# Balls are maximizers of the Riesz-type functionals with supermodular
integrands

Hichem Hajaiej  Justus-Liebig-Universität Giessen
Mathematisches Institut
Arnd Str 2, 35392 Giessen
Germany<EMAIL_ADDRESS>
###### Abstract.

For a large class of supermodular integrands, we establish conditions under
which balls are the unique (up to translations) maximizers of the Riesz-type
functionals with constraints.

##  1\. Introduction

Over the last decades, one field of intense research activity has been the
study of extremals of integral functionals. The Riesz-type kind has attracted
growing attention and played a crucial role in the resolution of Choquard’s
conjecture in a breakthrough paper by E. H. Lieb [1]. The determination of
cases of equality in the Riesz-rearrangement inequality has also received a
large amount of interest from mathematicians due to its connection with many
other functional inequalities and its several applications to physics [2, 3,
4]. Variational problems for steady axisymmetric vortex-rings in which kinetic
energy is maximized subject to prescribed impulse involves Riesz-type
functionals with constraints. In [5], G. R. Burton has proved the existence of
maximizers in an extended constraint set, he has also showed that the
maximizer is Schwarz symmetric (up to translations). His method hinges on a
resolution of an optimization of a Riesz-type functional under constraint [5,
Proposition 8]. The purpose of this paper is to answer the more general
question: When do maximizers of the Riesz-type functional inherit the symmetry
and monotonicity properties of the integrand involved in it?

The method of G. R. Burton [5] cannot apply to solve the above problem. In
this paper, we develop a self-contained approach. Let us give here a foretaste
of our ideas. First, we recall that:

A Riesz-type functional is a functional of the form:

$R(f,g)=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\Psi\left(f(x),g(y)\right)\,r(x,y)\,dx\,dy.$

In this paper, we will consider $r(x,y)=j\left(|x-y|\right)$. We are
interested in the following maximization problem:
(P1) $\sup\limits_{(f,g)\in C}J(f,g)$
where

(1.1)
$J(f,g)=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\Psi\left(f(x),g(y)\right)\,j\left(|x-y|\right)\,dx\,dy.$

and

(1.2) $C=(f,g):\begin{cases}f:&\mathbb{R}^{n}\rightarrow\mathbb{R};0\leq f\leq
k_{1}\mbox{ and }\int_{\mathbb{R}^{n}}f\leq\ell_{1}\\\
g:&\mathbb{R}^{n}\rightarrow\mathbb{R};0\leq g\leq k_{2}\mbox{ and
}\int_{\mathbb{R}^{n}}g\leq\ell_{2}\end{cases}$

$\ell_{1},k_{1},\ell_{2},k_{2}$ are positive numbers.

For supermodular operators $\Psi$ and nonincreasing functions $j$, we know
that $J(f,g)\leq J(f^{*},g^{*})$ [4, Theorem 1], where $u^{*}$ denotes the
Schwarz symmetrization of $u$. Hence the problem reduces to:
(P2) $\sup\limits_{(f^{*},g^{*})\in C}J\left(f^{*},g^{*}\right)$
For continuous integrands $\Psi$ having the N-Luzin property (for any subset N
having Lebesgue measure zero, $\Psi$(N) has the same property), lemma 2.6
enables us to assert that (P2) is equivalent to an optimization of a Hardy-
Littlewood type functionals where balls are maximizers. We will then extend
this study to supermodular non-continuous bounded functions $\Psi$ thanks to
the decomposition of these functions into
$\tilde{\Psi}\left(\varphi_{1}(s_{1}),\varphi_{2}(s_{2})\right)$ in the spirit
of [4, 6]. The approximation of unbounded supermodular functions by bounded
ones inheriting the monotonicity properties will enable us to prove that balls
are maximizers in the general case.

Main Result:

Let $\Psi:\mathbb{R}_{+}\times\mathbb{R}_{+}\rightarrow\mathbb{R}$ be a
H-Borel function satisfying:
($\Psi$1) $\Psi$ vanishes at hyperplanes;
($\Psi$2) $\Psi(b,d)-\Psi(b,c)-\Psi(a,d)+\Psi(a,c)\geq 0$ for all $0\leq a<b$
and $0\leq c<d$;
($\Psi$3)(i) $\Psi(tx,b_{2})-t\Psi(x,b_{2})-\Psi(tx,b_{1})+t\Psi(x,b_{1})\leq
0$ for all $x\geq 0$, $0\leq b_{1}<b_{2}$ and $0<t<1$;
($\Psi$3)(ii) $\Psi(a_{2},ty)-t\Psi(a_{2},y)-\Psi(a_{1},ty)+t\Psi(a_{1},y)\leq
0$ for all $y\geq 0$, $0\leq a_{1}<a_{2}$ and $0<t<1$;
(j1) $j$ is nonincreasing.

Suppose in addition that $\Psi$ is continuous with respect to each variable
and has the N-Luzin property, then for all $(f_{1},f_{2})\in C$

$J\left(f_{1},f_{2}\right)\leq
J\left(k_{1}\mathrm{1}_{B_{1}},k_{2}\mathrm{1}_{B_{2}}\right)$

where $B_{1}$ and $B_{2}$ are centered in the origin, $\mathrm{1}_{B}$ is the
characteristic function of $B$, and $\mu(B_{1})=\ell_{1}/k_{1}$,
$\mu(B_{2})=\ell_{2}/k_{2}$. Moreover, if ($\Psi$2) and ($\Psi$3) hold with
strict inequality, $j$ is strictly decreasing and
$J\left(f_{1},f_{2}\right)<\infty$ for any $(f_{1},f_{2})\in C$, then (P1) is
attained by exactly two couples
$\left(k_{1}\mathrm{1}_{B_{1}},k_{2}\mathrm{1}_{B_{2}}\right)$ and
$(h_{1},h_{2})$ where $h_{1}$ and $h_{2}$ are translates by the same vector of
$k_{1}\mathrm{1}_{B_{1}}$ and $k_{2}\mathrm{1}_{B_{2}}$ (respectively).

##  2\. Notations and preliminaries

######  Definition 2.1.

If $A\subset\mathbb{R}^{n}$ is a measurable set of finite Lebesgue measures
$\mu$, we define $A^{*}$, the symmetric rearrangement of the set $A$ to be the
open ball centered at the origin whose volume is that of $A$, thus
$A^{*}=\left\\{x\in\mathbb{R}^{n}:|x|<r\right\\}$ with $V_{n}r^{n}=\mu(A)$,
$V_{n}$ is a constant.

For a nonnegative measurable function $u$ on $\mathbb{R}^{n}$, we require $u$
to vanish at infinity in the sense that all its positive level sets
$\\{x\in\mathbb{R}^{n}:u(x)>t\\}$ having finite measure for $t>0$. The set of
these functions is denoted by $F_{n}$. The symmetric decreasing rearrangement
$u^{*}$ of $u$ is the unique upper semicontinuous, nonincreasing radial
function that is equimeasurable with $u$. Explicitly,
$u^{*}(x)=\int\limits_{0}^{\infty}{\mathbf{1}}^{*}_{\\{u>t\\}}(x)\,dt$ where
${\mathbf{1}}^{*}_{A}={\mathbf{1}}_{A^{*}}$. We say that $u$ is Schwarz
symmetric if $u\equiv u^{*}$.

######  Definition 2.2.

A reflexion $\sigma$ on $\mathbb{R}^{n}$is an isometry with the properties:

  * (i)

$\sigma^{2}_{x}=\sigma_{x}\circ\sigma_{x}=x$ for all $x\in\mathbb{R}^{n}$;

  * (ii)

the fixed point set of $H_{0}$ of $\sigma$ separates $\mathbb{R}^{n}$ into two
half spaces $H_{+}$ and $H_{-}$ that are interchanged by $\sigma$;

  * (iii)

$|x-x^{\prime}|<|x-\sigma_{x^{\prime}}|$ for all $x,x^{\prime}\in H_{+}$.

$H_{+}$ is the half space containing the origin.

The two point rearrangement or polarization of a real valued function $u$ with
respect to a reflection $\sigma$ is defined by:

(2.1) $u^{\sigma_{x}}=\begin{cases}\max\\{u(x),u(\sigma_{x})\\},x\in H_{+}\cup
H_{0},\\\ \min\\{u(x),u(\sigma_{x})\\},x\in H_{-}.\end{cases}$

######  Lemma 2.3.

Let $j:[0,\infty)\rightarrow\mathbb{R}$ be a nonincreasing function then
$\nu(x)=\int_{{\mathbb{R}}^{n}}j\left(|x-y|\right)h(y)\,dy$ is radial and
radially decreasing for any Schwarz symmetric function $h$. If in addition $j$
is strictly radially decreasing then $\nu$ also inherits this property.

Proof: we will use [7, Lemma 2.8]: $u=u^{*}\Leftrightarrow u=u^{\sigma}$ for
all $\sigma$. It is sufficient to prove that $u(x)\geq u(\sigma_{x})$ for all
$x\in\mathbb{R}^{n}$, all $\sigma$.

$\displaystyle u(x)$ $\displaystyle=$
$\displaystyle\int_{H^{+}}j\left(|x-y|\right)h(y)+j\left(|x-\sigma_{y}|\right)h(\sigma_{y})\,dy$
$\displaystyle u(\sigma_{x})$ $\displaystyle=$
$\displaystyle\int_{H^{+}}j\left(|\sigma_{x}-y|\right)h(y)+j\left(|\sigma_{x}-\sigma_{y}|\right)h(\sigma_{y})\,dy$
$\displaystyle u(x)-u(\sigma_{x})$ $\displaystyle=$
$\displaystyle\int_{H_{+}}j\left(|x-y|\right)[h(y)-h(\sigma_{y})]-j\left(\sigma_{x}-y\right)[h(y)-h(\sigma_{y})]\,dy$
$\displaystyle=$
$\displaystyle\int_{H_{+}}\left(j\left(|x-y|\right)-j(\sigma_{x}-y)\right)\left(h(y)-h(\sigma_{y})\right)\,dy$

By (iii) $|x-y|<|\sigma_{x}-y|$, it follows that $j\left(|x-y|\right)\geq
j\left(|\sigma_{x}-y|\right)$. On the other hand $h$ is Schwarz symmetric,
hence $h(y)\geq h(\sigma_{y})$ for all $y\in H_{+}$, the conclusion follows.

######  Definition 2.4.

Let $\Psi:\mathbb{R}_{+}\times\mathbb{R}_{+}\rightarrow\mathbb{R}$:

  * (a)

$\Psi$ is supermodular if $(\Psi 2)$ holds.

  * (b)

We say that $\Psi$ vanishes at hyperplanes if $\Psi(s_{1},0)=\Psi(0,s_{2})=0$
for all $s_{1},s_{2}\geq 0$.

An important property of functions satisfying (c) is that the composition
$(x,y)\mapsto\Psi\left(f(x),g(y)\right)$ is measurable on $\mathbb{R}_{+}$ for
every $f,g\in F_{n}$. Hence $j\left(|x-y|\right)\Psi\left(f(x),g(y)\right)$ is
measurable on $\mathbb{R}_{+}\times\mathbb{R}_{+}$.

In the spirit of [4] and [6], we obtain:

######  Lemma 2.5.

Assume that $\Psi:\mathbb{R}_{+}\times\mathbb{R}_{+}\rightarrow\mathbb{R}$ is
a supermodular bounded function vanishing at hyperplanes. Then there exist two
bounded nondecreasing functions $\varphi_{1}$ and $\varphi_{2}$ on
$\mathbb{R}_{+}$ with $\varphi_{i}(0)=0$ and a Lipschitz continuous function
$\tilde{\Psi}$ on $\mathbb{R}_{+}^{2}$ such that
$\Psi(u,v)=\tilde{\Psi}\left(\varphi(u),\varphi(v)\right)$.

Proof: First, we will prove the following: If $\varphi$ is a nondecreasing
real-valued function defined on an interval $I$, then for every $f$ on I
satisfying $|f(u)-f(v)|<c\left(\varphi(v)-\varphi(u)\right)$ where $u<v\in I$,
$c$ is a constant, there exists a Lipschitz continuous function
$\tilde{f}:\mathbb{R}\rightarrow[\inf{f},\sup{f}]$ such that
$f(x)=\tilde{f}\circ\varphi(x)$ (2.0). If $f$ is nondecreasing then
$\tilde{f}$ is nondecreasing also.

The result is obvious for $t=\varphi(v)$ and $s=\varphi(u)<t$ since we have

$|\tilde{f}(t)-\tilde{f}(s)|=\left|f\left(\varphi(v)\right)-f\left(\varphi(u)\right)\right|\leq
c\left(\varphi(v)\right)-\left(\varphi(u)\right)=c(t-s).$

Now $\tilde{f}$ has a unique extension to the closure of the image and the
complement consists of a countable number of disjoint bounded intervals, it is
sufficient to interpolate $\tilde{f}$ linearly between the values, that were
assigned to end-points. By construction $f=\tilde{f}\circ\varphi$ and
$\tilde{f}(\mathbb{R})=[\inf f,\sup f]$ the extension we have made by linear
interpolation preserves of course the modulus of continuity of $\tilde{f}$:
$|\tilde{f}(t)-\tilde{f}(s)|\leq c(t-s)$ for all $t>s$. If $f$ is
nondecreasing, it is easy to check that this property is inherited by
$\tilde{f}$.

Now we can prove our lemma:

First note that the fact that $\Psi$ is supermodular and vanishes at
hyperplanes imply that it is nondecreasing with respect to each variable and
it is nonnegative. Now set
$\varphi_{1}(u)=\lim\limits_{u\rightarrow+\infty}\Psi(u,v)$.

$\varphi_{1}$ is well-defined on $\mathbb{R}_{+}$ since $\Psi$ is bounded and
nondecreasing in the second variable. By the supermodularity of $\Psi$, it
follows that

$\Psi(u+h_{1},v+h_{2})-\Psi(u,v+h_{2})-\Psi(u+h_{1},v)+\Psi(u,v)\geq 0$

for any $u,v,h_{1}$ and $h_{2}\geq 0$.

Letting $h_{2}$ tend to infinity, we obtain

$\varphi_{1}(u+h_{1})-\varphi(u)\geq\Psi(u+h_{1},v)-\Psi(u,v)\geq 0$

for all $u,v,h_{1}\geq 0$.

For a fixed $v$, the last inequality enables us to apply (2.0) to
$\Psi(\cdot,v)$. Hence, there exists $\Psi^{1}$ such that:
$\Psi(u,v)=\Psi^{1}(\varphi_{1}(u),v)$. A moment’s consideration shows that
$\Psi^{1}$ inherits all the properties of $\Psi$. Now set
$\varphi_{2}(v)=\lim\limits_{u\rightarrow+\infty}\Psi(u,v)$, a similar
argument ensures us that there exists $\tilde{\Psi}$ such that
$\Psi^{1}\left(\varphi_{1}(u),u\right)=\tilde{\Psi}^{1}\left(\varphi_{1}(u),\varphi_{2}(v)\right)$.

$\tilde{\Psi}$ has the same monotonicity property as $\Psi^{1}$ and
consequently as $\Psi$. Note that $\varphi_{1}(0)=\varphi_{2}(0)=0$ and the
monotonicity properties of $\Psi$ imply that $\varphi_{1}$ and $\varphi_{2}$
are nondecreasing.

######  Lemma 2.6.

Let $l,k>0$, $D=\\{h:\mathbb{R}^{n}\rightarrow\mathbb{R}:0\leq h(x)\leq
k\mbox{ and }\int_{\mathbb{R}^{n}}h(x)\,dx\leq l\\}$. Suppose that
$\Gamma:\mathbb{R}_{+}\rightarrow\mathbb{R}$ is a function satisfying:

  1. (1)

$\Gamma(0)=0$,

  2. (2)

$\Gamma(tx)\leq t\Gamma(x)$ for all $x\geq 0$ and $0<t<1$.

  3. Assume also that

  4. (3)

$u:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is a Schwarz symmetric function. Then
for every $\nu\in
D:\int_{\mathbb{R}^{n}}u(x)\Gamma\left(\nu(x)\right)\,dx\leq\int_{\mathbb{R}^{n}}u(x)\Gamma\left(k{\mathbf{1}}_{B}(x)\right)\,dx$
where $B$ is the ball centered at the origin with $\mu(B)=\ell/k$.

Proof: (2) implies that

$\displaystyle\int_{\mathbb{R}^{n}}u(x)\Gamma\left(\nu(x)\right)\,dx\leq\int_{\mathbb{R}^{n}}u(x)\Gamma(k)\frac{\nu(x)}{k}\,dx$
$\displaystyle=$
$\displaystyle\Gamma(k)\left[\int_{B}u(x)\left[\frac{\nu(x)}{k}-1+1\right]\,dx+\int_{\mathbb{R}^{n}-B}\frac{u(x)\nu(x)}{k}\,dx\right]$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{n}}u(x)\Gamma\left(k{\mathbf{1}}_{B(x)}\right)\,dx$
$\displaystyle\quad+\Gamma(k)\left[\int_{B}u(x)\left[\frac{\nu(x)}{k}-1\right]\,dx+\int_{\mathbb{R}^{n}-B}\frac{u(x)\nu(x)}{k}\,dx\right]\,dx.$

Using (3), it follows that the above integrals are
$\leq\int_{\mathbb{R}^{n}}u(x)\Gamma\left(k{\mathbf{1}}_{B(x)}\right)\,dx+\Gamma(k)u(r)\left[\int_{\mathbb{R}^{n}}\frac{\nu(x)}{k}\,dx-\mu(B)\right]$
where $\mu(B)=V_{r}r^{n}$ (see definition 2.1). Thus
$\int_{\mathbb{R}^{n}}u(x)\Gamma\left(\nu(x)\right)\,dx\leq\int_{\mathbb{R}^{n}}u(x)\Gamma\left(k{\mathbf{1}}_{B(x)}\right)\,dx$,
since $\int_{\mathbb{R}^{n}}\frac{\nu(x)}{k}\,dx\leq\mu(B)=\ell/k$.

If additionally
$\int_{\mathbb{R}^{n}}u(x)\Gamma\left(\nu(x)\right)\,dx<\infty$ for any
$\nu\in D$, (2) holds with strict inequality and $u$ is strictly decreasing,
we can prove that for every $\nu\in D$:
$\int_{\mathbb{R}^{n}}u(x)\Gamma\left(\nu(x)\right)\,dx<\int_{\mathbb{R}^{n}}u(x)\Gamma\left(k{\mathbf{1}}_{B(x)}\right)\,dx$.

##  3\. Proof of the result

For the convenience of the reader, the proof will be divided in three parts.
First part: We suppose that: $\Psi(\cdot,s_{2})$ is absolutely continuous for
every $s_{2}\geq 0$, and $\Psi(s_{1},\cdot)$ is absolutely continuous for
every $s_{1}\geq 0$.

First note that $(\Psi 1)$ and $(\Psi 2)$ imply that $\Psi$ is a non-
decreasing function with respect to each variable and it is nonnegative.

Let $(f_{1},f_{2})\in C$, $(\Psi 2)$ and (j1) imply that

$\displaystyle J\left(f_{1},f_{2}\right)\leq
J\left(f_{1}^{*},f_{2}^{*}\right)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\Psi\left(f_{1}^{*}(x),f_{2}^{*}(y)\right)j\left(|x-y|\right)\,dx\,dy$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\left(\int_{0}^{f_{2}^{*}(y)}F\left(f_{1}^{*}(x),s\right)\,ds\right)j\left(|x-y|\right)\,dx\,dy$

where $\Psi(s_{1},s_{2})=\int_{0}^{s_{2}}F\left(s_{1},u\right)\,du$.

Applying Tonelli’s theorem (see (3.0)), we obtain:

$J\left(f_{1}^{*},f_{2}^{*}\right)=\int_{0}^{\infty}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}j\left(|x-y|\right){\mathbf{1}}_{\\{y\in\mathbb{R}^{n}:f_{2}^{*}(y)\geq
s\\}}F\left(f_{1}^{*}(x),s\right)\,dy\,dx\,ds.$

Setting
$u(x,s)=\int_{\mathbb{R}^{n}}{\mathbf{1}}_{\\{y\in\mathbb{R}^{n}:f_{2}^{*}(y)\geq
s\\}}j\left(|x-y|\right)\,dy$, it follows from lemma 2.3 that $u$ is radial
and radially decreasing with respect to $x$ for every fixed $s$.

$J\left(f_{1}^{*},f_{2}^{*}\right)=\int_{0}^{\infty}\int_{\mathbb{R}^{n}}u(x,s)F\left(f_{1}^{*}(x),s\right)\,dx\,ds.$

Now for a fixed $s_{1}\geq 0$,
$\Psi(s_{1},x_{2})-\Psi(s_{1},x_{1})=\int_{x_{1}}^{x_{2}}F(s_{1},t)\,dt\geq 0$
for $x_{2}\geq x_{1}$; from which we deduce that $F(s_{1},t)$ is nonnegative
for almost every $t\geq 0$. (3.0)

On the other hand, $0=\Psi(0,s_{2})=\int_{0}^{s_{2}}F(0,u)\,du$. By the
nonnegativity of $F$, we conclude that $F(0,s)=0$ for almost every $s\geq 0$.

Moreover ($\Psi$3) says that: $\Psi(tx,d)-t\Psi(x,d)-\Psi(tx,c)+t\Psi(x,c)\leq
0$ for every $x\geq 0$, $d\geq c\geq 0$.

Integrating this inequality, we have $\int_{c}^{d}F(tx,u)-tF(x,u)\,du\geq 0$
for every $x\geq 0$; $d\geq c\geq 0$.

Hence $F(tx,u)\leq tF(x,u)$ for all $x\geq 0$, $t\in]0,1[$ and almost every
$u\geq 0$.

This shows that for almost every $s\geq 0$, the function
$u(x,s)F\left(f_{1}^{*}(x),s\right)$ satisfies all the hypotheses of lemma
2.6, consequently:

For almost every $s\geq 0$
$\int_{\mathbb{R}^{n}}u(x,s)F\left(f_{1}^{*}(x),s\right)\,dx\leq\int_{\mathbb{R}^{n}}u(x,s)F\left(k_{1}{\mathbf{1}}_{B_{1}}(x),s\right)\,dx$
and

(3.1) $J\left(f_{1}^{*},f_{2}^{*}\right)\leq
J\left(k_{1}{\mathbf{1}}_{B_{1}},f_{2}^{*}\right).$

Using the same argument, we easily conclude that

(3.2) $J\left(k_{1}{\mathbf{1}}_{B_{1}},f_{2}^{*}\right)\leq
J\left(k_{1}{\mathbf{1}}_{B_{1}},k_{2}{\mathbf{1}}_{B_{2}}\right).$

By [4, Theorem 2] we know that:

(3.3) $J\left(f_{1},f_{2}\right)\leq J\left(f_{1}^{*},f_{2}^{*}\right).$

Combining these three inequalities, we obtain:

$J\left(f_{1},f_{2}\right)\leq J\left(f_{1}^{*},f_{2}^{*}\right)\leq
J\left(k_{1}{\mathbf{1}}_{B_{1}},f_{2}^{*}\right)\leq
J\left(k_{1}{\mathbf{1}}_{B_{1}},k_{2}{\mathbf{1}}_{B_{2}}\right).$

If in addition, we have strict inequality in ($\Psi$2) and ($\Psi$3), $j$ is
strictly decreasing and $J\left(f_{1},f_{2}\right)<\infty$ for any
$f_{1},f_{2}\in C$ then [4, Theorem 2] asserts that equality occurs in (3.3)
if and only if there exists $x_{0}\in\mathbb{R}^{n}$ such that
$f_{1}=f_{1}^{*}(\cdot-x_{0})$ and $f_{2}=f_{2}^{*}(\cdot-x_{0})$.

On the other hand, by lemma 2.6, equality occurs in (3.1) if and only if
$f_{1}^{*}=k_{1}{\mathbf{1}}_{B_{1}}$. Similarly equality holds in (3.2) if
and only if $f_{2}^{*}=k_{2}{\mathbf{1}}_{B_{2}}$.

Conclusion: we have proved that for any absolutely continuous function $\Psi$
satisfying ($\Psi$1), ($\Psi$2), ($\Psi$3) with a kernel $j$ satisfying (j1)
$\left(k_{1}{\mathbf{1}}_{B_{1}},k_{2}{\mathbf{1}}_{B_{2}}\right)$ is a
maximizer of $J$ under the constraint $C$. If additionally ($\Psi$2),
($\Psi$3) hold with strict inequality $j$ is strictly decreasing and
$J\left(f_{1},f_{2}\right)<\infty$ for all $\left(f_{1},f_{2}\right)\in C$
then $\left(k_{1}{\mathbf{1}}_{B_{1}},k_{2}{\mathbf{1}}_{B_{2}}\right)$ is the
unique maximizer of (P1) (up to a translation).

Remark 1: $\Psi$ is a nondecreasing function with respect to each variable, it
is then of bounded variations. The absolute continuity is then equivalent to
its continuity and the fact that it satisfies the N-Luzin property.

Remark 2: We can remove condition ($\Psi$1) from our theorem by modifying
($\Psi$3) and adding an integrability assumption in a same way as [8,
Proposition 3.2].

Part 2: $\Psi$ is bounded.

Applying lemma 2.5, we know that there exist $\varphi_{1},\varphi_{2}$ such
that
$\Psi(s_{1},s_{2})=\tilde{\Psi}\left(\varphi_{1}(s_{1}),\varphi_{2}(s_{2})\right)$,
where $\tilde{\Psi}$ is Lipschitz continuous with respect to each variable,
there exist a function $\tilde{F}$ defined on $\mathbb{R}_{+}$ such that
$\tilde{\Psi}(s_{1},s_{2})=\int_{0}^{s_{2}}\tilde{F}(s_{1},u)\,du$.

$\displaystyle J\left(f_{1},f_{2}\right)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\tilde{\Psi}\left(\varphi_{1}\left(f_{1}^{*}(x)\right),\varphi_{2}\left(f_{2}^{*}(x)\right)\right)j\left(|x-y|\right)\,dx\,dy$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\left(\int_{\mathbb{R}^{n}}\nu(x,s)\tilde{F}\left(\varphi_{1}\left(f_{1}^{*}(x)\right),s\right)\right)\,dx\,ds$

where
$\nu(x,s)=\int_{\mathbb{R}^{n}}{\mathbf{1}}_{\\{y\in\mathbb{R}^{n}:\varphi_{2}\left(f_{2}^{*}(y)\right)\geq
s\\}}j\left(|x-y|\right)\,dy$. The function
${\mathbf{1}}_{\\{y\in\mathbb{R}^{n}:\varphi_{2}\left(f_{2}^{*}(y)\right)\geq
s\\}}$ is Schwarz-symmetric for every $s$ since $\varphi_{2}$ is
nondecreasing. We can then apply Part 1 and the result follows.

Remark 3: Here we cannot obtain a uniqueness result since $\varphi_{1}$ and
$\varphi_{2}$ do not inherit the strict monotonicity properties of $\Psi$.

Part 3: $\Psi$ is not bounded.

For $L>0$, set
$\Psi^{L}\left(s_{1},s_{2}\right)=\Psi\left(\min(s_{1},L),\min(s_{2},L)\right)$.
It is easy to check that $\Psi^{L}$ inherits all the properties of $\Psi$
stated in our result. Moreover Part 2 applies to $\Psi^{L}$ since it is a
bounded function. Noticing that $\Psi^{L}\rightarrow\Psi$, the monotone
convergence theorem enables us to conclude.

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