# Some weighted isoperimetric inequalities in quantitative form

Nicola Fusco and Domenico Angelo La Manna

###### Abstract.

In this paper we study two different weighted isoperimetric inequalities.
First we prove a sharp stability result for the isoperimetric inequality with
a log-convex weight. Then we analize the behavior of a negative power weight
for the perimeter thus providing a complete picture of the isoperimetric
problem in this context.

##  1\. Introduction

In recent years weighted isoperimetric inequalities have attracted the
attention of many authors ([31],[36],[10],[32],[34],[35]) also in view of
their applications to different fields of Analysis. They play an important
role in dealing with Gamow type energies, see for instance [26], [5], [27],
[25], [20], [29] and in shape optimization problems involving eigenvalues, see
[7],[19],[24],[16] and the references therein.

Due to the relevance of the topic for applications, it would be important to
understand stability properties of such inequalities. Isoperimetric
inequalities in quantitative form have a long history, see [23], [21], [14],
and [22] for a complete overview on the subject.

In this paper we focus our attention on two types of weighted isoperimetric
problems: first we study the case of a log-convex density $e^{w(|x|)}$ and
then we consider a power density $|x|^{p}$ with $p$ negative. While for the
Gaussian isoperimetric inequality, where the density $e^{-|x|^{2}}$ is log-
concave, it is well known that isoperimetric sets are half spaces ([6]) and
that they are stable ([13],[2], see also [33, 18, 11] for the non local
extension), the case of a log-convex weight has been only recently settled by
G.R. Chambers in [12]. In that paper he proved that if $w$ is a $C^{3}$ even
convex function, then balls centered at the origin are the unique minimizers
of the weighted perimeter under a weighted volume constraint. Note that in the
Gaussian case it can be proved that balls with small weighted mass are local
minimizers while this property fails if the mass is sufficiently large, see
[28].

In this paper we study the stability of the isoperimetric problem with density
$e^{w(|x|)}$. For a set $E$ of locally finite perimeter we denote by $|E|_{w}$
and $P_{w}(E)$ respectively

$|E|_{w}=\int_{E}e^{w(|x|)}\,dx\;\text{ and
}\;P_{w}(E)=\int_{\partial^{*}E}e^{w(|x|)}\,d\mathcal{H}^{n-1},$

where $\partial^{*}E$ is the so called reduced boundary of $E$, see [30]. We
prove that the aforementioned isoperimetric inequality is actually stable. To
be precise, our main theorem reads as follows. Here and in the following we
denote the ball of radius $r$ centered at $x$ by $B_{r}(x)$ or simply by
$B_{r}$ when the center is the origin.

######  Theorem 1.1.

Let $n\geq 2$. Given an even convex function $w\in C^{3}(\mathbb{R})$ and
$r>0$ such that $w^{\prime\prime}(r)>0$, there exists a constant
$\kappa=\kappa(n,r,w)$ such that, for any measurable set
$E\subset\mathbb{R}^{n}$ with $|E|_{w}=|B_{r}|_{w}$,

(1.1) $P_{w}(E)-P_{w}(B_{r})\geq\kappa|E\triangle B_{r}|_{w}^{2}.$

We stress that although the constant in (1.1) might not be optimal, the
exponent is sharp as can be seen by computing the value of the weighted
perimeter on suitable ellipsoids, see for instance a similar example in
Section 4 of [22]. Note also that differently from the quantitative
isoperimetric inequality for the standard Euclidean perimeter, in our case we
have no scaling properties and this explains why on the right hand side of
(1.1) the constant depends on $r$ and we have $|E\triangle B_{r}|_{w}$ instead
of the usual Fraenkel asymmetry.

We now give a brief overview of the proof. Inspired by [14], we use the so
called selection principle. We first prove that (1.1) holds true for a special
class of sets, namely sets which are sufficiently close to the ball centered
at the origin with the same weighted volume. This first step of the proof is
achieved by a Fuglede type argument. Then we reduce by a compactness argument
to the case where the set $E$ is close to such a ball in the $L^{\infty}$
sense. Precisely we argue by contradiction assuming that there exists a
sequence $\\{E_{h}\\}_{h\in\mathbb{N}}$ such that $|E_{h}|_{w}=|B_{r}|_{w}$
for all $h$ and (1.1) fails for a suitably small constant. Following an idea
of [1] we construct a sequence of functionals $J_{h}$ whose minimizers $F_{h}$
also satisfy the opposite inequality in (1.1) with a small constant and
converge in $C^{1,\alpha}$ to $B_{r}$, thus getting a contradiction with the
estimate proved in the first part of the proof. Although this type of argument
has become more or less standard, one of the key difficulties here, due to the
fact that the density diverges at infinity, is to show that the functionals
$J_{h}$ do admit a minimizer and to get suitable a priori estimate ensuring
the $C^{1,\alpha}$ convergence of $F_{h}$.

Another difficulty in this context, maybe the most challanging one, comes from
the fact that neither the weighted volume nor the weighted perimeter are
invariant under translations. This would make a Fuglede type estimate for
nearly spherical sets $E$ useless, since it usually requires that the
barycenter of $E$ is at the origin. However, an interesting feature of our
problem is that the assumption $w^{\prime\prime}(r)>0$ yields such a Fuglede
type estimate without any further hypothesis on the barycenter, see (3.1).
Even more, this assumption turns out to be necessary for the validity of the
quantitative inequality (1.1). More precisely, the following result holds.

######  Proposition 1.2.

Let $w\in C^{2}(\mathbb{R})$ be a convex function such that
$w^{\prime\prime}(r)=0$ for some $r>0$. Then

$\lim_{\varepsilon\to 0+}\frac{P_{w}(B_{\rho(\varepsilon)}(\varepsilon
e_{1}))-P_{w}(B_{r})}{|B_{\rho(\varepsilon)}(\varepsilon e_{1})\triangle
B_{r}|_{w}^{2}}=0\,,$

where $\rho(\varepsilon)>0$ is such that $|B_{\rho(\varepsilon)}(\varepsilon
e_{1})|_{w}=|B_{r}|$.

The last part of the paper is devoted to another weighted isoperimetric
inequality. This time we do not deal with a log-convex density. Instead, the
weight is given by $|x|^{p}$, with $p$ negative. While for $p>0$ the
characterization of the balls centered at the origin as the unique
isoperimetric sets and their stability is well known, see [3], [17], [15],[8]
and the references therein, the case $p<0$ is less understood. First of all,
notice that when $-n+1<p<0$ then the problem becomes trivial since for any
fixed mass the infimum of the weighted perimeter under the mass constraint is
0. On the other hand, it is known (see [17]) that if $p\leq 1-n$ and $E$ is a
bounded open set with Lipschitz boundary containing the origin, then

(1.2) $\int_{\partial^{*}E}|x|^{p}\,d\mathcal{H}^{n-1}\geq\int_{\partial
B_{r}}|x|^{p}\,d\mathcal{H}^{n-1},$

where $B_{r}$ has the same volumeof $E$. Moreover, if equality holds in (1.2),
$E$ coincides with $B_{r}$. Note that the assumption that the origin belongs
to the interior of $E$ is crucial, since it can be easily checked that (1.2)
may fail when the origin belongs to the interior of the complement of $E$. In
the last section of the paper we extend (1.2) to the case of a set $E$ of
locally finite perimeter such that $0\in\operatorname{int}(E^{(1)})$, where
$E^{(1)}$ is the set of points where $E$ has density 1. Note that if
$0\in\partial^{*}E$ (1.2) becomes trivial because in this case the left hand
side is infinite, see Remark 6.2. Note also that the assumption that
$0\in\operatorname{int}(E^{(1)})$ is sharp in the sense that one may construct
a set of finite perimeter $E$ such that $0\in\partial
E^{(1)}\setminus\partial^{*}E$ for which the inequality fails. Finally we
prove that (1.2) also holds in a quantitative form.

##  2\. Notation and Preliminary Results

Throughout all the paper we will assume that $w:\mathbb{R}\to\mathbb{R}$ is a
$C^{3}$ even convex function. In the sequel by $C,c$ we denote positive
constants whose value may change from line to line and occasionally we
highlight the dependence of these constants by other parameters. The
dependence of the constants on $w$ will be always tacitly understood. For
$n\geq 2$ let $E\subset\mathbb{R}^{n}$ be a measurable set and
$\Omega\subset\mathbb{R}^{n}$ an open set. We say that $E$ has finite weighted
perimeter with respect to $e^{w(x)}$ in $\Omega$ if

$P_{w}(E;\Omega)=\sup_{\|X\|_{L^{\infty}(\mathbb{R}^{n})}\leq
1}\left\\{\int_{\Omega}\operatorname{div}(e^{w(x)}X(x))\,dx,\,\,X\in
C^{1}_{c}(\mathbb{R}^{n};\mathbb{R}^{n})\right\\}.$

From this definition it follows that if a set has finite weighted perimeter in
$\Omega$ then $P(E;\Omega)<\infty$, where $P(E;\Omega)$ denotes the standard
Euclidean perimeter in $\Omega$. If $\Omega=\mathbb{R}^{n}$ we simply write
$P_{w}(E)$ or $P(E)$ in place of $P_{w}(E;\mathbb{R}^{n})$ and
$P(E;\mathbb{R}^{n})$. For the definitions and properties of sets of finite
perimeter we refer to [30]. Note that if $\partial^{*}E$ is the reduced
boundary of $E$ from the definition above we have

$P_{w}(E;\Omega)=\int_{\partial^{*}E\cap\Omega}e^{w(x)}\,d\mathcal{H}^{n-1},$

where $\mathcal{H}^{n-1}$ denotes the $(n-1)$-dimensional Hausdorff measure.
We recall that at every point $x\in\partial^{*}E$ the exterior generalized
normal $\nu_{E}(x)$ is defined and the following generalized Gauss-Green
formula holds

$\int_{E}\operatorname{div}X\,dx=\int_{\partial^{*}E}\langle
X,\nu_{E}\rangle\,d\mathcal{H}^{n-1}$

for every vector field $X\in C_{c}^{1}(\Omega,\mathbb{R}^{n})$. We now
introduce the notion of quasiminimizer of the perimeter.

######  Definition 2.1.

Let $E\subset\mathbb{R}^{n}$ be a set of locally finite perimeter, $\omega\geq
0$ and let $\Omega$ be an open subset of $\mathbb{R}^{n}$. We say that $E$ is
an $\omega$-minimizer of the perimeter in $\Omega$ if for every ball
$B_{\rho}(x)\subset\subset\Omega$ with $\rho<1$ and for any set $F$ of locally
finite perimeter such that $E\triangle F\subset\subset B_{\rho}(x)$ it holds

(2.1) $P(E;B_{\rho}(x))\leq P(F;B_{\rho}(x))+\omega\rho^{n}$

The following theorem is a consequence of the classical De Giorgi’s
$\varepsilon$-regularity theorem, see for instance [37, Theorem 1.9] and also
the argument of the proof of [16, Lemma 3.6]. Before stating it we recall that
if $E_{h}$ and $E$ are measurable sets of $\mathbb{R}^{n}$ and $\Omega$ is an
open set, one says that $E_{h}\to E$ in measure in $\Omega$ if
$|E_{h}\triangle E\cap\Omega|\to 0$. The local convergence in measure is
defined in the obvious way.

######  Theorem 2.2.

Assume that $E_{h},E$ are equibounded $\omega$-minimizers of the perimeter in
$\mathbb{R}^{n}$ such that $E_{h}\to E$ in measure. If $E$ is of class $C^{2}$
then for $h$ large $E_{h}$ is of class $C^{1,\frac{1}{2}}$ and there exists a
function $v_{h}:\partial E\to\mathbb{R}$ such that

$\partial E_{h}=\\{x+v_{h}(x)\nu_{E}(x),\,\,x\in\partial E\\}.$

Moreover, $\|v_{h}\|_{C^{1,\alpha}(\partial E)}\to 0$ for
$0<\alpha<\frac{1}{2}$.

##  3\. A first stability estimate

In this section we give a Fuglede type result for nearly spherical sets under
the assumption that $w^{\prime\prime}(r)>0$. The interesting feature of this
result is that this assumption allows to prove the quantitative estimate (3.1)
without any assumption on the barycenter of $E$. We start with the definition
of nearly spherical sets.

######  Definition 3.1.

Let $n\geq 2$. We say that a set $E$ is nearly spherical if there exist $r>0$
and a Lipschitz function $u:\mathbb{S}^{n-1}\to(-1,1)$ such that

$E=\\{y:y=rx(1+u(x)),\,x\in\mathbb{S}^{n-1}\\}.$

######  Proposition 3.2.

Given $0<r_{1}<r_{2}$ such that $w^{\prime\prime}(r)>0$ for all
$r\in[r_{1},r_{2}]$, there exist $\varepsilon,c>0$ such that if $E$ is a
nearly spherical set as in Definition 3.1 with
$\|v\|_{W^{1,\infty}(\mathbb{S}^{n-1})}<\varepsilon$, $|B_{r}|_{w}=|E|_{w}$,
$r\in[r_{1},r_{2}]$, then

(3.1) $P_{w}(E)-P_{w}(B_{r})\geq
cr^{n-1}e^{w(r)}\|\nabla_{\tau}u\|_{L^{2}(\mathbb{S}^{n-1})}^{2},$

where $\nabla_{\tau}u$ stands for the tangential gradient of $u$.

######  Remark 3.3.

Note that by the Poincaré inequality on the sphere, (3.1) implies

(3.2) $P_{w}(E)-P_{w}(B_{r})\geq
cr^{n-1}e^{w(r)}\|u\|_{L^{2}(\mathbb{S}^{n-1})}^{2}\geq
c^{\prime}r^{n-1}e^{w(r)}|E\triangle B_{r}|^{2}_{w}.$

However (3.1) is clearly stronger than the quantitative inequality (1.1).

###### Proof of Lemma 3.2.

Given $r\in[r_{1},r_{2}]$ we consider the Lipschitz map $\Psi:\overline{B}\to
E$ defined by $\Psi(x)=rx\left(1+u\left(\frac{x}{|x|}\right)\right)$. A
straightforward computation shows that for every $x\in B$ the Jacobian $J\Psi$
at $x$ is given by

(3.3) $J\Psi(x)=r^{n}\left(1+u\left(\frac{x}{|x|}\right)\right)^{n}$

while for $x\in\mathbb{S}^{n-1}$ the tangential Jacobian is

$J_{\tau}\Psi(x)=r^{n-1}(1+u(x))^{n-2}\sqrt{(1+u)^{2}+|\nabla_{\tau}u|^{2}}.$

From this last equality, using the area formula, we get

$P_{w}(E)=r^{n-1}\int_{\mathbb{S}^{n-1}}(1+u(x))^{n-1}\sqrt{1+\frac{|\nabla_{\tau}u|^{2}}{(1+u)^{2}}}e^{w(r(1+u(x)))}\,d\mathcal{H}^{n-1}$

and recalling (3.3)

$|E|_{w}=r^{n}\int_{\mathbb{S}^{n-1}}(1+u(x))^{n}\int_{0}^{1}t^{n-1}e^{w(rt(1+u(x))}\,dt\,dx.$

Hence

$\begin{split}P_{w}(E)-P_{w}(B_{r})=&r^{n-1}\int_{\mathbb{S}^{n-1}}\left((1+u(x))^{n-1}\sqrt{1+\frac{|\nabla_{\tau}u|^{2}}{(1+u)^{2}}}e^{w(r(1+u(x)))}-e^{w(r)})\right)\,d\mathcal{H}^{n-1}\\\
=&r^{n-1}\int_{\mathbb{S}^{n-1}}(1+u(x))^{n-1}\left(\sqrt{1+\frac{|\nabla_{\tau}u|^{2}}{(1+u)^{2}}}-1\right)e^{w(r(1+u(x)))}\,d\mathcal{H}^{n-1}\\\
&+r^{n-1}\int_{\mathbb{S}^{n-1}}(1+u(x))^{n-1}e^{w(r(1+u(x)))}-e^{w(r)}\,d\mathcal{H}^{n-1}:=r^{n-1}(I_{1}+I_{2})\end{split}$

Using the hypothesis $\|u\|_{W^{1,\infty}}<\varepsilon$ we can control the
term $I_{1}$ by

$I_{1}\geq\left(\frac{1}{2}-C\varepsilon\right)e^{w(r)}\int_{\mathbb{S}^{n-1}}|\nabla_{\tau}u|^{2}d\mathcal{H}^{n-1},$

for some constant $C>0$ uniformly bounded for $r\in[r_{1},r_{2}]$. To estimate
$I_{2}$, by a second order Taylor expansion we have

$\begin{split}\frac{(1+u(x))^{n-1}e^{w(r(1+u(x)))}-e^{w(r)}}{e^{w(r)}}=(n-1+rw^{\prime}(r))u\\\
+\frac{1}{2}((n-1)(n-2)+2(n-1)rw^{\prime}(r)+r^{2}w^{\prime\prime}(r)+r^{2}w^{\prime}(r)^{2})u^{2}+o(u^{2}).\end{split}$

From this equality and the estimate on $I_{1}$ we get

(3.4)
$\begin{split}&\frac{P_{w}(E)-P_{w}(B_{r})}{r^{n-1}e^{w(r)}}\geq\left(\frac{1}{2}-C\varepsilon\right)\int_{\mathbb{S}^{n-1}}|\nabla_{\tau}u|^{2}d\mathcal{H}^{n-1}+(n-1+rw^{\prime}(r))\int_{\mathbb{S}^{n-1}}ud\mathcal{H}^{n-1}\\\
&\;\;\;\;\;\;+\frac{1}{2}((n-1)(n-2)+2(n-1)rw^{\prime}(r)+r^{2}w^{\prime\prime}(r)+r^{2}w^{\prime}(r)^{2}-C\varepsilon)\int_{\mathbb{S}^{n-1}}u^{2}d\mathcal{H}^{n-1},\end{split}$

for a constant $C$ uniformly bounded for $r\in[r_{1},r_{2}]$. Since
$|E|_{w}=|B_{r}|_{w}$ we have

$\int_{\mathbb{S}^{n-1}}(1+u(x))^{n}\int_{0}^{1}t^{n-1}e^{w(rt(1+u(x))}\,dt\,d\mathcal{H}^{n-1}=\int_{\mathbb{S}^{n-1}}\int_{0}^{1}t^{n-1}e^{w(rt)}\,dt\,d\mathcal{H}^{n-1}$

which in turn gives

$\int_{0}^{1}t^{n-1}\int_{\mathbb{S}^{n-1}}\left((1+u(x))^{n}e^{w(rt(1+u(x)))}-e^{w(rt)}\right)\,d\mathcal{H}^{n-1}\,dt=0.$

By a second order Taylor expansion of the functions $(1+\cdot)^{n}$ and
$e^{w(rt(1+\cdot))}$, using again the smallness assumption on $u$, we get

(3.5)
$\begin{split}(na_{n}+rb_{n})\int_{\mathbb{S}^{n-1}}u\,d\mathcal{H}^{n-1}\,dt\geq&-\frac{1}{2}(n(n-1)a_{n}+2nrb_{n}+r^{2}c_{n}+r^{2}d_{n})\int_{\mathbb{S}^{n-1}}u^{2}\,d\mathcal{H}^{n-1}\,dt\\\
&-C\varepsilon\int_{\mathbb{S}^{n-1}}u^{2}\,d\mathcal{H}^{n-1},\end{split}$

where

$a_{n}=\int_{0}^{1}t^{n-1}e^{w(rt)}\,dt,\;\;\;b_{n}=\int_{0}^{1}t^{n}w^{\prime}(rt)e^{w(rt)}\,dt,\;\;c_{n}=\int_{0}^{1}t^{n+1}w^{\prime\prime}(rt)e^{w(rt)}\,dt$

and

$d_{n}=\int_{0}^{1}t^{n+1}(w^{\prime}(rt))^{2}e^{w(rt)}\,dt.$

Integrating by parts we have the following identities

$rb_{n}=e^{w(r)}-na_{n},\;\;\;r^{2}(c_{n}+d_{n})=rw^{\prime}(r)e^{w(r)}-(n+1)(e^{w(r)}-na_{n}),$

which in turn imply that (3.5) can be rewritten as

(3.6)
$(n-1+rw^{\prime}(r))\int_{\mathbb{S}^{n-1}}u\,d\mathcal{H}^{n-1}\geq-\left(\frac{(n-1+rw^{\prime}(r))^{2}}{2}-C\varepsilon\right)\int_{\mathbb{S}^{n-1}}u^{2}\,d\mathcal{H}^{n-1},$

where again $C$ depends only on $r_{1},r_{2}$.

Note that if $\int_{\mathbb{S}^{n-1}}u\,d\mathcal{H}^{n-1}\geq 0$ then (3.1)
follows at once from (3.4) with $c=\frac{1}{4}$, provided $\varepsilon$ is
small enough. Assume instead that

(3.7) $\int_{\mathbb{S}^{n-1}}u\,d\mathcal{H}^{n-1}<0.$

Collecting all the previous inequalities we have

(3.8)
$\begin{split}\frac{P_{w}(E)-P_{w}(B)}{r^{n-1}e^{w(r)}}\geq&\left(\frac{1}{2}-C\varepsilon\right)\|\nabla
u\|_{L^{2}(\mathbb{S}^{n-1})}^{2}\\\
&+\frac{1}{2}\left(1-n+r^{2}w^{\prime\prime}(r)-C\varepsilon\right)\|u\|_{L^{2}(\mathbb{S}^{n-1})}^{2}.\end{split}$

For $k\in\mathbb{N}$ and $i\in\\{1,\dots,G(n,k)\\}$ let $Y_{k,i}$ be the
spherical harmonics of order $k$, i.e., the restrictions to $\mathbb{S}^{n-1}$
of the homogeneous harmonic polynomials of degree $k$, normalized so that
$\|Y_{k,i}\|_{L^{2}(\mathbb{S}^{n-1})}=1$. Note that
$\\{Y_{k,i}\\}_{k\in\mathbb{N},i\leq G(n,k)}$ forms an orthonormal system for
$L^{2}(\mathbb{S}^{n-1})$ and that for every $k,i$

$-\Delta_{\mathbb{S}^{n-1}}Y_{k,i}=k(k+n-2)Y_{k,i},$

where $\Delta_{\mathbb{S}^{n-1}}$ is the Laplace-Beltrami operator on
$\mathbb{S}^{n-1}$. Hence, we may represent $u$ with respect to this
orthonormal system as

$u=\sum_{k=0}^{\infty}\sum_{i=1}^{G(n,k)}a_{k,i}Y_{k,i},\quad\text{where
}\quad a_{k,i}=\int_{\mathbb{S}^{n-1}}uY_{k,i}\,d\mathcal{H}^{n-1},$

thus getting

$\|u\|_{L^{2}(\mathbb{S}^{n-1})}^{2}=\sum_{k\geq
0}\sum_{i=1}^{G(n,k)}a_{k,i}^{2}\quad\text{and}\quad\|\nabla_{\tau}u\|_{L^{2}(\mathbb{S}^{n-1})}=\sum_{k\geq
1}\sum_{i=1}^{G(n,k)}k(k+n-2)a^{2}_{k,i}.$

Note that condition (3.6), together with (3.7), implies that

$a_{0,1}^{2}\leq C\varepsilon\|u\|_{L^{2}(\mathbb{S}^{n-1})}^{2}.$

Observe that if $1-n+r^{2}w^{\prime\prime}(r)>0$ the conclusion holds with
$c=1/4$ and $\varepsilon$ sufficiently small. Otherwise,

$\|\nabla_{\tau}u\|^{2}_{L^{2}(\mathbb{S}^{n-1})}=\sum_{k\geq
1}\sum_{i=1}^{G(n,k)}k(k+n-2)a^{2}_{k,i}\geq(n-1)\|u\|^{2}_{L^{2}(\mathbb{S}^{n-1})}-C\varepsilon\|u\|_{L^{2}(\mathbb{S}^{n-1})}^{2}.$

Employing the latter in (3.8) we infer

$P_{w}(E)-P_{w}(B)\geq\frac{1}{2}r^{n-1}e^{w(r)}\left(r^{2}w^{\prime\prime}(r)-C\varepsilon\right)\|\nabla_{\tau}u\|_{L^{2}(\mathbb{S}^{n-1})}$

which concludes the proof of (3.1) taking $\varepsilon$ small enough. ∎

Note that a suitable modification of the above proof, see for instance the
proof of [22, Th. 3.1] immediately yields that if $w^{\prime\prime}(r)=0$ then
(3.1) still holds under the assumption that the barycenter of $E$ is at the
origin. However, if this condition is not satisfied and
$w^{\prime\prime}(r)=0$, then not only (3.1), but even (1.1) is not longer
true, see Proposition 1.2.

In order to prove this, given $r>0$, for any $\varepsilon>0$ we denote by
$\rho(\varepsilon)$ the unique positive number such that

$|B_{r}|_{w}=|B_{\varrho(\varepsilon)}(\varepsilon e_{1})|_{w}.$

Since $w$ is of class $C^{2}$, it is easily checked that $\rho\in
C^{2}(0,\infty)$.

###### Proof of Proposition 1.2.

To simplify the notation we assume, without loss of generality, that $r=1$ and
set $B=B_{1}$. For $\varepsilon>0$ let $\rho=\rho(\varepsilon)$ such that

(3.9) $|B|_{w}=|B_{\rho(\varepsilon)}(\varepsilon e_{1})|_{w}.$

Clearly, $\rho(0)=1$. Differentiating (3.9) with respect to $\varepsilon$ we
get

(3.10)
$\begin{split}0&=\frac{d}{d\varepsilon}|B_{\rho(\varepsilon)}(\varepsilon
e_{1})|_{w}=\frac{d}{d\varepsilon}\left(\rho(\varepsilon)^{n}\int_{B}e^{w(|\rho(\varepsilon)x+\varepsilon
e_{1}|)}\,dx\right)\\\
=&n\rho(\varepsilon)^{n-1}\rho^{\prime}(\varepsilon)\int_{B}e^{w(|\rho(\varepsilon)x+\varepsilon
x_{1}|)}\,dx\\\
&+\rho(\varepsilon)^{n}\int_{B}e^{w(|\rho(\varepsilon)x+\varepsilon
e_{1}|)}w^{\prime}(|\rho(\varepsilon)x+\varepsilon
e_{1}|)\Big{\langle}\frac{\rho(\varepsilon)x+\varepsilon
e_{1}}{|\rho(\varepsilon)x+\varepsilon
e_{1}|},\rho^{\prime}(\varepsilon)x+e_{1}\Big{\rangle}\,dx.\end{split}$

By symmetry

$\int_{B}e^{w(|x|)}w^{\prime}(|x|)\frac{x_{1}}{|x|}\,dx=0.$

Therefore, evaluating (3.10) at $\varepsilon=0$, we get

$\rho^{\prime}(0)\left(n|B|_{w}+\int_{B}e^{w(|x|)}|x|w^{\prime}(|x|)\,dx\right)=0$

which implies $\rho^{\prime}(0)=0$. Differentiating (3.10) again with respect
to $\varepsilon$ and evaluating the second derivative at $\varepsilon=0$ we
also get

$\begin{split}&\rho^{\prime\prime}(0)\bigg{(}n|B|_{w}+\int_{B}e^{w(|x|)}|x|w^{\prime}(|x|)\,dx\bigg{)}+\int_{B}e^{w(|x|)}\frac{x_{1}^{2}}{|x|^{2}}w^{\prime
2}(|x|)\,dx\\\
&\qquad+\int_{B}e^{w(|x|)}\frac{x_{1}^{2}}{|x|^{2}}w^{\prime\prime}(|x|)\,dx+\int_{B}e^{w(|x|)}\frac{w^{\prime}(|x|)}{|x|}\,dx-\int_{B}e^{w(|x|)}\frac{x_{1}^{2}}{|x|^{3}}w^{\prime}(|x|)\,dx=0.\end{split}$

Thus after some simplifications

(3.11)
$\rho^{\prime\prime}(0)=-\frac{1}{n}\frac{\displaystyle\int_{B}e^{w(|x|)}\Big{(}w^{\prime
2}(|x|)+w^{\prime\prime}(|x|)+(n-1)\frac{w^{\prime}(|x|)}{|x|}\Big{)}\,dx}{n|B|_{w}+\displaystyle\int_{B}e^{w(|x|)}|x|w^{\prime}(|x|)\,dx}.$

A similar calculation shows that
$\frac{d}{d\varepsilon}\Big{(}P_{w}(B_{\rho(\varepsilon)}(\varepsilon
e_{1}))\Big{)}_{\big{|}_{\varepsilon=0}}=0$ and

$\begin{split}&\frac{d^{2}}{d\varepsilon^{2}}\left(P_{w}(B_{\rho(\varepsilon)}(\varepsilon
e_{1}))\right)_{\big{|}_{\varepsilon=0}}\\\
&\qquad\qquad=P_{w}(B)\Big{[}\rho^{\prime\prime}(0)(n-1+w^{\prime}(1))+\frac{1}{n}\Big{(}w^{\prime
2}(1)+w^{\prime\prime}(1)+(n-1)w^{\prime}(1)\Big{)}\Big{]}.\end{split}$

From this equation, using (3.11), we get, recalling that
$w^{\prime\prime}(1)=0$,

(3.12)
$\begin{split}&\frac{1}{P_{w}(B)}\frac{d^{2}}{d\varepsilon^{2}}\left(P_{w}(B_{\rho(\varepsilon)}(\varepsilon
e_{1}))\right)_{\big{|}_{\varepsilon=0}}=\frac{1}{n}\Big{(}w^{\prime
2}(1)+(n-1)w^{\prime}(1)\Big{)}\\\
&-\frac{(n-1+w^{\prime}(1))}{n}\frac{\displaystyle\int_{B}e^{w(|x|)}\Big{(}w^{\prime
2}(|x|)+w^{\prime\prime}(|x|)+(n-1)\frac{w^{\prime}(|x|)}{|x|}\Big{)}\,dx}{n|B|_{w}+\displaystyle\int_{B}e^{w(|x|)}|x|w^{\prime}(|x|)\,dx}.\end{split}$

Observe that by divergence theorem

$\begin{split}(n-1)\int_{B}e^{w(|x|)}\frac{w^{\prime}(|x|)}{|x|}\,dx&=\int_{B}e^{w(|x|)}w^{\prime}(|x|)\operatorname{div}\Big{(}\frac{x}{|x|}\Big{)}\,dx\\\
&=w^{\prime}(1)P_{w}(B)-\int_{B}e^{w(|x|)}\big{(}w^{\prime
2}(|x|)+w^{\prime\prime}(|x|)\big{)}\,dx\end{split}$

and similarly

$\int_{B}e^{w(|x|)}|x|w^{\prime}(|x|)\,dx=\int_{B}\operatorname{div}(xe^{w(|x|)})\,dx-n|B|_{w}=P_{w}(B_{1})-n|B|_{w}.$

Plugging these identities in (3.12) we have that
$\frac{d^{2}}{d\varepsilon^{2}}\left(P_{w}(B_{\rho(\varepsilon)}(\varepsilon
e_{1}))\right)_{\big{|}_{\varepsilon=0}}=0$. Thus
$P_{w}(B_{\rho(\varepsilon)}(\varepsilon e_{1}))=P_{w}(B)+o(\varepsilon^{2})$
and the conclusion follows. ∎

##  4\. Preliminary Lemmas

We start this section by recalling the result of [12] on the uniqueness of
balls as isoperimetric sets for the log-convex isoperimetric inequality. We
recall this result for the reader’s convenience.

######  Theorem 4.1.

If $w$ is a convex even function of class $C^{3}$ with $w(r)>w(0)$ for $r>0$
the only isoperimetric regions are balls centered at the origin.

Next lemma shows the continuity of $P_{w}(\cdot)$ at $B_{r}$ with respect to
the convergence in measure.

######  Lemma 4.2.

Let $r>0$ such that $w(r)>w(0)$. Given $\varepsilon>0$ there exists $\delta>0$
such that for every set of finite perimeter $E$ with $|E|_{w}=|B_{r}|_{w}$, if
$P_{w}(E)-P_{w}(B_{r})<\delta$ then $|E\triangle B_{r}|_{w}<\varepsilon$.

###### Proof.

Assume by contradiction that there exists $\varepsilon_{0}>0$ such that for
every $k\in\mathbb{N}$ there exists a set $E_{k}$ with
$|E_{k}|_{w}=|B_{r}|_{w}=m$ and such that
$P_{w}(E_{k})-P_{w}(B_{r})\leq\frac{1}{k}$, but $|E_{k}\triangle
B_{r}|_{w}\geq\varepsilon_{0}$. Since $w$ is increasing on $\mathbb{R}^{+}$,
for $k$ sufficienlty large we have

$e^{w(0)}P(E_{k})\leq P_{w}(E_{k})\leq 2P_{w}(B_{r}).$

Hence $\\{E_{k}\\}_{k\in\mathbb{N}}$ is a sequence of sets with equibounded
perimeters and thus, up to a not relabeled subsequence, we have that there
exists a set $E$ such that $\chi_{E_{k}}\to\chi_{E}$ in
$L^{1}_{\text{loc}}(\mathbb{R}^{n})$ and

$P_{w}(E)\leq\liminf_{k}P_{w}(E_{k})=P_{w}(B_{r}).$

We claim that $|E|_{w}=m$.

To this aim it is enough to show that given $\sigma>0$ there exists $R>0$ such
that $|E_{k}\setminus B_{R}|_{w}<\sigma$ for all $k$. Indeed, if there exists
$k_{0}$ such that $|E_{k_{0}}\setminus B_{R}|>\sigma$ then

(4.1) $|E_{k_{0}}\setminus
B_{R}|_{w}=\int_{R}^{\infty}\mathcal{H}^{n-1}(E_{k_{0}}\cap\partial
B_{t})e^{w(t)}\,dt>\sigma.$

Recall that, for a.e. $t>0$, $E_{k_{0}}\cap\partial B_{t}$ is a set of finite
perimeter on the sphere such that $\partial^{*}E_{k}\cap\partial B_{t}$
coincides up to a set of zero $\mathcal{H}^{n-2}$-measure with the reduced
boundary of $E_{k_{0}}\cap\partial B_{t}$ relative to $\partial B_{t}$, see
for instance [9, Theorem 3.7]. If $G\subset\mathbb{S}^{n-1}$ is a set of
finite perimeter denote by $\partial_{\mathbb{S}^{n-1}}G$ the boundary of $G$
relative to $\mathbb{S}^{n-1}$ and by $\partial^{*}_{\mathbb{S}^{n-1}}G$ the
corresponding reduced boundary relative to $\mathbb{S}^{n-1}$. Then, the
isoperimetric inequality on the sphere (see [4]) states that

(4.2)
$\mathcal{H}^{n-2}(\partial^{*}_{\mathbb{S}^{n-1}}G)\geq\mathcal{H}^{n-2}(\partial_{\mathbb{S}^{n-1}}S_{\theta})$

where $S_{\theta}$ is the spherical cap with geodesic radius $\theta$ such
that $\mathcal{H}^{n-1}(S_{\theta})=\mathcal{H}^{n-1}(G)$. Since

$\mathcal{H}^{n-1}(S_{\theta})=(n-1)\omega_{n-1}\int_{0}^{\theta}\sin^{n-2}\varphi\,d\varphi,\qquad\mathcal{H}^{n-2}(\partial_{\mathbb{S}^{n-1}}S_{\theta})=(n-1)\omega_{n-1}\sin^{n-2}\theta,$

a straightforward computation shows that (4.2) implies that there exists
$c_{n}>0$ such that

(4.3) $\mathcal{H}^{n-2}(\partial^{*}_{\mathbb{S}^{n-1}}G)\geq
c_{n}\left(\mathcal{H}^{n-1}(G)\right)^{\frac{n-2}{n-1}}\quad\text{whenever}\,\,\mathcal{H}^{n-1}(G)\leq\frac{1}{2}\mathcal{H}^{n-1}(\mathbb{S}^{n-1}).$

Since for $R>0$ sufficiently large and for a.e. $t>R$ we have

(4.4) $\mathcal{H}^{n-1}(E_{k_{0}}\cap\partial B_{t})\leq
P(E_{k_{0}};\mathbb{R}^{n}\setminus
B_{t})\leq\frac{1}{e^{w(R)}}P_{w}(E_{k_{0}})\leq\frac{1}{2}\mathcal{H}^{n-1}(\mathbb{S}^{n-1}),$

from (4.3), using the coarea formula, we get

$\begin{split}P_{w}(E_{k_{0}})\geq&\int_{R}^{\infty}\mathcal{H}^{n-2}(\partial^{*}E_{k_{0}}\cap\partial
B_{t})\,dt\geq
c_{n}\int_{R}^{\infty}\left(\mathcal{H}^{n-1}(E_{k_{0}}\cap\partial
B_{t})\right)^{\frac{n-2}{n-1}}e^{w(t)}\,dt\\\
=&c_{n}\int_{R}^{\infty}\frac{\mathcal{H}^{n-1}(E_{k_{0}}\cap\partial
B_{t})}{\mathcal{H}^{n-1}(E_{k_{0}}\cap\partial
B_{t})^{\frac{1}{n-1}}}e^{w(t)}\,dt.\end{split}$

From this inequality, using (4.4) again and recalling (4.1) we conclude that

$P_{w}(E_{k_{0}})\geq
c_{n}\sigma\left(\frac{e^{w(R)}}{P_{w}(E_{k_{0}})}\right)^{\frac{1}{n-1}}$

that is $P_{w}(E_{k_{0}})\geq c\sigma^{\frac{n-1}{n}}e^{\frac{w(R)}{n}}$ which
is impossible if $R$ is sufficiently large. This proves the claim, hence by
Theorem 4.1 $E$ must coincide with a ball $B_{r}$, which is a contradiction
since $|E_{k}\triangle B_{r}|_{w}\to 0$. ∎

Next simple lemma is a useful tool in the proof of the main theorem.

######  Lemma 4.3.

Let $r>0$ such that $w(r)>w(0)$, $\Lambda_{1}\geq 0$ and $\Lambda_{2}\geq
2(4\frac{n+1}{r}+w^{\prime}(2r))$. Then $B_{r}$ is the only minimizer of the
functional defined for a measurable set $E\subset\mathbb{R}^{n}$ as

$J_{\Lambda_{1},\Lambda_{2}}(E)=P_{w}(E)+\Lambda_{1}\left||E|_{w}-|B_{r}|_{w}\right|+\Lambda_{2}|E\triangle
B_{r}|_{w}.$

The same conclusion holds if $\Lambda_{2}=0$ and $\Lambda_{1}\geq
n-1+rw^{\prime}(r)$.

###### Proof.

Let $\eta:\mathbb{R}\to[0,1]$ be a smooth cut-off function such that
$\eta(t)=1$ for $t\in\left[\frac{r}{2},\frac{3r}{2}\right]$, $\eta(t)=0$
outside of the interval $[\frac{r}{4},\frac{7r}{4}]$ and
$\|\eta^{\prime}(t)\|_{L^{\infty}}\leq 8/r$. Consider the smooth vector field
$X(x)=\eta(|x|)\frac{x}{|x|}$. It is readily checked that
$\|X\|_{L^{\infty}}=1$ and $\|\operatorname{div}X\|_{L^{\infty}}\leq(4n+4)/r$

By definition of reduced boundary we get

$\begin{split}\int_{\partial^{*}E}e^{w(|x|)}\,d\mathcal{H}^{n-1}\geq&\int_{\partial^{*}E}e^{w(|x|)}\langle
X,\nu_{E}\rangle\,d\mathcal{H}^{n-1}=\int_{E}\operatorname{div}(e^{w(|x|)}X)\,dx\\\
=&\int_{E}\left(\operatorname{div}X+w^{\prime}(|x|)\frac{\langle
X,x\rangle}{|x|}\right)e^{w(|x|)}dx\end{split}$

while for the ball it holds

$\int_{\partial
B_{r}}e^{w(|x|)}\,d\mathcal{H}^{n-1}=\int_{B_{r}}\left(\operatorname{div}X+w^{\prime}(|x|)\frac{\langle
X,x\rangle}{|x|}\right)e^{w(|x|)}\,dx.$

Hence we find

$\begin{split}J_{\Lambda_{1},\Lambda_{2}}(E)-J_{\Lambda_{1},\Lambda_{2}}(B_{r})\geq&\left(\Lambda_{2}-\|\operatorname{div}X\|_{L^{\infty}(\mathbb{R}^{n})}-\|w^{\prime}(|x|)X\|_{L^{\infty}(\mathbb{R}^{n})}\right)|E\triangle
B_{r}|_{w}\\\
\geq&\left(\Lambda_{2}-\frac{4n+4}{r}-w^{\prime}(2r)\right)|E\triangle
B_{r}|_{w}.\end{split}$

Taking in mind the definition of $\Lambda_{2}$ we immediately get the desired
result.

If $\Lambda_{2}=0$ by the uniqueness result stated in Theorem 4.1 we
immediately get that the minimizers of $J_{\Lambda_{1},\Lambda_{2}}$ are given
by balls centered at the origin. On such balls the value of the functional is
given by

$J_{\Lambda_{1},\Lambda_{2}}(B_{\varrho})=n\omega_{n}\varrho^{n-1}e^{w(\varrho)}+n\omega_{n}\Lambda_{1}\left|\int_{\varrho}^{r}e^{w(t)}t^{n-1}\,dt\right|=f(\varrho).$

By an elementary computation we get that under the assumption on $\Lambda_{1}$
the function $f(\varrho)$ attains its unique minimum when $\varrho=r$. ∎

##  5\. Proof of theorem 1.1

This section will be devoted to the proof of Theorem 1.1 which is achieved by
a contradiction argument which makes use of suitable energy functionals. One
problem here is to show the existence of minimizers for such functionals. This
fact is achieved by showing that there exists a minimizing sequence made up by
equibouded sets.

To this aim we introduce the functions
$\Phi,\Psi:\mathbb{R}_{+}\to\mathbb{R}_{+}$ defined for $s,t\geq 0$ as

(5.1)
$\Phi(s)=n\omega_{n}\int_{0}^{s}t^{n-1}e^{w(t)}\,dt,\qquad\Psi(t)=\Phi^{-1}(t).$

Note that $\Psi$ is well defined since $\Phi$ is a strictly increasing
function. Note also that $\Psi(t)$ is equal to the radius $r$ of the ball
centered at the origin such that $|B_{r}|_{w}=t$. The following lemma contains
a few useful properties of $\Psi$ whose elementary verification is left to the
reader.

######  Lemma 5.1.

Let $\Psi$ be the function defined in (5.1). Then $\Psi\in
C^{\infty}(0,\infty)$ and for $t>0$ we have

(5.2)
$\begin{split}\Psi^{\prime}(t)&=\frac{1}{n\omega_{n}\Psi^{n-1}(t)e^{w(\Psi(t))}},\\\
t&\leq n\omega_{n}\Psi(t)^{n}e^{w(\Psi(t))}.\end{split}$

######  Lemma 5.2.

Let $E\subset\mathbb{R}^{n}$ be a set of finite perimeter such that
$|E\setminus B_{r}|_{w}\leq\eta<1$. There exists $R_{E}\in[r,r+4\Psi(\eta)]$
such that

$P_{w}(E)\leq P_{w}(E\cap B_{R_{E}})-\frac{|E\setminus
B_{R_{E}}|_{w}}{2\Psi(\eta)}.$

###### Proof.

We argue by contradiction assumig that for any $r\leq t\leq r+4\Psi(\eta)$ it
holds

(5.3) $P_{w}(E\cap B_{t})>P_{w}(E)-\frac{|E\setminus
B_{t}|_{w}}{2\Psi(\eta)}.$

Set $v(t)=|E\setminus B_{t}|_{w}$. Then for a.e. $t>0$

$v^{\prime}(t)=-e^{w(t)}\mathcal{H}^{n-1}(E\cap\partial B_{t}).$

Since $P_{w}(E)\geq P_{w}(E\cap B_{t})+P_{w}(E\setminus
B_{t})+2v^{\prime}(t)$, inequality (5.3) implies that

$2v^{\prime}(t)+P_{w}(E\setminus B_{t})<\frac{v(t)}{2\Psi(\eta)}.$

The weighted isoperimetric inequality hence gives

$2v^{\prime}(t)+n\omega_{n}\Psi(v(t))^{n-1}e^{w(\Psi(v(t)))}<\frac{v(t)}{2\Psi(\eta)}.$

We now use the second inequality in (5.2) to infer

$\frac{v(t)}{\Psi(\eta)}\leq n\omega_{n}\Psi(v(t))^{n-1}e^{w(\Psi(v(t)))},$

which gives

$\Psi(v(t))^{n-1}e^{w\Psi(v(t))}<-\frac{4}{n\omega_{n}}v^{\prime}(t)\qquad\text{for
all $\,\,t\in[r,r+4\Psi(\eta)]$.}$

Integrating the latter inequality we get by a change of variable and using the
first equality in (5.2)

$\begin{split}4\Psi(\eta)&<-\frac{4}{n\omega_{n}}\int_{r}^{r+4\Psi(\eta)}\frac{v^{\prime}(t)}{\Psi(v(t))^{n-1}e^{\Psi(v(t))}}dt\\\
&=\frac{4}{n\omega_{n}}\int_{v(r+4\Psi(\eta))}^{v(r)}\frac{1}{\Psi(s)^{n-1}e^{\Psi(s)}}\,ds\\\
&=4(\Psi(v(r))-\Psi(v(r+4\Psi(\eta))),\end{split}$

which is impossible. ∎

We are now ready to state the following existence result.

######  Lemma 5.3.

Let $r>0$ such that $w(r)>w(0)$, $\Lambda_{1}\geq n-1+rw^{\prime}(r)$ and
$\Lambda_{2}>0$. There exist $0<\alpha_{1}<\frac{\Lambda_{2}}{2\Lambda_{2}+1}$
such that for any $\alpha\in[0,\alpha_{1}]$ the functional

$J_{\Lambda_{1},\Lambda_{2},\alpha}(F)=P_{w}(F)+\Lambda_{1}\left||F|_{w}-|B_{r}|_{w}\right|+\Lambda_{2}||F\triangle
B_{r}|_{w}-\alpha|,\qquad F\subset\mathbb{R}^{n},$

has always a minimizer $E\subset B_{R_{0}}$ where

(5.4) $R_{0}=r+4\Psi(1).$

###### Proof.

Let $F_{h}$ a minimizing sequence such that

$J_{\Lambda_{1},\Lambda_{2},\alpha}(F_{h})\leq\inf_{F\subset\mathbb{R}^{n}}J_{\Lambda_{1},\Lambda_{2},\alpha}(F)+\frac{\alpha_{1}}{h}\leq
P_{w}(B_{r})+\Lambda_{2}\alpha+\frac{\alpha_{1}}{h}.$

By the second part of Lemma 4.3 and from the previous inequality we have

$P_{w}(B_{r})+\Lambda_{2}||F_{h}\triangle B_{r}|_{w}-\alpha|\leq
J_{\Lambda_{1},\Lambda_{2},\alpha}(F_{h})\leq
P_{w}(B_{r})+\Lambda_{2}\alpha+\frac{\alpha_{1}}{h}.$

In turn this inequality implies that

$|F_{h}\setminus B_{r}|_{w}\leq|F_{h}\triangle
B_{r}|_{w}\leq\left(2+\frac{1}{h\Lambda_{2}}\right)\alpha_{1}.$

Set $\eta:=(\frac{2\Lambda_{2}+1}{\Lambda_{2}})\alpha_{1}<1$. Thus Lemma 5.2
implies that there exists $r_{h}\in[r,r+4\Psi(\eta)]$ such that

$P_{w}(F_{h}\cap B_{r_{h}})\leq P_{w}(F_{h})-\frac{|F_{h}\setminus
B_{r_{h}}|_{w}}{2\Psi(\eta)}$

Hence if we set $G_{h}=F_{h}\cap B_{r_{h}}$ we get

$\begin{split}J_{\Lambda_{1},\Lambda_{2},\alpha}(G_{h})\leq&P_{w}(F_{h})-\frac{|F_{h}\setminus
B_{r_{h}}|_{w}}{2\Psi(\eta)}+\Lambda_{1}\left||G_{h}|_{w}-|B_{r}|_{w}\right|+\Lambda_{2}||G_{h}\triangle
F_{h}|_{w}-\alpha|\\\
\leq&J_{\Lambda_{1},\Lambda_{2},\alpha}(F_{h})+\left(\Lambda_{1}-\frac{1}{2\Psi(\eta)}\right)|F_{h}\setminus
B_{r_{h}}|_{w}+\Lambda_{2}|G_{h}\triangle F_{h}|_{w}\\\
=&J_{\Lambda_{1},\Lambda_{2},\alpha}(F_{h})+\left(\Lambda_{1}+\Lambda_{2}-\frac{1}{2\Psi(\eta)}\right)|F_{h}\setminus
B_{r_{h}}|_{w}.\end{split}$

Therefore, taking $\eta$, hence $\alpha_{1}$, sufficiently small we have that
$G_{h}$ is a minimizing sequence such that $G_{h}\subset B_{R_{0}}$, where
$R_{0}=$ is as in (5.4). The conclusion then follows observing that the sets
$G_{h}$ have all equibounded perimeters and using the well known properties of
compactness and lower semicontinuity of the perimeter. ∎

######  Lemma 5.4.

Given $\Lambda_{1},\Lambda_{2}\geq 0$, there exists $\omega\geq 0$ such that
if $E\subset B_{R_{0}}$ is a minimizer of $J_{\Lambda_{1},\Lambda_{2},\alpha}$
with $\alpha\geq 0$, then $E$ is an $\omega$-minimizer of the perimeter in
$B_{2R_{0}}$.

###### Proof.

Let $F$ be a set of finite perimeter with $F\triangle E\subset\subset
B_{\rho}(x)\subset\subset B_{2R_{0}}$. If $|B_{\rho}(x)\cap E|=0$ then (2.1)
is trivially satisfied.

Hence we may assume without loss of generality that $|B_{\rho}(x)\cap E|>0$.
Since $F\triangle E\subset\subset B_{\rho}(x)$ we have that
$P_{w}(F;\mathbb{R}^{n}\setminus B_{\rho}(x))=P_{w}(E,\mathbb{R}^{n}\setminus
B_{\rho}(x))$. Moreover,

$\left||F|_{w}-|E|_{w}\right|\leq e^{w(2R_{0})}\omega_{n}\rho^{n}.$

Similarly,

$\left||F\triangle B_{r}|_{w}-|E\triangle B_{r}|_{w}\right|\leq|F\triangle
E|_{w}\leq e^{w(2R_{0})}\omega_{n}\rho^{n}.$

The above inequalities and the minimality of $E$ yield

(5.5)
$\begin{split}\min_{z\in\overline{B_{\rho}}(x)}e^{w(|z|)}P(E;B_{\rho}(x))\leq&P_{w}(E;B_{\rho}(x))\leq
P_{w}(F,B_{\rho}(x))+C_{0}\rho^{n}\\\
\leq&\max_{z\in\overline{B_{\rho}}(x)}e^{w(|z|)}P(F,B_{\rho}(x))+C_{0}\rho^{n}.\end{split}$

for a constant $C_{0}$ depending only on $\Lambda_{1},\Lambda_{2},r,n$.
Observe now that there exists another constant $C>0$, still indipendent of
$E$, $\alpha_{1}$ and $\rho$, such that

(5.6) $P(E,B_{\rho}(x))\leq C\rho^{n-1}.$

Indeed, if we first apply (5.5) with $F$ replaced by $E\cup
B_{\rho^{\prime}}(x)$ with $0<\rho^{\prime}<\rho$ such that
$\mathcal{H}^{n-1}(\partial^{*}E\cap\partial B_{\rho^{\prime}}(x))=0$ and then
let $\rho^{\prime}\uparrow\rho$, we get

$\min_{z\in\bar{B}_{\rho}(x)}e^{w(|z|)}P(E;B_{\rho}(x))\leq
n\omega_{n}\rho^{n-1}\max_{z\in\bar{B}_{\rho}(x)}e^{w(|z|)}+C_{0}\rho^{n}$

which gives (5.6) since $\rho\leq 2R_{0}$. Observe also that there exists
another constant, still denoted by $C$ and depending only on $R_{0}$, such
that

$\operatorname*{osc}_{z\in B_{\rho}(x)}e^{w(|z|)}\leq C\rho.$

The conclusion easily follows from this estimate using (5.5) and (5.6). ∎

######  Lemma 5.5.

Let $r>0$ such that $w(r)>w(0)$, let $\Lambda_{1}$, $\Lambda_{2}$ satisfy the
assumptions of Lemma 4.3 and let $\varepsilon_{i}\to 0$. Let $F_{i}$ be a
sequence of equibounded minimizers of
$J_{\Lambda_{1},\Lambda_{2},\varepsilon_{i}}$. Then, up to a not relabeled
subsequence, $F_{i}\to B_{r}$ in $C^{1,\alpha}$ for all $\alpha<\frac{1}{2}$.
Precisely, for all $i$ there exists $\psi_{i}\in
C^{1,\frac{1}{2}}(\mathbb{S}^{n-1})$ such that

$\partial F_{i}=\\{rx(1+\psi_{i}(x)),x\in\mathbb{S}^{n-1}\\}\quad\text{ with
}\quad\|\psi_{i}\|_{C^{1,\alpha}(\mathbb{S}^{n-1})}\to 0\quad\text{ for any
}\alpha\in(0,\frac{1}{2}).$

###### Proof.

By the minimality of $F_{i}$ we have

$e^{w(0)}P(F_{i})\leq P_{w}(F_{i})\leq
J_{\Lambda_{1},\Lambda_{2},\varepsilon_{i}}(F_{i})\leq
P_{w}(B_{r})+\varepsilon_{i}.$

Since $\varepsilon_{i}\to 0$ and the sets $F_{i}$ are equibounded, we get that
there exists a bounded set of finite perimeter $F$ such that up to a not
relabelled subsequence $|F_{i}\triangle F|\to 0$. Since for any set $E$ of
finite weighted perimeter and for every $i\in\mathbb{N}$ we have

$J_{\Lambda_{1},\Lambda_{2},\varepsilon_{i}}(F_{i})\leq
J_{\Lambda_{1},\Lambda_{2},\varepsilon_{i}}(E),$

sending $i$ to infinity and using the semicontinuity of the weighted perimeter
and the continuity of $\alpha_{w}(\cdot)$ with respect to the convergence in
meausure we infer

$J_{\Lambda_{1},\Lambda_{2}}(F)\leq J_{\Lambda_{1},\Lambda_{2}}(E).$

Hence, $F$ is a minimizer for the functional $J_{\Lambda_{1},\Lambda_{2}}$ and
thus from Lemma 4.3 $F_{i}\to B_{r}$ in measure. The conclusion then follows
from Lemma 5.4 and from Theorem 2.2.

∎

###### Proof of the Main Theorem.

In order to prove (1.1) it is enough to show that for any $r>0$ such that
$w^{\prime\prime}(r)>0$ there exists $\delta>0$ such that if $|B_{r}\triangle
E|_{w}<\delta$ and $|E|_{w}=|B_{r}|_{w}$ then

(5.7) $P_{w}(E)-P_{w}(B_{r})\geq c_{1}|B_{r}\triangle E|_{w}^{2},$

where $c_{1}$ is a constant whose explicit value will be given later. Indeed,
by Lemma 4.2 there exists $\sigma>0$ such that if $|E\triangle
B_{r}|_{w}\geq\delta$ then $P_{w}(E)-P_{w}(B_{r})\geq\sigma$ and thus we may
conclude that in this case

$P_{w}(E)-P_{w}(B_{r})\geq\frac{\sigma}{4|B_{r}|^{2}_{w}}|E\triangle
B_{r}|_{w}^{2}.$

In order to prove (5.7) we argue by contradiction assuming that there exists a
sequence $E_{i}$ such that $|E_{i}|_{w}=|B_{r}|_{w}$, $|E_{i}\triangle
B_{r}|_{w}\to 0$ as $i\to\infty$ and

$P_{w}(E_{i})\leq P_{w}(B_{r})+c_{1}|E_{i}\triangle B_{r}|_{w}^{2}.$

We now set $\varepsilon_{i}=|E_{i}\triangle B_{r}|_{w}$. Let $\Lambda_{1}\geq
n-1+rw^{\prime}(r)$ and $\Lambda_{2}>0$ to be chosen. By Lemma 5.3 we have
that for $i$ sufficiently large the functional
$J_{\Lambda_{1},\Lambda_{2},\varepsilon_{i}}$ has a minimizer $F_{i}$ with
$F_{i}\subset B_{R_{0}}$, $R_{0}=r+4\Psi(1)$. Note that by Lemma 5.5, passing
possibly to a subsequence, we have that $F_{i}\to B_{r}$ in $C^{1,\alpha}$ for
all $\alpha\in(0,1/2)$. By the minimality of $F_{i}$ we have that for $i$
large

(5.8) $J_{\Lambda_{1},\Lambda_{2},\varepsilon_{i}}(F_{i})\leq
J_{\Lambda_{1},\Lambda_{2},\varepsilon_{i}}(E_{i})=P_{w}(E_{i})\leq
P_{w}(B_{r})+c_{1}\varepsilon_{i}^{2}.$

From this inequality, if $\Lambda_{2}$ is chosen such that
$\Lambda_{2}>4(4\frac{n+1}{r}+w^{\prime}(2r))$, by applying Lemma 4.3 with
$\Lambda_{2}$ replaced by $\Lambda_{2}/2$ and $\Lambda_{1}=0$, we have

$\begin{split}P_{w}(F_{i})+\Lambda_{2}||F_{i}\triangle
B_{r}|_{w}-\varepsilon_{i}|&\leq P_{w}(B_{r})+c_{1}\varepsilon_{i}^{2}\\\
&\leq P_{w}(F_{i})+\frac{\Lambda_{2}}{2}|F_{i}\triangle
B_{r}|_{w}+c_{1}\varepsilon_{i}^{2}\end{split}$

from which it follows that for $i$ large

(5.9) $|F_{i}\triangle B_{r}|_{w}\geq\frac{\varepsilon_{i}}{2}.$

Assume now that $\Lambda_{1}\geq 2(n-1+rw^{\prime}(r))$. By (5.8) and Lemma
4.3 with $\Lambda_{1}$ replaced by $\Lambda_{1}/2$ and $\Lambda_{2}=0$ we have

$\begin{split}P_{w}(F_{i})+\Lambda_{1}||F_{i}|_{w}-|B_{r}|_{w}|&\leq
P_{w}(B_{r})+c_{1}\varepsilon_{i}^{2}\\\ &\leq
P_{w}(F_{i})+\frac{\Lambda_{1}}{2}||F_{i}|_{w}-|B_{r}|_{w}|+c_{1}\varepsilon_{i}^{2}.\end{split}$

From this in particular we deduce that

(5.10) $||F_{i}|_{w}-|B_{r}|_{w}|\leq 2c_{1}\varepsilon_{i}^{2}.$

Denote by $r_{i}$ the radius such that $|B_{r_{i}}|_{w}=|F_{i}|_{w}$. From the
inequality above we have

$|F_{i}\triangle B_{r}|_{w}\leq|F_{i}\triangle B_{r_{i}}|_{w}+|B_{r}\triangle
B_{r_{i}}|_{w}\leq|F_{i}\triangle B_{r_{i}}|_{w}+2c_{1}\varepsilon_{i}^{2}$

and thus for $i$ large, using (5.9), we have

$|F_{i}\triangle B_{r}|_{w}\leq 2|F_{i}\triangle B_{r_{i}}|_{w}.$

In turn, this inequality together with (5.10) and (5.9) implies

$\begin{split}P_{w}(B_{r})\leq P_{w}(B_{r_{i}})+C|r-r_{i}|\leq
P_{w}(B_{r_{i}})+Cc_{1}\varepsilon_{i}^{2}\leq
P_{w}(B_{r_{i}})+\tilde{C}c_{1}|F_{i}\triangle B_{r_{i}}|_{w}^{2}.\end{split}$

which is a contradiction to (3.8) if we choose $c_{1}<c_{0}/\tilde{C}$, where
$c_{0}$ is the constant provided by Proposition 3.2.

∎

##  6\. Negative power weights

Given a measurable set $E\subset\mathbb{R}^{n}$ and $a\in[0,1]$ we denote by
$E^{(a)}$ the set of points in $\mathbb{R}^{n}$ where $E$ has density equal to
$a$, that is the set of points $x\in\mathbb{R}^{n}$ such that

$\lim_{r\to 0}\frac{|E\cap B_{r}(x)|}{\omega_{n}r^{n}}=a\,.$

Note that $E^{(1)}$ and $E$ coincide up to a set of zero measure. If
$E\subset\mathbb{R}^{n}$ is a set of locally finite perimeter and
$p\in\mathbb{R}$ we set

$P_{p}(E)=\int_{\partial^{*}E}|x|^{p}\,d\mathcal{H}^{n-1}.$

As already mentioned in the introduction it is well known that for $p>0$ the
only isoperimetric sets with respect to the weight $|x|^{p}$ are balls
centered at the origin and moreover they are stable. On the contrary, when
$1-n<p<0$ there are no isoperimetric sets. We recall also that if $p\leq 1-n$
the isoperimetric inequality

(6.1) $P_{p}(E)\geq P_{p}(B_{r})$

where $|E|=|B_{r}|$ is true whenever $E$ is an open set containing the origin
(see for instance [17]). The condition $0\in E$ is clearly necessary for (6.1)
to holds since $P_{p}(B_{r}(x))\to 0$ as $|x|\to\infty$. Next theorem shows
that a quantitative version of (6.1) is also true.

######  Theorem 6.1.

Let $n\geq 2$ and $p<-n-1$. There exists a constant $c>0$ such that if $r>0$
and $E$ is a set of finite perimeter with $|E|=|B_{r}|$ such that the origin
belongs to the interior of $E^{(1)}$, then

$P_{p}(E)\geq P_{p}(B_{r})+c|E\triangle B_{r}|^{2}.$

###### Proof.

Since $0\in\operatorname{int}(E^{(1)})$ that there exists $r_{0}>0$ such that
$E^{(1)}\cap B_{r_{0}}=B_{r_{0}}$. Thus, we note that

$\begin{split}P_{p}(E)-P_{p}(B_{r})&=\int_{\partial^{*}E}|x|^{p}\,d\mathcal{H}^{n-1}-\int_{\partial
B_{r}}|x|^{p}\,d\mathcal{H}^{n-1}\\\
&\geq\int_{\partial^{*}E}|x|^{p}\langle\frac{x}{|x|},\nu_{E}\rangle\,d\mathcal{H}^{n-1}-\int_{\partial
B_{r}}|x|^{p}\langle\frac{x}{|x|},\nu_{B_{r}}\rangle\,d\mathcal{H}^{n-1}\\\
&=\int_{\partial^{*}(E\setminus
B_{r_{0}})}|x|^{p}\langle\frac{x}{|x|},\nu_{E}\rangle\,d\mathcal{H}^{n-1}-\int_{\partial^{*}(B_{r}\setminus
B_{r_{0}})}|x|^{p}\langle\frac{x}{|x|},\nu_{B_{r}}\rangle\,d\mathcal{H}^{n-1}.\end{split}$

By applying the divergence theorem to $E\setminus B_{r_{0}}$ and $B\setminus
B_{r_{0}}$ we get

$\begin{split}P_{p}(E)-P_{p}(B_{r})&\geq(n-1+p)\left(\int_{E\setminus
B_{r_{0}}}|x|^{p-1}\,dx-\int_{B_{r}\setminus B_{r_{0}}}|x|^{p-1}\,dx\right)\\\
&\geq(n-1+p)\left(\int_{B_{\bar{r}}\setminus
B_{r}}|x|^{p-1}\,dx\right)\end{split}$

where $\bar{r}$ is such that $|B_{\bar{r}}\setminus B_{r}|=|E\setminus
B_{r}|$. The conclusion then follows as in Lemma 6.1 in [29]. ∎

As mentioned before inequality (6.1) does not hold if
$0\not\in\mathbb{R}^{n}\setminus\overline{E^{(1)}}$. Thus in order to get a
complete picture it remains to analyze the case $0\in\partial E^{(1)}$. To
this aim we recall that if $E$ is a set of locally finite perimeter then

(6.2) $\overline{\partial^{*}E}=\partial E^{(1)}\,.$

The above inequality is well known to the experts, however for the reader’s
convenience we provide its simple proof. Recall that given any set of locally
finite perimeter $E$ the reduced boundary $\partial^{*}E$ is always contained
in the topological boundary $\partial E$ and does not change if one modifies
$E$ by a set of zero Lebesgue measure. Therefore

(6.3) $\overline{\partial^{*}E}=\overline{\partial^{*}E^{(1)}}\subset\partial
E^{(1)}\,.$

To show the opposite inclusion, let $x\not\in\overline{\partial^{*}E}$. Then,
there exists $B_{r}(x)$ such that $\partial^{*}E\cap B_{r}(x)=\emptyset$. Thus
$P(E;B_{r}(x))=\mathcal{H}^{n-1}(\partial^{*}E\cap B_{r}(x))=0$. Then by the
relative isoperimetric inequality in a ball we have

$\min\\{|E\setminus B_{r}(x)|,|E\cap B_{r}(x)|\\}^{\frac{n-1}{n}}\leq
c(n)P(E;B_{r}(x))=0\,.$

Therefore, if $|E\setminus B_{r}(x)|=0$ then $B_{r}(x)\subset E^{(1)}$ and so
$x$ belongs to the interior of $E^{(1)}$. If instead $|E\cap B_{r}(x)|=0$,
then $B_{r}(x)\subset E^{(0)}$ and so $x$ is in the interior of
$\mathbb{R}^{n}\setminus E^{(1)}$. In both cases $x\not\in\partial E^{(1)}$.
Therefore, recalling (6.3) we get (6.2).

######  Lemma 6.2.

Let $p<-n+1$ and $E$ a set of finite perimeter. If $0\in\partial^{*}E$ then
$P_{p}(E)=\infty.$

###### Proof.

Assume by contradiction that $P_{p}(E)<\infty$. Given a ball $B_{r}$ centered
at $0$ we would have

$P(E;B_{r})r^{p}\leq\int_{\partial^{*}E\cap
B_{r}}|x|^{p}\,d\mathcal{H}^{n-1}\leq P_{p}(E)$

and thus

$\frac{P(E;B_{r})}{r^{n-1}}\leq\frac{P_{p}(E)}{r^{n-1+p}}\,.$

Since $n-1+p<0$, from this inequality we get

$\lim_{r\to 0}\frac{P(E;B_{r})}{r^{n-1}}=0$

thus $0\not\in\partial^{*}E$ which is a contradiction. ∎

It remains to examine the case $0\in\partial E^{(1)}\setminus\partial^{*}E$.
Next example shows that in this case the isoperimetric inequality may be
false.

######  Example 6.3.

Fix $\alpha>1$. Let $\\{p_{h}\\}_{h=0,1,\dots,}$ a dense sequence in $B_{1}$,
with $p_{h}\not=0$ for all $h$ and set

$r_{i}=\frac{r}{2^{\frac{i}{n}}}\quad\text{for all $i=0,1,\dots$,
with}\,\,0<r<\frac{1}{2^{\alpha}}$

to be chosen later. We now rearrange the elements of the sequence
$\\{p_{h}\\}$ as follows. First we set $q_{0}=p_{h_{0}}$, where $h_{0}$ is the
smallest integer such that $|p_{h_{0}}|^{\alpha}>2^{\alpha}r_{0}$. Notice that
this is always possible since $2^{\alpha}r_{0}<1$. Then, for all $i=1,2,\dots$
we set $q_{i}=p_{h_{i}}$, where $h_{i}$ is the smallest integer different from
$h_{0},h_{1},\dots,h_{i-1}$ such that

(6.4) $|q_{i}|^{\alpha}=|p_{h_{i}}|^{\alpha}>2^{\alpha}r_{i}\,.$

Notice that since $r_{i}\to 0$ as $i\to\infty$, all the elements of the
sequence $\\{p_{h}\\}$ will be chosen once and only once. Finally we set

$E=\bigcup_{i=0}^{\infty}B_{r_{i}}(q_{i})\,.$

Then $|E|\leq\sum_{i}|B_{r_{i}}(q_{i})|=2\omega_{n}r^{n}$. Therefore, if
$B_{E}$ is the ball centered at the origin such that $|E|=|B_{E}|$ we have

$P_{p}(B_{E})\geq
P_{p}(B_{2^{\frac{1}{n}}r})=2^{\frac{n-1+p}{n}}n\omega_{n}r^{n-1+p}\,.$

Observe now that if $x\in\partial B_{r_{i}}(q_{i})$ then
$|x|\geq|q_{i}|-r_{i}$ and by (6.4)
$|q_{i}|-r_{i}>2r_{i}^{\frac{1}{\alpha}}-r_{i}>r_{i}^{\frac{1}{\alpha}}$.
Therefore for all $i$

$\int_{\partial B_{r_{i}}(q_{i})}|x|^{p}\,d\mathcal{H}^{n-1}\leq
n\omega_{n}r_{i}^{n-1+\frac{p}{\alpha}}$

and thus

$P_{p}(E)\leq\sum_{i=0}^{\infty}\int_{\partial
B_{r_{i}}(q_{i})}|x|^{p}\,d\mathcal{H}^{n-1}\leq
C(n,\alpha,p)r^{n-1+\frac{p}{\alpha}}<n\omega_{n}2^{\frac{n-1+p}{n}}r^{n-1+p}\leq
P_{p}(B_{E})\,,$

provided $r$ is sufficiently small.

Let us now show that $0\in\overline{\partial^{*}E}$. Since $B_{1}\subset
E^{(1)}\cup\partial E^{(1)}$ it is enough to show that $0\in E^{(0)}$. To this
end we estimate for $0<\varrho<r$

(6.5) $\frac{|E\cap
B_{\varrho}|}{\omega_{n}\varrho^{n}}\leq\frac{1}{\varrho^{n}}\sum_{\\{i:\,B_{r_{i}}(q_{i})\cap
B_{\varrho}\not=\emptyset\\}}r_{i}^{n}\,.$

Note that if $B_{r_{i}}(q_{i})\cap B_{\varrho}\not=\emptyset$ then
$|q_{i}|-r_{i}<\varrho$ and thus, recalling (6.4), $r_{i}<\varrho^{\alpha}$.
Thus, from (6.5) we have, denoting by $\lfloor\cdot\rfloor$ the integer part
of a real number,

$\displaystyle\frac{|E\cap B_{\varrho}|}{\omega_{n}\varrho^{n}}$
$\displaystyle\leq\frac{r^{n}}{\varrho^{n}}\sum_{\\{i:\,2^{\frac{i}{n}}>r/\varrho^{\alpha}\\}}\frac{1}{2^{i}}=\frac{r^{n}}{\varrho^{n}}\sum_{i=1+\lfloor
n\log_{2}(r/\varrho^{\alpha})\rfloor}^{\infty}\frac{1}{2^{i}}$
$\displaystyle=\frac{r^{n}}{\varrho^{n}}\frac{1}{2^{\lfloor
n\log_{2}(r/\varrho^{\alpha})\rfloor}}\leq\frac{r^{n}}{\varrho^{n}}\frac{2}{2^{n\log_{2}(r/\varrho^{\alpha})}}=2\frac{\varrho^{n\alpha}}{\varrho^{n}}\,.$

Then we conclude that

$\lim_{r\to 0}\frac{|E\cap B_{\varrho}|}{\omega_{n}\varrho^{n}}=0\,,$

thus proving that $0\in\overline{\partial^{*}E}$. Finally observe that, thanks
to Remark 6.2 we have indeed that
$0\in\overline{\partial^{*}E}\setminus\partial^{*}E$.

##  7\. Aknowledgment

The authors wish to thank Adi Adimurthi for informing them of a mistake in a
preliminary version of the paper.
This research was supported by MUR project PRIN2017TEXA3H.

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