A spectral shape optimization problem with a nonlocal competing term

We study the minimization of a spectral functional made as the sum of the first eigenvalue of the Dirichlet Laplacian and the relative strength of a Riesz-type interaction functional. We show that when the Riesz repulsion strength is below a critical value, existence of minimizers occurs. Then we prove, by means of an expansion analysis, that the ball is a rigid minimizer when the Riesz repulsion is small enough. We also show that if the Riesz repulsion is large, then minimizers do not exist.

Foreword. In this work we study the minimization under volume constraint of energies of the form where S is either the torsion energy E or the first eigenvalue of the Dirichlet-Laplacian λ 1 , N ≥ 2 and α ∈ (0, N ).
It is well-known that both the torsion energy and the first eigenvalue of the Dirichlet-Laplacian are minimized, among sets of fixed measure, by the ball. These results, obtained with symmetrization arguments, can be summarized in a scale invariant form as , where B is a generic ball and |Ω| denotes the Lebesgue measure in R N of the set Ω. In the literature, they are called Saint-Venant and Faber-Krahn inequalities, respectively. Both the inequalities are rigid, that is equality holds if and only if the Ω is a ball up to null capacity. We refer to [23] for a comprehensive background about these problems.
In sharp contrast, the Riesz Energy functional V α (Ω) := Ω Ω dx dy |x − y| N −α , which appears as the second addend in the definition of F , increases while symmetrizing the set Ω, and it is uniquely (up to a negligible set) maximized by balls [29,Theorem 3.7], leading to a competition while seeking to minimize F .
Motivation and background. In recents years the research of quantitative stability of various geometric, functional and spectral inequalities received a great attention, and this gave a strong impulse to the development of the field. In turn this led to a renewed interest in several variational models where a competition between a cohesive term is balanced by a repulsive term. A non-exhaustive list of papers in this field is [1,6,15,20,21,27,12,13,17,34,24,30,16,34].
Arguably the most famous instance of such variational models is the Gamow liquid drop model introduced in [19] to describe the stability of nuclear matter. Such a model is made up by the sum of a surface perimeter term and a Riesz energy term of a set Ω ⊂ R 3 J (Ω) := P (Ω) + Ω Ω dx dy |x − y| .
The usual mathematical questions about this class of models are: (1) To investigate existence and non existence of minimizers depending on the values of the mass of competitors, that is, depending on the choice of the volume constraint. (2) To study the regularity of minimizers, if existence holds.
(3) To characterize the ball as the unique minimizer as long as the mass is small enough. In particular, regarding the liquid drop model, in [13] Choksi and Peletier conjectured 1 that there exists a critical threshold mass m such that minimizers exist only if |Ω| ≤ m. Questions (1) and (3) above, as well as such a conjecture, follow the intuitive idea that because of the different scaling of the functionals, if the mass is small then the perimeter term is dominant, while if the mass is large then the Riesz term dominates, and disconnected configurations are favored. Since the Riesz energy decreases as the connected components of a set are pushed away from each other, this leads to non-existence. In fact, one can show that if the mass is approaching 0, then the problem reduces to the classical isoperimetric problem. The Choksi-Peletier conjecture, although being still open in its generality, was partially solved in [27,15,24] where the authors show that there are thresholds 0 < m small < m big such that the ball is the unique minimizer for m < m small and existence does not occur if m > m big . The scope of this paper is to begin this kind of analysis when the perimeter is replaced by a spectral functional.
In the case where S = E is the torsion energy, we obtain the following weaker result.
We stress that the value of the geometric constant R 0 can be explicitly computed from our proofs. A remark concerning the mass constraint is in order. as for all t > 0 we have λ 1 (tΩ) + V α (tΩ) = t −2 λ 1 (Ω) + t N +α+2 V α (Ω) .
In particular requiring the mass of competitors m ≈ t N to be small is equivalent to require ε ≈ t N +α+2 to be small. Therefore, Theorem 1.1 states that for small masses the only minimizer of λ 1 + V α is the ball, as long as α > 1, which is the analogous of the results obtained on the functional P + V α .
For the torsion energy the situation depends on the value of α. Indeed for any t > 0 one has so that small values of ε = t α−2 do correspond to small values of the mass only for α > 2.
The result stated in Theorem 1.1 is the spectral analog of the existence results in [27,24,15]. On the other hand, when dealing with the torsion energy, the result needs the additional assumption of equiboundedness of competitors. We believe such an hypothesis to be of technical nature, but its removal seems a challenging task and we do not solve it in this paper. We discuss this issue in the next remark.
Remark 1.4. The problem of proving the existence of minimizers among generic subsets of R N (instead of among equibounded sets) for spectral functionals has been a rather hot topic in the last years. Regarding the eigenvalues of the Dirichlet-Laplacian essentially two techniques are available in literature: one developed by Bucur in [9] is based on a concentration-compactness argument mixed together with regularity results for inward minimizing sets; the other, proposed by the first author and Pratelli in [32], is based on a De Giorgi type surgery argument. Seemingly none of these techniques works while tackling the case of the functional E + εV α . Even working with a more direct surgery-wise technique for the functional E as that used in [8,Section 5] seems to fail in our setting. Hence we are not able to get rid of the equiboundedness assumption in Theorem 1.2.
Restricting the class of Riesz energies to α ∈ (1, N ) seems a deep problem as well. In fact to show Theorems 1.1, 1.2 we need a fine regularity analysis of minimizers (see the discussion below) where the regularity of the Riesz potential v Ω (x) := Ω dy |x − y| N −α plays a crucial role. If α ≤ 1, then v Ω is at most of class C 0,γ , for some γ ∈ (0, 1], which is not enough for our proof to work. The third and last result we get is the following, in which we show that for big values of the mass, minimizers do not exist among sets satisfying uniform density constraints. Definition 1.5. We say that a set Ω has the internal δ−ball condition if for any x ∈ ∂Ω there exists a ball B δ ⊂ Ω tangent to ∂Ω in x. We call U(δ) the class of open sets Ω ⊂ R N that satisfy the internal δ-ball condition. Theorem 1.6. Let α ∈ (0, N ), ε > 0, δ > 0. Then there exists ε max = ε max (α, N ) such that for ε ≥ ε max both problems do not admit a minimizer.

1.2.
Outline of the proof and structure of the paper. The proofs of the main results of the paper, Theorem 1.1 and 1.2, are articulated in two main steps, which we briefly describe here below. The first step we discuss is the proof of Theorem 1.2, which covers most of the paper, then we describe a (completely independent) surgery argument for the functional λ 1 + εV α . Putting together these two steps, Theorem 1.1 follows. Strategy of the proof of Theorem 1.2. The proof of Theorem 1.2 is quite long and involved. It is inspired by many ideas proposed in [8,27].
First of all, we consider a problem without the mass constraint. This step is needed because the techniques from the free boundary regularity that we aim to apply do not work properly in presence of a measure constraint as perturbations become more difficult to manage. Whence we consider an auxiliary minimization problem of the form where f η is a suitable piecewise linear function which acts as a sort of Lagrange multiplier. This strategy in shape optimization problems was first proposed by Aguilera, Alt and Caffarelli in [3]. We point out that without the equiboundedness restriction, at least as long as α < 2, minimizers of problem (4) do not exist (see Section 2), and the infimum of G ε,η diverges toward minus infinity, which somewhat underlines one difficulty while trying to remove the equiboundedness of competitors in Theorem 1.2. Unfortunately the desired equivalence between (4) and (2) is not straightforward, and we first need to show existence of minimizers of problem (4), and some mild regularity (finiteness of the perimeter and density estimates). This permits us to show that for suitable values of η (again depending on R), the minimization of G ε,η and the measure constrained minimization of E + εV α are indeed equivalent, for ε small enough.
The next key point is therefore to prove a suitable regularity result on the free boundary of an optimal set for (4). To get such a regularity we switch to the following problem of Alt-Caffarelli type, with the idea of exploiting the regularity theory for ∂{u > 0} ∩ B R , where u is any minimizer of the above problem. Such an analysis is done in the spirit of the seminal work on free boundary regularity by Alt and Caffarelli [4] . The link between this regularity argument and the rigidity of the ball is then the quantitative version of the Saint-Venant inequality, stating that for any Ω ⊂ R N there exists a ball B r (x) of measure |Ω| such that This is a deep result, recently shown in [8] and -together with several of the ideas of its proof-plays a crucial role in our analysis. Indeed by comparing any candidate minimizer with a ball, we show that for ε small, minimizers are close in L 1 −topology to the ball. Whence, exploiting the free boundary regularity analysis, such an L 1 − proximity to a ball is improved to a nearly spherical one, stating that the boundary of any minimizer is a small C 2,γ −parametrization on a sphere. At this point, a perturbative analysis in the class of nearly spherical sets yields to the conclusion (again with the aid of the quantitative Saint-Venant inequality) that the ball is the only minimizer. Beside proving Theorem 1.2, this argument, together to the Kohler-Jobin inequality is enough to get the statement of Theorem 1.1 among equibounded sets. At this point we only need to show that any minimizing sequence can be chosen to be made up of equibounded sets. This, as mentioned above, is made by means of a surgery-wise argument.
The surgery argument. The strategy we follow is based on that proposed in [32] (see also [10]) in order to prove existence of minimizers under measure constraints for the k−th eigenvalue of the Dirichlet-Laplacian. Nevertheless some difference with respect to [32] occur. On the one hand the presence of the repulsive Riesz energy term forces us to work with connected sets. On the other hand we only deal with the first eigenvalue, thus we do not need to take care of the further difficulty about the orthogonality constraint of the higher eigenfunctions. Furthermore, up to choose ε small enough, we can deal with sets which are close to the ball in the L 1 topology which allows us to simplify the argument.
Plan of the paper. The paper is organized as follows: in Section 2 we give the basic definitions and we prove or recall some preliminary results. In Sections 3, 4, 5 and 6 we develop the proof of Theorem 1.2, as described above. Section 8 is devoted to a surgery argument for the functional λ 1 + εV α . Finally, Sections 9 and 10 contain the proof of Theorems 1.1 and 1.6, respectively.

Setting, notations and some preliminary results
The ambient space in this work is R N , where N ≥ 2 is an integer. With Ω we denote an open bounded set, unless otherwise stated. We write B r (x) to indicate the ball with radius r centered in x, and just B r if the center is x = 0, while by B we denote just a generic ball, unless otherwise stated. Moreover we set ω N the measure of the ball of unit radius in R N and the Lebesgue N -dimensional measure of a set D is denoted by |D|.
2.1. The functionals: definitions and properties. The problem we deal with is the minimization under volume constraint of the functional where E is the torsion energy E(Ω) := min and V α is the Riesz potential energy defined for α ∈ (0, N ) as Some features of these two functionals are in order. First, we remark that the minimum for the torsion energy functional is attained by a function w Ω , the torsion function, as long as Ω has finite measure. The Euler-Lagrange equation of the minimization problem defining E reads as −∆w Ω = 1 in Ω, w Ω ∈ H 1 0 (Ω). The definition of E together with the equation satisfied by w Ω leads to the following representation of the Torsion energy By the Pólya-Szëgo inequality (see [35]) it follows that where B is a ball of R N of measure |Ω|, that is: the torsion energy E is minimized by balls under volume constraint. Moreover the above inequality is rigid, in the sense that equality holds if and only if Ω is a ball, up to sets of null-capacity. This inequality is addressed as Saint-Venant inequality. The torsion energy E satisfies the following scaling law: and it is non-increasing with respect to set inclusion, i.e.
the inequality being strict as soon as |Ω 2 \ Ω 1 | > 0. Therefore we can rewrite the Saint-Venant inequality in the scale invariant form About the Riesz energy functional V α , we note that it scales as Moreover, we recall that by Riesz inequality (see [29,Theorem 3.7]) V α is maximized by balls, that is, Again the inequality is rigid, that is, equality holds if and only if Ω is a ball up to a negligible set. It is immediate to see that the Riesz potential energy is non-decreasing with respect to set inclusion, that is the inequality being strict if |Ω 2 \ Ω 1 | > 0. Alongside the Riesz energy we define the Riesz potential v Ω (x) := Notice that v Ω (0) satisfies The following result, which is a simple refinement of [27, Proof of Proposition 2.1] will be used several times in the paper. Lemma 2.1. Let α ∈ (0, N ), Ω, F ⊂ R N be two measurable sets, with finite perimeter, such that Ω∆F ⊂ B R (0), for some R > 0. Then it holds Proof. Let us set, for any couple of measurable sets A and B We begin by rewriting We have On the other hand, We note that (using also Fubini theorem), From (6), (7) and (8) we deduce We can now observe that, as a consequence of Riesz inequality (see [18,Lemma 2.3]) and the rescaling of v Ω (0), see (5), we get where we denote by B the ball of unit measure centered at the origin. The same computation holds also for We conclude this subsection recalling one of the main tool we exploit to solve problem (10): the sharp quantitative version of the Saint-Venant inequality, which was first proved as a intermediate result in [ is the Fraenkel asymmetry.
The last functional involved in our work is the first eigenvalue of the Dirichlet-Laplacian acting on an open and bounded set Ω ⊂ R N . We recall its variational definition given as the minimum of the so-called Rayleigh quotient: Ω |∇ϕ| 2 Ω ϕ 2 , we call u ∈ H 1 0 (Ω) the function attaining the minimum, which is the eigenfunction corresponding to λ 1 (Ω) and that solves the PDE −∆u = λ 1 (Ω)u, in Ω, u ∈ H 1 0 (Ω). The monotonicity and scaling properties of the eigenvalue follow immediately from its definition: for all t > 0, We finally recall the sharp quantitative Faber-Krahn inequality for the first eigenvalue of the Dirichlet-Laplacian, that was first proved in [8, Main Theorem].
. on an open set containing D .
We say that a property P(x) holds cap-quasi-everywhere in D, if it holds for all x ∈ D except at most a set of zero capacity, and in this case we write q.e. in D. A subset A of R N is said to be quasi-open if for every ε > 0 there exists an open subset ω ε of R N such that cap(ω ε ) < ε and A ∪ ω ε is open.
The notion of capacity is strictly related to spectral functionals such as the torsion energy and the first eigenvalue of the Dirichlet-Laplacian. In particular, one can not consider to be equivalent, a priori, two open (or quasi-open) sets which differ for a generic negligible set. Indeed for any open set Ω it is possible to construct a sequence of subsets Ω n ⊂ Ω of measure |Ω n | = |Ω| with E(Ω n ) < 1/n for all n ∈ N. For example take Ω = (0, 1) N and let {r i } i∈N be an enumeration of the rationals in (0, 1). Then, as cap((0, 1) N −1 ) > 0, it is possible to find k n so that and |Ω n | = |Ω|.
For every u ∈ H 1 (R N ), there exists a Borel and quasi-continuous representativeũ : R N → R of u and, ifũ andû are two quasi-continuous representatives of the same function u, then we haveũ =û q.e. in R N . From now on for every u ∈ H 1 (R N ), we consider only its quasi-continuous representative. In this setting, we are able to provide a more general definition of the space H 1 0 (Ω), which coincide with the usual one as soon as Ω is open, but that is suitable also for measurable sets (and quasi-open sets in particular).
It is nowadays standard to perform the minimization of functionals such as F α,ε in the class of quasi-open sets. As it can be noted, in the definition of the torsion energy and of the first eigenvalue of the Dirichlet-Laplacian, only the space H 1 0 (Ω) was really needed and therefore, once we have a definition which is suitable for quasi-open sets, we can work with them with no additional worries. On the other hand, the Riesz energy is well defined even for measurable sets, therefore there are no problems on its side.
As it is common in the Calculus of Variations, after finding a minimizer in the larger class of quasi-open sets, we will try later to restore the regularity of minimizers (and in particular, show that they are open).
We are now in position to properly define the problem we deal with in a large part of this paper. Let , so that a ball of radius R has measure greater than 1. Then we consider the problem From now on, we tacitly deal with quasi-open sets, unless otherwise stated.
2.3. Some notions of geometric measure theory. We give here some measure theoretic notions which will be used throughout the paper. Comprehensive references for this section are [2,31]. The measure theoretic perimeter (or De Giorgi perimeter) of a measurable set E is the quantity We say that E is a set of finite perimeter or Caccioppoli set if P (E) < +∞, that is if χ E is a function of bounded variation [2], and with ∇χ E we indicate the distributional derivative of χ E . Notice that if E is Lipschitz regular, by divergence theorem, where H k stands for the k−dimensional Hausdorff measure, k ∈ [0, N ]. For any Lebesgue measurable set E and t ∈ [0, 1] we define the quantities and the essential boundary of E as Beside the essential boundary we call reduced boundary the set exists and is a unit vector .
The quantity ν E (x) in the definition of ∂ * E is the measure theoretic normal of ∂E at the point x, whenever it is well defined. By results of Federer and De Giorgi [31] for sets of finite perimeter it holds In particular for a set of finite perimeter we have 3. An existence result for an auxiliary problem Let η ∈ (0, 1) and consider the function It is easy to check that, for all 0 ≤ s 2 ≤ s 1 , we have that We introduce then the functional G ε,η (Ω) := F α,ε (Ω) + f η (|Ω|), and, for R > ω −1/N N , the minimization problem In Section 5 we will show that such minimization problem and the minimization problem (10) are equivalent.
To do that we first have to prove existence and some mild regularity of minimizers of G ε,η . We begin by showing a lower bound for G ε,η on equibounded sets.
where B is any ball of measure 1.
Proof. By the Saint-Venant inequality and the positivity of V α we get On the other hand, if |Ω| ≥ 1 then and the conclusion easily follows.
The following existence result for the unconstrained functional G ε,η is an adaptation of [8,Lemma 4.6], which is in turn inspired by [9, Theorem 2.2 and Lemma 2.3].
There exists a minimizer in the class of quasi-open sets for problem (12). Moreover all minimizers have perimeter uniformly bounded by a constant depending on N, R, η.
Proof. Let (Ω n ) ⊂ B R be an minimizing sequence, with Let u n be the torsion function of Ω n , so that Ω n = {u n > 0} and let t n = 1/ √ n. We define which, since the torsion function of Ω n is precisely (u n − t n ) + , reads as Noting that recalling the property (11) of f η and the monotonicity of V α , the above inequality yields On the other hand, since η < 1, using coarea formula, the arithmetic geometric mean inequality and (14), we obtain Thanks to the choice of t n = 1/ √ n, we can find a level 0 < s n < 1/ √ n such that the sets W n := {u n > s n } satisfy It is easy to check that (W n ) is still a minimizing sequence for problem (12): where we have also used the monotonicity of V α and property (13) with s n in place of t n . Moreover, since the sets of the sequence (W n ) n∈N have equibounded perimeter, there exists a Borel set W ∞ such that (up to pass to subsequences) . On the other hand, the torsion function of W n , that is w n = (u n − s n ) + , is equibounded in H 1 (B R ). In fact, by Lemma 3.1, G ε,η is (uniformly) bounded from below and so Hence, up to subsequences, there is w ∈ H 1 0 (B R ) such that w n → w, strongly in L 2 (B R ) and weakly in H 1 0 (B R ). We set W := {w > 0}, and recall that we are identifying w with its quasi-continuous representative. Thus hence |W \ W ∞ | = 0, that is W ⊂ W ∞ up to a negligible set. We now observe that V α and f η are continuous with respect to the L 1 convergence of sets, while the first integral in the torsion energy is lower semicontinuous with respect to the weak H 1 and the second one with respect to the strong L 1 convergence. We can therefore pass to the limit in (15) and obtain On the other hand, using again the monotonicity of V α , we have e. and this is the desired minimizer for problem (12).
We conclude this section with a result concerning a property of the minimizers of G ε,η which will be useful later. N ) and B a ball of measure 1. There exist a constants ε 0 = ε 0 (N, α) > 0 and η 0 = η 0 (N, α) > such that, if η ≤ η 0 and ε ≤ ε 0 , then for any minimizer Ω of problem (12) we have Proof. The existence of an optimal set Ω follows from Lemma 3.2. If | Ω| ≥ 1, then we have, calling B a ball of unit measure, by minimality of Ω and as soon as we take ε ≤ ε 0 : On the other hand, if | Ω| < 1, using again the optimality of Ω we have

First regularity properties of minimizers of the unconstrained problem
In this Section we essentially follow the approach of [8,Section 4], which is in turn based on the seminal paper by Alt and Caffarelli [4], to prove density estimates, and Lipschitz regularity of the torsion function of minimizers for Problem (12).
The keystone idea is that we can pass from a functional defined on the class of quasi-open sets, to another defined on functions. In fact, for any Ω ⊂ B R quasi-open, calling u its torsion function, we have that Moreover, if Ω ε,η is optimal for problem (12), using the definition and minimality properties of its torsion function u ε,η , we have that, for Remark 4.1. In this section, we stress that instead of working on optimal sets for problem (12), we focus on functions optimal for problem (16). Clearly if u is optimal for problem (16), then it must be the torsion function of {u > 0}, therefore the two formulations are equivalent.
By Lemma 2.1 we get that u ε,η behaves like a quasi-minimizer 2 of a free boundary-type problem, that is for all v ∈ H 1 0 (B R ) and with a constant C depending only on N, α. Since v, u ε,η ∈ H 1 0 (B R ), from (17) ensues This quasi-minimality property does not provide any new information by itself and we need to take advantage of the (smallness of the) parameter ε, since the volume term is not in general of lower order term. We also (16) together with the monotonicity of V α entails that and we stress the fact that the parameter α does not appear in this formulation. Therefore, it should not be surprising that in the next Lemma 4.2 the constants ( as for example K 0 , ρ 0 ) do not depend on α.
We continue our analysis of the regularity of minimizers with the following non-degeneracy lemma. Its proof, which we provide for the sake of completeness, is basically a rewriting of [8,Lemma 4.9], in turn inspired by [4,Lemma 3.4].
and Ω be an optimal set for the problem we call u ∈ H 1 0 (Ω) its torsion function. For every κ ∈ (0, 1), there are positive constants K 0 , ρ 0 depending only on κ, η, N such that the following assertion holds: if ρ ≤ ρ 0 and x 0 ∈ B R , then Proof. Without loss of generality, we fix x 0 = 0. We also extend u to zero outside B R , so that it satisfies −∆u ≤ 1 in R N in weak sense. Then the function Thus, for every κ ∈ (0, 1), there exists c = c(κ, N ) such that (20) δ ρ := sup Let now w be the solution of We note that v = 0 in B κρ , therefore, using also (11), Thanks to the two inequalities above and the definition of v, we can infer On the other hand testing (21) with (u − w) + and integrating over B √ κρ \ B κρ , we obtain where ν denotes the outer unit normal exiting from B κρ and thanks to the fact that w = 0 on ∂B κρ and w ≥ u on ∂B √ κρ . We now observe that, since the torsion function on an annulus is explicit, with a direct computation one obtains We can now combine (22) and (23) to obtain Then, using the definition of δ ρ , the trace inequality in W 1,1 and the arithmetic geometric mean inequality we obtain for some β 2 = β 2 (N, κ) > 0. Putting together the above estimates, recalling again (20) we have, for all ρ ≤ ρ 0 Eventually, by choosing K 0 , ρ 0 such that up to possibly modify the constants K 0 , ρ 0 (but not their dependence only on N, κ, η).

Remark 4.4.
As it was first highlighted in [9], Lemma 4.2 holds for all sets that are optimal for a torsion energy-type functional only with respect to inward perturbations. These sets are referred to as shape subsolutions or inward minimizing sets and one can easily prove that if Ω is optimal for problem (12), then it is a shape subsolution for the torsion energy. Thus the non-degeneracy property of Lemma 4.2 follows from [9, Theorem 2.2]. Nevertheless we do not follow this approach since for our scope we need finer regularity properties of optimal sets that can not be deduced only by means of inward perturbations.
Remark 4.5. To obtain the regularity properties for minimizers we seek in this section, the previous lemma has to be paired with Lemma 4.6 below. Its proof is, as for the previous lemma, inspired by [8,Lemma 4.10], which is in turn based on [4]. One not completely obvious difference is that, contrary to the setting of [8], the parameter η is not fixed in our setting, thus we need to keep track of it in the proofs. This dependence on η will involve a dependence on R, the radius of the ball containing all competitors in Theorem 1.2. In particular the density estimates which ensue by the previous lemmata will depend on R, and this is a main obstacle in order to remove the equiboundedness hypothesis on competitors in (2).
Proof. First of all, we can reduce to the case when By maximum principle we have v > 0 in B ρ (x 0 ) and therefore Using this information, the quasi-minimality condition (17) of u and the property of the function f η , see (11), we obtain for some constant C = C(N, α, R). Now we can use the equation satisfied by v and the fact that ε < 1 < 1/η, to show   ii) For every x 0 ∈ ∂Ω and every ρ ≤ ρ 0 , we have This last result is the starting point of the higher regularity we need, that we treat in Section 6.

5.
Minima of the constrained problem are the same as the minima of the unconstrained problem In this section we show that unconstrained minima of G ε,η and volume constrained minima of F α,ε are actually the same. We begin by showing that for ε small, the minimizers of G ε,η in B R are close to a ball in L ∞ . To do that, we first start with an estimate that assures the L 1 −proximity of an optimal set for problem (12) to a ball with radius not too large.
Proof. Using Lemma 2.1 and the definition of f η , we get On the other hand, thanks to the quantitative version of the Saint-Venant inequality (Theorem 2.2), and since |Ω ε,η | = |B ε,η | we have (up to translations) that which proves the lemma.
A consequence almost immediate of the previous lemma is that the measure of the ball B ε,η is not too large Lemma 5.2. Let α and R be as in the previous lemma. There exists η 1 = η 1 (N, α, R) ≤ η 0 such that for all ε ∈ (0, 1) and η ≤ η 1 , we have that any optimal set for problem (12) satisfies Proof. Of course the statement of the lemma is trivial as long as |B R | ≤ 2. Thus we take R large enough so that |B R | > 2. Let us suppose for the sake of contradiction that |Ω ε,η | > 2. We are then going to reach a contradiction as long as for given constants C a (N, α) and C b (N, α) which will be precised later on in the proof. Since the functional is nondecreasing, we get sup where B is a ball of unit measure. On the other hand, using the Saint-Venant inequality, the positivity of V α , the fact that Ω ε,η ⊂ B R and since |Ω ε,η | > 2 we have By letting C a (N, α) = (−E(B))ω N and C b (N, α) = E(B) + V α (B), and by choosing η 1 = η 1 (N, α, R) such that η 1 ≤ η 0 and 1 we reach the desired contradiction.
We note that in the above lemma, η 1 depends on R and in particular η 1 ≈ 1 R N +2 .  (N, α) such that, for all ε ∈ (0, 1) and η ≤ η 1 , we have Proof. It is a direct consequence of Lemmas 5.2 and 5.1.
Next we show that, for ε small, the boundary of any optimizer Ω ε,η is close to the one of the corresponding optimal ball B ε,η in the definition of asymmetry, with respect to the Hausdorff distance d H (see [5,Definition 4.4.9] for the definition and properties of the Hausdorff distance).
It is worth noting that the constant ε δ in the lemma above depends also on R. This is one of the main difficulties in trying to get rid of the equiboundedness assumption of Theorem 1.2.
Remark 5.5. In view of the next result, we fix ε 1 (N, α, R) as the ε δ from Lemma 5.4 with the choice of δ := 1/2.
If ε ≤ ε 1 , then in the proof of Theorem 5.7, we will be allowed to inflate a set while remaining in a sufficiently big ball B R .
We can now show the equivalence between the constrained and the unconstrained problems. We will use the following elementary lemma.
Proof. It is easy to check that as the two functionals coincide on sets of measure 1. Then, if the first claim of the theorem holds, it follows that on the set of minimizers (of the first or of the second problem) the two functionals do coincide, that is, problems (10) and (12) are equivalent. We prove the first claim of the theorem by contradiction. Let and we also note that, since for all Ω ⊂ B R it holds F α,ε (Ω) ≤ εV α (B), then µ ≤ εV α (B). We moreover assume, without loss of generality, that Ω ε,η are minimizers for problem (12). We treat separately the case σ ε,η > 0 and σ ε,η < 0.
We recall from the previous sections that a minimizer Ω ε,η for G ε,η exists, and by Lemma 5.4, up to take ε 2 ≤ ε 1 as in Remark 5.5, and η 2 < η 1 as in Lemma 5.2, the rescaled set ρ ε,η Ω ε,η is still contained in B R , as soon as, for example, R 0 > 6.
We want to show that the minimum of the function g is attained at r = ρ := ρ ε,η . This is equivalent to show that for some η the inequality g(r) ≥ E(ρΩ ε,η ) + εV α (ρΩ ε,η ), for all r ∈ [1, ρ], holds true. Up to rearranging the terms, and by the homogeneity of the functionals E and V α such an inequality reads as Setting t := r ρ < 1, and observing that r N |Ω ε,η | = t N , the last inequality is equivalent to We recall now that V α (ρΩ ε,η ) ≤ V α (B) by the Saint Venant inequality, while E(ρΩ ε,η ) ≤ ρ N +2 E(B)

≤ E(B), by Lemma 3.3 and (25). Thus
Thus it is enough to show that for some η > 0 it holds To conclude that u ε > 0 in [0, 1) we directly apply Lemma 5.6 with u ε in place of u, −E(B) in place of P , and εV α in place of Q. Up to choose ε 2 small enough, depending only on N and α, we can satisfy the requirement of the Lemma. This concludes the proof.
We highlight that, from now on, we can fix an η > 0 so that Theorem 5.7 holds true, and therefore we have the equivalence of the constrained minimization problem for F α,ε and the unconstrained problem for G ε,η . It is then consistent to denote an optimal set for G ε,η or F α,ε by Ω ε (and u ε its torsion function), dropping the dependence on η.
On the other hand, we stress that this choice of η does depend on R!

Higher regularity of minimizers
In this section we show that the mild regularity proved in Section 4 can be improved to a higher regularity of minimizers for G ε,η or, equivalently, F α,ε . More precisely, we will show that minimizers of F α,ε are such that their boundary can be parametrized on the sphere so that the C 2,γ −norm of such a perturbation is arbitrarily small, up to choose ε small enough.
For this whole section, we fix R > R 0 and ε ≤ ε 2 (N, α, R) so that Theorem 5.7 holds. Then we denote Ω ε an optimal set for problem (12) and let u ε be its torsion function, extended to zero outside Ω ε . Hence u ε is optimal for problem (16).
We begin with a simple geometric result, whose proof is just a rephrasing of Lemma 5.4, since now we have the additional information that |Ω ε | = 1. Lemma 6.1. With the notations above, the sequence Ω ε converges to B in L 1 as ε → 0. Moreover, for any δ > 0 there exists ε δ > 0 such that if ε < ε δ , then To get the desired regularity of minimizers, we will apply results from [4], and techniques developed in [8], and later on in [14].
(i) There is a Borel function q uε : ∂Ω ε → R such that, in the sense of the distributions, one has (ii) There exist constants 0 < c < C < +∞, depending on R, N , α, such that c ≤ q uε ≤ C.
(iii) For all points x ∈ ∂ * Ω ε = ∂ * {u ε > 0}, the measure theoretic unit normal ν uε (x) is well defined and, as ρ → 0, Remark 6.3 (On the meaning of q uε ). For a regular set Ω, by means of a shape derivative argument, one can show that q uε (x) = |∂ ν u ε |(x) for x ∈ ∂Ω ε = ∂{u ε > 0}. The slightly more complicated arguments that follow are due since we only know, for the moment, that minimizers of problem (10) are open sets of finite perimeter. Namely, following ideas from [4] and [3], in order to show some higher regularity we first need to show some regularity results for q uε . Formally it is easy to see that the first variation of G ε,η reads as where u ε is the torsion function of Ω ε , v Ωε its Riesz potential and Λ some constant. Thus, since q uε stays far from zero and infinity (thanks to Theorem 6.2(ii)), then the regularity of q uε = ∂uε ∂ν is the same as that of v Ωε . Such relation on the other hand is not necessarily true, because of the lack of regularity of ∂Ω, but will turn out to be true on ∂ * Ω, the reduced boundary of Ω.
Before rigorously developing the argument described in the previous remark, we show a simple regularity result for the Riesz potentials. This is rather standard, but we give a proof for the sake of completeness.
Since α > 1, then N − α + 1 < N so that the w i are well defined. It is also clear that w ε is a smooth function. We define A 1 ε := {y ∈ A : |x − y| ≥ √ ε} and A 2 ε = A \ A 1 ε . Notice that by absolute continuity of the Lebesgue integral, it holds where o ε (1) does not depend on x, but only on the measure |A 2 ε |. Thanks to this, we have that (for a constant C depending only on N , α), Thus ∂ xi w ε (x) − w i (x) → 0 uniformly in R N . Since w ε converges pointwise to w, this implies that w is derivable and that It is now easy to show that ∂ xi w(x) is an Hölder continuous function. This concludes the proof.
In what follows we drop the subscript ε from Ω ε and u ε as here ε is fixed and there is no risk of confusion. The general strategy, and part of the details in the proof of the following theorem are inspired by an argument first proposed in [3] and readapted later on in [8].
Theorem 6.5. Let R > R 0 , α ∈ (1, N ) and ε ≤ ε 2 , and let Ω be a minimizer for G ε,η , u be its torsion function, v Ω = v = χ Ω * | · | α−N be its Riesz potential and q u be as in Theorem 6.2. Then the function x → q 2 Proof. Let us assume, for the sake of contradiction, that there are x 0 , x 1 ∈ ∂ * Ω such that . We construct a family of diffeomorphisms which preserves the volume at the first order by inflating Ω around x 0 , and deflating it around x 1 . Let κ < 1 and ρ < 1 be two parameters. Let ϕ ∈ C 1 0 (B 1 (0)) be a non-null, radially symmetric function supported in B 1 (0). Then we define The field τ is a diffeomorphism for ρ and κ small enough. Notice that τ (x)−x is null outside B ρ (x 0 )∪B ρ (x 1 ). A simple computation shows that We call Ω ρ = τ (Ω). We are going to show that for κ, ρ small enough it holds G ε,η (Ω ρ ) < G ε,η (Ω), contradicting the minimality of Ω. To do that we deal with the first variation of each term of the sum defining G ε,η . We stress that the computations regarding the volume and the torsion contributions are identical to those performed originally in [3] (see also [8] and [14], where the same idea is applied). We add them for the sake of completeness. Let us begin with the volume term. We claim that To see that, thanks to (11) we only have to show that Using the Area formula and the change of variables x = x i + ρy, we have that We can then deduce by Theorem 6.2 point (iii) that where the last equality is due to the radial symmetry of ϕ. Now that (27) is settled, we deal with the Torsion energy term. We claim that We note that ν can be any unit direction of R N : changing direction does not affect the value of C(ϕ), thanks to the radial symmetry of ϕ. To show (28) it suffices to prove that where u ρ = u • τ −1 , and that Indeed u ρ is a test function in the definition of E(Ω ρ ) so that (30) and (31) imply directly (28). The computation of (30) is exactly as in [3, Section 2] (it is done also in [8] and [14]), hence we do not repeat it here. To show (31), we compute where we performed the change of variable x = x i + ρy, and exploited the Taylor expansion det(∇(Id + ρκϕ(|y|)) = 1 + ρκϕ ′ (|y|) y |y| + o(ρ). Next we deal with the Riesz energy term V α . We are going to show that (32) 1 where C(ϕ) is the constant defined in (29). The proof of this variation is longer than the previous ones. Let us denote by v ρ (·) = χ Ωρ * | · | α−N the Riesz potential of Ω ρ , and by v(·) = χ Ω * | · | α−N the Riesz potential of Ω. We have We compute the last two addends of the previous formula separately: First of all we focus on the first term of the chain of inequalities above. By Lemma 6.4, and by Ascoli-Arzelà Theorem, v ρ uniformly converges in B 1 (0) to some function v as ρ → 0, and, since its pointwise limit is v, we have that v = v. As a consequence, using also Lemma 6.4 and the Dominate convergence theorem, we have uniformly on the compact sets and, again, the dominate convergence Theorem. A completely analogous computation shows that We wish to show now that the first addend on the right-hand side of (33) converges to 0 as ρ → 0. To this aim, we compute We remark that the last two addends converge to the same constant, with opposite sign. Thus in the limit they elide themselves: Now we notice that for any X, Y, Z ∈ R N it holds that .
Such an inequality can be proved easily by convexity, see for instance [18, formula 2.11]. By applying such an inequality in the first two addends of the right-hand side of (34) with X = x, Y = x i + ρy and Z = τ (x i + ρy) we get that i∈{0,1} B1(0)∩( In the second inequality we used the fact that Since the last two integrals are finite, being α > 1, we get the desired claim, that is (32).
The conclusion now readily follows: by minimality of Ω and thanks to (27), (28) and (32) we have that ) < 0, by hypothesis, by choosing ρ small enough, and then κ small enough, we get the desired contradiction. The proof is concluded.
An immediate consequence of Lemma 6.4 and Theorem 6.5 is the following.
Finally, to prove that the boundary of Ω ε is locally the graph of a C 2,γ function on the boundary of a ball, we only need to implement the improvement of flatness technique from [4, Section 7 and 8], which can be readapted with minimal changes to our setting as shown in [22,Appendix]. Definition 6.7. Let µ ± ∈ (0, 1] and k > 0. A weak solution u of (26) is of class F (µ − , µ + , k) in B ρ (x 0 ) with respect to direction ν ∈ S N −1 if (a) x 0 ∈ ∂{u > 0} and We note that if k = +∞, then condition (b) is void, that is, no bounds on the gradient are required. The fact that our minimizers are nearly spherical sets of class C 2,γ is now a direct consequence of the following regularity result, which was first proved in [4,Theorem 8.1] and [25, Theorem 2]. Theorem 6.8. Let u be a weak solution to (26) in B R and assume that q u is C 1,γ for some constant γ ∈ (0, 1) in a neighborhood of {u > 0}. Then there are constants µ and k, depending only on N , α, R, max q u , min q u , q u C 1,γ such that: If u is of class F (µ, 1, +∞) in B 4ρ (x 0 ) with respect to some direction ν ∈ S N −1 with µ ≤ µ and ρ ≤ kµ 2 , then there exists a C 2,γ function f : R N −1 → R with f C 2,γ ≤ C(N, α, R, q u C 1,γ ) such that, calling

Proof of Theorem 1.2
In the last section we have shown that any minimizer for problem (2) has boundary close to that of a ball (precisely, the ball which achieve the minimum in the definition of asymmetry), and is locally C 2,γ − regular. This, reasoning as in [8,Proof of Proposition 4.4], is enough to show that such a minimum is a nearly spherical set, and to conclude the proof of Theorem 1.2.
Proof of Theorem 1.2. Thanks to Theorem 5.7 and Lemma 3.2, for ε * small enough (depending on N, α, R), there is a minimizer Ω ε for (2) and we can assume without loss of generality that the barycenter of Ω ε is x Ωε = 0. It is not difficult to show that the sequence of the translated sets Ω ε with barycenter at the origin still converges in L 1 to the ball B of unit measure and centered at the origin, and thus the statement of Lemma 6.1 applies for them. We call u ε the torsion function of Ω ε , so that Ω ε = {u ε > 0}. We claim that Ω ε is a C 2,γ nearly spherical set. To see this, let k, µ be as in Theorem 6.8 and µ < µ to be fixed later. There exists ρ(µ) ≤ kµ 2 such that, for all ρ ≤ ρ(µ) and all x ∈ ∂B, we have where ν x is the outer unit normal to ∂B at x. By Lemma 6.1, up to take ε E small enough (depending possibly also on µ), there is a point Therefore, with the notation of Definition 6.7, u ε is of class F (µ, 1, +∞) in B 4ρ(µ) in direction ν x and hence, by Theorem 6.8 and Corollary 6.6, we infer that ∂Ω ε ∩ B ρ(µ) is the graph of a C 2,γ function with respect to ν x . Up to choose µ small enough, there are functions ϕ x ε with C 2,γ norm uniformly bounded such that As the balls {B ρ(µ) (x)} x∈∂B cover ∂B, by compactness there is a function ϕ ε ∈ C 2,γ (∂B) with bounded C 2,γ norm. Moreover, up to take ε E small enough, by Lemma 6.1, we can assume that ϕ ε C 2,γ ′ is as small as we wish. A direct application of [8, Theorem 3.3] (recalling also that Ω ε has barycenter in the origin) entails that By minimality of Ω ε and the two bounds above, we have Since the constants C and C ′ are independent of ε, we can take ε E small enough (depending on N, α, R) so that, for all ε ≤ ε E we have E(Ω ε ) = E(B), and by the rigidity of the Saint-Venant inequality, we conclude.

A surgery result for the functional involving the first eigenvalue
In this section we prove the following surgery result. Throughout this section, Ω is an open set of unit measure, B is the ball of unit measure centered at the origin and we define  The proof of the proposition is quite technical and is mostly inspired by [32] (see also [10]). We have skipped the proofs that are essentially identical, while we have detailed the points where substantial changes need to be made.
Remark 8.2. On the analogies and differences with respect to [32]. The connectedness assumption is a main difference with respect to the work in [32], though it does not change much the argument. The reason for which we need to impose it is the presence in our functional of the repulsive Riesz potential energy. On the other hand this difficulty is compensated by the fact that, by choosing ε small, we can arbitrarily impose that the sets we take into account have small Fraenkel asymmetry. Moreover, dealing with only the first eigenvalue simplifies many technical steps related to the orthogonality of the higher eigenfunctions. Let us introduce some notation. Let Ω be a connected set such that with λ 1 (Ω) − λ 1 (B) ≤ δ(N, α), so that, by the quantitative Faber-Krahn inequaity (see Theorem 2.3), up to translations we have From now on we fix Ω so that B is the ball of unit measure attaining the asymmetry and we will no more translate it. By defining K = K(N ) := λ 1 (B) + 1 ≥ λ 1 (B) + δ(N, α) we get immediately We then call t := 1 ωN 1/N the radius of the ball B and note that for all t ≥ t.
Let m ∈ (0, 1/4) be such that Moreover, we choose δ(N, α) small enough so that We first focus on the direction e 1 and detail the construction in this case. We shall denote z = (x, y) ∈ R×R N −1 and by z i the i-th component of z ∈ R N . For any t ∈ R, we define Ω t := y ∈ R N −1 : (t, y) ∈ Ω , and given any set Ω ⊆ R N , we define its 1-dimensional projections for 1 ≤ p ≤ N as Observe that We call u the first eigenfunction on Ω with unit L 2 norm. We define then also, for every t ≤t, which makes sense since u is smooth inside Ω. It is convenient to give the further notation Applying the Faber-Krahn inequality in R N −1 to the set Ω t , and using the rescaling property of eigenvalues on , calling B N −1 the unit ball in R N −1 . As a trivial consequence, we can estimate µ in terms of ε and δ: in fact, noticing that u(t, ·) ∈ H 1 0 (Ω t ) and writing ∇u = (∇ 1 u, ∇ y u), we have We can now present two estimates which assure that u and ∇u can not be too big in Ω − (t). , for some C 1 = C 1 (N ) (recalling that K for us is a precise constant depending only on N ).
The proof of the above Lemma follows exactly as in [32,Lemma 2.3]. Let us go further into the construction, giving some additional definitions. For any t ≤ −t and σ(t) > 0, we define the cylinder Q(t) as where for any t ≤ −t we set We let also Ω(t) = Ω + (t) ∪ Q(t), and we introduce u ∈ H 1 0 Ω(t) as The fact that u vanishes on ∂ Ω(t) is obvious; moreover, ∇u = ∇ u on Ω + (t), while on Q(t) one has

y) .
A simple calculation allows us to estimate the integrals of u and ∇ u on Q(t).
Lemma 8.4. For every t ≤ −t, one has for a suitable constant C 2 = C 2 (N ).
The proof of the above Lemma follows as [32,Lemma 2.4].
Another simple but useful estimate concerns the Rayleigh quotients of the functions u on the sets Ω(t).
Lemma 8.5. There exists a constant C 3 = C 3 (N ) such that for every t ≤ −t, one has The proof of the above Lemma follows as in [32,Lemma 2.5], but it is actually simpler since in our setting only the first eigenfunction is involved and we do not need to take care of orthogonality constraints.
We can now enter in the central part of our construction. Basically, we aim to show that either Ω already has bounded left "tail" in direction e 1 , or some rescaling of Ω(t) has energy lower than that of Ω. Lemma 8.6. Let Ω be as in the assumptions of Lemma 8.3, and let t ≤ −t. There exist ε = ε(N, α) and C 4 = C 4 (N, α) > 2 such that, for all ε ≤ ε exactly one of the three following conditions hold: (1) max ε(t), δ(t) > 1; (2) (1) does not hold and m(t) ≤ C 4 ε(t) + δ(t) ε(t) 1 N −1 ; (3) (1) and (2) do not hold and one has that λ 1 ( Ω(t)) ≤ λ 1 (Ω) and Proof. Assume (1) is false. Then it is possible to apply Lemma 8.5, to get By the scaling properties of the eigenvalue and the fact that Ω(t) = 1, we know that By construction, Ω(t) = Ω + (t) + Q(t) = 1 − m(t) + ε(t) N N −1 , hence the above estimates and (38) lead to At this point, defining then condition (2) holds true. Otherwise, we immediately have that for a constant C 5 > 0, therefore the first part of the third claim is verified. Moreover, we can compute, using Lemma 2.1, the Riesz inequality and noting that |Ω∆ Then, putting together (39) and (40), we deduce up to take ε ≤ ε(N, α) < C5 2C6 , so that in this case condition (3) holds and the proof is concluded.  with the usual convention that, if condition (3) is false for every t ≤ −t, thent = −∞. We introduce now the following subsets of (t, −t), A : = t ∈ (t, −t) : condition (1) of Lemma 8.6 holds for t , B : = t ∈ (t, −t) : condition (2) of Lemma 8.6 holds for t , and we further subdivide them as A 2 := t ∈ A : ε(t) < δ(t) , We aim to show that both A and B are uniformly bounded. Concerning A 1 , observe that so that |A 1 | ≤ 1. Concerning A 2 , in the same way and also recalling that λ 1 (Ω) ≤ K, we have so that |A 2 | ≤ K. Summarizing, we have proved that Let us then pass to the set B 1 . To deal with it, we need a further subdivision, namely, we write B 1 = ∪ n∈N B n 1 , where (42) B n 1 := t ∈ B 1 : We note that it is possible that some of the B n 1 are empty, in particular this happens for n < 2N − 1, because m(t) ≤ |Ω\[−t,t] N | ≤ m 2 2N , but this does not affect our argument. Keeping in mind (36), we know that t → m(t) is an increasing function, and that for a.e. t ∈ R one has m ′ (t) = ε(t). Moreover, for every t ∈ B 1 one has by construction that As a consequence, for every t ∈ B n 1 one has This readily implies 1 C m which in turn gives Finally, we deduce Concerning B 2 , we can almost repeat the same argument: in fact, thanks to (37), for every t ∈ B 2 we have which is the perfect analogous of the above setting with δ and φ in place of ε and m respectively. Since as already observed φ(t) ≤ Ω |∇u| 2 ≤ K, in analogy with (42) we can define B n 2 := t ∈ B 2 : thus the very same argument which leads to (43) now gives Putting (41), (43) and (44) together, we find We need now to distinguish two cases for Ω. Case I. One hast = −∞.
If this case happens, then condition (3) of Lemma 8.6 never holds true, i.e., for every t ≤ −t either condition (1) or (2) holds. Recalling the definition of A and B and (45), we deduce that, choosing simply U − 1 = Ω, 2t Therefore, the remaining parts of the claim of Lemma 8.7 is immediately obtained, noting that clearly In this case, let us notice the connectedness of Ω assures that it must be m(t) > 0, hence (t, −t) ⊆ A ∪ B and thus by (45)t ≥ −t − C 7 . Let us now pick some t ⋆ ∈ [t − 1,t] for which condition (3) holds, and define U − 1 := Ω(t ⋆ ). By definition, U − 1 has unit volume, and Observe now that by definition, for every 2 ≤ p ≤ N , one has π p Ω(t ⋆ ) = π p Ω + (t ⋆ ) , hence where we have used that Ω(t ⋆ ) ≥ 1/2. On the other hand, it is clear from the construction that As a consequence, being t ⋆ ≥ t − 1 ≥ −t − C 7 − 1, we deduce Concerning the last part of the claim, recalling again that we infer that so the proof is concluded also in this case.
In order to conclude our surgery result, we need to iterate Lemma 8.7. First, we apply it to U − 1 , in direction e 1 for t ≥ 2t =: t + 1 . Then we will recursively apply it to the new set that we obtain, in order to get a uniform boundedness in all the other N − 1 coordinate directions. We need to take care that, while rescaling, the diameter of the projections in the directions we already dealt with remains bounded.
For the first step, dealing with U − 1 in direction e 1 for t ≥ t + 1 , Lemma 8.6 can be repeated analogously with a suitable change of notation, for t ≥ t + 1 ≥t: We can then prove the following Lemma.
, and therefore diam(π 1 (U + 1 )) ≤ C + 1 (N, α) := 6C 7 + 12 + 6t + 1 , We argue exactly as in the proof of Lemma 8.7, noting that, since λ 1 (U − 1 ) ≤ λ 1 (Ω), the set still satisfies the condition λ 1 (U − 1 ) ≤ K and moreover Concerning the other bound, it is enough to observe, as in the proof of Lemma 8.7, that We can now iterate the argument in all the other coordinate directions to prove Proposition 8.1.
(Ω), and moreover diam(π 1 (U + 1 )) ≤ C + 1 (N, α), We now iterate the argument in all the remaining directions. Let us show it for e 2 . We call first t − 2 := 2t + 1 = 4t and repeat Lemma 8.7 to U + 1 in direction e 2 for z 2 ≤ −t − 2 so that we can find a new open and connected set of unit measure U − 2 such that Then we call t + 2 = 2t − 2 and repeat Lemma 8.8 to U − 2 in direction e 2 for z 2 ≥ t + 2 in order to find a new open and connected set of unit measure U + 2 such that λ 1 (U + 2 ) ≤ λ 1 (U − 2 ), F α,ε (U + 2 ) ≤ F α,ε (U − 2 ), diam(π 1 (U + 2 )) ≤ 2 2 C + 1 (N, α), diam(π 2 (U + 2 )) ≤ C + 2 (N, α), where we can take C + 2 (N, α) := 6C 7 + 12 + 6t + 2 . We note that the last condition is needed so that we can restart the cutting procedure in direction e 3 knowing that |U + 2 ∩ {z 3 ≤ −2t + 2 }| ≤ m, which is the condition required for Lemma 8.3. Iterating this procedure other N − 2 times, we obtain in the end an open and connected set of unit measure U + N such that λ 1 (U + N ) ≤ λ 1 (Ω), F α,ε (U + N ) ≤ F α,ε (Ω), diam(π p (U + N )) ≤ 2 2(N −p) C + p (N, α), for p = 1, . . . , N, where C + p (N, α) := 6C 7 + 12 + 6t + p and t + p = 2 2p−1t for all p = 1, . . . N . Clearly Ω = U + N is a good choice for proving the claim. 9. Proof of Theorem 1.1 As outlined in the introduction, the proof of Theorem 1.1 can be obtained as the juxtaposition of two independent results. Hence, we divide the section in two parts. In the first one, we prove the minimality of the ball for the functional F α,ε among equibounded sets. Then we use the surgery argument of Section 8 to conclude the proof of Theorem 1.1. 9.1. Rigidity of the ball in the class of equibounded sets. We aim to prove the following result.
The previous result is an easy consequence of Theorem 1.2 and the Kohler-Jobin inequality [28,7], which we recall. Proof of Proposition 9.1. We follow here a smart and easy technique proposed in [8]. Let Ω ⊂ B R be an (open) set with |Ω| = 1 and we call B a ball of unit measure centered for the sake of simplicity in the origin, as B R . By rewriting (47) as where we have used also the Saint-Venant inequality. Since, by Theorem 1.2 for all ε ≤ ε E (N, α, R), the ball of unit measure is the only minimizer for the functional Ω → E(Ω) + εV α (Ω), for Ω ⊂ B R , |Ω| = 1, we deduce that, On the other hand, if E(B) < 2E(Ω), we can still obtain from (48) ε(V α (B) − V α (Ω)).
In conclusion, we have proved that, for all we have that λ 1 (Ω) + εV α (Ω) ≥ λ 1 (B) + εV α (B), for all Ω ⊂ B R , |Ω| = 1, and the proof is concluded. 9.2. Conclusion of the proof of Theorem 1.1. We are now in position to prove the main result of the paper.
Clearly one can take a minimizing sequence (A n ) n∈N made of smooth sets. We show that the elements of the minimizing sequence (A n ) can be chosen to be also connected. Let us take a smooth open set of unit measure A, which is made of an at most countable number of connected components, For all ϑ > 0, we consider the segment S k connecting the components A k and A k+1 and we consider T k,ϑ = ∪ x∈S k B ζ (x), choosing ζ so that |T k,ϑ | ≤ ϑ 2 k . We call now A ϑ := ∪ k∈N T k,ϑ ∪ A ⊃ A, A ϑ := |A ϑ | −1/N A ϑ , and note that |A ϑ | ≤ 1 + ϑ, | A ϑ | = 1. By monotonicity and scaling of the eigenvalue, we immediately deduce On the other hand, by Lemma 2.1 and scaling of the Riesz energy, All in all, we have that, for all ϑ > 0, F α,ε ( A ϑ ) ≤ F α,ε (A) + C(N, α)ϑ, and by arbitrariety of ϑ, applying this procedure to all the elements of the minimizing sequence, we deduce that assuming all elements of the sequence to be connected is not restrictive.
At this point we can take (A n ) n∈N a minimizing sequence for problem (1), made of smooth, connected sets of unit measure. Up to take ε λ1 ≤ ε, for all n, we can apply Proposition 8.1 to A n and we find a new open and connected set, A n , of unit measure and with D(N, α).
Hence, ( A n ) n∈N is still a minimizing sequence for problem (46), made by sets with uniformly bounded diameter. It is eventually enough to restrict the minimization problem to a ball B R with R = D(N, α), and we can find that the unit ball is the unique optimal set for problem (46), thanks to Proposition 9.1. We note that, since now R has been fixed equal to D(N, α), then the constant ε R λ1 now depends only on N, α and we conclude taking ε λ1 = min{ε D(N,α) λ1 , ε, δ/V α (B)}.

The non-existence results
In this section we show Theorem 1.6.
Proof of Theorem 1.6. We give the proof just for the case of the torsion energy, since the other one is analogous. Notice that any set in U(δ) is bounded. Moreover any minimizer must be connected. Otherwise, if Ω is made up by the union of two disjoint open sets Ω 1 and Ω 2 we have that E(Ω) does not change by sending toward infinity Ω 1 while keeping Ω 2 fixed, while V α under such a translation strictly decreases, contradicting the minimality. Now, any connected open set lying in U(δ) has bounded diameter, with a bound depending only on δ and N : for all Ω ∈ U(δ).
Therefore, for all Ω ∈ U(δ), we can compute the following lower bound on F α,ε : where we have used the Saint-Venant inequality. On the other hand, let us consider Ω := B(x 1 ) ∪ B(x 2 ) the union of two disjoint balls of measure 1/2 each and centers x 1 , x 2 . We denote the distance between the two balls by d(x 1 , x 2 ), that is It is easy to check that, denoting by B the ball of unit measure, To show that there can not be minimizers for problem (3) it is enough to check that, the set Ω makes the functional F α,ε have less energy than the lower bound in (49), up to take d(x 1 , x 2 ) large enough. We obtain this as soon as E(B) + ε 1 d(δ) N −α > 2 −2/N E(B) + ε , up to choose x 1 , x 2 so that d(x 1 , x 2 ) > 2d(δ).