The nonlocal Almgren problem

Emanuel Indrei Department of Mathematics
Kennesaw State University
Marietta, GA 30060
USA.

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 some nonlocal perimeter and convex potential. In the paper, the main result is a quantitative stability theorem for the nonlocal free energy assuming symmetry on the potential. In addition, several results that involve uniqueness, non-existence, and moduli estimates from the theory for crystals are proven also in the nonlocal context.

1. Introduction

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

f(e)=f(e_{m})+f^{\prime}(e_{m})(e-e_{m})+\frac{f^{\prime\prime}(\alpha_{e,e_{m}})}{2}|e-e_{m}|^{2},

$\alpha_{e,e_{m}}\in(e_{m},e)$ . If $f^{\prime}(e_{m})=0$ , $f^{\prime\prime}(\alpha_{e,e_{m}})\geq a_{*}>0$ , then

(1)

f(e)-f(e_{m})\geq\frac{a_{*}}{2}|e-e_{m}|^{2}.

In particular, one may obtain sharp information from this: assume $f$ is a quantity which classifies an optimizer $e_{m}$ in the sense that $f(e)=f(e_{m})$ implies $e=e_{m}$ . Now, let $e$ satisfy $f(e)=f(e_{m})+\epsilon$ , with $\epsilon>0$ a small number; thus $f(e)\approx f(e_{m})$ and the expectation is that $e\approx e_{m}$ . The utility of (1) is to make this clear in the context that $e$ is at most, up to a constant, $\sqrt{\epsilon}$ from $e_{m}$ .

When instead of a real-valued function $f$ the object of investigation is an energy $\mathcal{E}$ which 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 $|E|=m$ . Thus at the minimum $E_{m}$ , the first variation is zero $\mathcal{E}^{\prime}(E_{m})=0$ , hence if some lower bound exists on the second variation, it is natural to anticipate that mod an invariance class

(2)

\mathcal{E}(E)-\mathcal{E}(E_{m})\geq a_{*}||\chi_{E}-\chi_{E_{m}}||^{2},

where $\chi_{E}$ is the characteristic function of $E$ . A natural norm is often chosen to be the $L^{1}$ norm [Fig13]. In applications, the free energy is of the form

\mathcal{E}(E)=\mathcal{E}_{s}(E)+\mathcal{E}_{a}(E),

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 $\partial^{*}E$ :

\mathcal{E}_{s}(E)=\mathcal{F}(E)=\int_{\partial^{*}E}f(\nu_{E})d\mathcal{H}^{n-1},

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

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

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

\mathcal{E}_{a}(E)=\mathcal{G}(E)=\int_{E}g(x)dx,

where $g\geq 0$ , $g(0)=0$ , $g\in L_{loc}^{\infty}$ [Ind24, IK23, IK24, FZ22, Tay78, TA87, Fon91, FM91, FMP10, DPV22, FM11]; see in addition many interesting references in [Ind24] that comprehensively discuss the history. In thermodynamics, to obtain a crystal, one minimizes the free energy

\mathcal{E}(E)=\mathcal{F}(E)+\mathcal{G}(E)

under a mass constraint. Gibbs and Curie independently discovered this physical principle [Gib78, Cur85].

1.2. The Binding Energy

The nonlocal Coulomb repulsion energy is given via

\mathcal{E}_{a}(E)=\mathcal{D}(E)=\alpha_{1}\int\int_{E\times E}\frac{1}{|z-y|^{\lambda}}dzdy,

$\lambda\in(0,n)$ , $\alpha_{1}>0$ . The binding energy of a set of finite perimeter $E\subset\mathbb{R}^{n}$ is the sum

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

In the classical context, $\lambda=1$ , $f(x)=|x|$ , $n=3$ , $\alpha_{1}=\frac{1}{2}$ [FN21]. The theory is historically attributed to Gamow via his 1930 paper [Gam30] and it successfully predicts the non-existence of nuclei with a large atomic number, cf. references in [Ind23, CR24, FFM⁺15].

1.3. The Nonlocal Free Energy

The nonlocal perimeter encodes a parameter $\alpha\in(0,1)$

\mathcal{E}_{s}(E)=P_{\alpha}(E)=\int_{E}\int_{E^{c}}\frac{1}{|x-y|^{\alpha+n}}dxdy

[FLS08, FFM⁺15, CRS10, BNO23]. Caffarelli, Roquejoffre, and Savin investigated the Plateau problem with respect to the nonlocal energy functionals [CRS10]. The nonlocal isoperimetric inequality has appeared in [FLS08]: assume $|E|=|B_{a}|$ , then

P_{\alpha}(E)\geq P_{\alpha}(B_{a})

with equality if and only if $E=B_{a}+x$ . Hence the nonlocal free energy is

\mathcal{E}(E)=P_{\alpha}(E)+\mathcal{G}(E),

[CN17, BNO23].

1.4. The Nonlocal Binding Energy

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

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

see [FFM⁺15] for a theorem on the minimizers when the mass is small.

1.5. The Main Problem

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

\mathcal{E}(E)=\mathcal{F}(E)+\mathcal{G}(E)

\mathcal{E}(E)=\mathcal{F}(E)+\mathcal{D}(E)

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

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

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

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

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

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

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

In this context, the nonlocal Almgren problem is to investigate the convexity of a minimizer $E_{m}$ under the assumption that $g$ is convex, refer to [FM11, p. 146] to understand the local Almgren problem. Several theorems may be shown without convexity on $g$ . In particular, a complete theory 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 $m>0$ [CN17].
2. Assuming that $m$ is sufficiently small, all minimizers are convex [BNO23].
3. One may construct a $g$ which is convex so that there are no minimizers if $m>0$ , see Theorem 3.6.
4. Assuming $g(x)=h(|x|)$ , $h:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is increasing, convex, $h(0)=0$ , there exists a stability estimate similar to (2), see Theorem 2.1.
5. Assuming $g\in L_{loc}^{\infty}$ and 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 3.1 and Theorem 3.7.
6. Upper bounds for the moduli are obtained with minimal assumptions, see Theorem 3.2

In the paper, the novelty mostly is in 4. Interestingly, 3, 5, 6 can be obtained, via minor changes, as in [Ind24, IK23] (in 6, one utilizes [BNO23]) and hence these proofs are in the appendix.

2. Stability for nonlocal free energy minimization

Theorem 2.1.

Suppose $g(x)=h(|x|)$ , $h:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is increasing, convex, $h(0)=0$ . Let $m>0$ , $|B_{a}|=|E|=m$ , then
(i)

\mathcal{E}(E)-\mathcal{E}(B_{a})\geq r(m,n,\alpha)||\chi_{E}-\chi_{B_{a}}||_{L^{1}}^{4}

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

(ii) supposing $\hat{a}>a$ , $E\subset B_{\hat{a}}$ , then

\mathcal{E}(E)-\mathcal{E}(B_{a})\geq r(m,\hat{a},n,\alpha)||\chi_{E}-\chi_{B_{a}}||_{L^{1}}^{2}

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

Remark 2.2.

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 [Hal92]. The conjecture was proven in [FMP08, FMP10].

Proof.

(i)
Assume

\mathcal{E}(E)-\mathcal{E}(B_{a})\geq\nu,

for $\nu>0$ . Then since

|E\Delta B_{a}|^{4}\leq 16m^{4},

	$\displaystyle\|E\Delta B_{a}\|^{4}$	$\displaystyle\leq 16m^{4}$
		$\displaystyle\leq\frac{16m^{4}}{\nu}\nu\leq\frac{16m^{4}}{\nu}(\mathcal{E}(E)-\mathcal{E}(B_{a})).$

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

\mathcal{E}(E)-\mathcal{E}(B_{a})\leq\nu,

then

(3)

w(|E\Delta B_{a}|)\leq\mathcal{E}(E)-\mathcal{E}(B_{a}),

where $w(t)=r(m,n,\alpha)t^{4}$ . To start, the existence of a modulus $w$ is proved so that (3) is true (observe this also follows from a compactness proof, cf. Proposition 3.1, 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 $w(t)=r(m,n,\alpha)t^{4}$ ). Suppose

T:E\setminus B_{a}\rightarrow B_{a}\setminus E

denotes the Brenier map between $\mu_{*}=\chi_{E\setminus B_{a}}dx$ and $\nu_{*}=\chi_{B_{a}\setminus E}dx$ [Bre91]. Since

T_{\#}\mu_{*}=\nu_{*},

one has

(4)

\int_{E\setminus B_{a}}h(|T(x)|)dx=\int_{B_{a}\setminus E}h(|x|)dx;

thus, (4) and monotonicity of $h$ yield

	$\displaystyle\int_{B_{a}}hdx$	$\displaystyle=\int_{E\cap B_{a}}hdx+\int_{B_{a}\setminus E}h(\|x\|)dx$
		$\displaystyle=\int_{E\cap B_{a}}hdx+\int_{E\setminus B_{a}}h(\|T(x)\|)dx$
		$\displaystyle\leq\int_{E\cap B_{a}}hdx+\int_{E\setminus B_{a}}h(\|x\|)dx$
(5)			$\displaystyle=\int_{E}hdx.$

Observe that via $|T(x)|\leq a$ and the above,

\int_{E\setminus B_{a}}[h(|x|)-h(a)]dx\leq\int_{E}hdx-\int_{B_{a}}hdx.

Hence the previous inequality and [FFM⁺15] imply

	$\displaystyle\mathcal{E}(E)-\mathcal{E}(B_{a})$	$\displaystyle=P_{\alpha}(E)-P_{\alpha}(B_{a})+\mathcal{G}(E)-\mathcal{G}(B_{a})$
		$\displaystyle\geq P_{\alpha}(B_{a})\frac{A^{2}(E)}{C(n,\alpha)}+\int_{E}hdx-\int_{B_{a}}hdx$
		$\displaystyle\geq P_{\alpha}(B_{a})\frac{A^{2}(E)}{C(n,\alpha)}+\int_{E\setminus B_{a}}[h(\|x\|)-h(a)]dx,$

A(E)=\inf\{\frac{|E\Delta(B_{a}+x)|}{|E|}:x\in\mathbb{R}^{n}\}.

Let $z_{E}$ achieve

A(E)=\frac{|E\Delta(B_{a}+z_{E})|}{|E|}.

Hence

(6)

\Big{(}\frac{|E\Delta(B_{a}+z_{E})|}{|E|}\Big{)}^{2}\leq\tilde{a}_{1}\Big{(}\mathcal{E}(E)-\mathcal{E}(B_{a})\Big{)},

$\tilde{a}_{1}=\tilde{a}_{1}(n,\alpha,a)>0.$ Assume

\mathcal{E}(E_{i})-\mathcal{E}(B_{a})\rightarrow 0,

$|E_{i}|=m=|B_{a}|$ . Observe

|E_{i}\Delta(B_{a}+z_{E_{i}})|\rightarrow 0.

Let $A(i)\subset B_{a}+z_{E_{i}}$ , $|A(i)|\geq p>0$ . Thus

\int|\chi_{E_{i}}(x)-\chi_{B_{a}+z_{E_{i}}}(x)|dx\rightarrow 0

yields

\int_{A(i)}|\chi_{E_{i}}(x)-1|dx\rightarrow 0.

By the triangle inequality,

	$\displaystyle\|\|A(i)\cap E_{i}\|-\|A(i)\|\|$	$\displaystyle=\|\int_{A(i)}(\chi_{E_{i}}(x)-1)dx\|$
		$\displaystyle\leq\int_{A(i)}\|\chi_{E_{i}}(x)-1\|dx\rightarrow 0.$

This thus implies

|A(i)\cap E_{i}|\geq\frac{p}{7}

assuming $i$ is large.

Claim 1: $\sup_{i}|z_{E_{i}}|<\infty$ .

Proof of Claim 1:
Assume not. Then modulo a subsequence,

|z_{E_{i}}|\rightarrow\infty.

Since $A(i)\subset B_{a}+z_{E_{i}}$ , one obtains

\inf_{A(i)\cap E_{i}}g\rightarrow\infty

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

	$\displaystyle\infty$	$\displaystyle=\lim_{i}\frac{p}{7}\inf_{A(i)\cap E_{i}}g$
		$\displaystyle\leq\int_{A(i)\cap E_{i}}g$
		$\displaystyle\leq\int_{E_{i}}g\rightarrow\int_{B}g<\infty,$

and this contradiction yields Claim 1.

Claim 2: $|z_{E_{i}}|\rightarrow 0$ .

Proof of Claim 2:
If one can find a subsequence (continued to have the same index) so that $\inf_{i}|z_{E_{i}}|>0$ , observe via the positive bound that up to possibly another subsequence,

z_{E_{i}}\rightarrow z\neq 0.

In particular,

	$\displaystyle\|E_{i}\Delta(B_{a}+z)\|$	$\displaystyle\leq\|(B_{a}+z_{E_{i}})\Delta(B_{a}+z)\|+\|E_{i}\Delta(B_{a}+z_{E_{i}})\|$
		$\displaystyle\leq a_{*}\|z_{E_{i}}-z\|+\|B_{a}\|\sqrt{\tilde{a}_{1}(\mathcal{E}(E_{i})-\mathcal{E}(B_{a}))}$
		$\displaystyle\rightarrow 0.$

Hence

\chi_{E_{i}}\rightarrow\chi_{B_{a}+z}\hskip 28.90755pt\text{in $L^{1}$}.

and this readily yields, mod a subsequence,

\chi_{E_{i}}\rightarrow\chi_{B_{a}+z}\hskip 28.90755pt\text{a.e.}.

Therefore via Fatou

\displaystyle\int_{B_{a}+z}g\leq\liminf_{i}\int_{\mathbb{R}^{n}}g\chi_{E_{i}}=\mathcal{G}(B_{a}),

Define $H=B_{a}+z$ . Consider $\{v_{1},v_{2},\ldots\}$ directions which generate via Steiner symmetrization with respect to the planes through the origin and with normal $v_{i}$ , $H_{v_{1}},H_{v_{2}},\ldots$ and

H_{v_{i}}\rightarrow B_{a}.

Set

H_{1}=\{(x_{2},\ldots,x_{n}):(x_{1},x_{2},\ldots,x_{n})\in H\}

H^{x_{2},x_{3},\ldots,x_{n}}=\{x_{1}:(x_{1},x_{2},\ldots,x_{n})\in H\}.

Note that one may let $\frac{x_{1}}{|x_{1}|}=v_{1}$ ; hence the monotonicity and complete symmetry of $g$ imply

	$\displaystyle\int_{H_{v_{1}}}gdx$	$\displaystyle=\int_{H_{1}}\Big{(}\int_{-\frac{\|H^{x_{2},x_{3},\ldots,x_{n}}\|}{2}}^{\frac{\|H^{x_{2},x_{3},\ldots,x_{n}}\|}{2}}gdx_{1}\Big{)}dx_{2}\ldots dx_{n}$
		$\displaystyle\leq\int_{H_{1}}\Big{(}\int_{H^{x_{2},x_{3},\ldots,x_{n}}}gdx_{1}\Big{)}dx_{2}\ldots dx_{n}$
		$\displaystyle=\int_{H}gdx.$

Moreover, via the radial property of $g$ , one may rotate the coordinate $x_{1}$ so that $\frac{x_{1}}{|x_{1}|}=v_{2}$ and iterate the argument above:

\int_{H_{v_{1}}}gdx\geq\int_{H_{v_{2}}}gdx;

in particular, note that via $H\neq B_{a}$ & 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

	$\displaystyle\int_{B_{a}}gdx$	$\displaystyle<\int_{H}g=\int_{B_{a}+z}g$
		$\displaystyle\leq\liminf_{i}\int_{\mathbb{R}^{n}}g\chi_{E_{i_{l}}}$
		$\displaystyle=\mathcal{G}(B_{a})=\int_{B_{a}}gdx.$

Note that this proves:

assuming

\mathcal{E}(E_{i})-\mathcal{E}(B_{a})\rightarrow 0,

it follows that

|E_{i}\Delta(B_{a}+z_{E_{i}})|\rightarrow 0,

(7)

z_{E_{i}}\rightarrow 0.

Now note

	$\displaystyle\|E_{i}\Delta B_{a}\|$	$\displaystyle\leq\|E_{i}\Delta(B_{a}+z_{E_{i}})\|+\|B_{a}\Delta(B_{a}+z_{E_{i}})\|$
		$\displaystyle\leq\|E_{i}\Delta(B_{a}+z_{E_{i}})\|+a_{*}\|z_{E_{i}}\|\rightarrow 0.$

Hence there is some modulus $w$ so that

\mathcal{E}(E)-\mathcal{E}(B_{a})\geq w(|E\Delta B_{a}|).

Supposing $\hat{a_{1}}>a_{1}>0$ , $E_{*}\subset B_{\hat{a_{1}}}$ , $|E_{*}|=|B_{a_{1}}|$ , then via strict monotonicity and convexity of $h$ , one obtains that the subdifferential

\partial_{+}h(a_{1})\neq\emptyset

is compact and

(8)

\inf_{\partial_{+}h(a_{1})}|x|>0:

assume $0\in\partial_{+}h(a_{1})$ , 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 ( $g(0)=0$ , $g\geq 0$ ), therefore this shows (8). Hence thanks to [IK23], there exists

(9)

r_{\hat{a_{1}},a_{1},\partial_{+}h(a_{1})}^{*}=\Big{(}\frac{1}{\inf_{\partial_{+}h(a_{1})}|x|A_{*}}\Big{)}^{\frac{1}{2}}>0,

$A_{*}=A_{*}(\hat{a_{1}},n,a)>0$ , so that

(10)

r_{\hat{a_{1}},a_{1},\partial_{+}h(a_{1})}^{*}\Big{[}\mathcal{G}(E_{*})-\mathcal{G}(B_{a_{1}})\Big{]}^{\frac{1}{2}}\geq|E_{*}\Delta B_{a_{1}}|.

Supposing $|E|=|B_{a}|$ , via the previous argument (cf. (7)) if

\mathcal{E}(E)-\mathcal{E}(B_{a})\rightarrow 0,

it follows that

z_{E}\rightarrow 0.

Hence if

\mathcal{E}(E)-\mathcal{E}(B_{a})\leq\nu

where $\nu$ small, then $|z_{E}|$ 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 (6)

(11)

|E\setminus B_{a+|z_{E}|}|\leq|E\setminus(B_{a}+z_{E})|\leq\sqrt{|B_{a}|^{2}\tilde{a}_{1}(\mathcal{E}(E)-\mathcal{E}(B_{a}))}.

Set

E_{*}=E\cap(B_{a+|z_{E}|});

then note

E=E_{*}\cup(E\setminus(B_{a+|z_{E}|}))

& thanks to (11)

	$\displaystyle m$	$\displaystyle=\|E_{*}\|+\|E\setminus(B_{a+\|z_{E}\|})\|$
		$\displaystyle\leq\|E_{*}\|+\sqrt{\|B_{a}\|^{2}\tilde{a}_{1}(\mathcal{E}(E)-\mathcal{E}(B_{a}))}.$

Therefore

(12)

m-|E_{*}|\leq\sqrt{|B_{a}|^{2}\tilde{a}_{1}(\mathcal{E}(E)-\mathcal{E}(B_{a}))};

next, consider $a_{1}>0$ via

|E_{*}|=|B_{a_{1}}|=a_{1}^{n}|B_{1}|.

Observe that (12) easily implies

m-\sqrt{|B_{a}|^{2}\tilde{a}_{1}(\mathcal{E}(E)-\mathcal{E}(B_{a}))}\leq|E_{*}|=a_{1}^{n}|B_{1}|,

hence

\Big{(}\frac{m-\sqrt{|B_{a}|^{2}\tilde{a}_{1}(\mathcal{E}(E)-\mathcal{E}(B_{a}))}}{|B_{1}|}\Big{)}^{\frac{1}{n}}\leq a_{1}.

In particular, since the energy difference is small $\mathcal{E}(E)-\mathcal{E}(B_{a})\leq\nu$ , there exists a lower bound on $a_{1}$ via $a,n,\alpha$ . Thanks to $|z_{E}|$ being small and (10), one may choose

5a>\hat{a_{1}}>a+|z_{E}|

such that

(13)

r_{\hat{a_{1}},a_{1},\partial_{+}h(a_{1})}^{*}\Big{[}\mathcal{G}(E_{*})-\mathcal{G}(B_{a_{1}})\Big{]}^{\frac{1}{2}}\geq|E_{*}\Delta B_{a_{1}}|,

where $r_{\hat{a_{1}},a_{1},\partial_{+}h(a_{1})}^{*}$ is bounded via $a,n,\alpha$ . Now since $a\geq a_{1}$ , (12) implies

	$\displaystyle\mathcal{G}(E_{*})-\mathcal{G}(B_{a_{1}})$	$\displaystyle=\mathcal{G}(E_{*})-\mathcal{G}(B_{a})+\mathcal{G}(B_{a})-\mathcal{G}(B_{a_{1}})$
		$\displaystyle\leq\mathcal{G}(E_{*})-\mathcal{G}(B_{a})+\big{(}\sup_{B_{a}}g\big{)}\|B_{a}\setminus B_{a_{1}}\|$
		$\displaystyle\leq\mathcal{G}(E_{*})-\mathcal{G}(B_{a})+\big{(}\sup_{B_{a}}g\big{)}(\|B_{a}\|-\|B_{a_{1}}\|)$
		$\displaystyle=\mathcal{G}(E_{})-\mathcal{G}(B_{a})+\big{(}\sup_{B_{a}}g\big{)}(m-\|E_{}\|)$
		$\displaystyle\leq\mathcal{E}(E)-\mathcal{E}(B_{a})+\big{(}\sup_{B_{a}}g\big{)}\sqrt{\|B_{a}\|^{2}\tilde{a}_{1}(\mathcal{E}(E)-\mathcal{E}(B_{a}))}.$

In particular, (13) and the above inequality imply

	$\displaystyle\|E_{*}\Delta B_{a_{1}}\|$	$\displaystyle\leq r_{\hat{a_{1}},a_{1},\partial_{+}h(a_{1})}^{}\Big{[}\mathcal{G}(E_{})-\mathcal{G}(B_{a_{1}})\Big{]}^{\frac{1}{2}}$
(14)			$\displaystyle\leq r_{\hat{a_{1}},a_{1},\partial_{+}h(a_{1})}^{*}\Big{[}\mathcal{E}(E)-\mathcal{E}(B_{a})+\sup_{B_{a}}g\sqrt{\|B_{a}\|^{2}\tilde{a}_{1}(\mathcal{E}(E)-\mathcal{E}(B_{a}))}\Big{]}^{\frac{1}{2}}.$

Also, (12) & (11) yield

(15)

|B_{a}\Delta B_{a_{1}}|=|B_{a}\setminus B_{a_{1}}|\leq\sqrt{|B_{a}|^{2}\tilde{a}_{1}(\mathcal{E}(E)-\mathcal{E}(B_{a}))}

(16)

|E_{*}\Delta E|=|E\setminus(B_{a+|z_{E}|})|\leq\sqrt{|B_{a}|^{2}\tilde{a}_{1}(\mathcal{E}(E)-\mathcal{E}(B_{a}))}.

Hence (15), (2), (16), (11), & the triangle inequality in $L^{1}$ imply

	$\displaystyle\|B_{a}\Delta(B_{a}+z_{E})\|$	$\displaystyle\leq\|B_{a}\Delta B_{a_{1}}\|+\|B_{a_{1}}\Delta E_{}\|+\|E\Delta E_{}\|+\|E\Delta(B_{a}+z_{E})\|$
		$\displaystyle\leq\alpha_{*}\Big{[}\mathcal{E}(E)-\mathcal{E}(B_{a})\Big{]}^{\frac{1}{4}}.$

Last,

	$\displaystyle\|E\Delta B_{a}\|$	$\displaystyle\leq\|E\Delta(B_{a}+z_{E})\|+\|(B_{a}+z_{E})\Delta B_{a}\|$
		$\displaystyle\leq\sqrt{\|B_{a}\|^{2}\tilde{a}_{1}(\mathcal{E}(E)-\mathcal{E}(B_{a}))}+\alpha_{*}\Big{[}\mathcal{E}(E)-\mathcal{E}(B_{a})\Big{]}^{\frac{1}{4}}$
		$\displaystyle\leq\overline{\alpha}_{*}\Big{[}\mathcal{E}(E)-\mathcal{E}(B_{a})\Big{]}^{\frac{1}{4}},$

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

(ii)
Supposing $\hat{a}>a$ , $E\subset B_{\hat{a}}$ , then one can show similarly to the proof in (i) (cf. (10), (9)) that there exists some constant $r_{m,\hat{a},n,\alpha}^{*}>0$ (that is explicit) so that

r_{m,\hat{a},n,\alpha}^{*}\Big{[}\mathcal{E}(E)-\mathcal{E}(B_{a})\Big{]}^{\frac{1}{2}}\geq r_{m,\hat{a},n,\alpha}^{*}\Big{[}\mathcal{G}(E)-\mathcal{G}(B_{a})\Big{]}^{\frac{1}{2}}\geq|E\Delta B_{a}|.

∎

Remark 2.3.

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

\displaystyle\mathcal{E}(B_{a}+x)-\mathcal{E}(B_{a})

\displaystyle=\int_{B_{a}}(2\langle y,x\rangle+|x|^{2})dy=|x|^{2}|B_{a}|;

supposing $|x|$ is small, since

|(B_{a}+x)\Delta B_{a}|\approx|x|,

|(B_{a}+x)\Delta B_{a}|^{2}\approx|x|^{2}.

Remark 2.4.

In the argument of the theorem, (2) implies that balls minimize the energy also when $g$ is non-decreasing, radial, and possibly non-convex.

Remark 2.5.

In the theorem, a quantitative inequality is proven without translation invariance. In particular, 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 [FI13, Ind21, Pog24, DPV22], translations are crucial: assuming the cone contains no line, the quantitative term is without translations.

Corollary 2.6.

Suppose $g(x)=h(|x|)$ , where $h$ is non-negative, non-decreasing, not identically zero, and homogeneous of degree $\nu$ . Let $m>0$ and assume $E_{m}$ is the minimizer with $|E_{m}|=m$ , set

m_{\alpha,\nu,g}=|B_{1}|\Big{[}\frac{\alpha(n-\alpha)}{\nu(n+\nu)}\frac{P_{\alpha}(B_{1})}{\int_{B_{1}}g(x)dx}\Big{]}^{\frac{n}{\nu+\alpha}}

it then follows that

m\mapsto\mathcal{E}(E_{m})

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

Proof.

Thanks to Remark 2.4, $E_{m}=B_{r}$ , $r=\Big{[}\frac{m}{|B_{1}|}\Big{]}^{\frac{1}{n}}$ , which therefore implies

\mathcal{E}(E_{m})=P_{\alpha}(B_{1})(\frac{1}{|B_{1}|^{\frac{n-\alpha}{n}}})m^{\frac{n-\alpha}{n}}+(\frac{1}{|B_{1}|^{\frac{\nu+n}{n}}})(\int_{B_{1}}h(|x|)dx)m^{\frac{n+\nu}{n}}.

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

3. Appendix

3.1. Compactness: existence of moduli

Proposition 3.1.

If $m>0$ , $g\in L_{loc}^{\infty}$ , and up to sets of measure zero, $g$ admits unique minimizers $E_{m}$ , then for $\epsilon>0$ there exists $w_{m}(\epsilon)>0$ such that if $|E|=|E_{m}|$ , $E\subset B_{R}$ , $R=R(m)$ , and

|\mathcal{E}(E)-\mathcal{E}(E_{m})|<w_{m}(\epsilon),

then

\frac{||\chi_{E}-\chi_{E_{m}}||_{L^{1}}}{|E_{m}|}<\epsilon.

Proof.

Assume the assumptions do not yield the conclusion, then there exists $\epsilon>0$ and for $a>0$ , there exist $E_{a}^{\prime}\subset B_{R}$ and $E_{m}$ , $|E_{a}^{\prime}|=|E_{m}|=m$ ,

|\mathcal{E}(E_{a}^{\prime})-\mathcal{E}(E_{m})|<a,

\frac{|E_{m}\Delta E_{a}^{\prime}|}{|E_{m}|}\geq\epsilon.

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}(E_{m})-\mathcal{E}(E_{\frac{1}{k}}^{\prime})|\leq\frac{1}{k},

\frac{|E_{m}\Delta E_{\frac{1}{k}}^{\prime}|}{|E_{m}|}\geq\epsilon.

In particular

	$\displaystyle P_{\alpha}(E_{\frac{1}{k}}^{\prime})$	$\displaystyle\leq\mathcal{E}(E_{\frac{1}{k}}^{\prime})$
		$\displaystyle\leq\frac{1}{k}+\mathcal{E}(E_{m}),$

E_{\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}\rightarrow E^{\prime}\hskip 21.68121ptin\hskip 5.78172ptL^{1}(B_{R});

thus $|E^{\prime}|=m$ because of the triangle inequality in $L^{1}(B_{R})$ . Next

\mathcal{E}(E^{\prime})\leq\liminf_{k}\mathcal{E}(E_{\frac{1}{k}}^{\prime})=\mathcal{E}(E_{m})

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

\frac{|E_{m}\Delta E^{\prime}|}{|E_{m}|}\geq\epsilon

via the uniqueness. ∎

3.2. Bounds on the moduli

Identifying the modulus $w_{m}(\epsilon)$ 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 3.2.

Suppose $m>0$ , and up to sets of measure zero, $g$ admits unique minimizers $E_{m}\subset B_{r(m)}$ in the collection of sets of finite perimeter, then
(i) if $g\in W_{loc}^{1,1}$ is locally uniformly differentiable, there exists $\alpha_{1}=\alpha_{1}(E_{m})>0$ such that if $\epsilon<\alpha_{1}$ , one has

w_{m}(\epsilon)\leq\lambda_{g,E_{m}}(m,\epsilon)=o_{g,E_{m}}(\frac{\epsilon m}{a_{m_{*}}}),

a_{m_{*}}=\alpha_{*}|D\boldsymbol{\chi}_{E_{m}}\cdot w_{*}|(\mathbb{R}^{n})

with $\alpha_{*}>0$ , $w_{*}\in\mathbb{R}^{n}$ ;
(ii) if $g\in W_{loc}^{1,\infty}$ , there exists $m_{a}>0$ such that if $m<m_{a}$ , $\epsilon<\alpha_{1}$ , then

\lambda_{g,E_{m}}(m,\epsilon)=\alpha_{1_{*}}m^{\frac{1}{n}}o_{g,E_{m}}(\epsilon),

o_{g,E_{m}}(\epsilon)=\epsilon\int_{E_{m}}o_{x,g}(1)dx,

\int_{E_{m}}o_{x,g}(1)dx\rightarrow 0

as $\epsilon\rightarrow 0^{+}$ , $\alpha_{1_{*}}>0$ ;
(iii) if $g\in C^{1}$ , there exist $m_{a},\alpha_{m}>0$ such that if $m<m_{a}$ , $\epsilon<\alpha_{m}$ , then

\lambda_{g,E_{m}}(m,\epsilon)=\alpha_{2_{*}}m^{1+\frac{1}{n}}o_{g}(\epsilon),

$\alpha_{2_{*}}=\alpha_{2_{*}}(||Dg||_{L^{\infty}(B_{r(m_{a})})})>0$ ;
(iv) if $g\in W_{loc}^{2,1}$ is twice differentiable, there exists $\alpha_{m}>0$ such that if $\epsilon<\alpha_{m}$ ,

w_{m}(\epsilon)\leq a_{m}\epsilon^{2},

a_{m}=\Big{(}||D^{2}g||_{L^{1}(E_{m})}+m\frac{1}{6}\Big{)}(\frac{m}{\alpha_{*}|D\boldsymbol{\chi}_{E_{m}}\cdot w_{*}|(\mathbb{R}^{n})})^{2}

with $\alpha_{*}>0$ ;
(v) if $g\in W_{loc}^{2,\infty}$ , there exists $m_{a}>0$ such that if $m<m_{a}$ , then

w_{m}(\epsilon)\leq a_{m}\epsilon^{2},

a_{m}=a_{*}m^{1+\frac{2}{n}},

$a_{*}=a_{*}(||D^{2}g||_{L^{\infty}(B_{r(m_{a})})})>0$ .

Proof.

Assuming $m$ is small, the convexity of $E_{m}$ is established in [BNO23] and utilized for $(ii),(iii),(v)$ similar to the proof in [IK23]. The argument for $(i)$ & $(iv)$ is the same as in [IK23]. ∎

Remark 3.3.

The existence of bounded $E_{m}$ was proven in [CN17] 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, 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 3.4.

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

Remark 3.5.

Theorem 2.1 encodes the modulus in an explicit way. In particular, supposing $m$ is small, if $g(|x|)=|x|^{2}$ , a possible modulus is $w_{m}(\epsilon)=a_{1}\epsilon^{2}m^{3}$ . Thus the $\epsilon$ 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 $a_{*}(||D^{2}g||_{L^{\infty}(B_{r(m_{a})})})$ .

3.3. Non-existence of minimizers

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

Theorem 3.6.

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

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

Proof.

The definition of $g$ via the construction in [Ind24] also works in this case:

g(x,y)=\begin{cases}x^{2}(1-y)+x^{2}y^{2}&\text{if }y\leq 0\\ \frac{x^{2}}{1+y}&\text{if }y>0.\end{cases}

Set $e_{2}=(0,1)$ , $a>0$ ; the potential is non-increasing in the $y$ -variable and strictly decreasing if $x\neq 0.$ in particular, note if a minimizer $E_{m}$ exists,

\int_{E_{m}+ae_{2}}g(x,y)dxdy<\int_{E_{m}}g(x,y)dxdy,

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

\mathcal{E}(E_{m}+ae_{2})<\mathcal{E}(E_{m}),

a contradiction.

∎

In particular, coercivity or another condition is necessary.

3.4. A product property

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

	$\displaystyle A\in\mathcal{A}_{m}=\mathcal{A}_{\alpha,g,m}$
	$\displaystyle=\{A:Ax=A_{a}x+x_{a},x_{a}\in\mathbb{R}^{n},\mathcal{E}(A_{a}E)=\mathcal{E}(E),\|A_{a}E\|=\|E\|=m\text{ for some minimizer $E$}\}.$

Uniqueness of minimizers can only be true up to $P_{\alpha}$ 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: 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 3.7.

Suppose $g\in L_{loc}^{\infty}(\{g<\infty\})$ admits minimizers $E_{m}\subset B_{R}$ for all $m$ small. There exists $m_{0}>0$ and a modulus of continuity $q(0^{+})=0$ such that for all $m<m_{0}$ there exists $\epsilon_{0}>0$ such that for all $0<\epsilon<\epsilon_{0}$ and for all minimizers $E_{m}\subset B_{R}$ , $E\subset B_{R}$ , $|E|=|E_{m}|=m<m_{0}$ , if

|\mathcal{E}(E_{m})-\mathcal{E}(E)|<a(m,\epsilon,\alpha)=q(\epsilon)m^{\frac{n-\alpha}{n}},

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

\frac{||\chi_{E}-\chi_{AE_{m}}||_{L^{1}}}{|E_{m}|}<\epsilon.

Also, $AE_{m}\approx E_{m}+\alpha_{m}$ , $\alpha_{m}\in\mathbb{R}^{n}$ cf. (3.4).

Remark 3.8.

If $g(x)=h(|x|)$ , $h:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is increasing, $h(0)=0$ , then $A$ is the identity in the conclusion of Theorem 3.7: if $m>0$ , then

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

Proof.

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 $\epsilon_{0}\in(0,2]$ there exists $\epsilon<\epsilon_{0}$ and $E_{m,\epsilon_{0}},E_{m,\epsilon_{0}}^{\prime}\subset B_{R}$ , $|E_{m,\epsilon_{0}}|=|E_{m,\epsilon_{0}}^{\prime}|=m$ ,

|\mathcal{E}(E_{m})-\mathcal{E}(E_{m}^{\prime})|<q(\epsilon)m^{\frac{n-\alpha}{n}},

and

(17)

\inf_{A\in\mathcal{A}_{m}}\frac{|E_{m,\epsilon_{0}}^{\prime}\Delta AE_{m,\epsilon_{0}}|}{|E_{m,\epsilon_{0}}|}\geq\epsilon>0.

In particular, let $m_{0}=\frac{1}{k}$ , $w_{k}\rightarrow 0^{+}$ , $\hat{q}$ define a modulus of continuity, and let

(18)

q_{k}=w_{k}\hat{q}(\epsilon).

Therefore there exists $m_{k}<\frac{1}{k}$ such that for a fixed $\epsilon_{0}\in(0,2]$ there exists $\epsilon<\epsilon_{0}$ and $E_{m_{k},\epsilon_{0}},E_{m_{k},\epsilon_{0}}^{\prime}\subset B_{R}$ , $|E_{m_{k},\epsilon_{0}}|=|E_{m_{k},\epsilon_{0}}^{\prime}|=m_{k}<\frac{1}{k}$ ,

|\mathcal{E}(E_{m_{k}})-\mathcal{E}(E_{m_{k}}^{\prime})|<q_{k}m_{k}^{\frac{n-\alpha}{n}},

and

(19)

\inf_{A\in\mathcal{A}_{m_{k}}}\frac{|E_{m_{k},\epsilon_{0}}^{\prime}\Delta AE_{m_{k},\epsilon_{0}}|}{|E_{m_{k},\epsilon_{0}}|}\geq\epsilon>0.

Let

(20)

a_{k}=q_{k}m_{k}^{\frac{n-\alpha}{n}},

$E_{m_{k}}=E_{m_{k},\epsilon_{0}}$ , $E_{m_{k}}^{\prime}=E_{m_{k},\epsilon_{0}}^{\prime}$ . Define $\gamma_{k}=(\frac{|B_{1}|}{m_{k}})^{\frac{1}{n}}$ ,

|\gamma_{k}E_{m_{k}}|=|B_{1}|.

Observe via the nonlocal isoperimetric inequality and the minimality,

	$\displaystyle P_{\alpha}(\frac{1}{\gamma_{k}}B_{1})+\int_{E_{m_{k}}}g$	$\displaystyle\leq P_{\alpha}(E_{m_{k}})+\int_{E_{m_{k}}}g$
		$\displaystyle\leq P_{\alpha}(\frac{1}{\gamma_{k}}B_{1})+\int_{\frac{1}{\gamma_{k}}B_{1}}g$

P_{\alpha}(E_{m_{k}})-P_{\alpha}(\frac{1}{\gamma_{k}}B_{1})\leq(\sup_{\frac{1}{\gamma_{k}}B_{1}}g)m_{k}

\gamma_{k}^{n-\alpha}\Big{(}P_{\alpha}(E_{m_{k}})-P_{\alpha}(\frac{1}{\gamma_{k}}B_{1})\Big{)}\leq\gamma_{k}^{n-\alpha}\Big{(}(\sup_{\frac{1}{\gamma_{k}}B_{1}}g)m_{k}\Big{)}

\Big{(}P_{\alpha}(\gamma_{k}E_{m_{k}})-P_{\alpha}(B_{1})\Big{)}\leq\Big{(}(\sup_{\frac{1}{\gamma_{k}}B_{1}}g)(\gamma_{k}^{n-\alpha}m_{k})\Big{)}

\Big{(}P_{\alpha}(\gamma_{k}E_{m_{k}})-P_{\alpha}(B_{1})\Big{)}\leq\Big{(}(\sup_{(\frac{1}{\gamma_{k}})B_{1}}g)(\frac{|B_{1}|}{\gamma_{k}^{\alpha}})\Big{)}.

Since $m_{k}\rightarrow 0$ as $k\rightarrow\infty$ , it thus follows that $\gamma_{k}\rightarrow\infty$ , hence

\displaystyle\delta(\gamma_{k}E_{m_{k}})

\displaystyle=\frac{P_{\alpha}(\gamma_{k}E_{m_{k}})}{P_{\alpha}(B_{1})}-1\rightarrow 0.

Next, by the triangle inequality,

	$\displaystyle\|P_{\alpha}(E_{m_{k}}^{\prime})$	$\displaystyle-P_{\alpha}(E_{m_{k}})\|$
		$\displaystyle=\|[\mathcal{E}(E_{m_{k}}^{\prime})-\mathcal{E}(E_{m_{k}})]+[\int_{E_{m_{k}}}g(x)dx-\int_{E_{m_{k}}^{\prime}}g(x)dx]\|$
		$\displaystyle\leq\|\mathcal{E}(E_{m_{k}}^{\prime})-\mathcal{E}(E_{m_{k}})\|+\int g(x)\|\chi_{E_{m_{k}}^{\prime}}-\chi_{E_{m_{k}}}\|dx$
		$\displaystyle<a_{k}+\int_{E_{m_{k}}^{\prime}\Delta E_{m_{k}}}g(x)dx.$

In particular,

	$\displaystyle\|P_{\alpha}(\gamma_{k}E_{m_{k}}^{\prime})$	$\displaystyle-P_{\alpha}(\gamma_{k}E_{m_{k}})\|$
		$\displaystyle<\gamma_{k}^{n-\alpha}a_{k}+2\gamma_{k}^{n-\alpha}m_{k}(\sup_{B_{R}\cap\{g<\infty\}}g)$
		$\displaystyle=\|B_{1}\|^{\frac{n-\alpha}{n}}\frac{a_{k}}{m_{k}^{\frac{n-\alpha}{n}}}+2\|B_{1}\|^{\frac{n-\alpha}{n}}(\sup_{B_{R}\cap\{g<\infty\}}g)m_{k}^{1-\frac{n-\alpha}{n}}$

and since from (18) and (20),

a_{k}=q_{k}m_{k}^{\frac{n-\alpha}{n}}=w_{k}\hat{q}(\epsilon)m_{k}^{\frac{n-\alpha}{n}},

one obtains

\frac{a_{k}}{m_{k}^{\frac{n-\alpha}{n}}}=w_{k}\hat{q}(\epsilon)\rightarrow 0

|P_{\alpha}(\gamma_{k}E_{m_{k}}^{\prime})-P_{\alpha}(\gamma_{k}E_{m_{k}})|\rightarrow 0

as $k\rightarrow\infty$ . Hence

(21)	$\displaystyle\delta(\gamma_{k}E_{m_{k}}^{\prime})$	$\displaystyle\leq\|\delta(\gamma_{k}E_{m_{k}}^{\prime})-\delta(\gamma_{k}E_{m_{k}})\|+\delta(\gamma_{k}E_{m_{k}})$
(22)		$\displaystyle=\frac{1}{P_{\alpha}(B_{1})}\|P_{\alpha}(\gamma_{k}E_{m_{k}}^{\prime})-P_{\alpha}(\gamma_{k}E_{m_{k}})\|+\delta(\gamma_{k}E_{m_{k}})$
(23)		$\displaystyle\hskip 10.84006pt\rightarrow 0$

as $k\rightarrow\infty$ . Next, by the sharp stability of the nonlocal isoperimetric inequality [FFM⁺15] or a compactness argument, there exist $x_{k},x_{k}^{\prime}\in\mathbb{R}^{n}$ such that

(24)

\frac{|(\gamma_{k}E_{m_{k}}+x_{k})\Delta B_{1}|}{|\gamma_{k}E_{m_{k}}|}\rightarrow 0,

(25)

\frac{|(\gamma_{k}E_{m_{k}}^{\prime}+x_{k}^{\prime})\Delta B_{1}|}{|\gamma_{k}E_{m_{k}}^{\prime}|}\rightarrow 0

as $k\rightarrow\infty$ .

Therefore (24) and (25) yield $k\in\mathbb{N}$ such that

	$\displaystyle\frac{\|(E_{m_{k}}^{\prime}+\frac{(x_{k}^{\prime}-x_{k})}{\gamma_{k}})\Delta E_{m_{k}}\|}{\|E_{m_{k}}\|}$	$\displaystyle=\frac{\|(\gamma_{k}E_{m_{k}}^{\prime}+x_{k}^{\prime})\Delta(\gamma_{k}E_{m_{k}}+x)\|}{\|\gamma_{k}E_{m_{k}}\|}$
		$\displaystyle\leq\frac{\|(\gamma_{k}E_{m_{k}}+x)\Delta B_{1}\|}{\|\gamma_{k}E_{m_{k}}\|}+\frac{\|B_{1}\Delta(\gamma_{k}E_{m_{k}}^{\prime}+x_{k}^{\prime})\|}{\|\gamma_{k}E_{m_{k}}\|}$
		$\displaystyle<\epsilon,$

a contradiction to

	$\displaystyle\frac{\Big{\|}\Big{(}E_{m_{k}}^{\prime}+\frac{(x_{k}^{\prime}-x_{k})}{\gamma_{k}}\Big{)}\Delta E_{m_{k}}\Big{\|}}{\|E_{m_{k}}\|}$	$\displaystyle=\frac{\Big{\|}E_{m_{k}}^{\prime}\Delta\Big{(}E_{m_{k}}-\frac{(x_{k}^{\prime}-x_{k})}{\gamma_{k}}\Big{)}\Big{\|}}{\|E_{m_{k}}\|}$
		$\displaystyle\geq\inf_{A\in\mathcal{A}_{m_{k}}}\frac{\|E_{m_{k}}^{\prime}\Delta AE_{m_{k}}\|}{\|E_{m_{k}}\|}$
		$\displaystyle\geq\epsilon>0,$

thanks to (19). To finish, assume $E_{m}$ is a minimizer, $m<m_{0}$ , and define $\gamma_{m}=(\frac{|B_{1}|}{m})^{\frac{1}{n}}$ ,

|\gamma_{m}E_{m}|=|B_{1}|.

Since

P_{\alpha}(\frac{1}{\gamma_{m}}B_{1})\leq P_{\alpha}(E_{m}),

	$\displaystyle P_{\alpha}(\frac{1}{\gamma_{m}}B_{1})+\int_{E_{m}}g(x)dx$	$\displaystyle\leq P_{\alpha}(E_{m})+\int_{E_{m}}g(x)dx$
		$\displaystyle\leq P_{\alpha}(\frac{1}{\gamma_{m}}B_{1})+\int_{\frac{1}{\gamma_{m}}B_{1}}g(x)dx,$

thus via subtracting

P_{\alpha}(\frac{1}{\gamma_{m}}B_{1})+\int_{E_{m}}g(x)dx,

	$\displaystyle P_{\alpha}(E_{m})-P_{\alpha}(\frac{1}{\gamma_{m}}B_{1})$	$\displaystyle\leq\int_{\frac{1}{\gamma_{m}}B_{1}}g(x)dx-\int_{E_{m}}g(x)dx$
(26)			$\displaystyle\leq(\sup_{B_{R_{m}}}g)m,$

\frac{1}{\gamma_{m}}B_{1}\subset B_{R_{m}}.

Next (26) implies

	$\displaystyle P_{\alpha}(\gamma_{m}E_{m})-P_{\alpha}(B_{1})$	$\displaystyle=(\gamma_{m})^{n-\alpha}(P_{\alpha}(E_{m})-P_{\alpha}(\frac{1}{\gamma_{m}}B_{1}))$
(27)			$\displaystyle\leq(\gamma_{m})^{n-\alpha}(\sup_{B_{R_{m}}}g)m=\|B_{1}\|^{\frac{n-\alpha}{n}}(\sup_{B_{R_{m}}}g)m^{\frac{\alpha}{n}}.$

Therefore, since

\delta(\gamma_{m}E_{m})\geq c(n,\alpha)\Big{(}\frac{|\gamma_{m}E_{m}\Delta(a_{m}+B_{1})|}{|B_{1}|}\Big{)}^{2},

thanks to (27) and the sharp quantitative nonlocal isoperimetric inequality one obtains

	$\displaystyle c(n,\alpha)\Big{(}\frac{\|\gamma_{m}E_{m}\Delta(a_{m}+B_{1})\|}{\|B_{1}\|}\Big{)}^{2}$	$\displaystyle\leq\delta(\gamma_{m}E_{m})$
		$\displaystyle=\frac{P_{\alpha}(\gamma_{m}E_{m})-P_{\alpha}(B_{1})}{P_{\alpha}(B_{1})}$
		$\displaystyle\leq\frac{\|B_{1}\|^{\frac{n-\alpha}{n}}}{P_{\alpha}(B_{1})}(\sup_{B_{R_{m}}}g)m^{\frac{\alpha}{n}}.$

Now

(28)

|\gamma_{m}E_{m}\Delta(a_{E_{m}}+B_{1})|\leq|B_{1}|\Big{(}\frac{1}{c(n,\alpha)}\frac{|B_{1}|^{\frac{n-\alpha}{n}}}{P_{\alpha}(B_{1})}(\sup_{B_{R_{m}}}g)\Big{)}^{\frac{1}{2}}m^{\frac{\alpha}{2n}}.

Observe this is valid for any minimizer $E_{m}$ , therefore for any minimizer $AE_{m}$ with $A\in\mathcal{A}_{m}$ :

(29)

|\gamma_{m}AE_{m}\Delta(a_{AE_{m}}+B_{1})|\leq|B_{1}|\Big{(}\frac{1}{c(n,\alpha)}\frac{|B_{1}|^{\frac{n-\alpha}{n}}}{P_{\alpha}(B_{1})}(\sup_{B_{R_{m}}}g)\Big{)}^{\frac{1}{2}}m^{\frac{\alpha}{2n}}.

Therefore (28), (29) yield

	$\displaystyle\|\gamma_{m}E_{m}\Delta\Big{(}\gamma_{m}AE_{m}+(a_{E_{m}}-a_{AE_{m}})\Big{)}\|$
	$\displaystyle\leq\|\gamma_{m}E_{m}\Delta(a_{E_{m}}+B_{1})\|+\|(a_{E_{m}}+B_{1})\Delta\Big{(}\gamma_{m}AE_{m}+(a_{E_{m}}-a_{AE_{m}})\Big{)}\|$
	$\displaystyle=\|\gamma_{m}E_{m}\Delta(a_{E_{m}}+B_{1})\|+\|(a_{AE_{m}}+B_{1})\Delta\Big{(}\gamma_{m}AE_{m}\Big{)}\|$
	$\displaystyle\leq 2\|B_{1}\|\Big{(}\frac{1}{c(n,\alpha)}\frac{\|B_{1}\|^{\frac{n-\alpha}{n}}}{P_{\alpha}(B_{1})}(\sup_{B_{R_{m}}}g)\Big{)}^{\frac{1}{2}}m^{\frac{\alpha}{2n}}.$

Hence

(30)

\frac{|B_{1}|}{m}|E_{m}\Delta\Big{(}AE_{m}+\frac{(a_{E_{m}}-a_{AE_{m}})}{\gamma_{m}}\Big{)}|\leq 2|B_{1}|\Big{(}\frac{1}{c(n,\alpha)}\frac{|B_{1}|^{\frac{n-\alpha}{n}}}{P_{\alpha}(B_{1})}(\sup_{B_{R_{m}}}g)\Big{)}^{\frac{1}{2}}m^{\frac{\alpha}{2n}}.

Set

\alpha_{m}:=\frac{a_{AE_{m}}-a_{E_{m}}}{\gamma_{m}},

thus via (30),

	$\displaystyle\|AE_{m}\Delta\Big{(}E_{m}+\alpha_{m}\Big{)}\|$	$\displaystyle\leq 2\Big{(}\frac{1}{c(n,\alpha)}\frac{\|B_{1}\|^{\frac{n-\alpha}{n}}}{P_{\alpha}(B_{1})}(\sup_{B_{R_{m}}}g)\Big{)}^{\frac{1}{2}}m^{1+\frac{\alpha}{2n}}$
(31)			$\displaystyle=2\Big{(}\frac{1}{c(n,\alpha)}\frac{\|B_{1}\|^{\frac{n-\alpha}{n}}}{P_{\alpha}(B_{1})}(\sup_{B_{R_{m}}}g)\Big{)}^{\frac{1}{2}}m^{1+\frac{\alpha}{2n}}.$

∎

Remark 3.9.

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

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	$\displaystyle\|E_{i}\Delta(B_{a}+z)\|$	$\displaystyle\leq\|(B_{a}+z_{E_{i}})\Delta(B_{a}+z)\|+\|E_{i}\Delta(B_{a}+z_{E_{i}})\|$
		$\displaystyle\leq a_{*}\|z_{E_{i}}-z\|+\|B_{a}\|\sqrt{\tilde{a}_{1}(\mathcal{E}(E_{i})-\mathcal{E}(B_{a}))}$
		$\displaystyle\rightarrow 0.$

	$\displaystyle\mathcal{G}(E_{*})-\mathcal{G}(B_{a_{1}})$	$\displaystyle=\mathcal{G}(E_{*})-\mathcal{G}(B_{a})+\mathcal{G}(B_{a})-\mathcal{G}(B_{a_{1}})$
		$\displaystyle\leq\mathcal{G}(E_{*})-\mathcal{G}(B_{a})+\big{(}\sup_{B_{a}}g\big{)}\|B_{a}\setminus B_{a_{1}}\|$
		$\displaystyle\leq\mathcal{G}(E_{*})-\mathcal{G}(B_{a})+\big{(}\sup_{B_{a}}g\big{)}(\|B_{a}\|-\|B_{a_{1}}\|)$
		$\displaystyle=\mathcal{G}(E_{})-\mathcal{G}(B_{a})+\big{(}\sup_{B_{a}}g\big{)}(m-\|E_{}\|)$
		$\displaystyle\leq\mathcal{E}(E)-\mathcal{E}(B_{a})+\big{(}\sup_{B_{a}}g\big{)}\sqrt{\|B_{a}\|^{2}\tilde{a}_{1}(\mathcal{E}(E)-\mathcal{E}(B_{a}))}.$

	$\displaystyle\|E_{*}\Delta B_{a_{1}}\|$	$\displaystyle\leq r_{\hat{a_{1}},a_{1},\partial_{+}h(a_{1})}^{}\Big{[}\mathcal{G}(E_{})-\mathcal{G}(B_{a_{1}})\Big{]}^{\frac{1}{2}}$
(14)			$\displaystyle\leq r_{\hat{a_{1}},a_{1},\partial_{+}h(a_{1})}^{*}\Big{[}\mathcal{E}(E)-\mathcal{E}(B_{a})+\sup_{B_{a}}g\sqrt{\|B_{a}\|^{2}\tilde{a}_{1}(\mathcal{E}(E)-\mathcal{E}(B_{a}))}\Big{]}^{\frac{1}{2}}.$

	$\displaystyle\|E\Delta B_{a}\|$	$\displaystyle\leq\|E\Delta(B_{a}+z_{E})\|+\|(B_{a}+z_{E})\Delta B_{a}\|$
		$\displaystyle\leq\sqrt{\|B_{a}\|^{2}\tilde{a}_{1}(\mathcal{E}(E)-\mathcal{E}(B_{a}))}+\alpha_{*}\Big{[}\mathcal{E}(E)-\mathcal{E}(B_{a})\Big{]}^{\frac{1}{4}}$
		$\displaystyle\leq\overline{\alpha}_{*}\Big{[}\mathcal{E}(E)-\mathcal{E}(B_{a})\Big{]}^{\frac{1}{4}},$

	$\displaystyle\|P_{\alpha}(E_{m_{k}}^{\prime})$	$\displaystyle-P_{\alpha}(E_{m_{k}})\|$
		$\displaystyle=\|[\mathcal{E}(E_{m_{k}}^{\prime})-\mathcal{E}(E_{m_{k}})]+[\int_{E_{m_{k}}}g(x)dx-\int_{E_{m_{k}}^{\prime}}g(x)dx]\|$
		$\displaystyle\leq\|\mathcal{E}(E_{m_{k}}^{\prime})-\mathcal{E}(E_{m_{k}})\|+\int g(x)\|\chi_{E_{m_{k}}^{\prime}}-\chi_{E_{m_{k}}}\|dx$
		$\displaystyle<a_{k}+\int_{E_{m_{k}}^{\prime}\Delta E_{m_{k}}}g(x)dx.$