LOCAL AND GLOBAL MINIMALITY RSULTS FOR A NONLOCAL ISOPRIMTRIC PROBLM ON R N M. BONACINI AND R. CRISTOFRI Abstract. We consier a nonlocal isoperimetric problem efine in the whole space R N, whose nonlocal part is given by a Riesz potential with exponent α 0, N. We show that critical configurations with positive secon variation are local minimizers an satisfy a quantitative inequality with respect to the L -norm. This criterion provies the existence of a explicitly etermine critical threshol etermining the interval of volumes for which the ball is a local minimizer, an allows to aress several global minimality issues.. Introuction In this paper we provie a escription of the energy lanscape of the family of functionals χ xχ y F := P + γ x y α xy, α 0, N, γ > 0. R N R N efine over finite perimeter sets R N, N 2. Here P enotes the perimeter of in R N an χ its characteristic function. In the following we will usually enote the secon term in the functional by N L α, an we will omit the subscript α when there is no risk of ambiguity. The nature of the energy. appears in the moeling of ifferent physical problems. The most physically relevant case is in three imensions with α =, where the nonlocal term correspons to a Coulombic repulsive interaction: one of the first examples is the celebrate Gamow s waterrop moel for the constitution of the atomic nucleus see [5], an energies of this kin are also relate via Γ-convergence to the Ohta-Kawasaki moel for iblock copolymers see [29]. For a more specific account on the physical backgroun of this kin of problems, we refer to [27]. From a mathematical point of view, functionals of the form. recently rew the attention of many authors see [, 7, 8,, 7, 8, 20, 22, 23, 25, 28, 33]. The main feature of the energy. is the presence of two competing terms, the sharp interface energy an the long-range repulsive interaction. Inee, while the first term is minimize by the ball by the isoperimetric inequality, the nonlocal term is in fact maximize by the ball, as a consequence of the Riesz s rearrangement inequality see [24, Theorem 3.7], an favours scattere or oscillating configurations. Hence, ue to the presence of this competition in the structure of the problem, the minimization of F is highly non trivial. We remark that, by scaling, minimizing. uner the volume constraint = m is equivalent to the minimization of the functional N α+ m N P + γ N L α B uner the constraint = B, where B is the unit ball in R N. It is clear from this expression that, for small masses, the perimeter is the leaing term an this suggests that in this case the ball shoul be the solution to the minimization problem; on the other han, for large masses the nonlocal term becomes ominant an causes the existence of a solution to fail. Inee it was prove, although not in full generality, that the functional F is uniquely minimize up to translations by the ball for every value of the volume below a critical threshol: in [22] for 200 Mathematics Subject Classification. 49Q0, 49Q20, 35R35, 82B24. Key wors an phrases. Nonlocal isoperimetric problem, Minimality conitions, Secon variation, Local minimizers, Global minimizers. Preprint SISSA 32/203/MAT.
2 M. BONACINI AND R. CRISTOFRI the planar case N = 2, in [23] for 3 N 7, an in [20] for any imension N but for α = N 2. Moreover, the existence of a critical mass above which the minimum problem oes not amit a solution was establishe in [22] in imension N = 2, in [23] for every N an for α 0, 2, an in [25] in the physical interesting case N = 3, α =. In this paper we aim at proviing a more etaile picture of the energy lanscape of the functional. by a totally ifferent approach, base on the positivity of the secon variation an mainly inspire by [] which eals with the same energy functional, with α = N 2, in a perioic setting. First, we reobtain an strengthen some of the above results, proving in full generality that the ball is the unique global minimizer for small masses, without restrictions on the parameters N an α Theorem 2.0. Moreover, for α small we also provie a complete characterization of the groun state, showing that the ball is the unique global minimizer, as long as a minimizer exists Theorem 2., an that in this regime we can write 0, = k m k, m k+ ], with m k+ > m k, in such a way that for m [m k, m k ] a minimizing sequence for the functional is given by a configuration of at most k isjoint balls with iverging mutual istance Theorem 2.2. The results state in Theorem 2. an Theorem 2.2 are completely new except for the first one, which was prove only in the special case N = 2 in [22]. Finally, we also investigate for the first time in this context the issue of local minimizers, that is, sets which minimize the energy with respect to competitors sufficiently close in the L -sense where we measure the istance between two sets by the quantity 2.5, which takes into account the translation invariance of the functional. For any N an α we show the existence of a volume threshol below which the ball is also an isolate local minimizer, etermining it explicitly in the three imensional case with a Newtonian potential Theorem 2.9. As anticipate above, our methos follow a secon variation approach which has been recently evelope an applie to ifferent variational problems, whose common feature is the fact that the energy functionals are characterize by the competition between bulk energies an surface energies see [4] in the context of epitaxially straine elastic films, [5, 4] for the Mumfor-Shah functional, [6] for a variational moel for cavities in elastic boies. In particular we stress the attention on [], which eals with energies in the form. in a perioic setting see also [2], where the same problem is consiere in an open set with Neumann bounary conitions. The basic iea is to associate with the secon variation of F at a regular critical set, a quaratic form efine on the functions ϕ H such that ϕ = 0, whose non-negativity is easily seen to be a necessary conition for local minimality. In fact, one of the main tools in proving the aforementione results is represente by Theorem 2.8, where we show that the strict positivity of this quaratic form is sufficient for local minimality with respect to L -perturbations. Although the general strategy to establish this theorem follows the one evelope in [4, ], we have to tackle the nontrivial technical ifficulties coming from working with a more general nonlocal term the exponent α is allowe to range in the whole interval 0, N an from the lack of compactness of the ambient space R N. The proof passes through an intermeiate step, which amounts to showing that a critical set with positive secon variation is a minimizer with respect to sets whose bounary is a normal graph on with W 2,p -norm sufficiently small Theorem 3.. Then a contraiction argument, which is mainly base on the regularity properties of quasi-minimizers of the area functional an which is close in spirit to the ieas introuce by Cicalese an Leonari [0] to provie an alternative proof of the stanar quantitative isoperimetric inequality leas to the conclusion that this weaker notion of local minimality implies the local minimality in the L -sense. The proofs of the global minimality results escribe above require aitional arguments an nontrivial refinements of the previous ieas. An issue which remains unsolve is concerne with the structure of the set of masses for which the problem oes not have a solution: is it always true that it has the form m, + for all the values of α an N? Notice that we provie a positive answer to this question in the case where α is small. Another interesting question asks if there are other global or local minimizers ifferent from the ball. Finally, our analysis leaves open the case of α [N, N, which seems to require ifferent techniques.
LOCAL AND GLOBAL MINIMALITY RSULTS FOR A NONLOCAL ISOPRIMTRIC PROBLM ON R N 3 This paper is organize as follows. In Section 2 we set up the problem an we list the main results of this work. The notion of secon variation of the functional F is introuce in Section 3, together with the first part of the proof of the main result of the paper, which is complete in Section 4. In Section 5 we compute explicitly the secon variation of the ball, an we iscuss its local minimality by applying our sufficiency criterion. Finally, Section 6 is evote to the proof of the results concerning the global minimality issues. 2. Statements of the results We start our analysis with some preliminary observations about the features of the energy functional., before listing the main results of this work. For a finite perimeter set, we will enote by ν the exterior generalize unit normal to, an we will not inicate the epenence on the set when no confusion is possible. Given a measurable set R N, we introuce an auxiliary function v by setting v x := x y α y for x RN. 2. The function v can be characterize as the solution to the equation s v = c N,s χ, s = N α 2.2 2 where s enotes the fractional laplacian an c N,s is a constant epening on the imension an on s see [2] for an introuctory account on this operator an the references containe therein. Notice that we are intereste in those values of s which range in the interval 2, N 2. We collect in the following proposition some regularity properties of the function v. Proposition 2.. Let R N be a measurable set with m. Then there exists a constant C, epening only on N, α an m, such that v W, R N C. Moreover, v C,β R N for every β < N α an v C,β R N C for some positive constant C epening only on N, α, m an β. Proof. The first part of the result is prove in [23, Lemma 4.4], but we repeat here the easy proof for the reaer s convenience. By 2., v x = x y α y + \B x x y α y y + m C. B y α B x By ifferentiating 2. in x an arguing similarly, we obtain v x α y α y + α m C. x y α+ B y α+ Finally, by aing an subtracting the term x y z y β z y x y β, we can write x y α+β+2 z y α+β+2 v x v z α x y x y α+2 z y z y α+2 y α x y α+β+ + x y β z y α+β+ z y β y 2.3 + α x y z y β z y x y β x y α+β+2 z y α+β+2 y Observe now that for every v, w R N \ {0} v v w α+2β+ w w v α+2β+ = v v α+2β w w α+2β C max{ v, w } α+2β v w C max{ v, w } α+β+ v w β
4 M. BONACINI AND R. CRISTOFRI where C epens on N, α an β. Using this inequality to estimate the secon term in 2.3 we euce v x v z α x z β x y α+β+ + z y α+β+ + C min{ x y, z y } α+β+ y which completes the proof of the proposition, since the last integral is boune by a constant epening only on N, α, m an β. Remark 2.2. In the case α = N 2, the function v solves the equation v = c N χ, an the nonlocal term is exactly N L N 2 = v x 2 x. R N By stanar elliptic regularity, v W 2,p loc RN for every p [, +. The following proposition contains an auxiliary result which will be use frequently in the rest of the paper. Proposition 2.3 Lipschitzianity of the nonlocal term. Given ᾱ 0, N an m 0, +, there exists a constant c 0, epening only on N, ᾱ an m such that if, F R N are measurable sets with, F m then N L α N L α F c 0 F for every α ᾱ, where enotes the symmetric ifference of two sets. Proof. We have that N L α N L α F = = R N \F F χ xχ y χ F y R x y α + χ F yχ x χ F x x y α xy N v x + v F x x v x + v F x x F \ v x + v F x x 2C F, where the constant C is provie by Proposition 2., whose proof shows also that it can be chosen inepenently of α ᾱ. The issue of existence an characterization of global minimizers of the problem min { F : R N, = m }, 2.4 for m > 0, is not at all an easy task. A principal source of ifficulty in applying the irect metho of the Calculus of Variations comes from the lack of compactness of the space with respect to L convergence of sets with respect to which the functional is lower semicontinuous. It is in fact well known that the minimum problem 2.4 oes not amit a solution for certain ranges of masses. Besies the notion of global minimality, we will aress also the stuy of sets which minimize locally the functional with respect to small L -perturbations. By translation invariance, we measure the L -istance of two sets moulo translations by the quantity α, F := min x R N x + F. 2.5 Definition 2.4. We say that R N is a local minimizer for the functional. if there exists δ > 0 such that F FF for every F R N such that F = an α, F δ. We say that is an isolate local minimizer if the previous inequality is strict whenever α, F > 0.
LOCAL AND GLOBAL MINIMALITY RSULTS FOR A NONLOCAL ISOPRIMTRIC PROBLM ON R N 5 The first orer conition for minimality, coming from the first variation of the functional see 3., an also [9, Theorem 2.3], requires a C 2 -minimizer local or global to satisfy the uler-lagrange equation H x + 2γv x = λ for every x 2.6 for some constant λ which plays the role of a Lagrange multiplier associate with the volume constraint. Here H := iv τ ν x enotes the sum of the principal curvatures of iv τ is the tangential ivergence on, see [3, Section 7.3]. Following [], we efine critical sets as those satisfying 2.6 in a weak sense, for which further regularity can be gaine a posteriori see Remark 2.6. Definition 2.5. We say that R N is a regular critical set for the functional. if is a boune set of class C an 2.6 hols weakly on, i.e., iv τ ζ H N = 2γ v ζ, ν H N for every ζ C R N ; R N such that ζ, ν H N = 0. Remark 2.6. By Proposition 2. an by stanar regularity see, e.g., [3, Proposition 7.56 an Theorem 7.57] a critical set is of class W 2,2 an C,β for all β 0,. In turn, recalling Proposition 2., by Schauer estimates see [6, Theorem 9.9] we have that is of class C 3,β for all β 0, N α. We collect in the following theorem some regularity properties of local an global minimizers, which are mostly known see, for instance, [23, 25, 33] for global minimizers, an [] for local minimizers in a perioic setting. The basic iea is to show that a minimizer solves a suitable penalize minimum problem, where the volume constraint is replace by a penalization term in the functional, an to euce that a quasi-minimality property is satisfie see Definition 4.. Theorem 2.7. Let R N be a global or local minimizer for the functional. with volume = m. Then the reuce bounary is a C 3,β -manifol for all β < N α, an the Hausorff imension of the singular set satisfies im H \ N 8. Moreover, is essentially boune. Finally, every global minimizer is connecte, an every local minimizer has at most a finite number of connecte components. Proof. We ivie the proof into three steps, following the ieas containe in [, Proposition 2.7 an Theorem 2.8] in the first part. Step. We claim that there exists Λ > 0 such that is a solution to the penalize minimum problem { min FF + Λ F : F R N, αf, δ }, 2 where δ is as in Definition 2.4 the obstacle αf, δ 2 is not present in the case of a global minimizer. To obtain this, it is in fact sufficient to show that there exists Λ > 0 such that if F R N satisfies αf, δ 2 an FF + Λ F F, then F =. Assume by contraiction that there exist sequences Λ h + an h R N such that α h, δ 2, F h + Λ h h F, an h. Notice that, since Λ h +, we have h. Here an in the rest of the paper connecteness is intene in a measure-theoretic sense: a connecte component of is efine as any subset F with F > 0, \ F > 0, such that P = PF + P \ F.
6 M. BONACINI AND R. CRISTOFRI We efine new sets F h := λ h h, where λ h = for h sufficiently large, that αf h, δ an FF h = F h + λ N h F + λ N h = F + λ N h h N h P h + γλ 2N α h N L α h, so that F h =. Then we have, P h + γλ 2N α h N L α h Λ h h λ N h P h λ N h + γ λ2n α h N L α h h λ N h h Λ h < F, which contraicts the local minimality of notice that the same proof works also in the case of global minimizers. Step 2. From the previous step, it follows that is an ω, r 0 -minimizer for the area functional for suitable ω > 0 an r 0 > 0 see Definition 4.. Inee, choose r 0 such that ω N r0 N δ 2 : then if F is such that F B r x with r < r 0, we clearly have that αf, δ 2 an by minimality of we euce that P PF + γ N L α F N L α + Λ F PF + γc 0 + Λ F using Proposition 2.3, an the claim follows with ω := γc 0 + Λ. Step 3. The C, 2 -regularity of, as well as the conition on the Hausorff imension of the singular set, follows from classical regularity results for ω, r 0 -minimizers see, e.g., [34, Theorem ]. In turn, the C 3,β -regularity follows from the uler-lagrange equation, as in Remark 2.6. To show the essential bouneness, we use the ensity estimates for ω, r 0 -minimizers of the perimeter, which guarantee the existence of a positive constant ϑ 0 > 0 epening only on N such that for every point y an r < min{r 0, /2Nω} P; B r y ϑ 0 r N 2.7 see, e.g., [26, Theorem 2.]. Assume by contraiction that there exists a sequence of points x n R N \ 0, where { } 0 := x R N B r x : lim sup r 0 r + N = 0, such that x n +. Fix r < min{r 0, /2Nω} an assume without loss of generality that x n x m > 4r. It is easily seen that for infinitely many n we can fin y n B r x n ; then P n P, B r y n n ϑ 0 r N = +, which is a contraiction. Connecteness of global minimizers follows easily from their bouneness, since if a global minimizer ha at least two connecte components one coul move one of them far apart from the others without changing the perimeter but ecreasing the nonlocal term in the energy see [25, Lemma 3] for a formal argument. Finally, let 0 be a connecte component of a local minimizer : then, enoting by B r a ball with volume B r = 0, using the isoperimetric inequality an the fact that is a ω, r 0 - minimizer for the area functional, we obtain P \ 0 + Nω N r N P \ 0 + P 0 = P P \ 0 + ω 0 = P \ 0 + ωω N r N, which is a contraiction if r is small enough. This shows an uniform lower boun on the volume of each connecte component of, from which we euce that can have at most a finite number of connecte components.
LOCAL AND GLOBAL MINIMALITY RSULTS FOR A NONLOCAL ISOPRIMTRIC PROBLM ON R N 7 We are now reay to state the main results of this paper. The central theorem, whose proof lasts for Sections 3 an 4, provies a sufficiency local minimality criterion base on the secon variation of the functional. Following [] see also [9], we introuce a quaratic form associate with the secon variation of the functional at a regular critical set see Definition 3.4; then we show that its strict positivity on the orthogonal complement to a suitable finite imensional subspace of irections where the secon variation egenerates, ue to translation invariance is a sufficient conition for isolate local minimality, accoring to Definition 2.4, by proving a quantitative stability inequality. The result reas as follows. Theorem 2.8. Assume that is a regular critical set for F with positive secon variation, in the sense of Definition 3.8. Then there exist δ > 0 an C > 0 such that FF F + C α, F 2 for every F R N such that F = an α, F < δ. The local minimality criterion in Theorem 2.8 can be applie to obtain information about local an global minimizers of the functional.. In orer to state the results more clearly, we will unerline the epenence of the functional on the parameters α an γ by writing F α,γ instea of F. We start with the following theorem, which shows the existence of a critical mass m loc such that the ball B R is an isolate local minimizer if B R < m loc, but is no longer a local minimizer for larger masses. We also etermine explicitly the volume threshol in the three-imensional case. The result, which to the best of our knowlege provies the first characterization of the local minimality of the ball, will be prove in Section 5. Theorem 2.9 Local minimality of the ball. Given N 2, α 0, N an γ > 0, there exists a critical threshol m loc = m loc N, α, γ > 0 such that the ball B R is an isolate local minimizer for F α,γ, in the sense of Definition 2.4, if 0 < B R < m loc. If B R > m loc, there exists R N with = B R an α, B R arbitrarily small such that F α,γ < F α,γ B R. In particular, in imension N = 3 we have 3 m loc 3, α, γ = 4 4 α 3 π 6 α4 α 2 3 α. γαπ Finally m loc N, α, γ as α 0 +. Our local minimality criterion allows us to euce further properties about global minimizers, which will be prove in Section 6. The first result states that the ball is the unique global minimizer of the functional for small masses. We provie an alternative proof of this fact which was alreay known in the literature in some particular cases, as explaine in the introuction which hols in full generality an removes the restrictions on the parameters N an α which were present in the previous partial results. Theorem 2.0 Global minimality of the ball. Let m glob N, α, γ be the supremum of the masses m > 0 such that the ball of volume m is a global minimizer of F α,γ in R N. Then m glob N, α, γ is positive an finite, an the ball of volume m is a global minimizer of F α,γ if m m glob N, α, γ. Moreover, it is the unique up to translations global minimizer of F α,γ if m < m glob N, α, γ. In the following theorems we analyze the global minimality issue for α close to 0, showing in particular that in this case the unique minimizer, as long as a minimizer exists, is the ball, an characterizing the infimum of the energy when the problem oes not have a solution. Theorem 2. Characterization of global minimizers for α small. There exists ᾱ = ᾱn, γ > 0 such that for every α < ᾱ the ball with volume m is the unique up to translations global minimizer of F α,γ if m m glob N, α, γ, while for m > m glob N, α, γ the minimum problem for F α,γ oes not have a solution. 2.8
8 M. BONACINI AND R. CRISTOFRI Theorem 2.2 Characterization of minimizing sequences for α small. Let α < ᾱ where ᾱ is given by Theorem 2. an let { k f k m := min FB i : B i ball, B i = µ i }. µ,...,µ k 0 µ +...+µ k =m j= There exists an increasing sequence m k k, with m 0 = 0, m = m glob, such that lim k m k = an inf F = f km for every m [m k, m k ], for all k N, 2.9 =m that is, for every m [m k, m k ] a minimizing sequence for the total energy is obtaine by a configuration of at most k isjoint balls with iverging mutual istance. Moreover, the number of non-egenerate balls tens to + as m +. Remark 2.3. Since m loc N, α, γ + as α 0 + an the non-existence threshol is uniformly boune for α 0, see Proposition 6., we immeiately euce that, for α small, m glob N, α, γ < m loc N, α, γ. Moreover, by comparing the energy of a ball of volume m with the energy of two isjoint balls of volume m 2, an sening to infinity the istance between the balls, we euce after a straightforwar computation an estimating N L α B ωn 2 2 α that the following upper boun for the global minimality threshol of the ball hols: 2 α N2 N N N+ α m glob N, α, γ < ω N. ω N γ 2 N α N Hence, by comparing this value with the explicit expression of m loc in the physical interesting case N = 3, α = see Theorem 2.9, we euce that m glob 3,, γ < m loc 3,, γ. Remark 2.4. In the planar case, one can also consier a Newtonian potential in the nonlocal term, i.e. log x y xy. It is clear that the infimum of the corresponing functional on R 2 is consier, for instance, a minimizing sequence obtaine by sening to infinity the istance between the centers of two isjoint balls. Moreover, also the notion of local minimality consiere in Definition 2.4 becomes meaningless in this situation, since, given any finite perimeter set, it is always possible to fin sets with total energy arbitrarily close to in every L -neighbourhoo of. Nevertheless, by reproucing the arguments of this paper one can show that, given a boune regular critical set with positive secon variation, an a raius R > 0 such that B R, there exists δ > 0 such that minimizes the energy with respect to competitors F B R with αf, < δ. 3. Secon variation an W 2,p -local minimality We start this section by introucing the notions of first an secon variation of the functional F along families of eformations as in the following efinition. Definition 3.. Let X : R N R N be a C 2 vector fiel. The amissible flow associate with X is the function Φ : R N, R N efine by the equations Φ t = XΦ, Φx, 0 = x. Definition 3.2. Let R N be a set of class C 2, an let Φ be an amissible flow. We efine the first an secon variation of F at with respect to the flow Φ to be t F t t=0 respectively, where we set t := Φ t. an 2 t 2 F t t=0
LOCAL AND GLOBAL MINIMALITY RSULTS FOR A NONLOCAL ISOPRIMTRIC PROBLM ON R N 9 Given a regular set, we enote by X τ := X X, ν ν the tangential part to of a vector fiel X. We recall that the tangential graient D τ is efine by D τ ϕ := Dϕ τ, an that B := D τ ν is the secon funamental form of. The following theorem contains the explicit formula for the first an secon variation of F. The computation, which is postpone to the Appenix, is performe by a regularization approach which is slightly ifferent from the technique use, in the case α = N 2, in [9] for a critical set, see also [28] an in [] for a general regular set: here we introuce a family of regularize potentials epening on a small parameter δ R to avoi the problems in the ifferentiation of the singularity in the nonlocal part, recovering the result by letting the parameter ten to 0. Theorem 3.3. Let R N be a boune set of class C 2, an let Φ be the amissible flow associate with a C 2 vector fiel X. Then the first variation of F at with respect to the flow Φ is F t = H + 2γv X, ν H N, 3. t t=0 an the secon variation of F at with respect to the flow Φ is 2 F t t 2 = Dτ X, ν 2 B 2 X, ν 2 H N t=0 + 2γ Gx, y Xx, ν x Xy, ν y H N xh N y + 2γ ν v X, ν 2 H N 2γv + H iv τ Xτ X, ν H N + 2γv + H ivx X, ν H N, where Gx, y := x y α is the potential in the nonlocal part of the energy. If is a regular critical set as in Definition 2.5 it hols 2γv + H iv τ Xτ X, ν H N = 0. Moreover if the amissible flow Φ preserves the volume of, i.e. if Φ t = for all t,, then see [9, equation 2.30] 0 = 2 t 2 t t=0 = ivx X, ν H N. Hence we obtain the following expression for the secon variation at a regular critical set with respect to a volume-preserving amissible flow: 2 F t t 2 = Dτ X, ν 2 B 2 X, ν 2 H N + 2γ ν v X, ν 2 H N t=0 + 2γ Gx, y Xx, ν x Xy, ν y H N xh N y. Following [], we introuce the space { H := ϕ H : } ϕ H N = 0 enowe with the norm ϕ H := ϕ L 2, an we efine on it the following quaratic form associate with the secon variation. Definition 3.4. Let R N be a regular critical set. We efine the quaratic form 2 F : H R by 2 F[ϕ] = Dτ ϕ 2 B 2 ϕ 2 H N + 2γ ν v ϕ 2 H N 3.2 + 2γ Gx, yϕxϕy H N xh N y.
0 M. BONACINI AND R. CRISTOFRI Notice that if is a regular critical set an Φ preserves the volume of, then 2 F[ X, ν ] = 2 F t t 2 t=0. 3.3 We remark that the last integral in the expression of 2 F is well efine for ϕ H, thanks to the following result. Lemma 3.5. Let be a boune set of class C. There exists a constant C > 0, epening only on, N an α, such that for every ϕ, ψ H Gx, yϕxψy H N xh N y C ϕ L 2 ψ L 2 C ϕ H ψ H. 3.4 Proof. The proof lies on [6, Lemma 7.2], which states that if Ω R n is a boune omain an µ 0, ], the operator f V µ f efine by V µ fx := x y nµ fy y maps L p Ω continuously into L q Ω provie that 0 δ := p q < µ, an δ δω µ V µ f L q Ω n Ω µ δ f L µ δ p Ω. Ω In our case, from the fact that our set has compact bounary, we can simply reuce to the above case using local charts an partition of unity notice that the hypothesis of compact bounary allows us to boun from above in the L -norm the area factor. In particular we have that µ = N α N, an applying this result with p = q = 2 we easily obtain the estimate in the statement by the Sobolev mbeing Theorem. Remark 3.6. Using the estimate containe in the previous lemma it is easily seen that 2 F is continuous with respect to the strong convergence in H an lower semicontinuous with respect to the weak convergence in H. Moreover, it is also clear from the proof that, given ᾱ < N, the constant C in 3.4 can be chosen inepenently of α 0, ᾱ. quality 3.3 suggests that at a regular local minimizer the quaratic form 3.2 must be nonnegative on the space H. This is the content of the following corollary, whose proof is analogous to [, Corollary 3.4]. Corollary 3.7. Let be a local minimizer of F of class C 2. Then 2 F[ϕ] 0 for all ϕ H. Now we want to look for a sufficient conition for local minimality. First of all we notice that, since our functional is translation invariant, if we compute the secon variation of F at a regular set with respect to a flow of the form Φx, t := x + tηe i, where η R an e i is an element of the canonical basis of R N, setting ν i := ν, e i we obtain that 2 F[ην i ] = 2 t 2 F t t=0 = 0. Following [], since we aim to prove that the strict positivity of the secon variation is a sufficient conition for local minimality, we shall exclue the finite imensional subspace of H generate by the functions ν i, which we enote by T. Hence we split H = T T, where T is the orthogonal complement to T in the L 2 -sense, i.e., { } T := ϕ H : ϕν i H N = 0 for each i =,..., N.
LOCAL AND GLOBAL MINIMALITY RSULTS FOR A NONLOCAL ISOPRIMTRIC PROBLM ON R N It can be shown see [, quation 3.7] that there exists an orthonormal frame ε,..., ε N such that ν, ε i ν, ε j H N = 0 for all i j, so that the projection on T of a function ϕ H is N π T ϕ = ϕ ϕ ν, ε i H N ν, ε i ν, ε i 2 L 2 i= notice that ν, ε i 0 for every i, since on the contrary the set woul be translation invariant in the irection ε i. Definition 3.8. We say that F has positive secon variation at the regular critical set if 2 F[ϕ] > 0 for all ϕ T \{0}. One coul expect that the positiveness of the secon variation implies also a sort of coercivity; this is shown in the following lemma. Lemma 3.9. Assume that F has positive secon variation at a regular critical set. Then an m 0 := inf { 2 F[ϕ] : ϕ T, ϕ H = } > 0, 2 F[ϕ] m 0 ϕ 2 H for all ϕ T. Proof. Let ϕ h h be a minimizing sequence for m 0. Up to a subsequence we can suppose that ϕ h ϕ 0 weakly in H, with ϕ 0 T. By the lower semicontinuity of 2 F with respect to the weak convergence in H see Remark 3.6, we have that if ϕ 0 0 m 0 = lim h 2 F[ϕ h ] 2 F[ϕ 0 ] > 0, while if ϕ 0 = 0 m 0 = lim h 2 F[ϕ h ] = lim D τ ϕ h 2 H N =. h The secon part of the statement follows from the fact that 2 F is a quaratic form. We now come to the proof of the main result of the paper, namely that the positivity of the secon variation at a critical set is a sufficient conition for local minimality Theorem 2.8. In the remaining part of this section we prove that a weaker minimality property hols, that is minimality with respect to sets whose bounaries are graphs over the bounary of with sufficiently small W 2,p -norm Theorem 3.. In orer to o this, we start by recalling a technical result neee in the proof, namely [, Theorem 3.7], which provies a construction of an amissible flow connecting a regular set R N with an arbitrary set sufficiently close in the W 2,p -sense. Theorem 3.0. Let R N be a boune set of class C 3 an let p > N. For all ε > 0 there exist a tubular neighbourhoo U of an two positive constants δ, C with the following properties: if ψ C 2 an ψ W 2,p δ then there exists a fiel X C 2 with ivx = 0 in U such that X ψν L2 ε ψ L2. Moreover the associate flow Φ Φx, 0 = 0, t = XΦ satisfies Φ, = {x + ψxν x : x }, an for every t [0, ] Φ, t I W 2,p C ψ W 2,p, where I enotes the ientity map. If in aition has the same volume as, then for every t we have t = an t X, ν t H N = 0.
2 M. BONACINI AND R. CRISTOFRI We are now in position to prove the following W 2,p -local minimality theorem, analogous to [, Theorem 3.9]. The proof containe in [] can be repeate here with minor changes, an we will only give a sketch of it for the reaer s convenience. Theorem 3.. Let p > max{2, N } an let be a regular critical set for F with positive secon variation, accoring to Definition 3.8. Then there exist δ, C 0 > 0 such that FF F + C 0 α, F 2, for each F R N such that F = an F = {x+ψxν x : x } with ψ W 2,p δ. Proof sketch. We just escribe the strategy of the proof, which is ivie into two steps. Step. There exists δ > 0 such that if F = {x + ψxν x : x } with F = an ψ W 2,p δ, then inf { 2 FF [ϕ] : ϕ H } F, ϕ H F =, ϕν F H N δ m 0 2, where m 0 is efine in Lemma 3.9. To prove this we suppose by contraiction that there exist a sequence F n n of subsets of R N such that F n = {x + ψ n xν x : x }, F n =, ψ n W 2,p 0, an a sequence of functions ϕ n H F n with ϕ n H F n =, F n ϕ n ν Fn H N 0, such that 2 FF n [ϕ n ] < m 0 2. We consier a sequence of iffeomorphisms Φ n : F n, with Φ n I in W 2,p, an we set ϕ n := ϕ n Φ n a n, a n := ϕ n Φ n H N. Hence ϕ n H, a n 0, an since ν Fn Φ n ν 0 in C 0,β for some β 0, an a similar convergence hols for the tangential vectors, we have that ϕ n ν, ε i H N 0 for every i =,..., N, so that π T ϕ n H. Moreover it can be prove that 2 FF n [ϕ n ] 2 F[ ϕ n ] 0. Inee, the convergence of the first integral in the expression of the quaratic form follows easily from the fact that B Fn Φ n B 0 in L p, an from the Sobolev mbeing Theorem recall that p > max{2, N }. For the secon integral, it is sufficient to observe that, as a consequence of Proposition 2., the functions v Fh are uniformly boune in C,β R N for some β 0, an hence they converge to v in C,γ B R for all γ < β an R > 0. Finally, the ifference of the last integrals can be written as Gx, yϕ n xϕ n y H N H N Gx, y ϕ n x ϕ n y H N H N F n F n = g n x, ygx, y ϕ n x ϕ n y H N H N + a n GΦ n x, Φ n yj n xj n y ϕ n x + ϕ n y + a n H N H N where J n z := J N Φ nz is the N -imensional jacobian of Φ n on, an g n x, y := F x y α Φ n x Φ n y α J nxj n y. Thus the esire convergence follows from the fact that g n 0 uniformly, a n 0, an from the estimate provie by Lemma 3.5.
LOCAL AND GLOBAL MINIMALITY RSULTS FOR A NONLOCAL ISOPRIMTRIC PROBLM ON R N 3 Hence m 0 2 lim n 2 FF n [ϕ n ] = lim n 2 F[ ϕ n ] = lim n 2 F[π T ϕ n ] m 0 lim π T ϕ n n H = m 0, which is a contraiction. Step 2. If F is as in the statement of the theorem, we can use the vector fiel X provie by Theorem 3.0 to generate a flow connecting to F by a family of sets t, t [0, ]. Recalling that is critical an that X is ivergence free, we can write FF F = F F 0 = t 2 0 t 2 F t t = t 2 F t [ X, ν t ] 2γv t + H t iv τt X τt X, ν t H N t, 0 t where iv τt stans for the tangential ivergence of t. It is now possible to boun from below the previous integral in a quantitative fashion: to o this we use, in particular, the result prove in Step for the first term, an we procee as in Step 2 of [, Theorem 3.9] for the secon one. In this way we obtain the esire estimate. 4. L -local minimality In this section we complete the proof of the main result of the paper Theorem 2.8, starte in the previous section. The main argument of the proof relies on a regularity property of sequences of quasi-minimizers of the area functional, which has been observe by White in [35] an was implicitly containe in [2] see also [3], [34]. Definition 4.. A set R N is sai to be an ω, r 0 -minimizer for the area functional, with ω > 0 an r 0 > 0, if for every ball B r x with r r 0 an for every finite perimeter set F R N such that F B r x we have Theorem 4.2. Let n R N P PF + ω F. be a sequence of ω, r 0 -minimizers of the area functional such that sup P n < + an χ n χ in L R N n for some boune set of class C 2. Then for n large enough n is of class C, 2 with ψ n 0 in C,β for all β 0, 2. n = {x + ψ n xν x : x }, Another useful result is the following consequence of the classical elliptic regularity theory see [, Lemma 7.2] for a proof. Lemma 4.3. Let be a boune set of class C 2 an let n be a sequence of sets of class C,β for some β 0, such that n = {x + ψ n xν x : x }, with ψ n 0 in C,β. Assume also that H n L p n for some p. If then ψ n 0 in W 2,p. H n + ψ n ν H in L p, We recall also the following simple lemma from [, Lemma 4.]. Lemma 4.4. Let R N be a boune set of class C 2. Then there exists a constant C > 0, epening only on, such that for every finite perimeter set F R N P PF + C F. an
4 M. BONACINI AND R. CRISTOFRI An intermeiate step in the proof of Theorem 2.8 consists in showing that the W 2,p -local minimality prove in Theorem 3. implies local minimality with respect to competing sets which are sufficiently close in the Hausorff istance. We omit the proof of this result, since it can be easily aapte from [, Theorem 4.3] notice, inee, that the ifficulties coming from the fact of working in the whole space R N are not present, ue to the constraint F I δ0. Theorem 4.5. Let R N be a boune regular set, an assume that there exists δ > 0 such that F FF 4. for every set F R N with F = an F = {x + ψxν x : x }, for some function ψ with ψ W 2,p δ. Then there exists δ 0 > 0 such that 4. hols for every finite perimeter set F with F = an such that I δ0 F I δ0, where for δ R we set enoting the signe istance to I δ := {x : x < δ}. We are finally reay to complete the proof of the main result of the paper. The strategy follows closely [, Theorem.], with the necessary technical moifications ue to the fact that here we have to eal with a more general exponent α an with the lack of compactness of the ambient space. Proof of Theorem 2.8. We assume by contraiction that there exists a sequence of sets h R N, with h = an α h, > 0, such that ε h := α h, 0 an F h < F + C 0 αh, 2, 4.2 4 where C 0 is the constant provie by Theorem 3.. By approximation we can assume without loss of generality that each set of the sequence is boune, that is, there exist R h > 0 which we can also take satisfying R h + such that h B Rh. We now efine F h R N as a solution to the penalization problem } αf, 2 min {J h F := FF + Λ εh + εh + Λ 2 F : F BRh, 4.3 where Λ an Λ 2 are positive constant, to be chosen notice that the constraint F B Rh guarantees the existence of a solution. We first fix Λ > C + c 0 γ. 4.4 Here C is as in Lemma 4.4, while c 0 is the constant provie by Proposition 2.3 corresponing to the fixe values of N an α an to m := +. We remark that with this choice Λ epens only on the set. We will consier also the sets F h obtaine by translating F h in such a way that αf h, = F h clearly J h F h = J h F h. Step. We claim that, if Λ 2 is sufficiently large epening on Λ, but not on h, then F h = for every h large enough. This can be euce by aapting an argument from [3, Section 2] see also [, Proposition 2.7]. Inee, assume by contraiction that there exist Λ h an F h solution to the minimum problem 4.3 with Λ 2 replace by Λ h such that F h < a similar argument can be performe in the case F h >. Up to subsequences, we have that F h F 0 in L loc an F h. As each set F h minimizes the functional FF + Λ αf, εh 2 + εh in B Rh uner the constraint F = F h, it is easily seen that F h is a quasi-minimizer of the perimeter with volume constraint, so that by the regularity result containe in [30, Theorem.4.4] we have that the N -imensional ensity of F h is uniformly boune from below by a constant inepenent of h. This observation implies that we can assume without loss of generality that the limit set F 0 is not empty an that there exists a point x 0 F 0, so that, by repeating
LOCAL AND GLOBAL MINIMALITY RSULTS FOR A NONLOCAL ISOPRIMTRIC PROBLM ON R N 5 an argument containe in [3], we obtain that given ε > 0 we can fin r > 0 an x R N that F h B r/2 x < εr N, F h B r x > ω Nr N 2 N+2 for every h sufficiently large an we assume x = 0 for simplicity. Now we moify F h in B r by setting G h := Φ h F h, where Φ h is the bilipschitz map σh 2 N x if x r 2, Φ h x := x + σ h r N x x if r N 2 < x < r, x if x r, such an σ h 0,. It can be shown see [3, Section 2], [, Proposition 2.7] for etails that ε 2 N an σ h can be chosen in such a way that G h =, an moreover there exists a imensional constant C > 0 such that J Λh F h J Λh G h σ h CΛ h r N 2 N N + Cγ + CΛ PF h ; B r where J Λh enotes the functional in 4.3 with Λ 2 replace by Λ h. This contraicts the minimality of F h for h sufficiently large. Step 2. We now show that Inee, by Lemma 4.4 we have that lim αf h, = 0. 4.5 h + P P F h + C F h, while by Proposition 2.3 N L N L F h c 0 F h. Combining the two estimates above, using the minimality of F h an recalling that F h = we euce which yiels P F h + γn L F h + Λ Fh ε h 2 + εh = J h F h J h = P + γn L + Λ ε 2 h + ε h P F h + γn L F h + C + c 0 γ F h + Λ ε 2 h + ε h, Λ Fh ε h 2 + εh C + c 0 γ F h + Λ ε 2 h + ε h. Passing to the limit as h +, we conclue that Λ lim sup h + F h C + c 0 γ lim sup F h, h + which implies F h 0 by the choice of Λ in 4.4. Hence 4.5 is prove, an this shows in particular that χ Fh χ in L R N. Step 3. ach set F h is an ω, r 0 -minimizer of the area functional see Definition 4., for suitable ω > 0 an r 0 > 0 inepenent of h. Inee, choose r 0 such that ω N r 0 N, an consier any ball B r x with r r 0 an any finite perimeter set F such that F F h B r x. We have N LF N LF h c 0 F F h
6 M. BONACINI AND R. CRISTOFRI by Proposition 2.3, where c 0 is the same constant as before since we can boun the volume of F by F F h + ω N r N 0 +. Moreover PF PF B Rh = H N x H N x F \B Rh F B Rh B Rh x x ν F H N x x x ν F B Rh H N x = F \B Rh F \B Rh x x ν F \B Rh H N x = F B Rh B Rh F \B Rh iv x x x 0. Hence, as F h is a minimizer of J h among sets containe in B Rh, we euce PF h PF B Rh + γ N LF B Rh N LF h + Λ 2 F B Rh + Λ αf BRh, ε h 2 + εh Λ αfh, ε h 2 + εh PF + c 0 γ + Λ + Λ 2 F BRh F h PF + c 0 γ + Λ + Λ 2 F Fh for h large enough. This shows that F h is an ω, r 0 -minimizer of the area functional with ω = c 0 γ + Λ + Λ 2 an the same hols obviously also for F h. Hence, by Theorem 4.2 an recalling that χ Fh χ in L, we euce that for h sufficiently large F h is a set of class C, 2 an F h = {x + ψ h xν x : x } for some ψ h such that ψ h 0 in C,β for every β 0, 2. We remark also that the sets F h are uniformly boune, an for h large enough F h B Rh : in particular, Fh solves the minimum problem 4.3. Step 4. We now claim that αf h, lim =. 4.6 h + ε h Inee, assuming by contraiction that αf h, ε h σε h for some σ > 0 an for infinitely many h, we woul obtain FF h + Λ σ 2 ε 2 h + ε h FF h + Λ αfh, ε h 2 + εh F h + Λ εh < F + C 0 4 ε2 h + Λ εh F F h + C 0 4 ε2 h + Λ εh where the secon inequality follows from the minimality of F h, the thir one from 4.2 an the last one from Theorem 4.5. This shows that Λ σ 2 ε 2 h + ε h C 0 4 ε2 h + Λ εh, which is a contraiction for h large enough. Step 5. We now show the existence of constants λ h R such that H Fh + 2γv Fh λ h L F h 4Λ εh 0. 4.7 We first observe that the function f h t := t ε h 2 + ε h satisfies f h t f h t 2 2 ε h t t 2 if t i ε h ε h. 4.8
LOCAL AND GLOBAL MINIMALITY RSULTS FOR A NONLOCAL ISOPRIMTRIC PROBLM ON R N 7 Hence for every set F R N with F =, F B Rh an αf, ε h ε h we have αf, F F 2 α 2 h FF + Λ εh + εh Fh, ε h + εh FF + 2Λ εh αf, α F h, 4.9 FF + 2Λ εh F F h where we use the minimality of Fh in the first inequality, an 4.8 combine with the fact that α F h, ε h ε h for h large which, in turn, follows by 4.6 in the secon one. Consier now any variation Φ t, as in Definition 3., preserving the volume of the set F h, associate with a vector fiel X. For t sufficiently small we can plug the set Φ t F h in the inequality 4.9: F F h FΦ t F h + 2Λ εh Φ t F h F h, which gives FΦ t F h F F h + 2Λ εh t X ν Fh H N + ot 0 F h for t sufficiently small. Hence, iviing by t an letting t 0 + an t 0, we get H + 2γv F Fh Fh X ν Fh H N 2Λ εh X ν Fh H N, h F h an by ensity H + 2γv F Fh Fh ϕ H N 2Λ εh ϕ H h N Fh for every ϕ C F h with F h ϕ H N = 0. In turn, this implies 4.7 by a simple functional analysis argument. Step 6. We are now close to the en of the proof. Recall that on for some constant λ, while by 4.7 H = λ 2γv 4.0 H Fh = λ h 2γv Fh + ρ h, with ρ h 0 uniformly. 4. Observe now that, since the functions v Fh are equiboune in C,β R N for some β 0, see Proposition 2. an they converge pointwise to v since χ Fh χ in L, we have that v Fh v in C B R for every R > 0. 4.2 We consier a cyliner C = B ] L, L[, where B R N is a ball centere at the origin, such that in a suitable coorinate system we have F h C = {x, x N C : x B, x N < g h x }, C = {x, x N C : x B, x N < gx } for some functions g h g in C,β B for every β 0, 2. By integrating 4. on B we obtain λ h L N B 2γ v Fh x, g h x L N x + ρ h x, g h x L N x B B g h = iv L N x g = h B + gh 2 + gh x 2 x HN 2, an the last integral in the previous expression converges as h 0 to g x g B + g 2 x HN 2 = iv L N x B + g 2 = λl N B 2γ v x, gx L N x, B B
8 M. BONACINI AND R. CRISTOFRI where the last equality follows by 4.0. This shows, recalling 4.2 an that ρ h tens to 0 uniformly, that λ h λ, which in turn implies, by 4.0, 4. an 4.2, H Fh + ψ h ν H in L. By Lemma 4.3 we conclue that ψ h W 2,p for every p an ψ h 0 in W 2,p. Finally, by minimality of Fh we have α F F h F F 2 h + Λ Fh, ε h + εh Λ εh F h < F + C 0 4 ε2 h F + C 0 α Fh, 2 2 where we use 4.2 in the thir inequality an 4.6 in the last one. This is the esire contraiction with the conclusion of Theorem 3.. Remark 4.6. It is important to remark that in the arguments of this section we have not mae use of the assumption of strict positivity of the secon variation: the quantitative L -local minimality follows in fact just from the W 2,p -local minimality. 5. Local minimality of the ball In this section we will obtain Theorem 2.9 as a consequence of Theorem 2.8, by computing the secon variation of the ball an stuying the sign of the associate quaratic form. 5.. Recalls on spherical harmonics. We first recall some basic facts about spherical harmonics, which are neee in our calculation. We refer to [9] for an account on this topic. Definition 5.. A spherical harmonic of imension N is the restriction to S N of a harmonic polynomial in N variables, i.e. a homogeneous polynomial p with p = 0. We will enote by H N the set of all spherical harmonics of imension N that are obtaine as restrictions to S N of homogeneous polynomials of egree. In particular H0 N is the space of constant functions, an H N is generate by the coorinate functions. The basic properties of spherical harmonics that we nee are liste in the following theorem. Theorem 5.2. It hols: for each N, H N is a finite imensional vector space. 2 If F H N, G HN e an e, then F an G are orthogonal in the L 2 -sense. 3 If F H N an 0, then S N F H N = 0. 4 If H,..., HimHN is an orthonormal basis of H N for every 0, then this sequence is complete, i.e. every F L 2 S N can be written in the form F = =0 imh N where c i := F, Hi L 2. 5 If H i are as in 4 an F, G L2 S N are such that i= c i H i, 5. then F = =0 imh N i= c i H i, G = F, G L 2 = =0 imh N i= =0 imh N c i e i. i= e i H i,
LOCAL AND GLOBAL MINIMALITY RSULTS FOR A NONLOCAL ISOPRIMTRIC PROBLM ON R N 9 6 Spherical harmonics are eigenfunctions of the Laplace-Beltrami operator S N. More precisely, if H H N then SN H = + N 2H. 7 If F is a C 2 function on S N represente as in 5., then S N D τ F 2 H N x = =0 imh N i= + N 2c i 2. We recall also the following important result in the theory of spherical harmonics. Theorem 5.3 Funk-Hecke Formula. Let f :, R such that Then if H H N an x 0 S N it hols ft t 2 N 3 2 t <. S N f x 0, x Hx H N x = µ Hx 0, where the coefficient µ is given by µ = N ω N P N, tft t 2 N 3 2 t. Here P N, is the Legenre polynomial of imension N an egree given by P N, t = Γ N 2 2 Γ + N 2 t2 N 3 2 t t 2 N 3 + 2, where Γx := 0 t x e t t is the Gamma function. 5.2. Secon variation of the ball. The quaratic form 3.2 associate with the secon variation of F at the ball B R, compute at a function ϕ H B R is 2 FB R [ ϕ] = D τ ϕx 2 N B R R 2 ϕ 2 x H N x + 2γ B R B R x y α ϕx ϕy HN x H N y BR x x y, x + 2γ α x y α+2 y ϕ 2 x H N x. B R Since we want to obtain a sign conition of 2 FB R [ ϕ] in terms of the raius R, we first make a change of variable: 2 FB R [ ϕ] = R B N 3 D τ ϕx 2 N ϕ 2 x H N x + 2γR 2N 2 α B B x y α ϕxϕx HN x H N y 5.2 + 2γR 2N 2 α x y, x α B x y α+2 y ϕ 2 x H N x, B where the function ϕ H S N is efine as ϕx := ϕrx. Since we are only intereste in the sign of the secon variation, which is continuous with respect to the strong convergence in H S N, we can assume ϕ C 2 S N T S N. The iea to compute the secon variation at the ball is to expan ϕ with respect to an orthonormal basis of spherical harmonics, as in 5.. First of all we notice that if ϕ T S N, then its harmonic expansion oes not contain spherical harmonics of orer 0 an. Inee, harmonics of orer 0 are constant functions, that are not allowe by the null average conition. Moreover
20 M. BONACINI AND R. CRISTOFRI H N = T S N, because ν S N x = x, an the functions x i form an orthonormal basis of H N. Hence we can write the harmonic expansion of ϕ C 2 S N T S N as follows: ϕ = =2 imh N i= c i H i, where H,..., HimHN is an orthonormal basis of H N for each N. We can now compute each term appearing in 5.2 as follows: the first term, by property 7 of Theorem 5.2, is B D τ ϕ 2 N ϕ 2 H N = =2 imh N i= + N 2 N c i 2. For the secon term we want to use the Funk-Hecke Formula to compute the inner integral; so we efine the function α 2 ft := 2 t an we notice that x y α = f x, y for x, y S N, an that, for α 0, N, f satisfies the integrability assumptions of Theorem 5.3. Hence for each y S N where the coefficient µ N,α B x y α ϕx HN x = := 2 N α N ω N 2 i=0 =2 imh N is obtaine by irect computation just integrating by parts. Therefore B i= B x y α ϕxϕy HN x H N y = For the last term of 5.2, noticing that the integral I N,α x y, x := y B x y α+2 is inepenent of x S N, we get B µ N,α c i H i y, α 2 + i Γ N α 2 Γ N 2 ΓN α 2 + 5.3 =2 x y, x α B x y α+2 y ϕ 2 x H N x = αi N,α Combining all the previous equalities with 5.2 we obtain 2 FB R [ ϕ] = =2 imh N i= imh N i= imh N =2 µ N,α c i 2. i= c i 2. R N 3 c i 2[ + N 2 N + 2γR N+ α µ N,α αi N,α]. 5.3. Local minimality of the ball. From the above expression we euce that the quaratic form 2 FB R is strictly positive on T B R, that is, the secon variation of F at B R is positive accoring to Definition 3.8, if an only if + N 2 N + 2γR N+ α µ N,α αi N,α > 0 5.4 for all 2, where the only if part is ue to the fact that H N T S N for each 2. On the contrary, 2 FB R [ ϕ] < 0 for some ϕ T B R if an only if there exists 2 such that the left-han sie of 5.4 is negative.