The Nonlocal Almgren Problem

Abstract

In the nonlocal Almgren problem, the goal is to investigate the convexity of a minimizer under a mass constraint via a nonlocal free energy generated with a nonlocal perimeter and convex potential. In this paper, the main result is a quantitative stability theorem for the nonlocal free energy under a symmetry assumption on the potential. In addition, several results that involve uniqueness, non-existence, and moduli estimates from the theory of crystals are also proven in the nonlocal context.

1. Introduction

A fundamental theorem in real analysis is Taylor’s theorem: assume $f:\mathbb{R}\to \mathbb{R}$ is twice differentiable,

$$f\left(e\right)=f\left(e_m\right)+f^{\prime }\left(e_m\right)(e-e_m)+{\displaystyle \frac{f^{″}\left({\alpha }_{e,e_m}\right)}{2}}{|e-e_m|}^2,$$

(1)

${\alpha }_{e,e_m}\in (e_m,e)$. If $f^{\prime }\left(e_m\right)=0$, $f^{″}\left({\alpha }_{e,e_m}\right)\ge a_{\ast }>0$, then

$$f\left(e\right)-f\left(e_m\right)\ge {\displaystyle \frac{a_{\ast }}{2}}{|e-e_m|}^2.$$

(2)

In particular, one may obtain sharp information from this: assume f is a quantity that classifies an optimizer $e_m$ in the sense that $f\left(e\right)=f\left(e_m\right)$ implies $e=e_m$. Now, let e satisfy $f\left(e\right)=f\left(e_m\right)+ϵ$, with $ϵ>0$ a small number; thus, $f\left(e\right)\approx f\left(e_m\right)$, and the expectation is that $e\approx e_m$. The utility of (2) is to make this clear in the context that e is at most, up to a constant, $\sqrt{ϵ}$ from $e_m$.

When instead of a real-valued function f the object of investigation is an energy $\mathcal{E}$ that is defined on measurable sets E, recent work has investigated analogous estimates. An application is to understand the perturbations of minimizers through the energy. The main problem is to minimize $\mathcal{E}$ subject to a mass constraint $\left|E\right|=m$. Thus, at the minimum $E_m$, the first variation is zero ${\mathcal{E}}^{\prime }\left(E_m\right)=0$; hence, if some lower bound exists on the second variation, it is natural to anticipate that modulo an invariance class,

$$\mathcal{E}\left(E\right)-\mathcal{E}\left(E_m\right)\ge a_{\ast }\Vert {\chi }_E-{\chi }_{E_m}{\Vert }^2,$$

(3)

where ${\chi }_E$ is the characteristic function of E. A natural norm is often chosen to be the $L^1$ norm [1]. In applications, the energy is of the form

$$\mathcal{E}\left(E\right)={\mathcal{E}}_s\left(E\right)+{\mathcal{E}}_a\left(E\right),$$

(4)

where ${\mathcal{E}}_s$ is a surface energy and ${\mathcal{E}}_a$ a potential/repulsion energy. In the next discussion, four physical and fundamental energies are underscored.

1.1. The Free Energy

The crystal theory starts with the anisotropic surface energy on sets of finite perimeter $E\subset {\mathbb{R}}^n$ with reduced boundary ${\partial }^{\ast }E$:

$${\mathcal{E}}_s\left(E\right)=\mathcal{F}\left(E\right)={\int }_{{\partial }^{\ast }E}f\left({\nu }_E\right)d{\mathcal{H}}^{n-1},$$

(5)

where f is a surface tension, i.e., a convex positively 1-homogeneous

$$f:{\mathbb{R}}^n\to [0,\infty )$$

with $f\left(x\right)>0$ if $\left|x\right|>0$. The potential energy of a set E is

$${\mathcal{E}}_a\left(E\right)=\mathcal{G}\left(E\right)={\int }_{E}g\left(x\right)dx,$$

(6)

where $g\ge 0$, $g\left(0\right)=0$, $g\in L_{loc}^{\infty }$ [2,3,4,5,6,7,8,9,10,11,12,13]; see in addition many interesting references in [3] that comprehensively discuss the history. In thermodynamics, to obtain a crystal, one minimizes the free energy

$$\mathcal{E}\left(E\right)=\mathcal{F}\left(E\right)+\mathcal{G}\left(E\right)$$

(7)

under a mass constraint. Gibbs and Curie independently discovered this physical principle [14,15]. Subject to the convexity of sub-level sets $\{g<t\}$, conjectures on the existence of convex minimizers appear in the literature [13]. Convexity results for all $m>0$ are rare even in two dimensions. In two recent papers, the author: (1) proved that, if the sub-level sets $\{g<t\}$ are convex, in one dimension there exist minimizers for all masses $m>0$ and all minimizers are intervals [2]; (2) proved that, if $n\ge 2$, there are convex functions $g\ge 0$, $g\left(0\right)=0$, so that there are no minimizers for $m>0$ [16]. Assuming $n=2$, under additional assumptions, the author proved convexity for all $m>0$ [3]. If $n=3$, the author and Karakhanyan recently proved a three-dimensional convexity theorem with a new maximum-principle approach [5]. Note the general partition of the problem into coercive (the convex strictly monotone radial potential) and non-coercive potentials (the gravitational potential).

1.2. The Binding Energy

The nonlocal Coulomb repulsion energy is given via

$${\mathcal{E}}_a\left(E\right)=\mathcal{D}\left(E\right)={\alpha }_1\int {\int }_{E\times E}{\displaystyle \frac{1}{{|z-y|}^{\lambda }}}dzdy,$$

(8)

$\lambda \in (0,n)$, ${\alpha }_1>0$. The binding energy of a set of finite perimeter $E\subset {\mathbb{R}}^n$ is the sum

$$\mathcal{E}\left(E\right)=\mathcal{F}\left(E\right)+\mathcal{D}\left(E\right).$$

(9)

In the classical context, $\lambda =1$, $f\left(x\right)=\left|x\right|$, $n=3$, ${\alpha }_1=\frac{1}{2}$ [17]. The theory is historically attributed to Gamow via his 1930 paper [18], and it successfully predicts the non-existence of nuclei with large atomic numbers; see references in [19,20,21,22]. A few entropic uncertainty principles appear in stability theorems for the logarithmic Sobolev inequality [23,24]. The entropic uncertainty is more general than Heisenberg’s uncertainty. Moreover, applications to electron bubbles show up in [25]. My main theorem connects to these types of results via quantitative stability of the minimizer of the nonlocal free energy defined below.

1.3. The Nonlocal Free Energy

The nonlocal perimeter (or $\alpha$-fractional perimeter) encodes a parameter $\alpha \in (0,1)$ ([21,26,27,28])

$${\mathcal{E}}_s\left(E\right)=P_{\alpha }\left(E\right)={\int }_E{\int }_{E^c}{\displaystyle \frac{1}{{|x-y|}^{\alpha +n}}}dxdy.$$

(10)

Caffarelli, Roquejoffre, and Savin investigated the Plateau problem with respect to the nonlocal energy functionals [27]. If one investigates a set E such that $P_{\alpha }\left(E\right)<\infty$, note that the set generates convergence of the integral, and hence the boundary of the set has a regularizing property. Applications include the structure of interphases arising in classical phase field models with long space correlations. The nonlocal perimeter is connected to a definition of the fractional derivative of the characteristic function of E and the $H^{\frac{\alpha }{2}}$ seminorm. There exists a more general definition of perimeter associated with a kernel $K:{\mathbb{R}}^n\to {\mathbb{R}}_{+}\cup \{\infty \}$:

$$P_K\left(E\right)={\int }_E{\int }_{E^c}K(x-y)dxdy.$$

(11)

Homogeneity of the kernel, $K\left(\nu x\right)={\nu }^{a}K\left(x\right)$ yields

$$P_K\left(\nu E\right)={\nu }^{2n+a}{P}_K\left(E\right).$$

(12)

In particular, one can consider $K\left(x\right)={\left|x\right|}^{-(\alpha +n)}$ as the kernel in (10). The restriction of $\alpha$ to $(0,1)$ is natural via

$$\underset{\alpha \to 1^{-}}{lim}(1-\alpha )P_{\alpha }\left(E\right)={\mathcal{H}}^{n-1}\left({\partial }^{\ast }E\right)$$

(13)

$$\underset{\alpha \to 0^{+}}{lim}\alpha P_{\alpha }\left(E\right)={\mathcal{H}}^{n-1}(\partial B_1)\left|E\right|,$$

(14)

therefore, one may interpolate between the n-dimensional Lebesgue measure and the $(n-1)$-dimensional Hausdorff measure with the nonlocal perimeter. An application to image processing and additional properties of the nonlocal perimeter associated with a kernel K are mentioned in [29]. The nonlocal isoperimetric inequality appears in [26]: assume $\left|E\right|=\left|B_a\right|$, then

$$P_{\alpha }\left(E\right)\ge P_{\alpha }\left(B_a\right)$$

with equality if and only if $E=B_a+x$. Hence, the nonlocal free energy is ([28,30])

$$\mathcal{E}\left(E\right)=P_{\alpha }\left(E\right)+\mathcal{G}\left(E\right).$$

(15)

1.4. The Nonlocal Binding Energy

The nonlocal Coulomb repulsion energy together with the nonlocal perimeter in the aforementioned provide the nonlocal binding energy

$$\mathcal{E}\left(E\right)=P_{\alpha }\left(E\right)+\mathcal{D}\left(E\right),$$

(16)

see [21] for a theorem on the minimizers when the mass is small and [31] for non-existence when the mass is large.

1.5. The Main Problem

Observe via the above that four main choices of $\mathcal{E}$ are

$$\mathcal{E}\left(E\right)=\mathcal{F}\left(E\right)+\mathcal{G}\left(E\right)$$

(17)

$$\mathcal{E}\left(E\right)=\mathcal{F}\left(E\right)+\mathcal{D}\left(E\right)$$

(18)

$$\mathcal{E}\left(E\right)=P_{\alpha }\left(E\right)+\mathcal{G}\left(E\right)$$

(19)

$$\mathcal{E}\left(E\right)=P_{\alpha }\left(E\right)+\mathcal{D}\left(E\right).$$

(20)

Therefore, the central problem is as follows: assume $m>0$ and solve

$$inf\left\{\mathcal{E}\right(E):|E|=m\}.$$

(21)

Naturally, the questions involve existence, uniqueness, convexity, and optimal stability. My paper investigates this for

$$\mathcal{E}\left(E\right)=P_{\alpha }\left(E\right)+\mathcal{G}\left(E\right).$$

(22)

Observe that two main ingredients define the nonlocal free energy of a set $E\subset {\mathbb{R}}^n$: $P_{\alpha }\left(E\right)$ and, $\mathcal{G}\left(E\right)={\int }_{E}g\left(x\right)dx$.

In this context, the nonlocal Almgren problem is to investigate the convexity of a minimizer $E_m$ under the assumption that g is convex (cf. [13] (p. 146)). In this way, a stability estimate encoding the nonlocal Almgren problem includes the minimizer and generates a general theorem. One can consider the minimizer in the classical Almgren problem as a solution to a PDE. Assume that f is a surface tension and that A is the second fundamental form of the boundary of the set; then, the anisotropic mean curvature is

$$H_f=\mathrm{trace}\left(D^{2}fA\right).$$

(23)

The formula for the first variation implies

$$H_f=\mu -g,$$

(24)

where

$$\mu ={\displaystyle \frac{(n-1)\mathcal{F}\left(E_m\right)+{\int }_{{\partial }^{\ast }{E}_m}g⟨x,{\nu }_{E_m}⟩d{\mathcal{H}}^{n-1}}{n|E_m|}}.$$

(25)

In particular, the PDE contains critical points of the local free energy. Naturally, not every critical point may generate a minimizer. Investigating solutions is a very classical initial attempt to classify all minimizers; however, in this paper, I opted to avoid the first variation for the nonlocal Almgren problem. Several theorems may be shown without convexity on g. In particular, a complete theory when $\mathcal{E}\left(E\right)=P_{\alpha }\left(E\right)+\mathcal{G}\left(E\right)$ begins via $g\in L_{loc}^{\infty }$ (in several contexts, one can assume $g\in L_{loc}^1$).

1.

Assuming coercivity, there are minima for $m>0$ [30].

2.

Assuming additionally that m is sufficiently small and g is locally Lipschitz, all minimizers are convex [28].

3.

One may construct a g that is convex, $g\ge 0$, $g\left(0\right)=0$ so that there are no minimizers if $m>0$; see Theorem 3.

4.

Assuming $g\left(x\right)=h\left(\right|x\left|\right)$, $h:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is increasing, convex, $h\left(0\right)=0$, there exists a stability estimate similar to (3); see Theorem 1.

5.

Assuming $g\in L_{loc}^{\infty }$ and that up to sets of measure zero g admits unique minimizers $E_m$, there exist energy moduli. In addition, assuming that m is sufficiently small, the energy modulus has a product structure; see Proposition 1 and Theorem 4.

6.

Upper bounds for the moduli are obtained with minimal assumptions; see Theorem 2.

In this paper, the novelty mostly concerns 4 in the context of the proof. Interestingly, 3, 5, and 6 are also new; however, they can be obtained as in [3,4] (in 6, one utilizes [28]); hence, the proofs are in Appendix A.

2. Stability for Nonlocal Free Energy Minimization

Theorem 1.

Suppose $g\left(x\right)=h\left(\right|x\left|\right)$, $h:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is increasing, convex, $h\left(0\right)=0$. Let $m>0$, $|B_a|=|E|=m$, then

(i) 

$$\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right)\ge r(m,n,\alpha ,g)\Vert {\chi }_E-{\chi }_{B_a}{\Vert }_{L^1}^4$$

for some $r(m,n,\alpha ,g)>0$;

(ii) 

supposing $\widehat{a}>a$, $E\subset B_{\widehat{a}}$, then

$$\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right)\ge r(m,\widehat{a},n,\alpha ,g)\Vert {\chi }_E-{\chi }_{B_a}{\Vert }_{L^1}^2$$

for some explicit $r(m,\widehat{a},n,\alpha ,g)>0$.

Remark 1.

The exponent 4 is appearing when considering the stability of the isoperimetric inequality. Hall proved a stability theorem with 4 and conjectured that the exponent can be replaced with 2 [32]. The conjecture was proven in [11,33].

Proof. 

(i) Assume

$$\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right)\ge \nu ,$$

for $\nu >0$. Then, since

$$|E\mathsf{\Delta }{B}_a{|}^4\le 16m^4,$$

$$\begin{array}{cc}\hfill |E\mathsf{\Delta }{B}_a{|}^4& \le 16m^4\hfill \\ \hfill & \le {\displaystyle \frac{16m^4}{\nu }}\nu \le {\displaystyle \frac{16m^4}{\nu }}(\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right)).\hfill \end{array}$$

Therefore, it is sufficient to prove that there exists $\nu >0$ so that, if

$$\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right)\le \nu ,$$

then

$$w\left(\right|E\mathsf{\Delta }{B}_a\left|\right)\le \mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right),$$

(26)

where $w\left(t\right)=r(m,n,\alpha ,g)t^4$. To start, the existence of a modulus w is proved so that (26) is true (observe this also follows from a compactness proof, cf. Proposition 1, as soon as one obtains that the ball is the unique minimizer; the new argument has advantages in the context of explicitly encoding estimates to identify $w\left(t\right)=r(m,n,\alpha ,g)t^4$). Suppose

$$T:E\setminus B_a\to B_a\setminus E$$

denotes the Brenier map between ${\mu }_{\ast }={\chi }_{E\setminus B_a}dx$ and ${\nu }_{\ast }={\chi }_{B_a\setminus E}dx$ [34]. Since

$$T_{\#}{\mu }_{\ast }={\nu }_{\ast },$$

one has

$${\int }_{E\setminus B_a}h\left(\right|T\left(x\right)\left|\right)dx={\int }_{B_a\setminus E}h\left(\right|x\left|\right)dx;$$

(27)

thus, (27) and monotonicity of h yield

$$\begin{array}{cc}\hfill {\int }_{B_a}hdx& ={\int }_{E\cap B_a}hdx+{\int }_{B_a\setminus E}h\left(\right|x\left|\right)dx\hfill \\ \hfill & ={\int }_{E\cap B_a}hdx+{\int }_{E\setminus B_a}h\left(\right|T\left(x\right)\left|\right)dx\hfill \\ \hfill & \le {\int }_{E\cap B_a}hdx+{\int }_{E\setminus B_a}h\left(\right|x\left|\right)dx\hfill \\ \hfill & ={\int }_{E}hdx.\hfill \end{array}$$

(28)

Observe that, via $\left|T\right(x\left)\right|\le a$ and the above,

$${\int }_{E\setminus B_a}\left[h\right(\left|x\right|)-h\left(a\right)]dx\le {\int }_{E}hdx-{\int }_{B_a}hdx.$$

Hence, the previous inequality and [21] imply

$$\begin{array}{cc}\hfill \mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right)& =P_{\alpha }\left(E\right)-P_{\alpha }\left(B_a\right)+\mathcal{G}\left(E\right)-\mathcal{G}\left(B_a\right)\hfill \\ \hfill & \ge P_{\alpha }\left(B_a\right){\displaystyle \frac{A^2\left(E\right)}{C(n,\alpha )}}+{\int }_{E}hdx-{\int }_{B_a}hdx\hfill \\ \hfill & \ge P_{\alpha }\left(B_a\right){\displaystyle \frac{A^2\left(E\right)}{C(n,\alpha )}}+{\int }_{E\setminus B_a}\left[h\right(\left|x\right|)-h\left(a\right)]dx,\hfill \end{array}$$

$$A\left(E\right)=inf\{{\displaystyle \frac{\left|E\mathsf{\Delta }\right(B_a+x\left)\right|}{\left|E\right|}}:x\in {\mathbb{R}}^n\}.$$

Let $z_E$ achieve

$$A\left(E\right)={\displaystyle \frac{\left|E\mathsf{\Delta }\right(B_a+z_E\left)\right|}{\left|E\right|}}.$$

Hence,

$${\left({\displaystyle \frac{\left|E\mathsf{\Delta }\right(B_a+z_E\left)\right|}{\left|E\right|}}\right)}^2\le {\tilde{a}}_1\left(\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right)\right),$$

(29)

${\tilde{a}}_1={\tilde{a}}_1(n,\alpha ,a)>0.$ Assume

$$\mathcal{E}\left(E_i\right)-\mathcal{E}\left(B_a\right)\to 0,$$

$\left|E_i|=m=|B_a\right|$. Observe

$$|E_i\mathsf{\Delta }(B_a+z_{E_i})|\to 0.$$

Let $A\left(i\right)\subset B_a+z_{E_i}$, $\left|A\right(i\left)\right|\ge p>0$. Thus,

$$\int |{\chi }_{E_i}\left(x\right)-{\chi }_{B_a+z_{E_i}}\left(x\right)|dx\to 0$$

yields

$${\int }_{A\left(i\right)}|{\chi }_{E_i}\left(x\right)-1|dx\to 0.$$

By the triangle inequality,

$$\begin{array}{cc}\hfill \Vert A\left(i\right)\cap E_i|-|A\left(i\right)\Vert & =|{\int }_{A\left(i\right)}({\chi }_{E_i}\left(x\right)-1)dx|\hfill \\ \hfill & \le {\int }_{A\left(i\right)}|{\chi }_{E_i}\left(x\right)-1|dx\to 0.\hfill \end{array}$$

This thus implies

$$|A\left(i\right)\cap E_i|\ge {\displaystyle \frac{p}{7}}$$

assuming i is large.

Claim 1.

${sup}_i\left|z_{E_i}\right|<\infty$.

Proof of Claim 1.

Assume not. Then, modulo a subsequence

$$|z_{E_i}|\to \infty .$$

Since $A\left(i\right)\subset B_a+z_{E_i}$, one obtains

$$\underset{A\left(i\right)\cap E_i}{inf}g\to \infty$$

thanks to the (strict) monotonicity of g. Thus,

$$\begin{array}{cc}\hfill \infty & =\underset{i}{lim}{\displaystyle \frac{p}{7}}\underset{A\left(i\right)\cap E_i}{inf}g\hfill \\ \hfill & \le {\int }_{A\left(i\right)\cap E_i}g\hfill \\ \hfill & \le {\int }_{E_i}g\to {\int }_{B_a}g<\infty ,\hfill \end{array}$$

and this contradiction yields Claim 1. □

Claim 2.

$|z_{E_i}|\to 0$.

Proof of Claim 2.

If one can find a subsequence (continued to have the same index) so that ${inf}_i\left|z_{E_i}\right|>0$, observe via the positive bound that, up to possibly another subsequence,

$$z_{E_i}\to z\ne 0.$$

In particular,

$$\begin{array}{cc}\hfill \left|E_i\mathsf{\Delta }(B_a+z)\right|& \le |(B_a+z_{E_i})\mathsf{\Delta }(B_a+z)|+|E_i\mathsf{\Delta }(B_a+z_{E_i})|\hfill \\ \hfill & \le a_{\ast }\left|z_{E_i}-z|+|B_a\right|\sqrt{{\tilde{a}}_1(\mathcal{E}\left(E_i\right)-\mathcal{E}\left(B_a\right))}\hfill \\ \hfill & \to 0.\hfill \end{array}$$

Hence,

$${\chi }_{E_i}\to {\chi }_{B_a+z}\mathrm{in}{L}^1.$$

and this readily yields, modulo a subsequence,

$${\chi }_{E_i}\to {\chi }_{B_a+z}\mathrm{a}.\mathrm{e}..$$

Therefore, via Fatou

$$\begin{array}{c}\hfill {\int }_{B_a+z}g\le \underset{i}{lim\; inf}{\int }_{{\mathbb{R}}^n}g{\chi }_{E_i}=\mathcal{G}\left(B_a\right),\end{array}$$

Define $H=B_a+z$. Consider $\{v_1,v_2,\dots \}$ directions that generate via Steiner symmetrization with respect to the planes through the origin and with normal $v_i$, $H_{v_1},H_{v_2},\dots$ and

$$H_{v_i}\to B_a.$$

Set

$$H_1=\{(x_2,\dots ,x_n):(x_1,x_2,\dots ,x_n)\in H\}$$

$$H^{x_2,x_3,\dots ,x_n}=\{x_1:(x_1,x_2,\dots ,x_n)\in H\}.$$

Note that one may let ${\displaystyle \frac{x_1}{\left|x_1\right|}}=v_1$; hence, the monotonicity and complete symmetry of g imply

$$\begin{array}{cc}\hfill {\int }_{H_{v_1}}gdx& ={\int }_{H_1}\left({\int }_{-\frac{\left|H^{x_2,x_3,\dots ,x_n}\right|}{2}}^{\frac{\left|H^{x_2,x_3,\dots ,x_n}\right|}{2}}gd{x}_1\right)d{x}_2\dots d{x}_n\hfill \\ \hfill & \le {\int }_{H_1}\left({\int }_{H^{x_2,x_3,\dots ,x_n}}gd{x}_1\right)d{x}_2\dots d{x}_n\hfill \\ \hfill & ={\int }_{H}gdx.\hfill \end{array}$$

Moreover, via the radial property of g, one may rotate the coordinate $x_1$ so that $\frac{x_1}{\left|x_1\right|}=v_2$ and iterate the argument above (via $H=B_a+z$, only one direction is sufficient, however the iteration could be useful in other problems when H is not a ball):

$${\int }_{H_{v_1}}gdx\ge {\int }_{H_{v_2}}gdx;$$

in particular, note that, via $H\ne B_a$ and the monotonicity of g (the strict monotonicity), there is one direction so that the inequality is strict; therefore,

$${\int }_{H}gdx>{\int }_{B_a}gdx.$$

Hence, one obtains a contradiction

$$\begin{array}{cc}\hfill {\int }_{B_a}gdx& <{\int }_{H}g={\int }_{B_a+z}g\hfill \\ \hfill & \le \underset{i}{lim\; inf}{\int }_{{\mathbb{R}}^n}g{\chi }_{E_{i_l}}\hfill \\ \hfill & =\mathcal{G}\left(B_a\right)={\int }_{B_a}gdx.\hfill \end{array}$$

□

Note that this proves: assuming

$$\mathcal{E}\left(E_i\right)-\mathcal{E}\left(B_a\right)\to 0,$$

it follows that

$$|E_i\mathsf{\Delta }(B_a+z_{E_i})|\to 0,$$

$$z_{E_i}\to 0.$$

(30)

Now note

$$\begin{array}{cc}\hfill \left|E_i\mathsf{\Delta }{B}_a\right|& \le |E_i\mathsf{\Delta }(B_a+z_{E_i})|+|B_a\mathsf{\Delta }(B_a+z_{E_i})|\hfill \\ \hfill & \le |E_i\mathsf{\Delta }(B_a+z_{E_i})|+a_{\ast }\left|z_{E_i}\right|\to 0.\hfill \end{array}$$

Hence, there is some modulus w so that

$$\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right)\ge w\left(\right|E\mathsf{\Delta }{B}_a\left|\right).$$

Supposing $\widehat{a_1}>a_1>0$, $E_{\ast }\subset B_{\widehat{a_1}}$, $\left|E_{\ast }|=|B_{a_1}\right|$, then, via strict monotonicity and convexity of h, one obtains that the subdifferential

$${\partial }_{+}h\left(a_1\right)\ne \varnothing$$

is compact and

$$\underset{{\partial }_{+}h\left(a_1\right)}{inf}\left|x\right|>0:$$

(31)

assume $0\in {\partial }_{+}h\left(a_1\right)$, one then can use $a_1>0$ and convexity to deduce that g has a global minimum at $a_1$, and this is a contradiction via the strict monotonicity ($g\left(0\right)=0$, $g\ge 0$); therefore, this shows (31). Hence, thanks to [4], there exists

$$r_{\widehat{a_1},a_1,{\partial }_{+}h\left(a_1\right)}^{\ast }={\left(\frac{1}{{inf}_{{\partial }_{+}h\left(a_1\right)}\left|x\right|A_{\ast }}\right)}^{\frac{1}{2}}>0,$$

(32)

$A_{\ast }=A_{\ast }(\widehat{a_1},n,a)>0$ so that

$$r_{\widehat{a_1},a_1,{\partial }_{+}h\left(a_1\right)}^{\ast }{\left[\mathcal{G}\left(E_{\ast }\right)-\mathcal{G}\left(B_{a_1}\right)\right]}^{{\displaystyle \frac{1}{2}}}\ge \left|E_{\ast }\mathsf{\Delta }{B}_{a_1}\right|.$$

(33)

Supposing $\left|E\right|=|B_a|$, via the previous argument (cf. (30)), if

$$\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right)\to 0,$$

it follows that

$$z_E\to 0.$$

Hence, if

$$\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right)\le \nu$$

where $\nu$ is small, then $\left|z_E\right|$ is small.

Next,

$$(B_a+z_E)\subset B_{a+|z_E|}$$

yields

$$E\setminus B_{a+|z_E|}\subset E\setminus (B_a+z_E)$$

and thus utilizing (29)

$$|E\setminus B_{a+|z_E|}|\le |E\setminus (B_a+z_E)|\le \sqrt{|B_a{|}^2{\tilde{a}}_1(\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right))}.$$

(34)

Set

$$E_{\ast }=E\cap \left(B_{a+|z_E|}\right);$$

then note

$$E=E_{\ast }\cup (E\setminus \left(B_{a+|z_E|}\right))$$

and thanks to (34)

$$\begin{array}{cc}\hfill m& =|E_{\ast }|+|E\setminus \left(B_{a+|z_E|}\right)|\hfill \\ \hfill & \le |E_{\ast }|+\sqrt{|B_a{|}^2{\tilde{a}}_1(\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right))}.\hfill \end{array}$$

Therefore,

$$m-|E_{\ast }|\le \sqrt{|B_a{|}^2{\tilde{a}}_1(\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right))};$$

(35)

next, consider $a_1>0$ via

$$|E_{\ast }|=|B_{a_1}|=a_1^n\left|B_1\right|.$$

Observe that (35) easily implies

$$m-\sqrt{|B_a{|}^2{\tilde{a}}_1(\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right))}\le |E_{\ast }|=a_1^n\left|B_1\right|,$$

hence,

$${\left({\displaystyle \frac{m-\sqrt{|B_a{|}^2{\tilde{a}}_1(\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right))}}{\left|B_1\right|}}\right)}^{\frac{1}{n}}\le a_1.$$

In particular, since the energy difference is small $\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right)\le \nu$, there exists a lower bound on $a_1$ via $a,n,\alpha$. Thanks to $\left|z_E\right|$ being small and (33), one may choose

$$5a>\widehat{a_1}>a+\left|z_E\right|$$

such that

$$r_{\widehat{a_1},a_1,{\partial }_{+}h\left(a_1\right)}^{\ast }{\left[\mathcal{G}\left(E_{\ast }\right)-\mathcal{G}\left(B_{a_1}\right)\right]}^{\frac{1}{2}}\ge \left|E_{\ast }\mathsf{\Delta }{B}_{a_1}\right|,$$

(36)

where $r_{\widehat{a_1},a_1,{\partial }_{+}h\left(a_1\right)}^{\ast }$ is bounded via $a,n,\alpha$. Now, since $a\ge a_1$, (35) implies

$$\begin{array}{cc}\hfill \mathcal{G}\left(E_{\ast }\right)-\mathcal{G}\left(B_{a_1}\right)& =\mathcal{G}\left(E_{\ast }\right)-\mathcal{G}\left(B_a\right)+\mathcal{G}\left(B_a\right)-\mathcal{G}\left(B_{a_1}\right)\hfill \\ \hfill & \le \mathcal{G}\left(E_{\ast }\right)-\mathcal{G}\left(B_a\right)+\left(\underset{B_a}{sup}g\right)|B_a\setminus B_{a_1}|\hfill \\ \hfill & =\mathcal{G}\left(E_{\ast }\right)-\mathcal{G}\left(B_a\right)+\left(\underset{B_a}{sup}g\right)\left(\right|B_a|-|B_{a_1}\left|\right)\hfill \\ \hfill & =\mathcal{G}\left(E_{\ast }\right)-\mathcal{G}\left(B_a\right)+\left(\underset{B_a}{sup}g\right)(m-|E_{\ast }\left|\right)\hfill \\ \hfill & \le \mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right)+\left(\underset{B_a}{sup}g\right)\sqrt{|B_a{|}^2{\tilde{a}}_1(\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right))}.\hfill \end{array}$$

In particular, (36) and the above inequality imply

$$\begin{array}{cc}\hfill \left|E_{\ast }\mathsf{\Delta }{B}_{a_1}\right|& \le r_{\widehat{a_1},a_1,{\partial }_{+}h\left(a_1\right)}^{\ast }{\left[\mathcal{G}\left(E_{\ast }\right)-\mathcal{G}\left(B_{a_1}\right)\right]}^{\frac{1}{2}}\hfill \\ \hfill & \le r_{\widehat{a_1},a_1,{\partial }_{+}h\left(a_1\right)}^{\ast }{\left[\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right)+\underset{B_a}{sup}g\sqrt{|B_a{|}^2{\tilde{a}}_1(\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right))}\right]}^{\frac{1}{2}}.\hfill \end{array}$$

(37)

Also, (34) and (35) yield

$$|B_a\mathsf{\Delta }{B}_{a_1}|=|B_a\setminus B_{a_1}|\le \sqrt{|B_a{|}^2{\tilde{a}}_1(\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right))}$$

(38)

$$|E_{\ast }\mathsf{\Delta }E|=|E\setminus \left(B_{a+|z_E|}\right)|\le \sqrt{|B_a{|}^2{\tilde{a}}_1(\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right))}.$$

(39)

Hence, (34), (37)–(39), and the triangle inequality in $L^1$ imply

$$\begin{array}{cc}\hfill \left|B_a\mathsf{\Delta }(B_a+z_E)\right|& \le |B_a\mathsf{\Delta }{B}_{a_1}|+|B_{a_1}\mathsf{\Delta }{E}_{\ast }|+|E\mathsf{\Delta }{E}_{\ast }|+|E\mathsf{\Delta }(B_a+z_E)|\hfill \\ \hfill & \le {\alpha }_{\ast }{\left[\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right)\right]}^{{\displaystyle \frac{1}{4}}}.\hfill \end{array}$$

Last,

$$\begin{array}{cc}\hfill |E\mathsf{\Delta }{B}_a|& \le |E\mathsf{\Delta }(B_a+z_E)|+|(B_a+z_E)\mathsf{\Delta }{B}_a|\hfill \\ \hfill & \le \sqrt{|B_a{|}^2{\tilde{a}}_1(\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right))}+{\alpha }_{\ast }{\left[\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right)\right]}^{{\displaystyle \frac{1}{4}}}\hfill \\ \hfill & \le {\overline{\alpha }}_{\ast }{\left[\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right)\right]}^{{\displaystyle \frac{1}{4}}},\hfill \end{array}$$

${\overline{\alpha }}_{\ast }={\overline{\alpha }}_{\ast }(a,n,\alpha ,g)>0$.

(ii) Supposing $\widehat{a}>a$, $E\subset B_{\widehat{a}}$, then one can show similarly to the proof in (i) (cf. (32) and (33)) that there exists some constant $r_{m,\widehat{a},n,\alpha ,g}^{\ast }>0$ (that is explicit) so that

$$r_{m,\widehat{a},n,\alpha ,g}^{\ast }{\left[\mathcal{E}\left(E\right)-\mathcal{E}\left(B_a\right)\right]}^{{\displaystyle \frac{1}{2}}}\ge r_{m,\widehat{a},n,\alpha ,g}^{\ast }{\left[\mathcal{G}\left(E\right)-\mathcal{G}\left(B_a\right)\right]}^{{\displaystyle \frac{1}{2}}}\ge \left|E\mathsf{\Delta }{B}_a\right|.$$

□

Remark 2.

Assuming $E\subset B_{\widehat{a}}$, the estimate is optimal: set $g\left(y\right)={\left|y\right|}^2$,

$$\begin{array}{cc}\hfill \mathcal{E}(B_a+x)-\mathcal{E}\left(B_a\right)& ={\int }_{B_a}{(2⟨y,x⟩+|x|}^2{)dy=|x|}^2\left|B_a\right|;\hfill \end{array}$$

supposing $\left|x\right|$ is small,

$$|(B_a+x)\mathsf{\Delta }{B}_a|\approx |x|,$$

$$|(B_a+x)\mathsf{\Delta }{B}_a{|}^2\approx {\left|x\right|}^2.$$

Remark 3.

In the argument of the theorem, (28) implies that balls minimize the energy also when g is non-decreasing, radial, and possibly non-convex.

Remark 4.

In the theorem, the invariance class is completely identified and stability is not modulo translations like the quantitative anisotropic isoperimetric inequality. Supposing a context where the set is in a convex cone [12,35,36,37], translations are crucial: assuming the cone contains no line, the quantitative term is without translations.

Corollary 1.

Suppose $g\left(x\right)=h\left(\right|x\left|\right)$, where h is non-negative, non-decreasing, not identically zero, and homogeneous of degree ν. Let $m>0$ and assume $E_m$ is the minimizer with $|E_m|=m$; set

$$m_{\alpha ,\nu ,g}=\left|B_1\right|{\left[{\displaystyle \frac{\alpha (n-\alpha )}{\nu (n+\nu )}}{\displaystyle \frac{P_{\alpha }\left(B_1\right)}{{\int }_{B_1}g\left(x\right)dx}}\right]}^{\frac{n}{\nu +\alpha }}$$

it then follows that

$$m\mapsto \mathcal{E}\left(E_m\right)$$

is concave on $(0,m_{\alpha ,\nu ,g})$ and convex on $(m_{\alpha ,\nu ,g},\infty )$.

Proof. 

Thanks to Remark 3, $E_m=B_r$, $r={\left[{\displaystyle \frac{m}{\left|B_1\right|}}\right]}^{{\displaystyle \frac{1}{n}}}$, which therefore implies

$$\mathcal{E}\left(E_m\right)=P_{\alpha }\left(B_1\right)\left({\displaystyle \frac{1}{|B_1{|}^{\frac{n-\alpha }{n}}}}\right)m^{{\displaystyle \frac{n-\alpha }{n}}}+\left({\displaystyle \frac{1}{|B_1{|}^{\frac{\nu +n}{n}}}}\right)({\int }_{B_1}h\left(\right|x\left|\right)dx)m^{{\displaystyle \frac{n+\nu }{n}}}.$$

The critical mass $m_{\alpha ,\nu ,g}$ is therefore calculated with the second derivative. □

Proposition 1.

If $m>0$, $g\in L_{loc}^{\infty }$, and up to sets of measure zero, g admits unique minimizers $E_m$; then, for $ϵ>0$, there exists $w_m\left(ϵ\right)>0$ such that, if $\left|E\right|=|E_m|$, $E\subset B_R$, $R=R\left(m\right)$, and

$$|\mathcal{E}\left(E\right)-\mathcal{E}\left(E_m\right)|<w_m\left(ϵ\right),$$

then

$${\displaystyle \frac{\Vert {\chi }_E-{\chi }_{E_m}{\Vert }_{L^1}}{\left|E_m\right|}}<ϵ.$$

Identifying the modulus $w_m\left(ϵ\right)$ is, in general, complex. Additional conditions illuminate interesting properties, as illustrated in the first theorem. In the subsequent theorem, upper bounds are illustrated with minimal assumptions.

Theorem 2.

Suppose $m>0$, and, up to sets of measure zero, g admits unique minimizers $E_m\subset B_{r\left(m\right)}$ in the collection of sets of finite perimeter; then,

(i) 

if $g\in W_{loc}^{1,1}$ is locally uniformly differentiable, there exists ${\alpha }_1={\alpha }_1\left(E_m\right)>0$ such that, if $ϵ<{\alpha }_1$, one has

$$w_m\left(ϵ\right)\le {\lambda }_{g,E_m}(m,ϵ)=o_{g,E_m}\left({\displaystyle \frac{ϵm}{a_{m_{\ast }}}}\right),$$

$$a_{m_{\ast }}={\alpha }_{\ast }|D{\chi }_{E_m}·w_{\ast }|\left({\mathbb{R}}^n\right)$$

with ${\alpha }_{\ast }>0$, $w_{\ast }\in {\mathbb{R}}^n$;

(ii) 

if $g\in W_{loc}^{1,\infty }$, there exists $m_a>0$ such that, if $m<m_a$, $ϵ<{\alpha }_1$, then

$${\lambda }_{g,E_m}(m,ϵ)={\alpha }_{1_{\ast }}{m}^{{\displaystyle \frac{1}{n}}}{o}_{g,E_m}\left(ϵ\right),$$

$$o_{g,E_m}\left(ϵ\right)=ϵ{\int }_{E_m}{o}_{x,g}\left(1\right)dx,$$

$${\int }_{E_m}{o}_{x,g}\left(1\right)dx\to 0$$

as $ϵ\to 0^{+}$, ${\alpha }_{1_{\ast }}>0$;

(iii) 

if $g\in C^1$, there exist $m_a,{\alpha }_m>0$ such that, if $m<m_a$, $ϵ<{\alpha }_m$, then

$${\lambda }_{g,E_m}(m,ϵ)={\alpha }_{2_{\ast }}{m}^{1+{\displaystyle \frac{1}{n}}}{o}_g\left(ϵ\right),$$

${\alpha }_{2_{\ast }}={\alpha }_{2_{\ast }}{(\Vert Dg\Vert }_{L^{\infty }\left(B_{r\left(m_a\right)}\right)})>0$;

(iv) 

if $g\in W_{loc}^{2,1}$ is twice locally uniformly differentiable, there exists ${\alpha }_m>0$ such that, if $ϵ<{\alpha }_m$,

$$w_m\left(ϵ\right)\le a_m{ϵ}^2,$$

$$a_m=\left(\Vert D^{2}g{\Vert }_{L^1\left(E_m\right)}+m{\displaystyle \frac{1}{6}}\right){\left({\displaystyle \frac{m}{{\alpha }_{\ast }|D{\chi }_{E_m}·w_{\ast }|\left({\mathbb{R}}^n\right)}}\right)}^2$$

with ${\alpha }_{\ast }>0$;

(v) 

if $g\in W_{loc}^{2,\infty }$, there exist $m_a,{\alpha }_m>0$ such that, if $m<m_a$, $ϵ<{\alpha }_m$, then

$$w_m\left(ϵ\right)\le a_m{ϵ}^2,$$

$$a_m=a_{\ast }{m}^{1+{\displaystyle \frac{2}{n}}},$$

$a_{\ast }=a_{\ast }(\Vert D^2{g\Vert }_{L^{\infty }\left(B_{r\left(m_a\right)}\right)})>0$.

Remark 5.

The existence of bounded minimizers $E_m$ was proven in [30] via assuming coercivity of g. Also, a minimizer $E_m$ is, up to a closed set with Hausdorff dimension at most $n-3$, a $C^{2,a}$ set; in particular, it is much more smooth than just a set of finite perimeter. One may in some contexts preclude translations (e.g., supposing that g is strictly convex). If g is zero on some small ball, then note that, if the mass is very small, uniqueness is only up to translations and sets of measure zero.

Remark 6.

The assumption that $g\in W_{loc}^{1,1}$ is locally uniformly differentiable in (i) can be weakened.

Remark 7.

Theorem 1 encodes the modulus in an explicit way. In particular, supposing m is small, if $g\left(\right|x\left|\right)={\left|x\right|}^2$, a possible modulus is $w_m\left(ϵ\right)=a_1{ϵ}^2{m}^3$. Thus, the ϵ dependence in (v) is optimal. Observe also that, via the quadratic, the m dependence is almost sharp in two dimensions. The optimal m-modulus also encodes $a_{\ast }\left(\Vert D^2{g\Vert }_{L^{\infty }\left(B_{r\left(m_a\right)}\right)}\right)$.

In the general case when g is convex, one does not have existence.

Theorem 3.

There exists $g\ge 0$ convex such that $g\left(0\right)=0$ and such that, if $m>0$, then there is no solution to

$$inf\left\{\mathcal{E}\right(E):|E|=m\}.$$

In particular, coercivity or another condition is necessary.

If $\alpha ,g$ are given, an invariance map of the nonlocal free energy is a transformation

$$\begin{array}{cc}\hfill & A\in {\mathcal{A}}_m={\mathcal{A}}_{\alpha ,g,m}\hfill \\ \hfill & =\{A:Ax=A_{a}x+x_a,x_a\in {\mathbb{R}}^n,\mathcal{E}\left(A_{a}E\right)=\mathcal{E}\left(E\right),|A_{a}E|=|E|=m\mathrm{for}\mathrm{some}\mathrm{minimizer}E\}.\hfill \end{array}$$

Uniqueness of minimizers can only be true up to sets of measure zero and an invariance map. Note that, in many classes of potentials, assuming m is small, $A\in {\mathcal{A}}_m$ is a translation; a simple case is to assume g is zero on a ball B; therefore, if m is small, $A_a=I_{n\times n}$, $x_a\in {\mathbb{R}}^n$ is such that $B_a+x_a\subset \{g=0\}$ when $B_a\subset B$.

Supposing g is locally bounded, the stability modulus, in the context of small mass, is a product in any dimension.

Theorem 4.

Suppose $g\in L_{loc}^{\infty }\left(\{g<\infty \}\right)$ admits minimizers $E_m\subset B_R$ for all m small. There exists $m_0>0$ and a modulus of continuity $q\left(0^{+}\right)=0$ such that, for all $m<m_0$, there exists ${ϵ}_0>0$ such that, for all $0<ϵ<{ϵ}_0$ and for all minimizers $E_m\subset B_R$, $E\subset B_R$, $\left|E\right|=|E_m|=m<m_0$; if

$$|\mathcal{E}\left(E_m\right)-\mathcal{E}\left(E\right)|<a(m,ϵ,\alpha )=q\left(ϵ\right)m^{\frac{n-\alpha }{n}},$$

there exists an invariance map $A\in {\mathcal{A}}_m$ such that

$${\displaystyle \frac{\Vert {\chi }_E-{\chi }_{A{E}_m}{\Vert }_{L^1}}{\left|E_m\right|}}<ϵ.$$

Also, $A{E}_m\approx E_m+{\alpha }_m$, ${\alpha }_m\in {\mathbb{R}}^n$ cf. (A15).

Remark 8.

If $g\left(x\right)=h\left(\right|x\left|\right)$, $h:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is increasing, $h\left(0\right)=0$, then A is the identity in the conclusion of Theorem 4: if $m>0$, then ${\mathcal{A}}_{\alpha ,h\left(\right|x\left|\right),m}$ may be replaced with

$${\overline{\mathcal{A}}}_{\alpha ,h\left(\right|x\left|\right),m}=\left\{I_{n\times n}\right\}.$$

Remark 9.

The theorem may also be extended to $g\in L_{loc}^1\left(\{g<\infty \}\right)$ with some assumptions via Lebesgue’s differentiation theorem.

3. Conclusions and Directions for Future Research

(1)

Assuming radial symmetry on the potential, a stability estimate is proven in Theorem 1 via the spherical geometry.

(2)

Assuming that the potential is—up to sets of measure zero—locally bounded, and also that—up to sets of measure zero—it admits unique minimizers, the existence of energy moduli is shown; see Proposition 1. Furthermore, assuming small mass, the energy modulus is proven to have a product structure; see Theorem 4.

(3)

An explicit convex potential $g\ge 0$, $g\left(0\right)=0$ is constructed, illuminating that there are no minimizers to the nonlocal free energy optimization under a mass constraint for positive masses; see Theorem 3.

(4)

Upper bounds for the moduli are obtained with minimal assumptions; see Theorem 2.

(5)

Future research directions: replacing 4 with 2 in Theorem 1 (i) and obtaining the explicit constant analogously to (ii); removing the symmetry assumption on g and imposing coercivity in Theorem 1; calculating the modulus in Theorem 4.

Appendix A.1. Compactness: Existence of Moduli

Proof of Proposition 1. 

Assume the assumptions do not yield the conclusion; then, there exists $ϵ>0$, and, for $a>0$, there exist $E_a^{\prime }\subset B_R$ and $E_m$, $|E_a^{\prime }|=|E_m|=m$,

$$|\mathcal{E}\left(E_a^{\prime }\right)-\mathcal{E}\left(E_m\right)|<a,$$

$$\frac{\left|E_m\mathsf{\Delta }{E}_a^{\prime }\right|}{\left|E_m\right|}\ge ϵ.$$

Define $a=\frac{1}{k}$, $k\in \mathbb{N}$; therefore, there exist $E_{\frac{1}{k}}^{\prime }$, $|E_{\frac{1}{k}}^{\prime }|=m$,

$$|\mathcal{E}\left(E_m\right)-\mathcal{E}(E_{\frac{1}{k}}^{\prime })|\le {\displaystyle \frac{1}{k}},$$

$$\frac{\left|E_m\mathsf{\Delta }{E}_{\frac{1}{k}}^{\prime }\right|}{\left|E_m\right|}\ge ϵ.$$

In particular,

$$\begin{array}{cc}\hfill P_{\alpha }(E_{\frac{1}{k}}^{\prime })& \le \mathcal{E}(E_{\frac{1}{k}}^{\prime })\hfill \\ \hfill & \le \frac{1}{k}+\mathcal{E}\left(E_m\right),\hfill \end{array}$$

$$E_{{\displaystyle \frac{1}{k}}}^{\prime }\subset B_R,$$

and via the compactness $H^{\frac{\alpha }{2}}\hookrightarrow L_{loc}^1$, up to a subsequence,

$$E_{\frac{1}{k}}^{\prime }\to E^{\prime }in{L}^1\left(B_R\right);$$

thus, $|E^{\prime }|=m$ because of the triangle inequality in $L^1\left(B_R\right)$. Next,

$$\mathcal{E}\left(E^{\prime }\right)\le \underset{k}{lim\; inf}\mathcal{E}\left(E_{\frac{1}{k}}^{\prime }\right)=\mathcal{E}\left(E_m\right)$$

implies $E^{\prime }$ is a minimizer, contradicting

$${\displaystyle \frac{\left|E_m\mathsf{\Delta }{E}^{\prime }\right|}{\left|E_m\right|}}\ge ϵ$$

via the uniqueness. □

Appendix A.2. Bounds on the Moduli

Proof of Theorem 2. 

Assuming m is small, the convexity of $E_m$ is established in [28] and utilized for $\left(ii\right),\left(iii\right),\left(v\right)$, similar to the proof in [4]. The argument for $\left(i\right)$ and $\left(iv\right)$ is the same as in [4]. □

Appendix A.3. Non-Existence of Minimizers

Proof of Theorem 3. 

One definition of g via the construction in [3] also works in this case: choose an $ϵ>0$ and let

$$g^{ϵ}(x,y)=\left\{\begin{array}{cc}{x}^2(1-y)+\frac{ϵ}{2}{y}^2\hfill & \mathrm{if}-\sqrt{\frac{ϵ}{2}}\le x\le \sqrt{\frac{ϵ}{2}},y\le 0\hfill \\ \frac{x^2}{1+y}\hfill & \mathrm{if}y>0;\hfill \end{array}\right.$$

in particular, if $y>0$, then note $(x,y)\mapsto \frac{x^2}{1+y}$ is convex and non-increasing in the y-variable and strictly decreasing if $x\ne 0$. Extend $g^{ϵ}$ to ${\mathbb{R}}^2$ by a convex envelope to construct g (e.g., $\overline{g}(x,y)=sup\{A(x,y):A\le g^{ϵ},\mathrm{A}\mathrm{is}\mathrm{affine}\}$ is a convex extension since it is a supremum of affine functions). Note that a finite-valued convex extension inherits the monotonicity in the y-variable: if one takes the hyperplane $\left\{\right(x,y,z):x=t\}$ and assumes $\left|t\right|>\sqrt{\frac{ϵ}{2}}$ and then intersects it with the graph when $y\ge 0$ and $\left|x\right|>\sqrt{\frac{ϵ}{2}}$, the intersection is a convex function of one variable and ${inf}_{y>0}\frac{x^2}{1+y}=0$; therefore, from the convexity of g, when x is fixed, observe $\frac{x^2}{1+y}$ has an extension to $y<0$ that cannot have horizontally flat sections. Let $e_2=(0,1)$, $a>0$; via monotonicity in the y-variable, if a minimizer $E_m$ exists,

$${\int }_{E_m+a{e}_2}g(x,y)dxdy<{\int }_{E_m}g(x,y)dxdy,$$

therefore, the translation invariance of $P_{\alpha }$ easily implies

$$\mathcal{E}(E_m+a{e}_2)<\mathcal{E}\left(E_m\right),$$

a contradiction. □

Appendix A.4. A Product Property

Proof of Theorem 4. 

Via contradiction, suppose the theorem is not true; then, for $m_0>0$, for all moduli q, there exists $m<m_0$ such that, for ${ϵ}_0\in (0,2]$, there exists $ϵ<{ϵ}_0$ and $E_{m,{ϵ}_0},E_{m,{ϵ}_0}^{\prime }\subset B_R$, $|E_{m,{ϵ}_0}|=|E_{m,{ϵ}_0}^{\prime }|=m$,

$$|\mathcal{E}\left(E_m\right)-\mathcal{E}\left(E_m^{\prime }\right)|<q\left(ϵ\right)m^{\frac{n-\alpha }{n}},$$

and

$$\underset{A\in {\mathcal{A}}_m}{inf}\frac{\left|E_{m,{ϵ}_0}^{\prime }\mathsf{\Delta }A{E}_{m,{ϵ}_0}\right|}{\left|E_{m,{ϵ}_0}\right|}\ge ϵ>0.$$

(A1)

In particular, let $m_0=\frac{1}{k}$, $w_k\to 0^{+}$, $\widehat{q}$ define a modulus of continuity, and let

$$q_k=w_k\widehat{q}\left(ϵ\right).$$

(A2)

Therefore, there exists $m_k<\frac{1}{k}$ such that, for a fixed ${ϵ}_0\in (0,2]$, there exists $ϵ<{ϵ}_0$ and $E_{m_k,{ϵ}_0},E_{m_k,{ϵ}_0}^{\prime }\subset B_R$, $|E_{m_k,{ϵ}_0}|=|E_{m_k,{ϵ}_0}^{\prime }|=m_k<\frac{1}{k}$,

$$|\mathcal{E}\left(E_{m_k}\right)-\mathcal{E}\left(E_{m_k}^{\prime }\right)|<q_k{m}_k^{\frac{n-\alpha }{n}},$$

and

$$\underset{A\in {\mathcal{A}}_{m_k}}{inf}{\displaystyle \frac{\left|E_{m_k,{ϵ}_0}^{\prime }\mathsf{\Delta }A{E}_{m_k,{ϵ}_0}\right|}{\left|E_{m_k,{ϵ}_0}\right|}}\ge ϵ>0.$$

(A3)

Let

$$a_k=q_k{m}_k^{{\displaystyle \frac{n-\alpha }{n}}},$$

(A4)

$E_{m_k}=E_{m_k,{ϵ}_0}$, $E_{m_k}^{\prime }=E_{m_k,{ϵ}_0}^{\prime }$. Define ${\gamma }_k={\left(\frac{\left|B_1\right|}{m_k}\right)}^{\frac{1}{n}}$,

$$|{\gamma }_k{E}_{m_k}|=|B_1|.$$

Observe via the nonlocal isoperimetric inequality and the minimality

$$\begin{array}{cc}\hfill P_{\alpha }\left({\displaystyle \frac{1}{{\gamma }_k}}{B}_1\right)+{\int }_{E_{m_k}}g& \le P_{\alpha }\left(E_{m_k}\right)+{\int }_{E_{m_k}}g\hfill \\ \hfill & \le P_{\alpha }({\displaystyle \frac{1}{{\gamma }_k}}{B}_1)+{\int }_{{\displaystyle \frac{1}{{\gamma }_k}}{B}_1}g\hfill \end{array}$$

$$P_{\alpha }(E_{m_k})-P_{\alpha }(\frac{1}{{\gamma }_k}{B}_1)\le (\underset{\frac{1}{{\gamma }_k}{B}_1}{sup}g)m_k$$

$${\gamma }_k^{n-\alpha }\left(P_{\alpha }\left(E_{m_k}\right)-P_{\alpha }(\frac{1}{{\gamma }_k}{B}_1)\right)\le {\gamma }_k^{n-\alpha }\left((\underset{\frac{1}{{\gamma }_k}{B}_1}{sup}g)m_k\right)$$

$$\left(P_{\alpha }({\gamma }_k{E}_{m_k})-P_{\alpha }\left(B_1\right)\right)\le \left((\underset{\frac{1}{{\gamma }_k}{B}_1}{sup}g)\left({\gamma }_k^{n-\alpha }{m}_k\right)\right)$$

$$\left(P_{\alpha }\left({\gamma }_k{E}_{m_k}\right)-P_{\alpha }\left(B_1\right)\right)\le \left((\underset{\left(\frac{1}{{\gamma }_k}\right)B_1}{sup}g)\left(\frac{\left|B_1\right|}{{\gamma }_k^{\alpha }}\right)\right).$$

Since $m_k\to 0$ as $k\to \infty$, it thus follows that ${\gamma }_k\to \infty$; hence,

$$\begin{array}{cc}\hfill \delta \left({\gamma }_k{E}_{m_k}\right)& ={\displaystyle \frac{P_{\alpha }\left({\gamma }_k{E}_{m_k}\right)}{P_{\alpha }\left(B_1\right)}}-1\to 0.\hfill \end{array}$$

Next, by the triangle inequality,

$$\begin{array}{cc}\hfill |P_{\alpha }\left(E_{m_k}^{\prime }\right)& -P_{\alpha }\left(E_{m_k}\right)|\hfill \\ \hfill & =|[\mathcal{E}\left(E_{m_k}^{\prime }\right)-\mathcal{E}\left(E_{m_k}\right)]+[{\int }_{E_{m_k}}g\left(x\right)dx-{\int }_{E_{m_k}^{\prime }}g\left(x\right)dx]|\hfill \\ \hfill & \le |\mathcal{E}\left(E_{m_k}^{\prime }\right)-\mathcal{E}\left(E_{m_k}\right)|+\int g\left(x\right)|{\chi }_{E_{m_k}^{\prime }}-{\chi }_{E_{m_k}}|dx\hfill \\ \hfill & <a_k+{\int }_{E_{m_k}^{\prime }\mathsf{\Delta }{E}_{m_k}}g\left(x\right)dx.\hfill \end{array}$$

In particular,

$$\begin{array}{cc}\hfill |P_{\alpha }\left({\gamma }_k{E}_{m_k}^{\prime }\right)& -P_{\alpha }\left({\gamma }_k{E}_{m_k}\right)|\hfill \\ \hfill & <{\gamma }_k^{n-\alpha }{a}_k+2{\gamma }_k^{n-\alpha }{m}_k(\underset{B_R\cap \{g<\infty \}}{sup}g)\hfill \\ \hfill & =|B_1{|}^{\frac{n-\alpha }{n}}\frac{a_k}{m_k^{\frac{n-\alpha }{n}}}+2{|B_1|}^{\frac{n-\alpha }{n}}(\underset{B_R\cap \{g<\infty \}}{sup}g)m_k^{1-\frac{n-\alpha }{n}}\hfill \end{array}$$

and since, from (A2) and (A4),

$$a_k=q_k{m}_k^{{\displaystyle \frac{n-\alpha }{n}}}=w_k\widehat{q}\left(ϵ\right)m_k^{{\displaystyle \frac{n-\alpha }{n}}},$$

one obtains

$${\displaystyle \frac{a_k}{m_k^{\frac{n-\alpha }{n}}}}=w_k\widehat{q}\left(ϵ\right)\to 0$$

$$|P_{\alpha }\left({\gamma }_k{E}_{m_k}^{\prime }\right)-P_{\alpha }\left({\gamma }_k{E}_{m_k}\right)|\to 0$$

as $k\to \infty$. Hence,

$$\delta \left({\gamma }_k{E}_{m_k}^{\prime }\right)\le |\delta \left({\gamma }_k{E}_{m_k}^{\prime }\right)-\delta \left({\gamma }_k{E}_{m_k}\right)|+\delta \left({\gamma }_k{E}_{m_k}\right)$$

(A5)

$$={\displaystyle \frac{1}{P_{\alpha }\left(B_1\right)}}|P_{\alpha }\left({\gamma }_k{E}_{m_k}^{\prime }\right)-P_{\alpha }\left({\gamma }_k{E}_{m_k}\right)|+\delta \left({\gamma }_k{E}_{m_k}\right)$$

(A6)

$$\to 0$$

(A7)

as $k\to \infty$. Next, by the sharp stability of the nonlocal isoperimetric inequality [21] or a compactness argument, there exist $x_k,x_k^{\prime }\in {\mathbb{R}}^n$ such that

$${\displaystyle \frac{\left|({\gamma }_k{E}_{m_k}+x_k)\mathsf{\Delta }{B}_1\right|}{\left|{\gamma }_k{E}_{m_k}\right|}}\to 0,$$

(A8)

and

$${\displaystyle \frac{\left|({\gamma }_k{E}_{m_k}^{\prime }+x_k^{\prime })\mathsf{\Delta }{B}_1\right|}{\left|{\gamma }_k{E}_{m_k}^{\prime }\right|}}\to 0$$

(A9)

as $k\to \infty$.

Therefore, (A8) and (A9) yield $k\in \mathbb{N}$ such that

$$\begin{array}{cc}\hfill {\displaystyle \frac{\left|(E_{m_k}^{\prime }+\frac{(x_k^{\prime }-x_k)}{{\gamma }_k})\mathsf{\Delta }{E}_{m_k}\right|}{\left|E_{m_k}\right|}}& ={\displaystyle \frac{\left|({\gamma }_k{E}_{m_k}^{\prime }+x_k^{\prime })\mathsf{\Delta }({\gamma }_k{E}_{m_k}+x)\right|}{\left|{\gamma }_k{E}_{m_k}\right|}}\hfill \\ \hfill & \le {\displaystyle \frac{\left|({\gamma }_k{E}_{m_k}+x)\mathsf{\Delta }{B}_1\right|}{\left|{\gamma }_k{E}_{m_k}\right|}}+{\displaystyle \frac{\left|B_1\mathsf{\Delta }({\gamma }_k{E}_{m_k}^{\prime }+x_k^{\prime })\right|}{\left|{\gamma }_k{E}_{m_k}\right|}}\hfill \\ \hfill & <ϵ,\hfill \end{array}$$

a contradiction to

$$\begin{array}{cc}\hfill {\displaystyle \frac{\left|\left(E_{m_k}^{\prime }+\frac{(x_k^{\prime }-x_k)}{{\gamma }_k}\right)\mathsf{\Delta }{E}_{m_k}\right|}{\left|E_{m_k}\right|}}& ={\displaystyle \frac{\left|E_{m_k}^{\prime }\mathsf{\Delta }\left(E_{m_k}-\frac{(x_k^{\prime }-x_k)}{{\gamma }_k}\right)\right|}{\left|E_{m_k}\right|}}\hfill \\ \hfill & \ge \underset{A\in {\mathcal{A}}_{m_k}}{inf}{\displaystyle \frac{\left|E_{m_k}^{\prime }\mathsf{\Delta }A{E}_{m_k}\right|}{\left|E_{m_k}\right|}}\hfill \\ \hfill & \ge ϵ>0,\hfill \end{array}$$

thanks to (A3). To finish, assume $E_m$ is a minimizer, $m<m_0$, and define ${\gamma }_m={\left({\displaystyle \frac{\left|B_1\right|}{m}}\right)}^{\frac{1}{n}}$,

$$|{\gamma }_m{E}_m|=|B_1|.$$

Since

$$P_{\alpha }\left({\displaystyle \frac{1}{{\gamma }_m}}{B}_1\right)\le P_{\alpha }\left(E_m\right),$$

$$\begin{array}{cc}\hfill P_{\alpha }\left({\displaystyle \frac{1}{{\gamma }_m}}{B}_1\right)+{\int }_{E_m}g\left(x\right)dx& \le P_{\alpha }\left(E_m\right)+{\int }_{E_m}g\left(x\right)dx\hfill \\ \hfill & \le P_{\alpha }\left({\displaystyle \frac{1}{{\gamma }_m}}{B}_1\right)+{\int }_{\frac{1}{{\gamma }_m}{B}_1}g\left(x\right)dx,\hfill \end{array}$$

thus, via subtracting

$$P_{\alpha }\left({\displaystyle \frac{1}{{\gamma }_m}}{B}_1\right)+{\int }_{E_m}g\left(x\right)dx,$$

$$\begin{array}{cc}\hfill P_{\alpha }\left(E_m\right)-P_{\alpha }\left({\displaystyle \frac{1}{{\gamma }_m}}{B}_1\right)& \le {\int }_{{\displaystyle \frac{1}{{\gamma }_m}}{B}_1}g\left(x\right)dx-{\int }_{E_m}g\left(x\right)dx\hfill \\ \hfill & \le (\underset{B_{R_m}}{sup}g)m,\hfill \end{array}$$

(A10)

$${\displaystyle \frac{1}{{\gamma }_m}}{B}_1\subset B_{R_m}.$$

Next, (A10) implies

$$\begin{array}{cc}\hfill P_{\alpha }\left({\gamma }_m{E}_m\right)-P_{\alpha }\left(B_1\right)& ={\left({\gamma }_m\right)}^{n-\alpha }(P_{\alpha }\left(E_m\right)-P_{\alpha }\left({\displaystyle \frac{1}{{\gamma }_m}}{B}_1\right))\hfill \\ \hfill & \le {\left({\gamma }_m\right)}^{n-\alpha }\left(\underset{B_{R_m}}{sup}g\right)m={\left|B_1\right|}^{{\displaystyle \frac{n-\alpha }{n}}}\left(\underset{B_{R_m}}{sup}g\right)m^{{\displaystyle \frac{\alpha }{n}}}.\hfill \end{array}$$

(A11)

Therefore, since

$$\delta \left({\gamma }_m{E}_m\right)\ge c(n,\alpha ){\left({\displaystyle \frac{\left|{\gamma }_m{E}_m\mathsf{\Delta }(a_m+B_1)\right|}{\left|B_1\right|}}\right)}^2,$$

thanks to (A11) and the sharp quantitative nonlocal isoperimetric inequality, one obtains

$$\begin{array}{cc}\hfill c(n,\alpha ){\left({\displaystyle \frac{\left|{\gamma }_m{E}_m\mathsf{\Delta }(a_m+B_1)\right|}{\left|B_1\right|}}\right)}^2& \le \delta \left({\gamma }_m{E}_m\right)\hfill \\ \hfill & ={\displaystyle \frac{P_{\alpha }\left({\gamma }_m{E}_m\right)-P_{\alpha }\left(B_1\right)}{P_{\alpha }\left(B_1\right)}}\hfill \\ \hfill & \le {\displaystyle \frac{|B_1{|}^{\frac{n-\alpha }{n}}}{P_{\alpha }\left(B_1\right)}}(\underset{B_{R_m}}{sup}g)m^{\frac{\alpha }{n}}.\hfill \end{array}$$

Now,

$$|{\gamma }_m{E}_m\mathsf{\Delta }(a_{E_m}+B_1)|\le |B_1|{\left({\displaystyle \frac{1}{c(n,\alpha )}}{\displaystyle \frac{|B_1{|}^{\frac{n-\alpha }{n}}}{P_{\alpha }\left(B_1\right)}}(\underset{B_{R_m}}{sup}g)\right)}^{\frac{1}{2}}{m}^{\frac{\alpha }{2n}}.$$

(A12)

Observe this is valid for any minimizer $E_m$; therefore, for any minimizer $A{E}_m$ with $A\in {\mathcal{A}}_m$:

$$\left|{\gamma }_{m}A{E}_m\mathsf{\Delta }(a_{A{E}_m}+B_1)\right|\le |B_1|{\left({\displaystyle \frac{1}{c(n,\alpha )}}{\displaystyle \frac{|B_1{|}^{\frac{n-\alpha }{n}}}{P_{\alpha }\left(B_1\right)}}(\underset{B_{R_m}}{sup}g)\right)}^{\frac{1}{2}}{m}^{\frac{\alpha }{2n}}.$$

(A13)

Therefore, (A12) and (A13) yield

$$\begin{array}{cc}\hfill & \left|{\gamma }_m{E}_m\mathsf{\Delta }\left({\gamma }_{m}A{E}_m+(a_{E_m}-a_{A{E}_m})\right)\right|\hfill \\ \hfill & \le \left|{\gamma }_m{E}_m\mathsf{\Delta }(a_{E_m}+B_1)\right|+\left|(a_{E_m}+B_1)\mathsf{\Delta }\left({\gamma }_{m}A{E}_m+(a_{E_m}-a_{A{E}_m})\right)\right|\hfill \\ \hfill & =\left|{\gamma }_m{E}_m\mathsf{\Delta }(a_{E_m}+B_1)\right|+\left|(a_{A{E}_m}+B_1)\mathsf{\Delta }\left({\gamma }_{m}A{E}_m\right)\right|\hfill \\ \hfill & \le 2|B_1|{\left({\displaystyle \frac{1}{c(n,\alpha )}}{\displaystyle \frac{|B_1{|}^{\frac{n-\alpha }{n}}}{P_{\alpha }\left(B_1\right)}}(\underset{B_{R_m}}{sup}g)\right)}^{\frac{1}{2}}{m}^{\frac{\alpha }{2n}}.\hfill \end{array}$$

Hence,

$${\displaystyle \frac{\left|B_1\right|}{m}}|E_m\mathsf{\Delta }\left(A{E}_m+{\displaystyle \frac{(a_{E_m}-a_{A{E}_m})}{{\gamma }_m}}\right)|\le 2|B_1|{\left({\displaystyle \frac{1}{c(n,\alpha )}}{\displaystyle \frac{|B_1{|}^{\frac{n-\alpha }{n}}}{P_{\alpha }\left(B_1\right)}}(\underset{B_{R_m}}{sup}g)\right)}^{\frac{1}{2}}{m}^{\frac{\alpha }{2n}}.$$

(A14)

Set

$${\alpha }_m:={\displaystyle \frac{a_{A{E}_m}-a_{E_m}}{{\gamma }_m}},$$

thus, via (A14),

$$\begin{array}{cc}\hfill |A{E}_m\mathsf{\Delta }\left(E_m+{\alpha }_m\right)|& \le 2{\left({\displaystyle \frac{1}{c(n,\alpha )}}{\displaystyle \frac{|B_1{|}^{\frac{n-\alpha }{n}}}{P_{\alpha }\left(B_1\right)}}(\underset{B_{R_m}}{sup}g)\right)}^{\frac{1}{2}}{m}^{1+\frac{\alpha }{2n}}\hfill \\ \hfill & =2{\left({\displaystyle \frac{1}{c(n,\alpha )}}{\displaystyle \frac{|B_1{|}^{\frac{n-\alpha }{n}}}{P_{\alpha }\left(B_1\right)}}(\underset{B_{R_m}}{sup}g)\right)}^{\frac{1}{2}}{m}^{1+\frac{\alpha }{2n}}.\hfill \end{array}$$

(A15)

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