A nonlocal isoperimetric problem with density perimeter

Abstract

We consider the minimization of an energy functional given by the sum of a density perimeter and a nonlocal interaction of Riesz type with exponent $\alpha$, under volume constraint, where the strength of the nonlocal interaction is controlled by a parameter $\gamma$. We show that for a wide class of density functions the energy admits a minimizer for any value of $\gamma$. Moreover these minimizers are bounded. For monomial densities of the form $|x|^p$ we prove that when $\gamma$ is sufficiently small the unique minimizer is given by the ball of fixed volume. In contrast with the constant density case, here the $\gamma \rightarrow 0$ limit corresponds, under a suitable rescaling, to a small mass $m=|\Omega |\rightarrow 0$ limit when $p<d-\alpha +1$, but to a large mass $m\rightarrow \infty$ for powers $p>d-\alpha +1$.

1 Introduction

We consider the nonlocal isoperimetric problems

$$\begin{aligned} e(\gamma ) :=\inf \Big \{ \mathcal {E}_\gamma (\Omega ) :|\Omega |=1 \Big \} \end{aligned}$$

(1.1)

over sets of finite perimeter $\Omega \subset \mathbb {R}^d$ with given volume, where $|\cdot |$ denotes the Lebesgue measure in $\mathbb {R}^d$, and the energy functional $\mathcal {E}_\gamma$ is defined as

$$\begin{aligned} \mathcal {E}_\gamma (\Omega ) :=\int _{\partial ^*\Omega } \mathbf {a}(x)\,\mathrm {d}\mathcal {H}^{d-1}+ \gamma \int _\Omega \int _\Omega \frac{1}{|x-y|^\alpha }\,\mathrm {d}x\,\mathrm {d}y \end{aligned}$$

(1.2)

for $\gamma >0$, $\alpha \in (0,d)$. Here $\partial ^*$ denotes the reduced boundary of a set. The first term in the energy functional is the perimeter of $\Omega$ with density $\mathbf {a}:\mathbb {R}^d\rightarrow [0,\infty )$, whereas the second term is a Riesz-type nonlocal interaction energy.

The minimization problem (1.1) is a variant of the classical liquid drop model introduced by Gamow in [26]. Gamow’s model is simply given by (1.1) with $\mathbf {a}\equiv 1$. The most important feature of this geometric variational problem is that the two terms present in the energy functional are in direct competition. For $\mathbf {a}\equiv 1$, the surface energy is minimized by a ball whereas the repulsive term does not admit a minimizer and prefers minimizing sequences with multiple vanishingly small components diverging infinitely apart in order to disperse the mass. The parameter of the problem, that is $\gamma$, sets a length scale between these competing forces and drives the competition between the short- and long-range interactions. This problem has generated considerable interest in the calculus of variations community (see e.g. [3, 7, 14, 20,21,22, 29, 30, 32,33,34, 38, 42] as well as [15] for a review) with several papers studying parameter regimes of existence and nonexistence of minimizers. Results of [21, 32, 34], for example, show that for large values of $\gamma$, the energy $\mathcal {E}_\gamma$ with $\mathbf {a}\equiv 1$ does not admit a minimizer. There are also several studies characterizing the minimizing sequences [3, 7, 33] even when minimizers fail to exist. In particular, in [3], the authors use a “regularization” of the energy by adding an attractive external potential which guarantees the existence of minimizers for all values of $\gamma$.

Also very recently there has been studies on the extensions of the liquid drop model to the anisotropic setting where the surface energy is replaced by an anisotropic surface tension [8, 13, 36]. In these models the surface energy is given by $\int _{\partial ^*\Omega } \psi (\nu _\Omega )\,\mathrm {d}\mathcal {H}^{d-1}$ for some convex, one-homogeneous function $\psi$ where $\nu _\Omega$ denotes the outward unit normal to the reduced boundary $\partial ^* \Omega$. Such anisotropic extensions do not annihilate the translation invariance of the liquid drop model and a simple scaling argument heuristically justifies that for large $\gamma$ values minimizers still fail to exist. In contrast, the inclusion of a translation variant density in the perimeter functional “regularizes” the liquid drop model in the sense that the problem admits a minimizer for all values of $\gamma$.

Isoperimetric problems defined via weighted perimeters

$$\begin{aligned} {\mathcal {P}_{\mathbf {a}}}(\Omega ) :=\int _{\partial ^*\Omega }\mathbf {a}(x) \,\mathrm {d}\mathcal {H}^{d-1}(x) \end{aligned}$$

(1.3)

have been studied for various choices of densities $\mathbf {a}$. Problems where the volume constraint is also weighted either by the density $\mathbf {a}$ or by some other function have especially attracted significant interest (see e.g. [5, 6, 10,11,12, 17,18,19, 24, 37, 40, 41, 44] and references therein). The main questions regarding these problems have been existence, boundedness and regularity of isoperimetric sets. These questions have been studied not only for specific densities (radial, monomial, Gauss-like) but also for rather general densities satisfying some boundedness and continuity conditions. To our knowledge, perturbations of density perimeter (either by long-range interactions or by external potentials) have not yet been considered in the literature.

As pointed above, the inclusion of a confining density in the perimeter functional provides a different type of “regularization” of the problem (1.1). Our first main result establishes the compactness of any minimizing sequence with global convergence to a minimizer for any $\gamma \geqslant 0$ and for a wide class of densities, satisfying a simple coercivity condition:

A1:

$\mathbf {a}\in C^0(\mathbb {R}^d)$, $\mathbf {a}(0)=0$, $\mathbf {a}(x)>0$ for all $x\ne 0$, and $\lim _{|x|\rightarrow \infty } \mathbf {a}(x) = +\infty$.

In order to state the existence result, we recall that sets $\Omega _n\rightarrow \Omega$ globally if $|\Omega _n\triangle \Omega |\rightarrow 0$, that is, their characteristic functions $\chi _{\Omega _n}\rightarrow \chi _\Omega$ in the $L^1(\mathbb {R}^d)$-norm.

Theorem 1.1

(Existence of minimizers) Let $\mathbf {a}$ be any density satisfying the assumption (A1), and fix any $\gamma \geqslant 0$. Then any minimizing sequence $\{\Omega _n\}_{n\in \mathbb {N}}$ for the problem (1.1) admits a subsequence which converges globally to a minimizer $\Omega _\gamma \subset \mathbb {R}^d$ with $|\Omega _\gamma |=1$ and $\chi _{\Omega _\gamma }\in BV_{{{\,\mathrm{loc}\,}}}(\mathbb {R}^d\backslash \{0\})$.

Remark 1.2

The proof of the existence result in Theorem 1.1 can be extended to a somewhat broader class of densities $\mathbf {a}$ satisfying

$$\begin{aligned} \mathbf {a}\in C^0(\mathbb {R}^d),\quad \mathbf {a}^{-1}(\{0\})={\mathsf {Z}}, \quad \mathbf {a}>0\,\text { in }\,\mathbb {R}^d\backslash {\mathsf {Z}},\, \text { and }\, \lim _{|x|\rightarrow \infty }\mathbf {a}(x)=+\infty , \end{aligned}$$

for any finite set ${\mathsf {Z}} =\{z_1,\ldots ,z_m\}\subset \mathbb {R}^d$ with $m\in \mathbb {N}$. We present the proof in the case ${\mathsf {Z}}=\{0\}$. In this general situation, any minimizer $\Omega _{{\mathsf {Z}}}\subset \mathbb {R}^d$ is such that $\chi _{\Omega _{{\mathsf {Z}}}}\in BV_{{{\,\mathrm{loc}\,}}}(\mathbb {R}^d\backslash {\mathsf {Z}})$.

We remark that the coercive nature of $\mathbf {a}$ at infinity ensures the existence of minimizers for (1.1), essentially because the splitting of mass off to infinity (the main reason for noncompactness in nonlocal isoperimetric problems) is rendered too costly. However it does not ensure that minimizers need be connected sets. Indeed, for large $\gamma$ the nonlocal interactions should become large enough to favor the fragmentation of sets, which will repel but be contained at finite distance. This behavior is also observed in nonlocal isoperimetric problems with a confining term [3].

Our existence result relies on a modified version of the relative isoperimetric inequality on annulli and requires only the minimal assumption (A1) on the densities $\mathbf {a}$. On the other hand, proving boundedness of minimizers is rather technical and we prove it under one of the following additional structural assumptions:

(A2a):

$\mathbf {a}\in C^{0,1}_{{{\,\mathrm{loc}\,}}}(\mathbb {R}^d)$ and there exists constants $C_{\mathbf {a}} \geqslant 1$ and $R_{\mathbf {a}} \geqslant 1$ such that for every $R \geqslant R_{\mathbf {a}}$ it holds

$$\begin{aligned} 0<\sup _{B_{2R}\backslash B_{R}}\mathbf {a}\leqslant C_{\mathbf {a}} \inf _{B_{2R}\backslash B_{R}}\mathbf {a}. \end{aligned}$$

(1.4)

(A2b):

$\mathbf {a}\in C^{0,1}_{{{\,\mathrm{loc}\,}}}(\mathbb {R}^d)$, $\mathbf {a}(x)=\mathbf {a}(|x|)$ and there exists $R_{\mathbf {a}} \geqslant 1$ such that $\mathbf {a}$ is non-decreasing for $|x| \geqslant R_{\mathbf {a}}$.

For densities satisfying either of the additional conditions (A2a) or (A2b), using regularity results of quasi-minimizers, such as density bounds, we obtain the boundedness of minimizers of the problem (1.1).

Theorem 1.3

(Boundedness of minimizers) For any density $\mathbf {a}$ satisfying (A1) and either (A2a) or (A2b), and for any $\gamma \geqslant 0$, any minimizer $\Omega _\gamma$ of (1.1) is essentially bounded.

Remark 1.4

We require the assumption (A1) in Theorem 1.3 only to obtain the existence of a minimizer to (1.1). If the existence of a minimizer could be obtained under some other conditions, either (A2a) or (A2b) would be sufficient to obtain the boundedness of minimizers.

Remark 1.5

(Almost polynomial densities) Let $\mathbf {a}\in C^{0,1}_{{{\,\mathrm{loc}\,}}}(\mathbb {R}^d)$ satisfy the following conditions:

• (i) $\mathbf {a}(0)=0$ and $\mathbf {a}(x)>0$ for $x\ne 0$.

• (ii) There exist $p>0$, $R_0>0$ and $C_1,C_2>0$ such that $C_1|x|^p \leqslant \mathbf {a}(x)\leqslant C_2|x|^p$ for all $|x|>R_0$.

Then $\mathbf {a}$ satisfies the assumptions (A1) and (A2a); hence, Theorems 1.1 and 1.3 hold for such densities.

For homogeneous densities, which satisfy the condition $\mathbf {a}(tx) = t^p\, \mathbf {a}(x)$ for some $p>0$, a simple scaling argument shows that the minimization problem (1.1) is equivalent to the problem

$$\begin{aligned} \inf \Big \{ {\mathcal {E}}_{1}(\Omega ) :|\Omega |=m \Big \} \end{aligned}$$

(1.5)

with the correspondence

$$\begin{aligned} \gamma = m^{-(p+\alpha -d-1)/d}, \qquad p\ne p_* :=d-\alpha +1. \end{aligned}$$

It is interesting to observe that for homogeneous weights the large mass/small mass behavior of the minimization problem depends on the specific power p. In particular, when $p>p_*$, the corresponding value of $\gamma$ varies inversely with mass. Thus, with $p>p_*$ the nonlocal energy is dominated by the perimeter term $\mathcal {P}_{\mathbf {a}}$ for large mass m, and the nonlocal term dominates for small m, exactly the opposite of the behavior for constant $\mathbf {a}$. For subcritical $p<p_*$ the opposite is true, and the energy is perimeter-dominated for small m. At the critical value $p=p_*$ the two problems (1.1) and (1.5) are not equivalent, and (1.5) is scale invariant: the minimizers at any mass m are all rescaled copies of the same set. Since Theorem 1.1 guarantees the existence of a minimizer for all values of m (or $\gamma$), an interesting question is the characterization of minimizers for a range of values of the parameters. We provide a partial answer to this question in the next theorem.

Theorem 1.6

(Global minimizers in the small $\gamma$ regime) Let $\mathbf {a}(x)=|x|^p$ with $p>0$. For $\gamma$ sufficiently small the ball $B\subset \mathbb {R}^d$ of volume one, centered at the origin is the unique minimizer of $e(\gamma )$.

For $p<p_*$, the $\gamma \rightarrow 0$ limit is equivalent to the small m regime in (1.5), and the optimality of the spherical ball for small mass is well-known for the unweighted $\mathbf {a}\equiv 1$ case (see [7, 29,30,31,32]). With $p>p_*$, the situation is reversed and Theorem 1.6 shows that the ball minimizes for all sufficiently large m. Such $\mathbf {a}$ are very coercive at infinity, and the situation is similar to the case studied by Généraux and Oudet in [27], where the problem (1.5) with constant density is supplemented with a confining term. In that case they also prove the minimality of the ball for very large m.

The proof of Theorem 1.6 relies on a penalization technique, similar to those used in several geometric variational problems involving the perimeter functional (see [1, 7, 16, 20, 39]). Utilizing results from the regularity theory for density perimeters [18, 41] we reduce the minimizers of the nonlocal problem to nearly spherical or isoperimetric sets in the small $\gamma$ regime. The novelty here is, though, that we cannot directly apply the results from the literature due to the degeneracy of the density $\mathbf {a}$ at the origin and the possibility of small nonsmooth components of $\partial ^*\Omega _\gamma$ near the origin. Once we reduce the problem to nearly spherical sets we use a Fuglede-type argument (see [23]) to control the isoperimetric and nonlocal deficits between minimizers and the ball and show that for small $\gamma >0$ these quantities have to be identically zero.

The regime of large $\gamma$ is also very interesting, but its analysis requires a very different approach. While the existence of minimizers is guaranteed by Theorem 1.1, in this regime, the nonlocal term is dominant and prefers the minimizer to break into smaller pieces distributed in a compact set whose size is determined by the confining term $\mathbf {a}$. Hence, the characterization of minimizers (i.e., the shape of the disconnected components as well as their locations) depends on the delicate balance between the preferred shapes dictated by the density perimeter and the inter-component interactions. A similar phenomenon is also observed in models of copolymer/homopolymer blends. While existence of minimizers is obtained for all values of m in [9], only in the small m regime the minimizers are uniquely characterized leaving the question of the precise morphology of minimizing configurations for large m open.

1.1 Structure of the paper

The paper is organized as follows. In Sects. 2 and 3 we prove Theorems 1.1 and 1.3, respectively. Section 4 is devoted to the proof of Theorem 1.6.

1.2 Notation

Throughout the paper $\omega _d$ denotes the volume of the unit ball $B_1(0)$ in $\mathbb {R}^d$ and we write $B_r :=B_r(0)$ to denote the ball of radius r centered at zero. Constants, denoted by C, can change from line to line (unless otherwise noted). We will denote the $\mathbf {a}$-volume measure and the $\mathbf {a}$-surface area measure, respectively, by

$$\begin{aligned} |\Omega |_{\mathbf {a}}:=\int _{\Omega }\mathbf {a}(y)\,\mathrm {d}y, \quad \text { and } \quad \mathcal {H}^{d-1}_{\mathbf {a}}(\Omega ) :=\int _{\Omega }\mathbf {a}(y)\,\,\mathrm {d}\mathcal {H}^{d-1}(y). \end{aligned}$$

The relative weighted perimeter of E in F will be denoted by either ${\mathcal {P}_{\mathbf {a}}}(E,F)$ or $\int _F \mathbf {a}(x)|\nabla \chi _E|$, where $\chi _E$ is the characteristic function of the set E and $|\nabla \chi _E|$ is the total variation of $\chi _E$. The weighted perimeter of E in F (or the $\mathbf {a}$-perimeter) is defined as

$$\begin{aligned} {\mathcal {P}_{\mathbf {a}}}(E,F) :=\sup \left\{ \int _{E} {{\,\mathrm{div}\,}}\big (\mathbf {a}(x) X(x)\big ) \,\mathrm {d}x:X\in C_c^\infty (F;\mathbb {R}^d), \ \Vert X\Vert _{L^\infty } \leqslant 1 \right\} . \end{aligned}$$

In particular, ${\mathcal {P}_{\mathbf {a}}}(E,F)=\mathcal {H}^{d-1}_{\mathbf {a}}(\partial ^* E \cap F)$. We will say that $\chi _E \in BV_{\mathbf {a}}(F)$ if ${\mathcal {P}_{\mathbf {a}}}(E,F) < +\infty$. Perimeters of sets in the whole space (i.e., when $F=\mathbb {R}^d$) are denoted by $\mathcal {P}_{\mathbf {a}}(E)$ or $\int _{\mathbb {R}^d} \mathbf {a}(x)|\nabla \chi _E|$. We will denote the Euclidean perimeter (when $\mathbf {a}\equiv 1$) by simply $\mathcal {P}$. Finally, we will denote the nonlocal term by $\mathcal {V}$, i.e.,

$$\begin{aligned} {\mathcal {V}}(\Omega ) :=\int _\Omega \int _\Omega \frac{\,\mathrm {d}x\,\mathrm {d}y}{|x-y|^{\alpha }}, \end{aligned}$$

for any $\alpha \in (0,d)$.

2 Existence of minimizers

In this section we present the proof of Theorem 1.1. It relies on the following modified version of the relative isoperimetric inequality on annulli.

Lemma 2.1

Let $A_{r,R}=\big \{x\in \mathbb {R}^d: r\leqslant |x|< R\big \}=B_R\backslash B_r$ denote the half-open annulus of inner radius r and outer radius R. Then, there exists a dimensional constant $c_d>0$, so that

$$\begin{aligned} \min \Big \{|\Omega \cap A_{1,2}|^{(d-1)/d},\, |A_{1,2}\backslash \Omega |^{(d-1)/d}\Big \} \leqslant c_d\,\mathcal {P}(\Omega ,A_{1,2}) \end{aligned}$$

(2.1)

for every set of finite perimeter $\Omega \subset \mathbb {R}^d$.

Remark 2.2

Inequality (2.1) is still valid with the same constant $c_d$ over any annulus $A_{2^j,2^{j+1}}$ with $j\geqslant 1$, as the inequality is invariant under scalings.

The relative isoperimetric inequality is typically stated for balls in $\mathbb {R}^d$ (see e.g. [47, Thm 5.4.3],) but in fact the same proof verifies that it holds in any domain for which one can prove the validity of the Poincaré inequality,

$$\begin{aligned} \left( \int _{A_{1,2}}|u-{\bar{u}}_{r,R}|^{\frac{d}{d-1}}\,\,\mathrm {d}x\right) ^{\frac{d-1}{d}}\leqslant C_{1,2}\int _{A_{1,2}}|\nabla u|\,\mathrm {d}x, \end{aligned}$$

where is the average on annuli. The latter can be found in [2].

We now turn to the proof of the existence of minimizers for $e(\gamma )$. For translation-invariant nonlocal isoperimetric problems existence is a delicate issue, as minimizing sequences can split, with pieces diverging to infinity. The increasing weight $\mathbf {a}(x)$ raises the cost of splitting, an effect which is quantified in our proof via the relative isoperimetric inequality, Lemma 2.1.

Proof of Theorem 1.1

Let $\{\Omega _n\}\subset \mathbb {R}^d$ be a minimizing sequence of (1.1):

$$\begin{aligned} \mathcal {E}_\gamma (\Omega _n)\rightarrow e(\gamma )\;\text { as }\;n\rightarrow \infty ,\;\;\text { and }\;\; |\Omega _n|=1\;\; \text {for any }n\geqslant 1. \end{aligned}$$

We first show that $\{\Omega _n\}$ is uniformly bounded in $BV(\mathbb {R}^d\backslash {\overline{B}}_{\varepsilon })$ for all $\varepsilon >0$ sufficiently small. To see this, note that $\lim _{|x|\rightarrow \infty }\mathbf {a}(x)=+\infty$ implies that there exists $R_1>0$ such that $\mathbf {a}\geqslant 1$ in $\mathbb {R}^d\backslash B_{R_1}$. Define $\delta _{\varepsilon } :=\min \{\mathbf {a}(x) :\varepsilon \leqslant |x|\leqslant R_1\}$ for $0<\varepsilon <1$ small enough so that $\delta _{\varepsilon }\leqslant 1$ (as $\lim _{|x|\rightarrow 0 }\mathbf {a}(x)=0$). Then, for any such choice of $\varepsilon$, $\mathbf {a}(x)\geqslant \delta _{\varepsilon }>0$ for all $x\in \mathbb {R}^d\backslash \overline{B}_{\varepsilon }$, and we deduce

$$\begin{aligned} \delta _{\varepsilon }\,{\mathcal {P}}(\Omega _n,\mathbb {R}^d\backslash {\overline{B}}_{\varepsilon })<{\mathcal {P}_{\mathbf {a}}}(\Omega _n,\mathbb {R}^d\backslash {\overline{B}}_{\varepsilon })<\mathcal {E}_\gamma (\Omega _n)=e(\gamma )+o_n(1), \end{aligned}$$

which confirms uniform boundedness in $BV(\mathbb {R}^d\backslash {\overline{B}}_{\varepsilon })$.

We next show local convergence of $\{\Omega _n\}$ to a limiting set $\Omega _\gamma$. Invoking compactness results of sets with uniformly bounded perimeter, there exists $\Omega ^{\varepsilon }\subset \mathbb {R}^d$ so that $\Omega _{n_{\ell }}\rightarrow \Omega ^{\varepsilon }$ locally in $\mathbb {R}^d\backslash {\overline{B}}_{\varepsilon }$, as $\ell \rightarrow \infty$. Running a diagonalization argument over a sequence $\varepsilon _k=1/k\rightarrow 0^+$, there exists a subsequence ${n_\ell }\rightarrow \infty$ such that for all $k\geqslant 1$, $\Omega _{n_\ell } \rightarrow \Omega ^{1/k}$ locally in $\mathbb {R}^d\backslash {\overline{B}}_{1/k}$, as $\ell \rightarrow \infty$. In particular, $\Omega ^{1/(k+j)}\backslash {\overline{B}}_{1/k}=\Omega ^{1/k}\backslash {\overline{B}}_{1/k}$ for any $j\geqslant 1$. Defining the limit set as $\Omega _\gamma :=\bigcup ^{\infty }_{k=1} (\Omega ^{1/k}\backslash {\overline{B}}_{1/k})$, we claim that up to subsequence,

$$\begin{aligned} \begin{aligned} \Omega _n\rightarrow \Omega _\gamma \;\text { locally in }\mathbb {R}^d\;\text { as }\; n\rightarrow \infty ,\text { and } \\ \chi _{\Omega _n}\rightarrow \chi _{\Omega _\gamma } \;\text { pointwise a.e. in }\mathbb {R}^d\;\text { as }\; n\rightarrow \infty . \end{aligned} \end{aligned}$$

(2.2)

Assuming the claim, we may conclude that ${\mathcal {P}}(\Omega _\gamma ,\mathbb {R}^d\backslash {\overline{B}}_{\varepsilon })\leqslant \liminf \limits _{n\rightarrow \infty }{\mathcal {P}}(\Omega _n,\mathbb {R}^d\backslash {\overline{B}}_{\varepsilon })$ for $0<\varepsilon \ll 1$, which shows $\chi _{\Omega _\gamma }\in BV_{{{\,\mathrm{loc}\,}}}(\mathbb {R}^d\backslash \{0\})$.

To verify (2.2), let $K\subset \mathbb {R}^d$ be a compact set and for $\varepsilon >0$, fix $k \geqslant 1$ so that $|B_{1/k}|\leqslant \varepsilon$. Then

$$\begin{aligned} |(\Omega _{n}\triangle \Omega _\gamma )\cap K|&= |(\Omega _{n}\triangle \Omega ^{1/k})\cap (K\backslash B_{1/k})|+ |(\Omega _{n}\triangle \Omega _\gamma )\cap K\cap B_{1/k}| \\&\leqslant o_n(1)+\varepsilon . \end{aligned}$$

Since $\varepsilon >0$ is arbitrary small, we conclude (2.2). The aforementioned convergence along with the lower semicontinuity of the $\mathbf {a}$-perimeter functional, together with Fatou’s Lemma shows that

$$\begin{aligned} \mathcal {E}_\gamma (\Omega _\gamma )={\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma )+\gamma {\mathcal {V}}(\Omega _\gamma )\leqslant \liminf _{n\rightarrow \infty }({\mathcal {P}_{\mathbf {a}}}(\Omega _n)+\gamma {\mathcal {V}}(\Omega _n))=e(\gamma ). \end{aligned}$$

We are only left to show that $\Omega _\gamma$ is admissible in (1.1), from which it will follow that $\mathcal {E}_\gamma (\Omega _\gamma )\geqslant e(\gamma )$; thus obtaining the existence of a minimizer.

We observe that $|\Omega _\gamma |\leqslant 1$ in view of Fatou’s Lemma, once again. We claim that in fact $|\Omega _\gamma |=1$. Suppose, on the contrary, that $|\Omega _\gamma |<\beta$ for some $\beta \in (0,1)$.

The local convergence (2.2) shows that for all $R>0$, $|\Omega _n\cap B_R|=|\Omega _\gamma \cap B_R|+o_n(1)<\beta$ for all but finitely many n. Thus, the sets $\Omega _n$ have very thick tails, which will introduce huge energy cost via the relative isoperimetric inequality. By running a diagonalization argument over $\{R_k=2^k\}$, there exists an increasing subsequence $n_k\rightarrow +\infty$ such that for all $k\geqslant 1$,

$$\begin{aligned} \inf _{n\geqslant n_k}|\Omega _n\backslash B_{2^k}|> 1-\beta . \end{aligned}$$

(2.3)

On the other hand, as $\mathbf {a}(x)\rightarrow +\infty$ as $|x|\rightarrow \infty$, for $M>1$ arbitrarily large, for all $n\geqslant 1$, and every $k\geqslant k_M$ sufficiently large,

$$\begin{aligned} \mathcal {E}_\gamma (\Omega _n)\geqslant {\mathcal {P}_{\mathbf {a}}}(\Omega _n,\mathbb {R}^d\backslash B_{2^k}) \geqslant M {\mathcal {P}}(\Omega _n,\mathbb {R}^d\backslash B_{2^k}). \end{aligned}$$

(2.4)

Intuitively, when j is large we expect $|\Omega _n\cap A_{2^{j},2^{j+1}}|$ to be much smaller than its complement $|A_{2^j,2^{j+1}}\backslash \Omega _n|$ in the annulus. Indeed, we claim that there exists $j_0\in \mathbb {N}$ such that for all $j\geqslant j_0$ and for all but finitely many n we have

$$\begin{aligned} |\Omega _n\cap A_{2^{j},2^{j+1}}|<(2^{j+1})^{-d}|A_{2^j,2^{j+1}}\backslash \Omega _n|. \end{aligned}$$

(2.5)

For otherwise, there would exist increasing sequences $j_{\ell }\rightarrow +\infty$ and $n_{\ell }\rightarrow +\infty$ for which $|\Omega _{n_{\ell }}\cap A_{2^{j_{\ell }},2^{j_{\ell }+1}}|\geqslant (2^{j_{\ell }+1})^{-d}|A_{2^{j_{\ell }},2^{j_{\ell }+1}}\backslash \Omega _{n_{\ell }}|$, for every $\ell \geqslant 1$. This would imply that

$$\begin{aligned} |\Omega _n\cap A_{2^{j},2^{j+1}}| \geqslant {\frac{1}{1+2^{(j+1)d}}} |A_{2^{j_{\ell }},2^{j_{\ell }+1}}|, \end{aligned}$$

and hence,

$$\begin{aligned} 1=|\Omega _{n_{\ell }}|&\geqslant \sum \limits ^{\infty }_{\ell =1}|\Omega _{n_{\ell }}\cap A_{2^{j_{\ell }},2^{j_{\ell }+1}}| \\&\geqslant \sum \limits ^{\infty }_{\ell =1}\frac{1}{1+(2^{j_{\ell }+1})^d}|A_{2^{j_{\ell }},2^{j_{\ell }+1}}| >\sum \limits ^{\infty }_{\ell =1}\frac{1}{2}\biggl (1-\frac{1}{2^{d}}\biggr )\omega _d=+\infty , \end{aligned}$$

establishing the claim.

We now fix any $k\geqslant \max \{k_M,j_0\}$ and $n \gg 1$ sufficiently large to obtain the validity of (2.4) and of (2.5) for all $j\geqslant k$. Utilizing the relative isoperimetric inequality on $\{A_{2^j,2^{j+1}}: j\geqslant k\}$ (see Remark 2.2), we get the lower bound

$$\begin{aligned} c_d\, {\mathcal {P}}(\Omega _n,\mathbb {R}^d\backslash B_{2^{k}})&=\sum ^{\infty }_{j=k}c_d\, {\mathcal {P}}(\Omega _n,A_{2^j,2^{j+1}}) \\&\geqslant \sum ^{\infty }_{j=k}|\Omega _n\cap A_{2^j,2^{j+1}}|^{\frac{d-1}{d}}\\&\geqslant \biggl (\sum ^{\infty }_{j=k}|\Omega _n\cap A_{2^j,2^{j+1}}|\biggr )^{\frac{d-1}{d}}=|\Omega _n\cap (\mathbb {R}^d\backslash B_{2^k})|^{\frac{d-1}{d}} \end{aligned}$$

Increasing the value of $n\geqslant n_k$ if necessary, it follows from (2.3) and (2.4) that

$$\begin{aligned} +\infty >e(\gamma )+o_n(1) \geqslant M {\mathcal {P}}(\Omega _n,\mathbb {R}^d\backslash B_{2^{k}}) \geqslant \frac{M}{c_d}(1-\beta )^{\frac{d-1}{d}}, \end{aligned}$$

with $M>1$ arbitrarily large. Thus, we reach a contradiction. Hence, $|\Omega _\gamma |=1$, and we have proven that $\Omega _\gamma$ attains the minimum in the nonlocal isoperimetric problem (1.1).

Finally, by the identity $|\Omega _n\triangle \Omega _\gamma |= 2|\Omega _\gamma \backslash \Omega _n|+|\Omega _n|-|\Omega _\gamma |$ together with $|\Omega _n|= 1=|\Omega _\gamma |$ and (2.2), we deduce the global convergence of the subsequence $\{\Omega _n\}$, as (by local convergence,)

$$\begin{aligned} |\Omega _\gamma \backslash \Omega _n|\leqslant |(\Omega _\gamma \backslash \Omega _n)\cap B_r|+|\Omega _\gamma \backslash B_r|\leqslant |(\Omega _\gamma \backslash \Omega _n)\cap B_r|+o_r(1). \end{aligned}$$

Thus, every minimizing sequence $\Omega _n$ for $e(\gamma )$ contains a subsequence which converges globally (in $L^1(\mathbb {R}^d)$) to a minimizer of $e(\gamma )$. $\square$

3 Boundedness of minimizers

In this section we prove Theorem 1.3. Since the assumptions (A2a) and (A2b) characterize different types of densities, the proof of the theorem requires two different approaches. For densities which are polynomial-like and have bounded oscillations (i.e., densities satisfying (A2a)) we make use of a series of technical lemmas establishing uniform density bounds for quasi-minimizers of the weighted perimeter functional where the density is measured with respect to weighted volumes. For radial and monotone densities (i.e., densities satisfying (A2b)), on the other hand, we utilize a regularity result, called $\varepsilon -\varepsilon ^{(d-1)/d}$ property, which basically says that a set of finite perimeter can be locally modified where one increases its volume by $\varepsilon$, while the perimeter increases at most by a constant multiple of $\varepsilon ^{(d-1)/d}$. We present the proof in two subsections.

3.1 Densities with bounded oscillations

We start with densities satisfying the assumptions (A1) and (A2a). First, we prove that any minimizer $\Omega _\gamma$ of (1.1) has finite $\mathbf {a}$-volume.

Lemma 3.1

For any density $\mathbf {a}$ satisfying (A1) and (A2a) any minimizer $\Omega _\gamma$ of (1.1) has finite $\mathbf {a}$-volume, i.e., $|\Omega _\gamma |_{\mathbf {a}}<+\infty$.

Proof

By passing to the limit $\Omega _n\rightarrow \Omega _\gamma$ in (2.5),

$$\begin{aligned} |\Omega _\gamma \cap A_{2^{j},2^{j+1}}|\leqslant (2^{j+1})^{-d}|A_{2^j,2^{j+1}}\backslash \Omega _\gamma |, \end{aligned}$$

(3.1)

for all $j\geqslant j_0$, for some $j_0$. Fix $j_{\mathbf {a}}\in \mathbb {N}$ with $j_{\mathbf {a}}\geqslant j_0$, such that $R_{\mathbf {a}}<2^{j_{\mathbf {a}}}$ where $R_{\mathbf {a}}$ is given as in (1.4). Using  (1.4), (3.1), and the relative isoperimetric inequality (see Remark 2.2) we have:

$$\begin{aligned} \int _{\Omega _\gamma \backslash B_{2^{j_{\mathbf {a}}}}}\mathbf {a}(x)\,\mathrm {d}x&=\sum ^{\infty }_{j=j_{\mathbf {a}}}\int _{\Omega _\gamma \cap A_{2^j,2^{j+1}}}\mathbf {a}(x)\,\mathrm {d}x\\&\leqslant \sum ^{\infty }_{j=j_{\mathbf {a}}}\big (\sup _{A_{2^j,2^{j+1}}}\mathbf {a}\big )|\Omega _\gamma \cap A_{2^j,2^{j+1}}|\\&\leqslant \sum ^{\infty }_{j=j_{\mathbf {a}}}C_{\mathbf {a}}\big (\inf _{A_{2^j,2^{j+1}}}\mathbf {a}\big )|\Omega _\gamma \cap A_{2^j,2^{j+1}}|^{\frac{d-1}{d}}|\Omega _\gamma \backslash B_{2^{j}}|^{\frac{1}{d}}\\&\leqslant C_{\mathbf {a}}\,|\Omega _\gamma \backslash B_{2^{j_{\mathbf {a}}}}|^{\frac{1}{d}}\sum ^{\infty }_{j=j_{\mathbf {a}}}\big ( \inf _{A_{2^j,2^{j+1}}}\mathbf {a}\big ) c_d\int _{A_{2^j,2^{j+1}}}|\nabla \chi _{\Omega _\gamma }|\\&\leqslant c_d\,C_{\mathbf {a}}\,|\Omega _\gamma \backslash B_{2^{j_{\mathbf {a}}}}|^{\frac{1}{d}}\sum ^{\infty }_{j=j_{\mathbf {a}}}\int _{A_{2^j,2^{j+1}}}\mathbf {a}(x)|\nabla \chi _{\Omega _\gamma }|. \end{aligned}$$

Hence, as $|\Omega _\gamma |\leqslant 1$, we conclude that

$$\begin{aligned} \int _{\Omega _\gamma \backslash B_{2^{j_{\mathbf {a}}}}}\mathbf {a}(x)\,\mathrm {d}x\leqslant c_d\,C_{\mathbf {a}}\;\int _{\mathbb {R}^d\backslash B_{2^{j_{\mathbf {a}}}}}\mathbf {a}(x)|\nabla \chi _{\Omega _\gamma }|. \end{aligned}$$

(3.2)

Since $\Omega _\gamma$ has finite $\mathbf {a}$-perimeter, we obtain that $|\Omega _\gamma |_{\mathbf {a}}<+\infty$. $\square$

At the heart of the proof of Theorem 1.3 lies the regularity of quasi-minimal sets with a volume constraint. In order to establish this we will largely follow the argument carried out by Rigot in [43, Chapter 2], where the author studies the case of standard perimeter functional $\mathbf {a}(x)\equiv 1$. As in [43, Chapter 1] and [35, Chapter 21], given a function $g:(0,+\infty )\rightarrow (0,+\infty )$ with $g(x)=o(x^{(d-1)/d})$ for x close to 0, we will say that $\Omega _\gamma$ is a volume constrained quasi-minimal set for $\mathbf {a}$-perimeter if

$$\begin{aligned} {\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ) \leqslant {\mathcal {P}_{\mathbf {a}}}(F)+g(|F\triangle \Omega _\gamma |) \end{aligned}$$

for any $F\subset \mathbb {R}^d$ with $\chi _F\in BV_{\mathbf {a}}(\mathbb {R}^d)$, $|F|=1$ and $F\triangle \Omega _\gamma \subset \!\subset \mathbb {R}^d$.

Minimizers of isoperimetric problems with a Riesz-type nonlocal term are also volume constrained quasi-minimizers for the perimeter functional with the choice $g_{\gamma }(x)\simeq \gamma \,x$. Indeed, as argued in [32, Proposition 2.1], we define the potential of a Borel set $E\subset \mathbb {R}^d$ by

$$\begin{aligned} v_E(x) :=\int _{E}\frac{1}{|x-y|^{\alpha }}\,\mathrm {d}y\end{aligned}$$

and consider a minimizer $\Omega _\gamma$ of (1.1) together with a set F with prescribed mass and $F\triangle \Omega _\gamma \subset B_r(0)$ for some $r>0$. A simple argument shows that the interaction energy $\mathcal {V}$ is Lipschitz with respect to symmetric difference:

$$\begin{aligned} \begin{aligned} {\mathcal {V}}(F)-{\mathcal {V}}(E)&=\int _{F}\int _{F}\frac{1}{|x-y|^{\alpha }}\,\mathrm {d}x\,\mathrm {d}y-\int _{E}\int _{E}\frac{1}{|x-y|^{\alpha }}\,\mathrm {d}x\,\mathrm {d}y\\&\leqslant \int _{E\triangle F}(v_{E}+v_{F})\,\mathrm {d}x\\&\leqslant C |E\triangle F| \end{aligned} \end{aligned}$$

(3.3)

with $C=2\int _{B_1}\frac{\,\mathrm {d}y}{|y|^{\alpha }}+2$.

Hence, any minimizer $\Omega _\gamma$ of $\mathcal {E}_\gamma$ must satisfy

$$\begin{aligned} {\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ) \leqslant {\mathcal {P}_{\mathbf {a}}}(F) + C\gamma \, |F\triangle \Omega _\gamma |, \end{aligned}$$

(3.4)

for any suitable competitor F as above.

We will now present some technical lemmas essentially studied in [43, Chapter 2] and adapt these results to the case of weighted perimeters. The next lemma, proven for $\mathbf {a}=1$ by Giusti (see [28, Lemma 2.1]), shows that any set of positive perimeter can be approximated in $L^1$ by another set without substantially increasing the weighted perimeter. We denote by $[\, \cdot \,]_{1,D}$ the Lipschitz seminorm in D.

Lemma 3.2

Let $D\subset \mathbb {R}^d$ be a bounded domain, $L\subset \mathbb {R}^d$ with $\chi _L\in BV_{\mathbf {a}}(\mathbb {R}^d)$ for a density $\mathbf {a}\in C^{0,1}(D)$, $\inf _{D}\mathbf {a}>0$, and such that

$$\begin{aligned} \int _D\mathbf {a}(x)|\nabla \chi _L|>0. \end{aligned}$$

Then there exist $\varepsilon >0$ and $Q_{\mathbf {a}}>0$, depending on $L\cap D$ and D only, such that, $Q_{\mathbf {a}}\lesssim \big (\sup _{D}\mathbf {a}\big ) \big (1+[\mathbf {a}]_{1,D}\big )$, and for every $v\in (-\varepsilon ,\varepsilon ),$ there exists $F\subset \mathbb {R}^d$ with $F=L$ in a neighborhood of $\mathbb {R}^d\backslash D$ satisfying

$$\begin{aligned} |F|= & {} |L|+v, \\ \int _D\mathbf {a}(x)|\nabla \chi _F|\leqslant & {} \int _D\mathbf {a}(x)|\nabla \chi _L|+Q_{\mathbf {a}}|v|,\\ \int _D|\chi _F-\chi _L|\leqslant & {} Q_{\mathbf {a}}|v|. \end{aligned}$$

Proof

By definition of $\mathbf {a}$-perimeter, there exists $w\in C^1_c(D;\mathbb {R}^d)$, $|w(x)|\leqslant \mathbf {a}(x)$ a.e. in D, such that

$$\begin{aligned} \int _{D}\chi _L{\text {div}}w\,\mathrm {d}x\geqslant \frac{1}{2}\int _{D}\mathbf {a}(x)|\nabla \chi _L|>0. \end{aligned}$$

(3.5)

Note that, since $\mathbf {a}>0$ in D, $BV_{\mathbf {a}}(D)\subset BV(D)$. For $t\in (0,1)$ we put $\eta _t=x+tw(x)$ and $K :={\text {spt}}w\subset \!\subset D$. Then $\eta _t\equiv I$ in $\mathbb {R}^d\backslash K$, and for |t| small enough, $\eta _t :D\rightarrow D$ is a diffeomorphism. Letting $L_t :=\eta _t(L)$, we claim:

$$\begin{aligned} \begin{aligned}&|L_t|=\int _L|\det D\eta _t|\,\mathrm {d}x\\&\quad \int _{D}\mathbf {a}(x)|\nabla \chi _{L_t}| \leqslant \int _{D}\mathbf {a}(x)f_t(x)|\nabla \chi _L|+|t| \big (\sup _K\mathbf {a}\big )[\mathbf {a}]_{1,K}\int _{K}f_t(x)|\nabla \chi _{L}| \end{aligned} \end{aligned}$$

(3.6)

where $f_t(x) :=|\det D\eta _t(x)||(D\eta _t)^{-1}(x)|$. Indeed, the first equality is clear, and the inequality below it is obtained by noting,

$$\begin{aligned} \int _{D}\mathbf {a}(x)|\nabla \chi _{L_t}|&\leqslant \int _{D}(\mathbf {a}\circ \eta _t)(x)\,f_t(x)|\nabla \chi _L| \\&\leqslant \int _{D}\mathbf {a}(x)\,f_t(x)|\nabla \chi _L|+\int _{K}f_t(x)\big |\mathbf {a}\circ \eta _t(x)-\mathbf {a}(x)\big |\,|\nabla \chi _L| \end{aligned}$$

and estimating, for $x\in K$,

$$\begin{aligned} |\mathbf {a}\circ \eta _t(x)-\mathbf {a}(x)| = |\mathbf {a}(x+tw(x))-\mathbf {a}(x)| \leqslant [\mathbf {a}]_{1,K}\,|tw(x)|\leqslant |t|\, [\mathbf {a}]_{1,K}\big (\sup _{K}\mathbf {a}\big ). \end{aligned}$$

Also, $\det D\eta _t=1+t{\text {div}}w+t^2A(x,t)$ and $(D\eta _t)^{-1}=I-tH(x,t)$, with |A| and |H| bounded uniformly by a constant, which depends exclusively on $L\cap D$ and D. For |t| small enough, $f_t(x)\leqslant 1+t({\text {div}}w+|H|)+O(t^2)$, and so (3.6) shows

$$\begin{aligned} |L_t|=|L|+t\int _{D}\chi _L{\text {div}}w\,\mathrm {d}x+t^2\int _{D}\chi _LA(x,t)\,\mathrm {d}x, \end{aligned}$$

(3.7)

and

$$\begin{aligned} \begin{aligned}&\int _{D}\mathbf {a}(x)|\nabla \chi _{L_t}|\\&\quad \leqslant \big (1+|t|\cdot \Vert {\text {div}}w+|H|\Vert _{L^{\infty }(D)}\big )\int _{D}\mathbf {a}(x)|\nabla \chi _L| \\&\qquad + |t| [\mathbf {a}]_{1,D} \big (\sup _{D}\mathbf {a}\big ) \int _{D}|\nabla \chi _L|+O(t^2)\\&\quad \leqslant \int _{D}\mathbf {a}(x)|\nabla \chi _L| +|t| \big (\sup _{D}\mathbf {a}\big ) (C+\,[\mathbf {a}]_{1,D})\int _{D}|\nabla \chi _L|+O(t^2). \end{aligned} \end{aligned}$$

(3.8)

In view of (3.5), there exists $\varepsilon ^\prime >0$ sufficiently small so that for any choice of $v\in (-\varepsilon ',\varepsilon ')$ the relation $t\,\int _{D}\chi _{L}{\text {div}}w\,\mathrm {d}x+t^2\int _{D}\chi _{L}A(x,t)=|v|$ in (3.7) holds true for some $t_v$, and moreover $|t_v|\leqslant C'|v|$, with $C'$ depending on $L\cap D$ and D only. We take $F :=L_{t_v}$ and observe that F satisfies the first two statements of the lemma, in light of (3.7)–(3.8), for the value $Q_{\mathbf {a}}=2(\sup _{D}\mathbf {a})(C+[\mathbf {a}]_{1,D})\int _{D}|\nabla \chi _{L}|$, by decreasing the value of $\varepsilon '$ if necessary, in order that $C'\varepsilon '\leqslant 2$.

To verify the final statement, for $g\in C^1(D)$ let $g_t :=g\circ \eta ^{-1}_t$, so $g_t-g = g_t-g_t\circ \eta _t$. Then

$$\begin{aligned} \int _{D}|g_t-g|\,\mathrm {d}x&=-\int _{D}\int ^1_0tw(x)\cdot \nabla g_t(x+tsw(x))\,\mathrm {d}s\,\mathrm {d}x\nonumber \\&\leqslant |t|\int ^1_0\int _{D}\mathbf {a}(x)|\nabla g_t\circ \eta _{ts}|\,\mathrm {d}x\,\mathrm {d}s\nonumber \\&\leqslant |t|\int _{D}\mathbf {a}(x)|\nabla g|\,\mathrm {d}x+|t|^2\,[\mathbf {a}]_{1,D}\int _{D}|\nabla g|+O(|t|^3), \end{aligned}$$

(3.9)

where the last inequality will be derived below. Observe the third bound in the statement of Lemma 3.2 holds for $Q_{\mathbf {a}}=2(\sup _D\mathbf {a})(1+[\mathbf {a}]_{1,D})\int _{D}|\nabla \chi _{L}|$, upon decreasing the value of $\varepsilon '$ if necessary. An approximation argument justifies estimate (3.9) for $g\in BV_{\mathbf {a}}(\mathbb {R}^d)$ and so in particular for $g=\chi _L$ and $g_t=\chi _{L}\circ \eta ^{-1}_t=\chi _{L_t}$. First, note that

$$\begin{aligned} \int _{D}\mathbf {a}(x)|\nabla g_t\circ \eta _{ts}|\,\mathrm {d}x=&\int _{D}(\mathbf {a}\circ \eta ^{-1}_{ts})(x)|\nabla g_t|\,|\det D(\eta ^{-1}_{ts})|\,\mathrm {d}x\\ \leqslant&(1+|t|\,\Vert H\Vert _{\infty }) \int _{D}(\mathbf {a}\circ \eta ^{-1}_{ts})(x)|\nabla (g\circ \eta ^{-1}_t)|\,\mathrm {d}x\\ \leqslant&(1+|t|\,\Vert H\Vert _{\infty }) \int _{D}\mathbf {a}(\eta ^{-1}_{ts}\circ \eta _{t})(x)|(D\eta _{t})^{-1}|\,|\nabla g|\,|\det D\eta _{t}|\,\mathrm {d}x\\ \leqslant&(1+|t|\,\Vert H\Vert _{\infty })^2 \Big (1+|t|\,\Vert {\text {div}}w\Vert _{\infty } \\&+|t|\,\Vert H\Vert _{\infty }+O(t^2)\Big )\int _{D}\mathbf {a}(\eta ^{-1}_{ts}\circ \eta _{t})(x)|\nabla g|\,\mathrm {d}x. \end{aligned}$$

Also, it can be checked that $|\mathbf {a}(\eta ^{-1}_{ts}\circ \eta _t)(x)-\mathbf {a}(x)|\lesssim [\mathbf {a}]_{1,K} |t|$ for $x\in K$, since we have that $|(\eta ^{-1}_{ts}\circ \eta _t)(x)-x|\lesssim |t|(1+|s|)\Vert w\Vert _{L^{\infty }(K)}$. Hence,

$$\begin{aligned} \int _{D}\mathbf {a}(\eta ^{-1}_{ts}\circ \eta _{t})(x)|\nabla g|\,\mathrm {d}x\leqslant \int _{D}\mathbf {a}(x)|\nabla g|\,\mathrm {d}x+C\,|t|\,[\mathbf {a}]_{1,K}\int _{D}|\nabla g|, \end{aligned}$$

where the constant C depends on D only. Recalling (3.7)–(3.8)–(3.9), and the fact that $|t_v|\leqslant C'|v|$, we can choose

$$\begin{aligned} Q_{\mathbf {a}}=2(\sup _D\mathbf {a})(\max \{1,C\}+[\mathbf {a}]_{1,D})\int _{D}|\nabla \chi _{L}| \end{aligned}$$

and this concludes the proof. $\square$

Remark 3.3

We note that in the proof of Lemma 3.2 above we only use the Lipschitzianity of $\mathbf {a}$.

Let us continue with an adaptation of a classical notion in geometric measure theory, to our setting with weight function $\mathbf {a}$. Given $x\in \mathbb {R}^d$ and $r>0$ let us define the weighted relative density function of the set $\Omega _\gamma$ as

$$\begin{aligned} h_{\mathbf {a}}(x,r) :=\min \left\{ \frac{|\Omega _\gamma \cap B_r(x)|_{\mathbf {a}}}{|B_r(x)|_{\mathbf {a}}},\;\frac{|B_r(x)\backslash \Omega _\gamma |_{\mathbf {a}}}{|B_r(x)|_{\mathbf {a}}} \right\} . \end{aligned}$$

The rest of the proof is devoted to establishing a uniform lower bound of the form $h_{\mathbf {a}}(x,r)\geqslant \varepsilon _0>0$ for any point $x\in \partial ^*\Omega _\gamma$, as long as r is taken sufficiently small. From here we will conclude the boundedness of the minimizer $\Omega _\gamma$ of $\mathcal {E}_\gamma$.

Now, in view of the behavior of the density at infinity, $\lim _{|x|\rightarrow \infty }\mathbf {a}(x)=+\infty$, the constant $R_{\mathbf {a}}\geqslant 1$ in condition (1.4) can be chosen large enough so that $\mathbb {R}^d\backslash B_{R_{\mathbf {a}}}\subset \{x\in \mathbb {R}^d:\mathbf {a}(x)\geqslant 1\}$. If $\Omega _\gamma \subset B_{2R_{\mathbf {a}}}$, then the minimizer is essentially bounded and we are done. Therefore, in the following we may assume that the total variation measure is nonvanishing in $\mathbb {R}^d\backslash B_{R_{\mathbf {a}}}$. We may also fix a constant $t_0\in (0,1)$ and balls $B^{1}$ and $B^{2}$, each of radius $t_0$, for which

$$\begin{aligned} 3B^{1}\cap B^{2}=\emptyset ,\quad 3B^{1}\cup B^{2}\subset \!\subset \mathbb {R}^d\backslash B_{R_{\mathbf {a}}} \quad \text { and }\quad \int _{B^i}\mathbf {a}(x)|\nabla \chi _{\Omega _\gamma }|>0\;\text { for }\;i=1,2.\nonumber \\ \end{aligned}$$

(3.10)

In what follows, $B^1$ and $B^2$ are to be used as reference sets, inside of which we will perform small deformations of our minimizer $\Omega _\gamma$ in order to create a competitor set F with $|F|=|\Omega _\gamma |$ (see Lemma 3.2 above with D being $B^1$ or $B^2$, by analyzing two cases). This will allow us to exploit the volume constrained quasi-minimality of $\Omega _\gamma$ with respect to the $\mathbf {a}$-perimeter, to derive a delicate growth estimate for the weighted relative density function of $\Omega _\gamma$ as a function of the radius r, which will ultimately justify the uniform lower bound on $h_{\mathbf {a}}$ that was claimed above.

Before we continue, let us remark that for densities $\mathbf {a}$ satisfying the assumptions (A1) and (A2a) (hence, in particular, the condition (1.4)), the $\mathbf {a}$-volume of any two sets $F_1, F_2\subset \!\subset B_{2{\bar{R}}}\backslash B_{{\bar{R}}}$ are uniformly comparable:

$$\begin{aligned} C_{\mathbf {a}}^{-1}\;\frac{|F_1|}{|F_2|}\leqslant \frac{\inf _{F_1}\mathbf {a}}{\sup _{F_2}\mathbf {a}}\,\frac{|F_1|}{|F_2|}\leqslant \frac{|F_1|_{\mathbf {a}}}{|F_2|_{\mathbf {a}}}\leqslant \frac{\sup _{F_1}\mathbf {a}}{\inf _{F_2}\mathbf {a}}\,\frac{|F_1|}{|F_2|} \leqslant C_{\mathbf {a}}\;\frac{|F_1|}{|F_2|} \end{aligned}$$

(3.11)

for any ${\bar{R}}>R_{\mathbf {a}}$. In particular, for any set $F\subset \!\subset \mathbb {R}^d\backslash B_{R_{\mathbf {a}}}$,

$$\begin{aligned} |F|=\int _{F}1\,\mathrm {d}x\leqslant \int _{F}\mathbf {a}(x)\,\mathrm {d}x= |F|_{\mathbf {a}} \end{aligned}$$

(3.12)

These facts will be used in the following technical results.

The first lemma establishes a bound on the growth rate of the weighted relative density function, for any minimizer $\Omega _\gamma$ of $\mathcal {E}_\gamma$, as a function of the radius r on balls having small $\mathbf {a}$-volume, provided that the set $\Omega _\gamma$ or its complement $\mathbb {R}^d\backslash \Omega _\gamma$ have small density on that ball.

Lemma 3.4

For every $\gamma >0$ there exist $0<\varepsilon '<1$ with $\max \{1,\gamma ^d\}\varepsilon ' \ll 1$, and $0<t_0<1$, such that, for any minimizer $\Omega _\gamma$ of $\mathcal {E}_\gamma$ and for any ball $B_r(x)\subset \!\subset \mathbb {R}^d\backslash B_{R_{\mathbf {a}}}$ with $0<r<t_0$ and $R_{\mathbf {a}}$ as in condition (1.4), there holds: If $h_{\mathbf {a}}(x,r)<\varepsilon '\,\text { and }\,0<|B_r(x)|_{\mathbf {a}}\leqslant 1$, then

$$\begin{aligned} h_{\mathbf {a}}\left( x,\frac{r}{2}\right) \leqslant \frac{1}{2}h_{\mathbf {a}}(x,r). \end{aligned}$$

Proof

Let $x\in \mathbb {R}^d$ and $0<r<t_0$. Loosely speaking, the main strategy is the following. If $h_{\mathbf {a}}(x,r)=|\Omega _\gamma \cap B_r(x)|_{\mathbf {a}}/|B_r(x)|_{\mathbf {a}}$ is small, we would like to delete the portion of $\Omega _\gamma$ that is inside of $B_{t_*}(x)$ for some $t_*\in (r/2,r)$ appropriately chosen. In a similar fashion, if $h_{\mathbf {a}}(x,r)=|B_r(x)\backslash \Omega _\gamma |_{\mathbf {a}}/|B_r(x)|_{\mathbf {a}}$ is small, we would like to append the ball $B_{t_*}(x)$ to $\Omega _\gamma$. In these two cases, the resulting set has an additional portion of its boundary located inside $\overline{B}_{t_*}(x)$, when compared to $\Omega _\gamma \cap B_r(x)$ or $B_r(x)\backslash \Omega _\gamma$. As $h_{\mathbf {a}}(x,r)$ is assumed small, we lose in the volume term in (3.4) less than what we are adding on the boundary (F in (3.4) is our resulting set). We must then analyze the contribution of the boundary term in $\overline{B}_{t_*}(x)$. We choose $t_{*}$ in such a way that we can control $\mathcal {H}^{d-1}_{\mathbf {a}}(\Omega _\gamma \cap \partial B_{t_*}(x))$ in the first case, and $\mathcal {H}^{d-1}_{\mathbf {a}}(\partial B_{t_*}(x)\backslash \Omega _\gamma )$ on the second case, in terms of $|B_r(x)|_{\mathbf {a}}^{(d-1)/d}\,h_{\mathbf {a}}(x,r)$.

We distinguish four cases.

Case 1: Assume that $h_{\mathbf {a}}(x,r)=|\Omega _\gamma \cap B_r(x)|_{\mathbf {a}}/|B_r(x)|_{\mathbf {a}}$ and $B_r(x)\cap B^{1}=\emptyset$, where $B^1$ denotes the fixed ball in (3.10). By Fubini and Chebyshev inequalities one can find $t_*\in (r/2,r)$ and $C=C(d)>0$ such that

$$\begin{aligned} \mathcal {H}^{d-1}_{\mathbf {a}}(\Omega _\gamma \cap \partial B_{t_*}(x))\leqslant C\, \frac{ (\sup _{B_r(x)}\mathbf {a})^{1/d}}{|B_r(x)|_{\mathbf {a}}^{1/d}}\;|\Omega _\gamma \cap B_r(x)|_{\mathbf {a}}, \end{aligned}$$

(3.13)

Indeed, using Chebyshev’s inequality with $M_{\theta } :=\big (\theta \,|B_{r}(x)|_{\mathbf {a}}^{1/d}\big )^{-1}|\Omega _\gamma \cap B_r(x)|_{\mathbf {a}}$, for $\theta >0$, yields

$$\begin{aligned} \Big |\{t\in (r/2,r) :\mathcal {H}^{d-1}_{\mathbf {a}}(\Omega _\gamma \cap \partial B_t(x))>M_{\theta }\}\Big |&\leqslant \frac{1}{M_{\theta }}\int ^r_{r/2}\mathcal {H}^{d-1}_{\mathbf {a}}(\Omega _\gamma \cap \partial B_t(x)) \,\mathrm {d}t \\&\leqslant \frac{1}{M_{\theta }}|\Omega _\gamma \cap (B_{r}(x)\backslash B_{r/2}(x))|_{\mathbf {a}} \\&\leqslant \theta \,|B_r(x)|_{\mathbf {a}}^{1/d} \leqslant \theta \,\biggl (\sup _{B_r(x)}\mathbf {a}\biggr )^{1/d} \omega ^{1/d}_d\,r< \frac{r}{2} \end{aligned}$$

for $\theta :=(4^d\omega _d\sup _{B_r(x)}\mathbf {a})^{-1/d}$. On the other hand, note that

$$\begin{aligned} \int _{\mathbb {R}^d}\mathbf {a}(y)|\nabla \chi _{\Omega _\gamma \backslash B_{t_*}(x)}| =\int _{\mathbb {R}^d\backslash B_{t_*}(x)}\mathbf {a}(y)|\nabla \chi _{\Omega _\gamma }| +\mathcal {H}^{d-1}_{\mathbf {a}}(\Omega _\gamma \cap \partial B_{t_*}(x)). \end{aligned}$$

With $D=B^{1}$, as defined in (3.10), and $L=\Omega _\gamma$, we apply Lemma 3.2 to obtain values $\varepsilon$ and $Q_{\mathbf {a}}$ satisfying the conclusions of that lemma. The constants $\varepsilon$ and $Q_{\mathbf {a}}$ depend on $\Omega _\gamma \cap B^{1}$ and $B^{1}$ only, so, in particular, they are independent of x and r (and $t_*$). Fix $\varepsilon '\in (0,\varepsilon )$ for the time being, and choose $r\in (0,t_0)$ sufficiently enough so that $h_{\mathbf {a}}(x,r)<\varepsilon '$ and $|B_r(x)|<\varepsilon$. Later on $\varepsilon '$ will be reduced accordingly (independent of r).

Let $\Omega ' :=\Omega _\gamma \backslash B_{t_*}(x)$ and observe, by hypothesis, that $\Omega '\cap B^{1}=\Omega _\gamma \cap B^{1}$. This means that the same constants $\varepsilon$ and $Q_{\mathbf {a}}$ still work for Lemma 3.2 when applying it with $L=\Omega '$ and $D=B^{1}$. Applying Lemma 3.2 to $L=\Omega '$, $D=B^{1}$ and $v=|\Omega _\gamma \cap B_{t_*}(x)|$ (note that $|v|<|B_r(x)|<\varepsilon$), we find a set $F\subset \mathbb {R}^d$ such that $F=\Omega '$ in a neighborhood of $\mathbb {R}^d\backslash B^{1}$, $|F|=|\Omega '|+|\Omega _\gamma \cap B_{t_*}(x)|=1$ and for which

$$\begin{aligned} \begin{aligned} \int _{B^{1}}\mathbf {a}(y)|\nabla \chi _{F}| \leqslant \int _{B^{1}}\mathbf {a}(y)|\nabla \chi _{\Omega '}|+Q_{\mathbf {a}}|\Omega _\gamma \cap B_{t_*}(x)| \\ \text { and } \quad |F\triangle \Omega '| \leqslant Q_{\mathbf {a}}|\Omega _\gamma \cap B_{t_*}(x) |, \end{aligned} \end{aligned}$$

(3.14)

where $Q_{\mathbf {a}}\lesssim \big ( \sup _{B^{1}}\mathbf {a}\big )\big (1+[\mathbf {a}]_{1,B^{1}}\big )$ is fixed and independent of x and r.

In light of (3.12), we have $|\Omega _\gamma \cap B_{t_*}(x)| \leqslant |\Omega _\gamma \cap B_{t_*}(x)|_{\mathbf {a}}$, and in addition, using $h_{\mathbf {a}}(x,r)<\varepsilon '$ and $|B_r(x)|_{\mathbf {a}}\leqslant 1$, we deduce that

$$\begin{aligned} |\Omega _\gamma \cap B_{t_*}(x)|_{\mathbf {a}}\leqslant & {} |\Omega _\gamma \cap B_{r}(x)|_{\mathbf {a}}\nonumber \\= & {} |B_r(x)|_{\mathbf {a}}\;h_{\mathbf {a}}(x,r)\leqslant |B_r(x)|_{\mathbf {a}}^{(d-1)/d}\;h_{\mathbf {a}}(x,r)< \varepsilon '. \end{aligned}$$

(3.15)

Since $F=\Omega '$ in a neighborhood of $\mathbb {R}^d\backslash B^{1}$, we have

$$\begin{aligned} \int _{\mathbb {R}^d\backslash B^{1}}\mathbf {a}(y)|\nabla \chi _F|=\int _{\mathbb {R}^d\backslash B^{1}} \mathbf {a}(y)|\nabla \chi _{\Omega '}|. \end{aligned}$$

Combining these facts, we obtain

$$\begin{aligned} \int _{\mathbb {R}^d}\mathbf {a}(y)|\nabla \chi _{F}|&\,\,\, \leqslant \int _{\mathbb {R}^d}\mathbf {a}(y)|\nabla \chi _{\Omega '}|+Q_{\mathbf {a}} |\Omega _\gamma \cap B_{t_*}(x)|_{\mathbf {a}}\\&{\mathop {\leqslant }\limits ^{{(3.15)}}} \int _{\mathbb {R}^d}\mathbf {a}(y)|\nabla \chi _{\Omega '}| +Q_{\mathbf {a}}|B_r(x)|_{\mathbf {a}}^{(d-1)/d}\,h_{\mathbf {a}}(x,r). \end{aligned}$$

In addition,

$$\begin{aligned} \int _{\mathbb {R}^d}\mathbf {a}(y)|\nabla \chi _{\Omega '}|&\,\,\, =\int _{\mathbb {R}^d\backslash B_{t_*}(x)}\mathbf {a}(y)|\nabla \chi _{\Omega _\gamma }| +\mathcal {H}^{d-1}_{\mathbf {a}}(\Omega _\gamma \cap \partial B_{t_*}(x))\\&{\mathop {\leqslant }\limits ^{{(3.13)}}} \int _{\mathbb {R}^d\backslash B_{t_*}(x)}\mathbf {a}(y)|\nabla \chi _{\Omega _\gamma }|+C\,\frac{(\sup _{B_r(x)}\mathbf {a})^{1/d}}{|B_r(x)|_{\mathbf {a}}^{1/d}}\;|\Omega _\gamma \cap B_r(x)|_{\mathbf {a}}\\&{\mathop {\leqslant }\limits ^{{(3.15)}}} \int _{\mathbb {R}^d\backslash B_{t_*}(x)}\mathbf {a}(y)|\nabla \chi _{\Omega _\gamma }|+C\,(\sup _{B_r(x)}\mathbf {a})^{1/d}|B_r(x)|_{\mathbf {a}}^{(d-1)/d}\;h_{\mathbf {a}}(x,r). \end{aligned}$$

Assuming, without loss of generality, that $\sup _{B_r(x)}\mathbf {a}\gg 1$, the above estimate yields

$$\begin{aligned} \int _{\mathbb {R}^d}\mathbf {a}(y)|\nabla \chi _{F}| \leqslant \int _{\mathbb {R}^d\backslash B_{t_*}(x)}\mathbf {a}(y)|\nabla \chi _{\Omega _\gamma }| +C \big (\sup _{B_r(x)}\mathbf {a}\big )^{1/d}|B_r(x)|_{\mathbf {a}}^{(d-1)/d}\;h_{\mathbf {a}}(x,r) \end{aligned}$$

(3.16)

On the other hand, recalling (3.12), (3.14), and (3.15), we have

$$\begin{aligned} |F\triangle \Omega _\gamma |&\leqslant |F\triangle \Omega '|+|\Omega '\triangle \Omega _\gamma | \leqslant (1+Q_{\mathbf {a}})|\Omega _\gamma \cap B_{t_*}(x)| \\&\leqslant (1+Q_{\mathbf {a}})|\Omega _\gamma \cap B_{t_*}(x)|_{\mathbf {a}} \leqslant (1+Q_{\mathbf {a}})|B_r(x)|_{\mathbf {a}}\,h_{\mathbf {a}}(x,r)\leqslant C\,\varepsilon '. \end{aligned}$$

Recall now that $\Omega _\gamma$ is a volume constrained quasi-minimal set for $\mathbf {a}$-perimeter with $g_{\gamma }(x)\simeq \gamma \,x=o(x^{(d-1)/d})$; see (3.4). By reducing the value of $\varepsilon '$ in such a way that $\max \{1,\gamma \}{\varepsilon '}^{\frac{1}{d}} \ll 1$, we obtain

$$\begin{aligned} g_{\gamma }(|F\triangle \Omega _\gamma |)\leqslant \eta |B_r(x)|_{\mathbf {a}}^{\frac{d-1}{d}}h_{\mathbf {a}}(x,r)^{\frac{d-1}{d}}, \end{aligned}$$

where $\eta$ will be chosen below (independent of $\varepsilon '$). Since $\Omega _\gamma$ is a volume constrained quasi-minimizer for the $\mathbf {a}$-perimeter, the above inequality and (3.16) yield the following estimate:

$$\begin{aligned} \int _{B_{t_*}(x)}\mathbf {a}(y)|\nabla \chi _{\Omega _\gamma }|\leqslant \biggl (C \big ( \sup _{B_r(x)}\mathbf {a}\big )^{1/d}\,h_{\mathbf {a}}(x,r)+\eta \, h_{\mathbf {a}}(x,r)^{\frac{d-1}{d}}\biggr )|B_r(x)|_{\mathbf {a}}^{\frac{d-1}{d}}. \end{aligned}$$

(3.17)

In view of (3.17), using the standard isoperimetric inequality on balls, we obtain that

$$\begin{aligned}&|B_{r/2}(x)|_{\mathbf {a}}\, h_{\mathbf {a}}\left( x,r/2\right) \\&\quad =\min \left\{ |\Omega _\gamma \cap B_{r/2}(x)|_{\mathbf {a}},|B_{r/2}(x)\backslash \Omega _\gamma |_{\mathbf {a}}\right\} \\&\quad \leqslant \sup _{B_{r/2}(x)}\mathbf {a}\cdot \min \left\{ |\Omega _\gamma \cap B_{r/2}(x)|,|B_{r/2}(x)\backslash \Omega _\gamma |\right\} \\&\quad {\mathop {\leqslant }\limits ^{{(1.2)}}}\big (C_{\mathbf {a}}\inf _{B_{r/2}(x)}\mathbf {a}\big )\cdot \; C\left( \int _{B_{r/2}(x)}|\nabla \chi _{\Omega _\gamma }|\right) ^{\frac{d}{d-1}}\\&\quad \leqslant C\,C_{\mathbf {a}}\big (\inf _{B_{r/2}(x)}\mathbf {a}\big )^{-\frac{1}{d-1}}\left( \int _{B_{t_*}(x)}\mathbf {a}(y)|\nabla \chi _{\Omega _\gamma }|\right) ^{\frac{d}{d-1}}\\&\quad \leqslant C\,C_{\mathbf {a}}\;(\inf _{B_{r/2}(x)}\mathbf {a})^{-\frac{1}{d-1}}\biggl (C\big (\sup _{B_r(x)}\mathbf {a}\big )^{1/d}\,h_{\mathbf {a}}(x,r)+\eta \, h_{\mathbf {a}}(x,r)^{\frac{d-1}{d}}\biggr )^{\frac{d}{d-1}}|B_r(x)|_{\mathbf {a}}\\&\quad \leqslant C\,C_{\mathbf {a}}\;(\inf _{B_{r/2}(x)}\mathbf {a})^{-\frac{1}{d-1}}\biggl (\big (\sup _{B_r(x)}\mathbf {a}\big )^{\frac{1}{d-1}}\,h_{\mathbf {a}}(x,r)^{\frac{1}{d-1}}+\eta ^{\frac{d}{d-1}}\biggr )h_{\mathbf {a}}(x,r)|B_r(x)|_{\mathbf {a}}\\&\quad \leqslant C\,C_{\mathbf {a}}\;\left( \frac{\sup _{B_{r}(x)}\mathbf {a}}{\inf _{B_{r/2}(x)}\mathbf {a}}\right) ^{\frac{1}{d-1}}\biggl ({\varepsilon '}^{\frac{1}{d-1}}+\eta ^{\frac{d}{d-1}}\biggr )\;h_{\mathbf {a}}(x,r)|B_r(x)|_{\mathbf {a}}\\&\quad {\mathop {\leqslant }\limits ^{{(3.11)}}} C\,C_{\mathbf {a}}^2\;\biggl ({\varepsilon '}^{\frac{1}{d-1}}+\eta ^{\frac{d}{d-1}}\biggr )\;h_{\mathbf {a}}(x,r)\big (2^d\,C_{\mathbf {a}}\,|B_{r/2}(x)|_{\mathbf {a}}\big ). \end{aligned}$$

In other words, we conclude that

$$\begin{aligned} h_{\mathbf {a}}\left( x,\frac{r}{2}\right) \leqslant C\;C_{\mathbf {a}}^3\;\Big ({\varepsilon '}^{\frac{1}{d-1}}+\eta ^{\frac{d}{d-1}}\Big )\;h_{\mathbf {a}}(x,r) \leqslant \frac{1}{2}h_{\mathbf {a}}(x,r), \end{aligned}$$

provided $0<\eta <(4CC_{\mathbf {a}}^3)^{-\frac{d-1}{d}}$ is chosen small enough; and possibly reducing the value of $\varepsilon '\lesssim \min \{\varepsilon ,(4CC_{\mathbf {a}}^3)^{-(d-1)}\}$, independently of x and r.

Case 2: Suppose $h_{\mathbf {a}}(x,r)=|\Omega _\gamma \cap B_r(x)|_{\mathbf {a}}/|B_r(x)|_{\mathbf {a}}$ and $B_r(x)\cap B^{1}\ne \emptyset$. Under the hypotheses, $B_{r}(x)\subset 3B^{1}$, and (3.10) ensures that $B_r(x)\cap B^{2}=\emptyset$. We proceed exactly as in the argument in the first case, with $B^{1}$ replaced by $B^{2}$.

Case 3: Assume that $h_{\mathbf {a}}(x,r)=|B_r(x)\backslash \Omega _\gamma |_{\mathbf {a}}/|B_r(x)|_{\mathbf {a}}$ and $B_r(x)\cap B^{1}=\emptyset$. We will proceed analogously as the first case, but with

$$\begin{aligned} \Omega ' :=\Omega _\gamma \cup B_{t_*}(x) \end{aligned}$$

for a certain $t_{*}$ to be determined below. Again, by Fubini and Chebyshev inequalities, we can find $t_{*}\in (r/2,r)$ in such a way that

$$\begin{aligned} \mathcal {H}^{d-1}_{\mathbf {a}}(\partial B_{t_*}(x)\backslash \Omega _\gamma ) \leqslant C\, \frac{(\sup _{B_r(x)}\mathbf {a})^{1/d}}{|B_r(x)|_{\mathbf {a}}^{1/d}}|B_r(x)\backslash \Omega _\gamma |_{\mathbf {a}}. \end{aligned}$$

(3.18)

Also, we have that

$$\begin{aligned} \int _{\mathbb {R}^d}\mathbf {a}(y)|\nabla \chi _{\Omega _\gamma \cup B_{t_*}(x)}|=\int _{\mathbb {R}^d\backslash B_{t_*}(x)}\mathbf {a}(y)|\nabla \chi _{\Omega _\gamma }|+\mathcal {H}^{d-1}_{\mathbf {a}}(\partial B_{t_*}(x)\backslash \Omega _\gamma ). \end{aligned}$$

(3.19)

Fix now $\varepsilon '$ as in the previous cases so that $h_{\mathbf {a}}(x,r)<\varepsilon '<\varepsilon$ and $|B_r(x)|<\varepsilon$. Note that (3.12) yields $|\Omega '\backslash \Omega _\gamma |\leqslant |\Omega '\backslash \Omega _\gamma |_{\mathbf {a}}$. Also, recalling that $|B_r(x)|_{\mathbf {a}}\leqslant 1$, we have

$$\begin{aligned} |\Omega '\backslash \Omega _\gamma |_{\mathbf {a}}&\leqslant |B_{t_*}(x)\backslash \Omega _\gamma |_{\mathbf {a}}\leqslant |B_{r}(x)\backslash \Omega _\gamma |_{\mathbf {a}}\\&\leqslant |B_r(x)|_{\mathbf {a}}\;h_{\mathbf {a}}(x,r)\leqslant |B_r(x)|_{\mathbf {a}}^{(d-1)/d}\;h_{\mathbf {a}}(x,r)< \varepsilon '. \end{aligned}$$

Because of the choice of $\varepsilon '$, we can apply Lemma 3.2 to $L=\Omega '$, $D=B^{1}$ and $v=-|\Omega '\backslash \Omega _\gamma |$ (with $|v|\leqslant |B_r(x)|<\varepsilon$) to obtain the existence of a set $F\subset \mathbb {R}^d$ such that $F=\Omega '$ in a neighborhood of $\mathbb {R}^d\backslash B^{1}$ and $|F|=|\Omega '|-|\Omega '\backslash \Omega _\gamma |=1$. Moreover, F satisfies the estimates

$$\begin{aligned} \begin{aligned} \int _{B^{1}}\mathbf {a}(y)|\nabla \chi _F|&\leqslant \int _{B^{1}}\mathbf {a}(y)|\nabla \chi _{\Omega '}|+Q_{\mathbf {a}}|\Omega '\backslash \Omega _\gamma |\\&\leqslant \int _{B^{1}}\mathbf {a}(y)|\nabla \chi _{\Omega '}|+Q_{\mathbf {a}}\,|B_r(x)|_{\mathbf {a}}^{(d-1)/d}\;h_{\mathbf {a}}(x,r), \end{aligned} \end{aligned}$$

and

$$\begin{aligned} |F\triangle \Omega '| \leqslant Q_{\mathbf {a}}\,|\Omega '\backslash \Omega _\gamma |, \end{aligned}$$

where $Q_{\mathbf {a}}\lesssim \big (\sup _{B^{1}}\mathbf {a}\big ) \big (1+[\mathbf {a}]_{1,B^{1}}\big )$ is fixed, and independent of x and r.

Then, just like in the first case, we deduce

$$\begin{aligned} \int _{\mathbb {R}^d}\mathbf {a}(y)|\nabla \chi _F|\leqslant \int _{\mathbb {R}^d}\mathbf {a}(y)|\nabla \chi _{\Omega '}| +C|B_r(x)|_{\mathbf {a}}^{(d-1)/d}\;h_{\mathbf {a}}(x,r). \end{aligned}$$

(3.20)

Thus, utilizing (3.19) and (3.18), we can estimate

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^d}\mathbf {a}(y)|\nabla \chi _{\Omega '}|&\leqslant \int _{\mathbb {R}^d\backslash B_{t_*}(x)}\mathbf {a}(y)|\nabla \chi _{\Omega _\gamma }|+ C\,\frac{(\sup _{B_r(x)}\mathbf {a})^{1/d}}{|B_r(x)|_{\mathbf {a}}^{1/d}}|B_r(x)\backslash \Omega _\gamma |_{\mathbf {a}}\\&\leqslant \int _{\mathbb {R}^d\backslash B_{t_*}(x)}\mathbf {a}(y)|\nabla \chi _{\Omega _\gamma }|+ C\,(\sup _{B_r(x)}\mathbf {a})^{1/d}|B_r(x)|_{\mathbf {a}}^{(d-1)/d}\,h_{\mathbf {a}}(x,r), \end{aligned} \end{aligned}$$

which can be combined with (3.20), in turn, to obtain

$$\begin{aligned} \int _{\mathbb {R}^d}\mathbf {a}(y)|\nabla \chi _F|\leqslant \int _{\mathbb {R}^d\backslash B_{t_*}(x)}\mathbf {a}(y)|\nabla \chi _{\Omega _\gamma }| +C\,(\sup _{B_r(x)}\mathbf {a})^{1/d}\,|B_r(x)|_{\mathbf {a}}^{(d-1)/d}\;h_{\mathbf {a}}(x,r). \end{aligned}$$

On the other hand,

$$\begin{aligned} \begin{aligned} |F\triangle \Omega _\gamma |&\leqslant |F\triangle \Omega '|+|\Omega ' \triangle \Omega _\gamma |\\&\leqslant (Q_{\mathbf {a}}+1)|\Omega '\backslash \Omega _\gamma |\leqslant (Q_{\mathbf {a}}+1)|\Omega '\backslash \Omega _\gamma |_{\mathbf {a}}\\&\leqslant (Q_{\mathbf {a}}+1)\,|B_r(x)|_{\mathbf {a}}\;h_{\mathbf {a}}(x,r)\leqslant C\,\varepsilon '. \end{aligned} \end{aligned}$$

We reach the conclusion like in the first case, utilizing the volume constrained quasi-minimality of $\Omega _\gamma$, the relative isoperimetric inequality on balls, and the fact that the weight has uniformly bounded oscillation (condition (1.4)).

Case 4: Suppose $h_{\mathbf {a}}(x,r)=|B_r(x)\backslash \Omega _\gamma |_{\mathbf {a}}/|B_r(x)|_{\mathbf {a}}$ and $B_r(x)\cap B^{1}\ne \emptyset$. We employ the same construction as in the third case, with $B^{1}$ replaced by $B^{2}$, to conclude the proof of this lemma. $\square$

Lemma 3.5

Given $\gamma >0$, there exist $0<\varepsilon _0<1$ with $\max \{1,\gamma ^d\}\varepsilon _0 \ll 1$, and $0<t_0<1$, such that for any minimizer $\Omega _\gamma$ of $\mathcal {E}_\gamma$ and any ball $B_{r}(x)\subset \!\subset \mathbb {R}^d\backslash B_{R_{\mathbf {a}}}$, for which $0<r\leqslant t_0$, and for $R_{\mathbf {a}}$ as in (1.4), the following holds: If $h_{\mathbf {a}}(x,r)<\varepsilon _0\quad \text { and }\quad 0<|B_{r}(x)|_{\mathbf {a}}\leqslant 1$, then

$$\begin{aligned} \text { either }\;\; \left| \Omega _\gamma \cap B_{r/2}(x)\right| =0\;\;\text { or }\;\; \left| B_{r/2}(x)\backslash \Omega _\gamma \right| =0. \end{aligned}$$

Remark 3.6

We note that, in particular, for any point $x\in \partial ^*\Omega _\gamma$ and any ball $B_{r}(x)$ as above, one has

$$\begin{aligned} h_{\mathbf {a}}(x,r)=\frac{|\Omega _\gamma \cap B_r(x)|_{\mathbf {a}}}{|B_r(x)|_{\mathbf {a}}}\geqslant \varepsilon _0>0. \end{aligned}$$

Proof

Under the hypotheses of this lemma, for any $y\in B_{r/2}(x)$ we write $d_k :=h_{\mathbf {a}}(y,2^{-k}r)$. Choose

$$\begin{aligned} \varepsilon _0 :=(2^dC_{\mathbf {a}})^{-1}\min \{\varepsilon ',1/2\}, \end{aligned}$$

with $\varepsilon '>0$ given as in Lemma 3.4. If $h_{\mathbf {a}}(x,r)=|B_r(x)|_{\mathbf {a}}^{-1}\;|\Omega _\gamma \cap B_{r}(x)|_{\mathbf {a}}$, then

$$\begin{aligned} d_1 :={|B_{r/2}(x)|_{\mathbf {a}}}^{-1}|\Omega _\gamma \cap B_{r/2}(y)|_{\mathbf {a}}\leqslant 2^dC_{\mathbf {a}}\;h_{\mathbf {a}}(x,r)<\min \{\varepsilon ',1/2\} \end{aligned}$$

where we used the estimate (3.11). Since $d_1<\varepsilon '$, by Lemma 3.4 we deduce that

$$\begin{aligned} d_2=h_{\mathbf {a}}\left( y,\frac{r}{4}\right) \leqslant \frac{1}{2}h_{\mathbf {a}}\left( y,\frac{r}{2}\right) =\frac{d_1}{2}. \end{aligned}$$

In particular, $d_2<\min \{\varepsilon ',1/2\}$. Hence, by induction it follows that, for all $k\geqslant 1$, we have $d_{k+1}\leqslant d_1/2^k$.

Hence, for any $y\in B_{r/2}(x)$, and by virtue of condition (1.4), for all $k \geqslant 1$, we have

$$\begin{aligned} \frac{\varepsilon '}{2^{k-1}} \geqslant \frac{|\Omega _\gamma \cap B_{2^{-k}r}(y)|_{\mathbf {a}}}{|B_{2^{-k}r}(y)|_{\mathbf {a}}} \geqslant \frac{\inf _{B_r(y)}\mathbf {a}}{\sup _{B_r(y)}\mathbf {a}} \, \frac{|\Omega _\gamma \cap B_{2^{-k}r}(y)|}{|B_{2^{-k}r}(y)|}\geqslant C_{\mathbf {a}}^{-1}\,\frac{|\Omega _\gamma \cap B_{2^{-k}r}(y)|}{|B_{2^{-k}r}(y)|} \end{aligned}$$

Thus, y is not a point of density of $\Omega _\gamma$, which yields $|\Omega _\gamma \cap B_{r/2}(x)|=0$.

Otherwise, if $h_{\mathbf {a}}(x,r)=r^{-d}|B_{r}(x)\backslash \Omega _\gamma |_{\mathbf {a}}$, it can be shown in the same fashion that, for any $k\geqslant 1$ and any $y\in B_{r/2}(x)$,

$$\begin{aligned} d_{k}=|B_{2^{-k}r}(y)|_{\mathbf {a}}^{-1}|B_{r}(y)\backslash \Omega _\gamma |_{\mathbf {a}}<2^d\,C_{\mathbf {a}}\;\varepsilon <\min \{\varepsilon ', 1/2\}\quad \text { and }\quad d_{k+1}\leqslant \frac{d_k}{2}. \end{aligned}$$

Therefore,

$$\begin{aligned} C_{\mathbf {a}}^{-1}\cdot \frac{|B_{2^{-k}r}(y)\backslash \Omega _\gamma |}{|B_{2^{-k}r}(y)|}\leqslant \frac{|B_{2^{-k}r}(y)\backslash \Omega _\gamma |_{\mathbf {a}}}{|B_{2^{-k}r}(y)|_{\mathbf {a}}} \leqslant \frac{\varepsilon '}{2^{k-1}}\underset{k\rightarrow \infty }{\longrightarrow } 0. \end{aligned}$$

and so y is not a density point of $\mathbb {R}^d\backslash \Omega _\gamma$. This shows that $|B_{r/2}(x)\backslash \Omega _\gamma |=0$. $\square$

Proof of Theorem 1.3 (Part 1)

We now finish the argument to establish the essential boundedness of the minimizer $\Omega _\gamma$ in Theorem 1.3 under the hypotheses (A1) and (A2a). Let $\varepsilon _0$ and $t_0>0$ be given as in Lemma 3.5, and let $j_{\mathbf {a}}\in \mathbb {N}$ be such that $R_{\mathbf {a}}<2^{j_{\mathbf {a}}}$.

Let $\{R_1,\ldots , R_N\}\subset (2^{j_{\mathbf {a}}},+\infty )$ be any finite collection for which

$$\begin{aligned} \partial ^*\Omega _\gamma \cap (B_{R_{i+1}}\backslash {\overline{B}}_{R_i})\ne \emptyset \;\;\text { and }\;\; R_{i+1}-R_i\geqslant 1,\quad \text { for all }i. \end{aligned}$$

Then we can select balls $B_{r_i}(x^i)\subset \!\subset B_{R_{i+1}}\backslash B_{R_i}$ with $x^i\in \partial ^*\Omega _\gamma$ and $r_i\leqslant t_0< 1$ taken suffieciently small so that

$$\begin{aligned} \frac{1}{2}\leqslant |B_{r_i}(x^i)|_{\mathbf {a}}< 1. \end{aligned}$$

In view of (3.2), and employing Remark 3.6 over each ball, we derive the following bound

$$\begin{aligned} \begin{aligned} c_d\,C_{\mathbf {a}}\int _{\mathbb {R}^d\backslash B_{2^{j_{\mathbf {a}}}}}\mathbf {a}(x)|\nabla \chi _{\Omega _\gamma }|&\geqslant |\Omega _\gamma \backslash B_{2^{j_{\mathbf {a}}}}|_{\mathbf {a}}\geqslant \sum ^N_{i=1}|\Omega _\gamma \cap (B_{R_{i+1}}\backslash B_{R_i})|_{\mathbf {a}}\\&\geqslant \sum ^N_{i=1}|\Omega _\gamma \cap B_{r_i}(x^i)|_{\mathbf {a}}\geqslant \sum ^N_{i=1}\varepsilon _0\,|B_{r_i}(x^i)|_{\mathbf {a}}\\&\geqslant \tfrac{1}{2}\varepsilon _0\, N \end{aligned} \end{aligned}$$

thus proving that the maximal number of such radii $\{R_i\}$ must be finite. This shows that there exists $R_\mathrm{max}\in (R_{\mathbf {a}},+\infty )$, depending on $\mathbf {a}$, for which $\partial ^*\Omega _\gamma \cap (\mathbb {R}^d\backslash B_{R_\mathrm{max}})=\emptyset$. Since $\Omega _\gamma$ is assumed to be essentially closed, then

$$\begin{aligned} \Omega _\gamma \subset B_{R_{\max }}. \end{aligned}$$

This establishes Theorem 1.3 under the hypotheses (A1) and $\square$ (A2a).

3.2 Radial and monotone densities

Now we turn to densities $\mathbf {a}$ satisfying the assumptions (A1) and (A2b). A key ingredient in the second part of the proof of Theorem 1.3 is the $\varepsilon -\varepsilon ^{(d-1)/d}$ property recently proved by Pratelli and Saracco [41]. We state this result as a lemma here for the convenience of the reader.

Lemma 3.7

(cf. Theorem A in [41]) For any set of finite perimeter $E\subset \mathbb {R}^d$ and for any ball B such that $\mathcal {H}^{d-1}(B \cap \partial ^*E)>0$ there exists $\overline{\varepsilon }>0$ and $C_E>0$ such that for any $|\varepsilon |<\overline{\varepsilon }$ there is a set $F\subset \mathbb {R}^d$ satisfying

$$\begin{aligned} F \triangle E \subset \!\subset B, \qquad |F|-|E| = \varepsilon , \qquad {\mathcal {P}_{\mathbf {a}}}(F)-{\mathcal {P}_{\mathbf {a}}}(E) \leqslant C_E|\varepsilon |^{(d-1)/d}. \end{aligned}$$

(3.21)

Moreover the constant $C_E$ can be chosen arbitrarily small up to possibly choosing $\overline{\varepsilon }$ smaller.

Now we will finish the proof of Theorem 1.3 following an argument similar to [37, Theorem 5.9] and [32, Lemma 5.1].

Proof of Theorem 1.3 (Part 2)

Assume, for a contradiction, that $\Omega _\gamma$ is not bounded. Then $|\Omega _\gamma \backslash B_r|>0$ for all $r>0$. In particular, for $r \geqslant R_\mathbf {a}$,

$$\begin{aligned} |\Omega _\gamma \backslash B_r|_{\mathbf {a}} \geqslant \mathbf {a}(R_{\mathbf {a}})|\Omega _\gamma \backslash B_r| >0. \end{aligned}$$

Now let

$$\begin{aligned} \Omega (r) :=\Omega _\gamma \cap B_r, \quad \Omega _r :=\Omega _\gamma \cap \partial B_r, \quad {\mathcal {P}_{\mathbf {a}}}(r) :=\mathcal {H}^{d-1}_{\mathbf {a}}(\partial ^* \Omega _\gamma \backslash B_r), \text { and } \ V_\mathbf {a}(r) :=|\Omega _\gamma \backslash B_r|_\mathbf {a}. \end{aligned}$$

By the proof of Theorem 4.3 of [37],

$$\begin{aligned} {\mathcal {P}_{\mathbf {a}}}(\Omega (r)) < {\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ) \quad \text { for }r \geqslant R_\mathbf {a}. \end{aligned}$$

This follows from a nifty argument involving projection onto the sphere $B_r$ and for the reader’s convenience we include the details here. Consider the projection map $\Pi :\partial ^*\Omega _\gamma \backslash B_r \rightarrow \partial B_r$. Clearly, $\Pi$ is strictly 1-Lipschitz and its image ${{\,\mathrm{Im}\,}}(\Pi )$ satisfies

$$\begin{aligned} \partial ^* \Omega (r) \backslash \partial ^*\Omega _\gamma \subset {{\,\mathrm{Im}\,}}(\Pi ). \end{aligned}$$

This inclusion would be trivially true for bounded $\Omega _\gamma$; however, since we assumed that $\Omega _\gamma$ is unbounded it contains the whole cone $C=\{\lambda x :\lambda \geqslant 1,\ x\in H\}$ where $H=\left( \partial ^*\Omega (r)\backslash \partial ^*\Omega _\gamma \right) \backslash {{\,\mathrm{Im}\,}}(\Pi )$. But since $\mathbf {a}$ is increasing for large values of |x|, the cone C, and hence the set $\Omega _\gamma$, have infinite volume, unless $\mathcal {H}^{d-1}(H)=0$. Thus $H=\emptyset$ up to $\mathcal {H}^{d-1}$-measure zero.

Since $\mathbf {a}$ is eventually increasing, for $r \geqslant R_\mathbf {a}$, by the co-area formula we get that

$$\begin{aligned} {\mathcal {P}_{\mathbf {a}}}(\Omega (r))&= \int _{\partial ^*\Omega _\gamma \cap B_r} \mathbf {a}(|x|) \,\mathrm {d}\mathcal {H}^{d-1}(x) + \int _{\partial ^*\Omega (r) \backslash \partial ^*\Omega _\gamma } \mathbf {a}(\theta )\,d\mathcal {H}^{d-1}(\theta ) \\&< \int _{\partial ^*\Omega _\gamma \cap B_r} \mathbf {a}(|x|) \,\mathrm {d}\mathcal {H}^{d-1}(x) + \int _{\partial ^*\Omega _\gamma \backslash \partial B_r} \mathbf {a}(\Pi (x)) \,\mathrm {d}\mathcal {H}^{d-1}(x) \\&\leqslant \int _{\partial ^*\Omega _\gamma } \mathbf {a}(|x|)\,\mathrm {d}\mathcal {H}^{d-1}(x) = {\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ). \end{aligned}$$

Also, note that

$$\begin{aligned} {\mathcal {P}_{\mathbf {a}}}(\Omega (r)) = {\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ) - {\mathcal {P}_{\mathbf {a}}}(r) + \mathcal {H}^{d-1}_{\mathbf {a}}(\Omega _r). \end{aligned}$$

This, combined with ${\mathcal {P}_{\mathbf {a}}}(\Omega (r)) < {\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma )$, implies that

$$\begin{aligned} {\mathcal {P}_{\mathbf {a}}}(r) > \mathcal {H}^{d-1}_{\mathbf {a}}(\Omega _r). \end{aligned}$$

(3.22)

For $r \geqslant 1$, the standard isoperimetric inequality in the sphere yields that for any subset $E_r$ of the sphere $\partial B_r$ having area at most half of the sphere, we have $\mathcal {H}^{d-2}(\partial ^* E_r) \geqslant c \big ( \mathcal {H}^{d-1}(E_r) \big )^{(d-2)/(d-1)}$ for some constant $c>0$ where $\partial ^* E_r$ denotes the reduced boundary of $E_r$ inside $\partial B_r$. In fact, as $\Omega _\gamma$ has finite volume, for r sufficiently large, $\mathcal {H}^{d-1}(\Omega _r) \leqslant \frac{1}{2} \mathcal {H}^{d-1}(\partial B_r)$. Since the density $\mathbf {a}$ is constant on $\partial B_r$, we get

$$\begin{aligned} \mathcal {H}^{d-2}_\mathbf {a}(\partial ^* \Omega _r) \geqslant c \Big (\mathcal {H}^{d-1}_{\mathbf {a}}(\Omega _r)\Big )^{\frac{d-2}{d-1}} \big (\mathbf {a}(r)\big )^{\frac{1}{d-1}}. \end{aligned}$$

(3.23)

Combining (3.22) and (3.23), we obtain

$$\begin{aligned} \big ({\mathcal {P}_{\mathbf {a}}}(r)\big )^{\frac{1}{d-1}}\, \mathcal {H}^{d-2}_\mathbf {a}(\partial ^* \Omega _r) > \big (\mathcal {H}^{d-1}_{\mathbf {a}}(\Omega _r)\big )^{\frac{1}{d-1}}\, \mathcal {H}^{d-2}_\mathbf {a}(\partial ^*\Omega _r) \geqslant c\, \mathcal {H}^{d-1}_{\mathbf {a}}(\Omega _r) \big (\mathbf {a}(r)\big )^{\frac{1}{d-1}}. \end{aligned}$$

Since $\mathbf {a}(r) \geqslant \mathbf {a}(R_\mathbf {a})$ for $r \geqslant R_\mathbf {a}$, the estimate above becomes

$$\begin{aligned} \mathcal {H}^{d-2}_\mathbf {a}(\partial ^* \Omega _r) \geqslant c\, \big ( {\mathcal {P}_{\mathbf {a}}}(r) \big )^{-\frac{1}{d-1}} \, \mathcal {H}^{d-1}_{\mathbf {a}}(\Omega _r). \end{aligned}$$

(3.24)

Also, observe that

$$\begin{aligned} -\frac{d}{dr}{\mathcal {P}_{\mathbf {a}}}(r) = \left| \frac{d}{dr}{\mathcal {P}_{\mathbf {a}}}(r)\right| \geqslant \mathcal {H}^{d-2}_\mathbf {a}(\partial ^*\Omega _r) \quad \text { and } \quad -\frac{d}{dr}V_\mathbf {a}(r) = \mathcal {H}^{d-1}_{\mathbf {a}}(\Omega _r). \end{aligned}$$

(3.25)

Hence, by (3.24), we get

$$\begin{aligned} -\frac{d}{dr}\Big ( \big ({\mathcal {P}_{\mathbf {a}}}(r)\big )^{\frac{d}{d-1}} \Big ) \geqslant - c\, \frac{d}{dr}V_\mathbf {a}(r). \end{aligned}$$

Since both ${\mathcal {P}_{\mathbf {a}}}(r)$, $V_\mathbf {a}(r) \rightarrow 0$ as $r\rightarrow \infty$, integrating both sides from r to $\infty$ yields

$$\begin{aligned} \big ( {\mathcal {P}_{\mathbf {a}}}(r) \big )^{\frac{d}{d-1}} \geqslant c\, V_\mathbf {a}(r). \end{aligned}$$

(3.26)

Now, let $R\in \mathbb {R}$ be such that $\Omega _\gamma \cap B_R \ne \emptyset$. Then by Lemma 3.7, there exists $\overline{\varepsilon }$ and $C_{\Omega _\gamma }>0$ such that for all $|\varepsilon |<\overline{\varepsilon }$ there exists $\Omega _\varepsilon$ satisfying

$$\begin{aligned} \Omega _\varepsilon \triangle \Omega _\gamma \subset \!\subset B_R, \qquad |\Omega _\varepsilon |-|\Omega _\gamma | = \varepsilon , \qquad {\mathcal {P}_{\mathbf {a}}}(\Omega _\varepsilon )-{\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ) \leqslant C_{\Omega _\gamma }|\varepsilon |^{(d-1)/d}.\nonumber \\ \end{aligned}$$

(3.27)

By choosing $\varepsilon$ smaller, if necessary, by Lemma 3.2 and Remark 3.3, we can take $\Omega _\varepsilon$ such that

$$\begin{aligned} |\Omega _\varepsilon \triangle \Omega _\gamma | < C\varepsilon . \end{aligned}$$

(3.28)

Let $r>R$ be such that $\varepsilon :=V_\mathbf {a}(r) < \overline{\varepsilon }$ and define $\Omega _\varepsilon (r) = \Omega _\varepsilon \cap B_r$. Then $|\Omega _\varepsilon (r)|=|\Omega _\gamma |=1$ and $\Omega _\varepsilon (r)$ is competitor in (1.1). So,

$$\begin{aligned} {\mathcal {P}_{\mathbf {a}}}(\Omega _\varepsilon (r))&= {\mathcal {P}_{\mathbf {a}}}(\Omega _\varepsilon ) - {\mathcal {P}_{\mathbf {a}}}(r) + \mathcal {H}^{d-1}_{\mathbf {a}}(\Omega _r) \\&{\mathop {\leqslant }\limits ^{{(3.26)},{(3.27)}}} {\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ) + C_{\Omega _\gamma } \varepsilon ^{\frac{d-1}{d}} - c\,\varepsilon ^{\frac{d-1}{d}} + \mathcal {H}^{d-1}_{\mathbf {a}}(\Omega _r) \\&\leqslant {\mathcal {P}_{\mathbf {a}}}(\Omega _\varepsilon (r)) + \gamma \, \bigg (\mathcal {V}(\Omega _\varepsilon (r)) - \mathcal {V}(\Omega _\gamma ) \bigg ) + C_{\Omega _\gamma } \varepsilon ^{\frac{d-1}{d}} - c\,\varepsilon ^{\frac{d-1}{d}} + \mathcal {H}^{d-1}_{\mathbf {a}}(\Omega _r) \\&{\mathop {\leqslant }\limits ^{{(3.3)}}} {\mathcal {P}_{\mathbf {a}}}(\Omega _\varepsilon (r)) + c \gamma \, |\Omega _\varepsilon (r) \triangle \Omega _\gamma | + C_{\Omega _\gamma } \varepsilon ^{\frac{d-1}{d}} - c\,\varepsilon ^{\frac{d-1}{d}} + \mathcal {H}^{d-1}_{\mathbf {a}}(\Omega _r). \end{aligned}$$

Since, as noted in Lemma 3.7, we can choose $C_{\Omega _\gamma }$ arbitrarily small, and since (3.27) and (3.28) hold, for $\varepsilon$ sufficiently small, we have

$$\begin{aligned} \mathcal {H}^{d-1}_{\mathbf {a}}(\Omega _r) \geqslant C \varepsilon ^{\frac{d-1}{d}} \end{aligned}$$

for some $C>0$. Then, by (3.25), $-\frac{d}{dr}V_\mathbf {a}(r) \geqslant C\varepsilon ^{(d-1)/d} = C\big (V_\mathbf {a}(r)\big )^{(d-1)/d}$, i.e.,

$$\begin{aligned} \frac{d}{dr} \left( \big (V_\mathbf {a}(r)\big )^{1/d} \right) \leqslant -C. \end{aligned}$$

This contradicts the fact that $V_\mathbf {a}(r)>0$ for all $r>R_\mathbf {a}$ and we establish Theorem 1.3 under the hypotheses (A1) and (A2b). $\square$

4 Global minimizers in the small $\gamma$ regime

In this section we will prove Theorem 1.6. The proof of this theorem relies on the regularity of almost-minimizers of the perimeter functional. As in [1, Definition 2.5], we will call a set in $E\subset \mathbb {R}^d$ an $\omega$-minimizer of the Euclidean perimeter functional $\mathcal {P}$ in an open set $O\subset \mathbb {R}^d$ with $\omega >0$ if for every ball $B_r(x)\subset O$ and for every set of finite perimeter $F \subset \mathbb {R}^d$ such that $E \triangle F \subset \!\subset B_r(x)$ we have

$$\begin{aligned} \mathcal {P}(E) \leqslant \mathcal {P}(F) + \omega r^d. \end{aligned}$$

This definition also coincides with the definition of almost-minimizers according to [45, Section 1.5] with $\alpha (r)=\omega r$, although it is weaker than the $(\omega ,r)$-minimizers as defined in [35]. Nevertheless, by the regularity theory developed in [45], it will be sufficient to justify passage to a limit as $\gamma \rightarrow 0$.

First we show that any minimizer of the energy $\mathcal {E}_\gamma$ is an $\omega$-minimizer of the Euclidean perimeter functional $\mathcal {P}$ in any open set that does not contain the origin.

Lemma 4.1

Let $\Omega _\gamma \subset \mathbb {R}^d$ be a solution of the problem (1.1). Then $\Omega _\gamma$ is $\omega$-minimal for the Euclidean perimeter functional $\mathcal {P}$ in $O=\mathbb {R}^d\backslash \overline{B_\delta }$ for any $\delta >0$.

Proof

We first note that the constraint $|\Omega _\gamma |=1$ may be replaced by a penalization term in the energy. Namely, arguing as in Step 1 of the proof of [7, Theorem 2.7] we obtain that there exists a constant $\lambda >0$ such that

$$\begin{aligned} \min \mathcal {F}_\gamma ^\lambda = \mathcal {F}_\gamma ^\lambda (\Omega _\gamma ) = \mathcal {E}_\gamma (\Omega _\gamma ) \end{aligned}$$

where

$$\begin{aligned} \mathcal {F}_\gamma ^\lambda (E) = \mathcal {E}_\gamma (E) + \lambda \big | |E|-1 \big | \end{aligned}$$

for any $E\subset \mathbb {R}^d$.

Fix $r>0$ such that $B_r(x) \cap B_\delta = \emptyset$ and let $F \subset \mathbb {R}^d$ be any set such that $\Omega _\gamma \triangle F \subset \!\subset B_r(x)$. Since $\mathcal {E}_\gamma (\Omega _\gamma ) = \mathcal {F}_\gamma ^\lambda (\Omega _\gamma ) \leqslant \mathcal {F}_\gamma ^\lambda (F)$, using (3.3) we obtain that

$$\begin{aligned} {\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ) \leqslant {\mathcal {P}_{\mathbf {a}}}(F) + \omega |\Omega _\gamma \triangle F| \end{aligned}$$

for some constant $\omega$ depending only on $\gamma$ and $\lambda$.

Let $m :=\max \{ \mathbf {a}(y) :y\in B_r(x) \}$. Since $\mathbf {a}$ is Lipschitz there exists a constant $C>0$ such that $\mathbf {a}(y) \geqslant m-Cr$ for all $y\in B_r(x)$. Then

$$\begin{aligned} (m-Cr) \mathcal {P}(\Omega _\gamma ,B_r(x))&\leqslant {\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ,B_r(x)) \leqslant {\mathcal {P}_{\mathbf {a}}}(F,B_r(x)) + \omega |\Omega _\gamma \triangle F| \\&\leqslant m \mathcal {P}(F,B_r(x)) + \omega |\Omega _\gamma \triangle F|. \end{aligned}$$

Hence,

$$\begin{aligned} \mathcal {P}(\Omega _\gamma ,B_r(x)) \leqslant \mathcal {P}(F,B_r(x)) + Cr^d \end{aligned}$$

provided $\mathcal {P}(\Omega _\gamma ,B_r(x)) \leqslant C r^{d-1}$. This establishes that $\Omega _\gamma$ is an $\omega$-minimizer of the Euclidean perimeter in $\mathbb {R}^d\backslash \overline{B_\delta }$.

It now remains to prove that $\mathcal {P}(\Omega _\gamma ,B_r(x)) \leqslant C r^{d-1}$. Note that $\Omega _\gamma$ satisfies the $\varepsilon -\varepsilon ^{(d-1)/d}$ property given by (3.21) in Lemma 3.7. Let $B^1$ and $B^2$ be two disjoint balls, intersecting $\partial ^* \Omega _\gamma$ in a set of positive $\mathcal {H}^{d-1}$-measure. Let $C_1,\,C_2>0$ and $0<\varepsilon _1,\varepsilon _2<1$ be the corresponding constants satisfying the conditions in (3.21). Take $C=\max \{C_1,C_2\}$, $\overline{\varepsilon } = \min \{\varepsilon _1,\varepsilon _2\}$, and $\overline{r} = \min \big \{ (\overline{\varepsilon }/\omega _d)^{1/d}, {{\,\mathrm{dist}\,}}(B^1,B^2) \big \}$.

If $r < \overline{r}$, then $\varepsilon :=|B_r(x) \cap \Omega _\gamma | < \omega _d r^d \leqslant \overline{\varepsilon }$, and $B_r(x)$ cannot intersect both $B^1$ and $B^2$. Without loss of generality, assume that $B_r(x) \cap B^1 = \emptyset$. For this $\varepsilon$, let $F\subset \mathbb {R}^d$ be the set of finite perimeter satisfying (3.21). Clearly, we also have

$$\begin{aligned} {\mathcal {P}_{\mathbf {a}}}(F) \leqslant {\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ) + C\varepsilon \leqslant {\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ) + C\omega _d r^d. \end{aligned}$$

Now, let $G=F \backslash B_r(x)$. Then $|G|=|\Omega _\gamma |=1$ and G is admissible for (1.1). By minimality of $\Omega _\gamma$ and Lipschitzianity of $\mathcal {V}$ (see (3.3)) we obtain

$$\begin{aligned} {\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma )&\leqslant {\mathcal {P}_{\mathbf {a}}}(G) + \gamma \big ( {\mathcal {V}}(G) - {\mathcal {V}}(\Omega _\gamma ) \big ) \\&\leqslant {\mathcal {P}_{\mathbf {a}}}(F) - {\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ,B_r(x)) + d\omega _d R_{\max }^p r^{d-1} + C\gamma \, |G \triangle \Omega _\gamma | \\&\leqslant {\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ) + C\omega _d r^d - {\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ,B_r(x)) + d\omega _d R_{\max }^p r^{d-1} + C\gamma \, r^d \\&\leqslant {\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ) - {\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ,B_r(x)) + C\,r^{d-1} \end{aligned}$$

with a constant depending only on d, $R_{\max }$, and $\gamma$, where $R_{\max }$ is the constant obtained in the proof of Theorem 1.3. Recalling that $\mathbf {a}(x)\geqslant \delta ^p$ in $\mathbb {R}^d\backslash \overline{B_\delta }$ we have that

$$\begin{aligned} \mathcal {P}(\Omega _\gamma ,B_r(x)) \leqslant \delta ^{-p} {\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ,B_r(x)) \leqslant C r^{d-1}. \end{aligned}$$

If $r \geqslant \overline{r}$, on the other hand, then

$$\begin{aligned} \mathcal {P}(\Omega _\gamma ,B_r(x)) \leqslant \delta ^{-p} {\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ,B_r(x)) \leqslant \delta ^{-p} {\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ) \leqslant \frac{\delta ^{-p}{\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma )}{\overline{r}^{d-1}} \, r^{d-1}. \end{aligned}$$

Hence, in either case we obtain that $\mathcal {P}(\Omega _\gamma ,B_r(x)) \leqslant C r^{d-1}$, completing the proof of the lemma. $\square$

Since the density $\mathbf {a}(x)=|x|^p$ is vanishing at zero we cannot conclude that $\Omega _\gamma$ is an $\omega$-minimizer of the perimeter in $\mathbb {R}^d$. Nevertheless, combining [45, Sections 1.9 and 1.10] (see also [4, 46]), we obtain the following regularity result for any $\omega$-minimizer of perimeter in open sets.

Lemma 4.2

Let $O\subset \mathbb {R}^d$ be an open set.

• (i) If $E\subset \mathbb {R}^d$ is an $\omega$-minimizer of Euclidean perimeter in O then $\partial ^*E\cap O$ is a $C^{1,\alpha }$ hypersurface for any $\alpha \in (0,1/2)$.

• (ii) If $E_n\subset \mathbb {R}^d$ is a sequence of uniformly $\omega$-minimizers of Euclidean perimeter in O and if $E_n \rightarrow E$ in $L^1(O)$, then $\partial E_n \rightarrow \partial E$ in $C^{1,\alpha }$ and $\partial E_n$ is a $C^{1,\alpha }$ graph over $\partial E$.

We are now ready to present the proof of global minimality of balls in the small $\gamma$ regime.

Proof of Theorem 1.6

We fix $r_0=(1/\omega _d)^{1/d}$ with $|B_{r_0}|=1$, and recall (see e.g. [5, 10]) that $B_{r_0}$ is the unique minimizer of the local weighted isoperimetric problem e(0). By Theorem 1.1, for any $\gamma >0$, there exists $\Omega _\gamma \subset \mathbb {R}^d$, $|\Omega _\gamma |=1$, minimizing $\mathcal {E}_\gamma$. We first claim that $\Omega _\gamma \rightarrow B_{r_0}$ in $L^1(\mathbb {R}^d)$. Indeed, clearly $e(0)\leqslant \mathcal {E}_0(\Omega _\gamma )\leqslant \mathcal {E}_\gamma (\Omega _\gamma )$ for all $\gamma >0$, while on the other hand,

$$\begin{aligned} \limsup _{\gamma \rightarrow 0}\mathcal {E}_\gamma (\Omega _\gamma )\leqslant \limsup _{\gamma \rightarrow 0}\mathcal {E}_\gamma (B_{r_0}) = \mathcal {E}_0(B_{r_0}) = e(0). \end{aligned}$$

Thus, $\{\Omega _\gamma \}_{\gamma >0}$ give minimizing sequences for e(0). By Theorem 1.1 it follows that for every sequence $\gamma _n\rightarrow 0$ there is a subsequence along which $\{\Omega _{\gamma _n}\}$ is compact in $L^1(\mathbb {R}^d)$, and converges to the minimizer $B_{r_0}$ of e(0). As the minimizer of the limit problem is unique, we conclude $\Omega _\gamma \rightarrow B_{r_0}$ in $L^1(\mathbb {R}^d)$ as claimed.

Next, by Lemma 4.1$\Omega _\gamma$ is an $\omega$-minimizer of the Euclidean perimeter in $O=\mathbb {R}^d\backslash \overline{B_\delta }$ for any $\delta >0$; hence, by Lemma 4.2(ii) $\partial ^* \Omega _\gamma \cap O$ is a $C^{1,\alpha }$-graph over the limit set $B_{r_0}$. Note in particular that Lemma 4.2(ii) assures that for any $\delta >0$, there exists $\gamma _\delta >0$ so that

$$\begin{aligned} \partial ^*\Omega _\gamma \cap (B_{r_0-\delta }\backslash \overline{B_\delta }) = \emptyset \end{aligned}$$

(4.1)

for all $0<\gamma <\gamma _\delta$. However, since we cannot conclude quasi-minimality (hence, regularity) of $\Omega _\gamma$ in the whole space we cannot a priori assume that $\partial \Omega _\gamma \cap B_\delta$ is empty for all $\delta >0$. Assume for a contradiction that for some $\delta >0$, $\partial \Omega _\gamma \cap B_\delta$ is non-empty; from (4.1) we may conclude that $|\Omega _\gamma ^c \cap B_\delta |>0$ for all $0<\gamma <\gamma _\delta$, and hence $\Omega _\gamma$ is multiply-connected, and its boundary is disconnected into disjoint components. That is, $\Omega _\gamma = \widetilde{\Omega }_\gamma \backslash \Omega _\gamma ^0$, where $\widetilde{\Omega }_\gamma$ is simply connected, with smooth boundary, and $\Omega _\gamma ^0 :=\Omega _\gamma ^c\cap B_\delta$, and thus $\partial ^*\Omega _\gamma = \partial \widetilde{\Omega }_\gamma \cup \partial ^* \Omega _\gamma ^0$, a disjoint union. We define $m_\gamma :=|\Omega _\gamma ^0|$. As $|\Omega _\gamma \triangle B_{r_0}|\rightarrow 0$ it follows that $m_\gamma \rightarrow 0$ and $|\widetilde{\Omega }_\gamma | = 1 + m_\gamma$. Moreover, let $R_\gamma = \left( \frac{1+m_\gamma }{\omega _d}\right) ^{1/d}$ so that $|B_{R_\gamma }|=1+m_\gamma$. In fact, using a Fuglede-type argument (cf. [23]), we will prove that $m_\gamma =0$ for $\gamma$ sufficiently small, and therefore $\Omega _\gamma = B_{r_0}$ for all small $\gamma$.

Since $\partial \widetilde{\Omega }_\gamma$ is a graph over $\partial B_{r_0}$ we can think of it as a graph over $\partial B_{R_\gamma }$ given by

$$\begin{aligned} \partial \widetilde{\Omega }_\gamma = \big \{ \widetilde{x} (1+\psi _\gamma (\widetilde{x})) :\widetilde{x}\in \partial B_{R_\gamma } \big \} \end{aligned}$$

for some $\psi _\gamma \in C^{1,\alpha }(\partial B_{R_\gamma })$ such that $\int _{\partial B_{R_\gamma }} \psi _\gamma \,\mathrm {d}\mathcal {H}^{d-1}=0$. Now, we can write both $\partial \widetilde{\Omega }_\gamma$ and $\partial B_{R_\gamma }$ over the unit sphere $\mathbb {S}^{d-1}$. Namely,

$$\begin{aligned} \partial \widetilde{\Omega }_\gamma = \big \{ R_\gamma x (1+u_\gamma (x)) :x\in \mathbb {S}^{d-1}\big \} \quad \text { and } \quad \partial B_{R_\gamma } =\big \{ R_\gamma x :x\in \mathbb {S}^{d-1}\big \}, \end{aligned}$$

where $u_\gamma (x) = \psi _\gamma (R_\gamma x)$. Note that $\int _{\mathbb {S}^{d-1}} u_\gamma \,\mathrm {d}\mathcal {H}^{d-1}= R_{\gamma }^{1-d} \int _{\partial B_{R_\gamma }} \psi _\gamma \,\mathrm {d}\mathcal {H}^{d-1}=0$. Computing the perimeter, we have

$$\begin{aligned} {\mathcal {P}_{\mathbf {a}}}(\widetilde{\Omega }_\gamma ) = R_\gamma ^{d-1} \int _{\mathbb {S}^{d-1}} \mathbf {a}\big ( R_\gamma x (1+u_\gamma (x))\big ) (1+u_\gamma )^{d-1} \sqrt{1+\frac{|\nabla _{\tau } u_\gamma |^2}{(1+u_\gamma )^2}} \,\mathrm {d}\mathcal {H}^{d-1}, \end{aligned}$$

where $\nabla _\tau$ denotes the gradient with respect to $\mathbb {S}^{d-1}$. Likewise,

$$\begin{aligned} {\mathcal {P}_{\mathbf {a}}}(B_{R_\gamma }) = R_\gamma ^{d-1} \int _{\mathbb {S}^{d-1}} \mathbf {a}(R_\gamma x) \,\mathrm {d}\mathcal {H}^{d-1}. \end{aligned}$$

Putting these two together,

$$\begin{aligned} {\mathcal {P}_{\mathbf {a}}}(\widetilde{\Omega }_\gamma ) - {\mathcal {P}_{\mathbf {a}}}(B_{R_\gamma })&= R_\gamma ^{d-1} \int _{\mathbb {S}^{d-1}} (1+u_\gamma )^{d-1} \mathbf {a}\big ( R_\gamma x (1+u_\gamma (x))\big ) \left[ \sqrt{1+\frac{|\nabla _{\tau } u_\gamma |^2}{(1+u_\gamma )^2}}-1 \right] \,\mathrm {d}\mathcal {H}^{d-1}\\&\quad + R_\gamma ^{d-1} \int _{\mathbb {S}^{d-1}} \Big [ (1+u_\gamma )^{d-1} \mathbf {a}\big (R_\gamma x(1+u_\gamma (x))\big ) - \mathbf {a}(R_\gamma x) \Big ] \,\mathrm {d}\mathcal {H}^{d-1}\\&=: R_\gamma ^{d-1} I_1 + R_\gamma ^{d-1} I_2. \end{aligned}$$

Now we will estimate $I_1$ and $I_2$. Let $\varepsilon >0$ be small so that $1+u_\gamma \geqslant 1/2$ and $\Vert u_\gamma \Vert _{C^1(\mathbb {S}^{d-1})}\leqslant \varepsilon$ for $\gamma$ sufficiently small. Using the estimate $\sqrt{1+t} \geqslant 1+\frac{t}{2}-\frac{t^2}{8}$, we get that

$$\begin{aligned} I_1&\geqslant \int _{\mathbb {S}^{d-1}} R_\gamma ^p(1+u_\gamma )^{p+d-1}\left[ \frac{1}{2}\frac{|\nabla _\tau u_\gamma |^2}{(1+u_\gamma )^2} -\frac{1}{8}\frac{|\nabla _\tau u_\gamma |^4}{(1+u_\gamma )^4} \right] \,\mathrm {d}\mathcal {H}^{d-1}\\&\geqslant \left( \frac{1}{2} - C\varepsilon \right) \int _{\mathbb {S}^{d-1}} R_\gamma ^p (1+u_\gamma )^{p+d-3} |\nabla _\tau u_\gamma |^2 \,\mathrm {d}\mathcal {H}^{d-1}\\&\geqslant C \int _{\mathbb {S}^{d-1}} |\nabla _\tau u_\gamma |^2 \,\mathrm {d}\mathcal {H}^{d-1}\end{aligned}$$

for $\varepsilon$ sufficiently small.

Since $\int _{\mathbb {S}^{d-1}} u_\gamma \,\mathrm {d}\mathcal {H}^{d-1}=0$ and $(1+t)^q \geqslant 1+qt$, we have

$$\begin{aligned} I_2 = \int _{\mathbb {S}^{d-1}} R_\gamma ^p \big ( (1+u_\gamma )^{p+d-1}-1 \big ) \,\mathrm {d}\mathcal {H}^{d-1}\geqslant 0. \end{aligned}$$

These two estimates imply that

$$\begin{aligned} {\mathcal {P}_{\mathbf {a}}}(\widetilde{\Omega }_\gamma ) - {\mathcal {P}_{\mathbf {a}}}(B_{R_\gamma }) \geqslant C \Vert u_\gamma \Vert _{H^1(\mathbb {S}^{d-1})}^2 \end{aligned}$$

(4.2)

for some constant $C>0$ independent of $\gamma$.

As for the nonlocal term, [20, Lemma 5.3] (see also [25, Remark 3.2]) implies that

$$\begin{aligned} {\mathcal {V}}(B_{R_\gamma }) - {\mathcal {V}}(\widetilde{\Omega }_\gamma ) \leqslant C \Vert u_\gamma \Vert _{H^1(\mathbb {S}^{d-1})}^2. \end{aligned}$$

Furthermore,

$$\begin{aligned} \mathcal {V}(B_{R_\gamma }\backslash \Omega _\gamma ^0)-\mathcal {V}(\Omega _\gamma ) \leqslant \mathcal {V}(B_{R_\gamma }) - \mathcal {V}(\widetilde{\Omega }_\gamma ) + Cm_\gamma \leqslant C \big ( \Vert u_\gamma \Vert _{H^1(\mathbb {S}^{d-1})}^2 + m_\gamma \big ). \end{aligned}$$

(4.3)

To see the first estimate above, let $\overline{R}>0$ be large enough such that $\Omega _\gamma$, $B_{R_\gamma }\subset B_{\overline{R}}$ for all $\gamma >0$. For $\mathcal {I}(E,F)=\int _E\!\int _F |x-y|^{-\alpha }\,\mathrm {d}x \,\mathrm {d}y$, we have

$$\begin{aligned} \mathcal {V}(B_{R_\gamma }\backslash \Omega _\gamma ^0)-\mathcal {V}(\Omega _\gamma ) = \mathcal {V}(B_{R_\gamma }) - \mathcal {V}(\widetilde{\Omega }_\gamma ) + 2 \mathcal {I}(\Omega _\gamma ^0,\widetilde{\Omega }_\gamma )- 2 \mathcal {I}(\Omega _\gamma ^0,B_{R_\gamma }). \end{aligned}$$

Since

$$\begin{aligned} \big | \mathcal {I}(\Omega _\gamma ^0,\widetilde{\Omega }_\gamma )- \mathcal {I}(\Omega _\gamma ^0,B_{R_\gamma }) \big |&\leqslant \int _{\Omega _\gamma ^0} \! \int _{\widetilde{\Omega }_\gamma \triangle B_{R_\gamma }} \frac{1}{|x-y|^\alpha }\,\mathrm {d}x \,\mathrm {d}y \\&\leqslant |\Omega _\gamma ^0| \, \sup _{x\in B_{\overline{R}}} \int _{\widetilde{\Omega }_\gamma \triangle B_{R_\gamma }} \frac{1}{|x-y|^\alpha }\,\mathrm {d}x \,\mathrm {d}y \leqslant C m_\gamma , \end{aligned}$$

the first estimate in (4.3) follows.

Again, using minimality of $\Omega _\gamma$, i.e. $\mathcal {E}_\gamma (\Omega _\gamma ) \leqslant \mathcal {E}_\gamma (B_{R_\gamma } \backslash \Omega _\gamma ^0)$, and the lower and upper bounds (4.2) and (4.3) we get that

$$\begin{aligned} C \Vert u_\gamma \Vert _{H^1(\mathbb {S}^{d-1})}^2&\leqslant {\mathcal {P}_{\mathbf {a}}}(\widetilde{\Omega }_\gamma ) - {\mathcal {P}_{\mathbf {a}}}(B_{R_\gamma }) = {\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ) - {\mathcal {P}_{\mathbf {a}}}(B_{R_\gamma }\backslash \Omega _\gamma ^0) \\&\leqslant \gamma \, \big ( \mathcal {V}(B_{R_\gamma }\backslash \Omega _\gamma ^0) - \mathcal {V}(\Omega _\gamma ) \big ) \leqslant C\gamma \, \big ( \Vert u_\gamma \Vert _{H^1(\mathbb {S}^{d-1})}^2 + m_\gamma \big ). \end{aligned}$$

Thus, for $\gamma$ sufficiently small,

$$\begin{aligned} \Vert u_\gamma \Vert _{H^1(\mathbb {S}^{d-1})}^2 \leqslant C\gamma \, m_\gamma . \end{aligned}$$

(4.4)

This, in turn, implies that

$$\begin{aligned} \mathcal {E}_\gamma (B_{R_\gamma }\backslash \Omega _\gamma ^0)-\mathcal {E}_\gamma (\Omega _\gamma ) \leqslant C\gamma \,m_\gamma . \end{aligned}$$

(4.5)

The last step in the argument is a lower bound on the above difference. In fact, we will show that replacing the domain $\Omega _\gamma =\widetilde{\Omega }_\gamma \backslash \Omega _\gamma ^0$ by the sphere $B_{r_0}$ results in a much larger energy difference, due to the reduction of the radius $R_\gamma$ to $r_0$. We observe this right away, in estimating the difference in perimeter, ${\mathcal {P}_{\mathbf {a}}}(B_{R_\gamma })+{\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ^0)-{\mathcal {P}_{\mathbf {a}}}(B_{r_0})$. Let $\rho _\gamma = (m_\gamma /\omega _d)^{1/d}$ and note that

$$\begin{aligned} R_\gamma =(r_0^d+\rho _\gamma ^d)^{1/d} = r_0 \left( 1+\left( \frac{\rho _\gamma }{r_0}\right) ^d \right) ^{1/d}. \end{aligned}$$

Now, for $m_\gamma$ (hence, for $\rho _\gamma /\omega _d$) small, we have

$$\begin{aligned} {\mathcal {P}_{\mathbf {a}}}(B_{R_\gamma })+{\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ^0)-{\mathcal {P}_{\mathbf {a}}}(B_{r_0})&\geqslant {\mathcal {P}_{\mathbf {a}}}(B_{R_\gamma })+{\mathcal {P}_{\mathbf {a}}}(B_{\rho _\gamma })-{\mathcal {P}_{\mathbf {a}}}(B_{r_0}) \\&= d\omega _d (R_\gamma ^{d-1+p}+\rho _\gamma ^{d-1+p}-r_0^{d-1+p}) \\&= d\omega _d r_0^{d-1+p} \left( \left[ 1+\left( \frac{\rho _\gamma }{r_0}\right) ^d \right] ^{(d-1+p)/d} + \left( \frac{\rho _\gamma }{r_0}\right) ^{d-1+p} -1 \right) \\&=d\omega _d r_0^{p-1} \left( 1 + \frac{p-1}{d} \right) \rho _\gamma ^d + O(\rho _\gamma ^{d-1+p})\\&= {\bar{C}}\,m_\gamma + o(m_\gamma ). \end{aligned}$$

Combining this estimate with (4.3) and (4.4), and recalling that the ball maximizes the Riesz potential, for sufficiently small $\gamma$, we get

$$\begin{aligned} \mathcal {E}_\gamma (B_{R_\gamma }\backslash \Omega _\gamma ^0)-\mathcal {E}_\gamma (B_{r_0})&= {\mathcal {P}_{\mathbf {a}}}(B_{R_\gamma })+{\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ^0)-{\mathcal {P}_{\mathbf {a}}}(B_{r_0}) + \gamma \big ( \mathcal {V}(B_{R_\gamma }\backslash \Omega _\gamma ^0)-\mathcal {V}(B_{r_0}) \big ) \nonumber \\&\geqslant ({\bar{C}}-\gamma C) m_\gamma \geqslant C m_\gamma . \end{aligned}$$

(4.6)

Now (4.5) and (4.6) imply that

$$\begin{aligned} Cm_\gamma + \mathcal {E}_\gamma (B_{r_0}) \leqslant \mathcal {E}_\gamma (B_{R_\gamma }\backslash \Omega _\gamma ^0) \leqslant \mathcal {E}_\gamma (\Omega _\gamma ) + C\gamma \, m_\gamma \leqslant \mathcal {E}_\gamma (B_{r_0}) + C\gamma \, m_\gamma . \end{aligned}$$

Hence, for $\gamma$ sufficiently small, $Cm_\gamma \leqslant 0$, i.e., $m_\gamma =0$. This implies that $\Omega _\gamma ^0=\emptyset$ for $\gamma$ small and that $\partial \Omega _\gamma$ is a graph over $\partial B_{r_0}$.

Therefore, running through the Fuglede-type argument one more time, we get

$$\begin{aligned} C \Vert u_\gamma \Vert _{H^1(\mathbb {S}^{d-1})}^2 \leqslant {\mathcal {P}_{\mathbf {a}}}(\Omega _\gamma ) - {\mathcal {P}_{\mathbf {a}}}(B_{r_0}) \leqslant \gamma \big ({\mathcal {V}}(B_{r_0}) - {\mathcal {V}}(\Omega _\gamma )\big )\leqslant C\gamma \,\Vert u_\gamma \Vert _{H^1(\mathbb {S}^{d-1})}^2. \end{aligned}$$

Hence, for $\gamma$ sufficiently small $u_\gamma \equiv 0$, i.e., $\Omega _\gamma = B_{r_0}$. $\square$