1. Introduction
The goal of the classical cluster problem in 
$\mathbb R^d$ is to find a configuration of 
$\textit{N}$ regions with fixed finite 
$\textit{d}$-dimensional volumes and an exterior region of infinite volume so that the total surface measure of the interfaces between the regions is minimal (see, for example, [Reference Maggi19, Part IV]). For 
$N \leqslant d + 1$ and for any given collection of 
$\textit{N}$ finite volumes, the standard N-bubble is conjectured to be the unique minimizer (up to isometries) of the cluster problem. If 
$\textit{N}$ = 1, then the problem reduces to the isoperimetric problem. For any given pair of volumes, the double bubble problem (i.e. when 
$\textit{N}$ = 2) has been studied extensively: Foisy, Alfaro, Brock, Hodges, and Zimba proved the minimality of the standard double bubble in 
$\mathbb R^2$ [Reference Foisy, Alfaro, Brock, Hodges and Zimba12]; this was extended to 
$\mathbb R^3$ by Hutchings, Morgan, Ritoré, and Ros [Reference Hutchings, Morgan, Ritoré and Ros16]; to 
$\mathbb R^4$ by Reichardt, Heilmann, Lai, and Spielman [Reference Reichardt, Heilmann, Lai and Spielman26], and to any 
$\mathbb R^d$ by Reichardt [Reference Reichardt27]. For 
$\textit{N}$ = 3 Wichiramala [Reference Wichiramala28] proves that triple bubbles are the isoperimetric clusters in 
$\mathbb R^2$. Recently, Milman and Neeman confirmed the double bubble conjectures for 
$d \geqslant 2$, the triple bubble conjectures for 
$d \geqslant 3$, and the quadruple bubble conjectures for 
$d \geqslant 4$ in [Reference Milman and Neeman20].
A variant of the classical cluster problem is to characterize locally isoperimetric 
$\textit{N}$-partitions with more than one region having infinite volume. Such partitions divide the space into 
$\textit{N}$ regions with prescribed (finite or infinite) volume and locally minimize the surface measure of the interfaces with respect to all compactly supported variations that also preserve the volume of each region. When two (or more) regions have infinite volume the measure of their interface is also infinite. Therefore, for such partitions, one needs to consider locally minimizing configurations in the following sense: For every ball 
${B}_{R}$ of radius R the perimeter of the partition in the interior of 
${B}_{R}$ is minimal among all partitions with regions of the same prescribed volumes as the original partition, but whose difference, in the set-theoretic sense, with the corresponding regions of the original partition, is compactly contained in 
${B}_{R}$.
The study of locally isoperimetric partitions with more than one infinite region has only recently been initiated by Alama, Bronsard, and Vriend [Reference Alama, Bronsard and Vriend5]. They characterize the unique locally isoperimetric partition of the plane into three regions with one region of given fixed area and the other two having infinite area, as the standard lens partition. Novaga, Paolini, and Tortorelli [Reference Novaga, Paolini and Tortorelli21] further this study by obtaining a general closure theorem for limits of sequences of locally isoperimetric partitions, showing that they are themselves local minimizers, provided that they have flat interfaces outside some compact set. This enables them to identify several locally isoperimetric partitions in 
$\mathbb R^d$. In two dimensions they prove that any planar locally isoperimetric partition has at most 3 chambers with infinite area. They also give a complete characterization of planar local minimizers in the case the total number of finite and infinite regions does not exceed 4 as the lens partition (1 finite, 2 infinite regions), the peanut (2 finite, 2 infinite regions), or the Reuleaux triangle (1 finite, 3 infinite regions), where the last two were conjectured to be local minimizers in [Reference Alama, Bronsard and Vriend5] (see Figure 1). Finally, in [Reference Bronsard and Novack6], Bronsard and Novack study a partitioning of 
$\mathbb R^d$ into 1 finite and 2 infinite regions where the surface measure between different pairs of regions is computed with respect to some given weights. After establishing that the standard weighted lens cluster is locally minimizing under the standard positivity and triangularity conditions on the weights, they also prove its uniqueness under some additional symmetry and growth assumptions on the weights.
Some planar locally isoperimetric partitions: the standard double bubble, the standard lens, the peanut, and the Reuleaux triangle. All the highlighted angles are 120 degree angles.
In this paper we are interested in the stability of locally isoperimetric partitions in the spirit of the quantitative isoperimetric inequality, where a suitable distance of a set from a ball of the same volume is controlled in terms of the difference in the perimeter of the set and the perimeter of the ball (see [Reference Fusco13, Reference Maggi18] for two excellent reviews). For clusters, the only stability result that we are aware of is by Cicalese, Leonardi, and Maggi [Reference Cicalese, Leonardi and Maggi11] where they obtain the stability of the planar standard double bubble. Closely related is the proof by Caroccia and Maggi in [Reference Caroccia and Maggi7] of a quantitative version of the minimality of the honeycomb tiling of the plane. The approach adopted in both results is based on the selection principle devised by Cicalese and Leonardi [Reference Cicalese and Leonardi9] and utilizes, as an essential tool, an improved convergence theorem for bubble clusters by Cicalese, Leonardi and Maggi [Reference Cicalese, Leonardi and Maggi10, Reference Leonardi and Maggi23]. Similar ideas were also developed independently by Fusco and Morini in [Reference Fusco and Morini15].
We follow a similar strategy and prove in our main result (Theorem 2.5) the stability of the planar standard lens cluster when the interfaces between regions are weighted equally. The core idea behind the proof strategy is a contradiction argument, where one assumes the existence of a sequence of partitions which violate the stability inequality and converge locally in 
$\textit{L}^1$ to the lens partition. Then one proceeds in two steps. First, the selection principle allows one to replace the previous sequence by a sequence of quasi-minimizing partitions (see Definition 3.3), which still violate the stability inequality and also have better regularity properties. Then the improved convergence theorem yields that these new partitions converge in a stronger sense and in particular that they are small 
$\textit{C}^1$-perturbations of the lens partition. Therefore this reduces the proof of the stability inequality to a class of smooth perturbations of the lens. This is precisely the content of the second step of the proof, which is obtained by a ‘Fuglede-type argument’.
We highlight that only in the second step of this strategy we make essential use of the specific structure of the lens partition, whereas the first step is carried out for any planar locally isoperimetric partition (in the case of a single region with infinite volume, this was already established in [Reference Cicalese, Leonardi and Maggi11]). This paves the way for the proof of the stability of other locally isoperimetric partitions such as the peanut and the Reuleaux triangle, which will be the object of future investigation.
Finally, we would like to mention that the standard lens cluster is directly related to the classical problem of finding the equilibrium shapes of liquid drops confined in a half-space in the absence of gravity (see [Reference Maggi19, Chapter 19] as well as [Reference Alama, Bronsard and Vriend4]). The stability of minimizing shapes for liquid drops has only recently been established by Pascale and Pozzetta [Reference Pascale and Pozzetta25]. Furthermore, in [Reference Pascale24], Pascale studies classical capillarity problems with the inclusion of nonlocal repulsion and gravity terms and obtains the existence and nonexistence of minimizers. In the final section of this paper, we also study a nonlocal perturbation of an isoperimetric problem where the perimeter term is related to both capillarity problems and to the partitioning problem studied here. Exploiting the stability result for the standard lens cluster, we show that the minimizers of this nonlocal problem are close (in the 
$\textit{L}^1$-sense) to the standard lens cluster in certain parameter regimes.
The paper is organized as follows. 
$\S$2 contains the general formulation of the problem of locally isoperimetric partitions and the statement of our main result on the stability of the lens partition, Theorem 2.5. The proof is carried out in two steps as described in 
$\S$3 and 
$\S$4, respectively. Eventually, in 
$\S$5 we discuss an application of the stability to a partitioning problem perturbed by a nonlocal interaction.
2. Definitions and main result
2.1. Locally isoperimetric partitions: definitions
We start by fixing the notation and by formulating in any dimension the notion of locally isoperimetric partitions introduced in [Reference Alama, Bronsard and Vriend5]; we follow in particular the presentation in [Reference Novaga, Paolini and Tortorelli21].
Given a measurable set 
$E\subset\mathbb R^d$, 
$d \geqslant 2$, we denote by 
$|E|$ its 
$\textit{d}$-dimensional Lebesgue measure. The open ball of 
$\mathbb R^d$ of radius 
$\textit{r}$ > 0 centred at 
$x_0\in\mathbb R^d$ is denoted by 
$B_r(x_0)$, and we simply write 
$\textit{B}_r$ when the centre is at the origin. If 
$\textit{E}$ is a set of locally finite perimeter (in the sense of Caccioppoli–De Giorgi), we denote by 
$\partial^*E$ its reduced boundary, by 
$\unicode{x03BD}_{E}$ the measure-theoretic outer unit normal, and by 
$\mathcal{P}(E;\Omega)=\mathcal{H}^{d-1}(\partial^*E\cap\Omega)$ its relative perimeter in a Borel set 
$\Omega\subset\mathbb R^d$, with 
$\mathcal{P}(E) := \mathcal{P}(E;\mathbb R^d)$. For a set of finite perimeter 
$E\subset\mathbb R^d$ we adopt the convention
\begin{equation}
\partial E = \bigl\{x\in\mathbb R^d \;:\; 0 \lt |E\cap B_r(x)| \lt |B_r| \text{ for all }r \gt 0 \bigr\},
\end{equation} which can be always assumed up to modifying 
$\textit{E}$ in a Lebesgue-negligible set, see for instance [Reference Maggi19, Proposition 12.19]. The symmetric difference of two sets 
$E,F\subset\mathbb R^d$ is denoted by 
$E\triangle F:= (E\backslash F)\cup(F\backslash E)$. Given an open set 
$\Omega\subset\mathbb R^d$, we say that a sequence of measurable sets 
$(E_n)_{n\in\mathbb N}$ converge to a set 
$\textit{E}$ in Ω if 
$|(E_n\triangle E)\cap\Omega|\to0$, that is, if the characteristic functions 
$\chi_{E_n}$ converge to χE in 
$L^1(\Omega)$. We say that 
$E_n\to E$ locally in 
$\mathbb R^d$ if 
$E_n\to E$ in 
${B}_{R}$, for every R > 0.
Definition 2.1. (Partition)
An N-partition of 
$\mathbb R^d$, 
$N \geq 2$, is an 
$\textit{N}$-tuple 
$\mathcal{E}=(\mathcal{E}(1),\ldots,\mathcal{E}(N))$ of sets of locally finite perimeter in 
$\mathbb R^d$ such that 
$0 \lt |\mathcal{E}(i)| \leqslant \infty$, 
$|\mathcal{E}(i)\cap\mathcal{E}(j)|=0$ for all 
$i,j\in\{1,\ldots,N\}$, i ≠ j, and 
$|\mathbb R^d\backslash\bigcup_{i=1}^N\mathcal{E}(i)|=0$.
We denote the interfaces between the different regions of a partition by
\begin{equation}
\mathcal{E}(i,j):= \partial^*\mathcal{E}(i)\cap\partial^*\mathcal{E}(j), \qquad i,j\in\{1,\ldots,N\},
\end{equation}and the boundary, the reduced boundary, and the singular set of the partition, respectively, by
\begin{equation}
\partial\mathcal{E}:= \bigcup_{i=1}^N \partial\mathcal{E}(i), \qquad \partial^*\mathcal{E}:= \bigcup_{i=1}^N \partial^*\mathcal{E}(i), \qquad \Sigma(\mathcal{E}):= \partial\mathcal{E}\backslash\partial^*\mathcal{E}.
\end{equation} The perimeter of a partition 
$\mathcal{E}$ relative to a Borel set 
$\Omega\subset\mathbb R^d$ is defined as
\begin{equation}
\mathcal{P}(\mathcal{E};\Omega) := \frac12\sum\limits_{i=1}^N \mathcal{P}(\mathcal{E}(i);\Omega) = \sum\limits_{1 \leqslant i \lt j \leqslant N} \mathcal{H}^{d-1}(\mathcal{E}(i,j)\cap\Omega).
\end{equation} We say that a sequence of partitions 
$(\mathcal{E}_n)_{n\in\mathbb N}$ locally converge to a partition 
$\mathcal{E}$, and we write 
$\mathcal{E}_n\stackrel{\mathrm{loc}}{\to}\mathcal{E}$, if 
$\mathcal{E}_n(i)\to\mathcal{E}(i)$ locally in 
$\mathbb R^d$ for all 
$i\in\{1,\ldots,N\}$, that is, if 
$|(\mathcal{E}_n(i)\triangle\mathcal{E}(i))\cap B_R|\to0$ for all R > 0 and for all 
$i\in\{1,\ldots,N\}$.
Definition 2.2. (Locally isoperimetric partition)
A partition 
$\mathcal{E}_0=(\mathcal{E}_0(1),\ldots,\mathcal{E}_0(N))$ of 
$\mathbb R^d$ is a locally isoperimetric partition if for every R > 0
\begin{equation}
\mathcal{P}(\mathcal{E}_0;B_R) \leqslant \mathcal{P}(\mathcal{E};B_R)
\end{equation} whenever 
$\mathcal{E}=(\mathcal{E}(1),\ldots,\mathcal{E}(N))$ is a partition satisfying
\begin{equation}
|\mathcal{E}(i)|=|\mathcal{E}_0(i)| \quad\text{and}\quad
\mathcal{E}(i)\triangle\mathcal{E}_0(i) \subset\!\subset B_R
\quad\text{for all }i\in\{1,\ldots,N\}.
\end{equation} A locally isoperimetric partition 
$\mathcal{E}_0$ is said to be uniquely minimizing if the following property holds: whenever 
$\mathcal{E}$ is an 
$\textit{N}$-partition satisfying (2.6) for some 
$\textit{R}$ > 0, equality in (2.5) implies the existence of an isometry 
$T:\mathbb R^d\to\mathbb R^d$ such that 
$\mathcal{E}(i)=T(\mathcal{E}_0(i))$ for all 
$i\in\{1,\ldots,N\}$.
Notice that at least one region of a partition 
$\mathcal{E}$ must have infinite Lebesgue measure. As a particular case, 
$(N+1)$-partitions such that all regions have finite measure except one (the exterior region) are usually referred to as 
$\textit{N}$-clusters; we refer to [Reference Maggi19, Part IV] for a presentation of the key ideas about existence and regularity of minimizing clusters. In this paper we are mostly concerned with partitions with at least two regions with infinite measure (in short, infinite regions). In this case the perimeter in the full space is necessarily infinite, so that the minimality condition (2.5) has to be formulated locally. It is convenient to introduce a notation for the indices of the regions with finite measure: for a 
$\textit{N}$-partition 
$\mathcal{E}$ we set
\begin{equation}
I_{\mathcal{E}} := \bigl\{i\in\{1,\ldots,N\} \,:\, |\mathcal{E}(i)| \lt \infty\bigr\}.
\end{equation}The basic regularity properties of locally isoperimetric partitions are given in [Reference Novaga, Paolini and Tortorelli21, Theorem 2.4]. Planar locally isoperimetric partitions have a rigid structure that we recall from [Reference Novaga, Paolini and Tortorelli21, Theorem 4.1 and Theorem 4.2].
Theorem 2.3. (Structure of planar locally isoperimetric partitions)
Let 
$\textit{d}$ = 2, and let 
$\mathcal{E}_0$ be a locally isoperimetric partition in 
$\mathbb R^2$. Then 
$\partial\mathcal{E}_0$ is connected, and there exist a finite family of points 
$\{p_i\}_{i\in I}$ (vertices) and a finite family 
$\{\gamma_j\}_{j\in J}$ of closed curves with boundary such that
\begin{equation*}
\partial\mathcal{E}_0 = \bigcup_{j\in J}\gamma_j, \qquad \partial^*\mathcal{E}_0 = \bigcup_{j\in J}\mathrm{int}(\gamma_j), \qquad \Sigma(\mathcal{E}_0) = \bigcup_{j\in J}\mathrm{bd(\gamma_j)}=\bigcup_{i\in I}\{p_i\},
\end{equation*} where 
$\mathrm{int}(\gamma)$ and 
$\mathrm{bd}(\gamma)$ denote the interior and the boundary points of the curve 
$\unicode{x03B3}$, respectively. Moreover,
(i) each vertex
${p}_{i}$ is a boundary point of exactly three of the curves
$\{\gamma_j\}_{j\in J}$, forming 120 degree angles at ${p}_{i}$,(ii) each curve
$\unicode{x03B3}_j$ is a circular arc, a segment, or a half-line,(iii) the three signed curvatures of the arcs meeting in a vertex have zero sum,
(iv) all the regions with finite area are bounded,
(v) there are at most three regions with infinite area. If there are two infinite regions, the interface between them coincides with a straight line outside a sufficiently large ball; if there are three infinite regions, the interfaces between them coincide, outside a sufficiently large ball, with three half-lines whose prolongations define angles of 120 degrees with each other (but not necessarily passing through a single point).
Given a uniquely minimizing locally isoperimetric partition 
$\mathcal{E}_0=(\mathcal{E}_0(1),\ldots,\mathcal{E}_0(N))$, we define for R > 0 the class of volume-constrained competitors obtained by perturbing 
$\mathcal{E}_0$ in a ball of radius R (up to isometries):
\begin{equation}
\begin{array}{rcl}
\mathfrak{M}_R(\mathcal{E}_0) &:= & \Bigl\{\mathcal{E}=(\mathcal{E}(1),\ldots,\mathcal{E}(N)) \,:\, |\mathcal{E}(i)|=|\mathcal{E}_0(i)| \,\text{ and }\\
&&\quad T(\mathcal{E}(i))\triangle\mathcal{E}_0(i)\subset\!\subset B_R(x_0)\,\text{for all }i\in\{1,\ldots,N\},\\
&&\quad \text{for some }T:\mathbb R^d\to\mathbb R^d\text{ isometry and } x_0\in\mathbb R^d\Bigr\}.
\end{array}
\end{equation} We set 
$\mathfrak{M}(\mathcal{E}_0):= \bigcup_{R \gt 0}\mathfrak{M}_R(\mathcal{E}_0)$. We measure the distance of a partition 
$\mathcal{E}\in\mathfrak{M}(\mathcal{E}_0)$ from 
$\mathcal{E}_0$ by the quantity
\begin{equation}
\Delta(\mathcal{E},\mathcal{E}_0) := \inf \Bigl\{\mathrm{d}(\mathcal{E},T(\mathcal{E}_0)) \,:\, T:\mathbb R^d\to\mathbb R^d \text{ isometry} \Bigr\},
\end{equation} where 
$T(\mathcal{E}_0)$ is the partition defined by 
$T(\mathcal{E}_0)(i):= T(\mathcal{E}_0(i))$, and for every two partitions 
$\mathcal{E}=(\mathcal{E}(1),\ldots,\mathcal{E}(N))$ and 
$\mathcal{F}=(\mathcal{F}(1),\ldots,\mathcal{F}(N))$ we have set
\begin{equation}
\mathrm{d}(\mathcal{E},\mathcal{F}):= \frac12\sum\limits_{i=1}^N |\mathcal{E}(i)\triangle\mathcal{F}(i)|.
\end{equation} For a partition 
$\mathcal{E}\in\mathfrak{M}(\mathcal{E}_0)$ we introduce its perimeter deficit (with respect to 
$\mathcal{E}_0$) as
\begin{equation}
\delta(\mathcal{E},\mathcal{E}_0) := \mathcal{P}(T(\mathcal{E});B_R(x_0)) - \mathcal{P}(\mathcal{E}_0;B_R(x_0)),
\end{equation} where 
$\textit{R}$ > 0 is any radius such that 
$\mathcal{E}\in\mathfrak{M}_R(\mathcal{E}_0)$, and 
$\textit{T}$ and 
$\textit{x}$0 are as in (2.8).
A natural question is, then, whether it is possible to strengthen the minimality inequality (2.5), which can be rephrased as 
$\delta(\mathcal{E},\mathcal{E}_0) \geq 0$ for every 
$\mathcal{E}\in\mathfrak{M}(\mathcal{E}_0)$, in a quantitative form, namely, whether it is possible to find a constant 
$\unicode{x03BA}$ > 0 such that
\begin{equation}
\delta(\mathcal{E},\mathcal{E}_0) \geq \kappa \Delta(\mathcal{E},\mathcal{E}_0)^2
\end{equation} for all partitions 
$\mathcal{E}\in\mathfrak{M}(\mathcal{E}_0)$. In the case 
$\textit{N}$ = 2 (i.e., one region with finite volume and one infinite region, with 
$\mathcal{E}_0=(B_r,\mathbb R^d\backslash B_r)$), the stability inequality (2.12) reduces to the celebrated quantitative isoperimetric inequality [Reference Fusco, Maggi and Pratelli14]. For minimizing clusters, to the best of our knowledge the only stability inequality available is for the planar standard double bubble (two regions with finite area and one infinite region in 
$\mathbb R^2$), proved by Cicalese, Leonardi, and Maggi in [Reference Cicalese, Leonardi and Maggi11].
For partitions with at least two infinite regions, however, one quickly realizes that an estimate of the form (2.12) cannot hold in the full class 
$\mathfrak{M}(\mathcal{E}_0)$: indeed, one can construct perturbations 
$\mathcal{E}_R$ of 
$\mathcal{E}_0$ in large balls 
${B}_{R}$, 
$R\gg 1$, such that 
$\sup_R\delta(\mathcal{E}_R,\mathcal{E}_0) \lt \infty$ and 
$\Delta(\mathcal{E}_R,\mathcal{E}_0)\to\infty$ as 
$R\to+\infty$. In dimension 
$\textit{d}$ = 2 it is enough to construct 
$\mathcal{E}_R$ by replacing a portion of length 
$\textit{R}$ of one of the infinite boundaries of 
$\mathcal{E}_0$ (which are half-lines by Theorem 2.3) by another segment of the same length 
$\textit{R}$, parallel to and at distance 1 from the first and connect its end points to the rest of the boundary by two segments of length 1 each (see Figure 2). Then 
$\delta(\mathcal{E}_R,\mathcal{E}_0)=2$, but 
$\Delta(\mathcal{E}_R,\mathcal{E}_0)=R$.
Two-dimensional construction showing that the constant 
$\unicode{x03BA}$ in (2.12) should depend on the diameter 
$\textit{R}$ of the perturbation.
For this reason we restrict to the class 
$\mathfrak{M}_R(\mathcal{E}_0)$, that is, we impose an upper bound on the diameter of the symmetric difference of 
$\mathcal{E}$ and 
$\mathcal{E}_0$ (up to isometries), and we wish to prove the stability inequality (2.12) in 
$\mathfrak{M}_R(\mathcal{E}_0)$, with a constant 
$\unicode{x03BA}$ also depending on 
$\textit{R}$.
2.2. Main result: stability of the planar standard lens
We now leave the general setting considered in the previous subsection and we consider planar partitions (
$\textit{d}$ = 2) into 
$\textit{N}$ = 3 regions, one of which has finite area and the remaining two have infinite measure; that is, we consider partitions 
$\mathcal{E}=(\mathcal{E}(1),\mathcal{E}(2),\mathcal{E}(3))$ of 
$\mathbb R^2$ such that 
$|\mathcal{E}(1)|=m$, where m > 0 is a fixed parameter, and 
$|\mathcal{E}(2)|=|\mathcal{E}(3)|=+\infty$. In this case the perimeter of a partition 
$\mathcal{E}$ in a ball 
${B}_{R}$ is simply given by
\begin{equation}
\mathcal{P}(\mathcal{E};B_R) = \mathcal{P}(\mathcal{E}(1);B_R) + \mathcal{H}^1(\mathcal{E}(2,3)\cap B_R).
\end{equation} It has been proved in [Reference Alama, Bronsard and Vriend5] (see also [Reference Novaga, Paolini and Tortorelli21]) that in this case the only locally minimizing partition is given by the standard lens partition defined below, see Figure 3. This result has been extended to general dimension 
$d \geq 2$ and to possibly weighted perimeter in [Reference Bronsard and Novack6].
Definition 2.4. (Standard lens partition)
Let 
$\textit{m}$ > 0. The standard lens with area 
$\textit{m}$ is the set
\begin{equation}
L_{m} := \Bigl\{(x,y)\in\mathbb R^2 \,:\, |x| \lt {\textstyle\frac{\sqrt{3}}{2}} r_{m}, \; |y| \lt \sqrt{r_{m}^2-x^2}- {\textstyle\frac12} r_{m} \Bigr\},
\end{equation} where the radius 
$r_{m} \gt 0$ is such that 
$|L_{m}|=m$, that is,
\begin{equation}
r_{m} = \sqrt{m}\biggl(\frac23\pi-\frac{\sqrt{3}}{2}\biggr)^{-1/2}.
\end{equation} The standard lens partition with area m is the 3-partition of 
$\mathbb R^2$ given by
\begin{equation}
\mathcal{L}_{m} := (L_{m}, H^+\backslash L_{m}, H^-\backslash L_{m}),
\end{equation} where 
$H^+:= \{(x,y)\in\mathbb R^2 \,:\, y \gt 0 \}$, 
$H^-:= \{(x,y)\in\mathbb R^2 \,:\, y \lt 0 \}$.
The standard lens partition 
$\mathcal{L}_{m}$ as in Definition 2.4.
The boundary of 
$\textit{L}_m$ is made of two symmetric circular arcs meeting at the two points 
$p_1=(-{\textstyle\frac{\sqrt{3}}{2}} r_{m},0)$, 
$p_2=({\textstyle\frac{\sqrt{3}}{2}} r_{m},0)$ on the x-axis and forming an angle 
$\frac23\pi$ with the x-axis. The two circular arcs have radius 
$\textit{r}_m$ and subtend an angle 
$\frac23\pi$ at the centre. The interface between the regions 
$\mathcal{L}_{m}(2)$ and 
$\mathcal{L}_{m}(3)$ of the lens partition is flat and is given by
\begin{equation*}
\mathcal{L}_{m}(2,3) = \bigl\{(x,0)\in\mathbb R^2 \,:\, |x| \gt {\textstyle\frac{\sqrt{3}}{2}} r_{m}\bigr\}.
\end{equation*} By [Reference Alama, Bronsard and Vriend5, Theorem 1.9] and [Reference Bronsard and Novack6, Theorem 2.9], the lens partition 
$\mathcal{L}_{m}$ is a uniquely minimizing locally isoperimetric partition, in the sense of Definition 2.2: 
$\delta(\mathcal{E},\mathcal{L}_{m}) \geq 0$ for any other partition 
$\mathcal{E}\in\mathfrak{M}(\mathcal{L}_{m})$. Our main result is the following sharp stability inequality for the standard lens, which strengthens the local minimality of the lens partition in a quantitative form. The proof of the theorem is given at the end of §4.
Theorem 2.5. (Stability of the standard lens)
Let 
$\textit{m}$ > 0. For every 
$\textit{R}$ > 0 there exists a constant 
$\kappa_{m,R} \gt 0$, depending on 
$\textit{m}$ and 
$\textit{R}$, such that
\begin{equation}
\delta(\mathcal{E},\mathcal{L}_{m}) \geq \kappa_{m,R}\Delta(\mathcal{E},\mathcal{L}_{m})^2 \qquad\text{for all }\mathcal{E}\in\mathfrak{M}_R(\mathcal{L}_{m}).
\end{equation}3. Selection principle and improved convergence
In this section we set up the general strategy for the proof of the stability of planar locally isoperimetric partitions, following the approach via improved convergence due to Cicalese, Leonardi, and Maggi [Reference Cicalese, Leonardi and Maggi10] and, in turn, based on the selection principle devised by Cicalese and Leonardi [Reference Cicalese and Leonardi9].
We point out that in this section we do not make use of the specific geometry of the lens partition. The main results of this part (Theorem 3.2, Theorem 3.5, and Theorem 3.8) are indeed valid for any uniquely minimizing locally isoperimetric partition in 
$\mathbb R^2$ and might be instrumental in proving the stability of different planar locally isoperimetric partitions, other than the standard lens. Notice, however, the restriction to two dimensions, which we require in order to exploit the structure of the boundaries among infinite regions in 
$\mathbb R^2$ (see Theorem 2.3): indeed we have to formulate the selection principle including a Dirichlet boundary condition outside of a large ball. For clusters (i.e. partitions with a single infinite region) these results have been proved in [Reference Cicalese, Leonardi and Maggi11, Appendix A] in any dimension; however, we focus here on the case of at least two infinite regions.
For the rest of this section we work in the general setting introduced in 
$\S$ 2.1 and in dimension 
$\textit{d}$ = 2.
3.1. Selection principle
We let 
$\mathcal{E}_0=(\mathcal{E}_0(1),\ldots,\mathcal{E}_0(N))$ be a uniquely minimizing locally isoperimetric partition in 
$\mathbb R^2$, according to Definition 2.2. For R > 0 we define the quantity
\begin{align}
\begin{array}{ll}
\kappa_R(\mathcal{E}_0) &:= \inf\biggl\{\liminf\limits_{k\to\infty} \frac{\delta(\mathcal{E}_k,\mathcal{E}_0)}{\Delta(\mathcal{E}_k,\mathcal{E}_0)^2} \;:\; (\mathcal{E}_k)_{k\in\mathbb N}\subset\mathfrak{M}_R(\mathcal{E}_0), \\
&\qquad\Delta(\mathcal{E}_k,\mathcal{E}_0) \gt 0\quad \text{for all}\enspace k\in\mathbb N,\lim\limits_{k\to\infty}\mathrm{d}(\mathcal{E}_k,\mathcal{E}_0)=0 \biggr\},
\end{array}
\end{align} where the deficit 
$\delta(\cdot,\mathcal{E}_0)$, the distance 
$\Delta(\cdot,\mathcal{E}_0)$, and the class of competitors 
$\mathfrak{M}_R(\mathcal{E}_0)$ are defined in (2.11), (2.9), and (2.8), respectively.
Remark 3.1. If 
$\mathcal{E}_0$ has at least two infinite regions, it is easy to see that 
$\kappa_R(\mathcal{E}_0)$ is finite: indeed, since there is no need to preserve the volume of the infinite regions, it is enough to slightly perturb one of the flat boundaries between two infinite regions (for instance, removing a segment and replacing it by two segments forming a sawtooth of small height t > 0) and construct partitions 
$\mathcal{E}_t$ such that the ratio 
$\delta(\mathcal{E}_t,\mathcal{E}_0)/\Delta(\mathcal{E}_t,\mathcal{E}_0)^2$ remains bounded as t → 0.
From this construction it also follows that the quadratic decay on the right-hand side of (2.12) is sharp, in the following sense (see also [Reference Cicalese, Leonardi and Maggi10, Remark 1.2]): if 
$\delta(\mathcal{E},\mathcal{E}_0) \geq \phi(\Delta(\mathcal{E},\mathcal{E}_0))$ for some function 
$\phi:[0,\infty)\to[0,\infty)$ and for all 
$\mathcal{E}\in\mathfrak{M}_R(\mathcal{E}_0)$, then there exist C > 0 and 
$t_0 \gt 0$ such that 
$\phi(t) \leqslant Ct^2$ for all 
$t\in(0,t_0)$.
In the following theorem we show that the proof of the stability inequality for 
$\mathcal{E}_0$ can be reduced to the case of partitions 
$\mathcal{E}$ with small distance 
$\Delta(\mathcal{E},\mathcal{E}_0)$ from 
$\mathcal{E}_0$.
Theorem 3.2. Let 
$\mathcal{E}_0=(\mathcal{E}_0(1),\ldots,\mathcal{E}_0(N))$ be a uniquely minimizing locally isoperimetric partition in 
$\mathbb R^2$, and let R > 0. Then for every ɛ > 0 there exists δ > 0 such that for every 
$\mathcal{E}\in\mathfrak{M}_R(\mathcal{E}_0)$, if 
$\delta(\mathcal{E},\mathcal{E}_0) \lt \delta$ then 
$\Delta(\mathcal{E},\mathcal{E}_0) \lt \varepsilon$. Furthermore, the condition
\begin{equation}
\kappa_R(\mathcal{E}_0) \gt 0
\end{equation} is equivalent to the existence of a constant 
$\kappa_{R,\mathcal{E}_0} \gt 0$, depending on R and 
$\mathcal{E}_0$, such that
\begin{equation}\delta(\mathcal E,{\mathcal E}_0)\geqslant\kappa_{R,{\mathcal E}_0}\Delta(\mathcal E,{\mathcal E}_0)^2\ \text{for all}\ \mathcal E\in{\mathfrak M}_R({\mathcal E}_0).\end{equation}Proof. We argue by contradiction, and we assume that, for some R > 0, there exist 
$\varepsilon_* \gt 0$ and a sequence of partitions 
$(\mathcal{E}_k)_{k\in\mathbb N}\subset\mathfrak{M}_{R}(\mathcal{E}_0)$ such that
\begin{equation}\lim_{k\rightarrow\infty}\delta({\mathcal E}_k,{\mathcal E}_0)=0,\qquad\Delta({\mathcal E}_k,{\mathcal E}_0)\geqslant\varepsilon_\ast.\end{equation} By definition of 
$\mathfrak{M}_{R}(\mathcal{E}_0)$ in (2.8), and up to replacing each partition 
$\mathcal{E}_k$ by an isometric copy, we can find points 
$x_k\in\mathbb R^2$ such that 
$\mathcal{E}_k(i)\triangle\mathcal{E}_0(i)\subset\!\subset B_R(x_k)$.
We consider first the case 
$\sup_k|x_k| \lt \infty$, so that 
$x_k\to x_0\in\mathbb R^2$ up to a (not relabeled) subsequence and 
$B_R(x_k)\subset B_{R+1}(x_0)$ for all k large enough. For every 
$i\in\{1,\ldots,N\}$ the sets 
$\mathcal{E}_k(i)\cap B_{R+1}(x_0)$ have uniformly bounded perimeter in 
$B_{R+1}(x_0)$, and by standard compactness results (see [Reference Maggi19, Theorem 12.26]) we can assume that, up to further subsequences, 
$\mathcal{E}_k(i)\cap B_{R+1}(x_0)\to F_i$ as 
$k\to\infty$, for some set of finite perimeter 
$F_i\subset B_{R+1}(x_0)$. Since 
$\mathcal{E}_k(i)\triangle\mathcal{E}_0(i)\subset\!\subset B_R(x_k)$, we also have that Fi coincides with 
$\mathcal{E}_0(i)$ in a uniform neighbourhood of 
$\partial B_{R+1}(x_0)$. We then define the partition 
$\mathcal{E}_\infty$ with regions 
$\mathcal{E}_\infty(i):= F_i\cup (\mathcal{E}_0(i)\backslash B_{R+1}(x_0))$, and by construction it is immediate to check that 
$\mathcal{E}_\infty\in\mathfrak{M}_{R+1}(\mathcal{E}_0)$. Furthermore by lower semicontinuity of the perimeter we have that
\begin{equation*}\delta({\mathcal E}_\infty,{\mathcal E}_0)\leqslant\liminf_{k\rightarrow\infty}\delta({\mathcal E}_k,{\mathcal E}_0)=0,\qquad\Delta({\mathcal E}_\infty,{\mathcal E}_0)=\lim_{k\rightarrow\infty}\Delta({\mathcal E}_k,{\mathcal E}_0)\geqslant\varepsilon_\ast \gt 0.\end{equation*} These two properties contradict the assumption that 
$\mathcal{E}_0$ is uniquely minimizing.
If instead 
$\sup_k|x_k|=\infty$, recalling that the regions of 
$\mathcal{E}_0$ with finite area are bounded (Theorem 2.3), we have 
$\mathcal{E}_0(i)\cap B_R(x_k)=\emptyset$ for all 
$i\in I_{\mathcal{E}_0}$ and for all k sufficiently large, where 
$I_{\mathcal{E}_0}$ is the set of the indices corresponding to the finite regions as in (2.7). Hence 
$B_R(x_k)$ has nonempty intersection only with some of the infinite regions of 
$\mathcal{E}_0$. By the structure of the interfaces among the infinite regions in Theorem 2.3, we have that for all k sufficiently large the interface 
$\partial\mathcal{E}_0\cap B_R(x_k)$ is either empty or a segment. We can then find new centres 
$y_k\in\mathbb R^2$, with 
$\sup_k|y_k| \lt \infty$, so that 
$\partial\mathcal{E}_0\cap B_R(y_k)$ coincides with a translation of 
$\partial\mathcal{E}_0\cap B_R(x_k)$, and ‘copy and paste’ 
$\mathcal{E}_k\cap B_R(x_k)$ into 
$B_R(y_k)$. This way we obtain a new sequence of partitions which satisfy the same properties as 
$\mathcal{E}_k$ and are perturbations of 
$\mathcal{E}_0$ inside 
$B_R(y_k)$. Since the new centres yk are uniformly bounded, the same compactness argument as in the previous case allows us to conclude by contradiction. This completes the proof of the first part of the statement.
Concerning the equivalence between (3.2) and (3.3), it is immediate to see that (3.3) implies (3.2). Conversely, assume that (3.2) holds and by contradiction that (3.3) fails, that is, there exists a sequence 
$(\mathcal{E}_k)_{k\in\mathbb N}\subset\mathfrak{M}_{R}(\mathcal{E}_0)$, with 
$\Delta(\mathcal{E}_k,\mathcal{E}_0) \gt 0$ such that
\begin{equation}
\frac{\delta(\mathcal{E}_k,\mathcal{E}_0)}{\Delta(\mathcal{E}_k,\mathcal{E}_0)^2} \to 0 \qquad \text{as }k\to\infty.
\end{equation} We first observe that 
$\Delta(\mathcal{E}_k,\mathcal{E}_0)\to0$ as 
$k\to\infty$. Indeed, if not then 
$\Delta(\mathcal{E}_k,\mathcal{E}_0) \geq \varepsilon$ for some ɛ > 0 and in turn, by the first part of the statement, also 
$\delta(\mathcal{E}_k,\mathcal{E}_0) \geq \delta \gt 0$. Moreover, since 
$\mathcal{E}_k\in\mathfrak{M}_{R}(\mathcal{E}_0)$ we also have 
$\Delta(\mathcal{E}_k,\mathcal{E}_0) \leqslant |B_R|$. These inequalities contradict (3.5), showing that 
$\Delta(\mathcal{E}_k,\mathcal{E}_0)\to0$.
Let now 
$T:\mathbb R^2\to\mathbb R^2$ be an isometry such that 
$\Delta(\mathcal{E}_k,\mathcal{E}_0)=\mathrm{d}(T(\mathcal{E}_k),\mathcal{E}_0)$. The sequence of partitions 
$\mathcal{F}_k:= T(\mathcal{E}_k)$ is such that 
$\mathcal{F}_k\in\mathfrak{M}_R(\mathcal{E}_0)$ and 
$\mathrm{d}(\mathcal{F}_k,\mathcal{E}_0)\to0$ and is therefore admissible in the definition of 
$\kappa_R(\mathcal{E}_0)$ in (3.1). It follows
\begin{equation*}
0 \lt \kappa_R(\mathcal{E}_0) \leqslant \liminf_{k\to\infty}\frac{\delta(\mathcal{F}_k,\mathcal{E}_0)}{\Delta(\mathcal{F}_k,\mathcal{E}_0)^2} = \liminf_{k\to\infty}\frac{\delta(\mathcal{E}_k,\mathcal{E}_0)}{\Delta(\mathcal{E}_k,\mathcal{E}_0)^2} = 0,
\end{equation*}which is a contradiction.
Thanks to Theorem 3.2, our final goal will be to show the strict inequality 
$\kappa_R(\mathcal{L}_{m}) \gt 0$ for the standard lens partition. We next show in Theorem 3.5 the existence of a recovery sequence for 
$\kappa_R(\mathcal{E}_0)$ made of 
$(\Lambda,r_0)$-minimizing partitions (with uniform constants), according to the following definition.
Definition 3.3. Given 
$\Lambda \gt 0$ and 
$r_0 \gt 0$, a partition 
$\mathcal{E}=(\mathcal{E}(1),\ldots,\mathcal{E}(N))$ is said to be a 
$(\Lambda,r_0)$-minimizing partition if
\begin{equation}
\mathcal{P}(\mathcal{E};B_{r_0}(x)) \leqslant \mathcal{P}(\mathcal{F};B_{r_0}(x)) + \Lambda \mathrm{d}(\mathcal{E},\mathcal{F})
\end{equation} whenever 
$x\in\mathbb R^2$ and 
$\mathcal{F}$ is a partition such that 
$\mathcal{F}(i)\triangle\mathcal{E}(i)\subset\!\subset B_{r_0}(x)$.
In the proof of Theorem 3.5 we need the following construction of a suitable neighbourhood OR of a ball 
${B}_{R}$, depending on the structure of the infinite regions of 
$\mathcal{E}_0$ (see Figure 4).
Definition 3.4. Let 
$\mathcal{E}_0=(\mathcal{E}_0(1),\ldots,\mathcal{E}_0(N))$ be a uniquely minimizing locally isoperimetric partition in 
$\mathbb R^2$ with k regions with infinite area, where k is either 2 or 3.
(i) Let
$R\gg1$ be such that
$\partial\mathcal{E}_0\backslash B_R$ is the union of k half-lines, according to Theorem 2.3. By possibly taking a larger R, we can assume that each half-line intersects
$\partial B_R$ with an angle close to
$\frac\pi2$ (say, between
$\frac\pi3$ and
$\frac{2\pi}{3}$).(ii) Let L be any of these k half-lines, meeting
$\partial B_R$ at a point p. Let
$\partial B_R \cap \partial B_{1/2}(p)=\{q_1,q_2\}$, and let
$(L\cap\partial B_{1/2}(p))\backslash B_R=\{q_0\}$. For
$i=1,2$, connect the point qi to q 0 by a smooth curve
$\Gamma_i\subset \overline{B_{1/2}(p)}\backslash B_R$, meeting
$\partial B_R$ at qi and L at q 0 in a C 2-way and intersecting L only at q 0. Let EL be the region enclosed by the curves Γ1, Γ2, and
$\partial B_{R}\cap B_{1/2}(p)$.(iii) Let OR be the open set obtained by the union of
${B}_{R}$ with each of the k sets EL constructed at the previous point.
Notice that 
$B_R\subset O_R \subset\!\subset B_{R+1}$, and the boundary of OR is a curve of class C 2, except for k cusp points at the intersection 
$\partial O_R\cap\partial\mathcal{E}_0$.
The set OR constructed in Definition 3.4, depending on whether 
$\mathcal{E}_0$ has two infinite regions (left) or three infinite regions (right).
Theorem 3.5. Let 
$\mathcal{E}_0=(\mathcal{E}_0(1),\ldots,\mathcal{E}_0(N))$ be a uniquely minimizing locally isoperimetric partition in 
$\mathbb R^2$ with at least two infinite regions, and let 
$\overline{R} \gt 0$.
There exist a radius 
$R\geqslant\overline R$, positive constants 
$\Lambda \gt 0$ and 
$r_0 \gt 0$ (all depending only on 
$\mathcal{E}_0$ and 
$\overline{R}$) and a sequence 
$(\mathcal{E}_k)_{k\in\mathbb N}\subset\mathfrak{M}_{R+1}(\mathcal{E}_0)$ of 
$(\Lambda,r_0)$-minimizing partitions, such that 
$\mathcal{E}_k(i)\triangle\mathcal{E}_0(i)\subset\!\subset B_{R+1}$ for all i, and
\begin{align}
&\Delta(\mathcal{E}_k,\mathcal{E}_0) \gt 0 \quad\text{for all }k\in\mathbb N, \qquad \lim_{k\to\infty}\Delta(\mathcal{E}_k,\mathcal{E}_0)=0,\nonumber\\
&\lim_{k\to\infty}\frac{\delta(\mathcal{E}_k,\mathcal{E}_0)}{\Delta(\mathcal{E}_k,\mathcal{E}_0)^2} = \kappa_{\overline{R}}(\mathcal{E}_0).
\end{align}Proof. We divide the proof into three steps.
Step 1: localization in a large ball. Let 
$(\mathcal{F}_k)_{k\in\mathbb N}\subset\mathfrak{M}_{\overline{R}}(\mathcal{E}_0)$ be a recovery sequence for 
$\kappa_{\overline{R}}(\mathcal{E}_0)$ in (3.1), that is,
\begin{align}
& \Delta(\mathcal{F}_k,\mathcal{E}_0) \gt 0 \quad\text{for all }k\in\mathbb N, \qquad \lim_{k\to\infty}\Delta(\mathcal{F}_k,\mathcal{E}_0)=0,\nonumber\\ &\lim_{k\to\infty}\frac{\delta(\mathcal{F}_k,\mathcal{E}_0)}{\Delta(\mathcal{F}_k,\mathcal{E}_0)^2} = \kappa_{\overline{R}}(\mathcal{E}_0).
\end{align} By definition of the class 
$\mathfrak{M}_{\overline{R}}(\mathcal{E}_0)$, up to replacing 
$\mathcal{F}_k$ by an isometric copy, we can find points 
$x_k\in\mathbb R^2$ such that 
$\mathcal{F}_k(i)\triangle\mathcal{E}_0(i)\subset\!\subset B_{\overline{R}}(x_k)$ for all 
$i\in\{1,\ldots,N\}$. By finiteness of 
$\kappa_{\overline{R}}(\mathcal{E}_0)$, it follows from (3.8) that as 
$k\to\infty$
\begin{equation}
\mathcal{P}(\mathcal{F}_k;B_{\overline{R}}(x_k))-\mathcal{P}(\mathcal{E}_0;B_{\overline{R}}(x_k)) = \delta(\mathcal{F}_k,\mathcal{E}_0) = \kappa_{\overline{R}}(\mathcal{E}_0)\Delta(\mathcal{F}_k,\mathcal{E}_0)^2 + o(\Delta(\mathcal{F}_k,\mathcal{E}_0)^2),
\end{equation}so that by assuming k large enough we can bound
\begin{equation}
\mathcal{P}(\mathcal{F}_k;B_{\overline{R}}(x_k))-\mathcal{P}(\mathcal{E}_0;B_{\overline{R}}(x_k)) \leqslant \bigl(\kappa_{\overline{R}}(\mathcal{E}_0)+1\bigr)\Delta(\mathcal{F}_k,\mathcal{E}_0)^2.
\end{equation} We claim that there exists 
$R \geq {\overline{R}}$, depending only on 
${\overline{R}}$ and 
$\mathcal{E}_0$, such that we can assume
\begin{equation}
B_{\overline{R}}(x_k)\subset B_{R} \quad\text{for all } k,
\qquad
\mathcal{E}_0(i)\subset\!\subset B_{R} \quad\text{for all } i\in I_{\mathcal{E}_0}.
\end{equation} Indeed, since by Theorem 2.3 the regions of 
$\mathcal{E}_0$ with finite area are bounded, there exists a radius 
$R_0 \gt 0$ depending only on 
$\mathcal{E}_0$ such that 
$\mathcal{E}_0(i)\subset\!\subset B_{R_0}$ for all 
$i\in I_{\mathcal{E}_0}$, so that the second condition in (3.11) is satisfied for all 
$R \gt R_0$. Regarding the first condition, we have that either 
$B_{\overline{R}}(x_k)\cap \mathcal{E}_0(i)\neq\emptyset$ for some 
$i\in I_{\mathcal{E}_0}$ (and in this case it is enough to take 
$R \gt R_0+2{\overline{R}}$ to guarantee that 
$B_{\overline{R}}(x_k)\subset B_{R}$) or 
$B_{\overline{R}}(x_k)\cap \mathcal{E}_0(i)=\emptyset$ for all 
$i\in I_{\mathcal{E}_0}$. In this second case we can use the same argument as in the proof of Theorem 3.2 to ‘copy and paste’ 
$\mathcal{F}_k\cap B_{\overline{R}}(x_k)$ into balls 
$B_{\overline{R}}(y_k)$ that are contained in a uniform ball 
${B}_{R}$, so that we can assume without loss of generality that the first condition in (3.11) also holds.
Step 2: Construction of 
$\mathcal{E}_k$. Let OR be the set introduced in Definition 3.4, depending on the structure of the infinite regions of 
$\mathcal{E}_0$, see in particular Figure 4 (notice that, up to taking a larger R, the assumption in Definition 3.4 is satisfied). We define 
$\mathcal{E}_k$ as a solution to the following minimum problem:
\begin{equation}
\begin{array}{ll}
&\min\biggl\{\mathcal{P}(\mathcal{E};B_{R+1}) + \big| \Delta(\mathcal{E},\mathcal{E}_0)-\Delta(\mathcal{F}_k,\mathcal{E}_0)\big|^{3/2} \,:\, |\mathcal{E}(i)|=|\mathcal{E}_0(i)| \text{ and } \\
& \quad|(\mathcal{E}(i)\triangle\mathcal{E}_0(i))\backslash O_{R}|=0 \text{ for all }i\in\{1,\ldots,N\} \biggr\},
\end{array}
\end{equation} whose existence follows by the direct method of the calculus of variations. Notice that, since 
$O_{R}\subset\!\subset B_{R+1}$, we have 
$\mathcal{E}_k(i)\triangle\mathcal{E}_0(i)\subset\!\subset B_{R+1}$, and 
$\mathcal{E}_k$ satisfies the volume constraint as well; hence 
$\mathcal{E}_k\in\mathfrak{M}_{R+1}(\mathcal{E}_0)$ for all k.
Furthermore, 
$\mathcal{F}_k$ is admissible as a competitor in (3.12), hence by minimality of 
$\mathcal{E}_0$ and by (3.10) we have
\begin{align}
&\mathcal{P}(\mathcal{E}_k;B_{R+1}) + \big| \Delta(\mathcal{E}_k,\mathcal{E}_0)-\Delta(\mathcal{F}_k,\mathcal{E}_0)\big|^{3/2} \leqslant \mathcal{P}(\mathcal{F}_k;B_{R+1}) \nonumber\\
&\quad\leqslant \mathcal{P}(\mathcal{E}_0;B_{R+1}) + \bigl(\kappa_{\overline{R}}(\mathcal{E}_0)+1\bigr)\Delta(\mathcal{F}_k,\mathcal{E}_0)^2.
\end{align} Since 
$\mathcal{P}(\mathcal{E}_k;B_{R+1})-\mathcal{P}(\mathcal{E}_0;B_{R+1}) \geq 0$ by local minimality of 
$\mathcal{E}_0$, it follows from (3.13)
\begin{equation}
\big| \Delta(\mathcal{E}_k,\mathcal{E}_0)-\Delta(\mathcal{F}_k,\mathcal{E}_0)\big|^{3/2} \leqslant \bigl(\kappa_{\overline{R}}(\mathcal{E}_0)+1\bigr)\Delta(\mathcal{F}_k,\mathcal{E}_0)^2.
\end{equation} In turn, since 
$0 \lt \Delta(\mathcal{F}_k,\mathcal{E}_0)\to0$, dividing by 
$\Delta(\mathcal{F}_k,\mathcal{E}_0)^{3/2}$ in (3.14) we find
\begin{equation}
\lim_{k\to\infty}\frac{\Delta(\mathcal{E}_k,\mathcal{E}_0)}{\Delta(\mathcal{F}_k,\mathcal{E}_0)}=1,
\qquad
\Delta(\mathcal{E}_k,\mathcal{E}_0) \gt 0 \quad\text{for all }k\in\mathbb N,
\qquad \lim_{k\to\infty}\Delta(\mathcal{E}_k,\mathcal{E}_0)=0.
\end{equation}Again by (3.13) and (3.9) we have
\begin{align}
\delta(\mathcal{E}_k,\mathcal{E}_0)
= \mathcal{P}(\mathcal{E}_k;B_{R+1})-\mathcal{P}(\mathcal{E}_0;B_{R+1})
& \leqslant \mathcal{P}(\mathcal{F}_k;B_{R+1})-\mathcal{P}(\mathcal{E}_0;B_{R+1}) \nonumber\\
& = \kappa_{\overline{R}}(\mathcal{E}_0)\Delta(\mathcal{F}_k,\mathcal{E}_0)^2 + o(\Delta(\mathcal{F}_k,\mathcal{E}_0)^2)\nonumber\\
& = \kappa_{\overline{R}}(\mathcal{E}_0)\Delta(\mathcal{E}_k,\mathcal{E}_0)^2 + o(\Delta(\mathcal{E}_k,\mathcal{E}_0)^2),
\end{align} where the last identity follows from (3.15). In view of (3.15) and (3.16), the sequence 
$(\mathcal{E}_k)_{k\in\mathbb N}$ satisfies the conditions in (3.7).
We further notice for later use that we can find an isometry 
$T:\mathbb R^2\to\mathbb R^2$ such that
\begin{equation}
\lim_{k\to\infty}\mathrm{d}(\mathcal{E}_k,T(\mathcal{E}_0))=0,
\quad
T(\mathcal{E}_0(i))\subset O_{R} \quad\text{for all }i\in I_{\mathcal{E}_0},
\end{equation}and
\begin{equation}
|(T(\mathcal{E}_0(i))\triangle\mathcal{E}_0(i))\backslash O_{R}| = 0 \quad\text{for all }i\in\{1,\ldots,N\}.
\end{equation} Indeed, let 
$T_k:\mathbb R^2\to\mathbb R^2$ be isometries such that 
$\mathrm{d}(\mathcal{E}_k,T_k(\mathcal{E}_0))=\Delta(\mathcal{E}_k,\mathcal{E}_0)\to0$. Notice that 
$\sup_k|T_k(0)| \lt \infty$ in view of the condition 
$\mathcal{E}_k(i)\triangle\mathcal{E}_0(i)\subset\!\subset B_{R+1}$, so that, up to a subsequence, Tk converges to some isometry T. By triangle inequality the first condition in (3.17) is satisfied. The second condition holds since, if by contradiction 
$|T(\mathcal{E}_0(i))\backslash O_{R}| \gt 0$ for some 
$i\in I_{\mathcal{E}_0}$, then we would have 
$|\mathcal{E}_k(i)\backslash O_{R}| \gt 0$ for k sufficiently large and in turn also 
$|\mathcal{E}_0(i)\backslash O_{R}| \gt 0$ since 
$\mathcal{E}_k$ coincides with 
$\mathcal{E}_0$ outside OR, contradicting (3.11). For the same reason, (3.18) also holds.
Step 3: 
$(\Lambda,r_0)$-minimality. We are left to prove that 
$\mathcal{E}_k$ is a 
$(\Lambda,r_0)$-minimizing partition for all k according to Definition 3.3, for suitable uniform constants 
$\Lambda \gt 0$, 
$r_0 \gt 0$. We choose 
$r_0 \gt 0$ given by Lemma A.1 corresponding to 
$T(\mathcal{E}_0)$ (notice that 
$\textit{r}$0 depends ultimately only on 
$\mathcal{E}_0$ and 
${\overline{R}}$).
Fix 
$k\in\mathbb N$, and let 
$\mathcal{F}=(\mathcal{F}(1),\ldots,\mathcal{F}(N))$ be a partition such that 
$\mathcal{E}_k(i)\triangle\mathcal{F}(i)\subset\!\subset B_{r_0}(x)$ for some 
$x\in\mathbb R^2$. We shall prove the inequality (3.6) for a suitable 
$\Lambda \gt 0$.
Assume first that 
$B_{r_0}(x)\cap O_{R}=\emptyset$. Recalling (3.11), 
$\partial\mathcal{E}_0\cap B_{r_0}(x)$ is either empty or a segment (the boundary among two infinite regions, say 
$\mathcal{E}_0(i_1)$ and 
$\mathcal{E}_0(i_2)$). Then, since 
$|(\mathcal{E}_k(i)\triangle\mathcal{E}_0(i))\backslash O_{R}|=0$ for all i, we have
\begin{align*}
\mathcal{P}(\mathcal{E}_k; B_{r_0}(x))
&= \mathcal{P}(\mathcal{E}_0; B_{r_0}(x))
= \mathcal{P}(\mathcal{E}_0(i_1); B_{r_0}(x))\\
&\leqslant \mathcal{P}(\mathcal{F}(i_1);
B_{r_0}(x))\leqslant \mathcal{P}(\mathcal{F}; B_{r_0}(x)),
\end{align*} where the first inequality follows from the fact that 
$\partial\mathcal{E}_0(i_1)\cap B_{r_0}(x)$ is either empty or a segment, and 
$\mathcal{F}(i_1)\triangle\mathcal{E}_0(i_1)\subset\!\subset B_{r_0}(x)$. The previous estimate proves (3.6) in this case (with 
$\Lambda=0$).
Consider next the case 
$B_{r_0}(x)\cap O_{R}\neq\emptyset$, and let us also prove (3.6) in this case. By taking 
$r_0 \lt \frac12$ we can assume that 
$B_{r_0}(x)\subset\!\subset B_{R+1}$. We modify the partition 
$\mathcal{F}$ in order to obtain an admissible competitor for the minimum problem (3.12). We have two constraints to satisfy, and we proceed in two steps:
• Let
$\mathcal{F}'=(\mathcal{F}'(1),\ldots,\mathcal{F}'(N))$ be obtained from
$\mathcal{F}$ by setting
\begin{equation*}
\mathcal{F}'(i) := \bigl( \mathcal{F}(i)\cap O_{R} \bigr) \cup \bigl( \mathcal{E}_0(i) \backslash O_{R} \bigr) \qquad\text{for all }i\in\{1,\ldots,N\}.
\end{equation*}Notice that
$|(\mathcal{F}'(i)\triangle\mathcal{E}_0(i))\backslash O_{R}|=0$ and
$\mathcal{F}'(i)\triangle\mathcal{E}_k(i)\subset\!\subset B_{r_0}(x)$.• Let
$\mathcal{F}^{\prime\prime} =(\mathcal{F}^{\prime\prime} (1),\ldots,\mathcal{F}^{\prime\prime} (N))$ be obtained from
$\mathcal{F}'$ by applying the volume-fixing variation Lemma A.1, with
$\mathcal{E}=\mathcal{E}_k$ and
$\mathcal{G}=\mathcal{F}'$ (that is, we set
$\mathcal{F}^{\prime\prime} =\widetilde{\mathcal{G}}$ obtained by the lemma with the previous choices). The lemma can be applied since
$\mathrm{d}(\mathcal{E}_k,T(\mathcal{E}_0)) \lt \varepsilon_0$ for k sufficiently large by (3.17) and
$\mathcal{F}'(i)\triangle\mathcal{E}_k(i)\subset\!\subset B_{r_0}(x)$.
Let C 0 be the constant given by Lemma A.1, and let 
$C_1 \gt 0$ be such that 
$\mathcal{P}(\mathcal{E}_k;B_{R+1}) \leqslant C_1$ for all k, which exists by (3.13). By the third property in Lemma A.1 the partition 
$\mathcal{F}^{\prime\prime} $ satisfies the estimate
\begin{align}
|\mathcal{P}(\mathcal{F}^{\prime\prime} ;B_{R+1})-\mathcal{P}(\mathcal{F}';B_{R+1})|
& \leqslant C_0\mathcal{P}(\mathcal{E}_k;B_{R+1})\sum\limits_{i\in I_{\mathcal{E}_0}}\big| |\mathcal{F}'(i)|-|\mathcal{E}_k(i)| \big| \nonumber\\
& \leqslant C_0C_1\sum\limits_{i\in I_{\mathcal{E}_0}}\big| \mathcal{F}'(i)\triangle\mathcal{E}_k(i) \big| \nonumber\\
& \leqslant C_0C_1\sum\limits_{i\in I_{\mathcal{E}_0}}\big| \mathcal{F}(i)\triangle\mathcal{E}_k(i) \big|,
\end{align}where the last passage follows since
\begin{equation}
\big|\mathcal{F}(i)\triangle\mathcal{E}_k(i)\big|
\geq \big| \bigl(\mathcal{F}(i)\triangle\mathcal{E}_k(i)\bigr)\cap O_{R} \big|
= \big| \bigl(\mathcal{F}'(i)\triangle\mathcal{E}_k(i)\bigr)\cap O_{R} \big|
= \big| \mathcal{F}'(i)\triangle\mathcal{E}_k(i) \big|.
\end{equation}Similar to (3.19), by the fourth property in Lemma A.1 we also have
\begin{equation}
|\mathrm{d}(\mathcal{F}^{\prime\prime} ,\mathcal{E}_k)-\mathrm{d}(\mathcal{F}',\mathcal{E}_k)| \leqslant C_0C_1\sum\limits_{i\in I_{\mathcal{E}_0}}\big| \mathcal{F}(i)\triangle\mathcal{E}_k(i) \big|.
\end{equation} Notice that 
$|(\mathcal{F}^{\prime\prime} (i)\triangle\mathcal{E}_0(i))\backslash O_{R}|=0$ for all i, since 
$\mathcal{F}'$ satisfies this property and 
$\mathcal{F}^{\prime\prime} $ is obtained by perturbing 
$\mathcal{F}'$ inside OR (first property in Lemma A.1). Moreover, 
$|\mathcal{F}^{\prime\prime} (i)|=|\mathcal{E}_k(i)|=|\mathcal{E}_0(i)|$ for all i (by the second property in Lemma A.1). Hence 
$\mathcal{F}^{\prime\prime} $ obeys both constraints in the minimum problem (3.12). Defining
\begin{equation*}
\Delta_k := \big| \Delta(\mathcal{F}^{\prime\prime} ,\mathcal{E}_0)-\Delta(\mathcal{F}_k,\mathcal{E}_0)\big|^{3/2} - \big| \Delta(\mathcal{E}_k,\mathcal{E}_0)-\Delta(\mathcal{F}_k,\mathcal{E}_0)\big|^{3/2},
\end{equation*} we have, by minimality of 
$\mathcal{E}_k$ in (3.12),
\begin{align}
\mathcal{P}(\mathcal{E}_k;B_{R+1}) & \leqslant \mathcal{P}(\mathcal{F}^{\prime\prime} ;B_{R+1}) + \Delta_k \nonumber\\
&\mathop{\leqslant }\limits^{(3.19)} \mathcal{P}(\mathcal{F}';B_{R+1}) + C_0C_1\sum\limits_{i\in I_{\mathcal{E}_0}}\big| \mathcal{F}(i)\triangle\mathcal{E}_k(i) \big| + \Delta_k \nonumber\\
&= \mathcal{P}(\mathcal{F};B_{R+1}) + \Bigl( \mathcal{P}(\mathcal{F}';B_{R+1})-\mathcal{P}(\mathcal{F};B_{R+1}) \Bigr)\nonumber\\
&\quad+ C_0C_1\sum\limits_{i\in I_{\mathcal{E}_0}}\big| \mathcal{F}(i)\triangle\mathcal{E}_k(i) \big| + \Delta_k.
\end{align} We now estimate 
$\Delta_k$. By using the elementary inequality 
$|a^{3/2}-b^{3/2}| \leqslant \frac32\sqrt{\max\{a,b\}}|a-b|$ for all 
$a,b \geq 0$ and observing that 
$\Delta(\mathcal{F}_k,\mathcal{E}_0)\to0$, 
$\Delta(\mathcal{E}_k,\mathcal{E}_0)\to0$, and 
$\Delta(\mathcal{F}^{\prime\prime} ,\mathcal{E}_0) \leqslant |O_{R}|$, we have, for a constant C 2 depending on R,
\begin{align*}
|\Delta_k| & \leqslant C_2 \big| \Delta(\mathcal{F}^{\prime\prime} ,\mathcal{E}_0) - \Delta(\mathcal{E}_k,\mathcal{E}_0) \big| \leqslant C_2 \, \mathrm{d}(\mathcal{F}^{\prime\prime} ,\mathcal{E}_k) \nonumber\\
& \mathop{\leqslant }\limits^{(3.21)} C_2\,\mathrm{d}(\mathcal{F}',\mathcal{E}_k) + C_0C_1C_2\sum\limits_{i\in I_{\mathcal{E}_0}}\big| \mathcal{F}(i)\triangle\mathcal{E}_k(i) \big| \nonumber\\
& \leqslant\frac{C_2}{2}\sum\limits_{i=1}^N \big| \mathcal{F}(i)\triangle\mathcal{E}_k(i) \big| + C_0C_1C_2\sum\limits_{i\in I_{\mathcal{E}_0}}\big| \mathcal{F}(i)\triangle\mathcal{E}_k(i) \big|,
\end{align*} where the last passage follows from (3.20). By inserting this estimate into (3.22) we get, for a constant C 3 depending on 
$\mathcal{E}_0$ and R,
\begin{equation}
\mathcal{P}(\mathcal{E}_k;B_{R+1}) \leqslant \mathcal{P}(\mathcal{F};B_{R+1}) + \Bigl( \mathcal{P}(\mathcal{F}';B_{R+1})-\mathcal{P}(\mathcal{F};B_{R+1}) \Bigr) + C_3\sum\limits_{i=1}^N |\mathcal{E}_k(i)\triangle\mathcal{F}(i)|.
\end{equation} It remains to estimate the change in perimeter between 
$\mathcal{F}$ and 
$\mathcal{F}'$. In the following computations, in view of the regularity of 
$\mathcal{E}_0$ given by Theorem 2.3, we assume that the regions 
$\mathcal{E}_0(i)$ are open sets. By definition of 
$\mathcal{F}'$ we have
\begin{equation}
\mathcal{P}(\mathcal{F}';B_{R+1}) - \mathcal{P}(\mathcal{F};B_{R+1})
= \mathcal{P}(\mathcal{E}_0; B_{R+1}\backslash\overline{O_{R}}) + \mathcal{P}(\mathcal{F}';\partial O_{R}) - \mathcal{P}(\mathcal{F};B_{R+1}\backslash O_{R}).
\end{equation} To compute the perimeter of 
$\mathcal{F}'$ on 
$\partial O_{R}$, we recall that by (3.11) the finite regions of 
$\mathcal{E}_0(i)$ are compactly contained in OR and that 
$\mathcal{P}(\mathcal{E}_0;\partial O_{R})=0$, and hence we can decompose 
$\partial O_{R}$, up to a negligible set, into the disjoint union of the sets 
$\partial O_{R}\cap\mathcal{E}_0(i)$ for 
$i\in I_{\mathcal{E}_0}^c:= \{1,\ldots,N\}\backslash I_{\mathcal{E}_0}$ (the indices corresponding to the infinite regions). Therefore
\begin{align*}
\mathcal{P}(\mathcal{F}';\partial O_{R}) & = \sum\limits_{i\in I_{\mathcal{E}_0}^c} \mathcal{P}(\mathcal{F}';\partial O_{R}\cap \mathcal{E}_0(i)) \\
& = \sum\limits_{i\in I_{\mathcal{E}_0}^c} \mathcal{H}^1\bigl(\partial O_{R}\cap \mathcal{E}_0(i)\cap \mathcal{F}(i)^{(0)}\bigr)\\
&+ \sum\limits_{i\in I_{\mathcal{E}_0}^c} \mathcal{H}^1\bigl( \{\nu_{\mathcal{F}(i)}=-\nu_{O_{R}}\}\cap \mathcal{E}_0(i) \bigr).
\end{align*} Here 
$A^{(\theta)}$ denotes the set of points where A has Lebesgue density 
$\theta\in[0,1]$, and we write 
$\{\nu_A=\pm\nu_B\}:= \{x\in\partial^*A\cap\partial^*B \,:\, \nu_A(x)=\pm\nu_B(x)\}$ for any two sets of finite perimeter A and B. Similarly,
\begin{equation*}
\begin{array}{rcl}
\mathcal{P}(\mathcal{F};B_{R+1} \backslash O_{R}) &=& \sum\limits_{i\in I_{\mathcal{E}_0}^c} \mathcal{P}(\mathcal{F};\mathcal{E}_0(i)\cap B_{R+1}\backslash O_{R})\\
&&+ \frac12\sum\limits_{i\in I_{\mathcal{E}_0}^c} \mathcal{P}(\mathcal{F};\partial^*\mathcal{E}_0(i)\cap B_{R+1}\backslash O_{R}) \\
& \geq & \sum\limits_{i\in I_{\mathcal{E}_0}^c} \mathcal{P}(\mathcal{F}(i);\mathcal{E}_0(i)\cap B_{R+1}\backslash O_{R})\\
&&+ \frac12\sum\limits_{i\in I_{\mathcal{E}_0}^c} \mathcal{H}^1\bigl( \{\nu_{\mathcal{F}(i)}=\nu_{\mathcal{E}_0(i)}\}\cap B_{R+1}\backslash\overline{O_{R}} \bigr).
\end{array}
\end{equation*}By inserting the last two identities into (3.24) we obtain
\begin{equation}
\begin{array}{l}
\mathcal{P} (\mathcal{F}'; B_{R+1}) - \mathcal{P}(\mathcal{F};B_{R+1}) \leqslant \mathcal{P}(\mathcal{E}_0; B_{R+1}\backslash\overline{O_{R}}) \\
\qquad + \sum\limits_{i\in I_{\mathcal{E}_0}^c} \mathcal{H}^1\bigl(\partial O_{R}\cap \mathcal{E}_0(i)\cap \mathcal{F}(i)^{(0)}\bigr)\\
\qquad+ \sum\limits_{i\in I_{\mathcal{E}_0}^c} \mathcal{H}^1\bigl( \{\nu_{\mathcal{F}(i)}=-\nu_{O_{R}}\}\cap \mathcal{E}_0(i) \bigr) \\
\qquad - \sum\limits_{i\in I_{\mathcal{E}_0}^c} \mathcal{P}(\mathcal{F}(i);\mathcal{E}_0(i)\cap B_{R+1}\backslash O_{R})\\
\qquad- \frac12\sum\limits_{i\in I_{\mathcal{E}_0}^c} \mathcal{H}^1\bigl( \{\nu_{\mathcal{F}(i)}=\nu_{\mathcal{E}_0(i)}\}\cap B_{R+1}\backslash\overline{O_{R}} \bigr).
\end{array}
\end{equation} Define now for 
$i\in I_{\mathcal{E}_0}^c$ the sets
\begin{equation*}
\mathcal{G}(i) := \bigl(\mathcal{E}_0(i) \cap \mathcal{F}(i)\bigr) \backslash O_{R} .
\end{equation*}By [Reference Maggi19, Theorem 16.3] we have
\begin{align*}
\mathcal{P}(\mathcal{G}(i);B_{R+1}) &= \mathcal{P}(\mathcal{F}(i); \mathcal{E}_0(i)\cap B_{R+1}\backslash\overline{O_{R}}) + \mathcal{P}(\mathcal{E}_0(i); \mathcal{F}(i)^{(1)}\cap B_{R+1}\backslash\overline{O_R}) \\
&\quad+ \,\mathcal{H}^1\bigl( \partial O_R \cap \mathcal{E}_0(i) \cap \mathcal{F}(i)^{(1)} \bigr) \\
&\quad+ \,\mathcal{H}^1\bigl( \{\nu_{\mathcal{F}(i)}=\nu_{\mathcal{E}_0(i)}\}\cap B_{R+1}\backslash\overline{O_{R}} \bigr)\\
&\quad+ \,\mathcal{H}^1\bigl( \{\nu_{\mathcal{F}(i)}=-\nu_{O_{R}}\}\cap\mathcal{E}_0(i)\bigr).
\end{align*}Inserting this identity into (3.25) we find
\begin{equation*}
\begin{array}{l}
\mathcal{P}(\mathcal{F}'; B_{R+1}) - \mathcal{P}(\mathcal{F};B_{R+1}) \leqslant \mathcal{P}(\mathcal{E}_0; B_{R+1}\backslash\overline{O_{R}}) \\
\qquad + \sum\limits_{i\in I_{\mathcal{E}_0}^c} \mathcal{H}^1\bigl(\partial O_{R}\cap \mathcal{E}_0(i)\cap \mathcal{F}(i)^{(0)}\bigr) + \sum\limits_{i\in I_{\mathcal{E}_0}^c} \mathcal{H}^1\bigl( \{\nu_{\mathcal{F}(i)}=-\nu_{O_{R}}\}\cap \mathcal{E}_0(i) \bigr) \\
\qquad - \sum\limits_{i\in I_{\mathcal{E}_0}^c} \mathcal{P}(\mathcal{F}(i);\mathcal{E}_0(i)\cap\partial O_{R}) + \frac12\sum\limits_{i\in I_{\mathcal{E}_0}^c} \mathcal{H}^1\bigl( \{\nu_{\mathcal{F}(i)}=\nu_{\mathcal{E}_0(i)}\}\cap B_{R+1}\backslash\overline{O_{R}} \bigr) \\
\qquad + \sum\limits_{i\in I_{\mathcal{E}_0}^c}\mathcal{P}(\mathcal{E}_0(i); \mathcal{F}(i)^{(1)}\cap B_{R+1}\backslash\overline{O_R}) + \sum\limits_{i\in I_{\mathcal{E}_0}^c} \mathcal{H}^1\bigl( \partial O_R \cap \mathcal{E}_0(i) \cap \mathcal{F}(i)^{(1)} \bigr) \\
\qquad + \sum\limits_{i\in I_{\mathcal{E}_0}^c}\mathcal{H}^1\bigl( \{\nu_{\mathcal{F}(i)}=-\nu_{O_{R}}\}\cap\mathcal{E}_0(i)\bigr) - \sum\limits_{i\in I_{\mathcal{E}_0}^c}\mathcal{P}(\mathcal{G}(i);B_{R+1}).
\end{array}
\end{equation*} We can group together all the terms on 
$\partial O_R$, whose combination is controlled by 
$\mathcal{H}^1(\partial O_R)$:
\begin{align*}
\mathcal{H}^1\bigl(\partial O_{R}&\cap \mathcal{E}_0(i)\cap \mathcal{F}(i)^{(0)}\bigr)+ 2\mathcal{H}^1\bigl( \{\nu_{\mathcal{F}(i)}=-\nu_{O_{R}}\}\cap \mathcal{E}_0(i) \bigr) \\
&\quad - \mathcal{P}(\mathcal{F}(i);\mathcal{E}_0(i)\cap\partial O_{R}) + \mathcal{H}^1\bigl( \partial O_R \cap \mathcal{E}_0(i) \cap \mathcal{F}(i)^{(1)} \bigr) \\
&\quad \leqslant \mathcal{H}^1\bigl(\partial O_{R}\cap \mathcal{E}_0(i)\cap \mathcal{F}(i)^{(0)}\bigr) + \mathcal{H}^1\bigl( \partial O_R \cap \mathcal{E}_0(i) \cap \mathcal{F}(i)^{(1)} \bigr) \\
&\quad+ \mathcal{H}^1\bigl( \partial O_R \cap \mathcal{E}_0(i) \cap \partial^*\mathcal{F}(i) \bigr) = \mathcal{H}^1(\partial O_R\cap\mathcal{E}_0(i)).
\end{align*}Therefore
\begin{equation*}
\begin{array}{rcl}
&&\mathcal{P}(\mathcal{F}'; B_{R+1}) - \mathcal{P}(\mathcal{F};B_{R+1})\\
&&\quad\leqslant \frac12\sum\limits_{i\in I_{\mathcal{E}_0}^c}\mathcal{P}(\mathcal{E}_0(i);B_{R+1}\backslash O_R) + \sum\limits_{i\in I_{\mathcal{E}_0}^c}\mathcal{H}^1\bigl(\partial O_R \cap \mathcal{E}_0(i)\bigr) \\
&&\qquad+\frac12\sum\limits_{i\in I_{\mathcal{E}_0}^c} \mathcal{H}^1\bigl( \{\nu_{\mathcal{F}(i)}=\nu_{\mathcal{E}_0(i)}\}\cap B_{R+1}\backslash\overline{O_{R}} \bigr) \\
&&\qquad+\sum\limits_{i\in I_{\mathcal{E}_0}^c}\mathcal{P}(\mathcal{E}_0(i); \mathcal{F}(i)^{(1)}\cap B_{R+1}\backslash\overline{O_R}) - \sum\limits_{i\in I_{\mathcal{E}_0}^c}\mathcal{P}(\mathcal{G}(i);B_{R+1}).
\end{array}
\end{equation*}Observe now that
\begin{equation*}
\begin{array}{l}
\sum\limits_{i\in I_{\mathcal{E}_0}^c}\biggl[ \frac12\mathcal{H}^1\bigl( \{\nu_{\mathcal{F}(i)}=\nu_{\mathcal{E}_0(i)}\}\cap B_{R+1}\backslash\overline{O_{R}} \bigr) + \mathcal{P}(\mathcal{E}_0(i); \mathcal{F}(i)^{(1)}\cap B_{R+1}\backslash\overline{O_R}) \biggr] \\
\qquad\qquad\quad= \sum\limits_{i\in I_{\mathcal{E}_0}^c}\frac12\mathcal{P}(\mathcal{E}_0(i);B_{R+1}\backslash O_R),
\end{array}
\end{equation*}which yields
\begin{align*}
\mathcal{P}(\mathcal{F}'; B_{R+1}) - \mathcal{P}(\mathcal{F};B_{R+1}) &\leqslant \sum\limits_{i\in I_{\mathcal{E}_0}^c} \biggl( \mathcal{P}(\mathcal{E}_0(i); B_{R+1}\backslash O_R) + \mathcal{H}^1\bigl(\partial O_R \cap \mathcal{E}_0(i)\bigr)\\
&\qquad- \mathcal{P}(\mathcal{G}(i);B_{R+1})\biggr) \\
& = \sum\limits_{i\in I_{\mathcal{E}_0}^c} \biggl( \mathcal{P}(\mathcal{E}_0(i)\backslash O_R; B_{R+1}) - \mathcal{P}(\mathcal{G}(i);B_{R+1})\biggr).
\end{align*} For each 
$i\in I_{\mathcal{E}_0}^c$ the set 
$\mathcal{E}_0(i)\backslash O_R$ has boundary of class C 2. By a standard result, that we recall in Lemma 3.6, any set with boundary of class C 2 is a quasi-minimizer of the perimeter for a suitably large constant, depending on the set itself: that is, we can find a constant 
$C_4 \gt 0$, depending ultimately only on 
$\mathcal{E}_0$ and 
$\overline{R}$, such that
\begin{align}
\mathcal{P}(\mathcal{F}'; B_{R+1}) - \mathcal{P}(\mathcal{F};B_{R+1}) & \leqslant C_4\sum\limits_{i\in I_{\mathcal{E}_0}^c}\big| (\mathcal{E}_0(i)\backslash O_R)\triangle\mathcal{G}(i)\big| \nonumber\\
& = C_4\sum\limits_{i\in I_{\mathcal{E}_0}^c}\big| |(\mathcal{E}_0(i)\backslash\mathcal{F}(i)) \cap (B_{R+1}\backslash O_R) \big| \nonumber\\
& \leqslant C_4 \sum\limits_{i\in I_{\mathcal{E}_0}^c}\big| \mathcal{E}_k(i)\triangle\mathcal{F}(i) \big|,
\end{align} where we used the fact that 
$|(\mathcal{E}_k(i)\triangle\mathcal{E}_0(i))\backslash O_R|=0$ in the last inequality.
By inserting (3.26) into (3.23) we finally obtain the quasi-minimality inequality (3.6) with 
$\Lambda:= C_3+C_4$ depending only on 
$\mathcal{E}_0$ and 
$\overline{R}$.
The following property, used in the proof of Theorem 3.5, is well-known to the experts (see for instance [Reference Acerbi, Fusco and Morini1, Lemma 4.1]). Here we state and prove it in the setting of the relative perimeter for the convenience of the reader.
Lemma 3.6. Let 
$E\subset\mathbb R^d$ be an open set with boundary of class C 2, and let ρ > 0. Then there exists a constant C > 0, depending only on 
$\textit{E}$ and ρ, such that for every set of finite perimeter 
$F\subset\mathbb R^d$ with 
$E\triangle F\subset\!\subset B_\rho$ one has
\begin{equation*}
\mathcal{P}(E;B_\rho) \leqslant \mathcal{P}(F;B_\rho) + C |E\triangle F|.
\end{equation*}Proof. Let 
$X\in C^1(\mathbb R^d;\mathbb R^d)$ be a vector field such that 
$X=\nu_E$ on 
$\partial E$ and 
$\|X\|_\infty \leqslant 1$. Then
\begin{align*}\mathcal P(E;B_\rho)-\mathcal P(F;B_\rho)&\leqslant\int_{\partial E\cap B_\rho}X\cdot\nu_E\,\mathrm d\mathcal H^{d-1}-\int_{\partial^\ast F\cap B_\rho}X\cdot\nu_F\,\mathrm d\mathcal H^{d-1}\\ &=\int_{E\cap B_\rho}\!\!\!\!\operatorname{div}X\,\mathrm dx-\int_{F\cap B_\rho}\operatorname{div}X\,\mathrm dx\leqslant\Arrowvert\operatorname{div}X\Arrowvert_{L^\infty(B_\rho)}\vert E\triangle F\vert,\end{align*} where we used the fact that, in applying the divergence theorem, all the terms on 
$\partial B_\rho$ cancel out due to the assumption 
$E\triangle F\subset\!\subset B_{\rho}$.
3.2. Improved convergence
The next step of the strategy exploits the improved convergence theorem for clusters by Cicalese, Leonardi, and Maggi [Reference Cicalese, Leonardi and Maggi10], which allows us to conclude that the partitions 
$\mathcal{E}_k$ of the recovery sequence for 
$\kappa_R(\mathcal{E}_0)$ constructed in Theorem 3.5 are actually smooth perturbations of 
$\mathcal{E}_0$. Combined with Theorem 3.2, this reduces the proof of the stability inequality for 
$\mathcal{E}_0$ to a suitable class of smooth perturbations of 
$\mathcal{E}_0$. We premise some notation for smooth planar partitions, following [Reference Cicalese, Leonardi and Maggi10].
Definition 3.7. A partition 
$\mathcal{E}=(\mathcal{E}(1),\ldots,\mathcal{E}(N))$ in 
$\mathbb R^2$ is a 
$C^{k,\alpha}$-partition (
$k\in\mathbb N$, 
$\alpha\in[0,1]$) if there exist a finite set of points 
$\{p_i\}_{i\in I}$ and a finite family 
$\{\gamma_j\}_{j\in J}$ of closed, connected 
$C^{k,\alpha}$-curves with boundary such that
\begin{equation*}
\partial\mathcal{E} = \bigcup_{j\in J}\gamma_j,
\qquad
\partial^*\mathcal{E} = \bigcup_{j\in J}\mathrm{int}(\gamma_j),
\qquad
\Sigma(\mathcal{E}) = \bigcup_{j\in J}\mathrm{bd(\gamma_j)}=\bigcup_{i\in I}\{p_i\}.
\end{equation*} For a 
$C^{k,\alpha}$-partition 
$\mathcal{E}=(\mathcal{E}(1),\ldots,\mathcal{E}(N))$, with 
$\{p_i\}_{i\in I}$ and 
$\{\gamma_j\}_{j\in J}$ as in Definition 3.7, we say that 
$f\in C^{k,\alpha}(\partial\mathcal{E};\mathbb R^2)$ if 
$f\colon \partial\mathcal{E} \to \mathbb R^2$ is continuous, 
$f\in C^{k,\alpha}(\gamma_j;\mathbb R^2)$ for every 
$j\in J$, and
\begin{equation*}
\| f \|_{C^{k,\alpha}(\partial\mathcal{E})} := \sup_{j\in J} \| f \|_{C^{k,\alpha}(\gamma_j)} \lt \infty.
\end{equation*} Moreover, given two 
$C^{k,\alpha}$-partitions 
$\mathcal{E}$ and 
$\mathcal{F}$, we say that a map 
$f\colon \partial\mathcal{E} \to \partial\mathcal{F}$ is a 
$C^{k,\alpha}$-diffeomorphism between 
$\partial\mathcal{E}$ and 
$\partial\mathcal{F}$ if f is a homeomorphism such that 
$f \in C^{k,\alpha}(\partial \mathcal{E};\mathbb R^2)$, 
$f^{-1}\in C^{k,\alpha}(\partial\mathcal{F};\mathbb R^2)$, and 
$f(\Sigma(\mathcal{E}))=\Sigma(\mathcal{F})$.
Given a partition 
$\mathcal{E}$ in 
$\mathbb R^2$ and a map 
$f:\mathbb R^2\to\mathbb R^2$, we also define the tangential component of f with respect to 
$\mathcal{E}$ as the map 
$\tau_{\mathcal{E}}f:\partial^*\mathcal{E}\to\mathbb R^2$ given by
\begin{equation*}
\tau_{\mathcal{E}}f (x) := f(x) - \bigl(f(x)\cdot\nu_{\mathcal{E}}(x)\bigr)\nu_{\mathcal{E}}(x), \qquad x\in\partial^*\mathcal{E},
\end{equation*} where 
$\nu_{\mathcal{E}}:\partial^*\mathcal{E}\to\mathbb{S}^1$ is any Borel function such that either 
$\nu_{\mathcal{E}}(x)=\nu_{\mathcal{E}(i)}(x)$ or 
$\nu_{\mathcal{E}}(x)=\nu_{\mathcal{E}(j)}(x)$ for 
$x\in\mathcal{E}(i,j)$, i ≠ j.
With these positions, we can state the main result of this section, which is a direct consequence of the improved convergence theorem in [Reference Cicalese, Leonardi and Maggi10].
Theorem 3.8 Let 
$\mathcal{E}_0=(\mathcal{E}_0(1),\ldots,\mathcal{E}_0(N))$ be a uniquely minimizing locally isoperimetric partition in 
$\mathbb R^2$ with at least two infinite regions, and fix 
$\overline{R} \gt 0$.
Then there exist constants R > 0, 
$\mu_0 \gt 0$, 
$C_0 \gt 0$ (depending only on 
$\mathcal{E}_0$ and 
$\overline{R}$), and a sequence of 
$C^{1,1}$-partitions 
$(\mathcal{F}_k)_{k\in\mathbb N}\subset\mathfrak{M}_{R}(\mathcal{E}_0)$ such that
\begin{equation}
\kappa_{\overline{R}}(\mathcal{E}_0) = \lim_{k\to\infty}\frac{\delta(\mathcal{F}_k,\mathcal{E}_0)}{\Delta(\mathcal{F}_k,\mathcal{E}_0)^2}
\qquad\text{and}\qquad
\lim_{k\to\infty}\mathrm{d}(\mathcal{F}_k,\mathcal{E}_0)=0,
\end{equation} and for every 
$\mu\in(0,\mu_0)$ there exist 
$k(\mu)\in\mathbb N$ and a sequence of 
$C^{1,1}$-diffeomorphisms 
$(f_k)_{k \geq k(\mu)}$ between 
$\partial\mathcal{E}_0$ and 
$\partial\mathcal{F}_k$ with the following properties:
(i)
$\operatorname{supp}(f_k-\operatorname{Id}) \subset\!\subset B_R$,(ii)
$\| f_k \|_{C^{1,1}(\partial \mathcal{E}_0)} \leqslant C_0$,(iii)
$\| f_k - \operatorname{Id} \|_{C^1(\partial \mathcal{E}_0)} \to 0$ as
$k\to\infty$,(iv)
$\tau_{\mathcal{E}_0}(f_k - \operatorname{Id}) =0$ on
$\partial \mathcal{E}_0 \backslash I_{\mu}(\Sigma(\mathcal{E}_0))$, where
$I_{\mu}(\Sigma(\mathcal{E}_0)):= \{x \in \mathbb R^2 \colon {\mathrm{dist}}(x,\Sigma(\mathcal{E}_0)) \lt \mu\}$,(v)
$\| \tau_{\mathcal{E}_0}(f_k-\operatorname{Id})\|_{C^1(\partial^*\mathcal{E}_0)} \leqslant \frac{C_0}{\mu}\| f-\operatorname{Id} \|_{C^0(\Sigma(\mathcal{E}_0))}$.
Proof. Let R > 0 and 
$(\mathcal{E}_k)_{k\in\mathbb N}\subset\mathfrak{M}_{R+1}(\mathcal{E}_0)$ be the sequence of 
$(\Lambda,r_0)$-minimizing partitions given by Theorem 3.5. We can find isometries 
$T_k:\mathbb R^2\to\mathbb R^2$ such that 
$\mathcal{F}_k:= T_k(\mathcal{E}_k)$ satisfy the conditions (3.27). Notice that, since 
$\mathcal{E}_k(i)\triangle\mathcal{E}_0(i)\subset\!\subset B_{R+1}$ for all i, up to taking a larger R we can assume that the same condition is satisfied by 
$\mathcal{F}_k$.
Since 
$\mathcal{E}_0$ is in particular a 
$C^{2,1}$-partition by Theorem 2.3, and 
$(\mathcal{F}_k)_k$ are 
$(\Lambda,r_0)$-minimizing partitions such that 
$\mathrm{d}(\mathcal{F}_k,\mathcal{E}_0)\to0$, the conclusion follows from [Reference Cicalese, Leonardi and Maggi10, Theorem 1.5] by noticing that all the arguments there are local and can be adapted to our case.
4. Stability of the lens among smooth perturbations
In this section we prove Theorem 2.5 among smooth perturbations of the standard lens partition. In view of Theorem 3.8, we introduce the following class of smooth perturbations of a locally isoperimetric partition 
$\mathcal{E}_0$.
Definition 4.1. Let 
$\mathcal{E}_0$ be a locally isoperimetric partition in 
$\mathbb R^2$. Given constants 
$\varepsilon_0 \gt 0$ and R > 0, a 
$\textit{C}^1$-partition 
$\mathcal{E}$ is said to be an ɛ0-perturbation of 
$\mathcal{E}_0$ in R if 
$|\mathcal{E}(i)|=|\mathcal{E}_0(i)|$ for all i, and there exists a 
$\textit{C}^1$-diffeomorphism Ψ between 
$\partial\mathcal{E}_0$ and 
$\partial \mathcal{E}$ such that
\begin{equation*}
\operatorname{supp}(\Psi-\operatorname{Id}) \subset\!\subset B_R
\qquad\text{and}\qquad
\| \Psi - \operatorname{Id} \|_{C^1(\partial \mathcal{E}_0)} \leqslant \varepsilon_0.
\end{equation*} Taking 
$\mathcal{E}_0=\mathcal{L}_{m}$ to be lens partition, its stability among these smooth perturbations is given by the following result.
Theorem 4.2. (Stability among ɛ 0-perturbations)
Let m > 0 and R > 0 be fixed. There exist 
$\varepsilon_0 \gt 0$ and 
$\kappa_0 \gt 0$ (depending on m and R) such that if 
$\mathcal{E}$ is an ɛ 0-perturbation of 
$\mathcal{L}_{m}$ in 
${B}_{R}$, then
\begin{equation*}
\delta(\mathcal{E},\mathcal{L}_{m}) \geq \kappa_0\Delta(\mathcal{E},\mathcal{L}_{m})^2.
\end{equation*}In order to prove this theorem we will use a geometric fact which states that the image of a graph by a diffeomorphism close to the identity is again a graph. We state this fact as a separate lemma since it could be of independent interest.
Lemma 4.3. Let 
$f\in C^{1,1}([a,b])$, with 
$-\infty \lt a \lt b \lt +\infty$. Then, there exist 
$\lambda_0 \gt 0$ and C > 0, depending only on the 
$C^{1,1}$-norm of f and on 
$|b-a|$, with the following property: if Ψ is a C 1-diffeomorphism between 
$\mathrm{graph}(f)$ and its image with 
$\|\Psi - \mathrm{Id}\|_{C^1(\mathrm{graph}(f))} \lt \lambda_0$, then there exist 
$-\infty \lt c \lt d \lt +\infty$ and 
$\widetilde{f}\in C^1([c,d])$ such that
\begin{equation}\Psi(\mathrm{graph}(f))=\mathrm{graph}(\widetilde f)\ \text{and}\ \Arrowvert\widetilde f-f\circ\eta\Arrowvert_{C^1(\lbrack c,d\rbrack)}\leqslant C\Arrowvert\Psi-\mathrm{Id}\Arrowvert_{C^1(\mathrm{graph}(f))},\end{equation} where 
$\eta:[c,d]\to [a,b]$ is defined as
\begin{equation*}
\eta(x):= a + \frac{b-a}{d-c}(x-c).
\end{equation*}Proof. Using Whitney’s Extension Theorem (see [Reference Cicalese, Leonardi and Maggi10, Theorem 2.3]), we can extend Ψ to a diffeomorphism defined in the entire 
$\mathbb R^2$ with
\begin{equation}
\|\Psi-\mathrm{Id}\|_{C^1(\mathbb R^2;\mathbb R^2)} \leqslant C \|\Psi-\mathrm{Id}\|_{C^1(\mathrm{graph}(f))} \leqslant C\lambda_0
\end{equation} for a constant C > 0 depending only on 
$|b-a|$ and on the C 0-norm of f. Consider the function 
$G:[a,b]\to\mathbb R$ defined as
\begin{equation*}
G(x):= \Pi_1\left( \Psi(x,f(x)) \right),
\end{equation*} where 
$\Pi_1:\mathbb R^2\to\mathbb R$ is the projection on the first coordinate. Note that 
$G\in C^1([a,b])$. By writing 
$\Psi(x,y) = (\Psi_1(x,y), \Psi_2(x,y))$, so that 
$G(x)=\Psi_1(x,f(x))$, we easily obtain the following estimates:
\begin{equation*}
|G(x) - x| \leqslant |(\Psi-\mathrm{Id})(x,f(x))| \leqslant \|\Psi-\mathrm{Id}\|_{C^0(\mathbb R^2;\mathbb R^2)}
\end{equation*}and
\begin{align*}
|G'(x)-1|& = |\partial_x\Psi_1(x,f(x)) + \partial_y\Psi_1(x,f(x)) f'(x) -1 | \\
& \leqslant |\partial_x\Psi_1(x,f(x))-1| + |\partial_y\Psi_1(x,f(x))| |f'(x)| \\
& \leqslant \bigl(1+\|f\|_{C^1([a,b])}\bigr) \|\Psi - \mathrm{Id}\|_{C^1(\mathbb R^2;\mathbb R^2)}.
\end{align*}It follows in particular that
\begin{equation}
\| G - \mathrm{Id} \|_{C^1([a,b])} \leqslant \bigl(1+\|f\|_{C^1([a,b])}\bigr) \|\Psi - \mathrm{Id}\|_{C^1(\mathbb R^2;\mathbb R^2)},
\end{equation}so that choosing
\begin{equation*}
\lambda_0 := \frac{1}{2C\bigl(1+\|f\|_{C^1([a,b])}\bigr)}
\end{equation*} and recalling (4.2) can guarantee that G is invertible on 
$[a,b]$ with a 
$\textit{C}^1$-inverse 
$G^{-1}:[c,d]\to[a,b]$, where 
$c:= G(a)$ and 
$d:= G(b)$.
Define 
$\widetilde{f}:[c,d]\to\mathbb R$ as
\begin{equation*}
\widetilde{f}(x) := \Psi_2\left( G^{-1}(x), f(G^{-1}(x)) \right),
\end{equation*} so that 
$\widetilde{f}$ is of class 
$\textit{C}^1$ and, by definition, 
$\Psi(\mathrm{graph}(f)) = \mathrm{graph}(\widetilde{f})$, proving the first condition in (4.1). It remains to estimate 
$\|\widetilde{f} - f\circ\eta \|_{C^1([c,d])}$. In order to do so, we first observe that, by (4.3), we also have
\begin{equation}
\|G^{-1} - \mathrm{Id} \|_{C^1([c,d])}
\leqslant 2 \|G - \mathrm{Id} \|_{C^1([a,b])}
\leqslant 2 \bigl(1+\|f\|_{C^1([a,b])}\bigr)\|\Psi-\mathrm{Id}\|_{C^1(\mathbb R^2;\mathbb R^2)}.
\end{equation} Moreover, for 
$x\in[c,d]$ we have
\begin{align*}
|\eta(x) - x|& = \Big| a + \frac{b-a}{G(b)-G(a)}(x-G(a)) -x \Big| \\
& = \Big| a - G(a) + \Bigl(\frac{b-a}{G(b)-G(a)}-1\Bigr)(x-G(a)) \Big| \\
& \leqslant |a-G(a)| + \bigg| \frac{b-a}{G(b)-G(a)}-1 \bigg| |G(b)-G(a)| \\
& \leqslant \|G-\mathrm{Id}\|_{C^0([a,b])} + \|G^{-1}-\mathrm{Id}\|_{C^1([c,d])} \|G\|_{C^1([a,b])}|b-a|\\
& \leqslant c_0 \|\Psi-\mathrm{Id}\|_{C^1(\mathbb R^2;\mathbb R^2)},
\end{align*} where last step follows from (4.3) and (4.4), for a constant c 0 depending on 
$|b-a|$ and on 
$\|f\|_{C^1([a,b])}$. Similarly,
\begin{equation*}
|\eta'(x) - 1|= \bigg| \frac{b-a}{G(b)-G(a)} - 1 \bigg| \leqslant \|G^{-1}-\mathrm{Id}\|_{C^1([c,d])} \leqslant c_0 \|\Psi-\mathrm{Id}\|_{C^1(\mathbb R^2;\mathbb R^2)}.
\end{equation*}Thus,
\begin{equation}
\|\eta - \mathrm{Id}\|_{C^1([a,b])} \leqslant c_0 \|\Psi-\mathrm{Id}\|_{C^1(\mathbb R^2;\mathbb R^2)}.
\end{equation} We are now in a position to obtain the desired estimate. For 
$x\in[c,d]$ we write
\begin{align*}
\widetilde{f}(x) - f\circ\eta(x) &= \widetilde{f}(x) - f(G^{-1}(x)) + f(G^{-1}(x)) - f(\eta(x)) \\
&= \Pi_2\bigl( (\Psi-\mathrm{Id}) \bigl(G^{-1}(x), f(G^{-1}(x)) \bigl) \bigr) + f(G^{-1}(x)) - f(\eta(x)),
\end{align*}from which we get that
\begin{align*}
\| \widetilde{f} - f\circ\eta \|_{C^0([c,d])} &\leqslant \|\Psi-\mathrm{Id}\|_{C^0(\mathbb R^2;\mathbb R^2)}+ \|f'\|_{C^0([a,b])} \bigl( \|G^{-1} - \mathrm{Id}\|_{C^0([c,d])}\\
&\qquad+ \|\eta - \mathrm{Id}\|_{C^0([c,d])} \bigr)
\end{align*}and
\begin{align*}
\| \widetilde{f}' - (f\circ\eta)' \|_{C^0([c,d])}& \leqslant \|G^{-1}\|_{C^1([c,d])} \bigl(1 + \|f'\|_{C^0([a,b])}\bigr) \|\Psi-\mathrm{Id}\|_{C^1(\mathbb R^2;\mathbb R^2)} \\
& \quad+ \| (f'\circ G^{-1}) ((G^{-1})'-1)\|_{C^0([c,d])}\\
& \quad+ \| (f'\circ\eta)(\eta'-1)\|_{C^0([c,d])} \\
& \quad+ \| f'\circ G^{-1} - f'\circ\eta \|_{C^0([c,d])} \\
& \leqslant \|G^{-1}\|_{C^1([c,d])} \bigl(1 + \|f'\|_{C^0([a,b])}\bigr) \|\Psi-\mathrm{Id}\|_{C^1(\mathbb R^2;\mathbb R^2)} \\
& \quad+ \|f\|_{C^{1,1}([a,b])} \bigl( \|G^{-1} - \mathrm{Id} \|_{C^1([c,d])}\\
& \quad+ \|\eta - \mathrm{Id}\|_{C^1([c,d])} \bigr).\end{align*}Thus, combining (4.4) and (4.5), and recalling (4.2), we get the second estimate in (4.1).
Proof of Theorem 4.2
Note that by scaling, we can assume that
\begin{equation}
r_m = 1,
\end{equation} which, by (2.15), corresponds to taking 
$m=\frac{2\pi}{3}-\frac{\sqrt{3}}{2}$. The singular set 
$\Sigma(\mathcal{L}_{m})$ is given by the two points 
$p_1=(-\sqrt{3}/2,0)$ and 
$p_2=(\sqrt{3}/2,0)$. We can also assume without loss of generality that 
$R \geq R_0$, where 
$L_{m}\subset\!\subset B_{R_0}$, since any perturbation in a smaller ball is also a perturbation in a larger ball. We write 
$\partial \mathcal{E}_0=\gamma_0 \cup \gamma_1 \cup \gamma_2$ with
\begin{equation*}
\gamma_0 = \Big\{(x,0) \colon |x| \geq \sqrt{3}/2\Big\}, \qquad \gamma_i = \Big\{(x,u_i(x)) \colon |x| \leqslant \sqrt{3}/2 \Big\}, \quad i=1,2,
\end{equation*} where 
$u_1(x)=\sqrt{1-x^2}-1/2$ and 
$u_2(x)=-u_1(x)$.
Step 1: Graph representation. Let 
$\mathcal{E}$ be an ɛ 0-perturbation of 
$\mathcal{L}_{m}$ in 
${B}_{R}$, with 
$\varepsilon_0\in(0,1)$ to be chosen later, and let 
$(x_i,y_i)=\Psi(p_i)$, where Ψ is a diffeomorphism as in Definition 4.1. We apply a small horizontal translation to 
$\mathcal{E}$ by 
$\tau=(-\frac{1}{2}(x_1+x_2),0)$ (notice that 
$|\tau| \leqslant \varepsilon_0$) and define 
$\mathcal{F}=\mathcal{E}+\tau$ so that
\begin{align*}
|\mathcal{F}(1)|=|\mathcal{E}(1)| = m, \quad \mathcal{F}(i)\triangle\mathcal{L}_{m}(i)\subset\!\subset B_{R+1},\end{align*}
\begin{align*}
\partial\mathcal{F} = \bigcup_{i=0}^2 \big(\Psi(\gamma_i)+\tau\big),\end{align*}
\begin{align*}\Sigma(\mathcal{F}) = \{q_1,q_2\}\ \text{with }\ q_i=\Psi(p_i)+\tau.
\end{align*} Note that with this translation 
$q_1\cdot e_1 = -q_2\cdot e_1$ (see Figure 5). We can then write
\begin{equation*}q_1=\left(-\frac{\sqrt3}2(1+\sigma),y_1\right)\ \text{and} \ q_2=\left(\frac{\sqrt3}2(1+\sigma),y_2\right)\end{equation*} for some 
$\sigma\in[-2\varepsilon_0,2\varepsilon_0]$. Notice that 
$\mathcal{F}$ is a 
$(2\varepsilon_0)$-perturbation of 
$\mathcal{L}_{m}$ in 
$B_{R+1}$.
A translated smooth perturbation of the standard lens partition 
$\mathcal{L}_{m}$.
We write
\begin{equation*}
\partial\mathcal{F} = \tilde{\gamma}_0 \cup \tilde{\gamma}_1 \cup \tilde{\gamma}_2,
\end{equation*} where the former is the boundary between the two unbounded regions, while the other two are the boundaries between the finite region and the upper and the lower unbounded regions, respectively. By Lemma 4.3 we can find ɛ 0 sufficiently small and 
$C_0 \gt 0$, depending on R, such that for all 
$\mathcal{F}$ as before there exist
\begin{equation*}
\begin{array}{l}
g_0 \in C^1_c\Big((-R-1,R+1)\backslash\big(-\tfrac{\sqrt{3}}{2}(1+\sigma),\tfrac{\sqrt{3}}{2}(1+\sigma)\big)\Big), \\
\qquad g_1,g_2 \in C^1\Big(\big[-\tfrac{\sqrt{3}}{2}(1+\sigma),\tfrac{\sqrt{3}}{2}(1+\sigma)\big]\Big),
\end{array}
\end{equation*} with 
$g_1 \geq g_2$,
\begin{equation}
g_0\big(\pm(1+\sigma)\tfrac{\sqrt{3}}{2}\big)=g_1\big(\pm(1+\sigma)\tfrac{\sqrt{3}}{2}\big)=g_2\big(\pm(1+\sigma)\tfrac{\sqrt{3}}{2}\big),
\end{equation}
\begin{equation}
\int_{-(1+\sigma)\frac{\sqrt{3}}{2}}^{(1+\sigma)\frac{\sqrt{3}}{2}} (g_1-g_2) \,\mathrm{d} x = m = \int_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} (u_1-u_2)\,\mathrm{d} x,
\end{equation}such that
\begin{align*}
\tilde{\gamma}_0 &= \Big\{(x,g_0(x)) \colon |x| \geq (1+\sigma)\tfrac{\sqrt{3}}{2} \Big\},\\
\tilde{\gamma}_i &= \Big\{(x,g_i(x)) \colon |x| \leqslant (1+\sigma)\tfrac{\sqrt{3}}{2} \Big\} \quad\text{for } i=1,2,
\end{align*}and
\begin{equation}
\|g_0\|_{C^1\big(\big[-\tfrac{\sqrt{3}}{2}(1+\sigma),\tfrac{\sqrt{3}}{2}(1+\sigma)\big]^c\big)} + \sum\limits_{i=1}^2 \| g_i - \tilde{u}_i \|_{C^1\big(\big[-\tfrac{\sqrt{3}}{2}(1+\sigma),\tfrac{\sqrt{3}}{2}(1+\sigma)\big]\big)} \leqslant C_0\varepsilon_0,
\end{equation}where we define
\begin{equation*}
\tilde{u}_i(x) := (1+\sigma) u_i\left(\frac{x}{1+\sigma}\right), \quad i=1,2.
\end{equation*} Step 2: Perimeter deficit. We now estimate the perimeter deficit between 
$\mathcal{E}$ and 
$\mathcal{E}_0$, by exploiting the graph representation obtained in the previous step. For simplicity of notation we call 
$\ell = \frac{\sqrt{3}}{2}$ and denote by Iσ the interval 
$[-(1+\sigma)\ell,(1+\sigma)\ell]$ so that 
$I_0=[-\ell,\ell]$. Then we have
\begin{align*}
\delta(\mathcal{E},\mathcal{L}_{m}) & = \delta(\mathcal{F},\mathcal{L}_{m}) = \mathcal{P}(\mathcal{F};B_{R+1})-\mathcal{P}(\mathcal{L}_{m};B_{R+1}) \\
& = \int_{I_\sigma^c} \left(\sqrt{1+(g_0^\prime)^2}-1\right) \,\mathrm{d} x + \sum\limits_{i=1}^2 \int_{I_\sigma} \sqrt{1+(g_i^\prime)^2} \,\mathrm{d} x\\
& \quad- \sum\limits_{i=1}^2 \int_{I_0} \sqrt{1+(u_i^\prime)^2} \,\mathrm{d} x - 2\sigma\ell \\
& = \int_{I_\sigma^c} \left(\sqrt{1+(g_0^\prime)^2}-1\right) \,\mathrm{d} x\\
& \quad+ \sum\limits_{i=1}^2 \int_{I_\sigma} \left(\sqrt{1+(g_i^\prime)^2}-\sqrt{1+(\tilde{u}_i^\prime)^2}\right) \,\mathrm{d} x \\
& \quad+ \sum\limits_{i=1}^2 \int_{I_\sigma} \sqrt{1+\left(u_i^\prime\left(\frac{y}{1+\sigma}\right)\right)^{2}}\,\mathrm{d} y\\
& \quad- \sum\limits_{i=1}^2 \int_{I_0} \sqrt{1+(u_i^\prime)^2} \,\mathrm{d} x - 2\sigma\ell \\
& = \int_{I_\sigma^c} \left(\sqrt{1+(g_0^\prime)^2}-1\right) \,\mathrm{d} x\\
& \quad+ \sum\limits_{i=1}^2 \int_{I_\sigma} \left(\sqrt{1+(g_i^\prime)^2}-\sqrt{1+(\tilde{u}_i^\prime)^2}\right)\,\mathrm{d} x \\
& \quad+ \sigma \sum\limits_{i=1}^2 \int_{I_0} \sqrt{1+(u_i^\prime)^2}\,\mathrm{d} x - 2\sigma \ell \\
& \geq \frac{1}{2^{5/2}} \int_{I_\sigma^c} (g_0^\prime)^2 \,\mathrm{d} x + \sum\limits_{i=1}^2 \int_{I_\sigma} \frac{\tilde{u}_i^\prime}{\sqrt{1+(\tilde{u}_i^\prime)^2}}(g_i^\prime - \tilde{u}_i^\prime) \,\mathrm{d} x \\
& \quad+ c_1 \sum\limits_{i=1}^2 \int_{I_\sigma} (g_i^\prime - \tilde{u}_i^\prime)^2 \,\mathrm{d} x + \sigma \sum\limits_{i=1}^2 \int_{I_0} \sqrt{1+(u_i^\prime)^2}\,\mathrm{d} x - 2\sigma \ell,
\end{align*} where 
$c_1 \gt 0$ is a numeric constant and, in the last inequality, we used the basic estimates
\begin{align*}\sqrt{1+t^2}-1&\geqslant\frac1{2^{5/2}}t^2\quad\text{for all}\ \vert t\vert\leqslant1,\\\sqrt{1+t^2}-\sqrt{1+s^2}\geqslant\frac s{\sqrt{1+s^2}}&(t-s)+\frac1{2\cdot5^{3/2}}(t-s)^2\quad\text{for all}\ \vert s\vert,\vert t\vert\leqslant2,\end{align*} combined with the fact that 
$\|g_0^\prime\|_{C^0(I_\sigma^c)} \leqslant 1$, 
$\|\tilde{u}_i^\prime\|_{C^0(I_\sigma)} \leqslant \sqrt{3}$ and 
$\| g_i'\|_{C^0(I_\sigma)} \leqslant 2$ for 
$i=1,2$ by (4.9) (up to taking a smaller ɛ 0, if needed).
Observe now that from the explicit form of the lens partition and recalling that 
$m=\frac{2\pi}{3}-\frac{\sqrt{3}}{2}$ in view of the normalization assumption (4.6), we easily find
\begin{equation*}
\sigma \sum\limits_{i=1}^2 \int_{I_0} \sqrt{1+(u_i^\prime)^2}\,\mathrm{d} x - 2\sigma \ell = 2m\sigma.
\end{equation*} Therefore, using this identity, the explicit form of 
$\tilde{u}_i^\prime$, integrating by parts, and using the boundary conditions (4.7) for g 1 and g 2, we obtain
\begin{align*}
\delta(\mathcal{E},\mathcal{L}_{m}) & \geq \frac{1}{2^{5/2}} \int_{I_\sigma^c} (g_0^\prime)^2\,\mathrm{d} x + c_1 \sum\limits_{i=1}^2 \int_{I_\sigma} (g_i^\prime - \tilde{u}_i^\prime)^2 \,\mathrm{d} x \\
& \quad-\int_{I_\sigma} \frac{x}{1+\sigma} (g_1^\prime - \tilde{u}_1^\prime) \,\mathrm{d} x + \int_{I_\sigma} \frac{x}{1+\sigma} (g_2^\prime - \tilde{u}_2^\prime) \,\mathrm{d} x + 2m\sigma \\
& \geq \frac{1}{2^{5/2}} \int_{I_\sigma^c} (g_0^\prime)^2\,\mathrm{d} x + c_1 \sum\limits_{i=1}^2 \int_{I_\sigma} (g_i^\prime - \tilde{u}_i^\prime)^2 \,\mathrm{d} x \\
& \quad+\frac{1}{1+\sigma} \left[ \int_{I_\sigma} (g_1-g_2) \,\mathrm{d} x - (1+\sigma)^2 \int_{I_0}(u_1-u_2) \,\mathrm{d} x \right] + 2m\sigma \\
& \mathop{=}\limits^{(4.8)} \frac{1}{2^{5/2}} \int_{I_\sigma^c} (g_0^\prime)^2\,\mathrm{d} x + c_1 \sum\limits_{i=1}^2 \int_{I_\sigma} (g_i^\prime - \tilde{u}_i^\prime)^2 \,\mathrm{d} x - m\sigma\frac{2+\sigma}{1+\sigma} + 2m\sigma \\
&= \frac{1}{2^{5/2}} \int_{I_\sigma^c} (g_0^\prime)^2\,\mathrm{d} x + c_1 \sum\limits_{i=1}^2 \int_{I_\sigma} (g_i^\prime - \tilde{u}_i^\prime)^2 \,\mathrm{d} x + \frac{m\sigma^2}{(1+\sigma)}.
\end{align*} Now we extend 
$\tilde{u}_i$ to zero outside of the interval Iσ, and define
\begin{equation*}
\tilde{g}_i(x) =
\begin{cases}
g_i(x) &\text{if }x\in I_\sigma,\\
g_0(x) &\text{if }x\in I_\sigma^c
\end{cases}
\end{equation*} for 
$i=1,2$. Then, by Poincaré inequality, we have for 
$c_2:= \min\{\frac{1}{2^{7/2}},c_1\}$
\begin{align}
\delta(\mathcal{E},\mathcal{L}_{m}) &\geq c_2\sum\limits_{i=1}^2 \int_{-R-1}^{R+1} \big[ (\tilde{g}_i-\tilde{u}_i)^\prime\big]^2 \,\mathrm{d} x + \frac{m\sigma^2}{1+\sigma}\nonumber\\
&\geq C_R \sum\limits_{i=1}^2 \int_{-R-1}^{R+1} \big(\tilde{g}_i-\tilde{u}_i \big)^2 \,\mathrm{d} x + \frac{m\sigma^2}{1+\sigma}
\end{align} for some constant 
$C_R \gt 0$ depending on R.
Step 3: Asymmetry estimate. Let
\begin{equation*}
\tilde{L}_m := \Bigl\{(x,y)\in \mathbb R^2 \colon |x| \leqslant (1+\sigma)\ell, \ \ |y| \leqslant (1+\sigma)u_1\Bigl(\frac{x}{1+\sigma}\Bigr) \Bigr\}
\end{equation*} so that 
$|\tilde{L}_m| = (1+\sigma)^2 |L_{m}|$. Define the rescaled lens partition 
$\tilde{\mathcal{L}}_m = (\tilde{L}_m, H^+\backslash\tilde{L}_m,H^-\backslash\tilde{L}_m)$. Then
\begin{align*}
\Delta(\mathcal{E},\mathcal{L}_{m}) &= \Delta(\mathcal{F},\mathcal{L}_{m}) \leqslant \frac{1}{2} \sum\limits_{i=1}^3 |\mathcal{F}(i)\triangle\tilde{\mathcal{L}}_m(i)| + \frac{1}{2} \sum\limits_{i=1}^3 |\tilde{\mathcal{L}}_m(i)\triangle\mathcal{L}_{m}(i)| \\
& \leqslant \int_{I_\sigma^c} |g_0|\,\mathrm{d} x + \sum\limits_{i=1}^2 \int_{I_\sigma} |g_i - \tilde{u}_i|\,\mathrm{d} x + \big| |\tilde{L}_m|-|L_{m}| \big| \\
& \leqslant \sum\limits_{i=1}^2 \int_{-R-1}^{R+1} |\tilde{g}_i - \tilde{u}_i| \,\mathrm{d} x + |\sigma|(\sigma+2)m.
\end{align*}Therefore
\begin{equation}
\Delta(\mathcal{E},\mathcal{L}_{m})^2 \leqslant C_R' \left( \sum\limits_{i=1}^2 \int_{-R-1}^{R+1} \big(\tilde{g}_i - \tilde{u}_i\big)^2 \,\mathrm{d} x + |\sigma|^2 m^2 \right)
\end{equation} for a constant 
$C_R' \gt 0$ depending only on R.
Step 4: Conclusion. Since 
$|\sigma| \leqslant 2\varepsilon_0$, combining the estimates (4.10) and (4.11) we obtain the result of the theorem.
Proof of Theorem 2.5
In view of Theorem 3.2, it is enough to show that the constant 
$\kappa_R(\mathcal{L}_{m})$ defined in (3.1) is strictly positive, for all m and R. Consider the sequence of 
$C^{1,1}$-partitions 
$(\mathcal{F}_k)_k$ constructed in Theorem 3.8, satisfying in particular
\begin{equation}
\kappa_R(\mathcal{L}_{m}) = \lim_{k\to\infty}\frac{\delta(\mathcal{F}_k,\mathcal{E}_0)}{\Delta(\mathcal{F}_k,\mathcal{E}_0)}.
\end{equation} For every ɛ > 0, each partition 
$\mathcal{F}_k$ is an ɛ-perturbation of 
$\mathcal{L}_{m}$ in 
$B_{R_0}$ for all k sufficiently large (depending on ɛ), according to Definition 4.1, for a suitable radius 
$R_0 \gt 0$ depending only on m and R.
In particular we can apply Theorem 4.2 to 
$\mathcal{F}_k$ to deduce that the right-hand side of (4.12) is uniformly bounded from below by a positive constant 
$\kappa_0$, as desired.
5. An application: small-mass minimizers for an isoperimetric problem with nonlocal perturbation
Let 
$\alpha\in(0,2)$ and 
$\gamma \geq 0$ be fixed parameters. We consider for m > 0 the following area-constrained nonlocal isoperimetric problem in 
$\mathbb R^2$:
\begin{equation}
\min\left\{\mathcal{F}_\gamma(E) \,:\, E\subset\mathbb R^2,\, |E| = m \right\},
\end{equation} where for every set of finite perimeter 
$E\subset\mathbb R^2$ we define the functional
\begin{equation}
\mathcal{F}_\gamma(E):= \mathcal{P}(E) - \mathcal{H}^1\bigl((\partial^*E\cup E^{(1)})\cap H\bigr) + \gamma \int_E\int_E \frac{1}{|x-y|^\alpha}\,\mathrm{d} x \,\mathrm{d} y.
\end{equation} Here 
$H:= \{(x,y)\in\mathbb R^2:y=0\}$, and we recall that 
$E^{(\theta)}$ denotes the set of points of Lebesgue density θ of 
$\textit{E}$. For simplicity of notation we set 
$E_*:= \partial^*E\cup E^{(1)}$ for any set of finite perimeter 
$\textit{E}$.
Although this problem can be formulated in any dimension, we restrict here to the case of dimension 
$\textit{d}$ = 2 since our main goal is to show an application of the stability theorem for the lens partition. Nonlocal isoperimetric problems where the perimeter functional is perturbed by a nonlocal repulsive interaction have received great attention in the last decade, particularly in connection with Gamow’s liquid drop model for the atomic nucleus (see [Reference Choksi, Muratov and Topaloglu8] for a review) and with the Ohta–Kawasaki model for diblock copolymers. In particular the functional (5.2) is expected to emerge in the scaling limit of a three-phase system for triblock copolymers, in a regime where two majority phases equally occupy nearly all the space forming lamellar structures and the third minority phase organizes in small droplets on the lamellar flat interfaces, see for instance [Reference Alama, Bronsard, Lu and Wang2, Reference Alama, Bronsard, X. Lu and C. Wang3].
The connection with the partitioning problem studied in this paper is made clear in the following remark. As a consequence, we obtain that the standard lens 
$\textit{L}_m$ (see (2.14)) minimizes the local functional 
$\mathcal{F}_0$, see Remark 5.2.
Remark 5.1. For 
$\unicode{x03B3}$ = 0, the local functional 
$\mathcal{F}_0$ computed on any bounded set 
$F\subset\!\subset B_R$, R > 0, coincides up to a constant (depending on R) with the perimeter of the partition 
$\mathcal{F}:= (F,H^+\backslash F, H^-\backslash F)$ associated with F (
$H^+$ and 
$H^-$ denote the upper and lower half-planes, respectively):
\begin{equation}
\mathcal{F}_0(F) = \mathcal{P}(\mathcal{F};B_R) - \mathcal{H}^1(B_R\cap H) \qquad\text{for all }F\subset\!\subset B_R.
\end{equation}Indeed,
\begin{equation*}
\begin{array}{rcl}
\mathcal{F}_0(F) + \mathcal{H}^1(B_R\cap H)& = & \mathcal{P}(F) + \mathcal{H}^1(F^{(0)}\cap H\cap B_R) \\
& =& \mathcal{P}(F) + \mathcal{H}^1(\partial^*(H^+\backslash F)\cap\partial^*(H^-\backslash F)\cap H\cap B_R)\\
&=& \mathcal{P}(\mathcal{F};B_R).
\end{array}
\end{equation*} We also notice that in view of the identity (5.3), the local functional 
$\mathcal{F}_0$ is lower semicontinuous along any sequence of sets converging in 
$\textit{L}^1$ and all contained in a ball of fixed radius, see [Reference Alama, Bronsard and Vriend5, Lemma 2.4].
Remark 5.2. For all m > 0, the standard lens 
$\textit{L}_m$ with area m (see (2.14)) minimizes the local functional 
$\mathcal{F}_0$:
\begin{equation}
\mu_0 := \min\bigl\{\mathcal{F}_0(E) \,:\, |E|=1 \bigr\} = \mathcal{F}_0(L_1)
\end{equation}and, by scaling,
\begin{equation}
\min\bigl\{\mathcal{F}_0(E) \,:\, |E|=m \bigr\} = \mathcal{F}_0(L_{m}) = \mu_0 \sqrt{m}.
\end{equation} Indeed, for every bounded set F with 
$|F|=m$, we immediately obtain that 
$\mathcal{F}_0(F) \geq \mathcal{F}_0(L_{m})$ in view of the relation (5.3) and of the minimality of the lens partition 
$\mathcal{L}_{m}$. Since we can take a minimizing sequence 
$(F_k)_k$ for the minimum problem (5.5) made of bounded sets, we have that 
$\mathcal{F}_0(L_{m}) \leqslant \mathcal{F}_0(F_k)$ for every k, so that the minimality of 
$\textit{L}_m$ follows.
In the following theorem we prove the existence of minimizers of (5.1) for small values of m, and that (rescaled) minimizers converge in measure to the standard lens as m → 0, as a consequence of the stability property of the lens partition proved in Theorem 2.5. Notice that by a standard scaling argument, using the homogeneity of the nonlocal kernel, the ‘small mass regime’ 
$m\to0^+$ corresponds to small values of the coefficient 
$\unicode{x03B3}$ multiplying the nonlocal term; this observation, combined with Remark 5.2, provides a heuristic explanation of the following result.
Theorem 5.3. There exists a threshold 
$m_0 \gt 0$, depending only on α and 
$\unicode{x03B3}$, such that for every 
$m\in(0,m_0)$ the minimum problem (5.1) admits a solution Em. Moreover
\begin{equation}
\big|\bigl({\textstyle\frac{1}{\sqrt{m}}}E_m\bigr) \triangle L_1\big|\to 0\ \text{as }m\to 0^+,
\end{equation}where L 1 denotes the unit-area lens defined in (2.14).
Proof. This type of result is by now quite standard for nonlocal isoperimetric problems. We adapt in particular the 2-dimensional existence argument by Knüpfer and Muratov for Gamow’s liquid drop model, see [Reference Knüpfer and Muratov17, Theorem 2.2]. We assume in the following 
$m \leqslant 1$, and we denote along the proof by C a generic, positive constant, depending only on α and 
$\unicode{x03B3}$, which might change from line to line.
Step 1: Existence for small m. Let 
$(E_k)_k$ be a minimizing sequence for the minimum problem (5.1), and assume without loss of generality that each Ek is the union of finitely many disjoint, open, and smooth connected components 
$E_k = \bigcup_{i=1}^{N_k}E_{k,i}$, 
$N_k\in\mathbb N$, ordered so that 
$\vert E_{k,1}\vert\geqslant\vert E_{k,2}\vert\geqslant\dots\geqslant\vert E_{k,N_k}\vert \gt 0$. We can assume that, for k large,
\begin{align}
\mathcal{F}_\gamma(E_k) \leqslant \mathcal{F}_{\gamma}(L_{m})
&= \mu_0\sqrt{m} + \gamma m^{\frac{4-\alpha}{2}}\int_{L_1}\int_{L_1}\frac{1}{|x-y|^\alpha}\,\mathrm{d} x \,\mathrm{d} y\nonumber\\
&= \mu_0\sqrt{m} \Bigl( 1 + C m^{\frac{3-\alpha}{2}}\Bigr)
\end{align} (or else 
$\textit{L}_m$ would already be a minimizer), where we used (5.5).
Suppose now 
$N_k \gt 1$, so that 
$|E_{k,i}| \leqslant \frac{m}{2}$ for all 
$i=2,\ldots,N_k$. By minimality of 
$\textit{L}_m$ for the local functional 
$\mathcal{F}_0$ and positivity of the nonlocal term we have for all 
$i\in\{1,\ldots,N_K\}$
\begin{equation}
\mathcal{F}_\gamma(E_k) \geq \mathcal{F}_0(E_{k,i}) + \mathcal{F}_0(E_k\backslash E_{k,i})
\mathop{\geq }\limits^{(5.5)} \mu_0\bigl(|E_{k,i}|^{1/2} + (m-|E_{k,i}|)^{1/2}\bigr).
\end{equation} By combining (5.7) and (5.8), squaring both sides, we find for 
$i=2,\ldots,N_k$
\begin{equation*}
2|E_{k,i}|^{1/2}(m-|E_{k,i}|)^{1/2} \leqslant Cm^{4-\alpha} + C m ^{\frac{5-\alpha}{2}} \leqslant C m^{\frac{5-\alpha}{2}},
\end{equation*} from which it follows, using also 
$|E_{k,i}| \leqslant \frac{m}{2}$,
\begin{equation}
|E_{k,i}| \leqslant C m^{4-\alpha} \text{for all }i=2,\ldots, N_k.
\end{equation}Define the set
\begin{equation*}
F_k:= \lambda (E_k\backslash E_{k,N_k}), \qquad\text{with }\lambda:= \biggl(\frac{m}{m-|E_{k,N_k}|}\biggr)^\frac12 \in (1,\sqrt{2}],
\end{equation*} so that 
$|F_k|=m$. We have
\begin{align*}
\mathcal{F}_{\gamma}(F_k) - \mathcal{F}_{\gamma}(E_k) & = \lambda\Bigl[\mathcal{P}(E_k)-\mathcal{H}^1((E_k)_*\cap H)\Bigr]\\
& \quad- \lambda\Bigl[\mathcal{P}(E_{k,N_k})-\mathcal{H}^1((E_{k,N_k})_*\cap H)\Bigr] \\
& \quad+ \gamma\lambda^{4-\alpha}\int_{F_k}\int_{F_k}\frac{1}{|x-y|^\alpha}\,\mathrm{d} x \,\mathrm{d} y - \mathcal{F}_{\gamma}(E_k) \\
& \leqslant (\lambda^{4-\alpha}-1)\mathcal{F}_{\gamma}(E_k) - \lambda \mathcal{F}_0(E_{k,N_k}) \\
& \leqslant (\lambda^{4}-1)\mathcal{F}_{\gamma}(E_k) - \mathcal{F}_0(E_{k,N_k}) \\
& \leqslant \frac{6}{m}|E_{k,N_k}|\mathcal{F}_{\gamma}(E_k) - \mu_0 |E_{k,N_k}|^{1/2} \\
& \leqslant \mu_0 |E_{k,N_k}|^{1/2}\Bigl(C\mathcal{F}_\gamma(E_k)m^{\frac{2-\alpha}{2}} - 1 \Bigr),
\end{align*} where we used in particular (5.5) in the third inequality and (5.9) in the last one. Since 
$\mathcal{F}_{\gamma}(E_k)$ is uniformly bounded by a constant depending only on α and 
$\unicode{x03B3}$ (recall that 
$m \leqslant 1$), we can find 
$m_0\in(0,1)$, also depending only on α and 
$\unicode{x03B3}$, such that the previous quantity is negative for all 
$m\in(0,m_0)$ and for all k. Therefore 
$\mathcal{F}_{\gamma}(F_k) \lt \mathcal{F}_{\gamma}(E_k)$, that is, by removing the last connected component of Ek and rescaling we reduce the energy. By iterating this argument 
$N_k-1$ times, removing a connected component at each step, we replace Ek by a connected set Gk such that 
$|G_k|=m$ and 
$\mathcal{F}_{\gamma}(G_k) \leqslant \mathcal{F}_{\gamma}(E_k)$.
In particular, 
$(G_k)_k$ is a minimizing sequence for (5.1) made of connected sets. Moreover, the sets Gk have equibounded perimeter: this follows from the estimate
\begin{equation}
\mathcal{F}_0(E) \geq \frac12\mathcal{P}(E)
\end{equation} for every set of finite perimeter 
$\textit{E}$, which can be proved as follows. By [Reference Maggi19, Proposition 19.22] one has 
$\mathcal{P}(F;H^\pm) \gt \mathcal{P}(F;H)$ for all 
$F\subset H^\pm$ with finite perimeter and finite measure. Hence, using also [Reference Maggi19, Theorem 16.3],
\begin{align*}
\mathcal{P}(E) & = \mathcal{P}(E;H^+) + \mathcal{P}(E;H^-) + \mathcal{H}^1(\partial^*E\cap H) \\
& \geq \mathcal{P}(E\cap H^+;H) + \mathcal{P}(E\cap H^-;H) + \mathcal{H}^1(\partial^*E\cap H) \\
& = 2\mathcal{H}^1(E^{(1)}\cap H) + 2\mathcal{H}^1(\partial^*E\cap H) = 2 \mathcal{H}^1(E_*\cap H),
\end{align*}from which the estimate (5.10) follows.
Since we are in dimension 
$\textit{d}$ = 2 and 
$(G_k)_k$ are connected sets with uniformly bounded perimeter, we have that 
$\sup_k\mathrm{diam} (G_k) \lt \infty$ [Reference Maggi19, Remark 12.28]. By applying horizontal translations (notice that the functional 
$\mathcal{F}_\gamma$ is invariant with respect to horizontal translations) we can then assume that the sets Gk are contained in a fixed ball of large radius. A standard compactness argument, combined with the lower semicontinuity of 
$\mathcal{F}_{\gamma}$ with respect to 
$\textit{L}^1$-convergence (see Remark 5.1), yields the existence of a minimizer in (5.1) for all 
$m\in(0,m_0)$.
Step 2: Convergence to the lens. Let Em be a minimizer in (5.1) for 
$m\in(0,m_0)$, and define 
$\widetilde{E}_m:= \frac{1}{\sqrt{m}}E_m$, so that 
$|\widetilde{E}_m|=1$.
Notice that every minimizer Em is necessarily connected, in the sense that it cannot be written as disjoint union 
$E_m=A\cup B$ of two sets of positive Lebesgue measure in such a way that 
$\mathcal{P}(E_m)=\mathcal{P}(A)+\mathcal{P}(B)$. Indeed, if not then one could horizontally translate one of the two components far apart from the other, without changing the local energy 
$\mathcal{F}_0$ but strictly decreasing the nonlocal energy.
The sets 
$\widetilde{E}_m$ have an equibounded perimeter, since by scaling and comparing with 
$\textit{L}_m$
\begin{align*}
\mathcal{P}(\widetilde{E}_m) &= \frac{1}{\sqrt{m}}\mathcal{P}(E_m) \mathop{\leqslant }\limits^{(5.10)} \frac{2}{\sqrt{m}}\mathcal{F}_0(E_m) \leqslant \frac{2}{\sqrt{m}}\mathcal{F}_\gamma(E_m)\\
&\leqslant \frac{2}{\sqrt{m}}\mathcal{F}_\gamma(L_{m}) \leqslant 2\mathcal{F}_\gamma(L_1).
\end{align*} Therefore, by connectedness, there exists 
$R_0 \gt 0$ such that 
$\sup_{m \lt m_0}\mathrm{diam}(\widetilde{E}_m) \leqslant R_0$. Associate with 
$\widetilde{E}_m$ the partition 
$\mathcal{E}_m:= (\widetilde{E}_m,H^+\backslash\widetilde{E}_m,H^-\backslash\widetilde{E}_m)$, and notice that 
$\mathcal{E}_m\in\mathfrak{M}_{R_0}(\mathcal{L}_1)$. By applying a horizontal translation, we can assume that
\begin{equation*}
\Delta(\mathcal{E}_m,\mathcal{L}_1) = \mathrm{d}(\mathcal{E}_m,\mathcal{L}_1) = |\widetilde{E}_m\triangle L_1|.
\end{equation*} Then by the stability of 
$\mathcal{L}_1$ proved in Theorem 2.5 we have
\begin{align*}
\kappa_{1,R_0}|\widetilde{E}_m\triangle L_1|^2 & \leqslant \mathcal{P}(\mathcal{E}_m;B_{R_0}) - \mathcal{P}(\mathcal{L}_1;B_{R_0}) \mathop{=}\limits^{(5.3)} \mathcal{F}_0(\widetilde{E}_m) - \mathcal{F}_0(L_1) \\
& = \frac{1}{\sqrt{m}} \Bigl( \mathcal{F}_0(E_m) - \mathcal{F}_0(L_{m}) \Bigr) \\
& \leqslant \frac{\gamma}{\sqrt{m}} \biggl( \int_{L_{m}}\int_{L_{m}}\frac{1}{|x-y|^\alpha}\,\mathrm{d} x \,\mathrm{d} y - \int_{E_m}\int_{E_m}\frac{1}{|x-y|^\alpha}\,\mathrm{d} x \,\mathrm{d} y \biggr) \\
& \leqslant \frac{C\gamma}{\sqrt{m}}|E_m\triangle L_{m}| = C\gamma\sqrt{m}|\widetilde{E}_m\triangle L_1|,
\end{align*}where the second inequality follows by minimality of Em, and the third inequality is a standard Lipschitz continuity estimate of the nonlocal energy, see for instance [Reference Knüpfer and Muratov17, Equation (3.2)]. Hence (5.6) follows.
Remark 5.4. For values of m above a suitable threshold, it is expected that minimizers of (5.1) fail to exist, since the nonlocal part of the energy becomes dominant and can be decreased by splitting a set into two parts and moving them far apart from each other (by horizontal translations). However, it is possible to prove the existence of generalized minimizers for all m, following for instance the approach in [Reference Novaga and Pratelli22] for a general Gamow’s model. By the generalized minimizer of (5.1) we mean a collection of sets of finite perimeter 
$(E_1,\ldots,E_M)$, 
$M\in\mathbb N$, such that
\begin{equation*}
\sum\limits_{i=1}^M|E_i|=m\ \text{and}\ \inf\Bigl\{\mathscr{F}_\gamma(E) \,:\, E\subset\mathbb R^2,\, |E| = m \Bigr\}=\sum\limits_{i=1}^M\mathscr{F}_\gamma(E_i).
\end{equation*}We also remark that one can prove analogous results for more general kernels in the nonlocal energy (as those considered in [Reference Novaga and Pratelli22]). We will not further investigate these problems here.
Acknowledgements
M.B. is a member of the GNAMPA group of INdAM. M.B. is partially supported by the project PRIN 2022PJ9EFL ``Geometric Measure Theory: Structure of Singular Measures, Regularity Theory and Applications in the Calculus of Variations’’, funded by the European Union - Next Generation EU, Mission 4 Component 2 - CUP:E53D23005860006. R.C. was partially supported under NWO-OCENW.M.21.336, MATHEMAMI - Mathematical Analysis of phase Transitions in HEterogeneous MAterials with Materials Inclusions. I.T. was partially supported by the Simons Foundation (851065), by the NSF-DMS (2306962), and by the Alexander von Humboldt Foundation.
Appendix A. A ‘volume-fixing variation’ lemma
We here state a variant of a key result in the theory of minimizing clusters (see for instance [Reference Maggi19, Section 29.5]), which allows us to exchange volumes between the different chambers of a given partition through local deformations, with a control on the corresponding perimeter variation. For our purposes (see in particular the proof of Theorem 3.5), we need to make sure that the perturbation is compactly supported in a fixed open set.
Lemma A.1. Volume-fixing variation
Let 
$\mathcal{E}_0=(\mathcal{E}_0(1),\ldots,\mathcal{E}_0(N))$ be a locally isoperimetric partition in 
$\mathbb R^2$, and let OR be the set constructed in Definition 3.4. Let also 
$T:\mathbb R^2\to\mathbb R^2$ be an isometry such that 
$T(\mathcal{E}_0(i))\subset O_{R}$ for all 
$i\in I_{\mathcal{E}_0}$.
Then there exist constants 
$C_0 \gt 0$, 
$\varepsilon_0 \gt 0$, and 
$r_0 \gt 0$ (depending on 
$T(\mathcal{E}_0)$ and R) with the following property.
If 
$\mathcal{E}$ and 
$\mathcal{G}$ are 
$\textit{N}$-partitions such that 
$\mathrm{d}(\mathcal{E},T(\mathcal{E}_0)) \lt \varepsilon_0$ and 
$\mathcal{G}(i)\triangle\mathcal{E}(i)\subset\!\subset B_{r_0}(x)\subset\!\subset B_{R+1}$ for some 
$x\in\mathbb R^2$, then there exists a 
$\textit{N}$-partition 
$\widetilde{\mathcal{G}}$ such that
(i)
$\widetilde{\mathcal{G}}(i)\triangle\mathcal{G}(i) \subset\!\subset O_{R} \backslash\overline{B_{r_0}(x)}$ for all
$i\in\{1,\ldots,N\}$,(ii)
$|\widetilde{\mathcal{G}}(i)|=|\mathcal{E}(i)|$ for all
$i\in\{1,\ldots,N\}$,(iii)
$|\mathcal{P}(\widetilde{\mathcal{G}};B_{R+1})-\mathcal{P}(\mathcal{G};B_{R+1})| \leqslant C_0\mathcal{P}(\mathcal{E};B_{R+1})\sum\limits_{i\in I_{\mathcal{E}_0}}\big| |\mathcal{G}(i)|-|\mathcal{E}(i)| \big|$,(iv)
$|\mathrm{d}(\widetilde{\mathcal{G}},\mathcal{E})-\mathrm{d}(\mathcal{G},\mathcal{E})| \leqslant C_0\mathcal{P}(\mathcal{E};B_{R+1})\sum\limits_{i\in I_{\mathcal{E}_0}}\big| |\mathcal{G}(i)|-|\mathcal{E}(i)| \big|$.
Proof. This result is proved in [Reference Maggi19, Corollary 29.17] for clusters in 
$\mathbb R^d$. The presence of multiple regions with an infinite measure in our case does not affect the proof, as their volumes do not need to be preserved and, furthermore, everything is localized in a large ball 
$B_{R+1}$. We need only to enforce the condition 
$\widetilde{\mathcal{G}}(i)\triangle\mathcal{G}(i) \subset\!\subset O_{R}$.
The proof is based on [Reference Maggi19, Theorem 29.14], where one selects two finite families 
$\{y_\alpha\}_{\alpha=1}^M$ and 
$\{z_\alpha\}_{\alpha=1}^M$ of interface points of 
$T(\mathcal{E}_0)$ and constructs the required perturbation by modifying 
$\mathcal{G}$ either in 
$\bigcup_{\alpha=1}^M B_{\varepsilon_1}(y_\alpha)$ or in 
$\bigcup_{\alpha=1}^M B_{\varepsilon_1}(z_\alpha)$, for some 
$\varepsilon_1 \gt 0$, by means of suitable diffeomorphisms.
Therefore, to ensure that the part outside OR is unchanged, we only need to make sure that it is possible to choose the points yα and zα inside OR. This is guaranteed by [Reference Maggi19, Remark 29.15], which only requires that 
$|T(\mathcal{E}_0(i))\cap O_R| \gt 0$ for all 
$i\in\{1,\ldots,N\}$.
With these considerations, the proof can be adapted to deal with our situation. The only condition not addressed in [Reference Maggi19] is the estimate (iv). Its proof is discussed in [Reference Cicalese, Leonardi and Maggi10, Appendix B].
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