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Existence and uniqueness of maximizers of a class of functionals under constraints: optimal conditions
Journal of Inequalities and Applications volume 2012, Article number: 151 (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.
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}.
From now on: F:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+} is a Borel measurable function,

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 HardyLittlewood type functional and
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:
For u Schwarz symmetric, {k}_{1},{\ell}_{1},{k}_{2},{\ell}_{2}>0;
and
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.
H2 The function F satisfies the monotonicity properties:
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}).
Hence I(u,v)\le \frac{1}{{k}_{1}}{\int}_{{\mathbb{R}}^{n}}F(u(x),{k}_{1})v(x)\phantom{\rule{0.2em}{0ex}}dx. Thus it suffices to prove the inequality:
Decomposing the lefthand side of the above inequality into integrals over {B}_{r} and {B}_{r}^{\mathrm{\prime}}={\mathbb{R}}^{n}\mathrm{\setminus}{B}_{r}, we can rewrite it as:
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}}.
Therefore, it remains to verify that:
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}.
Therefore I(u,{k}_{1}{1}_{{\mathcal{B}}_{1}})=\mu ({\mathcal{B}}_{0})F({x}_{2},y).
But
implying that
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}).
But:
from which we deduce that
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 thatF:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}satisfies:
(F1)

(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.
(j1) j:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}is nonincreasing, then for any({f}_{1},{f}_{2})\in C({k}_{1},{\ell}_{1},{k}_{2},{\ell}_{2}):
Part 2 Uniqueness of the maximizers (up to translations)
If in addition (F 1), (F 2), and (j 1) hold true with strict sign then for any(f,g)\in C({k}_{1},{\ell}_{1},{k}_{2},{\ell}_{2})
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).
Proof First note that (F2) together with the fact that F(x,0)=F(0,y)=0 imply that F is nondecreasing with respect to each variable, and consequently it is nonnegative. Let (f,g)\in C({k}_{1},{\ell}_{1},{k}_{2},{\ell}_{2}). (F2) together with (j1) imply that: J(f,g)\le J({f}^{\ast},{g}^{\ast}) ({f}^{\ast} denotes the Schwarz symmetrization of f) by [8], Theorem 1]. Thanks to (F1)(i):
where u(x)={\int}_{{\mathbb{R}}^{n}}F({k}_{1},{g}^{\ast}(y))j(xy)\phantom{\rule{0.2em}{0ex}}dy.
By Theorem 1,
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}}).
In conclusion, for any (f,g)\in C({k}_{1},{\ell}_{1},{k}_{2},{\ell}_{2}), we obtain
Using equality cases established in [8], Theorem 1] and Part 2 of Theorem 1, we can easily prove Part 2 of Theorem 2. □
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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.
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Hajaiej, H. Existence and uniqueness of maximizers of a class of functionals under constraints: optimal conditions. J Inequal Appl 2012, 151 (2012). https://doi.org/10.1186/1029242X2012151
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DOI: https://doi.org/10.1186/1029242X2012151
Keywords
 Strict Inequality
 Monotonicity Property
 Unique Maximizer
 Borel Measurable Function
 Powerful Result