\documentclass[reqno]{amsart} \usepackage{amssymb} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 92, pp. 1--26.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/92\hfil $N$-point vortex system] {Statistical mechanics of the $N$-point vortex system with random intensities on $\mathbb{R}^2$} \author[C. Neri \hfil EJDE-2005/92\hfilneg] {Cassio Neri} \address{Cassio Neri \hfill\break Instituto de Matem\'atica \\ Universidade Federal do Rio de Janeiro \\ Caixa Postal 68530\\ CEP: 21945-970, Rio de Janeiro, Brazil} \email{cassio@labma.ufrj.br} \date{} \thanks{Submitted March 3, 2005. Published August 24, 2005.} \thanks{Supported by grant 200491/97-0 from CNPq Brazil} \subjclass[2000]{76F55, 82B5} \keywords{Statistical mechanics; N-point vortex system; Onsager theory; \hfill\break\indent mean field equation} \begin{abstract} The system of $N$-point vortices on $\mathbb{R}^2$ is considered under the hypothesis that vortex intensities are independent and identically distributed random variables with respect to a law $P$ supported on $(0,1]$. It is shown that, in the limit as $N$ approaches $\infty$, the 1-vortex distribution is a minimizer of the free energy functional and is associated to (some) solutions of the following non-linear Poisson Equation: $-\Delta u(x) = C^{-1}\int_{(0,1]} r\mathrm{e}^{-\beta ru(x)- \gamma r|x|^2}P(\mathrm{d}r), \quad\forall x\in \mathbb{R}^2,$ where $\displaystyle C = \int_{(0,1]}\int_{\mathbb{R}^2}\mathrm{e}^{-\beta ru(y) - \gamma r|y|^2}\mathrm{d} yP(\mathrm{d}r)$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \renewcommand{\theenumi}{\roman{enumi}} \renewcommand{\labelenumi}{(\theenumi)} \section{Introduction} In a previous work \cite{Neri} we have studied the system of $N$ point vortices on a bounded domain of $\mathbb{R}^2$ with random-vortice intensities identically distributed with respect to a law $P$. We generalized some results of Cagliotti et al. \cite{MR93i:82060} in which all the vortices have intensity equal to 1, and thus, $P$ is a Dirac measure concentrated on $1$. Here we will study the same problem on the whole plane. We shall have some technical difficulties which did not arise on the case of bounded domain \cite {Neri}, since $\mathbb{R}^2$ has infinite Lebesgue measure. However, the presence of factors like $\mathrm{e}^{-r|x|^2}$ inside integrals are sufficient to fix most of these problems. Often, the proofs will be analogous to those in \cite{Neri} just replacing $\mathrm{d} x$ by $\mathrm{e}^{-r|x|^2}\mathrm{d} x$. Related to this, we should suppose also that vortex intensities (which correspond to $r$ in $\mathrm{e}^{-r|x|^2}$) are positive and the law $P$ decreases'' fast enough near $0$. The phase space of this Hamiltonian system is, essentially, $\mathbb{R}^2$. But despite its infinity Lebesgue measure $\mathrm{d} x$, the exponential term acts in such a way that the phase space has $\mathrm{e}^{-r|x|^2}\mathrm{d} x$ finite measure. Therefore, similar to the bounded case, we shall find for this system, negative temperature states as noticed by Onsager \cite{MR12:60f}. These states have been studied by several authors \cite{MR2000e:60040,Book,MR93i:82060,MR96k:82059,Edwards,Joyce1,Joyce2, MR93k:82003,Lundgren1,Lundgren2,MR84k:82019,Montgomery1,Montgomery2,Neri, MR12:60f} since they arises naturally on some physical systems. We emphasize the work of Lundgren and Pointin \cite{Lundgren2} which also considered the system on the plane. In their work all the vortices have the same intensities. We weaked this assumption by modeling intensities as random variables. But, as explained before, we consider only positive intensities. Our strategy is the following: we introduce the Gibbs measure $\mu^N$ (where $N$ is the number of vortices) and its marginal density of the first $k$ coordinates $\mu^N_k$. Taking the limit as $N\to \infty$, we observe the same factorization property (the so called propagation of chaos'') found in the bounded case. Hence, $\mu^N_k$ behaves like product measures $\mu^{\otimes k}$ (or in better terms, as an average of product measures) of $k$ copies of the $1$-vortex distribution $\mu$. Since the Gibbs measure is, naturally, a solution of a variational problem, we can characterize the $1$-vortex distributions as a solution of a limit variational problem. The Newtonian potentials associated to this $1$-vortex distributions are solutions of $$\label{Eq: MFE in abstract} \begin{gathered} -\Delta u(x) = C^{-1}\int_{(0,1]} r\mathrm{e}^{-\beta ru(x)- \gamma r|x|^2}P(\mathrm{d}r), \quad\forall x\in \mathbb{R}^2,\\ C = \int_{(0,1]}\int_{\mathbb{R}^2}\mathrm{e}^{-\beta ru(y)- \gamma r|y|^2}\mathrm{d} yP(\mathrm{d}r). \end{gathered}$$ (the Mean Field Equation, MFE, for short). The propagation of chaos is related to the uniqueness of solution for MFE. Even in the easiest case of positive temperature states the functional minimized by the $1$-vortex distribution is not convex and hence we do not have general results of uniqueness. \subsection*{Notation} We introduce some notation which will be used in the sequel. Set $\Omega=\mathbb{R}^2$ and $\tilde{\Omega}=\Omega\times(0,1]$. $\tilde{X}=(\tilde{x}_1,\dots ,\tilde{x}_N)$ denotes an arbitrary point in $\tilde{\Omega}^N$, where $\tilde{x}_i=(x_i,r_i)$ ($x_i\in\Omega$ and $r_i\in(0,1]$). All $r_i$'s are random variables identically distributed with respect to a Borelian probability measure $P$ on $(0,1]$. On $\tilde{\Omega}$ we consider the product measure \emph{Lebesgue}$\times P$. By $\text{a.e. }$ we mean \emph{almost everywhere} with respect to \emph{Lebesgue}, $P$, or \emph{Lebesgue}$\times P$ measures without precising which one we are considering. For $\tilde{X}\in\tilde{\Omega}^N$ and $1\le k\le n$ we set $X=(x_1,\dots x_N)$ and define $\tilde{X}_k= (\tilde{x}_1,\dots ,\tilde{x}_k)$ and $\tilde{X}^{N-k}=(\tilde{x}_{k+1},\dots ,\tilde{x}_N)$ ($X_k$ and $X^{N-k}$ are analogous defined.) For the purpose of integration we set $\mathrm{d}\tilde{x}_i=\mathrm{d}x_i P(\mathrm{d}r_i)$, $\mathrm{d}\tilde{X}=\mathrm{d}\tilde{x}_1 \cdots\mathrm{d}\tilde{x}_N$, and $\mathrm{d}X=\mathrm{d}x_1\cdots\mathrm{d}x_N$. In an obvious way we define $\mathrm{d}\tilde{X}_k$, $\mathrm{d}\tilde{X}^{N-k}$, $\mathrm{d}X_k$, and $\mathrm{d}X^{N-k}$. The Hamiltonian of the $N$-point vortex system is given by $H^N (\tilde{X})=\frac12\sum_{i\neq j}^Nr_ir_jV(x_i,x_j),$ where $V$ is the Green function of the Poisson equation in $\mathbb{R}^2$, that is, $$\label{Eq: Bounds for V} V(x_1,x_2)=-\frac1{2\pi}\log|x_1-x_2|.$$ For this system we have other integrals beyond $H^N$ named the center of vorticity $M^N$ and the moment of inertia defined by $M^N(\tilde{X})=\sum_{i=1}^Nr_ix_i$ which is supposed to be null, and $I^N(\tilde{X})=\sum_{i=1}^Nr_i|x_i|^2$. Given $\beta\in\mathbb{R}$ and $\gamma>0$ we define the canonical Gibbs measure, with inverse temperature $\beta/N$, by $\mu^N(\tilde{X})=\frac1{Z(N,\beta,\gamma)}\mathrm{e}^{-\frac\beta NH^N (\tilde{X})-\gamma I^N(\tilde{X})},$ where $Z$ is the partition function given by $Z(N,\beta,\gamma)=\int_{\tilde{\Omega}^N}\mathrm{e}^{-\frac\beta NH^N (\tilde{X})-\gamma I^N(\tilde{X})}\mathrm{d}\tilde{X}.$ For simplicity, we denote $H=H^{2}$ and $I=I^{1}$. For $\rho\in L^1(\tilde{\Omega}^N)$ symmetric, that is, for which $\rho(\tilde{x}_1,\dots ,\tilde{x}_i,\dots ,\tilde{x}_j,\dots ,\tilde{x}_N)=\rho(\tilde{x}_1,\dots , \tilde{x}_j,\dots ,\tilde{x}_i,\dots , \tilde{x}_N),$ we define the family of correlation functions of $\rho$, $(\rho_k)_{1\leq k0$ and $x_1\in\tilde{\Omega}$ we set $\tilde{B}_t(x_1)=\{\tilde{x}_2\in\tilde{\Omega}: |x_2-x_1|0. For that reason we suppose that vortex intensities and \gamma are strictly positive. Moreover, we shall suppose that the decay'' of P near 0 is fast enough. More precisely, we assume $$\label{Eq: Hypothesis on P} \int_{\tilde{\Omega}}\mathrm{e}^{-\gamma I(\tilde{x}_1)}\mathrm{d}\tilde{x}_1=\frac\pi\gamma\int_{(0,1]}\frac1{r_1}P (\mathrm{d}r_1)<\infty.$$ In the sequel we set \[ |\tilde{\Omega}|_\gamma=\int_{\tilde{\Omega}}\mathrm{e}^{-\gamma I(\tilde{x}_1)}\mathrm{d}\tilde{x}_1.$ \section{Bounds for the partition function} We start with a proposition giving the range of $\beta$ and $\gamma$ for which the partition function is well defined (and thus, also the Gibbs measure.) \begin{proposition} \label{Prp: Upper bound for Z(N,beta,gamma)} Let $\beta>-8\pi$ and $\gamma>0$. There exists some constant $C=C(\beta,\gamma)$ such that $Z(N,\beta,\gamma)\le C^N.$ Moreover, $C$ is bounded in $\beta$ for $\beta$ on compact subsets of $(-8\pi, \infty)$. In particular, the Gibbs measure $\mu^N$ is well defined for $\beta>-8 \pi$ and $\gamma>0$. \end{proposition} \begin{proof} We take $a$ and $b$ such that $-8\pi0$, then we have \begin{align*} Z(N,\beta,\gamma)&=\int_{\tilde{\Omega}^N}\Big[\prod_{i\neq j}^N|x_i-x_j|^{\beta r_ir_j/4 \pi N}\Big]\mathrm{e}^{-\gamma I^N(\tilde{X})}\mathrm{d}\tilde{X}\\ &\le\int_{\tilde{\Omega}^N}\Big[\prod_{i\neq j}^N(|x_i|+1)^{br_i/4\pi N}(|x_j|+1)^{br_j/4 \pi N}\Big]\mathrm{e}^{-\gamma I^N(\tilde{X})}\mathrm{d}\tilde{X}\\ &=\int_{\tilde{\Omega}^N}\prod_{i=1}^N(|x_i|+1)^{br_i(N-1)/2\pi N}\mathrm{e}^{-\gamma I(\tilde{x}_i)} \mathrm{d}\tilde{X}\\ &\le\Big[\int_{\tilde{\Omega}}(|x_1|+1)^{br_1/2\pi}\mathrm{e}^{-\frac\gamma2I(\tilde{x}_1)}e^{-\frac\gamma2 I(\tilde{x}_1)}\mathrm{d}\tilde{x}_1\Big]^N. \end{align*} Since the map $\tilde{x}_1\ni\tilde{\Omega}\mapsto(|x_1|+1)^{br_1/2\pi}\mathrm{e}^{-\frac\gamma2 I(\tilde{x}_1)}$ is bounded from above by some constant $C=C(b,\gamma)$, the conclusion follows from (\ref{Eq: Hypothesis on P}). Now, if $-8\pi<\beta\le0$, then we have \begin{align*} Z(N,\beta,\gamma)&=\int_{\tilde{\Omega}^N}\prod_{i=1}^N\mathrm{e}^{-\frac\gamma NI(\tilde{x}_i)}\prod _{\substack{j=1\\j\neq i}}^N|x_i-x_j|^{\beta r_ir_j/4\pi N}\mathrm{e}^{-\frac \gamma NI(\tilde{x}_j)}\mathrm{d}\tilde{X}\\ &\le\prod_{i=1}^N\Big[\ \int_{\tilde{\Omega}^N}\mathrm{e}^{-\gamma I(\tilde{x}_i)}\prod_{\substack{j=1\\ j\neq i}}^N|x_i-x_j|^{\beta r_ir_j/4\pi}\mathrm{e}^{-\gamma I(\tilde{x}_j)} \mathrm{d}\tilde{X}\Big]^{1/N}\\ &=\int_{\tilde{\Omega}}\mathrm{e}^{-\gamma I(\tilde{x}_1)}\Big[\ \int_{\tilde{\Omega}}|x_1-x_2|^{\beta r_1r_2/4\pi}\mathrm{e}^{-\gamma I(\tilde{x}_2)}\mathrm{d}\tilde{x}_2\Big]^{N-1}\mathrm{d}\tilde{x}_1. \end{align*} Hence it is enough to show that there exists some constant $C=C(a,\gamma)$ which is an upper bound for the integral inside the brackets. We have, $\int_{\tilde{B}_1(\tilde{x}_1)}|x_1-x_2|^{\beta r_1r_2/4\pi}\mathrm{e}^{-\gamma I(\tilde{x}_2)} \mathrm{d}\tilde{x}_2\le\int_{B_1(x_1)}|x_1-x_2|^{a/4\pi}dx_2=\frac{8\pi^2}{8\pi+a}$ and $\int_{\tilde{\Omega}\setminus\tilde{B}_1(\tilde{x}_1)}|x_1-x_2|^{\beta r_1r_2/4\pi}\mathrm{e}^{-\gamma I(\tilde{x}_2)}\mathrm{d}\tilde{x}_2\le\int_{\tilde{\Omega}\setminus\tilde{B}_1(x_1)}\mathrm{e}^{-\gamma I(\tilde{x}_2)} \mathrm{d}\tilde{x}_2\leq|\tilde{\Omega}|_\gamma.$ \end{proof} \begin{remark} \label{Rem: He^(-gamma Ix2) in L^p} \rm From Proposition \ref{Prp: Upper bound for Z(N,beta,gamma)} with $N=2$ and $\beta=\pm 2$ it follows that the function $\mathrm{e}^{\pm H-\gamma I^2}$ is in $L^1 (\tilde{\Omega}^2)$. Hence, $He^{-\gamma I^2}\in L^p(\tilde{\Omega}^2)$, for all $p\in[1,\infty)$, which follows from the fact that there exists some constant $C=C(p)$ such that $|t|^p\le C(\mathrm{e}^t+\mathrm{e}^{-t})$. \end{remark} \begin{lemma} \label{Lem: Lower bound for Z(N,beta,gamma)} Let $\beta>-8\pi$ and $\gamma>0$. There exists some constant $C=C(\beta,\gamma)$ such that $C^N\le Z(N,\beta,\gamma).$ Moreover, $C$ is bounded in $\beta$ for $\beta$ on bounded sets of $(-8\pi, \infty)$. \end{lemma} \begin{proof} Let $\alpha>|\beta|$. By Jensen's inequality we have \begin{align*} Z(N,\beta,\gamma)&\geq|\tilde{\Omega}|_\gamma^N\exp\Big(-\frac\beta{2N|\tilde{\Omega}|_\gamma^N}\sum _{i\neq j}^N\int_{\tilde{\Omega}^N} H(\tilde{x}_i,\tilde{x}_j)\mathrm{e}^{-\gamma I^N(\tilde{X})}\mathrm{d}\tilde{X}\Big)\\ &\geq C^N\exp\Big(-\frac{\alpha(N-1)}{2|\tilde{\Omega}|_\gamma^2}\int_{\tilde{\Omega}^2} |H(\tilde{X}_2)|\mathrm{e}^{-\gamma I^{2}(\tilde{X}_2)}\mathrm{d}\tilde{X}_2\Big)\\ &\geq C^N\exp(-C(N-1))\geq C^N \end{align*} with $C=C(\alpha,\gamma)$. \end{proof} For the rest of this article, $\beta$ and $\gamma$ will be fixed in $(-8\pi,\infty)$ and $(0, \infty)$, respectively. \section{Existence of weak cluster points of Gibbs measures} The elements of the Gibbs sequence $(\mu^N)_{N>1}$ are functions defined on different domains. They are points in different functional spaces. This leads to a problem when looking for limits of this sequence. To overcome this problem we proceed as in \cite {Neri} by introducing the family of correlation functions $(\rho_k)_ {1\le k\le N}$ of a function $\rho\in L^1(\tilde{\Omega}^N)$, defined by $\rho_k(\tilde{X}_k)=\int_{\tilde{\Omega}^{N-k}}\rho(\tilde{X})\mathrm{d}\tilde{X}^{N-k}.$ Now, for each $k\in\mathbb{N}$, $(\mu_k^N)_{N>k}$ is a sequence on $L^1(\tilde{\Omega}^k)$ and thus we can look for its cluster points. Before finding $L^p$ estimates for these sequences we find pointwise ones. First we have the following lemma. \begin{lemma} \label{Lem: Relation between Z(k) and Z(N)} There exists some constant $C=C(\beta,\gamma)$ such that $Z\Big(k,\frac{\beta k}N,\gamma\Big)\le C^{N-k}Z(N,\beta,\gamma)\quad\forall N>k.$ Moreover, $C$ is bounded in $\beta$ for $\beta$ on bounded subsets of $(-8\pi, \infty)$. \end{lemma} \begin{proof} Let $N>k$ and fix $a>\beta$. It is easy too see that $$\label{Eq: b} Z\Big(k+1,\frac{\beta(k+1)}N,\gamma\Big)=\int_{\tilde{\Omega}^k}\mathrm{e}^{-\frac\beta NH^k(\tilde{X}_k)- \gamma I^k(\tilde{X}_k)}f(\tilde{X}_k)\mathrm{d}\tilde{X}_k,$$ where $f(\tilde{X}_k)=\int_{\tilde{\Omega}}\mathrm{e}^{-\frac\beta N\sum_{i=1}^kr_ir_{k+1}V(x_i,x_{k+1})-\gamma I(\tilde{x}_{k+1})}\mathrm{d}\tilde{x}_{k+1}.$ It follows from Jensen's inequality that $$\label{Eq: c} f(\tilde{X}_k)\geq|\tilde{\Omega}|_\gamma\exp\Big(\frac\beta{2\pi|\tilde{\Omega}|_\gamma N}\sum_{i=1} ^k\int_{\tilde{\Omega}} r_ir_{k+1}\log|x_i-x_{k+1}|\mathrm{e}^{-\gamma I(\tilde{x}_{k+1})}\mathrm{d}\tilde{x}_{k+1} \Big).$$ Consider $\beta\geq0$. From (\ref{Eq: c}) it follows that \begin{align*} f(\tilde{X}_k)&\geq C\exp\Big(\frac\beta{2\pi|\tilde{\Omega}|_\gamma N}\sum_{i=1}^k\int _{B_1(x_i)}\log|x_i-x_{k+1}|\mathrm{d}x_{k+1}\Big)\\ &\geq C\exp\Big(-\frac{aCk}N\Big)\geq C\exp(-aC)=C, \end{align*} and thus, from (\ref{Eq: b}), we conclude that $Z\Big(k,\frac{\beta k}N,\gamma\Big)\le CZ\Big(k+1,\frac{\beta(k+1)}N,\gamma \Big),$ with $C=C(a,\gamma)$. The result follows by induction on $k$. Now, we suppose $-8\pi<\beta<0$. From $r_ir_{k+1}\log|x_i-x_{k+1}|\le r_ir_{k+1}|x_i-x_{k+1}|^2\le2(r_i|x_i|^2+r_{k+1} |x_{k+1}|^2)$ we conclude that \begin{align*} \int_{\tilde{\Omega}} r_ir_{k+1}\log|x_i-x_{k+1}|\mathrm{e}^{-\gamma I(\tilde{x}_{k+1})}\mathrm{d}\tilde{x}_{k+1}&\le2I(\tilde{x}_i) |\tilde{\Omega}|_\gamma+\theta(\gamma), \end{align*} where $\theta(\gamma)=2\int_{\tilde{\Omega}} I(\tilde{x}_{k+1})\mathrm{e}^{-\gamma I(\tilde{x}_{k+1})} \mathrm{d}\tilde{x}_{k+1}=2\pi\gamma^{-1}|\tilde{\Omega}|_\gamma.$ From (\ref{Eq: c}) and (\ref{Eq: Hypothesis on P}) it follows that \begin{align*} f(\tilde{X}_k)&\geq|\tilde{\Omega}|_\gamma\exp\Big(\frac\beta{2\pi|\tilde{\Omega}|_\gamma N}\sum_{i= 1}^k\left[2I(\tilde{x}_i)|\tilde{\Omega}|_\gamma+2\pi\gamma^{-1}|\tilde{\Omega}|_\gamma\right]\Big)\\ &\geq C\gamma^{-1}\exp\Big(\frac\beta{\pi N}I^k(\tilde{X}_k)\Big) \exp\Big(\frac{\beta k}{\gamma N}\Big)\\ &\geq C\gamma^{-1}\exp\Big(-\frac8N I^k(\tilde{X}_k)\Big) \exp\Big(-\frac{8\pi}\gamma\Big). \end{align*} Note that the constant $C$ depends neither on $\beta$ nor on $\gamma$, and thus, (\ref{Eq: b}) yields $Z\Big(k,\frac{\beta k}N,\gamma+\frac8N\Big)\le\varphi(\gamma)Z\Big(k+1,\frac {\beta(k+1)}N,\gamma\Big),$ where $\varphi(\gamma)=C\gamma\mathrm{e}^{8\pi/\gamma}$ is continuous from $(0,\infty)$ in $(0,\infty)$. By repeating $N-k$ times this argument and replacing $\gamma$ by $$\label{Eq: j} \gamma+\frac{8(N-k-1)}N,\dots ,\quad\gamma+\frac8N,\quad\gamma,$$ we obtain $Z\Big(k,\frac{\beta k}N,\gamma+\frac{8(N-k)}N\Big)\le\varphi\Big(\gamma+ \frac{8(N-k-1)}N\Big)\cdots\varphi(\gamma)Z(N,\beta,\gamma).$ All numbers in (\ref{Eq: j}) are in $[\gamma,\gamma+8]$. Since $\varphi$ is continuous, it is bounded from above on this interval by some constant $C=C (\gamma)$. Therefore, $Z\Big(k,\frac{\beta k}N,\gamma+8\Big)\le Z\Big(k,\frac{\beta k}N,\gamma+ \frac{8(N-k)}N\Big)\le C^{N-k}Z(N,\beta,\gamma).$ The sequence $(\beta k/N)_{N>k}$ is in a compact subset of $(-8\pi,\infty)$. Hence, Proposition \ref{Prp: Upper bound for Z(N,beta,gamma)} and Lemma \ref {Lem: Lower bound for Z(N,beta,gamma)} give the existence of constants $C_1=C_1 (\alpha,\gamma)$ and $C_2=C_2(\alpha,\gamma)$ such that $C_1^k\le Z\Big(k,\frac{\beta k}N,\gamma+8\Big) \le Z\Big(k,\frac{\beta k}N, \gamma\Big)\le C_2^k.$ Hence, $Z\Big(k,\frac{\beta k}N,\gamma\Big)\le\Big[\frac{C_2}{C_1}\Big]^kZ \Big(k, \frac{\beta k}N,\gamma+8\Big)\le C^{N-k}Z(N,\beta,\gamma).$ \end{proof} \begin{proposition} \label{Prp: Bound for muN_k} There exists some constant $C=C(\beta,\gamma)$ such that for $N$ large enough, $\mu_k^N(\tilde{X}_k)\le C^k\mathrm{e}^{-\frac\beta NH^k(\tilde{X}_k)-\frac\gamma2 I^k(\tilde{X}_k)}\,.$ \end{proposition} \begin{proof} Let $k\geq2,\ N_0,N\in\mathbb{N}$, and $r,p,p'\in\mathbb{R}$ such that \begin{itemize} \item $r>1$ with $\beta r\in(-8\pi,\infty)$; \item $N_0=\min\left\{N\in\mathbb{N}: N>2k\text{ and }N/(N-2k)-8\pi$, from Proposition \ref{Prp: Upper bound for Z(N,beta,gamma)} and Lemma \ref{Lem: Lower bound for Z(N,beta,gamma)}, it follows that there exist constants $C_1=C_1(\beta,\gamma)$ and $C_2=C_2 (\beta,\gamma)$ such that $Z(N,\beta r,\gamma)^{\theta/r}\le C_2^{N\theta/r}\le C^k\quad\text{and}\quad Z(N,\beta,\gamma)^{-\theta}\le C_1^{-N\theta}\le C^k.$ \end{proof} \begin{corollary} \label{Cor: Existence of a weak cluster point} Let $p\in[1,\infty)$. Thus $\mu_k^N\in L^p(\tilde{\Omega}^k)$ for all $k\in\mathbb{N}$ and for all $N$ large enough. Moreover, there exists a constant $C=C(\beta,\gamma,p)$ such that $\|\mu_k^N\|_{L^p}\le C^k,\quad\forall k\in\mathbb{N},\quad\text{for N large enough}.$ Hence, if $p>1$, then there exists $\mu_k\in L^p(\tilde{\Omega}^k)$ and a subsequence $(\mu^{N_j}_k)_{j\in\mathbb{N}}$ such that $\mu^{N_j}_k\rightharpoonup\mu_k$ weakly in $L^p(\tilde{\Omega}^k)$. \end{corollary} For the proof of the above corollary, see \cite[Corollary 4, p. 386.]{Neri}. \begin{remark} \rm A priori the index choice $(N_j)_{j\in\mathbb{N}}$ depends on $p$ and $k$. But we can always, by a diagonalization process, suppose that $\mu^{N_j}_k\rightharpoonup\mu_k\quad\text{weakly in }L^p(\tilde{\Omega}^k),\quad\forall k\in \mathbb{N},\quad\forall p\in[1,\infty).$ This holds even for $p=1$ by Proposition \ref{Prp: muN_k converges better than weakly} (by taking $f\in L^\infty(\tilde{\Omega}^k)$.) In the sequel, we shall say that ${\mu_*}=(\mu_k)_{k\in\mathbb{N}}$ is a weak cluster point of $(\mu^N)_{N>1}$ in that sense and we shall denote the index sequence always by $(N_j)_{j\in\mathbb{N}}$. \end{remark} \begin{lemma} \label{Lem: B has finite measure} There exists some constant $C$ such that $\int_{B^N_r}\mathrm{d}\tilde{X}\le C^Nr^N,\quad\forall r>0,\; \forall N\in\mathbb{N},$ where $B^N_r=\{\tilde{X}\in\tilde{\Omega}^N: I^N(\tilde{X})0$. We proceed by induction on $N$. We have \begin{align*} \int_{B^1_r}\mathrm{d}\tilde{x}_1&=\int_{(0,1]}\Big[\int_{\left\{|x_1|^2 0$. Then,$f\mu^N_k\in L^1(\tilde{\Omega}^k)$for$N$large enough. Moreover, if$\mu^{N_j}_k\rightharpoonup\mu_k$weakly in$L^2 (\tilde{\Omega}^k)$, then$f\mu_k\in L^1(\tilde{\Omega}^k)$and $\int_{\tilde{\Omega}^k} f(\tilde{X}_k)\mu^{N_j}_k(\tilde{X}_k)\mathrm{d}\tilde{X}_k\to \int_{\tilde{\Omega}^k} f(\tilde{X}_k)\mu_k(\tilde{X}_k)\mathrm{d}\tilde{X}_k.$ \end{proposition} \begin{proof} Let$r>1$be such that$\beta r>-8\pi$and let$\phi\in L^\infty(\tilde{\Omega})$with$0\le\phi\le1$. From the bound on$f$and Proposition \ref{Prp: Bound for muN_k}, there exists a constant$C=C(\beta,\gamma,k)$such that $\phi|f|\mu^N_k\le C\phi\mathrm{e}^{-\frac\beta NH^k-\frac\gamma4 I^k}.$ In particular, by taking$\phi=1$, we have$f\mu_k^N\in L^1(\tilde{\Omega}^k)$for$Nlarge enough. By H\"older's inequality we have \begin{align*} \int_{\tilde{\Omega}^k}\phi|f|\mu^N_k\mathrm{d}\tilde{X}_k &\le C\Big[\ \int_{\tilde{\Omega}^k}\phi\mathrm{e}^{-\frac\gamma4 I^k}\mathrm{d}\tilde{X}_k\Big]^ {1/r'}\Big[\int_{\tilde{\Omega}^k}\mathrm{e}^{-\frac{\beta r}NH^k-\frac\gamma4 I^k}\mathrm{d}\tilde{X}_k\Big]^{1/r}\\ &=CZ\left(k,\frac{\beta rk}N,\frac\gamma4\right)^{1/r} \Big[\int_{\tilde{\Omega}^k}\phi\mathrm{e}^ {-\frac\gamma4 I^k}\mathrm{d}\tilde{X}_k\Big]^{1/r'}. \end{align*} Since(\beta rk/N_j)_{N_j>k}$is in a compact subset of$(-8\pi,\infty)$, Proposition \ref{Prp: Upper bound for Z(N,beta,gamma)} yields a constant$C=C(k, \beta,\gamma)$such that $$\label{Eq: g} \int_{\tilde{\Omega}^k}\phi|f|\mu^N_k\mathrm{d}\tilde{X}_k\le C\Big[\int_{\tilde{\Omega}^k}\phi\ \mathrm{e}^{-\frac\gamma4 I^k} \mathrm{d}\tilde{X}_k\Big]^{1/r'}.$$ We have shown that there is a constant$C=C(\beta,\gamma,k)$such that $$\label{Eq: h} \int_{\tilde{\Omega}^k}\phi|f|\mu^N_k\mathrm{d}\tilde{X}_k\le C,\quad\forall\phi\in L^\infty(\tilde{\Omega}^k) \text{ such that }0\le\phi\le1.$$ For$r>0$we set$g_r=1-f_r$, where$f_r$is given by $f_r(\tilde{X}_k)= \begin{cases} 1,&\text{if }I^k(\tilde{X}_k)0. In the same way, \[ \int_{\tilde{\Omega}^k} f_r|f|\mu^{N_j}_k\mathrm{d}\tilde{X}_k\to \int_{\tilde{\Omega}^k} f_r|f|\mu_k\mathrm{d}\tilde{X}_k\quad\text{when }j \to \infty,\quad\forall r>0.$ By taking$\phi=f_r$in (\ref{Eq: h}) we conclude that the above sequence is bounded from above by a constant$C=C(\beta,\gamma,k)$. By taking limits we find $\int_{\tilde{\Omega}^k} f_r|f|\mu_k\mathrm{d}\tilde{X}_k\le C\quad\forall r>0.$ But$f_r|f|\mu_k\nearrow|f|\mu_k$when$r\to \infty$thus, Monotone Convergence Theorem yields$|f|\mu_k\in L^1(\tilde{\Omega}^k)$. Hence$(f_rf\mu_k)_{r>0} \subset L^1(\tilde{\Omega}^k)$is bounded from above, in absolute value, by$|f|\mu_k\in L^1 (\tilde{\Omega}^k)$. From Dominated Convergence Theorem it follows that $$\label{Eq: int f_r f mu_k converges to int f mu_k} \int_{\tilde{\Omega}^k} f_rf\mu_k\mathrm{d}\tilde{X}_k\to \int_{\tilde{\Omega}^k} f\mu_k\mathrm{d}\tilde{X}_k,\quad\text{when }r\to \infty.$$ It is not difficult to see that$(g_re^{-\frac\gamma4 I^k})_{r>0}\subset L^1 (\tilde{\Omega}^k)$is convergent a.e. to$0$and is bounded from above, in absolute value, by$\mathrm{e}^{-\frac\gamma4 I^k}\in L^1(\tilde{\Omega}^k)$. Again, by Dominated Convergence Theorem, this sequence converges to$0$in$L^1(\tilde{\Omega}^k)$. Hence, by taking$\phi= g_r$in (\ref{Eq: g}), we show that $$\label{Eq: int g_r f muN_k <= epsilon} \int_{\tilde{\Omega}^k} g_r|f|\mu^{N_j}_k\mathrm{d}\tilde{X}_k\to 0\quad\text{when }r\to \infty, \quad\text{uniformly on }j.$$ By writing$f=f_rf+g_rfwe have \begin{align*} &\Big|\ \int_{\tilde{\Omega}^k} f\mu^{N_j}_k\mathrm{d}\tilde{X}_k-\int_{\tilde{\Omega}^k} f\mu_k\mathrm{d}\tilde{X}_k\Big|\\ &\le\Big|\ \int_{\tilde{\Omega}^k} f_rf \mu^{N_j}_k\mathrm{d}\tilde{X}_k-\int_{\tilde{\Omega}^k} f_rf\mu_k\mathrm{d}\tilde{X}_k\Big|\\ &+\Big|\ \int_{\tilde{\Omega}^k} f_rf\mu_k\mathrm{d}\tilde{X}_k-\int_{\tilde{\Omega}^k} f\mu_k\mathrm{d}\tilde{X}_k\Big| +\Big|\ \int_{\tilde{\Omega}^k} g_rf\mu^{N_j}_k\mathrm{d}\tilde{X}_k\Big|. \end{align*} Finally, the result follows from (\ref{Eq: int f_r f muN_k converges to int f_r f mu_k}), (\ref{Eq: int f_r f mu_k converges to int f mu_k}) and (\ref{Eq: int g_r f muN_k <= epsilon}). \end{proof} \section{Variational problems} ForN\in\mathbb{N}$we set $D(F^N)=\{\rho\in L^1(\tilde{\Omega}^N): \rho\log\rho\in L^1(\tilde{\Omega}^N), I^N\rho\in L^1 (\tilde{\Omega}^N)\}.$ For$\rho\in D(F^N)$we define the following functionals \begin{gather*} S^N(\rho)=\int_{\tilde{\Omega}^N}\rho(\tilde{X})\log\rho(\tilde{X})\mathrm{d}\tilde{X}\quad\text{(entropy)},\\ E^N(\rho)=\frac1N\int_{\tilde{\Omega}^N} H^N (\tilde{X})\rho(\tilde{X})\mathrm{d}\tilde{X}\quad\text{(energy)},\\ J^N(\rho)=\int_{\tilde{\Omega}^N} I^N(\tilde{X})\rho(\tilde{X})\mathrm{d}\tilde{X}\quad\text{(moment of inertia)},\\ F^N(\rho)=S^N(\rho)+\beta E^N(\rho)+\gamma J^N(\rho)\quad\text{(free energy)}. \end{gather*} We shall see that$D(F^N)$is convex. The functional$F^N$is convex, since$S^N$is convex and$E^N$and$J^N$are linear. \begin{lemma} \label{Lem: FN is well defined} Let$N\ge2$and$\rho\in D(F^N)$. Then$H^N \rho\in L^1(\tilde{\Omega}^N). \end{lemma} \begin{proof} We have \begin{align*} -\frac1N\rho(\tilde{X})H^N (\tilde{X})&=\frac1{4\pi N}\rho(\tilde{X})\sum_{i\neq j}^Nr_ir_j \log|x_i-x_j|\\ &\le\frac1{2\pi N}\rho(\tilde{X})\sum_{i\neq j}^N(r_i|x_i|^2+r_j|x_j|^2)\le \frac1\pi\rho(\tilde{X}) I^N(\tilde{X}). \end{align*} Hence,H^N \rho$is bounded from below by some function in$L^1(\tilde{\Omega}^N)$. Apply the following inequality $$\label{Eq: Young} sr\leq r\log r+\frac1{\mathrm{e}} \mathrm{e}^s,\quad\forall r\geq0,\;\forall s\in\mathbb{R},$$ with$r=\rho\mathrm{e}^{I^N}$and$s=\frac1NH^N $and, then multiply by$\mathrm{e}^{-I^N}$to find $\frac1N\rho H^N \le\rho\log\rho+\rho I^N+\frac1{\mathrm{e}}\mathrm{e}^{\frac1NH^N -I^N}.$ Since all the terms on the right hand side are in$L^1(\tilde{\Omega}^N)$we have$H^N \rho \in L^1(\tilde{\Omega}^N)$. \end{proof} \begin{remark} \label{Rem: Symmetric energy} \rm Let$N\geq2$and$\rho\in D(F^N)$. If$\rho$is symmetric, then we have simpler expressions for the energy and moment of inertia, which are \begin{gather*} E^N(\rho)=\frac{N-1}2\int_{\tilde{\Omega}} \int_{\tilde{\Omega}} H(\tilde{x}_1,\tilde{x}_2)\rho_2(\tilde{x}_1\tilde{x}_2)\mathrm{d}\tilde{x}_1\mathrm{d}\tilde{x}_2,\\ J^N(\rho)=N\int_{\tilde{\Omega}} I(\tilde{x}_1)\rho_1(\tilde{x}_1)\mathrm{d}\tilde{x}_1. \end{gather*} \end{remark} \begin{proposition} \label{Prp: Subadditivity of SN} Let$1\le k0\}=\{\tilde{X}_2\in \tilde{\Omega}^2: |x_1-x_2|<1\}. \] We write $H=g+f$, where $g=\frac121_A|H|^2\mathrm{e}^{-\frac\gamma4 I^2}\quad\text{and}\quad f=1_AH-g+1_{A^ \complement}H.$ We shall show that $f$ and $g$ satisfy the stated properties. By Remark \ref{Rem: He^(-gamma Ix2) in L^p}, we have $g\in L^p(\tilde{\Omega}^2)$ for all $p\in[1,\infty)$. From Young's inequality, it follows that $1_AH=1_AH\mathrm{e}^{-\frac\gamma8 I^2}\mathrm{e}^{\frac\gamma8 I^2}\le\frac121_A|H|^2\mathrm{e}^{-\frac \gamma4 I^2}+\frac12\mathrm{e}^{\frac\gamma4 I^2}=g+\frac12\mathrm{e}^{\frac\gamma4 I^2},$ and thus, $1_AH-g\le\frac12\mathrm{e}^{\frac\gamma4 I^2}$. Finally, for $\tilde{X}_2\in\tilde{\Omega}^2$ we have \begin{align*} |1_{A^\complement}(\tilde{X}_2)H(\tilde{X}_2)| &=1_{A^\complement}(\tilde{X}_2)\frac1{2\pi}r_1r_2\log|x_1-x_2| \le\frac1{2\pi}r_1r_2|x_1-x_2|^2\\ &\le\frac1\pi r_1r_2(|x_1|^2+|x_2|^2) \le\frac1\pi(r_1|x_1|^2+r_2|x_2|^2)\\ &=\frac1\pi I^2(\tilde{X}_2)\le C\mathrm{e}^{\frac\gamma4 I^2(\tilde{X}_2)}, \end{align*} for some constant $C=C(\gamma)$. \end{proof} \begin{remark} \label{Rem: muN in D(FN)} \rm If $N$ is large enough, then $\mu^N\in D(F^N)$. Indeed, $\mu^N\log\mu^N=-\frac\beta NH^N \mu^N-\gamma I^N\mu^N-\mu^N\log Z(N,\beta,\gamma).$ Hence, it suffices to show that $H^N \mu^N,\ I^N\mu^N\in L^1(\tilde{\Omega}^N)$. By symmetries of $H^N$ and $\mu^N$, it is enough to show that $H\mu^N_2\in L^1(\tilde{\Omega}^2)$ and $I\mu ^N_1\in L^1(\tilde{\Omega})$. Using the decomposition of $H$ (Lemma \ref{Lem: Decomposition of H}) we write $H\mu^N_2=g\mu^N_2+f\mu^N_2$, where $g\in L^2(\tilde{\Omega}^2)$ and $|f|\le C \mathrm{e}^{\frac\gamma4 I^2}$. Hence, $g\mu^N_2\in L^1(\tilde{\Omega}^2)$ since $\mu^N_2\in L^2(\tilde{\Omega} ^2)$ (Corollary \ref{Cor: Existence of a weak cluster point}.) By Proposition \ref{Prp: muN_k converges better than weakly} we have $f\mu^N_2\in L^1(\tilde{\Omega}^2)$ and $I\mu^N_1\in L^1(\tilde{\Omega})$ since $I\le C\mathrm{e}^{\frac\gamma 4I}$ for some constant $C=C(\gamma)$. \end{remark} \begin{lemma} \label{Lem: Bound on rho log rho-} Let $\rho:\tilde{\Omega}^N\to \mathbb{R}$ be a positive measurable function such that $I^N \rho\in L^1(\tilde{\Omega}^N)$. Then $[\rho\log\rho]^-\in L^1(\tilde{\Omega}^N)$ and there exists a constant $C=C(N)$ such that $\int_{\tilde{\Omega}^N}\left[\rho(\tilde{X})\log\rho(\tilde{X})\right]^-\mathrm{d}\tilde{X}\le C+J^N(\rho).$ \end{lemma} \begin{proof} We write \begin{align*} &\int_{\tilde{\Omega}^N}\left[\rho(\tilde{X})\log\rho(\tilde{X})\right]^-\mathrm{d}\tilde{X}\\ &=-\int_{\{\rho\le\mathrm{e}^{-I^N} \}}\rho(\tilde{X})\log\rho(\tilde{X})\mathrm{d}\tilde{X}-\int_{\{\mathrm{e}^{-I^N}<\rho\le1\}}\rho(\tilde{X})\log \rho(\tilde{X})\mathrm{d}\tilde{X}. \end{align*} Since $-t\log t\le C\sqrt{t}$ for all $t\ge 0$ and for some constant $C>0$, we have $-\int_{\{\rho\le\mathrm{e}^{-I^N}\}}\rho(\tilde{X})\log\rho(\tilde{X})\mathrm{d}\tilde{X}\le C\int_{\tilde{\Omega}^N}\mathrm{e}^ {-\frac12 I^N(\tilde{X})}\mathrm{d}\tilde{X}=C.$ We have also $-\int_{\{e^{-I^N}<\rho\le1\}}\rho(\tilde{X})\log\rho(\tilde{X})d\tilde{X}\le\int_{\tilde{\Omega}^N} I^N(\tilde{X} )\rho(\tilde{X})d\tilde{X}=J^N(\rho)$ which completes the proof. \end{proof} It follows immediately from Lemma \ref{Lem: Bound on rho log rho-} that $D(F^N)=\{\rho\in L^1(\tilde{\Omega}^N): [\rho\log\rho]^+\in L^1(\tilde{\Omega}^N),\ I^N\rho\in L^1 (\tilde{\Omega}^N)\}.$ Hence $D(F^N)$ is convex since the map $t\in[0,\infty)\mapsto[t\log t]^+$ is convex and $J^N$ is linear. \begin{lemma} \label{Lem: M_C is weakly compact} For $C>0$, the set $M_C=\{\rho\in D(F^N): \int_{\tilde{\Omega}^N}[\rho(\tilde{X})\log\rho(\tilde{X})]^+\mathrm{d}\tilde{X}\le C\text{ and } J^N(\rho)\le C\}$ is weakly compact on $L^1(\tilde{\Omega}^N)$. \end{lemma} \begin{proof} We shall show that $M_C$ is closed in the strong topology of $L^1(\tilde{\Omega}^N)$. Since $M_C$ is convex, it will follow that $M_C$ is weakly closed on $L^1(\tilde{\Omega}^N)$. Let $(\rho_n)_{n\in\mathbb{N}}$ be a strongly convergent sequence on $M_C$ to $\rho\in L^1(\tilde{\Omega}^N)$. We can take a subsequence $(\rho_{n_j})_{j\in\mathbb{N}}$ such that $\rho_ {n_j}\to \rho$ almost everywhere on $\tilde{\Omega}^N$. The sequences $([\rho_ {n_j}\log\rho_{n_j}]^+)_{j\in\mathbb{N}}$ and $(I^N\rho_{n_j})_{n\in \mathbb{N}}$ are bounded on $L^1(\tilde{\Omega}^N)$, almost everywhere convergent to $[\rho\log \rho]^+$ and $I^N\rho$, respectively, and composed by positive functions. From Fatou's Lemma we conclude that $\rho\in M_C$. We show now that every sequence on $M_C$ has a weakly convergent subsequence on $L^1(\tilde{\Omega}^N)$. Let $(\rho_n)_{n\in\mathbb{N}}\subset M_C$. Given $\varepsilon>0$, take $r>0$ and $M>1$ such that $\frac1rJ^N(\rho_n)\le\varepsilon\text{ \ and \ }\frac1{\log M}\int_{\tilde{\Omega}^N}\left[ \rho_n(\tilde{X})\log\rho_n(\tilde{X})\right]^+\mathrm{d}\tilde{X}\le\varepsilon,\quad\forall n\in\mathbb{N}.$ We set $B=\{\tilde{X}\in\tilde{\Omega}^N : I^N(\tilde{X})-8\pi. From inequality (\ref{Eq: Young}), applied to r=\frac1t\rho and s=-(\beta t/N)H^N -(\gamma t/2)I^N, it follows that \[ -\frac\beta NH^N \rho-\frac\gamma2 I^N\rho\le\frac1t\rho\log\left(\frac\rho t \right)+\frac1{\mathrm{e}}\mathrm{e}^{-\frac{\beta t}NH^N -\frac{\gamma t}2 I^N}.$ Therefore, $$\label{Eq: Application of Youg} \rho\log\rho+\frac\beta NH^N \rho+\gamma I^N\rho\geq\left(1-\frac1t\right)\rho \log\rho+\frac1t\rho\log t-\frac1{\mathrm{e}}\mathrm{e}^{-\frac{\beta t}NH^N -\frac{\gamma t}2 I^N}+ \frac\gamma2 I^N\rho.$$ In particular, for $t=1$ one has $$\label{Eq: h is positive} \rho\log\rho+\frac\beta NH^N \rho+\gamma I^N\rho+\frac1{\mathrm{e}}\mathrm{e}^{-\frac\beta NH^N - \frac\gamma2 I^N}\geq\frac\gamma2 I^N\rho\ge0.$$ Let us show that $F^N$ is a l.s.c. in the strong topology of $L^1(\tilde{\Omega}^N)$. Hence, by convexity, $F^N$ is also l.s.c. in the weakly topology of $L^1(\tilde{\Omega}^N)$. Let $(\rho_n)_{n\in\mathbb{N}}\subset D(F^N)$ be a convergent sequence to $\rho\in L^1 (\tilde{\Omega}^N)$ in the strong topology. We can take a subsequence $(\rho_{n_j})_{j\in \mathbb{N}}$ such that \begin{align*} \rho_{n_j}&\to \rho\quad\text{a.e. }\text{on }\tilde{\Omega}^N\\ F^N(\rho_{n_j})&\to \liminf_{n\to \infty}F^N(\rho_n)\quad \text{(which is supposed to be finite.)} \end{align*} The sequence $(h_j)_{j\in\mathbb{N}}$, given by $h_j=\rho_{n_j}\log\rho_{n_j}+\frac\beta NH^N \rho_{n_j}+\gamma I^N\rho_{n_j}+ \frac1ee^{-\frac\beta NH^N -\frac\gamma2 I^N},$ satisfies \begin{itemize} \item $h_j\in L^1(\tilde{\Omega}^N),\quad\forall j\in\mathbb{N}$; \item $\|h_j\|_{L^1}=F^N(\rho_{n_j})+\frac1{\mathrm{e}} Z\left(N,\beta,\frac\gamma2 \right)\le C,\quad\forall j\in\mathbb{N}$; \item $h_j\geq0$ (by (\ref{Eq: h is positive})),\quad$\forall j\in\mathbb{N}$; \item $h_j\to \rho\log\rho+\frac\beta NH^N \rho+\gamma I^N\rho+\frac1{\mathrm{e}} \mathrm{e}^{-\frac\beta NH^N -\frac\gamma2 I^N}\quad\text{a.e. }$on $\tilde{\Omega}^N$. \end{itemize} From Fatou's Lemma, it follows that $F^N(\rho)\le\liminf_{j\to \infty}F^N(\rho_{n_j})=\liminf_ {n\to \infty}F^N(\rho_n).$ Now, suppose that $(\rho_n)_{n\in\mathbb{N}}$ is a minimizing sequence for the problem. Taking $t>1$ in (\ref{Eq: Application of Youg}) and integrating on $\tilde{\Omega}^N$ we obtain \begin{align*} C\ge F^N(\rho_n)&\geq\Big(1-\frac1t\Big)S^N(\rho_n)+\frac1t\log t+\frac\gamma2 J^N(\rho_n)-\frac1{\mathrm{e}} Z\Big(N,\beta t,\frac{\gamma t}2\Big). \end{align*} Hence, $S^N(\rho_n)\le C,\quad\forall n\in\mathbb{N},$ and from (\ref{Eq: h is positive}) it follows that $J^N(\rho_n)\le C,\quad\forall n\in\mathbb{N}.$ From the last two estimates and from Lemma \ref{Lem: Bound on rho log rho-}, we conclude that there exists $C$ such that $\int_{\tilde{\Omega}^N}[\rho_n(\tilde{X})\log\rho_n(\tilde{X})]^+\mathrm{d}\tilde{X}\le C,\quad\forall n\in\mathbb{N}.$ Hence, we have shown that there exists $C>0$ such that $(\rho_n)_{n\in\mathbb{N}}$ is in a set $M_C$ as in Lemma \ref{Lem: M_C is weakly compact} and thus it has a subsequence weakly convergent to $\tilde\mu\in D(F^N)$. It is clear that $\|\tilde\mu\|_{L^1}=1$. Hence, by the lower semi-continuity of $F^N$ in the weak topology of $L^1(\tilde{\Omega}^N)$, $\tilde\mu$ is a solution for the problem. \smallskip \noindent Step 2: We are going to show that $\tilde\mu=\mu^N$. For $\delta>0$ we set $\Lambda_\delta=\{\tilde{X}\in\tilde{\Omega}^N: \tilde\mu(\tilde{X})>\delta\}\quad \text{and}\quad U_\delta=\{\varphi\in C_c(\tilde{\Omega}^N): \|\varphi\|_{L^\infty} <\frac\delta2\}.$ Consider the following functionals $\begin{array}{rclcrcl} J_\delta:U_\delta&\to &\mathbb{R}&\quad\text{and}\quad&G_\delta:U_\delta& \to &\mathbb{R}\\ \varphi&\mapsto&F^N(\tilde\mu+1_{\Lambda_\delta}\varphi)&\quad &\varphi&\mapsto&\int_{\tilde{\Omega}^N}1_{\Lambda_\delta}\varphi. \end{array}$ Take $\varphi\in U_\delta$ and $\rho=\tilde\mu+1_{\Lambda_\delta}\varphi$. First, we can easily see that $0\le\rho\le2\tilde\mu$. Thus $I^N\rho\in L^1 (\tilde{\Omega}^N)$ and $\left[\rho\log\rho\right]^+\in L^1(\tilde{\Omega}^N)$. From Lemma \ref{Lem: Bound on rho log rho-} we deduce that $\left[\rho\log\rho\right]^-\in L^1 (\tilde{\Omega}^N)$. Hence, $\rho\in D(F^N)$ and thus $J_\delta$ is a real valued functional defined on $U_\delta$. Since $\tilde\mu$ is a minimizer of $F^N$ under the constraint $\|\rho\|_{L^1}= 1$ we know that $J_\delta(0)=\min_{G_\delta(\varphi)=0}J_\delta(\varphi).$ By the Lagrange Multiplier Theorem, there exists $\lambda_\delta$ such that $J'_\delta(0)=\lambda_\delta G'_\delta(0)$, that is, for all $\varphi\in C_c (\tilde{\Omega}^N)$ we have $\int_{\tilde{\Omega}^N}[\log\tilde\mu+1]1_{\Lambda_\delta}\varphi+\frac\beta N\int_{\tilde{\Omega}^N} H^N 1_{\Lambda_ \delta}\varphi+\gamma\int_{\tilde{\Omega}^N} I^N1_{\Lambda_\delta}\varphi=\lambda_\delta\int_{\tilde{\Omega}^N}1_{ \Lambda_\delta}\varphi.$ Therefore, $\log\tilde\mu+1+\frac\beta NH^N +\gamma I^N=\lambda_\delta\quad\text{a.e. }\text{on } \Lambda_\delta.$ It follows that $\tilde\mu=C_\delta\mathrm{e}^{-\frac\beta NH^N -\gamma I^N}$ almost everywhere on $\Lambda_\delta$. If $\delta_1<\delta_2$, then $\Lambda_{\delta_2}\subset\Lambda_{\delta_1}$. Since $\tilde\mu=C_{\delta_2}\mathrm{e}^{-\frac\beta NH^N -\gamma I^N}$ on $\Lambda_ {\delta_2}$ and $\tilde\mu=C_{\delta_1}\mathrm{e}^{-\frac\beta N H^N -\gamma I^N}$ on $\Lambda_{\delta_1}$ we have $C_{\delta_1}=C_{\delta_2}=C$ (independent on $\delta$). We set $\Lambda=\{\tilde{X}\in\tilde{\Omega}^N: \tilde\mu>0\}=\bigcup_{\delta>0} \Lambda_\delta.$ Hence, $\tilde\mu=0$ on $\Lambda^\complement$ and $\tilde\mu=C\mathrm{e}^{-\frac\beta N H^N -\gamma I^N}$ on $\Lambda$, where $C=\left[\int_\Lambda\mathrm{e}^{-\frac\beta NH^N (\tilde{X})-\gamma I^N(\tilde{X})}\mathrm{d}\tilde{X} \right]^{-1}.$ A simple calculus shows that \begin{gather*} F^N(\tilde\mu)=-\log\Big(\ \int_\Lambda e^{-\frac\beta NH^N -\gamma I^N} \Big) \\ F^N(\mu^N)=-\log\Big(\ \int_{\tilde{\Omega}^N} e^{-\frac\beta NH^N -\gamma I^N}\Big). \end{gather*} Since $F^N(\tilde\mu)\le F^N(\mu^N)$ we have $|\Lambda^\complement|=0$, and thus $\tilde\mu=\mu^N$. \end{proof} \begin{remark} \label{Rem: FN is lsc} \rm We emphasize that in the last proof we have shown that $F^N$ is a l.s.c. functional on the weak topology of $L^1(\tilde{\Omega}^N)$. \end{remark} We consider now the limit problem. We define the set $D(F^*)$ of all ${\rho_*}=(\rho_ k)_{k\in\mathbb{N}}\in\prod_{k=1}^{\infty}D(F^k)$ which verify, for all $k\in\mathbb{N}$, \begin{enumerate} \item $\|\rho_k\|_{L^1}=1$; \item $\rho_k$ is symmetric; \item $\rho_k(\tilde{X}_k)=\int_{\tilde{\Omega}}\rho_{k+1}(\tilde{X}_{k+1})\mathrm{d}\tilde{x}_{k+1}$; \item there exits $C=C({\rho_*})$ such that $\|\rho_k\|_{L^\infty}\le C^k$. \end{enumerate} \begin{remark} \rm If ${\mu_*}$ is a weak cluster point of $(\mu^N)_{N>1}$, then ${\mu_*}\in D(F^*)$. Indeed, the first three properties are easily verified. The fourth property follows from Proposition \ref{Prp: Bound for muN_k}. To verify that $\mu_k\in D(F^k)$ we note, again by Proposition \ref{Prp: muN_k converges better than weakly}, that $I^k \mu_k\in L^1(\tilde{\Omega}^k)$. Hence, from Lemma \ref{Lem: Bound on rho log rho-} it follows that $[\mu_k\log\mu_k]^-\in L^1(\tilde{\Omega}^k)$. Finally, from $(iv)$ we obtain $[\mu_k\log\mu_k]^+\le[\mu_k\log C^k]^+=k[\log C]^+\mu_k\in L^1(\tilde{\Omega}^k)$. \end{remark} For ${\rho_*}\in D(F^*)$ we define the following functionals \begin{gather*} S^*({\rho_*})=\lim_{k\to \infty}\frac1k\int_{\tilde{\Omega}^k}\rho_k(\tilde{X}_k) \log\rho_k(\tilde{X}_k)\mathrm{d}\tilde{X}_k=\lim_{k\to \infty}\frac1kS^k(\rho_k),\\ E^*({\rho_*})=\frac12\int_{\tilde{\Omega}} \int_{\tilde{\Omega}} H(\tilde{x}_1,\tilde{x}_2)\rho_2(\tilde{x}_1,\tilde{x}_2)\mathrm{d}\tilde{x}_1\mathrm{d}\tilde{x}_2,\\ J^*({\rho_*})=\int_{\tilde{\Omega}} I(\tilde{x}_1)\rho_1(\tilde{x}_1)\mathrm{d}\tilde{x}_1,\\ F^*({\rho_*})=S^*({\rho_*})+\beta E^*({\rho_*})+ \gamma J^*({\rho_*}). \end{gather*} For ${\rho_*}\in D(F^*)$ is not difficult to see that $E^*({\rho_*})\in\mathbb{R}$ (apply Lemma \ref{Lem: FN is well defined} with $N=2$) and $J^*({\rho_*})\in\mathbb{R}$ (since $\rho_1 \in D(F^1)$). From property $(iii)$, by induction, it follows that $\rho_k(\tilde{X}_k)=\int_{\tilde{\Omega}^{N-k}}\rho_N(\tilde{X})\mathrm{d}\tilde{X}^{N-k}.$ By Proposition \ref{Prp: Subadditivity of SN}, $(S^k(\rho_k))_{k\in\mathbb{N}}$ is sub-additive. Thus the limit which defines $S^*$ exists but it can be infinity. However, by property $(iv)$, he have the following bounds $\frac1k\int_{\tilde{\Omega}^k}\rho_k(\tilde{X}_k)\log\rho_k(\tilde{X}_k)\mathrm{d}\tilde{X}_k\le\frac1k\int_{\tilde{\Omega}^k}\rho_N(\tilde{X}_k)\log C^k\mathrm{d}\tilde{X}_k=\log C.$ Therefore, $S^*({\rho_*})\in\mathbb{R}$ and $F^*$ are real valued. \begin{proposition} \label{Prp: Convergences} Let ${\rho_*}\in D(F^*)$ and ${\mu_*}$ be a weak cluster point of $(\mu^N)_{N>1}$. We have \begin{enumerate} \item$\frac1NE^N(\rho_N)\to E^*({\rho_*})$ as $N\to \infty$; \item$\frac1N J^N(\rho_N)=J^*({\rho_*})\ \forall N\in\mathbb{N}$; \item$\frac1N S^N(\rho_N)\to S^*({\rho_*})$ as $N\to \infty$; \item$\frac1N F^N(\rho_N)\to F^*({\rho_*})$ as $N\to \infty$; \item$\frac1{N_j} E^{N_j}(\mu^{N_j})\to E^*({\mu_*})$ as $j\to \infty$; \item$\frac1{N_j} J^{N_j}(\mu^{N_j})\to J^*({\mu_*})$ as $j\to \infty$; \item$S^k(\mu_k)\le\liminf_{j\to \infty}S^k(\mu^{N_j}_k)$; \item$\frac1{N_j} F^{N_j}(\mu^{N_j})\to F^*({\mu_*})$ as $j\to \infty$; \item$\frac1{N_j} S^{N_j}(\mu^{N_j})\to S^*({\mu_*})$ as $j\to \infty$; \item$\frac1k S^k(\mu_k)\le\liminf_{j\to \infty}\frac1k S^k(\mu^ {N_j}_k)\le\limsup_{j\to \infty}\frac1k S^k(\mu^{N_j}_k)\le S^* ({\mu_*})$. \end{enumerate} \end{proposition} \begin{proof} $(i)$, $(ii)$ and $(iii)$ have trivial proofs. $(iv)$. It follows from $(i)$, $(ii)$ and $(iii)$. Now we prove $(v)$. By the symmetry of $\mu^{N_j}$ (see Remark \ref{Rem: Symmetric energy}) it is sufficient to show that $\int_{\tilde{\Omega}} \int_{\tilde{\Omega}} H(\tilde{x}_1,\tilde{x}_2)\mu^{N_j}_2(\tilde{x}_1,\tilde{x}_2)\mathrm{d}\tilde{x}_1\mathrm{d}\tilde{x}_2\to \int_{\tilde{\Omega}} \int_{\tilde{\Omega}} H (\tilde{x}_1,\tilde{x}_2)\mu_2(\tilde{x}_1,\tilde{x}_2)\mathrm{d}\tilde{x}_1\mathrm{d}\tilde{x}_2,$ as $j\to \infty$. But this is a consequence of Proposition \ref{Prp: muN_k converges better than weakly}, Lemma \ref{Lem: Decomposition of H} and the weak convergence $\mu^{N_j}_2\rightharpoonup\mu_2$ in $L^2(\tilde{\Omega}^2)$. \noindent $(vi)$. This point is a consequence of the symmetry of $\mu^{N_j}$ (see Remark \ref{Rem: Symmetric energy}) and Proposition \ref{Prp: muN_k converges better than weakly} (with $k=1$ and $f=I$.) \noindent $(vii)$. By hypothesis, $\mu^{N_j}_k\rightharpoonup\mu_k$ weakly in $L^1(\tilde{\Omega}^k)$. Here, we suppose $\beta=0$. Thus, from the weak lower semicontinuity of $F^k$ in $L^1(\tilde{\Omega}^k)$ (Remark \ref{Rem: FN is lsc}) it follows that $S^k(\mu_k)+\gamma J^k(\mu_k)\le\liminf_{j\to \infty}\left(S^k (\mu^{N_j}_k)+\gamma J^k(\mu^{N_j}_k)\right).$ Hence, it is enough to show that $J^k(\mu^{N_j}_k)\to J^k(\mu_k)$ as $j\to \infty$. We note (see Remark \ref{Rem: Symmetric energy}) that $\frac1kJ^k(\mu^{N_j}_k)=\int_{\tilde{\Omega}} I(\tilde{x}_1)\mu^{N_j}_1 (\tilde{x}_1)\mathrm{d}\tilde{x}_1=\frac1{N_j} J^{N_j}(\mu^{N_j}_k).$ The result follows now from $(ii)$ and $(vi)$. \noindent $(viii)$. Fix $k\in\mathbb{N}$. For each $j\in\mathbb{N}$, large enough, we find integers $m_j$ and $n_j$ such that $N_j=m_jk+n_j$ and $01}$. Then ${\mu_*}$ is a solution of $\min\{F^*({\rho_*}): {\rho_*}\in D(F^*)\}.$ \end{theorem} \begin{proof} Take ${\rho_*}\in D(F^*)$ and $j\in\mathbb{N}$. By Theorem \ref{Thm: muN minimizes FN}, we have that $\mu^{N_j}$ minimizes $F^{N_j}$ and thus $F^{N_j}(\mu^{N_j})\le F^{N_j}(\rho_{N_j}).$ The result follows from Proposition \ref{Prp: Convergences} $(iv)$ and $(viii)$. \end{proof} \begin{definition} \rm By $\mathcal{P}(\tilde{\Omega})$ we denote the space of Borelian probabilities on $\tilde{\Omega}$ endowed with the weak topology. We denote $\mathcal{Q}(\tilde{\Omega})$ the set of all Borelian probabilities $\nu$ on $\mathcal{P}(\tilde{\Omega})$ such that for $\nu$-almost all $\rho$ in the support of $\nu$ we have \begin{itemize} \item $\rho\in L^\infty(\tilde{\Omega})\cap D(F)$ (where $D(F)=D(F^1)$); \item There exists $C=C(\nu)$ such that $\|\rho\|_{L^\infty}\le C$. \end{itemize} \end{definition} \begin{remark} \label{Rem: Equivalent definition of QOt} \rm In the previous definition, we can take the set $N(\tilde{\Omega})=\Big\{\rho\in L^\infty (\tilde{\Omega}): I\rho\in L^1(\tilde{\Omega})\Big\}$ instead of $L^\infty(\tilde{\Omega})\cap D(F)$. Indeed, it is clear that $L^\infty(\tilde{\Omega})\cap D(F)\subset N(\tilde{\Omega})$. On the other hand, if $\rho\in N(\tilde{\Omega})\cap\mathcal{P}(\tilde{\Omega})$, then $\rho\in L^1(\tilde{\Omega})$ and $[\rho\log\rho]^+\le\rho \log(1+\|\rho\|_{L^\infty})\in L^1(\tilde{\Omega})$. From Lemma \ref {Lem: Bound on rho log rho-} we deduce that $\rho\log\rho\in L^1(\tilde{\Omega})$. Hence $\rho\in D(F)$. \end{remark} \begin{theorem} \label{Thm: Hewitt-Savage} The application which maps $\nu\in\mathcal{Q}(\tilde{\Omega})$ to ${\rho_*}\in D(F^*)$ given by $\rho_k(\tilde{X}_k)=\int_\mathcal{P}(\tilde{\Omega})\rho(\tilde{x}_1)\cdots\rho(\tilde{x}_k)\nu(\mathrm{d}\rho)\quad\forall k\in\mathbb{N},$ or, equivalently, by $$\label{Eq: Hewitt-Savage 2} \int_{\tilde{\Omega}^k} f(\tilde{X}_k)\rho_k(\tilde{X}_k)\mathrm{d}\tilde{X}_k=\int_\mathcal{P}(\tilde{\Omega})\int_{\tilde{\Omega}^k} f(\tilde{X}_k)\rho(\tilde{x}_1)\cdots \rho(\tilde{x}_k)\mathrm{d}\tilde{X}_k\nu(\mathrm{d}\rho),$$ for all $f$ such that $f\rho_k\in L^1(\tilde{\Omega}^k)$, is onto. \end{theorem} \begin{proof} Let ${\rho_*}\in D(F^*)$. By Hewitt-Savage's Theorem (see \cite{MR17:863g}, Theorem 7.4) there exists a (unique) Borelian probability $\nu$ on $\mathcal{P}(\tilde{\Omega})$ such that $$\label{Eq: Hewitt-Savage} \int_{\tilde{\Omega}^k} f(\tilde{X}_k)\rho_k(\tilde{X}_k)\mathrm{d}\tilde{X}_k=\int_{\mathcal{P}(\tilde{\Omega})}\int_{\tilde{\Omega}^k} f(\tilde{X}_k)\rho(\mathrm{d}\tilde{x}_1)\cdots \rho(\mathrm{d}\tilde{x}_k)\nu(\mathrm{d}\rho)$$ for all $f$ such that $f\rho_k\in L^1(\tilde{\Omega}^k)$. By taking $f(\tilde{X}_k)=g(\tilde{x}_1)\cdots g(\tilde{x}_k)$ ($g\in L^1(\tilde{\Omega})$) and recalling that $\|\rho_k\|_{L^\infty}\le C^k$ we deduce $\int_\mathcal{P}(\tilde{\Omega})\Big[\int_{\tilde{\Omega}} g(\tilde{x}_1)\rho(\mathrm{d}\tilde{x}_1)\Big]^k\nu(\mathrm{d}\rho)\le C^k\|g \|_{L^1}^k,\quad\forall g\in L^1(\tilde{\Omega}).$ Hence, $\Big|\int_{\tilde{\Omega}} g(\tilde{x}_1)\rho(\mathrm{d}\tilde{x}_1)\Big|\le C\|g\|_{L^1}\quad\nu-\text{a.e. }\rho\in\mathcal{P}(\tilde{\Omega}).$ This means that the support of $\nu$ is included in the ball of $L^\infty(\tilde{\Omega})$ with center $0$ and radius $C$. Thus, the relation (\ref{Eq: Hewitt-Savage}) becomes (\ref{Eq: Hewitt-Savage 2}). To conclude, we should show that $\nu\in \mathcal{Q}(\tilde{\Omega})$. Since $I\rho_1\in L^1(\tilde{\Omega})$, by taking $f=I$ in (\ref{Eq: Hewitt-Savage 2}), Fubini's Theorem gives $I\rho\in L^1(\tilde{\Omega})\quad\nu-\text{a.e. }\rho\in\mathcal{P}(\tilde{\Omega}),$ that is, $\nu$ is supported in $\{\rho\in L^\infty(\tilde{\Omega}): I \rho\in L^1 (\tilde{\Omega})\}$. Therefore, $\nu\in\mathcal{Q}(\tilde{\Omega})$ (see Remark \ref{Rem: Equivalent definition of QOt}). \end{proof} Let ${\rho_*}\in D(F^*)$ and $\nu\in\mathcal{Q}(\tilde{\Omega})$ for which (\ref{Eq: Hewitt-Savage 2}) holds. We apply this relation with $f=H$ ($H\rho_2\in L^1(\tilde{\Omega}^2)$ by Lemma \ref{Lem: FN is well defined}) and also with $f=I$ to obtain $E^*({\rho_*})=\int_\mathcal{P}(\tilde{\Omega}) E(\rho)\nu(\mathrm{d}\rho),\quad\text{where}\quad E(\rho)= \frac12\int_{\tilde{\Omega}} \int_{\tilde{\Omega}} H(\tilde{x}_1,\tilde{x}_2)\rho(\tilde{x}_1)\rho(\tilde{x}_2)\mathrm{d}\tilde{x}_1\mathrm{d}\tilde{x}_2,$ and $J^*({\rho_*})=\int_\mathcal{P}(\tilde{\Omega}) J(\rho)\nu(\mathrm{d}\rho),\quad\text{where}\quad J(\rho)= \int_{\tilde{\Omega}} I(\tilde{x}_1)\rho(\tilde{x}_1)\mathrm{d}\tilde{x}_1.$ In \cite{MR44:6279} it is shown that $S^*({\rho_*})=\int_\mathcal{P}(\tilde{\Omega}) S(\rho)\nu(\mathrm{d}\rho),\quad\text{where}\quad S(\rho)= \int_{\tilde{\Omega}}\rho(\tilde{x}_1)\log\rho(\tilde{x}_1)\mathrm{d}\tilde{x}_1.$ Hence, by setting $F=S+\beta E+\gamma J$ we find $$\label{Eq: F((rho_k))=int F(rho)dnu(rho) 2} F^*({\rho_*})=\int_\mathcal{P}(\tilde{\Omega}) F(\rho)\nu(\mathrm{d}\rho).$$ All the functionals $E$, $J$, $S$ and $F$ are defined on $D(F)$. For the rest of this article, $\xi$ stands for an element of $\mathcal{Q}(\tilde{\Omega})$ associated (as stated in Theorem \ref{Thm: Hewitt-Savage}) to a weak cluster point ${\mu_*}=(\mu_k)_{k\in \mathbb{N}}$ of $(\mu^N)_{N>1}$. That is, ${\mu_*}$ and $\xi$ are related by $$\label{Eq: mu and xi} \mu_k(\tilde{X}_k)=\int_\mathcal{P}(\tilde{\Omega})\mu(\tilde{x}_1)\cdots\mu(\tilde{x}_k)\xi(\mathrm{d}\mu),\quad\forall k \in\mathbb{N}.$$ Using (\ref{Eq: F((rho_k))=int F(rho)dnu(rho) 2}) one can rewrite the claim of Theorem \ref{Thm: (mu_k) minimizes Fi} to find that $\xi$ is a solution of $\min\Big\{\ \int_\mathcal{P}(\tilde{\Omega}) F(\rho)d\nu(\rho), \nu\in\mathcal{Q}(\tilde{\Omega})\Big\}.$ Thus we obtain easily the following theorem. \begin{theorem} \label{Thm: mu minimizes F} The functional $F$ is $\xi$-almost everywhere constant on the support of $\xi$ and equal to its minimum value. In other words, $\xi$-almost all $\mu\in\mathop{\rm supp} \xi$ is a solution of $\min\left\{F(\rho): \rho\in \mathcal{P}(\tilde{\Omega}) \cap L^\infty(\tilde{\Omega})\cap D(F)\right\}.$ \end{theorem} Any solution of this problem will be called minimizer of $F$. \section{The mean field equation} For each cluster point ${\mu_*}$ of $(\mu^N)_{N>1}$ we have a measure $\xi\in\mathcal{Q}(\tilde{\Omega})$ such that (\ref{Eq: mu and xi}) holds. Theorem \ref{Thm: mu minimizes F}, essentially, says that the minimality of ${\mu_*}$ is carried to $\xi$ in the sense that in the support of $\xi$ we should have only minimizers of $F$. Thus to each weak cluster point of $(\mu^N)_{N>1}$ corresponds an average'' (with respect to $\xi$) of minimizers of $F$. In this section we look for such minimizers and we shall see that they are solutions of a certain partial differential equation. We recall that $N(\tilde{\Omega})=\Big\{\rho\in L^\infty(\tilde{\Omega}): I\rho\in L^1(\tilde{\Omega})\Big\}$. The potential of $\rho\in L^1(\tilde{\Omega})\cap N(\tilde{\Omega})$, given by $v(x_1)=\int_{\tilde{\Omega}} r_2V(x_1,x_2)\rho(\tilde{x}_2)\mathrm{d}\tilde{x}_2=-\frac1{2\pi}\int_{\tilde{\Omega}} r_2\log|x_1-x_2| \rho(\tilde{x}_2)\mathrm{d}\tilde{x}_2,$ is in $L^\infty_\mathrm{loc}(\tilde{\Omega})$. Indeed, for $|x_1|0\}$ and $C=\Big[\int_{\tilde{\Omega}}\mathrm{e}^{-\beta r_2 u(x_2)-\gamma I(\tilde{x}_2)}\mathrm{d}\tilde{x}_2\Big]^{-1}$. We should show that $|\Lambda^\complement|=0$. Suppose, by contradiction, that $|\Lambda^\complement|>0$. Then there exists a bounded measurable set $A\subset \Lambda^\complement$ such that $|A|=a>0$. For $\delta>0$ we set $\rho=\frac{\mu+\delta1_A}{1+\delta a}.$ It is easy to see that $\rho\in \mathcal{P}(\tilde{\Omega}) \cap L^\infty(\tilde{\Omega})\cap D(F) ,\, \|\rho\|_{L^1}=1$ and, by simple but tedious computations, we find a constant $C$, not depending on $\delta$, such that $F(\rho)\le F(\mu)+C\delta(1+\delta+\log\delta).$ Hence, for $\delta$ small enough we have $F(\rho)-8\pi$ and $\gamma>0$, the MFE has a unique solution $u$, then $(\mu_k^N)_{N>k}$ converges in $L^p(\tilde{\Omega}^k)$ to $\mu^{\otimes k}$ for all $k\in\mathbb{N}$ and for all $p\in[1,\infty)$, where $\mu$ is the distribution associated to $u$. \end{proposition} \begin{proof} Let $k\in\mathbb{N}$ and $p\in[1,\infty)$. From Theorem \ref{Thm: u is a solution of MFE}, we have that for $\xi$-almost all $\mu\in\mathop{\rm supp}\xi$ the associated potential is a solution of MFE, ant thus, by uniqueness, equals to $u$. By Proposition \ref{Prp: mu and u} $\mu$ is the distribution associated to $u$. Thus, $\xi$ is a Dirac measure concentrated on $\mu$. It follows that $\mu_k (\tilde{X}_k)=\mu(\tilde{x}_1)\cdots\mu(\tilde{x}_k)$. Hence, $\frac1kS^k(\mu_k)=\frac1k\int_{\tilde{\Omega}^k}\mu_k(\tilde{X}_k)\log\mu_k(\tilde{X}_k)\mathrm{d}\tilde{X}_k=\int_{\tilde{\Omega}}\mu(\tilde{x}_1) \log\mu(\tilde{x}_1)\mathrm{d}\tilde{x}_1=S(\mu).$ We know also that $S^*({\mu_*})=\int_\mathcal{P}(\tilde{\Omega}) S(\rho)\mathrm{d}\xi(\rho)=S(\mu).$ From Proposition \ref{Prp: Convergences}, item $(x)$, we have $S^k(\mu^{N_j}_k)\to S^k(\mu_k)$. Since $S$ is strictly convex we conclude that $\mu^{N_j}_k\to \mu_k$ strongly in $L^p(\tilde{\Omega}^k)$. We have shown that every weakly cluster point of $(\mu_k^N)_{N>k}$ is a strongly one and unique. \end{proof} \section{An alternative to the study of MFE} One may think that it is straightforward to proceed as we did in \cite{Neri} by introducting a functional $G$ on $H^1(\mathbb{R}^2)$ and for which the Euler-Lagrange equation is the MFE. Thus changing our point of view to a variational problem on potentials. But there are many technical difficulties which arises. For example: \begin{enumerate} \item\label{It: a} We need an inequality of Poincar\'e type over $\mathbb{R}^2$. In fact, this is not a problem because we work with the measure $\mathrm{e}^{-\gamma r_1|x_1|^2}\mathrm{d} x_1$. \item We need also an $\mathbb{R}^2$ version of Trudinger-Moser inequality. It may exist by the same reason as in (\ref{It: a}). \item The potential associated to $\mu\in D(F)$ is not always on $H^1(\mathbb{R}^2)$! Indeed, even, for example, if $\mu$ is compactly supported we have that $|\nabla u(x_1)|$ decreases at infinity as fast as $|x_1|^{-1}$ which is not in $L^2(\mathbb{R}^2)$. \end{enumerate} Another inconvenience to study the MFE: in general, the hypothesis of Proposition \ref{Prp: Unique solution} does not hold! Take, for example, $P= \delta_1$. The MFE becomes $-\Delta u(x_1)=\Big[\int_{\mathbb{R}^2}\mathrm{e}^{-\beta u(x_2)-\gamma|x_2|^2}\mathrm{d} x_2\Big]^{-1}\mathrm{e}^{-\beta u(x_1)-\gamma|x_1|^2}.$ It is clear that, if $u$ is a solution, then $u+C$ is also a solution. However the conclusion of Proposition \ref{Prp: Unique solution} holds by modifying the argument. The fundamental idea of the proof of Proposition \ref{Prp: Unique solution} was to follow the statistical approach backwards: each weakly cluster point of $(\mu^N)_{N>1}$ gives a solution of MFE. Hence, if this equation has a unique solution, then we have the uniqueness of minimizers of $F$. We conclude that $(\mu^N)_{N>1}$ has a unique weak cluster point and the convergence is strong. Now we do not have the uniqueness of MFE's solution anymore, but we can start our argument from the uniqueness of minimizers of $F$. The previous remarks show us that it might be more convenient to forget the potentials and MFE and study the problem by means of distributions and minimizers of $F$. Theorem \ref{Thm: mu minimizes F} gave us the existence now we have a uniqueness result. \begin{proposition} \label{Prp: Unique minimizer} If $P=\delta_1$ and $\beta>0$, then $F$ has a unique minimizer. \end{proposition} \begin{proof} Considering this measure $P$, we can identify $\tilde{\Omega}=\mathbb{R}^2\times(0,1]$ to $\mathbb{R}^2 \times\{1\}$ and thus to $\mathbb{R}^2$. Let $\mu$ be a minimizer of $F$. We shall show that $\int_{\mathbb{R}^2}x_1\mu(x_1)\mathrm{d} x_1=0.$ First note that the integral above is convergent since $\mu\in L^1(\mathbb{R}^2)$ and $|x_1|^2\mu(x_1)\in L^1(\mathbb{R}^2)$. Suppose, by contradiction, that the integral above equals to $x_0\ne 0$. Consider the function $\tilde\mu(x_1)=\mu(x_1+x_0)$. It is easy to see that $\tilde\mu\in\mathcal{P}(\tilde{\Omega}) \cap L^\infty(\tilde{\Omega})\cap D(F)$, $S(\tilde\mu)=S(\mu)$ and $E(\tilde \mu)=E(\mu)$. By a change of variables we have \begin{align*} J(\tilde\mu)&=\int_{\mathbb{R}^2}|x_1|^2\mu(x_1+x_0)\mathrm{d} x_1=\int_{\mathbb{R}^2}|x_1 -x_0|^2\mu(x_1)\mathrm{d} x_1\\ &=\int_{\mathbb{R}^2}|x_1|^2\mu(x_1)\mathrm{d} x_1-2x_0\cdot\int_{\mathbb{R}^2}x_1\mu(x_1) \mathrm{d} x_1+|x_0|^2\int_{\mathbb{R}^2}\mu(x_1)\mathrm{d} x_1\\ &=J(\mu)-2|x_0|^2+|x_0|^2=J(\mu)-|x_0|^20$and$E$is quadratic, it is enough to show that$E(\rho)$is positive if$\rho\in L^1(\mathbb{R}^2)\cap L^ \infty(\mathbb{R}^2)$,$|x_2|^2\rho(x_2)\in L^1(\mathbb{R}^2)$, $$\label{Eq: Constraints} \int_{\mathbb{R}^2}\rho(x_2)\mathrm{d} x_2=0\quad\text{and}\quad \int_{\mathbb{R}^2}x_2\rho(x_2)\mathrm{d} x_2=0.$$ Take a such$\rho$and let$v\in L^\infty_\mathrm{loc}(\mathbb{R}^2)$be its potential. We shall show that$v\in L^2(\mathbb{R}^2)$. For$x_1\in\mathbb{R}^2$, we denote$A=A(x_1)=\{x_2 \in\mathbb{R}^2: |x_2|\le |x_1|/2\}$and write$v=-\frac1{2\pi}(v_1+v_2)$, with $v_1(x_1)=\int_A\log|x_1-x_2|\rho(x_2)\mathrm{d} x_2\quad\text{and}\quad v_2(x_1)=\int_{A^\complement}\log|x_1-x_2|\rho(x_2)\mathrm{d} x_2.$ Hence, it suffices to show that$v_1,v_2\in L^2(\mathbb{R}^2)$. From Taylor's expansion of$\log|x|$we have $\log|x+y|=x\cdot y+{\mathcal O}(|y|^2)\quad\text{if }|x|=1,\quad |y|\le\frac12,$ uniformly on$x$since$\log|x|$is$C^2$on the compact$\{0.5\le |x|\le1\}$. Hence, take$x=x_1/|x_1|$and$y=-x_2/|x_1|to obtain $\log|x_1-x_2|=\log|x_1|-\frac{x_1}{|x_1|^2}\cdot x_2+\frac1{|x_1|^2}{\mathcal O} (|x_2|^2)\quad\text{if }x_2\in A.$ Thus, from (\ref{Eq: Constraints}) it follows that \begin{align*} &v_1(x_1)\\ &=\log|x_1|\int_A\rho(x_2)\mathrm{d} x_2 -\frac{x_1}{|x_1|^2}\cdot\int_Ax_2\rho(x_2)\mathrm{d} x_2 +\frac1{|x_2|}\int_A{\mathcal O}(|x_2|^2)\rho(x_2)\mathrm{d} x_2\\ &=-\log|x_1|\int_{A^\complement}\rho(x_2)\mathrm{d} x_2 +\frac{x_1}{|x_1|^2}\cdot\int_{A^\complement}x_2\rho(x_2)\mathrm{d} x_2 +\frac1{|x_2|}\int_A{\mathcal O}(|x_2|^2)\rho(x_2)\mathrm{d} x_2. \end{align*} Since|x_2|^2\rho(x_2)\in L^1(\rho)$, for$|x_1|>1, we have \begin{align*} |v_1(x_1)|&\le\log|x_1|\int_{A^\complement}|\rho(x_2)|\mathrm{d} x_2 +\frac1{|x_1|}\int_{A^\complement}|x_2||\rho(x_2)|\mathrm{d} x_2 +{\mathcal O}\left(\frac1{|x_1|^2}\right)\\ &\le\frac{4\log|x_1|}{|x_1|^2}\int_{\mathbb{R}^2}|x_2|^2|\rho(x_2)|\mathrm{d} x_2 +\frac2{|x_1|^2}\int_{\mathbb{R}^2}|x_2|^2|\rho(x_2)|\mathrm{d} x_2 +{\mathcal O}\left(\frac1{|x_1|^2}\right)\\ &={\mathcal O}\left(\frac{\log|x_1|}{|x_1|^2}\right). \end{align*} Thereforev_1\in L^2(\mathbb{R}^2)$. To show that$v_2\in L^2(\mathbb{R}^2)$, we write$v_2=w_1+w_2$, where$w_1=v_21_{B_1 (x_1)}$and$w_2=v_21_{B_1(x_1)^\complement}. Holder's Inequality yields \begin{align*} |w_1(x_1)|&\le\Big[\int_{A^\complement\cap B_1(x_1)}\Big|\log|x_1-x_2| \Big|^3|\rho(x_2)|\mathrm{d} x_2\Big]^\frac13\Big[\int_{A^\complement\cap B_1(x_1)}|\rho(x_2)|\mathrm{d} x_2\Big]^\frac23\\ &\le\|\rho\|_{L^\infty}\Big[\int_{B_1(0)}\Big|\log|x_2|\Big|^3\mathrm{d} x_2 \Big]^\frac13\Big[4\int_{A^\complement}\frac{|x_2|^2}{|x_1|^2}\ |\rho(x_2)|\mathrm{d} x_2\Big]^\frac23\le\frac C{|x_1|^{4/3}}. \end{align*} Therefore,w_1\in L^2(\mathbb{R}^2)$. We take$C>0$such that$\log r\le Cr^{1/4}$for all$r>1. It follows that \begin{align*} |w_2(x_1)|&\le C\int_{A^\complement\cap B_1(x_1)^\complement}|x_1-x_2|^ {1/4}|\rho(x_2)|\mathrm{d} x_2\\ & \le C\int_{A^\complement}|x_2|^{1/4}\frac{|x_2|^{7/4}}{|x_1|^{7/4}} |\rho(x_2)|\mathrm{d} x_2\le\frac C{|x_1|^{7/4}}. \end{align*} Therefore,w_2\in L^2(\mathbb{R}^2). From Plancherel's Theorem, we obtain \begin{align*} \int_{\mathbb{R}^2}\hat v(\eta)\overline{\hat\rho(\eta)}\mathrm{d}\eta &=\int_{\mathbb{R}^2} v(x_1)\rho(x_1)\mathrm{d} x_1\\ &=\int_{\mathbb{R}^2}\int_{\mathbb{R}^2}V(x_1,x_2)\rho(x_1) \rho(x_2)\mathrm{d} x_1\mathrm{d} x_2=2E(\rho), \end{align*} where\hat v$and$\hat\rho$are the Fourier transforms of$v$and$\rho$, resp. But,$-\Delta v=\rho$and thus$|\eta|^2{\hat v}(\eta)=\hat\rho(\eta)$, hence $E(\rho)=\frac12\int_{\mathbb{R}^2}|\eta|^2|\hat v(\eta)|^2\mathrm{d}\eta\ge 0.$ \end{proof} The uniqueness of minimizer, as often, has followed from strict convexity of$F$. The previous proof can be adapted to a Dirac measure supported on any point. Unfortunately, without the strict convexity we don't know whether uniqueness holds or not. As a bad news, we have that the convexity is a particularity of the case of a Dirac measure. \begin{proposition} If$P$is not a Dirac measure and$\beta>0$, then the functional$F$is not convex on $M=\Big\{\rho\in\mathcal{P}(\tilde{\Omega}) \cap L^\infty(\tilde{\Omega})\cap D(F): \int_{\mathbb{R}^2}x_1\rho(x_1)\mathrm{d} x_1=0\Big\}.$ \end{proposition} \begin{proof} Since$P$is not a Dirac measure, we can take$00$and$P([a,b])>0$. Taking$a$and$b$close enough, we may suppose$b_0b2$(to be chosen later) and consider a regular polygon of$N$sides and radius$t$centered at origin. We denote by$B_i$the unity ball centered at its$i$-th vertex. Let$A=\bigcup_{i=1}^N B_i$($t$is large enough to have a disjoint union). We define also$\tilde B_0=B_0\times [a_0,b_0],\ \tilde B_i=B_i\times[a,b]$($i=1,\dots,N$) and$\tilde A=\bigcup_{i= 1}^N\tilde B_i$. Finally, we define$\rho_0=\alpha_01_{\tilde B_0}$and$\rho_1=\alpha_11_{\tilde A}$, where$\alpha_0,\alpha_1>0$are such that $\alpha_0\int_{\tilde B_0}\mathrm{d}\tilde{x}_1=\int_{\tilde{\Omega}}\rho_0(\tilde{x}_1)\mathrm{d}\tilde{x}_1=1$ and $N\alpha_1\int_{\tilde B_i}\mathrm{d}\tilde{x}_1=\alpha_1\int_{\tilde A}\mathrm{d}\tilde{x}_1= \int_{\tilde{\Omega}}\rho_1(\tilde{x}_1)\mathrm{d}\tilde{x}_1=1.$ By the symmetries of$\tilde B_0$and$\tilde A$with respect to origin, it is easy too see that$\rho_0,\rho_1\in M$. Simple computations give $S\left(\frac12(\rho_0+\rho_1)\right)=\frac12S(\rho_0)+\frac12S(\rho_1)-\log 2$ and $J\left(\frac12(\rho_0+\rho_1)\right)=\frac12J(\rho_0)+\frac 12J(\rho_1).$ Since$E$is quadratic, we have $E\left(\frac12(\rho_0+\rho_1)\right)=\frac12E(\rho_0)+\frac12E(\rho_1)-\frac14E (\rho_0-\rho_1).$ Hence, $F\left(\frac12(\rho_0+\rho_1)\right)=\frac12F(\rho_0)+\frac12F(\rho_1)-\log2- \frac\beta4E(\rho_0-\rho_1).$ We shall show that$E(\rho_0-\rho_1)\to -\infty$as$t\to \infty$. Therefore, for$t$large enough, we shall have $F\left(\frac12(\rho_0+\rho_1)\right)>\frac12F(\rho_0)+\frac12F(\rho_1).$ The result follows. Since$\tilde B_0,\tilde B_1\dots,\tilde B_N$are disjoints and$\rho_0$and$\rho_1are constants and supported on these balls, we have \begin{align*} E(\rho_0-\rho_1) &=-\frac{\alpha_0^2}{4\pi}\int_{\tilde{B}_0}\int_{\tilde{B}_0} r_1r_2\log|x_1-x_2|\mathrm{d}\tilde{x}_1\mathrm{d}\tilde{x}_2\\ &\quad -\frac{\alpha_1^2}{4\pi}\sum_{i=1}^N\int_{\tilde{B}_i}\int_{\tilde{B}_i} r_1r_2\log|x_1-x_2|\mathrm{d}\tilde{x}_1\mathrm{d}\tilde{x}_2\\ &\quad +\frac{\alpha_0\alpha_1}{4\pi}\sum_{i=1}^N\int_{\tilde{B}_0}\int_ {\tilde{B}_i}r_1r_2\log|x_1-x_2|\mathrm{d}\tilde{x}_1\mathrm{d}\tilde{x}_2\\ &\quad -\frac{\alpha_1^2}{4\pi}\sum_{i\ne j}^N\int_{\tilde{B}_i}\int_{\tilde{B}_j} r_1r_2\log|x_1-x_2|\mathrm{d}\tilde{x}_1\mathrm{d}\tilde{x}_2. \end{align*} Clearly, the first term on the RHS does not depend ont$. The second neither, since by a translation on$x$variables, the integrand does not change and the domain becomes, for example,$(B_0\times[a,b])^2$. For$t$large enough the last two terms behave like $\Phi(t)=\frac{\alpha_0\alpha_1}{4\pi}\sum_{i=1}^N\int_{\tilde{B}_0} \int_{\tilde{B}_i}r_1r_2\log t\mathrm{d}\tilde{x}_1\mathrm{d}\tilde{x}_2 -\frac{\alpha_1^2}{4\pi}\sum_{i\ne j}^N\int_{\tilde{B}_i}\int_{\tilde{B}_j} r_1r_2\log(\theta t)\mathrm{d}\tilde{x}_1\mathrm{d}\tilde{x}_2,$ where$\theta$is a constant (the ratio side/radius for the regular polygon of$N$sides). But,$\log(\theta t)=\log\theta+\log t$and, by the same translation on$x$variables as before, we conclude that the factor multiplying$\log\theta$does not depend on$t$. Therefore, we may suppose$\theta=1. Then \begin{align*} \Phi(t)&=\frac{\log t}{4\pi}\Big[\alpha_0\alpha_1\sum_{i=1}^N \int_{\tilde{B}_0}\int_{\tilde{B}_i}r_1r_2\mathrm{d}\tilde{x}_1\mathrm{d}\tilde{x}_2 -\alpha_1^2\sum_{i\ne j}^N\int_{\tilde{B}_i}\int_{\tilde{B}_j} r_1r_2\mathrm{d}\tilde{x}_1\mathrm{d}\tilde{x}_2\Big]\\ &\le\frac{\log t}{4\pi}\Big[\alpha_0\alpha_1bb_0\sum_{i=1}^N \int_{\tilde{B}_0}\int_{\tilde{B}_i}\mathrm{d}\tilde{x}_1\mathrm{d}\tilde{x}_2 -\alpha_1^2a^2\sum_{i\ne j}^N \int_{\tilde{B}_i}\int_{\tilde{B}_j}\mathrm{d}\tilde{x}_1\mathrm{d}\tilde{x}_2\Big]\\ &=\frac{\log t}{4\pi}\Big[bb_0-a^2\frac{(N-1)}N\Big] \end{align*} We conclude that\Phi(t)\to -\infty$as$t\to \infty$. \end{proof} \subsection*{Acknowledgment} The author would like to thank P.-L. Lions who introduced him to this subject during his doctoral studies at Universit\'e Paris IX-Dauphine. 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