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Lille, CNRS, Inria, UMR 8524 - Laboratoire Paul Painlevé\\ F-59000 Lille, France} \email{benoit.merlet@univ-lille.fr} \urladdr{https://math.univ-lille1.fr/~merlet/} \author[\initial{M.} \lastname{Pegon}]{\firstname{Marc} \lastname{Pegon}} \address{Univ. Lille, CNRS, Inria, UMR 8524 - Laboratoire Paul Painlevé\\ F-59000 Lille, France} \email{marc.pegon@univ-lille.fr} \urladdr{https://mpegon.perso.math.cnrs.fr/} \thanks{B.M.\ and M.P.\ are partially supported by the Labex CEMPI (ANR-11-LABX-0007-01)} \begin{abstract} Motivated by Gamow's liquid drop model in the large mass regime, we consider an isoperimetric problem in which the standard perimeter $P(E)$ is replaced by $P(E)-\gamma P_\varepsilon(E)$, with $0<\gamma<1$ and $P_\varepsilon$ a nonlocal energy such that $P_\varepsilon(E)\to P(E)$ as $\varepsilon$ vanishes. We prove that unit area minimizers are disks for $\varepsilon>0$ small enough. More precisely, we first show that in dimension $2$, minimizers are necessarily convex, provided that $\varepsilon$ is small enough. In turn, this implies that minimizers have nearly circular boundaries, that is, their boundary is a small Lipschitz perturbation of the circle. Then, using a Fuglede-type argument, we prove that (in~arbitrary dimension $n\geq 2$) the unit ball in $\mathbb{R}^n$ is the unique unit-volume minimizer of the problem among centered nearly spherical sets. As a consequence, up to translations, the unit disk is the unique minimizer. This isoperimetric problem is equivalent to a generalization of the liquid drop model for the atomic nucleus introduced by Gamow, where the nonlocal repulsive potential is given by a radial, sufficiently integrable kernel. In that formulation, our main result states that if the first moment of the kernel is smaller than an explicit threshold, there exists a critical mass $m_0$ such that for any $m>m_0$, the disk is the unique minimizer of area $m$ up to translations. This is in sharp contrast with the usual case of Riesz kernels, where the problem does not admit minimizers above a critical mass. \end{abstract} \subjclass{28A75, 49Q05, 49Q10, 49Q20, 52A10} \keywords{Geometric variational problems, nonlocal isoperimetric problems, nonlocal perimeters, regularity, liquid drop model} \altkeywords{Problèmes variationnels géométriques, problèmes isopérimétriques non locaux, périmètres non locaux, régularité, modèle de goutte liquide} \alttitle{Rigidité pour un modèle de goutte liquide en 2D avec des noyaux de moments finis et dans le régime des grandes masses} \begin{altabstract} Motivé par l’étude du modèle de goutte liquide de Gamow dans le régime des grandes masses, nous considérons un problème isopérimétrique dans lequel le périmètre classique $P(E)$ est remplacé par $P(E)-\gamma P_\varepsilon(E)$, où $0<\gamma<1$ et $P_\varepsilon$ est une énergie non locale telle que $P_\varepsilon(E)\to P(E)$ lorsque $\varepsilon$ tend vers zéro. Nous montrons que pour $\varepsilon$ assez petit les minimiseurs à aire fixée sont les disques. Pour cela, nous établissons d'abord qu'en dimension $2$, les minimiseurs sont convexes dès que $\varepsilon$ est suffisamment petit. Ceci implique que le bord d’un minimiseur est une petite perturbation Lipschitz d’un cercle. Puis, par un argument à la Fuglede, nous prouvons (en dimension arbitraire $n\geq 2$) que si un minimiseur à volume fixé est une perturbation d’une boule au sens précédent, alors c’est une boule. Ce problème isopérimétrique est équivalent à une généralisation du modèle de goutte liquide pour le noyau atomique introduit par Gamow lorsque le potentiel répulsif non local est donné par un noyau suffisamment intégrable. Dans cette formulation, notre résultat principal indique que si le premier moment du noyau est inférieur à un seuil explicite, il existe une masse critique $m_0$ telle que les minimiseurs de masse prescrite $m>m_0$ sont les disques. Ceci contraste fortement avec le cas classique des noyaux de Riesz, où le problème n'admet pas de minimiseur au-delà d'une masse critique. \end{altabstract} \maketitle \tableofcontents \mainmatter \section{Introduction} Given a positive, radial, measurable kernel $G:\Ren\to [0,+\infty)$ with finite first moment (that is, $\abs{x}G(x)\in L^1(\Ren)$), we consider the nonlocal perimeter functional $\Per_G$ (see \eg \cite{CN2018,BP2019}) defined on measurable sets $E\subsq\Ren$ by \begin{equation}\label{eq:defPG} \begin{split} \Per_G(E)&\coloneqq 2\iint_{E \times \left(\Ren\setminus E\right)} G(x-y) \dx\dy\\ &\hphantom{:}=\iint_{\Ren\times\Ren} \abs{\ind_E(x)-\ind_E(y)}G(x-y)\dx\dy. \end{split} \end{equation} Here $\ind_E$ denotes the indicator function of $E$. For $\eps>0$, we introduce the rescaled kernel $G_\eps(x)\coloneqq\eps^{-(n+1)}G(\eps^{-1}x)$, $x\in\Ren$. As will be justified later, the first moment of $G$ is fixed to an explicit dimensional constant (see \cref{Hint}) so that $\Per_{G_\eps}(E)$ converges to $P(E)$ as $\eps$ vanishes. Given $\gamma\in (0,1)$ and $\eps>0$, we study the minimization problem \begin{equation}\label{minrp}\tag{$\mathcal{P}_{\gamma,\eps}$} \min \Bigl\{ P(E)-\gamma\Per_{G_\eps}(E)\sep \abs{E}=\abs{B_1}\Bigr\}, \end{equation} over sets of finite perimeter $E$ in $\Ren$, where $\abs{E}$ denotes the volume of $E$ (which we often call its mass), that is, its Lebesgue measure, and $B_1$ is the open unit ball of $\Ren$. Let us emphasize the competition between the two terms. The perimeter is an attractive term minimized by balls under volume constraint. On the contrary, if $G$ is radially decreasing, due to the negative sign, the nonlocal term is maximized by balls,\footnote{This can be seen by Riesz' symmetric rearrangement, using \eg \cite[Chap.\,3.7]{LL2001} and the fact that~$G$ is equal to its symmetric rearrangement in that case. In fact, even if $G$ is not radially decrea\-sing, $G_\eps$ \enquote{concentrates} near the origin when $\eps$ is small, heuristically making $(-\Per_{G_\eps})$ a repulsive term whenever $G$ is not identically equal to $0$.} and there exists no minimizer for the functional $(-\Per_{G_\eps})$ under volume constraint. This competition makes the minimization problem nontrivial, even when it comes to existence of minimizers. Problem \cref{minrp} is closely linked with variations of Gamow's liquid drop model for the atomic nucleus in the large mass regime. Indeed, thanks to the $\eps^{-(n+1)}$ factor in~$G_\eps$, changing variables in \cref{eq:defPG}, we have $\Per_{G_\eps}(E)=\eps^{n-1}\Per_G(\eps^{-1}E)$, so that \[ P(E)-\gamma \Per_{G_\eps}(E) = \eps^{n-1}\big( P(\eps^{-1}E)-\gamma \Per_G(\eps^{-1}E)\big), \] and \cref{minrp} is equivalent to the problem \begin{equation}\label{minpb}\tag{$\mathcal{P}'_{\gamma,\eps}$} \min \Bigl\{ P(F)-\gamma\Per_{G}(F)\sep \abs{F}=\sfrac{\abs{B_1}}{\eps}\Bigr\}, \end{equation} in the sense that $E_\eps$ is a minimizer of \cref{minrp} if and only if $F_\eps\coloneqq \eps^{-1}E_\eps$ is a minimizer of \cref{minpb}. Then, if in addition we assume that~$G$ is integrable in $\Ren$, we may write \begin{equation}\label{eq:equivgamow} \frac{1}{2}\Per_{G}(F) = \norm{G}_{L^1(\Ren)}\abs{F}-\iint_{F\times F} G(x-y)\dx\dy, \end{equation} thus \cref{minpb} is in fact equivalent to \begin{equation}\label{mingam}\tag{$\mathcal{G}_{m_\eps}$} \min \biggl\{ P(F)+\iint_{F\times F} \wt{G}(x-y)\sep \abs{F}=m_\eps\biggr\}, \end{equation} where we have set $m_\eps\coloneqq \eps^{-1}\abs{B_1}$ and $\wt{G}\coloneqq 2\gamma G$. When $n=3$ and $\wt{G}(x)=1/(8\pi\abs{x})$, this is Gamow's liquid drop model (see \cite{CMT2017} for a general overview); note however that in that case, the minimized functional cannot be rewritten as the difference between the perimeter and a nonlocal perimeter, since $\wt{G}$ is not integrable at infinity. As a prototypical model for various physical systems involving the competition between short-range attractive forces and long-range repulsive ones, generalizations of this model have gained increasing interest during the past decade, in particular generalizations in higher dimensions, where the Coulomb potential is replaced with Riesz potentials, that is, $\wt{G}(x)=\abs{x}^{\alpha-n}$, $\alpha\in(0,n)$. In particular, it was shown that for every Riesz kernel, in the small mass regime, the unique minimizer of the liquid drop model~\cref{mingam} is the ball, up to translations (see~\cite{KM2013,KM2014,Jul2014,BC2014,FFM+2015}). Conversely, for $\alpha\in [n-2,n)$, the problem admits no minimizer above a critical mass (see \cite{BC2014,KM2013,KM2014,LO2014,FKN2016,FN2021}; see also \cite{FL2015}). More general kernels of Riesz-type were studied \eg in \cite{CFP2020,NP2021,MW2021}, where the unit ball is shown to be the unique minimizer in the small mass regime. Although the small mass regime has been extensively studied, the literature on large mass minimizers for Gamow-type problems is still sparse, since existence is rather unexpected in that case, and is usually only recovered by adding an extra attractive potential, such as in \cite{ABCT2017,ABCT2018,GO2018}, or by adding a density to the perimeter, as in \cite{ABTZ2021}, where the authors show that if the density is a power-law growing sufficiently fast at infinity, then minimizers always exist, and are balls for large masses. It is worth mentioning that in the case of general kernels \emph{with compact support}, the author of \cite{Rig2000} shows that minimizers exist \emph{for all masses}. Between Riesz kernels, which are not integrable at infinity, where \cref{mingam} does not admit minimizers above a critical mass, and compactly supported kernels, where \cref{mingam} always admits minimizers, it is natural to wonder what happens with non-compactly supported but reasonably decaying kernels, such as Bessel kernels. These kernels beha\-ve as Riesz potentials near the origin, but decrease exponentially at infinity. They were suggested in \cite{KMN2016} as a replacement for Riesz kernels for modeling diblock copolymers when long-range interactions are partially screened by fluctuations in the background nuclear fluid density. In \cite{MW2021} (see Section 1.2 therein), motivated by some model for cell motility, the authors suggest to study problem \cref{mingam} for rescalings of the kernel~$G$, where $G$ is the fundamental solution of $\left(\id +(-\Delta)^{s/2}\right)G=\delta_0$, for $s\in (0,2)$. As~pointed out in \cite[Rem.\,1.3]{MW2021}, for $s\in(0,1)$ their asymptotic rescalings correspond to the small mass regime, while for $s\in(1,2)$ they correspond to the large mass regime. The authors focus on the case $s\in(0,1)$. Our work actually addresses the case $s\in(1,2)$ (and with more general kernels). The study of the liquid drop model in the \emph{large mass regime} for integrable kernels with finite first moment (such as Bessel kernels or the ones just mentioned for $s\in(1,2)$), which is equivalent to the study of \cref{minrp} when $\eps$ is small, has been started by the second author in \cite{Peg2021}. The existence of minimizers for any $\gamma\in(0,1)$ was established therein for $\eps$ small enough, as well as the convergence of minimizers to the unit ball as $\eps$ vanishes. It was conjectured there that the ball is actually the unique minimizer up to translations, for $\eps$ small enough. In this paper, we give a positive answer to this conjecture in dimension $n=2$ under reasonable assumptions on the first moment of $G$ and the second moment of $\grad G$, which are still satisfied by Bessel kernels. The conjecture remains open in higher dimensions. Let us introduce the \enquote{critical energies} \[ \calE_{G_\eps}:=P-\Per_{G_\eps}=\calF_{1,G_\eps}, \] and define the energies \begin{equation}\label{eq:defF} \calF_{\gamma,G_\eps}(E) \coloneqq (1-\gamma) P(E)+\gamma\calE_{G_\eps}(E) = P(E)-\gamma\Per_{G_\eps}(E). \end{equation} Although the paper mostly deals with the \enquote{subcritical case} $\gamma<1$, we focus on the critical energies in \cref{subsec:convexification}, and show that they decrease by convexification. Finally, for any $k\in\N\setminus\{0\}$, we denote by \begin{equation}\label{defmom} I_G^{k} \coloneqq \int_{\Ren} \abs{x}^{k}\abs{\partial_r^{k-1} G(x)}\dx \end{equation} the $k$-th moment of the $(k-1)$-th radial derivative of the kernel $G$, whenever it is well-defined. In the paper, starting from \cref{sec:prelim}, \emph{we shall always implicitly assume} that the kernel $G$ satisfies the following general assumptions: \begin{enumerate}\renewcommand{\theenumi}{H.\arabic{enumi}} \item\label{Hrad} $G$ is nonnegative and radial, that is, there exists a measurable function $g:(0,+\infty)\to[0,+\infty)$ such that $G(x)=g(\abs{x})$ for almost every $x\in\Ren$; \item\label{Hint} the first moment of $G$ is finite and set to be \[ I_G^1 = \frac{1}{\bfK_{1,n}}, \] where, for any $\nu\in\bbS^{n-1}$, the constant $\bfK_{1,n}$ is defined by \[ \bfK_{1,n}=\dashint_{\bbS^{n-1}} \abs{\sigma\cdot \nu}\dH^{n-1}_\sigma. \] \end{enumerate} Starting from \cref{sec:fuglede}, we may explicitly use the extra assumption: \begin{enumerate}\renewcommand{\theenumi}{H.\arabic{enumi}} \setcounter{enumi}{2} \item\label{Hhighermoments} $G\in W^{1,1}_\loc(\Ren\setminus\{0\})$, $I_G^2<\infty$, and $g'(r)=O(r^{-(n+1)})$ at infinity. \end{enumerate} These assumptions are in particular satisfied by the Bessel kernels $\mathcal{B}_{\kappa,\alpha}$, that is, fundamental solutions of the operators $(I-\kappa\Delta)^{\alpha/2}$, for $\kappa,\alpha>0$ (see \eg \cite[\S 3.2]{Peg2021} or \cite{Gra2014} for their definition and properties). Even though our main result is in dimension $n=2$, where we prove that the unit disk is the only minimizer up to translations, provided that $\eps$ is small enough, note that the intermediate results of \cref{subsec:convnearlysph} and \cref{sec:fuglede} are obtained in arbitrary dimension. That is, convex minimizers are nearly spherical sets, and the unit ball of~$\Ren$ is the unique minimizer among nearly spherical sets whenever $\eps$ is small enough. Note also that the kernel $G$ is assumed to be radial but not necessarily radially nonincreasing, as is often the case. Let us emphasize that, contrarily to the small mass regime for Riesz-type potentials, here the nonlocal perimeter term does not vanish in the limit but rather converges to a fraction of the standard perimeter. We shall now state the main result of the paper. \begin{mainres}[Minimality of the unit disk]\label{mainthm:mindisk} Assume $n=2$, $\gamma\in(0,1)$ and $G$ satisfies \crefrange{Hrad}{Hhighermoments}. Then there exists $\eps_A=\eps_A(G,\gamma)>0$, such that, for every $\eps<\eps_A$, the unit disk is the unique minimizer of \cref{minrp}, up to translations and Lebesgue-negligible sets. \end{mainres} In terms of Gamow's problem \cref{mingam}, this means that if $I_{\wt{G}}^1<2/\bfK_{1,2}=\pi$ (here~$I_{\wt{G}}^1$ is the first moment of the kernel $\wt{G}$ defined as in \cref{defmom}) and $\wt{G}$ is in addition integrable, then there exists a critical mass $m_*$ such that the only solutions of \cref{mingam} with $m_\eps>\nobreak m_*$ are the disks of area $m_\eps$. In the particular case of the Bessel kernels $\wt{G}=\mathcal{B}_{\kappa,\alpha}$ with $\kappa,\alpha>0$, such a critical mass exists whenever \[ \kappa<\pi\Bigl(\frac{\Gamma\left(\sfrac{\alpha}{2}\right)} {\Gamma\left(\psfrac{1+\alpha}{2}\right)}\Bigr)^2, \] (see \cite[Cor.\,3.9 \& Prop.\,3.10]{Peg2021}). The proof of \cref{mainthm:mindisk} decomposes as follows. First we establish: \begin{mainthm}[2D minimizers are convex]\label{mainthm:convexmin} Assume $n=2$, $\gamma\in(0,1)$ and $G$ satisfies \cref{Hrad,Hint}. Then there exists $\eps_1=\eps_1(G,\gamma)>0$ such that, for every $\eps$ such that $0<\eps<\eps_1$, \cref{minrp} admits a minimizer, and every minimizer is convex, up to a Lebesgue-negligible set. \end{mainthm} The existence of minimizers for small $\eps$ was shown in \cite{Peg2021}, where the second author also proved that they are necessarily connected whenever $\eps$ is small enough. The idea for proving the convexity of minimizers is to study the critical energy on the real line, and show that it decreases by convexification and by expansion of segments, so that, by a slicing argument, the critical energy of a \emph{connected set in dimension~$2$} decreases after convexification. As a consequence, since the perimeter of a connected set is also reduced by convexification, so is $\calF_{\gamma,G_\eps}$. This slicing argument is specific to the dimension $2$, where a line intersects a connected set if and only if it intersects its convex hull. This fails in higher dimension. Note that this is not enough to conclude that minimizers are convex. Indeed, although for every minimizer $E_\eps\subset\R^2$ with $\eps$ small enough, we have $\calF_{\gamma,G_\eps}(\co(E_\eps))\leq\calF_{\gamma,G_\eps}(E_\eps)$, where $\co(E_\eps)$ denotes the convex hull of $E_\eps$, the volume of $\co(E_\eps)$ is larger than $\abs{B_1}$ if $E_\eps$ is not convex. However, using the fact that a minimizer $E_\eps$ is already close to the unit ball by \cite{Peg2021} and the convexity of $\co(E_\eps)$, we prove that, if $E_\eps$ is not convex, scaling down $\co(E_\eps)$ to make its volume equal to $\abs{B_1}$ strictly decreases the energy~$\calF_{\gamma,G_\eps}$, which contradicts the minimality of $E_\eps$. The convexity of minimizers $E_\eps$ allows us to improve the convergence of $\partial E_\eps$ towards $\partial B_1$ as $\eps$ goes to $0$, from the previously known Hausdorff convergence to Lipschitz convergence. We deduce that minimizers are nearly spherical sets, whose definition is given just below. \begin{maindfn}[Nearly spherical sets]\label{maindfn:nearlysph} For $t\in (0,1/2)$, we say that $E\subsq\Ren$ is a centered $t$-nearly spherical set if \[ \int_E x\dx=0, \] and if there exists $u\in \Lip(\bbS^{n-1})$ with $\norm{u}_{L^\infty(\bbS^{n-1})}+\norm{\grad_\tau\, u}_{L^\infty(\bbS^{n-1})}\leq 1$ such that \[ \partial E=\bigl\{ \big(1+tu(x)\big)x\sep x\in\bbS^{n-1}\bigl\}. \] In dimension $n=2$, we use the terminology \enquote{$t$-nearly circular set} for \enquote{$t$-nearly spherical set}. \end{maindfn} \begin{mainthm}[2D minimizers have nearly circular boundaries; see {\cref{prp:minsph}}]\label{mainthm:minnearlycirc} Assume $n=2$, $\gamma\in(0,1)$ and~$G$ satisfies \cref{Hrad,Hint}. There exist $\eps_2=\eps_2(G,\gamma)>0$ and a function $t:(0,\eps_2)\to [0,1/2)$ depending only on $G$ and $\gamma$ such that \begin{itemize} \item $t(\eps)\to 0$ as $\eps\to 0$; \item for every $\eps<\eps_2$, any minimizer $E$ of \cref{minrp} is, up to a translation and a Lebesgue-negligible set, a centered $t(\eps)$-nearly circular set. \end{itemize} \end{mainthm} \cref{mainthm:minnearlycirc} is a direct consequence of \cref{mainthm:convexmin} and the uniform convergence of minimizers already shown in \cite{Peg2021}, using the geometric fact that the normal vectors to the boundary of a convex set lying between two balls~$B_r$ and $B_R$, $r< 1< R$ converge to those of the unit sphere as $r,R\to 1$ (see \cite{Oss1987,Fug1989}). We end the proof by showing that for $\eps$ and $t$ small enough, any centered $t$-spherical minimizer of~\cref{minrp} is the unit ball. This last result is \emph{not} specific to dimension $n=2$. \begin{mainthm}[Minimality of the unit ball among nearly spherical sets; see {\cref{prp:fuglede2}}]\label{mainthm:minballsph} Assume that $\gamma\in(0,1)$ and $G$ satisfies \crefrange{Hrad}{Hhighermoments}. Then there exist $t_*=t_*(n,G,\gamma)>0$ and $\eps_3=\eps_3(n,G,\gamma)>0$ such that, for every $t0. \] \end{prp} We also have the following convergence result, which is a consequence of \cite{Dav2002} and our choice of $I_G^1$. \begin{prp}\label{prp:limPerG} For any set of finite perimeter $E$ in $\Ren$, we have \begin{equation}\label{limPerG:res} \Per_{G_\eps}(E)~\xrightarrow{\eps\to 0}~ P(E). \end{equation} \end{prp} We will use the following computation obtained in \cite[Lem.\,3.5]{Peg2021}, which clarifies the behavior of the nonlocal perimeter under scaling. \begin{lem}\label{lem:derivpg} For any set of finite perimeter $E\subsq\Ren$, the function $t\mapsto \Per_{G_\eps}(tE)$ is locally Lipschitz continuous in $(0,+\infty)$, and for almost every $t$, we have \[ \begin{aligned} \frac{\dd}{\dt} \left[\Per_{G_\eps}(tE)\right] &= \frac{n}{t}\Per_{G_\eps}(tE)-\frac{1}{t}\,\wt{P}_{G_\eps}(tE), \end{aligned} \] where $\wt{P}_{G_\eps}(E)$ is defined by \begin{equation}\label{defReps} \wt{P}_{G_\eps}(E):= 2\int_E \int_{\partial^* E} G_\eps(x-y)\,(y-x)\cdot\nu_E(y)\dH^{n-1}_y\dx. \end{equation} \end{lem} Let us remark that in \cite{Peg2021}, $G$ is assumed to be in addition integrable in $\Ren$, but \cref{lem:derivpg} can be deduced by approximating $G$ with maps $G_k:x\mapsto \chi_k(\abs{x})G(x) \in L^1(\Ren)$. Indeed, \mbox{let $\chi_k\in C^\infty(\R^+,[0,1])$} be cutoff functions with $\chi_k(r)=0$ for \hbox{$r\leq 1/k$}, $\chi_k(r)=1$ for $r\geq \sfrac{2}{k}$, so that~$I_{G_k}^1\leq I_G^1$, $x\mapsto \abs{x}G_k(x)$ converges to $x\mapsto \abs{x}G(x)$ in $L^1(\Ren)$, and notice that \[ \abs{\Per_G(E)-\Per_{G_k}(E)} \leq P(E)\int_{\Ren} \abs{x}\abs{(G-G_k)(x)}\dx, \] and \[ \abs{\wt{\Per}_G(E)-\wt{\Per}_{G_k}(E)} \leq 2P(E)\int_{\Ren} \abs{x}\abs{(G-G_k)(x)}\dx. \] In order to study the minimality of the unit ball among nearly spherical sets, we will use the following Bourgain-Brezis-Mironescu-type result (see \cite{BBM2001}) for approximating the $H^1$ seminorm on the sphere by nonlocal seminorms. \begin{lem}\label{lem:cvsphunif} Let us define the $(n-1)$-dimensional approximation of identity $(\eta_\eps)_{\eps>0}$ by \[ \eta(t)\coloneqq 2t^2g(t),\qquad\text{ and }\qquad \eta_\eps(t)\coloneqq\eps^{-(n-1)}\eta(\eps^{-1}t), \qquad\forall t>0,~\forall \eps>0. \] When $n=2$, we assume in addition that $g$ is such that the family $(\eta_\eps)_{\eps>0}$ satisfies \begin{equation}\label{eq:Tcond} \sup_{r\in (R,2)} \eta_\eps(r)\,\xrightarrow{\eps\to 0}\, 0,\qquad\forall R\in(0,2). \end{equation} Then for any $u\in H^1(\bbS^{n-1})$, we have \begin{multline*} \iint_{\bbS^{n-1}\times\bbS^{n-1}} \frac{(u(x)-u(y))^2}{\abs{x-y}^2} \eta_\eps(\abs{x-y}) \dH_x^{n-1}\dH_y^{n-1}\\[-5pt] \leq \big(1+q_\eta(\eps)\big) \int_{\bbS^{n-1}} \abs{\grad_\tau\, u}^2\dH^{n-1}, \end{multline*} where $q_\eta(\eps)$ vanishes as $\eps$ goes to $0$, and depends only on $n$ and $G$. In addition, for any $u\in H^1(\bbS^{n-1})$, \[ \begin{aligned} \iint_{\bbS^{n-1}\times\bbS^{n-1}} \frac{(u(x)-u(y))^2}{\abs{x-y}^2} \eta_\eps(\abs{x-y}) \dH_x^{n-1}\dH_y^{n-1} \,\xrightarrow{\eps\to 0}\, \int_{\bbS^{n-1}} \abs{\grad_\tau\, u}^2\dH^{n-1}. \end{aligned} \] \end{lem} \begin{proof} One easily checks that assumptions \cref{Hrad,Hint} ensure that the family $(\eta_\eps)_{\eps>0}$ is a $(n-1)$-dimensional approximation of identity, up to multiplication by the constant $\bfK_{2,n-1}=1/(n-1)$, \ie \begin{enumeratei} \item $\displaystyle \abs{\bbS^{n-2}}\int_0^{+\infty} \eta_{\eps}(r)r^{n-2}\dr = \frac{1}{\bfK_{2,n-1}}$; \smallskip \item$\displaystyle\lim_{\eps \to 0}\, \int_\delta^{+\infty} \eta_{\eps}(r)r^{n-2} \dr= 0,\quad \forall \delta > 0$. \end{enumeratei} These properties (together with \cref{eq:Tcond} when $n=2$) allow us to apply \cite[Prop.\,A.1~\&~A.4]{Peg2021}, which gives the result. \end{proof} \begin{rmk}\label{rmk:condTHinfty} If \cref{Hhighermoments} stands true, then the condition \cref{eq:Tcond} is satisfied, in particular when $n=2$. Indeed, with \cref{Hint}, \cref{Hhighermoments} implies that $g\in W^{1,1}_\loc(0,+\infty)$ and the functions $t\mapsto t^ng(t)$ and $t\mapsto t^{n+1}g'(t)$ are integrable on $(0,+\infty)$. In addition, integrating the function $(t^{n+1}g(t))'$ between $r$ and $R$, we have the relation \[ R^{n+1}g(R)-r^{n+1}g(r)=(n+1)\int_r^R t^ng(t)\dt+\int_r^R t^{n+1}g'(t)\dt. \] Since $t^ng(t)$ and $t^{n+1}g'(t)$ are integrable on $(0,+\infty)$, this implies that $r^{n+1}g(r)$ has a limit in $0^+$ and at infinity. By the integrability of $t^n g(t)$ on $(0,+\infty)$, these limits are necessarily $0$. In particular, $r^{n+1}g(r)\xrightarrow{r\to\infty} 0$, so that \[ \eta_\eps(r)=r^2\eps^{-(n+1)}g(\eps^{-1}r)=r^{1-n}(\eps^{-1}r)^{-(n+1)}g(\eps^{-1}r) \] vanishes uniformly on $(R,2)$ as $\eps\to 0$, for every $R\in(0,2)$. \end{rmk} Eventually, gathering results from \cite[Th.\,A \& B]{Peg2021} (see also Theorem~4.16 therein), we have existence and convergence results for minimizers of \cref{minrp}. We also know that minimizers are connected for small $\eps$. Here connectedness is to be understood in a measure-theoretic sense for sets of finite perimeter, often referred to as indecomposability, as defined below (see \cite{ACMM2001}). \begin{dfn} We say that a set of finite perimeter $E$ is decomposable if there exist two sets of finite perimeter $E_1$ and $E_2$ such that $E=E_1\sqcup E_2$, $\abs{E_1}>0$, $\abs{E_2}>0$ and $P(E)=P(E_1)+P(E_2)$. Naturally, we say that a set of finite perimeter is indecomposable if it is not decomposable. \end{dfn} Let us remark that by \cite[Th.\,2]{ACMM2001}, the notion of connectedness and indecomposability coincide whenever $E$ is an \emph{open} set of finite perimeter such that $\calH^{n-1}(\partial E)=\calH^{n-1}(\partial^* E)$. \begin{thm}\label{thm:existmin} There exist $\eps_0=\eps_0(n,G,\gamma)>0$ and a function $\delta=\delta(n,G,\gamma):(0,+\infty)\to (0,1/4)$ vanishing in $0^+$ such that the following holds. For every $\eps$ such that $0<\eps<\eps_0$, \cref{minrp} admits a minimizer. In addition, any such minimizer $E_\eps$ is indecomposable, and up to a translation and a Lebesgue-negligible set, it satisfies \begin{equation}\label{existmin:doubleincl} B_{1-\delta(\eps)} \subsq E_\eps \subsq B_{1+\delta(\eps)}. \end{equation} In dimension $n=2$, any minimizer $E_\eps$ with $0<\eps<\eps_0$ is Lebesgue-equivalent to a connected set which still satisfies \cref{existmin:doubleincl}. \end{thm} \begin{proof} In \cite{Peg2021}, the kernel $G$ is assumed to be integrable in $\Ren$. However, it is actually only required for the two following reasons: first, to be able to write \cref{eq:equivgamow} and obtain the equivalence with the Gamow-type minimization problem \cref{mingam}; second, by this equivalence, to deduce that minimizers of~\cref{minrp} are so-called quasi-minimizers of the perimeter, and thus are (non-uniformly in $\eps$) $C^{1,1/2}$-regular outside a \enquote{small} singular set. Here, we do not need the equivalence with \cref{mingam} nor the \emph{a priori} regularity of minimizers. In the end, apart from the $C^{1,1/2}$ partial regularity of minimizers, all the conclusions of \cite[Th.\,A \& B]{Peg2021} follow. More precisely, there exists $\eps_0=\eps_0(n,G,\gamma)$ such that, for any $\eps$ such that $0<\eps<\eps_0$, \cref{minrp} admits a minimizer. In addition, any such minimizer $E_\eps$ is indecomposable and, up to a translation and Lebesgue-negligible set, it satisfies \[ B_{1-\delta(\eps)} \subsq E_\eps \subsq B_{1+\delta(\eps)}, \] where $\delta:(0,+\infty)\to(0,1/4)$ is a function depending only on $n$, $G$, and $\gamma$ vanishing in $0^+$. To conclude, there remains to show that in dimension $n=2$, $E_\eps$ is equivalent to a connected set with~\cref{existmin:doubleincl}, that is, to link the indecomposability of $E_\eps$ with the topological notion of connectedness. It is not a trivial question, at least without (weak) regularity results on minimizers. However, \cite[Th.\,8]{ACMM2001} shows that in dimension $2$, $\wt{E}_\eps\coloneqq \mathring{E}^M_\eps\setminus\partial^S E_\eps$ is connected, where $\mathring{E}^M_\eps$ is the measure-theoretic interior of $E_\eps$, and \[ \partial^S E_\eps\coloneqq \Bigl\{ x\in\R^2\sep \limsup_{r\to 0^+}~ \frac{\calH^{1}(\partial^* E_\eps \cap B_r(x))}{r}>0\Bigr\}. \] Since $\calH^{1}(\partial^S E_\eps\setminus\partial^* E_\eps)=0$ and $\calL^2(E_\eps\triangle \mathring{E}_\eps^M)=0$, we have $\calL^2(E_\eps\triangle\wt{E}_\eps)=0$, and since $B_{1-\delta(\eps)}\subsq\mathring{E}^M_\eps\subsq B_{1+\delta(\eps)}$ and $\partial^S E_\eps \subsq \ol{B}_{1+\delta(\eps)}\setminus B_{1-\delta(\eps)}$, $\wt{E}_\eps$ satisfies \cref{existmin:doubleincl}. \end{proof} \begin{rmk} In higher dimensions $n\geq 3$, one could show as well that any minimizer of \cref{minrp} is equivalent to a connected set for small $\eps$, without assuming that $G\in L^1(\Ren)$. Indeed, proceeding \eg as in \cite[Lem.\,5.6]{MS2019}, it is possible to obtain \emph{uniform} (with respect to $\eps$) density estimates for minimizers, and with those to deduce that any minimizer is equivalent to an open set $E_\eps$ such that $\partial E_\eps=\spt \mu_\eps$. The indecomposability of $E_\eps$ then implies connectedness by \cite{ACMM2001}. \end{rmk} \begin{rmk}\label{rmk:distball} In view of \cite[Lem.\,4.1]{Peg2021} and the proofs of Section~4.1 therein, we see that \[ \delta(\eps)^2 = C(n)\Bigl(\frac{\gamma}{1-\gamma}\Bigr)\big(P(B_1)-\Per_{G_\eps}(B_1)\big), \] and that $\eps_0$ is actually chosen so that $\delta(\eps)\leq C(n)$ for every $\eps$ such that $0<\eps<\eps_0$, for some dimensional constant~$C(n)$. A computation shows that $P(B_1)-\Per_{G_\eps}(B_1)\sim C(n,G)\eps^2$ as $\eps$ vanishes, thus \[ \delta(\eps)^2 \sim C(n,G)\Bigl(\frac{\gamma}{1-\gamma}\Bigr)\eps^2, \] and we can choose \[ \eps_0=C(n,G)\Bigl(\frac{1-\gamma}{\gamma}\Bigr)^{\sfrac{1}{2}}. \] Hence, the closer $\gamma$ is to $1$, the smaller $\eps_0$ is. \end{rmk} \section{Minimizers are nearly circular sets in dimension \texorpdfstring{$2$}{2}} \label{sec:minnearlycirc} \subsection{Decrease of the critical energy by convexification}\label{subsec:convexification} We can recover the nonlocal perimeter of a measurable set $E\subsq\Ren$ by integrating the $1$-dimensional nonlocal perimeter of all $1$-dimensional slices of $E$ in a given direction, and averaging over all the directions. \begin{prp} For any measurable set $E\subsq\R^{n}$, we have \[ \begin{aligned} \Per_{G_\eps}(E) &=\int_{\bbS^{n-1}} \int_{\{\sigma\}^\perp} \biggl(\iint_{E_{\sigma,y}\times \left(\R\setminus E_{\sigma,y}\right)} \abs{s-t}^{n-1}g_\eps(\abs{s-t})\ds\dt\biggr) \dH^{n-1}_y\dH^{n-1}_\sigma\\ &=\frac{1}{2\om_{n-1}}\int_{\bbS^{n-1}}\biggl(\int_{\{\sigma\}^\perp} \Per^{1}_{\rho_\eps}(E_{\sigma,y}) \dH^{n-1}_y\biggr)\dH^{n-1}_\sigma, \end{aligned} \] where $g_\eps\coloneqq\eps^{-(n+1)}g(\eps^{-1}\,\cdot)$, \begin{equation}\label{eq:defEst} E_{\sigma,y} := \bigl\{ s\in\R\sep y+s\sigma\in E\bigr\}, \end{equation} $\om_{n-1}$ is the volume of the unit ball in $\R^{n-1}$, and $\Per^{1}_{\rho_\eps}$ is the $1$-dimensional nonlocal perimeter in $\R$ associated with the kernel $\rho_\eps$ defined by $\rho_\eps(r)\coloneqq \om_{n-1}\abs{r}^{n-1}g_\eps(\abs{r})$ for $r\in\R\setminus\{0\}$, that is, \[ \Per^1_{\rho_\eps}(J) \coloneqq \iint_{\R\times\R} \abs{\ind_J(s)-\ind_J(t)} \rho_\eps(s-t)\ds\dt, \] for every measurable set $J\subsq\R$. \end{prp} \begin{proof} By the change of variable $y=x+r\sigma$ with fixed $x$ and Fubini's theorem, we have \[ \begin{aligned} \Per_{G_\eps}(E) &= 2\iint_{E\times E^\compl} G_\eps(x-y)\dx\dy\\ &= 2\int_{\bbS^{n-1}} \int_0^{+\infty} \int_E \ind_{E^\compl}(x+r\sigma)\, r^{n-1}g_\eps(r) \dx\dr\dH^{n-1}_\sigma\\ &=\frac{1}{\om_{n-1}}\int_{\bbS^{n-1}} \int_\R \int_E \ind_{E^\compl}(x+r\sigma)\, \rho_\eps(r)\dx\dr\dH^{n-1}_\sigma, \end{aligned} \] where we have used the definition of $\rho_\eps$ for the last equality. Then, for $\sigma$ fixed, let us make the change of variable $x=y+s\sigma$, where $y=\pi_{\{\sigma\}^\perp}(x)$ is the orthogonal projection of $x$ on $\{\sigma\}^\perp$. This yields \[ \Per_{G_\eps}(E) = \frac{1}{2}\int_{\bbS^{n-1}}\int_{\{\sigma\}^\perp} \int_\R \int_\R \ind_E\big(s\sigma+y\big)\ind_{E^\compl}\big((s+r)\sigma+y\big) \rho_\eps(r) \ds\dr\dH^{n-1}_y\dH^{n-1}_\sigma. \] Finally, using Fubini's theorem and making the change of variable $t=s+r$, where $s$ is fixed, we obtain, by definition of $E_{\sigma,y}$ and $\Per_{\rho_\eps}^1$, \begin{multline*} \Per_{G_\eps}(E) = \frac{1}{\om_{n-1}}\int_{\bbS^{n-1}}\int_{\{\sigma\}^\perp}\int_{E_{\sigma,y}} \int_{\R\setminus E_{\sigma,y}} \rho_\eps(s-t) \ds\dt\dH^{n-1}_y\dH^{n-1}_\sigma\\ =\frac{1}{2\om_{n-1}}\int_{\bbS^{n-1}}\int_{\{\sigma\}^\perp}\Per^1_{\rho_\eps}(E_{\sigma,y}) \dH^{n-1}_y \dH^{n-1}_\sigma. \end{multline*} This concludes the proof. \end{proof} \begin{rmk}\label{rmk:PerGbound} Recall that by \cite[Lem.\,3.13]{Peg2021}, $\bfK_{1,n}=\sfrac{\Gamma\left(\sfrac{n}{2}\right)}{\sqrt{\pi}\,\Gamma\left(\psfrac{n+1}{2}\right)}$, so~that, by \cref{Hint}, the kernel $\rho_\eps$ satisfies \[ \int_{\R} \abs{t}\rho_\eps(t)\dt =2\om_{n-1}\int_0^\infty t^ng_\eps(t)\dt =\frac{2\om_{n-1}I_G^1}{\abs{\bbS^{n-1}}}=\frac{2\om_{n-1}}{\bfK_{1,n}\abs{\bbS^{n-1}}} =1 =\frac{1}{\bfK_{1,1}}. \] Hence by \cref{prp:boundPerG}, we have \[ \calE^1_{\rho_\eps}(E)= P(E)-\Per^1_{\rho_\eps}(E)\geq 0 \] for every measurable set $E\subsq\R$. \end{rmk} Similarly, as a straightforward consequence of \cite[Th.\,3.103]{AFP2000} (see also Crofton's formula \cite[\S 3.16]{Mor2016} or \cite[Th.\,3.2.26]{Fed1996}), for any set of finite perimeter $E\subsq\R^{n}$, $E_{\sigma,y}$ is a one-dimensional set of finite perimeter for $\calH^{n-1}$-almost every $\sigma$ and $y$, and we have \[ P(E)=\calH^{n-1}(\partial^* E)=\frac{1}{2\om_{n-1}}\int_{\bbS^{n-1}} \int_{\{\sigma\}^\perp} P^1(E_{\sigma,y})\dH_y^{n-1}\dH^{n-1}_\sigma, \] where $P^1(E_{\sigma,y})=\calH^0(\partial^* E_{\sigma,y})$ is the standard perimeter in dimension $1$. Hence, we~have the following representation of the critical energy $\calE_{G_\eps}$. \begin{cor}\label{cor:expecrit} For any set of finite perimeter $E\subsq\Ren$, we have \[ \begin{aligned} \calE_{G_\eps}(E) &=\frac{1}{2\om_{n-1}} \int_{\bbS^{n-1}} \int_{\{\sigma\}^\perp} \calE^1_{\rho_\eps}(E_{\sigma,y}) \dH_y^{n-1}\dH^{n-1}_\sigma, \end{aligned} \] where $\calE^1_{\rho_\eps}:=P^1-\Per^1_{\rho_\eps}$, and $E_{\sigma,y}$ is given by \cref{eq:defEst}. \end{cor} We give a simple expression of the one-dimensional critical energy. of a segment $(a,b)\subsq\R$. \begin{lem}\label{lem:critenergy1} For every $a,b\in \R$ such that $a0$, then $\calE^1_{\rho_\eps}(J)\leq\calE^1_{\rho_\eps}(E)$. \end{cor} \begin{proof} Let $E\subsq\R$ with $\calL^1(E)>0$, and let $J$ be an interval containing $E$. Let $a'\leq a0$. In addition, since $E$ is bounded, $E^\compl$ has two unbounded components~${C_1=(-\infty,a)}$ and $C_{k+1}=(b,+\infty)$ (up to renumbering). By \cref{lem:crit1d,lem:critenergy1}, we have \[ \calE^1_{\rho_\eps}(E)\geq 4\int_{C_1\times C_{k+1}} \rho_\eps(s-t)\ds\dt =\calE^1_{\rho_\eps}((a,b)) \geq \calE^1_{\rho_\eps}(J).\qedhere \] \end{proof} \begin{proof}[Proof of {\cref{lem:crit1d}}] Let \[ E=\bigsqcup_{i=1}^k~ (a_i,b_i), \] with $-\infty\leq a_1a_{i+1}\}} ~ + 2\int_{C_i} \int_{\{t\,\in\,E\sep t+\infty\}=\emptyset$. By \cref{eq:p1form1}, we have \begin{equation}\label{crit1d:eq2} P^1(E)=\sum_{i=1}^k \biggl(2\int_{-\infty}^{a_i} \int_{a_i}^{+\infty}+2\int_{-\infty}^{b_i}\int_{b_i}^{+\infty} \biggr). \end{equation} Note that this holds even if $a_1=-\infty$ or $b_k=+\infty$. Let us define \[ R_i\coloneqq 2\int_{-\infty}^{a_{i+1}} \int_{a_{i+1}}^{+\infty}~-2\int_{C_i} \int_{\{t\,\in\,E\sep t>a_{i+1}\}},\qquad\forall i\in\{0,\dotsc,k\}, \] and similarly \[ L_i\coloneqq 2\int_{-\infty}^{b_i} \int_{b_i}^{+\infty}~-2\int_{C_i} \int_{\{t\,\in\,E\sep ta_{i+1}\}}\\ &=2\int_{C_i} \int_{\{t\,\in\, E^\compl\sep t>a_{i+1}\}}~ + \frac{1}{2}\,\calE^1_{\rho_\eps}(C_i). \end{aligned} \end{equation} and similarly, \begin{equation}\label{crit1d:eq5} L_i=2\int_{C_i} \int_{\{t\,\in\, E^\compl\sep ta_{i+1}\}}~ +\int_{C_i} \int_{E^\compl\,\cap\,\{t0. \] \end{prp} \begin{proof} First, recall that a bounded convex set is a set of finite perimeter, since it is Lebesgue-equivalent to an open set with Lipschitz boundary, thus $\co(E)$ is a set of finite perimeter. Then, by \cref{cor:expecrit} we have \begin{equation}\label{decconvex:eq1} \begin{split} \calE_{G_\eps}(E) &=\frac{1}{4} \int_{\bbS^1} \int_\R \calE^1_{\rho_\eps}(E_{\sigma,t}) \dt\dH^1_\sigma, \\ \calE_{G_\eps}(\co(E)) &=\frac{1}{4} \int_{\bbS^1} \int_\R \calE^1_{\rho_\eps}(F_{\sigma,t}) \dt\dH^1_\sigma, \end{split} \end{equation} where \[ E_{\sigma,t} := \bigl\{ s\in\R\sep t\sigma^\perp+s\sigma\in E\bigr\}, \quad\text{ and }\quad F_{\sigma,t} := \bigl\{ s\in\R\sep t\sigma^\perp+s\sigma\in \co(E)\bigr\}. \] Since $E$ is connected, for every $\sigma\in\bbS^1$ and $t\in\R$, the slice $F_{\sigma,t}$ is empty if and only if $E_{\sigma,t}$ is empty (this is the argument which is valid in dimension $2$ only). In addition, since $E$ and $\co(E)$ are bounded sets of finite perimeter in $\R^2$, for $\calH^1$-almost every $\sigma\in\bbS^1$ and $\calL^1$-almost every $t\in\R$, $E_{\sigma,t}$ and $F_{\sigma,t}$ are bounded sets of finite perimeter in $\R$, and for every nonempty slice, $F_{\sigma,t}$ is an interval s.t. $E_{\sigma,t}\subsq F_{\sigma,t}$ (since $F_{\sigma,t}$ is a slice of a convex set). Hence, by \cref{lem:critenergy1,lem:crit1d}, for $\calH^1$-almost every $\sigma\in\bbS^1$ and $\calL^1$-almost every $t\in\R$, there holds \[ \calE_{\rho_\eps}^1(F_{\sigma,t})\leq\calE_{\rho_\eps}^1(E_{\sigma,t}). \] In view of \cref{decconvex:eq1}, this concludes the proof. \end{proof} \subsection{Convexity of minimizers in 2D}\label{subsec:minconv} In this part, we shall use for the sake of brevity the abbreviations $\calF_{\gamma,\eps}:=\calF_{\gamma,G_\eps}$ and $\calE_{\eps}:=\calE_{G_\eps}$. Consider a connected minimizer $E_\eps\subsq\R^2$ of \cref{minrp}. Recall that $\calF_{\gamma,\eps}=(1-\gamma)P+\gamma\calE_\eps$, thus by~\cref{prp:decconvex}, $\calF_{\gamma,\eps}(\co(E_\eps))\leq\calF_{\gamma,\eps}(E_\eps)$. However, if $E_\eps$ is not convex, $\abs{\co(E_\eps)}>\abs{E_\eps}$, so $\co(E_\eps)$ is not a valid competitor for \cref{minrp}. Recalling that $B_{1-\delta(\eps)} \subsq E_\eps \subsq B_{1+\delta(\eps)}$ for small $\eps$, we have \[ B_{1-\delta(\eps)} \subsq \co(E_\eps) \subsq B_{1+\delta(\eps)}, \] and defining $t_\eps\coloneqq \sqrt{\abs{B_1}/\abs{\co(E_\eps)}}$, we see that $t_\eps\co(E_\eps)$ is a valid competitor with $(1+\delta(\eps))^{-1}0$ such that the following holds. If $E\subsq\R^2$ is a convex set such that \[ B_{1-\delta(\eps)} \subsq E \subsq B_{1+\delta(\eps)}, \] with $0<\eps<\ol{\eps}_1$, and where $\delta$ is the function given by \cref{thm:existmin}, then \[ \calF_{\gamma,\eps}(tE)< \calF_{\gamma,\eps}(E),\qquad\forall t\in({\textstyle\sfrac{1}{2}},1). \] \end{lem} As a consequence, we obtain that minimizers of \cref{minrp} are necessarily convex for small $\eps$ in \mbox{dimension $n=2$}, that is \cref{mainthm:convexmin}: \begin{proof}[Proof of {\cref{mainthm:convexmin}}] Let $E_\eps\subsq\R^2$ be a minimizer of \cref{minrp} with $0<\eps<\eps_1$, where~$\eps_1$ is to be chosen later, smaller than $\eps_0(G,\gamma)$ and $\ol{\eps}_1(G,\gamma)$ given respectively by \cref{thm:existmin} and \cref{lem:decconvscale}. Thus, up to a Lebesgue-negligible set and a translation, $E_\eps$ is connected and satisfies \[ B_{1-\delta(\eps)}\subsq E_\eps\subsq \co(E_\eps)\subsq B_{1+\delta(\eps)}. \] By contradiction, let us assume that $E_\eps$ is not convex, so that $\abs{\co(E_\eps)}>\abs{B_1}$. We have ${\abs{t_\eps\co(E_\eps)}=\abs{B_1}}$ with $t_\eps\coloneqq\sqrt{\abs{B_1}/\abs{\co(E_\eps)}}$. Choosing $\eps_1=\eps_1(G,\gamma)$ small enough, we have $1/2\leq(1+\delta(\eps))^{-1}0$, there exists $\ol{\eps}_3=\ol{\eps}_3(G,\gamma,\tau)>0$ such that the following holds. If $E\subsq\R^2$ is a convex set such that \[ B_{1-\delta(\eps)} \subsq E \subsq B_{1+\delta(\eps)}, \] with $0<\eps<\ol{\eps}_3$, where the $\delta$ function is the one from \cref{thm:existmin}, then we have \[ (1-\tau)P(E)\leq\wt{P}_{G_\eps}(E)\leq P(E), \] where $\wt{P}_{G_\eps}(E)$ is defined by \cref{defReps}. \end{lem} \begin{proof} Let $E$ be as in the statement of the lemma, with $0<\eps<\ol{\eps}_3(G,\gamma,\tau)$, where~$\ol{\eps}_3$ is to be fixed later. Recall that since $E$ is a bounded convex set, it is a set of finite perimeter and its topological boundary is $\calH^1$-equivalent to $\partial^* E$. We proceed in two steps. \subsubsection*{Step 1. Upper bound} First, note that by convexity of $E$, we have \[ (x-y)\cdot\nu_E(x)\geq 0,\qquad\text{ for }\calH^1\text{-almost every }x\in\partial E,\text{ and every } y\in E, \] and, defining for $\calH^1$-almost every $x\in\partial E$, \[ H_x:= \bigl\{ y\in \Ren \sep (x-y)\cdot \nu_E(x)>0\bigr\}, \] it holds \[ \wt{P}_{G_\eps}(E)\leq 2\int_{\partial E} \int_{H_x} G_\eps(x-y) (x-y)\cdot \nu_E(x)\dH_x^1\dy. \] Then, letting $g_\eps=\eps^{-3}g(\eps^{-1}\,\cdot)$, by a change of variable and the coarea formula, we~find \[ \begin{aligned} \wt{P}_{G_\eps}(E) &\leq 2\int_{\partial E} \int_{\left\{z\,\in\,\R^2\sep z\cdot\nu_E(x)\,>\,0\right\}} G_\eps(z)\big(z\cdot\nu_E(x)\big)\dz\dH_x^1\\ &=2\int_{\partial E}\int_0^{+\infty} g_\eps(t)t^2\biggl(\int_{\left\{\sigma\,\in\,\bbS^1\sep \sigma \cdot\nu_E(x)\,>\,0\right\}} \sigma\cdot\nu_E(x)\dH^1_\sigma\biggr)\dt\dH^1_x\\ &=\int_{\partial E}\int_0^{+\infty} g_\eps(t)t^2\left(\int_{\bbS^{1}} \abs{\sigma\cdot\nu_E(x)}\dH_\sigma^1\right)\dt\dH_x^1\\ &=\bfK_{1,2}\int_{\partial E}\abs{\bbS^1}\int_0^{+\infty} g_\eps(t)t^2\dt\dH_x^1 =I_G^1\bfK_{1,2}P(E)=P(E), \end{aligned} \] where we also used the definition of $\bfK_{1,2}$ for the third equality, and \cref{Hint} for the last one. \subsubsection*{Step 2. Lower bound} Let us first recall that by assumption \cref{Hint}, \begin{equation}\label{id_PE} \begin{aligned} P(E) &=4\int_{\partial E}\int_0^{+\infty} t^2g_\eps(t)\dt\dH^1\\ &=2\int_{\partial E} \int_0^{+\infty} t^2g_\eps(t)\int_{\left\{\sigma\,\in\,\bbS^1\sep \sigma\cdot e\,>\,0\right\}} (\sigma\cdot e) \dH^1_\sigma\dt\dH^1, \end{aligned} \end{equation} for all $ e\in\bbS^1$. Let $\alpha\in(0,1)$ to be chosen later, and $\theta_0\coloneqq \arcsin(\alpha)$. Then let $r=r(\alpha)>0$ and~$\ol{\eps}_2(G,\gamma,\alpha)$ given by \cref{lem:minalmflat}. Assume that $\ol{\eps}_3<\ol{\eps}_2$. Let us write \[ \begin{aligned} \wt{P}_{G_\eps}(E) &=2\int_{\partial E} \int_{E\,\cap\,\{\abs{x-y}\,<\,r\}} G_\eps(x-y)\,(x-y)\cdot\nu_E(x)\dy\dH^1_x\\ &\hphantom{=\int_{\partial E} \int_E} +2\int_{\partial E} \int_{E\,\cap\,\{\abs{x-y}\,\geq\, r\}} G_\eps(x-y)\,(x-y)\cdot\nu_E(x)\dx\dH^1_y\\ &\eqqcolon \wt{P}_{G_\eps}^{(1)}(E)+\wt{P}_{G_\eps}^{(2)}(E). \end{aligned} \] Since $\eps<\ol{\eps}_2$, by \cref{lem:minalmflat}, we have \[ \begin{aligned} \wt{P}_{G_\eps}^{(1)}(E) &\geq 2\int_{\partial E} \int_{\left\{y\,\in\, B_r(x)\sep \frac{x-y}{\abs{x-y}}\cdot\nu_E(x)\,\geq\,\alpha\right\}} G_\eps(x-y)(x-y)\cdot\nu_E(x)\dy\dH_x^1\\ &=2\int_{\partial E} \int_0^{+\infty} g_\eps(t)t^2 \int_{\left\{\sigma\,\in\,\bbS^1\sep \alpha\,\leq\,\sigma\cdot \nu_E(x)\,\leq\, 1\right\}} \sigma\cdot\nu_E(x)\dH^1_\sigma\dt\dH^1_x. \end{aligned} \] We compute\vspace*{-3pt} \begin{multline*} \int_{\{\sigma\,\in\,\bbS^1\sep \alpha\,\leq\,\sigma\cdot\nu_E(x)\,\leq\, 1\}} \sigma\cdot\nu_E(x)\dH^1_\sigma =2\int_0^{\arccos\alpha} \cos\theta\dd\theta =2\sin(\arccos\alpha)\\[-5pt] =\sqrt{1-\alpha^2}. \end{multline*} Together with \cref{id_PE}, this leads do \[ \begin{aligned} \wt{P}_{G_\eps}^{(1)}(E) \geq \sqrt{1-\alpha^2}P(E). \end{aligned} \] Choosing $\alpha$ small enough, depending only on $\tau$, we then find \[ \wt{P}_{G_\eps}^{(1)}(E)\geq \left(1-\sfrac{\tau}{2}\right)P(E). \] Since $\alpha=\alpha(\tau)$, we have $r=r(\tau)$, and $\ol{\eps}_2=\ol{\eps}_2(G,\gamma,\tau)$. Now with $r$ fixed, up to choosing $\ol{\eps}_3<\ol{\eps}_2$ even smaller if needed, there holds \[ \abs{\wt{P}_{G_\eps}^{(2)}(E)} \leq P(E)\int_{B_r^\compl} \abs{x}G_\eps(x)\dx = P(E)\int_{B_{r/\eps}^\compl} \abs{x}G(x)\dx \leq \frac{\tau}{2}\,P(E), \] since the first moment of $G$ is finite. Hence, from our choice of $r$ and $\ol{\eps}_3$, we obtain \[ \wt{P}_{G_\eps}(E)\geq (1-\tau)P(E), \] which concludes the proof. \end{proof} We are now in position to prove \cref{lem:decconvscale}. \begin{proof}[Proof of {\cref{lem:decconvscale}}] Let $\tau>0$ to be chosen later, and let $\ol{\eps}_1=\ol{\eps}_3(G,\gamma,\tau)/4$, where~$\ol{\eps}_3$ is given by \cref{lem:mincone}. Assume that $0<\eps<\ol{\eps}_1$. Recall that by \cref{lem:derivpg}, for almost every $t$, we have \begin{equation}\label{decconvscale:eq1} \begin{aligned} \frac{\dd}{\dt} \left[\Per_{G_\eps}(tE)\right] &= \frac{2}{t}\Per_{G_\eps}(tE)-\frac{1}{t}\,\wt{P}_{G_\eps}(tE), \end{aligned} \end{equation} where $\wt{P}_{G_\eps}$ is defined by \cref{defReps}. Changing variables, we deduce \[ \begin{aligned} \frac{1}{t}\wt{P}_{G_\eps}(tE) &=2t^3\int_E\int_{\partial E} G_\eps(t(x-y))(y-x)\cdot\nu_E(y)\\ &=2\int_E\int_{\partial E} G_{(t^{-1}\eps)}(x-y)(y-x)\cdot\nu_E(y) \dH_y^1\dx =\wt{P}_{G_{t^{-1}\eps}}(E). \end{aligned} \] Let $\tau\coloneqq(1-\gamma)/(2\gamma)$. By our choice of $\ol{\eps}_1$, $t^{-1}\eps<\ol{\eps}_3$, thus by~\cref{lem:mincone}, we have \[ \frac{1}{t}\wt{P}_{G_\eps}(tE)\geq (1-\tau)P(E). \] With \cref{decconvscale:eq1} and \cref{prp:boundPerG}, this leads to \[ \frac{\dd}{\dt} \left[\Per_{G_\eps}(tE)\right] \leq \frac{2}{t}\,P(tE)-(1-\tau)P(E) = (1+\tau)P(E). \] Hence, for every $t\in(1/2,1)$, by our choice of $\tau$, it follows\vspace*{-3pt} \begin{multline*} \calF_{\gamma,\eps}(E)-\calF_{\gamma,\eps}(tE) =(1-t) P(E)-\gamma\int_t^1 \frac{\dd}{\dd s} \left[\Per_{G_\eps}(sE)\right] \dd s\\ \geq (1-t)P(E)\big[1-\gamma(1+\tau)\big]>0. \end{multline*} This proves the lemma. \end{proof} \subsection{Convex minimizers are nearly spherical sets}\label{subsec:convnearlysph} In arbitrary dimension, it is classical to improve Hausdorff convergence of the boundary of minimizers to Lipschitz convergence once convexity is established. Here, we show that convex minimizers of \cref{minrp} are centered $t(\eps)$-nearly spherical sets (see \cref{maindfn:nearlysph}), up to a translation, where the function $t$ vanishes in $0^+$. \begin{prp}[Convex minimizers are nearly spherical sets]\label{prp:minsph} There exists $\ol{\eps}_4=\ol{\eps}_4(n,G,\gamma)>0$ such that the following holds. If $E_\eps\subsq\Ren$ is a convex minimizer of \cref{minrp} with $0<\eps<\ol{\eps}_4$, then, up to a translation, we have \[ \partial E_\eps=\bigl\{ (1+u_\eps(x))x \sep x\in\bbS^{n-1} \bigr\} \] and \[ \int_{E_\eps} x\dx = 0, \] with $u_\eps\in \Lip(\bbS^{n-1})$, $\norm{u_\eps}_{L^\infty(\bbS^{n-1})}\leq C'\delta(\eps)$ and $\norm{\grad_\tau\, u_\eps}_{L^\infty(\bbS^{n-1})}\leq C''\delta(\eps)^{1/2}$, where $\delta=\delta(n,G,\gamma)$ is the function of \cref{thm:existmin} vanishing in $0^+$, and $C',C''>0$ only depend on $n$. \end{prp} \begin{proof} In the proof, we write $\norm{\cdot}_\infty$ for $\norm{\cdot}_{L^\infty(\bbS^{n-1})}$. Let $E_\eps$ be a convex minimizer \cref{minrp} with $0<\eps<\ol{\eps}_4$, where $\ol{\eps}_4$ is to be fixed later. If $\ol{\eps}_4<\eps_0$, where $\eps_0$ is given by \cref{thm:existmin}, up to a translation and a negligible set, $E_\eps$ lies between the balls $B_{1-\delta(\eps)}$ and $B_{1+\delta(\eps)}$. By convexity, this implies that the set $E_\eps$ itself (without the addition or the subtraction of a negligible set) satisfies, up to a translation, \[ B_{1-\delta(\eps)}\subsq E_\eps \subsq \ol{B}_{1+\delta(\eps)}. \] Setting $\displaystyle y_\eps:=-\frac{1}{\abs{B_1}}\int_{E_\eps}x\dx$, we have \[ \int_{y_\eps+E_\eps} x\dx =y_\eps\abs{E_\eps}+\int_{E_\eps} x\dx=y_\eps\abs{B_1}+\int_{E_\eps}x\dx=0. \] Notice that \[ -\int_{E_\eps} x\dx=-\int_{B_{1+\delta(\eps)}}x\dx+\int_{B_{1+\delta(\eps)}\setminus E_\eps} x\dx =\int_{B_{1+\delta(\eps)}\setminus E_\eps} x\dx, \] thus \begin{multline*} \abs*{\int_{E_\eps} x\dx}\leq (1+\delta(\eps))\abs{B_{1+\delta(\eps)}\setminus E_\eps} \leq \big(1+\delta(\eps)\big)\left[\big(1+\delta(\eps)\big)^n-\big(1-\delta(\eps)\big)^n\right] \abs{B_1}\\ \leq C(n)\delta(\eps) \end{multline*} provided that $\ol{\eps}_4$ is small enough depending only on $n$, $G$ and $\gamma$. Hence, up to translating $E_\eps$ by $y_\eps$, we may assume that it satisfies \begin{equation}\label{minspher:eq1} B_{1-C'\delta(\eps)}\subsq E_\eps \subsq B_{1+C'\delta(\eps)}, \end{equation} with $C'\coloneqq C(n)+1$ and is centered, that is, \[ \int_{E_\eps} x\dx=0. \] By convexity of $E_\eps$, for every $x\in\bbS^{n-1}$, there is a unique point of intersection $p_\eps(x)=t_\eps(x) x$ of~${\{tx\sep t>0\}}$ and $\partial E_\eps$, and by \cref{minspher:eq1}, $\abs{p_\eps(x)-x}\leq C'\delta(\eps)$. The map $p_\eps: \bbS^{n-1}\to\partial E_\eps$ is obviously onto, so that, setting $u_\eps(x)=t_\eps(x)-1$, we have \[ \partial E_\eps=\bigl\{ (1+u_\eps(x))x \sep x\in\bbS^{n-1} \bigl\}, \] and $\norm{u_\eps}_\infty\leq C'\delta(\eps)$. In addition, the fact that $E_\eps$ is convex implies $u_\eps\in \Lip(\bbS^{n-1})$. Moreover, for any~$x\in\bbS^{n-1}$, the distance between the normal vector of $\partial E_\eps$ (which exists for $\calH^{n-1}$-almost every $x\in\partial E_\eps$) at $p_\eps(x)$ and the vector $x$ (which is the normal vector of $\bbS^{n-1}$ at $x$) is controlled by~$\norm{p_\eps-\id}_{\infty}=\norm{u_\eps}_{\infty}$, in view of Step 1 of the proof of \cref{lem:minalmflat} (or simply by \cite[Cor.\,1]{Oss1987}), which gives a control of $\norm{\grad_\tau\, u_\eps}_{\infty}$ by~$\norm{u}_{\infty}$. More precisely, by \cite[Ineq.\,$(**)$]{Fug1989}, we have \[ \norm{\grad_\tau\, u_\eps}_{\infty} \leq 2\Bigl(\frac{1+\norm{u_\eps}_{\infty}}{1-\norm{u_\eps}_{\infty}} \Bigr) \norm{u_\eps}_{\infty}^{\sfrac{1}{2}} \leq C''\delta(\eps)^{\sfrac{1}{2}}, \] where we used the inequality $\norm{u_\eps}_\infty\leq C'\delta(\eps)\leq 1/2$ for the last inequality, provided that $\ol{\eps}_4(n,G,\gamma)$ is chosen small, and defined $C''\coloneqq 6\sqrt{C'}$. This concludes the proof. \end{proof} Observe that \cref{mainthm:minnearlycirc} is an immediate corollary of \cref{mainthm:convexmin,prp:minsph}. \begin{rmk}\label{rmk:depeps12} Let us comment on the dependence of the constants $\eps_1$ and $\eps_2$ respectively of \cref{mainthm:convexmin} and \cref{mainthm:minnearlycirc} with respect to the parameter $\gamma$. Tracking the constants, $\tau=(1-\gamma)/(2\gamma)$ in the proof of \cref{lem:decconvscale}, and $\alpha=C \min(1,\sqrt{\tau})$ in the proof of \cref{lem:mincone}, for some universal constant $C$. In \cref{lem:minalmflat}, $r=C\min(1,\alpha)$, and $\ol{\eps}_2$ is chosen so that $\delta(\eps)\leq C\min(1,r^2)$ for every $\eps$ such that $0<\eps<\ol{\eps}_2$. Combining all these with the asymptotic behavior of $\delta(\eps)$ given in \cref{rmk:distball}, we deduce that we can choose \[ \ol{\eps}_2=C(n,G)\min\Big(1,\Big(\frac{1-\gamma}{\gamma}\Big)^{\sfrac{3}{2}}\Big). \] Then, $\ol{\eps}_3$ of \cref{lem:mincone} must be smaller than $\ol{\eps}_2$ and satisfy \[ \int_{B_{r/\ol{\eps}_2^\compl}} \abs{x}G(x)\dx\leq \frac{\tau}{2}. \] Assuming that $G(x)=O(\abs{x}^{-(n+1+\beta)})$ at infinity for some positive constant $\beta$, and using that $r$ is chosen to be $C\min\big(1,\sqrt{(1-\gamma)/\gamma}\big)$, we can choose \[ \ol{\eps}_3=C(n,G)\min\Big(1,\Big(\frac{1-\gamma}{\gamma}\Big)^{\max\left(\sfrac{3}{2},\sfrac{1}{2} +\sfrac{1}{\beta}\right)}\Big). \] In \cref{lem:decconvscale}, $\ol{\eps}_1=\ol{\eps}_3/4$, and in \cref{mainthm:convexmin}, $\eps_1$ is simply chosen smaller than $\ol{\eps}_1$. Notice that in \cref{prp:minsph}, $\ol{\eps}_4=C(n,G)\sqrt{(1-\gamma)/\gamma}$ in view of \cref{rmk:distball}, and that $\eps_2$ of \cref{mainthm:minnearlycirc} must be smaller than $\min(\eps_1,\ol{\eps}_4)$ and such that $\delta(\eps)^{1/2}< C(n)$. In the end, $\eps_1$ and $\eps_2$ can be chosen to be \[ \eps_i=C(n,G)\min\Big(1,\Big(\frac{1-\gamma}{\gamma}\Big)^{\max\left(\sfrac{3}{2},\sfrac{1}{2} +\sfrac{1}{\beta}\right)}\Big),\quad i\in\{1,2\}. \] \end{rmk} \section{Minimality of the unit ball among nearly spherical sets}\label{sec:fuglede} This section is devoted to the proof of the minimality of the unit ball of $\Ren$ among $t$-nearly spherical sets, for small $t$ and $\eps$. \subsection{A Fuglede-type result for the nonlocal perimeter} For a centered nearly spherical set $E_t$, we wish to estimate the quantity $\calF_{\gamma,\eps}(E_t)-\calF_{\gamma,\eps}(B_1)$ from below. Thus, we need an estimate of $P(E_t)-P(B_1)$ from below, and an estimate of $P_\eps(E_t)-P(B_1)$ from above. For the former we use the following lower bound. \begin{lem}\label{lem:fugledeper} There exist positive constants $\ol{t}=\ol{t}(n)$ and $C=C(n)$ such that the following holds. For any centered $t$-nearly spherical set $E_t$ with $00$ and $\eps>0$. For $x,y\in\partial B$ fixed, making the changes of variables $a=u(y)+r(u(x)-u(y))$, and $b=u(x)+\rho(u(y)-u(x))$ in \cref{fuglede:eq3b} yields, with~\cref{fuglede:defPhi,fuglede:eq4d}, \begin{multline*} I_2(t) \leq Ct \int_0^1 \iint_{\partial B\times\partial B}\int_0^1\int_0^1 (u(x)-u(y))^2 k_\eps(\abs{\Phi_{stu}(x,y,r,\rho)})\\\dr\dd\rho\dH^{n-1}_x\dH^{n-1}_y\ds. \end{multline*} By \cref{Hhighermoments}, $I_{k}^1=I_G^2<\infty$ and $k(r)=O(r^{-(n+1)})$ at infinity, so the family $(k_\eps)_{\eps>0}$ satisfies the assumptions of~\cref{lem:techlem1}. Thus choosing $t_00$. For $v\in\Lip(\bbS^{n-1})$, we~define a map~$\Phi_v:\left(\bbS^{n-1}\right)^2\times (0,1)^2$ by \begin{equation}\label{eq:defPhi} \Phi_v(x,y,r,\rho)=(x-y)+\Big[\big(rv(x)+(1-r)v(y))\big)x-\big(\rho v(y)+(1-\rho)v(x)\big)y\Big]. \end{equation} Then there exist positive constants $\ol{t}_1=\ol{t}_1(n)$, $\ol{C}_0=\ol{C}_0(n,C_0,I_k^1)$ and $\ol{\eps}_6=\ol{\eps}_6(n,R_0)$ such that the following holds. For any $u\in\Lip(\bbS^{n-1})$ such that $\norm{u}_{L^\infty(\bbS^{n-1})}\leq 1$, $\norm{\grad_\tau\, u}_{L^\infty(\bbS^{n-1})}\leq 1$, and any $t\in(0,\ol{t}_1)$, we have\enlargethispage{3pt}% \begin{multline}\label{techlem1:skernel} \iint_{\bbS^{n-1}\times\bbS^{n-1}} \int_0^1\int_0^1 (u(x)-u(y))^2 k_\eps(\abs{\Phi_{tu}(x,y,r,\rho)})\dr\dd\rho\dH^{n-1}_x\dH^{n-1}_y\\ \leq \ol{C}_0\biggl(\int_{\bbS^{n-1}} \abs{\grad_\tau\, u}^2\dH^{n-1}+\int_{\bbS^{n-1}} \abs{u}^2\dH^{n-1} \biggr). \end{multline} \end{lem} \begin{proof} Let $\ol{t}_1>0$ to be chosen later. In the proof, we assume that $00$, provided $t<\ol{t}_1$, and $\ol{t}_1$ is chosen small enough depending only on $n$ and~$M$, and $C=C(n,I_k^1)$. Here, by a slight abuse of notation, $\Phi_v$ denotes the map from $\left(\R^{n-1}\right)^2\times(0,1)^2$ whose expression is given by \cref{eq:defPhi}. For $v\in\Lip(D_2)$, let us define the map $E_v:\left(\R^{n-1}\right)^2\times (0,1)^2$ by \[ E_v(x,y,r,\rho) \coloneqq \big(rv(x)+(1-r)v(y))\big)x-\big(\rho v(y)+(1-\rho)v(x)\big)y, \] so that $\Phi_v(x,y,r,\rho)=x-y+E_v(x,y,r,\rho)$. Due to the $L^\infty$ bound on $\wt{u}$ and $\grad \wt{u}$, we~easily see that the maps~$E_{t\wt{u}}$ converge uniformly to $0$ on $(D_2)^2\times (0,1)^2$ as $t$ vanishes. In fact, we have\vspace*{-1pt} \begin{equation}\label{techlem1:eq1} \begin{aligned} |E_{t\wt{u}}&(x,y,r,\rho)|\\ &\leq t\left(2\norm{\wt{u}}_{L^\infty(D_2)}+\norm{\grad\wt{u}}_{L^\infty(D_2)}\right) \abs{x-y}\\ &\leq 2M\ol{t}_1\abs{x-y} \end{aligned} \qquad\forall x,y\in D_2,\,\forall r,\rho\in(0,1). \end{equation} In particular, choosing $\ol{t}_1<1/(4M)$,\vspace*{-2pt} \begin{equation}\label{techlem1:eq2} \frac{\abs{x-y}}{2}\leq \abs{\Phi_{t\wt{u}}(x,y,r,\rho)} \leq 2\abs{x-y},\quad\forall x,y\in D_2,\,\forall r,\rho\in(0,1). \end{equation} Integrating on lines, using Cauchy-Schwarz inequality and Fubini's theorem, we have\vspace*{-5pt} \begin{multline}\label{techlem1:eq3} k_\eps(\abs{\Phi_{t\wt{u}}(x,y,r,\rho)})\dr\dd\rho\dx\dy\\ \quad\leq \int_0^1 \int_0^1\int_0^1 \int_{D_2}\int_{D_2} \abs{\grad\wt{u}(x+s(y-x))}^2 \abs{x-y}^2 k_\eps(\abs{\Phi_{t\wt{u}}(x,y,r,\rho)})\dx\dy\dd s\dr\dd\rho. \end{multline} Now let us focus on the integral on $D_2\times D_2$, fixing $s\in(0,1/2)$ and $r,\rho\in(0,1)$. By~\cref{techlem1:eq2}, our choice of~$\ol{t}_1$, and the fact that $01/4\right\}$ and $\left\{x,y\in\bbS^{n-1}\sep \abs{x-y}\leq 1/4\right\}$. We first treat the contribution of distant pairs $(x,y)$. Choosing $\ol{t}_1$ small enough depending only on $n$, we have \[ \abs{\Phi_{tu}(x,y,r,\rho)}\geq \frac{\abs{x-y}}{2},\qquad\forall x,y\in\bbS^{n-1},\,\forall r,\rho\in(0,1), \] so that, by the squared triangle inequality and by symmetry, \begin{multline}\label{techlem2:eq5b} \iint\limits_{\substack{\bbS^{n-1}\times\bbS^{n-1}\\\{\abs{x-y}>\sfrac{1}{4}\}}} \int_0^1\int_0^1 (u(x)-u(y))^2 k_\eps(\abs{\Phi_{tu}(x,y,r,\rho)})\dr\dd\rho\dH^{n-1}_x\dH^{n-1}_y\\\begin{aligned} &\leq 2\int_{\bbS^{n-1}} \abs{u(x)}^2 \biggl(\int\limits_{\substack{\bbS^{n-1}\\\{\abs{x-y}>\sfrac{1}{4}\}}} \int_0^1\int_0^1 k_\eps(\abs{\Phi_{tu}(x,y,r,\rho)})\dr\dd\rho\dH^{n-1}_y \biggr) \dH_x^{n-1}\\ &\leq 2\abs{\bbS^{n-1}} \biggl(\int_{\bbS^{n-1}} \abs{u}^2\dH^{n-1}\biggr)\Bigl(\sup_{r>\sfrac{1}{8}}~ k_\eps(r)\Bigr). \end{aligned} \end{multline} Then, choosing $\ol{\eps}_6=1/(8R_0)$, and using that $k(r)\leq C_0r^{-(n+1)}$ for all $r\in(R_0,+\infty)$, for any $\eps$ such that $0<\eps<\ol{\eps}_6$, \cref{techlem2:eq5b} implies \begin{multline}\label{techlem2:eq6} \iint\limits_{\substack{\bbS^{n-1}\times\bbS^{n-1}\\\{\abs{x-y}>\sfrac{1}{4}\}}} \int_0^1\int_0^1 (u(x)-u(y))^2 k_\eps(\abs{\Phi_{tu}(x,y,r,\rho)})\dr\dd\rho\dH^{n-1}_x\dH^{n-1}_y\\[-10pt] \leq C\int_{\bbS^{n-1}} \abs{u}^2\dH^{n-1}, \end{multline} for some $C=C(n,C_0)$. There remains to estimate the integral over $\mathcal{M}\coloneqq\left\{x,y\in\bbS^{n-1}\sep \abs{x-y}<1/4\right\}$. For this, we cover $\mathcal{M}$ by a finite number $N(n)$ of~$\mathcal{M}_i\coloneqq \bbS^{n-1}_+(x_i)\times \bbS^{n-1}_+(x_i)$, where for each $i\in\{1,\dotsc,N\}$, $x_i\in\bbS^{n-1}$, and $\bbS^{n-1}_+(x_i)$ denotes the hemisphere with center~$x_i$. Using the stereographic projection $\Pi_i$ with respect to $-x_i$, we map $\bbS^{n-1}_+(x_i)$ to $D_2\subsq \R^{n-1}$. By the changes of variables $\xi=\Pi_i(x)$, $\zeta=\Pi_i(y)$, setting $\wt{u}_i\coloneqq u\circ\Pi_i^{-1}$, we~have $\norm{\wt{u}_i}_{L^\infty(D_2)}+\norm{\grad\wt{u}_i}_{L^\infty(D_2)}\leq C(n)$ since $\norm{u}_{L^\infty(\bbS^{n-1})}\leq 1$ and $\norm{\grad_\tau\, u}_{L^\infty(\bbS^{n-1})}\leq 1$. Applying Step~1 with $\wt{u}=\wt{u}_i$, for any $\eps>0$, we obtain \[ \begin{aligned} &\iint_{\mathcal{M}_i}\int_0^1\int_0^1 (u(x)-u(y))^2 k_\eps(\abs{\Phi_{tu}(x,y,r,\rho)})\dr\dd\rho\dH_x^{n-1}\dH_y^{n-1}\\ &\hphantom{\iint_{\mathcal{M}_i}} \leq C\iint_{D_2\times D_2}\int_0^1\int_0^1 (\wt{u}_i(\xi)-\wt{u}_i(\zeta))^2 k_\eps(\abs{\Phi_{t\wt{u}_i}(\xi,\zeta,r,\rho)})\dr\dd\rho\dd\xi\dd\zeta\\ &\hphantom{\iint_{\mathcal{M}_i}} \leq C\int_{D_2} \abs{\grad \wt{u}_i}^2\dx \leq C\int_{\bbS^{n-1}_+(x_i)} \abs{\grad_\tau\, u}^2\dH^{n-1}, \end{aligned} \] whenever $t<\ol{t}_1$, where $\ol{t}_1=\ol{t}_1(n)$ and $C=C(n,I_k^1)$. Summing these estimates over $i\in\{1,\dotsc,N(n)\}$ and using \cref{techlem2:eq6}, we conclude the proof. \end{proof} \backmatter \bibliographystyle{jepplain+eid} \bibliography{merlet-pegon} \end{document}