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 Open Access
Existence and uniqueness of maximizers of a class of functionals under constraints: optimal conditions
 Hichem Hajaiej^{1}Email author
https://doi.org/10.1186/1029242X2012151
© Hajaiej; licensee Springer 2012
 Received: 8 February 2012
 Accepted: 11 May 2012
 Published: 3 July 2012
Abstract
In this article, we establish optimal assumptions under which general HardyLittlewood and Riesztype functionals are maximized by balls. We also determine additional hypotheses such that balls are the unique maximizers. In both cases, we prove that our assumptions are optimal.
Keywords
 Strict Inequality
 Monotonicity Property
 Unique Maximizer
 Borel Measurable Function
 Powerful Result
1 Introduction
Functional optimization under constraints plays a crucial role in many important areas like economics [1], physics, and engineering [3]. In many relevant cases, one needs very precise informations about maximizers or minimizers, especially in economics where the integrand represents the cost function, see [1] and references therein for a more detailed account. In this article, we will discuss such a case and a variational problem for steady axisymmetric vortexrings in which kinetic energy is maximized subject to a prescribed impulse involving Riesztype functionals under constraints [2, 3]. Burton [2] proved the existence of maximizers in a natural constraint set, and then showed that optimizers are unique (up to translations). In [3], we have extended his result to general Riesztype functionals, which enabled us to treat a much larger class of functionals.
In this article, we develop an innovative approach based on a powerful result in mass transportation theory [4], Theorem 1], which turns out to be very fruitful for our purpose. We are convinced that our method applies to other optimization problems. This approach is also very efficient to give some estimates [5]. First, we have established suitable conditions to prove that balls are maximizers of the HardyLittlewoodtype functionals (Part 1, Theorem 1). Note that this result improves [6], Proposition 3.1], in which Draghici and Hajaiej [7] needed the superfluous supermodularity of the integrand. It also improves a result of [7] in which the author used a mass transportation technique. We have then focused our attention to prove the optimality of our hypotheses. We also prove the optimality of these results. Characterization and uniqueness of optimizers of Riesztype functionals were established in [3], our new approach also allows us to weaken (Ψ3) of [3]; thanks to a subtle reduction of the Riesztype functionals to the HardyLittlewood ones (Theorem 2). This reduction is always possible; we will then give detailed proofs for HardyLittlewood functionals, and results concerning Riesztype functionals are immediate consequences.
2 Notations and preliminaries

For $n\in {\mathbb{N}}^{\ast}$, μ is Lebesgue measure in ${\mathbb{R}}^{n}$.

For a Lebesgue measurable set A in ${\mathbb{R}}^{n}$, $\mu (A)$ denotes its measure.

${\mathcal{B}}_{0}=\{x\in {\mathbb{R}}^{n}:x<1\}$ is the ball centered in the origin with radius 1.

For $a>0$, $a{\mathcal{B}}_{0}=\{x\in {\mathbb{R}}^{n}:x<a\}={B}_{a}$.

$M({\mathbb{R}}^{n})$ is the set of measurable functions in ${\mathbb{R}}^{n}$.

For $f,g\in M({\mathbb{R}}^{n})$:$I(f,g)={\int}_{{\mathbb{R}}^{n}}F(f(x),g(x))\phantom{\rule{0.2em}{0ex}}dx$(1)
is the Riesz type functional.

For ${\ell}_{1},{k}_{1},{\ell}_{2},{k}_{2}>0$:

${\mathcal{B}}_{1}$ is the ball centered in the origin such that $\mu ({\mathcal{B}}_{1})={\ell}_{1}/{k}_{1}$.

${\mathcal{B}}_{2}$ is the ball centered in the origin such that $\mu ({\mathcal{B}}_{2})={\ell}_{2}/{k}_{2}$.

$C({k}_{1},{\ell}_{1})=\{f\in M({\mathbb{R}}^{n}):0\le f\le {k}_{1}\text{and}\int f\le {\ell}_{1}\}$.

.

A function $u:{\mathbb{R}}^{n}\to {\mathbb{R}}_{+}$ is Schwarzsymmetric if it is radial and radially decreasing. We say that it is strictly Schwarzsymmetric if it is radial and strictly radially decreasing.
In this article, we are interested in the following maximization problems:
3 Main results
In Theorem 1, we will suppose without loss of generality that $F(x,0)=0$ for $x\ge 0$.
Theorem 1 Part 1 (Balls are maximizers of the HardyLittlewood type functionals)
The following assertions are equivalent:
H1 For any${\ell}_{1},{k}_{1}>0$, u Schwarz symmetric, (I1) admits the function${k}_{1}{\mathbf{1}}_{{\mathcal{B}}_{1}}$as a maximizer.
Part 2 (Balls are the unique maximizers of the HardyLittlewood type functionals) (A) = For every${\ell}_{1},{k}_{1}>0$, u Schwarz symmetric, if (I1) admits${k}_{1}{\mathbf{1}}_{{\mathcal{B}}_{1}}$as a unique maximizer, then; (B) = $F({x}_{1},y)<F({x}_{2},y)$$\mathrm{\forall}{x}_{1},{x}_{2},y\ge 0$, ${x}_{1}<{x}_{2}$. $F(x,ty)<tF(x,y)$$\mathrm{\forall}x,y\ge 0$, $t\in (0,1)$..Conversely, if(B^{′}) = $F({x}_{1},y)<F({x}_{2},y)$$\mathrm{\forall}{x}_{1},{x}_{2},y\ge 0$, ${x}_{1}<{x}_{2}$. $F(x,ty)<tF(x,y)$$\mathrm{\forall}x,y\ge 0$, $t\in (0,1)$. u is strictly Schwarz symmetric, then; (A^{′}) = for any${\ell}_{1},{k}_{1}>0$, u, ${k}_{1}{\mathbf{1}}_{{\mathcal{B}}_{1}}$is the unique maximizer of (I1)..
4 Proof
Part 1:
First note that $I(u,v)$ is welldefined for any $v\in C({k}_{1},{\ell}_{1})$ [6], Proposition 3.1]. Let us prove that (H2) implies (H1). Let $v\in C({k}_{1},{\ell}_{1})$, since $0\le v\le {k}_{1}$, we have that $F(u(x),v(x))\le \frac{v(x)}{{k}_{1}}F(u(x),{k}_{1})$.
But $F(u(x),{k}_{1})\ge F(u(r),{k}_{1})$$\mathrm{\forall}x\in {B}_{r}$, and $F(u(x),{k}_{1})\le F(u(r),{k}_{1})$$\mathrm{\forall}x\in {B}_{r}^{\mathrm{\prime}}$.
which holds true since $v\in C({k}_{1},{\ell}_{1})$.
Let us now prove that (H1) implies (H2).
$\text{(H1)}\Rightarrow \text{(H2)}$:
Let us assume that $I(u,{k}_{1}{1}_{{\mathcal{B}}_{1}})\ge I(u,f)$ for any ${k}_{1},{\ell}_{1}>0$ and $f\in C({k}_{1},{\ell}_{1})$, we want to prove (3) and (4).
Let ${x}_{1},{x}_{2},y\ge 0$ with ${x}_{1}<{x}_{2}$ fixed. Since $\mu (2{\mathcal{B}}_{0}\mathrm{\setminus}{\mathcal{B}}_{0})=({2}^{n}1)\mu ({\mathcal{B}}_{0})=({2}^{n}1)\mu ({\mathcal{B}}_{0})$, we can chose $E\subset 2{\mathcal{B}}_{0}\mathrm{\setminus}{\mathcal{B}}_{0}$ such that $\mu (E)=\mu ({\mathcal{B}}_{0})$. Let us set $u={x}_{2}{1}_{{\mathcal{B}}_{0}}+{x}_{1}{1}_{2{\mathcal{B}}_{0}\mathrm{\setminus}{\mathcal{B}}_{0}}$, ${f}_{1}=y{1}_{E}$, ${k}_{1}=y$, ${f}_{1}\in C({k}_{1},{\ell}_{1})$. Choose ${\ell}_{1}$ such that ${\mathcal{B}}_{1}\equiv {\mathcal{B}}_{0}$.
and (3) follows immediately.
Now we would like to prove (4).
For any $x,y\in {\mathbb{R}}^{+}$ and $t\in (0,1)$ fixed, define $u=x{\mathbf{1}}_{{\mathcal{B}}_{0}}$, ${k}_{1}=y$. Select ${\ell}_{1}$ such that ${\ell}_{1}=t{k}_{1}\mu ({\mathcal{B}}_{0})$. Therefore, ${k}_{1}{\mathbf{1}}_{{\mathcal{B}}_{1}}=y{1}_{{B}_{1}}$. Set ${f}_{1}=ty{\mathbf{1}}_{{\mathcal{B}}_{0}}$, ${f}_{1}\in C({k}_{1},{\ell}_{1})$ and by our assumption $I(u,{k}_{1}{\mathbf{1}}_{{\mathcal{B}}_{1}})\ge I(u,{f}_{1})$.
We now turn to the proof of Part 2 of our result: $\text{(A)}\Rightarrow \text{(B)}$.
We are assuming that $I(u,{k}_{1}{\mathbf{1}}_{{\mathcal{B}}_{1}})>I(u,f)$ for every Schwarz symmetric function u and ${k}_{1},{\ell}_{1}>0$, $f\in C({k}_{1},{\ell}_{1})$. Then (8) and (6) hold true with strict sign, it follows that (3) and (4) are strict.
${\text{(B}}^{\prime}\text{)}\Rightarrow {\text{(A}}^{\prime}\text{)}$:
Here all inequalities in the proof of Part 1, become strict inequalities which permit us to conclude.
Remark 1 Note that $\text{(A)}\Rightarrow \text{(B)}$ cannot be done if u is strictly Schwarz symmetric since in our construction, we need u to be constant on some domains. However for any u Schwarz symmetric function the condition $F({x}_{1},y)<F({x}_{2},y)$ for ${x}_{1},{x}_{2},y\ge 0$ with ${x}_{1}<{x}_{2}$ is necessary for the uniqueness of the maximizer. More precisely consider $F(x,y)={y}^{2}$. For any $f={k}_{1}{\mathbf{1}}_{A}$ where $\mu (A)=\mu ({\mathcal{B}}_{1})$, f is a maximizer of (I1) independently of the choice of u since $I(u,f)={k}_{1}^{2}\mu (A)={k}_{1}^{2}\mu ({\mathcal{B}}_{1})=I(u,{k}_{1}{\mathbf{1}}_{{\mathcal{B}}_{1}})$.
Theorem 2 Part 1 (Balls are maximizers of the Riesztype functionals)
Suppose that$F:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$satisfies:
 (i)
$F(t{x}_{1},y)\le tF({x}_{1},y)$$\mathrm{\forall}x,y\ge 0$, $t\in (0,1)$,
 (ii)
$F({x}_{1},ty)\le tF({x}_{1},y)$$\mathrm{\forall}x,y\ge 0$, $t\in (0,1)$,
(F2) $F(b,d)F(b,c)F(a,d)+F(a,c)\ge 0$$\mathrm{\forall}0\le a<b$, $0\le c<d$.
Part 2 Uniqueness of the maximizers (up to translations)
where${h}_{1}$and${h}_{2}$are translates by the same vector of${k}_{1}{\mathbf{1}}_{{\mathcal{B}}_{1}}$and${k}_{2}{\mathbf{1}}_{{\mathcal{B}}_{2}}$ (respectively).
where $u(x)={\int}_{{\mathbb{R}}^{n}}F({k}_{1},{g}^{\ast}(y))j(xy)\phantom{\rule{0.2em}{0ex}}dy$.
Similarly, using (F1)(ii), we deduce that $J({k}_{1}{\mathbf{1}}_{{\mathcal{B}}_{1}},{g}^{\ast})\le J({k}_{1}{\mathbf{1}}_{{\mathcal{B}}_{1}},{k}_{2}{\mathbf{1}}_{{\mathcal{B}}_{2}})$.
Using equality cases established in [8], Theorem 1] and Part 2 of Theorem 1, we can easily prove Part 2 of Theorem 2. □
Declarations
Acknowledgements
The author extend his appreciation to the Deanship of Scientific Research at King Saud University for funding the work through the research group project No RGPVPP124.
Authors’ Affiliations
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