{1}|^{n-1}}\\\
&=\frac{c_{0}}{2}\int_{-\pi}^{\pi}\\!dx_{1}\int_{-\pi}^{\pi}\\!dy_{1}\,\frac{|u_{k}(x_{1})^{n-1}-u_{k}(y_{1})^{n-1}|}{|x_{1}-y_{1}|^{1+s}}.\end{split}$
(2.14)

This gives a uniform bound on the $W^{s,1}(-\pi,\pi)$ seminorm of the
functions $u_{k}^{n-1}$. At the same time, thanks to our volume constraint, we
know that $|B_{1}^{n-1}|\int_{-\pi}^{\pi}dx_{1}\,u_{k}(x_{1})^{n-1}=\mu$ for
all $k$. Therefore, by the fractional Sobolev compactness theorem (see [18,
Theorem 7.1]), a subsequence of $\\{u_{k}^{n-1}\\}$ converges strongly in
$L^{1}(-\pi,\pi)$ to a nonnegative function $v\in W^{s,1}(-\pi,\pi)$. As a
consequence, if we set $u:=v^{1/(n-1)}$, then
$|B_{1}^{n-1}|\int_{-\pi}^{\pi}dx_{1}\,u(x_{1})^{n-1}=\mu$. Thus, extending
$u$ to be $2\pi$-periodic in ${\mathbb{R}}$, $u$ is an admissible competitor.
Finally, since the function $\phi$ in (2.3) is nonnegative, Fatou’s lemma
gives that $E:=\\{x\in{\mathbb{R}}^{n}:\,|x^{\prime}|<u(x_{1})\\}$ is a
minimizer.

The fact that $v=u^{n-1}\in W^{s,1}(-\pi,\pi)$ yields $u\in
W^{s/(n-1),n-1}(-\pi,\pi)$, by simply using that
$|t-s|^{n-1}\leq|t^{n-1}-s^{n-1}|$ for all nonnegative numbers $t$ and $s$.

Part 2 $($Symmetry of minimizers$).$ To prove $(i)$ in Theorem 1.1, let $E$ be
any minimizer. As in Part 1 above, by Lemma 3.1 we must have
$E=\\{|x^{\prime}|<u(x_{1})\\}$ for some nonnegative $2\pi$-periodic function
$u$. By (2.14) (with $u_{k}$ replaced by $u$), we see that $u^{n-1}\in
W^{s,1}(-\pi,\pi)$. As before, this leads to $u\in W^{s/(n-1),n-1}(-\pi,\pi)$.

Next, statement $(ii)$ in the theorem follows immediately from Theorem 1.2.

Part 3 $($Euler-Lagrange equation$)$. In this part we address the proof of
Theorem 1.1 $(iii)$. Firstly, we will establish that if
$E=\\{x\in{\mathbb{R}}^{n}:\,|x^{\prime}|<u(x_{1})\\}$ is a minimizer,
$x_{1}^{0}\in{\mathbb{R}}$ is such that $u$ is $C^{1,\alpha}$ in a
neighborhood $U(x_{1}^{0})$ of $x_{1}^{0}$ for some $\alpha>s$, and
$u(x_{1}^{0})>0$, then the nonlocal mean curvature of $\partial E$ is constant
in a neighborhood of $x_{1}^{0}$. Afterwards, we will show that the constant
does not depend on the point $x_{1}^{0}$ in whose neighborhood $u$ is
$C^{1,\alpha}$.

Note first that we may assume $x_{1}^{0}\in(-\pi,\pi)$, by the periodicity of
$E$. We take any smooth function $\xi=\xi(x_{1})$ with compact support in the
neighborhood $U(x_{1}^{0})$ of $x_{1}^{0}$, small enough to be contained in
$(-\pi,\pi)$, and we extend $\xi$ to be $2\pi$-periodic. The first variation
of the volume functional —that is, the derivative of
$|B_{1}^{n-1}|\int_{-\pi}^{\pi}dx_{1}\,(u(x_{1})+\epsilon\xi(x_{1}))^{n-1}$
with respect to $\epsilon$ evaluated at $\epsilon=0$— is

$(n-1)|B_{1}^{n-1}|\int_{-\pi}^{\pi}dx_{1}\,u(x_{1})^{n-2}\xi(x_{1}),$ (2.15)

while that of ${\mathcal{P}}_{s}$ is

$\begin{split}\frac{d}{d\epsilon}\biggr{\rvert}_{\epsilon=0}{\mathcal{P}}_{s}(u+\epsilon\xi)=\int_{-\pi}^{\pi}\\!dx_{1}\int_{-\infty}^{+\infty}\\!dy_{1}&\,{|x_{1}-y_{1}|^{n-2-s}}\Big{\\{}\phi_{p}\Big{(}\frac{u(x_{1})}{|x_{1}-y_{1}|},\frac{u(y_{1})}{|x_{1}-y_{1}|}\Big{)}\frac{\xi(x_{1})}{|x_{1}-y_{1}|}\\\
&\phantom{aaaaaaaaaa}+\phi_{q}\Big{(}\frac{u(x_{1})}{|x_{1}-y_{1}|},\frac{u(y_{1})}{|x_{1}-y_{1}|}\Big{)}\frac{\xi(y_{1})}{|x_{1}-y_{1}|}\Big{\\}}.\end{split}$
(2.16)

We next see that this expression leads to the nonlocal mean curvature of
$\partial E$ at $x=(x_{1},x^{\prime})$. In what follows, all expressions can
be checked to be well defined (as for the nonlocal mean curvature of general,
nonperiodic, $C^{1,\alpha}$ sets) using that $u$ is $C^{1,\alpha}$, with
$\alpha>s$, in the neighborhood of $x_{1}^{0}$ where $\xi$ has its support.

Since, by Lemma 2.1 (used with $m=1$ and $l=2$),

$\begin{split}\int_{-\pi}^{\pi}\\!dx_{1}&\int_{-\infty}^{+\infty}\\!dy_{1}\,\frac{\xi(y_{1})}{|x_{1}-y_{1}|^{3+s-n}}\,\phi_{q}\Big{(}\frac{u(x_{1})}{|x_{1}-y_{1}|},\frac{u(y_{1})}{|x_{1}-y_{1}|}\Big{)}\\\
&=\int_{-\infty}^{+\infty}\\!dx_{1}\int_{-\pi}^{\pi}\\!dy_{1}\,\frac{\xi(y_{1})}{|x_{1}-y_{1}|^{3+s-n}}\,\phi_{q}\Big{(}\frac{u(x_{1})}{|x_{1}-y_{1}|},\frac{u(y_{1})}{|x_{1}-y_{1}|}\Big{)}\\\
&=\int_{-\pi}^{\pi}\\!dx_{1}\int_{-\infty}^{+\infty}\\!dy_{1}\,\frac{\xi(x_{1})}{|x_{1}-y_{1}|^{3+s-n}}\phi_{q}\Big{(}\frac{u(y_{1})}{|x_{1}-y_{1}|},\frac{u(x_{1})}{|x_{1}-y_{1}|}\Big{)},\end{split}$
(2.17)

we deduce that (2.16) is equal to

$\begin{split}\int_{-\pi}^{\pi}\\!dx_{1}\int_{-\infty}^{+\infty}\\!dy_{1}\,\frac{\xi(x_{1})}{|x_{1}-y_{1}|^{3+s-n}}\Big{\\{}\phi_{p}\Big{(}\frac{u(x_{1})}{|x_{1}-y_{1}|},\frac{u(y_{1})}{|x_{1}-y_{1}|}\Big{)}+\phi_{q}\Big{(}\frac{u(y_{1})}{|x_{1}-y_{1}|},\frac{u(x_{1})}{|x_{1}-y_{1}|}\Big{)}\Big{\\}}.\end{split}$
(2.18)

Thus, by definition of $\phi$, this last expression becomes

$\begin{split}\int_{-\pi}^{\pi}\\!d&x_{1}\,\xi(x_{1})\int_{-\infty}^{+\infty}\\!dy_{1}\,{|x_{1}-y_{1}|^{n-3-s}}\Big{\\{}\phi_{p}\Big{(}\frac{u(x_{1})}{|x_{1}-y_{1}|},\frac{u(y_{1})}{|x_{1}-y_{1}|}\Big{)}+\phi_{q}\Big{(}\frac{u(y_{1})}{|x_{1}-y_{1}|},\frac{u(x_{1})}{|x_{1}-y_{1}|}\Big{)}\Big{\\}}\\\
&=\int_{-\pi}^{\pi}\\!dx_{1}\,\xi(x_{1})\int_{-\infty}^{+\infty}\\!dy_{1}\,{|x_{1}-y_{1}|^{n-3-s}}\\\
&\hskip
25.0pt\times\Big{\\{}\int_{\big{\\{}w^{\prime}\in{\mathbb{R}}^{n-1}:\,|w^{\prime}|=\frac{u(x_{1})}{|x_{1}-y_{1}|}\big{\\}}}\\!dw^{\prime}\int_{\big{\\{}z^{\prime}\in{\mathbb{R}}^{n-1}:\,|z^{\prime}|>\frac{u(y_{1})}{|x_{1}-y_{1}|}\big{\\}}}\\!dz^{\prime}\,(1+|w^{\prime}-z^{\prime}|^{2})^{-\frac{n+s}{2}}\\\
&\hskip
35.0pt-\int_{\big{\\{}w^{\prime}\in{\mathbb{R}}^{n-1}:\,|w^{\prime}|<\frac{u(y_{1})}{|x_{1}-y_{1}|}\big{\\}}}\\!dw^{\prime}\int_{\big{\\{}z^{\prime}\in{\mathbb{R}}^{n-1}:\,|z^{\prime}|=\frac{u(x_{1})}{|x_{1}-y_{1}|}\big{\\}}}\\!dz^{\prime}\,(1+|w^{\prime}-z^{\prime}|^{2})^{-\frac{n+s}{2}}\Big{\\}}\\\
&=|{\mathbb{S}}^{n-2}|\int_{-\pi}^{\pi}\\!dx_{1}\xi(x_{1})u(x_{1})^{n-2}\int_{-\infty}^{+\infty}\\!dy_{1}\,{|x_{1}-y_{1}|^{n-3-s}}\Big{(}\frac{1}{|x_{1}-y_{1}|}\Big{)}^{n-2}\\\
&\hskip
57.0pt\times\Big{\\{}\int_{\big{\\{}z^{\prime}\in{\mathbb{R}}^{n-1}:\,|z^{\prime}|>\frac{u(y_{1})}{|x_{1}-y_{1}|}\big{\\}}}\\!dz^{\prime}\,\Big{(}1+\Big{|}z^{\prime}-\frac{u(x_{1})e_{1}^{\prime}}{|x_{1}-y_{1}|}\Big{|}^{2}\Big{)}^{-\frac{n+s}{2}}\\\
&\hskip
67.0pt-\int_{\big{\\{}w^{\prime}\in{\mathbb{R}}^{n-1}:\,|w^{\prime}|<\frac{u(y_{1})}{|x_{1}-y_{1}|}\big{\\}}}\\!dw^{\prime}\Big{(}1+\Big{|}w^{\prime}-\frac{u(x_{1})e_{1}^{\prime}}{|x_{1}-y_{1}|}\Big{|}^{2}\Big{)}^{-\frac{n+s}{2}}\Big{\\}}.\end{split}$
(2.19)

The integral on $dy_{1}$ over $(-\infty,+\infty)$ is equal to

$\begin{split}\int_{-\infty}^{+\infty}&\\!dy_{1}\,{|x_{1}-y_{1}|^{-1-s}}\\\
&\hskip
10.0pt\times\Big{\\{}\int_{\\{\overline{z}^{\prime}\in{\mathbb{R}}^{n-1}:\,|\overline{z}^{\prime}|>{u(y_{1})}\\}}\\!d\overline{z}^{\prime}\,|x_{1}-y_{1}|^{1-n}\Big{(}1+\Big{|}\frac{\overline{z}^{\prime}}{|x_{1}-y_{1}|}-\frac{u(x_{1})e_{1}^{\prime}}{|x_{1}-y_{1}|}\Big{|}^{2}\Big{)}^{-\frac{n+s}{2}}\\\
&\hskip
30.0pt-\int_{\\{\overline{w}^{\prime}\in{\mathbb{R}}^{n-1}:\,|\overline{w}^{\prime}|<{u(y_{1})}\\}}\\!d\overline{w}^{\prime}\,|x_{1}-y_{1}|^{1-n}\Big{(}1+\Big{|}\frac{\overline{w}^{\prime}}{|x_{1}-y_{1}|}-\frac{u(x_{1})e_{1}^{\prime}}{|x_{1}-y_{1}|}\Big{|}^{2}\Big{)}^{-\frac{n+s}{2}}\Big{\\}}\\\
=&\int_{-\infty}^{+\infty}\\!dy_{1}\,\Big{\\{}\int_{\\{\overline{z}^{\prime}\in{\mathbb{R}}^{n-1}:\,|\overline{z}^{\prime}|>{u(y_{1})}\\}}\\!d\overline{z}^{\prime}\,(|x_{1}-y_{1}|^{2}+|u(x_{1})e_{1}^{\prime}-\overline{z}^{\prime}|^{2})^{-\frac{n+s}{2}}\\\
&\hskip
55.0pt-\int_{\\{\overline{w}^{\prime}\in{\mathbb{R}}^{n-1}:\,|\overline{w}^{\prime}|<{u(y_{1})}\\}}\\!d\overline{w}^{\prime}\,(|x_{1}-y_{1}|^{2}+|u(x_{1})e_{1}^{\prime}-\overline{w}^{\prime}|^{2})^{-\frac{n+s}{2}}\Big{\\}}.\end{split}$
(2.20)

This last expression is, by definition, the nonlocal mean curvature of
$\partial E$ at the point
$x=(x_{1},u(x_{1})e_{1}^{\prime})=(x_{1},u(x_{1}),0,\ldots,0)$, which is
defined by

$H_{s}[E](x)=\int_{{\mathbb{R}}^{n}}dy\,\frac{\chi_{E^{c}}(y)-\chi_{E}(y)}{|x-y|^{n+s}}.$
(2.21)

Thus, by (2.16), we have shown that

$\frac{d}{d\epsilon}\biggr{\rvert}_{\epsilon=0}{\mathcal{P}}_{s}(u+\epsilon\xi)=|{\mathbb{S}}^{n-2}|\int_{-\pi}^{\pi}\\!dx_{1}\,\xi(x_{1})u(x_{1})^{n-2}H_{s}[E](x_{1},u(x_{1})e_{1}^{\prime}).$
(2.22)

From this, (2.15), and the Lagrange multiplier rule,999Note that we are
actually using the Lagrange multiplier rule not for the original functional
${\mathcal{P}}_{s}$ but for another functional $\overline{{\mathcal{P}}}_{s}$
given by the restriction of ${\mathcal{P}}_{s}$ to functions which agree with
$u$ outside of the set where $u$ is regular. The Lagrange multiplier rule
applies to $\overline{{\mathcal{P}}}_{s}$, since $u$ is also a minimizer for
$\overline{{\mathcal{P}}}_{s}$ and $\overline{{\mathcal{P}}}_{s}$ is
differentiable. we see that, for some constant $\lambda\in{\mathbb{R}}$,

$H_{s}[E](x_{1},u(x_{1})e_{1}^{\prime})=\lambda\quad\text{ for all }x_{1}\in
U(x_{1}^{0}).$

It remains to show that $\lambda$ does not depend on $x_{1}^{0}$. For this
purpose, let $U(x_{1}^{0})$ and $U(y_{1}^{0})$ be two neighborhoods of two
different points $x^{0}_{1}$ and $y^{0}_{1}$ where $u$ is regular. Now, we can
simply repeat the previous argument by taking this time a smooth function
$\xi$ with compact support in the union $U(x_{1}^{0})\cup U(y_{1}^{0})$.
Taking such variations $\xi$ leads to the conclusion that
$H_{s}[E](x_{1},u(x_{1})e_{1}^{\prime})$ is constant in $U(x_{1}^{0})\cup
U(y_{1}^{0})$. This proves Theorem 1.1 $(iii)$.

Part 4 $($Small volumes$).$ We finally address the proof of Theorem 1.1
$(iv)$. That is, we prove (uniformly as $s\uparrow 1$ and $s\downarrow 0$)
that if the volume constraint $\mu$ is small enough, any minimizer $E$ of
${\mathcal{P}}_{s}$ is not a straight cylinder. To do so, we simply use an
energy comparison with the configuration ${\mathcal{B}}_{r}$ given by a
periodic array of disjoint balls

${\mathcal{B}}_{r}:=\\{x\in{\mathbb{R}}^{n}:\,|x-2k\pi e_{1}|<r\text{ for some
}k\in{\mathbb{Z}}\\},$ (2.23)

where $e_{1}=(1,0,\ldots,0)\in{\mathbb{R}}^{n}$ and $r<\pi$. To satisfy the
volume constraint, the radius $r=r(\mu)$ of the balls must be taken to satisfy
$|B_{1}|\,r(\mu)^{n}=\mu$.

Now, by expression (1.2) we have, for different constants $C_{n}$ depending
only on $n$,

$\begin{split}{\mathcal{P}}_{s}[{\mathcal{B}}_{r}]&=\int_{B_{r}}\\!dx\int_{{\mathbb{R}}^{n}\setminus{\mathcal{B}}_{r}}\frac{dy}{|x-y|^{n+s}}\\\
&\leq\int_{B_{r}}\\!dx\int_{{\mathbb{R}}^{n}\setminus
B_{r}}\frac{dy}{|x-y|^{n+s}}=r^{n-s}\int_{B_{1}}\\!dx\int_{{\mathbb{R}}^{n}\setminus
B_{1}}\frac{dy}{|x-y|^{n+s}}\\\ &\leq
r^{n-s}\int_{B_{1}}\\!dx\int_{{\mathbb{R}}^{n}\setminus
B_{1-|x|}(x)}\frac{dy}{|x-y|^{n+s}}\leq
C_{n}\,r^{n-s}\int_{B_{1}}\\!dx\int_{1-|x|}^{+\infty}\\!dt\,t^{-1-s}\\\
&=\frac{C_{n}}{s}\,r^{n-s}\int_{B_{1}}\\!dx\,(1-|x|)^{-s}=\frac{C_{n}}{s}\,r^{n-s}|{\mathbb{S}}^{n-1}|\int_{0}^{1}\\!d\rho\,\rho^{n-1}(1-\rho)^{-s}\\\
&\leq\frac{C_{n}}{s(1-s)}\,r^{n-s}=\frac{C_{n}}{s(1-s)}\,\mu^{\frac{n-s}{n}}.\end{split}$
(2.24)

Instead, for the straight cylinder satisfying the volume constraint, which is
given by
$E_{\mu}:=\\{x\in{\mathbb{R}}^{n}:\,|x^{\prime}|<c_{1}\,\mu^{1/(n-1)}\\}$ for
some constant $c_{1}$ depending only on $n$, we have

$\begin{split}{\mathcal{P}}_{s}[E_{\mu}]&=\int_{\\{x\in{\mathbb{R}}^{n}:\,|x_{1}|<\pi,\,|x^{\prime}|<c_{1}\,\mu^{1/(n-1)}\\}}\\!dx\,\int_{\\{y\in{\mathbb{R}}^{n}:\,|y^{\prime}|>c_{1}\,\mu^{1/(n-1)}\\}}\frac{dy}{|x-y|^{n+s}}\\\
&=c_{1}^{n-s}\,\mu^{\frac{n-s}{n-1}}\int_{\big{\\{}\overline{x}\in{\mathbb{R}}^{n}:\,|\overline{x}_{1}|<\frac{\pi}{c_{1}\,\mu^{1/(n-1)}},\,|\overline{x}^{\prime}|<1\big{\\}}}\\!d\overline{x}\,\int_{\\{\overline{y}\in{\mathbb{R}}^{n}:\,|\overline{y}^{\prime}|>1\\}}\frac{d\overline{y}}{|\overline{x}-\overline{y}|^{n+s}}\\\
&=c_{1}^{n-s}\,\mu^{\frac{n-s}{n-1}}\frac{2\pi}{c_{1}\,\mu^{1/(n-1)}}\int_{-\infty}^{+\infty}\\!dt\int_{\\{\overline{x}^{\prime}\in{\mathbb{R}}^{n-1}:\,|\overline{x}^{\prime}|<1\\}}\\!d\overline{x}^{\prime}\\!\int_{\\{\overline{y}^{\prime}\in{\mathbb{R}}^{n-1}:\,|\overline{y}^{\prime}|>1\\}}\\!d\overline{y}^{\prime}\,{(t^{2}+|\overline{x}^{\prime}-\overline{y}^{\prime}|^{2})^{-\frac{n+s}{2}}}\\\
&\geq
c_{1}^{n-s-1}\,\mu^{\frac{n-1-s}{n-1}}2\pi\,2^{-\frac{n+s}{2}}\int_{\\{\overline{x}^{\prime}\in{\mathbb{R}}^{n-1}:\,|\overline{x}^{\prime}|<1\\}}\\!d\overline{x}^{\prime}\int_{\\{\overline{y}^{\prime}\in{\mathbb{R}}^{n-1}:\,|\overline{y}^{\prime}|>1\\}}\\!d\overline{y}^{\prime}\int_{-|\overline{x}^{\prime}-\overline{y}^{\prime}|}^{|\overline{x}^{\prime}-\overline{y}^{\prime}|}\\!dt\,|\overline{x}^{\prime}-\overline{y}^{\prime}|^{-n-s}\\\
&\geq
c_{1}^{n-s-1}\,\mu^{\frac{n-1-s}{n-1}}2\pi\,2^{-\frac{n+s}{2}}2\int_{\\{\overline{x}^{\prime}\in{\mathbb{R}}^{n-1}:\,|\overline{x}^{\prime}|<1\\}}\\!d\overline{x}^{\prime}\int_{\\{\overline{y}^{\prime}\in{\mathbb{R}}^{n-1}:\,|\overline{y}^{\prime}|>1\\}}\\!d\overline{y}^{\prime}\,|\overline{x}^{\prime}-\overline{y}^{\prime}|^{1-n-s}.\end{split}$
(2.25)

To bound the last double integral from below, we denote
$\overline{y}^{\prime}=z^{\prime}=(z^{\prime}_{1},z^{\prime\prime})\in{\mathbb{R}}\times{\mathbb{R}}^{n-2}$
to get

$\begin{split}\int_{B_{1}^{n-1}}\\!d\overline{x}^{\prime}&\int_{{\mathbb{R}}^{n-1}\setminus
B_{1}^{n-1}}\\!d\overline{y}^{\prime}\,|\overline{x}^{\prime}-\overline{y}^{\prime}|^{1-n-s}\\\
&=\big{|}{\mathbb{S}}^{n-2}\big{|}\int_{0}^{1}\\!d\rho\,\rho^{n-2}\int_{{\mathbb{R}}^{n-1}\setminus
B_{1}^{n-1}}\\!dz^{\prime}\,|\rho\,e_{1}^{\prime}-z^{\prime}|^{1-n-s}\\\
&\geq\big{|}{\mathbb{S}}^{n-2}\big{|}\int_{0}^{1}\\!d\rho\,\rho^{n-2}\int_{\\{z\in{\mathbb{R}}^{n-1}:\,z^{\prime}_{1}>1\\}}\\!dz^{\prime}\,|\rho\,e_{1}^{\prime}-z^{\prime}|^{1-n-s}\\\
&=\big{|}{\mathbb{S}}^{n-2}\big{|}\int_{0}^{1}\\!d\rho\,\rho^{n-2}\int_{1}^{+\infty}\\!dz^{\prime}_{1}\int_{{\mathbb{R}}^{n-2}}\\!dz^{\prime\prime}\,\big{(}(\rho-z^{\prime}_{1})^{2}+|z^{\prime\prime}|^{2}\big{)}^{\frac{1-n-s}{2}}\\\
&\geq\big{|}{\mathbb{S}}^{n-2}\big{|}\int_{0}^{1}\\!d\rho\,\rho^{n-2}\int_{1}^{+\infty}\\!\\!dz^{\prime}_{1}\\!\int_{\\{z^{\prime\prime}\in{\mathbb{R}}^{n-2}:\,|z^{\prime\prime}|<|\rho-
z_{1}^{\prime}|\\}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!dz^{\prime\prime}\,\big{(}(\rho-z^{\prime}_{1})^{2}+|z^{\prime\prime}|^{2}\big{)}^{\frac{1-n-s}{2}}\\\
&\geq\big{|}{\mathbb{S}}^{n-2}\big{|}2^{\frac{1-n-s}{2}}\int_{0}^{1}\\!d\rho\,\rho^{n-2}\int_{1}^{+\infty}\\!dz^{\prime}_{1}\int_{\\{z^{\prime\prime}\in{\mathbb{R}}^{n-2}:\,|z^{\prime\prime}|<|\rho-z^{\prime}_{1}|\\}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!dz^{\prime\prime}\,|\rho-z^{\prime}_{1}|^{1-n-s}\\\
&=\big{|}{\mathbb{S}}^{n-2}\big{|}2^{\frac{1-n-s}{2}}\big{|}{\mathbb{S}}^{n-3}\big{|}\int_{0}^{1}\\!d\rho\,\rho^{n-2}\int_{1}^{+\infty}\\!dz^{\prime}_{1}\,|\rho-z^{\prime}_{1}|^{-1-s}\\\
&=\frac{1}{s}\big{|}{\mathbb{S}}^{n-2}\big{|}2^{\frac{1-n-s}{2}}\big{|}{\mathbb{S}}^{n-3}\big{|}\int_{0}^{1}d\rho\,\rho^{n-2}(1-\rho)^{-s}.\end{split}$
(2.26)

Here we implicitly assumed that $n\geq 3$. However, when $n=2$ we do not need
to introduce the integral over $z^{\prime\prime}\in{\mathbb{R}}^{n-2}$, and
the same bound (with the constant
$\frac{1}{s}\big{|}{\mathbb{S}}^{n-2}\big{|}2^{\frac{1-n-s}{2}}\big{|}{\mathbb{S}}^{n-3}\big{|}$
replaced by $\frac{2}{s}$) follows in an easier way. Observe now that

$\begin{split}\int_{0}^{1}\\!\rho^{n-2}(1-\rho)^{-s}\,d\rho\geq
2^{2-n}\int_{1/2}^{1}(1-\rho)^{-s}\,d\rho=\frac{2^{1-n+s}}{1-s}\geq\frac{2^{1-n}}{1-s}.\end{split}$
(2.27)

Hence, since $2^{-s}\geq 2^{-1}$ and $c_{1}^{-s}\geq\min(1,c_{1}^{-1})$, we
deduce that

$\begin{split}{\mathcal{P}}_{s}[E_{\mu}]\geq\frac{c_{n}}{s(1-s)}\mu^{\frac{n-1-s}{n-1}}\end{split}$
(2.28)

for some constant $c_{n}>0$ depending only on $n$.

From this and (2.24), we conclude that
${\mathcal{P}}_{s}[E_{\mu}]>{\mathcal{P}}_{s}[{\mathcal{B}}_{r(\mu)}]$ if
$\mu$ is small enough, since clearly $(n-1-s)/(n-1)<(n-s)/n$. ∎

##  3\. Cylindrical and periodic symmetric decreasing rearrangements

In this section we analyze the behavior of ${\mathcal{P}}_{s}$ under the two
rearrangements defined in Subsection 1.1, namely, the cylindrical
rearrangement and the symmetric decreasing periodic rearrangement.

In the previous section we have shown the existence of a cylindrically
symmetric minimizer using the Riesz rearrangement inequality (the key
ingredient in the proof of Lemma 3.1).101010The Riesz rearrangement inequality
was already used in [16, Proposition 13] for the functional
${\mathcal{P}}_{s}$ acting on periodic sets, to conclude that the cylindrical
rearrangement does not increase the functional ${\mathcal{P}}_{s}$. In the
next lemma we will see that characterizing the case of equality in the Riesz
rearrangement inequality will actually yield that every minimizer is
cylindrically symmetric. The same argument has been used in [23, Lemma 4.2] to
show the cylindrical symmetry of minimizers of a different functional.

Before stating the lemma we recall the notation introduced in Subsection 1.1.
For a measurable set $E\subset{\mathbb{R}}^{n}$, given $x_{1}\in{\mathbb{R}}$
consider

$E_{x_{1}}:=\\{x^{\prime}\in{\mathbb{R}}^{n-1}:\,(x_{1},x^{\prime})\in E\\}.$
(3.1)

Then, denoting balls in ${\mathbb{R}}^{n-1}$ by $B_{r}^{n-1}$ and the
$(n-1)$-dimensional Lebesgue measure by $\mathcal{L}^{n-1}$, we define

$E^{*\\!\operatorname{cyl}}:=\left\\{(x_{1},x^{\prime})\in{\mathbb{R}}^{n}:\,x^{\prime}\in
B_{r(x_{1})}^{n-1},\,\text{ where
}\mathcal{L}^{n-1}\big{(}B_{r(x_{1})}^{n-1}\big{)}=\mathcal{L}^{n-1}(E_{x_{1}}),\right\\},$
(3.2)

with the understanding that $B_{r(x_{1})}^{n-1}={\mathbb{R}}^{n-1}$ if
$\mathcal{L}^{n-1}(E_{x_{1}})=+\infty$. Notice that by definition

$\left(E^{*\\!\operatorname{cyl}}\right)_{x_{1}}=\left(E_{x_{1}}\right)^{*(n-1)},$
(3.3)

where $A^{*(n-1)}$ denotes the Schwarz rearrangement of a set
$A\subset{\mathbb{R}}^{n-1},$ i.e., the ball in ${\mathbb{R}}^{n-1}$ centered
at $0$ and of the same $(n-1)$-dimensional Lebesgue measure as $A$ —in case
that the measure of $A$ is infinite, we set $A^{*(n-1)}:={\mathbb{R}}^{n-1}$.

The following lemma concerns our periodic fractional perimeter, but its
statement does not require the periodicity of the set $E$. Observe that
${\mathcal{P}}_{s}[E]$ makes sense also for sets which are not periodic. Its
proof will use the classical Riesz rearrangement inequality, as well as its
strict version needed to discuss the case of equality.

######  Lemma 3.1.

Let $E\subset{\mathbb{R}}^{n}$ be a measurable set such that
$|E\cap((-\pi,\pi)\times{\mathbb{R}}^{n-1})|<+\infty$.

Then, ${\mathcal{P}}_{s}[E^{*\\!\operatorname{cyl}}]\leq{\mathcal{P}}_{s}[E]$.
Moreover, if in addition $|E\cap((-\pi,\pi)\times{\mathbb{R}}^{n-1})|>0$ and
${\mathcal{P}}_{s}[E]<+\infty$, the equality
${\mathcal{P}}_{s}[E^{*\\!\operatorname{cyl}}]={\mathcal{P}}_{s}[E]$ holds if
and only if $E=E^{*\\!\operatorname{cyl}}+(0,c)$ for some
$c\in{\mathbb{R}}^{n-1}$ up to a set of measure zero.

###### Proof.

For simplicity of notation, given $x_{1},y_{1}\in{\mathbb{R}}$ with $x_{1}\neq
y_{1}$, define $g_{x_{1},y_{1}}:{\mathbb{R}}^{n-1}\to{\mathbb{R}}$ by

$g_{x_{1},y_{1}}(z^{\prime}):=\left(|x_{1}-y_{1}|^{2}+|z^{\prime}|^{2}\right)^{-\frac{n+s}{2}}.$

Note that $g_{x_{1},y_{1}}\in L^{1}({\mathbb{R}}^{n-1})$ is a positive and
radially symmetric decreasing function.

To address the proof of the inequality
${\mathcal{P}}_{s}[E^{*\\!\operatorname{cyl}}]\leq{\mathcal{P}}_{s}[E]$, let
us first make some preliminary observations. Let $x_{1},y_{1}\in{\mathbb{R}}$
with $x_{1}\neq y_{1}$, and assume that
$\mathcal{L}^{n-1}(E_{x_{1}})<+\infty$. On the one hand, if
$\mathcal{L}^{n-1}(E_{y_{1}})<+\infty$, the Riesz rearrangement inequality
[22, Theorem 3.7] yields that

$\int_{E_{x_{1}}}dx^{\prime}\int_{E_{y_{1}}}dy^{\prime}\,g_{x_{1},y_{1}}(x^{\prime}-y^{\prime})\leq\int_{(E_{x_{1}})^{*(n-1)}}dx^{\prime}\int_{(E_{y_{1}})^{*(n-1)}}dy^{\prime}\,g_{x_{1},y_{1}}(x^{\prime}-y^{\prime}).$
(3.4)

In addition, the strict rearrangement inequality [6, Corollary 2.19] or [22,
Theorem 3.9]111111[22, Theorem 3.9] should include two hypotheses which are
needed for its validity —and which are explicitly stated in most of the
results of [22, Chapter 3]. Namely, that the functions $f$ and $h$ on its
statement must vanish at infinity and that neither $f$ nor $h$ is identically
zero. Indeed, there is always equality in the Riesz rearrangement inequality
if one of these two functions is constant.  shows that the equality in (3.4)
holds (when $E_{x_{1}}$ and $E_{y_{1}}$ have finite measure) if and only if
either $\mathcal{L}^{n-1}(E_{x_{1}})\mathcal{L}^{n-1}(E_{y_{1}})=0$ or

$E_{x_{1}}=B_{r(x_{1})}^{n-1}+c(x_{1},y_{1})\quad\text{and}\quad
E_{y_{1}}=B_{r(y_{1})}^{n-1}+c(x_{1},y_{1})$ (3.5)

for some $r(x_{1})$ and $r(y_{1})$ belonging to $(0,+\infty)$ and for some
$c(x_{1},y_{1})\in{\mathbb{R}}^{n-1}$.

On the other hand, if $\mathcal{L}^{n-1}(E_{y_{1}})=+\infty$ then
$(E_{y_{1}})^{*(n-1)}={\mathbb{R}}^{n-1}$ and, therefore,

$\begin{split}\int_{E_{x_{1}}}dx^{\prime}\int_{E_{y_{1}}}dy^{\prime}\,g_{x_{1},y_{1}}&(x^{\prime}-y^{\prime})\leq\int_{E_{x_{1}}}dx^{\prime}\int_{{\mathbb{R}}^{n-1}}dy^{\prime}\,g_{x_{1},y_{1}}(x^{\prime}-y^{\prime})\\\
&=\mathcal{L}^{n-1}(E_{x_{1}})\|g_{x_{1},y_{1}}\|_{L^{1}({\mathbb{R}}^{n-1})}=\mathcal{L}^{n-1}\big{(}(E_{x_{1}})^{*(n-1)}\big{)}\|g_{x_{1},y_{1}}\|_{L^{1}({\mathbb{R}}^{n-1})}\\\
&=\int_{(E_{x_{1}})^{*(n-1)}}dx^{\prime}\int_{{\mathbb{R}}^{n-1}}dy^{\prime}\,g_{x_{1},y_{1}}(x^{\prime}-y^{\prime})\\\
&=\int_{(E_{x_{1}})^{*(n-1)}}dx^{\prime}\int_{(E_{y_{1}})^{*(n-1)}}dy^{\prime}\,g_{x_{1},y_{1}}(x^{\prime}-y^{\prime}).\end{split}$
(3.6)

In addition, since $g_{x_{1},y_{1}}$ is positive and integrable in
${\mathbb{R}}^{n-1}$, equality in (3.6) holds if and only if either
$\mathcal{L}^{n-1}(E_{x_{1}})=0$ or $E_{y_{1}}={\mathbb{R}}^{n-1}$ up to a set
of measure zero. In particular, if $E_{x_{1}}$ has finite measure, $E_{y_{1}}$
has infinite measure, and equality in (3.6) holds, then either
$\mathcal{L}^{n-1}(E_{x_{1}})=0$ or

$E_{y_{1}}=(E_{y_{1}})^{*(n-1)}+c\quad\text{ for every
$c\in{\mathbb{R}}^{n-1}$.}$ (3.7)

Having these observations in mind, we can now proceed to the proof of the
first part of the lemma.

First of all, the assumption
$|E\cap((-\pi,\pi)\times{\mathbb{R}}^{n-1})|<+\infty$ leads to
$\mathcal{L}^{n-1}(E_{x_{1}})<+\infty$ for a.e. $x_{1}\in(-\pi,\pi)$. In
particular, for a.e. $x_{1}\in(-\pi,\pi)$ and all $y_{1}\in{\mathbb{R}}$ with
$x_{1}\neq y_{1}$, we have

$\int_{E_{x_{1}}}dx^{\prime}\int_{A}dy^{\prime}\,g_{x_{1},y_{1}}(x^{\prime}-y^{\prime})\leq|E_{x_{1}}|\|g_{x_{1},y_{1}}\|_{L^{1}({\mathbb{R}}^{n-1})}<+\infty$
(3.8)

for every measurable set $A\subset{\mathbb{R}}^{n-1}$. This justifies the
forthcoming computations. Using (3.4) and (3.6) (depending on whether
$\mathcal{L}^{n-1}(E_{y_{1}})<+\infty$ or
$\mathcal{L}^{n-1}(E_{y_{1}})=+\infty$), and recalling (3.3), we get

$\begin{split}{\mathcal{P}}_{s}[E]&=\int_{-\pi}^{\pi}dx_{1}\int_{{\mathbb{R}}}dy_{1}\int_{E_{x_{1}}}dx^{\prime}\int_{{\mathbb{R}}^{n-1}\setminus
E_{y_{1}}}dy^{\prime}\,g_{x_{1},y_{1}}(x^{\prime}-y^{\prime})\\\
&=\int_{-\pi}^{\pi}dx_{1}\int_{{\mathbb{R}}}dy_{1}\bigg{(}\int_{E_{x_{1}}}dx^{\prime}\int_{{\mathbb{R}}^{n-1}}dy^{\prime}\,g_{x_{1},y_{1}}(x^{\prime}-y^{\prime})-\int_{E_{x_{1}}}dx^{\prime}\int_{E_{y_{1}}}dy^{\prime}\,g_{x_{1},y_{1}}(x^{\prime}-y^{\prime})\bigg{)}\\\
&=\int_{-\pi}^{\pi}dx_{1}\int_{{\mathbb{R}}}dy_{1}\bigg{(}\mathcal{L}^{n-1}(E_{x_{1}})\int_{{\mathbb{R}}^{n-1}}dz^{\prime}\,g_{x_{1},y_{1}}(z^{\prime})-\int_{E_{x_{1}}}dx^{\prime}\int_{E_{y_{1}}}dy^{\prime}\,g_{x_{1},y_{1}}(x^{\prime}-y^{\prime})\bigg{)}\\\
&\geq\int_{-\pi}^{\pi}dx_{1}\int_{{\mathbb{R}}}dy_{1}\bigg{(}\mathcal{L}^{n-1}((E^{*\\!\operatorname{cyl}})_{x_{1}})\int_{{\mathbb{R}}^{n-1}}dz^{\prime}\,g_{x_{1},y_{1}}(z^{\prime})\\\
&\hskip
100.0pt-\int_{(E^{*\\!\operatorname{cyl}})_{x_{1}}}\\!\\!\\!\\!dx^{\prime}\int_{(E^{*\\!\operatorname{cyl}})_{y_{1}}}\\!\\!\\!\\!dy^{\prime}\,g_{x_{1},y_{1}}(x^{\prime}-y^{\prime})\bigg{)}\\\
&={\mathcal{P}}_{s}[E^{*\\!\operatorname{cyl}}],\end{split}$ (3.9)

which proves the first part of the lemma.

To prove its second part, we need to show that if $E$ has positive finite
measure in $(-\pi,\pi)\times{\mathbb{R}}^{n-1}$ and
${\mathcal{P}}_{s}[E]={\mathcal{P}}_{s}[E^{*\\!\operatorname{cyl}}]<+\infty$,
then there exists $c\in{\mathbb{R}}^{n-1}$ such that
$E_{y_{1}}=(E_{y_{1}})^{*(n-1)}+c$ for a.e. $y_{1}\in{\mathbb{R}}$. For this,
a key point is to use (3.9), already proven, as follows. Since
$\mathcal{L}^{n-1}(E_{x_{1}})=\mathcal{L}^{n-1}((E^{*\\!\operatorname{cyl}})_{x_{1}})<+\infty$
for a.e. $x_{1}\in(-\pi,\pi)$ and
${\mathcal{P}}_{s}[E]={\mathcal{P}}_{s}[E^{*\\!\operatorname{cyl}}]<+\infty$,
(3.9) shows that we must have

$\begin{split}\int_{-\pi}^{\pi}dx_{1}\int_{{\mathbb{R}}}dy_{1}\bigg{(}\int_{E_{x_{1}}}&dx^{\prime}\int_{E_{y_{1}}}dy^{\prime}\,g_{x_{1},y_{1}}(x^{\prime}-y^{\prime})\bigg{)}\\\
&=\int_{-\pi}^{\pi}dx_{1}\int_{{\mathbb{R}}}dy_{1}\bigg{(}\int_{(E^{*\\!\operatorname{cyl}})_{x_{1}}}\\!\\!\\!\\!dx^{\prime}\int_{(E^{*\\!\operatorname{cyl}})_{y_{1}}}\\!\\!\\!\\!dy^{\prime}\,g_{x_{1},y_{1}}(x^{\prime}-y^{\prime})\bigg{)}.\end{split}$
(3.10)

Using now (3.4) in case $\mathcal{L}^{n-1}(E_{y_{1}})<+\infty$, or (3.6) in
case $\mathcal{L}^{n-1}(E_{y_{1}})=+\infty$, we deduce that for a.e.
$(x_{1},y_{1})\in(-\pi,\pi)\times{\mathbb{R}}$ it holds

$\int_{E_{x_{1}}}dx^{\prime}\int_{E_{y_{1}}}dy^{\prime}\,g_{x_{1},y_{1}}(x^{\prime}-y^{\prime})=\int_{(E_{x_{1}})^{*(n-1)}}dx^{\prime}\int_{(E_{y_{1}})^{*(n-1)}}dy^{\prime}\,g_{x_{1},y_{1}}(x^{\prime}-y^{\prime}).$
(3.11)

Now, note that the assumption
$0<|E\cap((-\pi,\pi)\times{\mathbb{R}}^{n-1})|<+\infty$ yields that

$\Lambda:=\\{x_{1}\in(-\pi,\pi):\,0<\mathcal{L}^{n-1}(E_{x_{1}})<+\infty\\}$

has positive measure. Recall that we do not assume $E$ to be periodic, and
therefore there might exist a set $I\subset(-\infty,-\pi)\cup(\pi,+\infty)$ of
positive measure such that $\mathcal{L}^{n-1}(E_{y_{1}})=+\infty$ for all
$y_{1}\in I.$ In view of this fact we define

$\Sigma:=\\{y_{1}\in{\mathbb{R}}:\,0<\mathcal{L}^{n-1}(E_{y_{1}})<+\infty\\},$

which also has positive measure because $\Lambda\subset\Sigma$.

Let now $(x_{1},y_{1})\in\Lambda\times\Sigma$ be such that (3.11) holds. Then,
by the statement concluding (3.5) we have

$E_{x_{1}}=B_{r(x_{1})}^{n-1}+c(x_{1},y_{1})\quad\text{and}\quad
E_{y_{1}}=B_{r(y_{1})}^{n-1}+c(x_{1},y_{1}).$

Repeating this argument for any other
$(\tilde{x}_{1},y_{1})\in\Lambda\times\Sigma$ we also obtain
$E_{y_{1}}=B_{r(y_{1})}^{n-1}+c(\tilde{x}_{1},y_{1})$, and hence $c$ is
independent of $x_{1}$. Analogously, the same argument on any other
$({x}_{1},\tilde{y}_{1})\in\Lambda\times\Sigma$ yields that
$E_{x_{1}}=B_{r(x_{1})}^{n-1}+c(x_{1},\tilde{y}_{1})$. Hence, $c$ is also
independent of $y_{1}$. Thus, there exists $c\in{\mathbb{R}}^{n-1}$ such that
$E_{y_{1}}=B_{r(y_{1})}^{n-1}+c$ for a.e. $y_{1}\in{\mathbb{R}}$ satisfying
$0<\mathcal{L}^{n-1}(E_{y_{1}})<+\infty$.

For this constant $c\in{\mathbb{R}}^{n-1}$, we now claim that

$E_{y_{1}}=(E_{y_{1}})^{*(n-1)}+c$ (3.12)

for a.e. $y_{1}\in{\mathbb{R}}$. On the one hand, we have already proven
(3.12) for a.e. $y_{1}\in\Sigma$. On the other hand, if
$y_{1}\in{\mathbb{R}}\setminus\Sigma$, then we must have that either
$\mathcal{L}^{n-1}(E_{y_{1}})=0$ or $\mathcal{L}^{n-1}(E_{y_{1}})=+\infty$.
But if $\mathcal{L}^{n-1}(E_{y_{1}})=0$, then
$E_{y_{1}}=(E_{y_{1}})^{*(n-1)}+d$ for every $d\in{\mathbb{R}}^{n-1}$ up to a
set of $\mathcal{L}^{n-1}$-measure zero, since $(E_{y_{1}})^{*(n-1)}$ is the
empty set in ${\mathbb{R}}^{n-1}$. Thus, (3.12) holds in this case too. It
only remains to show (3.12) for a.e. $y_{1}\in{\mathbb{R}}$ such that
$\mathcal{L}^{n-1}(E_{y_{1}})=+\infty$. Recall that (3.11) holds for a.e.
$(x_{1},y_{1})\in(-\pi,\pi)\times{\mathbb{R}}$. Hence, since $|\Lambda|>0$,
for a.e. $y_{1}\in{\mathbb{R}}$ with $\mathcal{L}^{n-1}(E_{y_{1}})=+\infty$
there exists a point $x_{1}\in\Lambda$ such that (3.11) holds. Then the
statement concluding (3.7) yields that $E_{y_{1}}=(E_{y_{1}})^{*(n-1)}+d$ for
every $d\in{\mathbb{R}}^{n-1}$. Thus, (3.12) also follows in this last case.

In conclusion, there exists $c\in{\mathbb{R}}^{n-1}$ such that
$E_{y_{1}}=(E_{y_{1}})^{*(n-1)}+c$ for a.e. $y_{1}\in{\mathbb{R}}$.
Equivalently, $E=E^{*\\!\operatorname{cyl}}+(0,c)$ up to a set of measure
zero. ∎

We now address the proof of Theorem 1.2. Let us first recall some definitions
to facilitate the reading. For a measurable set $A\subset{\mathbb{R}}^{n}$ we
denote its Steiner symmetrization with respect to $\\{x_{1}=0\\}$ by $A^{*}.$
If $A\subset{\mathbb{R}}$ is measurable and of finite measure, then $A^{*}$ is
an interval centered at $0$ with the same measure as $A.$ The Steiner
symmetrization is well defined whenever the $1$-dimensional set
$A\cap\\{x^{\prime}=c\\}$ has finite length for all $c\in{\mathbb{R}}^{n-1}.$
Thus, the periodic rearrangement

$E^{*\\!\operatorname{per}}:=\bigcup_{k\in{\mathbb{Z}}}\left[\left(E\cap\\{-\pi<x_{1}<\pi\\}\right)^{*}+2k\pi
e_{1}\right]$

is well defined for every measurable set $E\subset{\mathbb{R}}^{n}$ which is
$2\pi$-periodic in the $x_{1}$ direction.

By periodicity, note that
$E^{*\\!\operatorname{per}}=(E+(c,0))^{*\\!\operatorname{per}}$ for all
$c\in{\mathbb{R}}.$

We denote the symmetric decreasing rearrangement of a real-valued function $f$
on ${\mathbb{R}}$ by $f^{*}$. It is defined by the following properties:
$f^{*}$ is nonnegative, equimeasurable with $|f|$, and its superlevel sets are
open intervals centered at $0$. Two explicit expressions for $f^{*}$ are

$f^{*}(x)=\int_{0}^{+\infty}dt\,\chi_{\\{|f|>t\\}^{*}}(x)=\sup\big{\\{}t:\,x\in\\{|f|>t\\}^{*}\big{\\}}.$
(3.13)

For a $2\pi$-periodic function $u:{\mathbb{R}}\to{\mathbb{R}}$, the periodic
symmetric deceasing rearrangement of $u$, denoted by
$u^{*\\!\operatorname{per}}$, is the unique $2\pi$-periodic function such that
$u^{*\\!\operatorname{per}}\chi_{(-\pi,\pi)}=(u\chi_{(-\pi,\pi)})^{*}$.

It follows from these definitions that if $E$ is as in (1.5), then

$E^{*\\!\operatorname{per}}=\\{x\in{\mathbb{R}}^{n}\,:\,|x^{\prime}|<u^{*\\!\operatorname{per}}(x_{1})\\}.$
(3.14)

The main ingredient of our proof of Theorem 1.2 will be the following Riesz
rearrangement inequality on the circle.

######  Theorem 3.2.

$($[7, Theorem 2], [19, Theorem 1], [8, Theorem 2], [6, Theorem 7.3]$)$ Let
$f,h,g:{\mathbb{R}}\to{\mathbb{R}}$ be three nonnegative $2\pi$-periodic
measurable functions. Assume that $g$ is even, as well as nonincreasing in
$(0,\pi)$.

Then,

$\int_{-\pi}^{\pi}dx\int_{-\pi}^{\pi}dy\,f(x)g(x-y)h(y)\leq\int_{-\pi}^{\pi}dx\int_{-\pi}^{\pi}dy\,f^{*}(x)g(x-y)h^{*}(y).$
(3.15)

In addition, if $g$ is decreasing in $(0,\pi)$ and the left-hand side of
(3.15) is finite, then equality holds in (3.15) if and only if at least one of
the following conditions holds:

  * $(i)$

either $f$ or $h$ is constant almost everywhere.

  * $(ii)$

there exists $z\in{\mathbb{R}}$ such that
$f(x)=f^{*\\!\operatorname{per}}(x+z)$ and
$h(x)=h^{*\\!\operatorname{per}}(x+z)$ for almost every $x\in{\mathbb{R}}$.

The inequality in Theorem 3.2 was first discovered, independently, in [7] and
[19]. Both references contain more general inequalities: [7] deals with the
sphere $\mathbb{S}^{n}$, whereas [19] deals with a product of more than three
functions. The result [6, Theorem 7.3] is also more general than Theorem 3.2,
as it deals with a Riesz rearrangement inequality on the sphere
$\mathbb{S}^{n}$ and not only on the circle. Moreover, [6] treats more general
functions of $f(x)$ and $h(y)$ than simply the product $f(x)h(y).$ The
inequality (3.15) can also be found in [5] in a more general form where $g$ is
also rearranged.

Instead, the statement in Theorem 3.2 concerning equality in (3.15) follows
from Burchard and Hajaiej [8, Theorem 2], who treated the case of equality in
$\mathbb{S}^{n}$ for the first time. For this result, we also cite [6, Theorem
7.3] since, being less general than [8], fits precisely with our setting.

We find reference [6] to be the simplest one for looking up all statements of
Theorem 3.2.

###### Proof of Theorem 1.2..

Let us first show that
${\mathcal{P}}_{s}[E^{*\\!\operatorname{per}}]\leq{\mathcal{P}}_{s}[E].$ In
contrast to the proof of Lemma 3.1, we now write ${\mathcal{P}}_{s}[E]$ as

${\mathcal{P}}_{s}[E]=\int_{{\mathbb{R}}^{n-1}}dx^{\prime}\int_{E_{x^{\prime}}\cap(-\pi,\pi)}dx_{1}\int_{{\mathbb{R}}^{n-1}}dy^{\prime}\int_{{\mathbb{R}}\setminus
E_{y^{\prime}}}dy_{1}\,\frac{1}{|x-y|^{n+s}},$

where we have set
$E_{x^{\prime}}:=\\{x_{1}\in{\mathbb{R}}:\,(x_{1},x^{\prime})\in E\\}$ (in
analogy with the sections $E_{x_{1}}$ defined above). Writing
${\mathcal{P}}_{s}[E^{*\\!\operatorname{per}}]$ in the same way (simply
replacing $E$ by $E^{*\\!\operatorname{per}}$) and interchanging the order of
integration $dx_{1}\,dy^{\prime}$, we see that it is sufficient to prove that

$\int_{E_{x^{\prime}}\cap(-\pi,\pi)}dx_{1}\int_{{\mathbb{R}}\setminus
E_{y^{\prime}}}\frac{dy_{1}}{|x-y|^{n+s}}\geq\int_{(E^{*\\!\operatorname{per}})_{x^{\prime}}\cap(-\pi,\pi)}dx_{1}\int_{{\mathbb{R}}\setminus(E^{*\\!\operatorname{per}})_{y^{\prime}}}\frac{dy_{1}}{|x-y|^{n+s}}$
(3.16)

for almost every
$(x^{\prime},y^{\prime})\in{\mathbb{R}}^{n-1}\times{\mathbb{R}}^{n-1}$.

For this, set

$a=|x^{\prime}-y^{\prime}|\quad\text{ and
}\quad\gamma=\frac{n+s}{2}\quad\text{ (and thus
}|x-y|^{n+s}=\big{(}(x_{1}-y_{1})^{2}+a^{2}\big{)}^{\gamma}\text{)},$

observe that the following integrals are all finite when $a\neq 0$, and that
the condition $a\neq 0$ holds for almost every
$(x^{\prime},y^{\prime})\in{\mathbb{R}}^{n-1}\times{\mathbb{R}}^{n-1}$. We
have

$\int_{{\mathbb{R}}\setminus
E_{y^{\prime}}}\frac{dy_{1}}{|x-y|^{n+s}}=\int_{{\mathbb{R}}}\frac{dy_{1}}{|x-y|^{n+s}}-\int_{E_{y^{\prime}}}\frac{dy_{1}}{|x-y|^{n+s}}$

and

$\int_{{\mathbb{R}}\setminus(E^{*\\!\operatorname{per}})_{y^{\prime}}}\frac{dy_{1}}{|x-y|^{n+s}}=\int_{{\mathbb{R}}}\frac{dy_{1}}{|x-y|^{n+s}}-\int_{(E^{*\\!\operatorname{per}})_{y^{\prime}}}\frac{dy_{1}}{|x-y|^{n+s}}.$

From the equality
$|E_{x^{\prime}}\cap(-\pi,\pi)|=|(E^{*\\!\operatorname{per}})_{x^{\prime}}\cap(-\pi,\pi)|,$
we infer that

$\int_{E_{x^{\prime}}\cap(-\pi,\pi)}dx_{1}\int_{{\mathbb{R}}}\frac{dy_{1}}{|x-y|^{n+s}}=\int_{(E^{*\\!\operatorname{per}})_{x^{\prime}}\cap(-\pi,\pi)}dx_{1}\int_{{\mathbb{R}}}\frac{dy_{1}}{|x-y|^{n+s}}.$

Therefore, to prove (3.16) it is enough to show that

$\int_{E_{x^{\prime}}\cap(-\pi,\pi)}dx_{1}\\!\int_{E_{y^{\prime}}}\\!\frac{dy_{1}}{((x_{1}-y_{1})^{2}+a^{2})^{\gamma}}\leq\int_{(E^{*\\!\operatorname{per}})_{x^{\prime}}\cap(-\pi,\pi)}dx_{1}\\!\int_{(E^{*\\!\operatorname{per}})_{y^{\prime}}}\\!\frac{dy_{1}}{((x_{1}-y_{1})^{2}+a^{2})^{\gamma}}$
(3.17)

for all $a\neq 0$.

In order to do this, recall that
$(E^{*\\!\operatorname{per}})_{y^{\prime}}=(E_{y^{\prime}})^{*\\!\operatorname{per}}$
and that both $E_{y^{\prime}}$ and $(E^{*\\!\operatorname{per}})_{y^{\prime}}$
are $2\pi$-periodic. Taking these facts in consideration, (3.17) will follow
if we show that, for every two measurable subsets $A$ and $B$ of $(-\pi,\pi)$,
it holds

$\begin{split}\int_{A}dx_{1}\int_{\bigcup_{k\in{\mathbb{Z}}}(B+2k\pi)}&\frac{dy_{1}}{((x_{1}-y_{1})^{2}+a^{2})^{\gamma}}\leq\int_{A^{*}}dx_{1}\int_{\bigcup_{k\in{\mathbb{Z}}}(B^{*}+2k\pi)}\frac{dy_{1}}{((x_{1}-y_{1})^{2}+a^{2})^{\gamma}}.\end{split}$
(3.18)

By the change of variables $\overline{y}_{1}=y_{1}-2k\pi$, this can be
equivalently written as

$\int_{A}dx_{1}\int_{B}d\overline{y}_{1}\,g(x_{1}-\overline{y}_{1})\leq\int_{A^{*}}dx_{1}\int_{B^{*}}d\overline{y}_{1}\,g(x_{1}-\overline{y}_{1})\text{
for all subsets $A$ and $B$ of $(-\pi,\pi)$},$ (3.19)

where we have set

$g(z):=\sum_{k\in{\mathbb{Z}}}\frac{1}{((z+2k\pi)^{2}+a^{2})^{\gamma}}.$
(3.20)

Clearly, the function $g$ is positive, $2\pi$-periodic, and even. We will next
show that $g$ is decreasing in $(0,\pi)$, and hence (3.19) will follow from
Theorem 3.2 applied to the characteristic functions $f=\chi_{A}$ and
$h=\chi_{B}$.

In order to show that $g$ is decreasing in $(0,\pi)$, we introduce the Laplace
transform of the function $t\mapsto t^{\gamma-1}$, which amounts to the
equality

$w^{-\gamma}=\frac{1}{\Gamma(\gamma)}\int_{0}^{+\infty}dt\,t^{\gamma-1}e^{-wt}\quad\text{for
all $w>0$,}$ (3.21)

where $\Gamma(\gamma):=\int_{0}^{+\infty}dr\,{r}^{\gamma-1}e^{-r}$ is the
Gamma function. The identity (3.21) follows simply by a change of variables
$r=wt$ in the definition of $\Gamma(\gamma)$. Using (3.21) in the definition
of $g$ yields

$g(z)=\frac{1}{\Gamma(\gamma)}\sum_{k\in{\mathbb{Z}}}\int_{0}^{+\infty}dt\,t^{\gamma-1}e^{-((z+2k\pi)^{2}+a^{2})t}=\frac{1}{\Gamma(\gamma)}\int_{0}^{+\infty}dt\,t^{\gamma-1}e^{-a^{2}t}G_{t}(z),$
(3.22)

where

$G_{t}(z):=\sum_{k\in{\mathbb{Z}}}e^{-(z+2k\pi)^{2}t}.$

We will now show that $G_{t}$ is decreasing in $(0,\pi)$ for every $t>0$,
yielding, in view of (3.22), that $g$ is also decreasing in $(0,\pi).$ The
claim on the strict monotonicity of $G_{t}$ follows directly from the fact
that the fundamental solution of the heat equation in $(-\pi,\pi)$ with
periodic boundary conditions (or, in other words, of the heat equation on the
circle), which is

$\Lambda_{t}(z):=\frac{1}{\sqrt{4\pi
t}}\sum_{k\in\mathbb{Z}}e^{-\frac{(z+2k\pi)^{2}}{4t}},$

is decreasing in $(0,\pi)$ for all $t>0$. For the sake of completeness, in
Theorem B.1 we give a simple proof of this well-known fact, based solely on
maximum principles for the heat equation.

We now prove that if
${\mathcal{P}}_{s}[E^{*\\!\operatorname{per}}]={\mathcal{P}}_{s}[E]<+\infty$,
then $E=E^{*\\!\operatorname{per}}+ce_{1}$ for some $c\in{\mathbb{R}}$, up to
a set of measure zero. Indeed, in view of (3.16) and the equalities following
it, if ${\mathcal{P}}_{s}[E^{*\\!\operatorname{per}}]={\mathcal{P}}_{s}[E]$
then there must be equality in (3.17) for almost every
$(x^{\prime},y^{\prime})\in{\mathbb{R}}^{n-1}\times{\mathbb{R}}^{n-1}$.
Equivalently, using the formulation of (3.19),

$\begin{split}\int_{E_{x^{\prime}}\cap(-\pi,\pi)}\\!\\!\\!dx_{1}\int_{E_{y^{\prime}}\cap(-\pi,\pi)}\\!\\!\\!dy_{1}\,g(x_{1}-y_{1})=\int_{(E^{*\\!\operatorname{per}})_{x^{\prime}}\cap(-\pi,\pi)}\\!\\!\\!dx_{1}\int_{(E^{*\\!\operatorname{per}})_{y^{\prime}}\cap(-\pi,\pi)}\\!\\!\\!dy_{1}\,g(x_{1}-y_{1})\end{split}$
(3.23)

for almost every
$(x^{\prime},y^{\prime})\in{\mathbb{R}}^{n-1}\times{\mathbb{R}}^{n-1}$.

Now, let $J\subset{\mathbb{R}}^{n-1}$ be the set of points
$x^{\prime}\in{\mathbb{R}}^{n-1}$ for which $E_{x^{\prime}}\cap(-\pi,\pi)$ is
neither $(-\pi,\pi)$ nor $\varnothing$ (up to zero measure sets). We may
assume that $J$ has positive $(n-1)$-dimensional Lebesgue measure —if $J$ has
measure zero then $E=E^{*\\!\operatorname{per}}$ up to a set of measure zero
in ${\mathbb{R}}^{n}$, and we would be done. Then, $J\times J$ has positive
measure in ${\mathbb{R}}^{n-1}\times{\mathbb{R}}^{n-1}$, and (3.23) holds for
almost every $(x^{\prime},y^{\prime})\in J\times J$. Note that, by definition
of $J$, we don’t fall in case $(i)$ of Theorem 3.2 when using (3.23) on points
$x^{\prime}$ and $y^{\prime}$ in $J$. Hence, by (3.23) and Theorem 3.2 $(ii)$
we deduce that, for almost every $(x^{\prime},y^{\prime})\in J\times J$, we
have

$E_{x^{\prime}}=(E^{*\\!\operatorname{per}})_{x^{\prime}}+c(x^{\prime},y^{\prime})\quad\text{
and }\quad
E_{y^{\prime}}=(E^{*\\!\operatorname{per}})_{y^{\prime}}+c(x^{\prime},y^{\prime})$

for some constant $c(x^{\prime},y^{\prime})\in{\mathbb{R}}$ depending, a
priori, on $x^{\prime}$ and $y^{\prime}$. This easily yields that
$c(x^{\prime},y^{\prime})=c$ is actually independent of $x^{\prime}$ and
$y^{\prime}$. Indeed, considering two different values $x^{\prime}\in J$ and
$\overline{x}^{\prime}\in J$ for the same $y^{\prime}$, we have

$(E^{*\\!\operatorname{per}})_{y^{\prime}}+c(x^{\prime},y^{\prime})=E_{y^{\prime}}=(E^{*\\!\operatorname{per}})_{y^{\prime}}+c(\overline{x}^{\prime},y^{\prime}).$
(3.24)

Since $y^{\prime}\in J$, $(E^{*\\!\operatorname{per}})_{y^{\prime}}$ is a
$2\pi$-periodic set for which
$(E^{*\\!\operatorname{per}})_{y^{\prime}}\cap(-\pi,\pi)$ is a nonempty open
interval different from $(-\pi,\pi)$. Therefore (3.24) yields that
$c(x^{\prime},y^{\prime})=c(\overline{x}^{\prime},y^{\prime})$ modulo $2\pi$.
By symmetry, $c(x^{\prime},y^{\prime})$ is also independent of $y^{\prime}$.
Therefore, $E_{y^{\prime}}=(E^{*\\!\operatorname{per}})_{y^{\prime}}+c$ for
some $c\in{\mathbb{R}}$ and all $y^{\prime}\in J$. By the definition of $J$,
we conclude that $E=E^{*\\!\operatorname{per}}+ce_{1}$ up to a set of measure
zero. ∎

######  Remark 3.3.

From the last part of the proof of Theorem 1.2, in particular in what regards
(3.21), we see that Theorem 1.2 easily extends to any kernel which is the
Laplace transform of a nonnegative function. More precisely, consider a
periodic nonlocal perimeter functional of the form

${\mathcal{P}}_{K}[E]:=\int_{E\cap\\{-\pi<x_{1}<\pi\\}}dx\int_{{\mathbb{R}}^{n}\setminus
E}dy\,K(|x-y|^{2}),$

where $K:(0,+\infty)\to(0,+\infty)$ is a kernel given by

$K(w):=\int_{0}^{+\infty}dt\,\kappa(t)e^{-wt},\quad w>0,$

for some measurable function $\kappa:(0,+\infty)\to[0,+\infty)$ which is not
identically zero. Then, Theorem 1.2 holds also for ${\mathcal{P}}_{K}$.

Indeed, arguing as in the proof of Theorem 1.2 up to (3.19), one sees that it
is enough to show that

$\int_{A}dx_{1}\int_{B}dy_{1}\,g_{K}(x_{1}-y_{1})\leq\int_{A^{*}}dx_{1}\int_{B^{*}}dy_{1}\,g_{K}(x_{1}-y_{1})\quad\text{for
all subsets $A$ and $B$ of $(-\pi,\pi)$},$ (3.25)

where

$g_{K}(z):=\sum_{k\in{\mathbb{Z}}}K((z+2k\pi)^{2}+a^{2}),\quad
z\in{\mathbb{R}}.$ (3.26)

But

$g_{K}(z)=\sum_{k\in{\mathbb{Z}}}\int_{0}^{+\infty}dt\,\kappa(t)e^{-((z+2k\pi)^{2}+a^{2})t}=\int_{0}^{+\infty}dt\,\kappa(t)e^{-a^{2}t}G_{t}(z)$
(3.27)

is a decreasing function in $(0,\pi)$ by Theorem B.1 and by the fact that
$\kappa\geq 0$ is not identically zero in $(0,+\infty)$. Hence, Theorem 3.2
still applies in this framework and, thus, we deduce that ${\mathcal{P}}_{K}$
is nonincreasing under periodic decreasing rearrangement. Moreover, as before,
if ${\mathcal{P}}_{K}[E^{*\\!\operatorname{per}}]={\mathcal{P}}_{K}[E]$ then
$E=E^{*\\!\operatorname{per}}+ce_{1}$ for some $c\in{\mathbb{R}}$, up to a set
of measure zero.

##  Appendix A Periodic minimizers in ${\mathbb{R}}^{2}$

In this appendix we present, when $n=2$, an expression for the functional
${\mathcal{P}}_{s}$ which is simpler than that of Lemma 2.2 in
${\mathbb{R}}^{n}$. As mentioned in the Introduction, this expression allowed
us to prove the existence of minimizer when $n=2$ in a simple way. We later
found the existence proof of Theorem 1.1 in all dimensions.

The formula for the periodic version of the fractional perimeter when $n=2$
reads as follows. If
$E=\\{(x_{1},x_{2})\in{\mathbb{R}}^{2}:\,|x_{2}|<u(x_{1})\\}$, for some
nonnegative $2\pi$-periodic function $u$, then ${\mathcal{P}}_{s}[E]$ can be
written in terms of $u$ as

${\mathcal{P}}_{s}[E]=\int_{-\pi}^{\pi}dx_{1}\\!\int_{-\infty}^{+\infty}dy_{1}\,\frac{2}{|x_{1}-y_{1}|^{s}}\Big{\\{}G\Big{(}\frac{u(x_{1})-u(y_{1})}{|x_{1}-y_{1}|}\Big{)}+H\Big{(}\frac{u(x_{1})+u(y_{1})}{|x_{1}-y_{1}|}\Big{)}\Big{\\}},$
(A.1)

where

$G(t):=\int_{0}^{t}d\tau\,(t-\tau)(1+\tau^{2})^{-\frac{2+s}{2}}\quad\text{and}\quad
H(t):=G^{\prime}(+\infty)t-G(t)\quad\text{for $t\in{\mathbb{R}}$.}$ (A.2)

Note that $G(0)=0$,

$0<G^{\prime}(t)=\int_{0}^{t}d\tau\,(1+\tau^{2})^{-\frac{2+s}{2}}<G^{\prime}(+\infty)<+\infty,$
(A.3)

and $G^{\prime\prime}(t)=(1+t^{2})^{-\frac{2+s}{2}}>0$ for all
$t\in{\mathbb{R}}$. Therefore, $G$ is a nonnegative even function which is
increasing in $(0,+\infty)$ and strictly convex on ${\mathbb{R}}$.
Furthermore, it behaves as a quadratic function near the origin and approaches
a linear one as $t\to\pm\infty$. On the other hand, $H(0)=0$,
$H^{\prime}=G^{\prime}(+\infty)-G^{\prime}>0$, and
$H^{\prime\prime}=-G^{\prime\prime}<0$ in $[0,+\infty).$ Thus, $H$ is
positive, increasing, and strictly concave in $[0,+\infty)$. Finally, by
Fubini’s theorem,

$\begin{split}H(+\infty)=\int_{0}^{+\infty}dt\,H^{\prime}(t)&=\int_{0}^{+\infty}dt\,(G^{\prime}(+\infty)-G^{\prime}(t))=\int_{0}^{+\infty}dt\int_{t}^{+\infty}d\tau\,(1+\tau^{2})^{-\frac{2+s}{2}}\\\
&=\int_{0}^{+\infty}d\tau\,\tau(1+\tau^{2})^{-\frac{2+s}{2}}<+\infty.\end{split}$
(A.4)

For the sake of completeness we will give the proof of (A.1), even though this
expression was not used to prove the results of this paper. But before, let us
explain how to use (A.1) to give a simple proof of the existence of a
constrained minimizer of ${\mathcal{P}}_{s}$ when $n=2$. The argument goes as
follows. Since $G(t)+C_{1}\geq C_{2}|t|$ for some constants $C_{1},C_{2}>0$,
the term in (A.1) involving $G$ bounds the $W^{s,1}(-\pi,\pi)$ norm of $u$ up
to an additive constant. In this way, since $H\geq 0$, a minimizing sequence
for ${\mathcal{P}}_{s}$ is a bounded sequence in $W^{s,1}(-\pi,\pi)$. By
compactness, this yields the existence of the desired constrained minimizer.
We refer to [2] for the details on this argument; see also the end of Part 1
in our proof of Theorem 1.1.

######  Lemma A.1.

Let $u:{\mathbb{R}}\to{\mathbb{R}}$ be nonnegative, $2\pi$-periodic, and such
that $u\in W^{s,1}(-\pi-\epsilon,\pi+\epsilon)$ for some (and hence for all)
$\epsilon>0$. Let
$E=\\{(x_{1},x_{2})\in{\mathbb{R}}^{2}:\,|x_{2}|<u(x_{1})\\}$. Then, (A.1)
holds.

###### Proof.

We prove (A.1) using (2.3). Let us first rewrite $\phi$ in (2.4) for $n=2$ in
a more convenient way by introducing the function $G^{\prime}$. For $p\geq 0$
and $q\geq 0$ we have (using the change of variables $z=\tau+w$ twice)

$\begin{split}\phi(p,q)&=\int_{-p}^{p}dw\,\Big{\\{}\int_{-\infty}^{-q}dz\,(1+|z-w|^{2})^{-\frac{2+s}{2}}+\int^{+\infty}_{q}dz\,(1+|z-w|^{2})^{-\frac{2+s}{2}}\Big{\\}}\\\
&=\int_{-p}^{p}dw\,\Big{\\{}G^{\prime}(-q-w)-G^{\prime}(-\infty)+G^{\prime}(+\infty)-G^{\prime}(q-w)\Big{\\}}.\end{split}$
(A.5)

Note that $G^{\prime}$ is odd,
$0<-G^{\prime}(-\infty)=G^{\prime}(+\infty)<+\infty$, and
$G^{\prime}(-q-\cdot)-G^{\prime}(q-\cdot)$ is an even function. Therefore,

$\begin{split}\phi(p,q)&=2\int_{0}^{p}dw\,\Big{\\{}G^{\prime}(-q-w)-G^{\prime}(q-w)+2G^{\prime}(+\infty)\Big{\\}}\\\
&=4G^{\prime}(+\infty)p+2\int_{0}^{p}dw\,G^{\prime}(-q-w)-2\int_{0}^{p}dw\,G^{\prime}(q-w)\\\
&=4G^{\prime}(+\infty)p-2\int_{-q}^{-q-p}dt\,G^{\prime}(t)+2\int_{q}^{q-p}dt\,G^{\prime}(t)\\\
&=2\big{(}2G^{\prime}(+\infty)p+G(p-q)-G(p+q)\big{)},\end{split}$ (A.6)

where we also used in the last equality that $G$ is even. Now, from (2.3) and
(A.6) we deduce that

$\begin{split}{\mathcal{P}}_{s}[E]&=2\int_{-\pi}^{\pi}dx_{1}\int_{-\infty}^{+\infty}dy_{1}\,|x_{1}-y_{1}|^{-s}\\\
&\phantom{aaaaaaaa}\times\Big{\\{}G\Big{(}\frac{u(x_{1})-u(y_{1})}{|x_{1}-y_{1}|}\Big{)}-G\Big{(}\frac{u(x_{1})+u(y_{1})}{|x_{1}-y_{1}|}\Big{)}+2G^{\prime}(+\infty)\,\frac{u(x_{1})}{|x_{1}-y_{1}|}\Big{\\}}.\end{split}$
(A.7)

Our goal is to show that in the term involving
$2G^{\prime}(+\infty)u(x_{1})|x_{1}-y_{1}|^{-1}$ we can replace $u(x_{1})$ by
$u(y_{1})$ and the identity (A.7) remains unchanged. Once this is proved, we
get that in this last term one can replace $2u(x_{1})$ by $u(x_{1})+u(y_{1})$.
Hence, using that $H(t)=G^{\prime}(+\infty)t-G(t)$, this gives (A.1).

Therefore, it only remains to show that we can replace $u(x_{1})$ by
$u(y_{1})$ in the last term on the right-hand side of (A.7). This follows
immediately from Lemma 2.1 (used with $m=l=1$) applied to the integral

$\int_{-\pi}^{\pi}dx\\!\int_{-\infty}^{+\infty}dy\,\frac{f(x,y)}{|x-y|^{s}},$
(A.8)

where

$f(x,y):=-G\Big{(}\frac{u(x)+u(y)}{|x-y|}\Big{)}+2G^{\prime}(+\infty)\,\frac{u(x)}{|x-y|}.$

To check the hypothesis of the lemma, we need to verify that

$\int_{-\pi}^{\pi}dx\\!\int_{-\infty}^{+\infty}dy\,\frac{|f(x,y)|}{|x-y|^{s}}<+\infty.$
(A.9)

For this purpose we write $f(x,y)=A+B,$ where

$A:=-G\Big{(}\frac{u(x)+u(y)}{|x-y|}\Big{)}+G\Big{(}\frac{2u(x)}{|x-y|}\Big{)}$

and

$B:=-G\Big{(}\frac{2u(x)}{|x-y|}\Big{)}+2G^{\prime}(+\infty)\,\frac{u(x)}{|x-y|}=H\Big{(}\frac{2u(x)}{|x-y|}\Big{)}.$

Using that $u$ is nonnegative, $H$ is nonnegative and increasing in
$(0,+\infty)$, and $H(+\infty)<+\infty$, we have

$0\leq B\leq H(+\infty)<+\infty.$ (A.10)

Concerning the quantity $A$, observe that

$|A|=\left|\int_{\frac{u(x)+u(y)}{|x-y|}}^{\frac{2u(x)}{|x-y|}}dt\,G^{\prime}(t)\right|\leq\sup_{t\in{\mathbb{R}}}|G^{\prime}(t)|\,\frac{|u(x)-u(y)|}{|x-y|}=G^{\prime}(+\infty)\frac{|u(x)-u(y)|}{|x-y|}.$
(A.11)

Recall that we are assuming $u\in W^{s,1}(-\pi-\epsilon,\pi+\epsilon)$ for
some $\epsilon>0$. According to this, to prove (A.9) we split the integral and
we use (A.10) and (A.11) to get

$\begin{split}&\int_{-\pi}^{\pi}dx\\!\int_{-\infty}^{+\infty}dy\,\frac{|f(x,y)|}{|x-y|^{s}}=\int_{-\pi}^{\pi}dx\\!\int_{|x-y|\leq\epsilon}dy\,\frac{|A|+|B|}{|x-y|^{s}}+\int_{-\pi}^{\pi}dx\\!\int_{|x-y|>\epsilon}dy\,\frac{|f(x,y)|}{|x-y|^{s}}\\\
&\leq
G^{\prime}(+\infty)\|u\|_{W^{s,1}(-\pi-\epsilon,\pi+\epsilon)}+H(+\infty)\int_{-\pi}^{\pi}dx\\!\int_{-\pi-\epsilon}^{\pi+\epsilon}\frac{dy}{|x-y|^{s}}+\int_{-\pi}^{\pi}dx\\!\int_{|x-y|>\epsilon}dy\,\frac{|f(x,y)|}{|x-y|^{s}}.\end{split}$
(A.12)

The first two terms on the right-hand side of (A.12) are finite. To estimate
the third one, we use that $0\leq G(t)\leq G^{\prime}(+\infty)t$ for all
$t\geq 0,$ and hence

$\begin{split}\int_{-\pi}^{\pi}dx\\!\int_{|x-y|>\epsilon}dy\,\frac{|f(x,y)|}{|x-y|^{s}}\leq&\,G^{\prime}(+\infty)\int_{-\pi}^{\pi}dx\\!\int_{|x-y|>\epsilon}dy\,\frac{3u(x)}{|x-y|^{1+s}}\\\
&+G^{\prime}(+\infty)\int_{-\pi}^{\pi}dx\\!\int_{|x-y|>\epsilon}dy\,\frac{u(y)}{|x-y|^{1+s}}.\end{split}$
(A.13)

The first double integral in the right-hand side of (A.13) is finite, since
$s>0$ and $u\in W^{s,1}(-\pi-\epsilon,\pi+\epsilon)\subset L^{1}(-\pi,\pi)$.
For the second double integral, since $u$ is $2\pi$-periodic and nonnegative,
using Lemma 2.1 we deduce

$\int_{-\pi}^{\pi}dx\\!\int_{|x-y|>\epsilon}dy\,\frac{u(y)}{|x-y|^{1+s}}=\int_{-\pi}^{\pi}dx\\!\int_{{\mathbb{R}}}dy\,\frac{\chi_{\\{|x-y|>\epsilon\\}}}{|x-y|^{1+s}}\,u(y)=\int_{-\pi}^{\pi}d\overline{y}\,u(\overline{y})\int_{|\overline{x}-\overline{y}|>\epsilon}\frac{d\overline{x}}{|\overline{x}-\overline{y}|^{1+s}}.$
(A.14)

As before, this is finite since $s>0$ and $u\in L^{1}(-\pi,\pi)$. Hence (A.9)
follows from (A.12) and the finiteness of the integrals in (A.13). ∎

##  Appendix B Strict monotonicity of the periodic heat kernel

In this section we address the proof of an important ingredient in the proof
of Theorem 1.2, namely, that the function defined in (3.20) is decreasing in
$(0,\pi)$. As shown there, this property follows from the fact that the
periodic heat kernel is decreasing with respect to the spatial variable in
$(0,\pi)$. This is a well-known fact [1, 3, 14, 15, 24], whose proofs are
commented next. Here, for the sake of completeness, we provide a simple proof
based on maximum principles for the heat equation.

Prior to this, let us briefly comment on the proofs given in the above-
mentioned references. The one in [1] follows exactly the same lines as the one
presented here, but it is carried out within the context of hypersurfaces of
revolution. The one in [15] is also based on the strong maximum principle for
the heat equation, but it is more technically involved than [1], since a more
general class of manifolds is considered there. The proof in [3] is based on
an explicit computation. The one in [14] follows by expressing the derivative
of the kernel in terms of the heat kernel in smaller dimensions and using an
induction argument; see [14, §6.3]. Finally, the one in [24] is a
straightforward application of the sharp estimates for the heat kernel on the
sphere $\mathbb{S}^{n}$ found there.

The fundamental solution of the heat equation in $(-\pi,\pi)$ with periodic
boundary conditions is given by

$\Gamma(z,t):=\frac{1}{\sqrt{4\pi
t}}\sum_{k\in{\mathbb{Z}}}e^{-\frac{(z+2k\pi)^{2}}{4t}}\quad\text{for
$z\in{\mathbb{R}}$ and $t>0$}.$ (B.1)

Given a $2\pi$-periodic initial data $g,$ the function

$u(x,t)=\int_{-\pi}^{\pi}dy\,\Gamma(x-y,t)g(y)$ (B.2)

satisfies $\partial_{t}u-\partial_{xx}u=0$ in ${\mathbb{R}}\times(0,+\infty)$,
that $u(\cdot,t)$ is $2\pi$-periodic for all $t>0$, and that $u(\cdot,0)=g$ in
${\mathbb{R}}$. This follows immediately from the properties of the standard
(nonperiodic) heat kernel and the fact that

$\begin{split}\sum_{k\in{\mathbb{Z}}}\int_{-\pi}^{\pi}dy\,e^{-\frac{(x-y+2k\pi)^{2}}{4t}}g(y)&=\sum_{k\in{\mathbb{Z}}}\int_{-\pi}^{\pi}dy\,e^{-\frac{(x-y+2k\pi)^{2}}{4t}}g(y-2k\pi)\\\
&=\sum_{k\in{\mathbb{Z}}}\int_{-\pi-2k\pi}^{\pi-2k\pi}d\overline{y}\,e^{-\frac{(x-\overline{y})^{2}}{4t}}g(\overline{y})=\int_{{\mathbb{R}}}d\overline{y}\,e^{-\frac{(x-\overline{y})^{2}}{4t}}g(\overline{y}).\end{split}$
(B.3)

We now state and prove the result on the periodic heat kernel that we used in
the proof of Theorem 1.2.

######  Theorem B.1.

For every $t>0$, the function

$z\mapsto\Gamma(z,t)\quad\text{is decreasing in }(0,\pi).$

###### Proof.

Let us first show that $\Gamma(\cdot,t)$ is nonincreasing in $(0,\pi)$. For
this, we take functions $g_{\epsilon}\in C_{c}^{\infty}(-\epsilon,\epsilon)$,
where $0<\epsilon<\pi/2$, approximating the Dirac delta as $\epsilon\downarrow
0$, and we consider their $2\pi$-periodic extensions from $[-\pi,\pi]$ to
${\mathbb{R}}$ (which we also denote by $g_{\epsilon}$). In particular, we
assume that

$\int_{-\epsilon}^{\epsilon}g_{\epsilon}=1\quad\text{for all
$0<\epsilon<\pi/2$}.$

We additionally assume that the $g_{\epsilon}$ are nonnegative, even, and
nonincreasing in $(0,\pi)$.

Let $u_{\epsilon}$ be the solution to

$\left\\{\begin{array}[]{ll}\partial_{t}u_{\epsilon}-\partial_{xx}u_{\epsilon}=0&\text{
in }{\mathbb{R}}\times(0,+\infty),\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
u_{\epsilon}(\cdot,0)=g_{\epsilon}&\text{ in
${\mathbb{R}}$}.\end{array}\right.$

Thus, $u_{\epsilon}(x,t)=\int_{-\pi}^{\pi}dy\,\Gamma(x-y,t)g_{\epsilon}(y)$
and $u_{\epsilon}(\cdot,t)$ is $2\pi$-periodic for all $t>0$. Notice that
$u_{\epsilon}(\cdot,t)$ is even with respect to $x=0$ and $x=\pi$ (by
uniqueness, since so is $g_{\epsilon}$). We deduce that the derivative
$v_{\epsilon}:=\partial_{x}u_{\epsilon}$ solves

$\left\\{\begin{array}[]{ll}\partial_{t}v_{\epsilon}-\partial_{xx}v_{\epsilon}=0&\text{
in }(0,\pi)\times(0,+\infty),\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
v_{\epsilon}(0,t)=v_{\epsilon}(\pi,t)=0&\text{ for all
$t>0$}.\end{array}\right.$

Moreover, by the assumptions on $g_{\epsilon}$, we have that
$v_{\epsilon}(x,0)=\partial_{x}u_{\epsilon}(x,0)=g_{\epsilon}^{\prime}(x)\leq
0$ for all $x\in[0,\pi]$. Hence, the maximum principle yields

$v^{\epsilon}\leq 0\quad\text{ in }[0,\pi]\times[0,+\infty).$ (B.4)

We now claim that, for every given $t>0$ and $m=0,1,2,\ldots$,

$u_{\epsilon}(\cdot,t)\text{ converges to }\Gamma(\cdot,t)\text{ in
$C^{m}([0,\pi])$ as $\epsilon\downarrow 0$}.$ (B.5)

This follows by combining the uniform continuity in $[0,\pi]$ of
$\Gamma(\cdot,t)$ and of all its derivatives with the fact that, for every
$\delta>0$, we have

$\big{|}\partial^{m}_{x}u_{\epsilon}(x,t)-\partial^{m}_{x}\Gamma(x,t)\big{|}\leq\int_{-\epsilon}^{\epsilon}dy\,\big{|}\partial^{m}_{x}\Gamma(x-y,t)-\partial^{m}_{x}\Gamma(x,t)\big{|}|g_{\epsilon}(y)|\leq\delta$

if $\epsilon$ is chosen small enough.

Now, from (B.4) and (B.5) we conclude that

$\partial_{x}\Gamma(x,t)=\lim_{\epsilon\downarrow
0}\partial_{x}u_{\epsilon}(x,t)=\lim_{\epsilon\downarrow
0}v_{\epsilon}(x,t)\leq 0$

for all $x\in[0,\pi]$ and $t>0$.

Finally, we show that $\Gamma(\cdot,t)$ is decreasing in $(0,\pi)$. For this,
notice that we already proved that, for all $t>0$, $\Gamma(\cdot,t)$ is even
with respect to $x=0$ and $x=\pi$. Therefore, given any $t_{0}>0$, we see that
$v:=\partial_{x}\Gamma$ solves

$\left\\{\begin{array}[]{ll}\partial_{t}v-\partial_{xx}v=0&\text{ in
}(0,\pi)\times(t_{0},+\infty),\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
v(0,t)=v(\pi,t)=0&\text{ for all $t>t_{0}$},\vskip 3.0pt plus 1.0pt minus
1.0pt\\\ v(\cdot,t_{0})=\partial_{x}\Gamma(\cdot,t_{0})\leq 0&\text{ in
$[0,\pi]$}.\end{array}\right.$ (B.6)

Now, the strong maximum principle yields that either $v<0$ in
$(0,\pi)\times(t_{0},+\infty)$ or $v\equiv 0$ in
$[0,\pi]\times[t_{0},+\infty)$. The proof is now complete, since $v\equiv 0$
is absurd —it would give that $\Gamma(\cdot,t)$ is constant in $x$, clearly
contradicting the fact that $\Gamma$ is the fundamental solution of the heat
equation with periodic boundary conditions, as shown in the beginning of this
appendix. ∎

## Acknowledgments

We thank M. Ritoré, E. Valdinoci, and T. Weth for sharing some references and
information on the topic of the paper.

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