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VARIATIONAL PROBLEMS WITH LONG-RANGE INTERACTION NICOLASOAVE,HUGOTAVARES,SUSANNATERRACINI,ANDALESSANDROZILIO 7 Abstract. We consider a class of variational problems for densities that repel each other at 1 distance. TypicalexamplesaregivenbytheDirichletfunctionalandtheRayleighfunctional 0 ˆ ´ n 2 D(u)=(cid:88)i=k1 Ω|∇ui|2 or R(u)=(cid:88)i=k1 Ω´|Ω∇uu2ii|2 a minimized in the class of H1(Ω,Rk) functions attaining some boundary conditions on ∂Ω, and J subjectedtotheconstraint 8 dist({ui>0},{uj >0})(cid:62)1 ∀i(cid:54)=j. 1 Fortheseproblems,weinvestigatetheoptimalregularityofthesolutions,proveafree-boundary ] condition,andderivesomepreliminaryresultscharacterizingthefreeboundary∂{(cid:80)ki=1ui>0}. P A 1. Introduction . h Theobjectofthispaperisthestudyofaclassofminimalconfigurationsforvariationalproblems t a involvingarbitrarilymanydensitiesrelatedbylong-rangerepulsiveinteractions. Themathematical m setting we consider is described by the following two archetypical situations. [ Problem (A) Let Ω be a bounded domain of RN, N (cid:62)2, and let 1 v Ω = (cid:91) B (x)={x∈RN : dist(x,Ω)<1}. 5 1 1 0 x∈Ω 0 Given k (cid:62)2 nonnegative nontrivial functions f ,...,f ∈H1(Ω )∩C(Ω ) satisfying 1 5 1 k 1 1 0 dist(suppf ,suppf )(cid:62)1 ∀i(cid:54)=j, i j . 1 we consider the minimization problem 0 7 inf J (u), 1 ∞ u∈H∞ : v where the set H and the functional J are defined by ∞ ∞ i arX (1.1) H∞ =(cid:26)u=(u1,...,uk)∈H1(Ω1,Rk)(cid:12)(cid:12)(cid:12)(cid:12) duiis=t(sfuipap.eu.i,isnuΩpp1u\jΩ)(cid:62)1 ∀i(cid:54)=j (cid:27), Date:January19,2017. Keywordsandphrases. Freeboundarycondition,Lipschitzregularity,Long-rangeinteraction,Optimalpartition problems,Segregationproblems,Variationalmethods. 1Hereandintherestofthepaper,thedistancebetweentwosetsAandB isunderstoodas dist(A,B):=inf{|x−y|: x∈A, y∈B}. 1 2 N.SOAVE,H.TAVARES,S.TERRACINI,ANDA.ZILIO and ˆ k (cid:88) J (u)= |∇u |2. ∞ i i=1 Ω The support of each component u is taken in the weak sense: it corresponds to the complement i in Ω of the largest open set ω ⊆ RN where u = 0 a.e. on ω (cf. [3, Proposition 4.17]). Notice 1 i also that the existence of f ,...,f with the above properties imposes some conditions on Ω (for 1 k instance,thediameterofΩcannotbetoosmall),andwesupposethatsuchconditionsaresatisfied. We are interested in existence and qualitative properties of minimizers. Problem (B) Let Ω be a bounded domain of RN, N (cid:62) 2, and let k (cid:62) 2. We consider the set of open partitions of Ω at distance 1, defined as (cid:26) (cid:12) (cid:27) Pk(Ω)= (ω1,...,ωk)(cid:12)(cid:12)(cid:12) aωnid⊂dΩisti(sωoip,ωenj)a(cid:62)nd1no∀ni-e(cid:54)=mjpty for every i, . Then, for a cost function F ∈C1((R+)k,R) satisfying • ∂ F(x) > 0 for all x ∈ (R+)k and i = 1,...,k, which in particular yields that F is i component-wise increasing; • for any given i=1,...,k, lim F(x¯ ,...,x¯ ,x ,x¯ ...,x¯ )=+∞ 1 i−1 i i+1 k xi→+∞ for all (x¯ ,...,x¯ ,x¯ ...,x¯ )∈(R+)k−1, 1 i−1 i+1 k we consider the minimization problem (1.2) inf F(λ (ω ),...,λ (ω )), 1 1 1 k (ω1,...,ωk)∈Pk(Ω) whereλ (ω)isthefirsteigenvalueoftheLaplaceoperatorinω withhomogeneousDirichletbound- 1 ary conditions. Problem (1.2) is a particular case of an optimal partition problem (cf. [1,4]). A typical case we have in mind is the cost function F(λ (ω ),...,λ (ω ))=(cid:80)k λ (ω ). 1 1 1 k i=1 1 i We are interested in existence and qualitative properties of an optimal partition. Our main results are, for problem (A): • the existence of a minimizer; • the optimal interior regularity of any minimizer; • the derivation of several properties of the positivity sets {u >0}; i • the derivation of a free boundary condition involving the normal derivatives of different components of any minimizers on the regular part of the free-boundary ∂{u >0}. i For problem (B): • the introduction of a weak formulation in terms of densities, and the existence of weak solutions; • theglobaloptimalregularityofanyweaksolution,whichleadsinparticulartotheexistence of a strong solution for the original problem; • the derivation of properties of the subsets ω , and of a free boundary condition on the i regular part of ∂ω . i In a forthcoming paper, we will study more in detail the regularity of the free-boundary. We stress that, both in problems (A) and (B), the interaction among different densities takes placeatdistance: inproblem(A)thepositivitysets{u >0},andinproblem(B)theopensubsets i ω , are indeed forced to stay at a fixed minimal distance from each other. i VARIATIONAL PROBLEMS WITH LONG-RANGE INTERACTION 3 When the interaction among the densities takes place point-wisely, segregation problems analogue to (A) and (B) have been studied intensively, in connection with optimal partition problems for Laplacian eigenvalues [5,9,10,11,21,25,26], with the regularity theory of harmonic maps into singular manifold [6,12,25], and with segregation phenomena for systems of elliptic equations arising in quantum mechanics driven by strong competition [6,13,18,22,23,24,30]. Incontrast,theonlyresultsavailablesofarregardingsegregationproblemsdrivenbylong-range competition are given in [7], where the authors analyze the spatial segregation for systems of type (cid:40)−∆u =−βu (cid:80) (1 (cid:63)|u |p) in Ω (1.3) i,β i,β j(cid:54)=i B1 j u =f ≥0 in Ω \Ω, i,β i 1 with 1(cid:54)p(cid:54)+∞. In the above equation, 1 denotes the characteristic function of B , the ball2 B1 1 of center 0 and radius 1, and (cid:63) stays for the convolution for p<+∞, so that ˆ (1 (cid:63)|u |p)(x)= |u (y)|pdy ∀x∈Ω, with 1(cid:54)p<+∞; B1 j j B1(x) incasep=+∞,weintendthattheintegralisreplacedbythesupremumoverB (x)of|u |. In[7], 1 j the authors prove the equi-continuity of families of viscosity solutions {u : β > 0} to (1.3), the β local uniform convergence to a limit configuration u, and then study the free-boundary regularity of the positivity sets {u > 0} in cases p = 1 and p = +∞, mostly in dimension N = 2. As we i shall see, our problem (A) is strictly related with the asymptotic study of the solutions to (1.3) in case p=2 (see the forthcoming Theorem 2.1); nevertheless, also in such a situation our approach isverydifferentwithrespecttotheonein[7],sinceweheavilyrelyonthevariationalnatureofthe problem. This gives differenti free boundary conditions which requires different techniques, and allows us to prove new results. Regarding problem (1.3), we also refer to [2], where the author proves uniqueness results in the cases p=1 and p=+∞. 1.1. Main results. We adopt the notation previously introduced. First of all, we have the following existence results for problems (A) and (B). Theorem 1.1 (Problem (A)). There exists a minimizer u=(u ,...,u ) for inf J . 1 k H∞ ∞ Theorem 1.2 (Problem (B)). There exists a minimizer (ω ,...,ω )∈P for (1.2). 1 k k Observe that, to each optimal partition (ω ,...,ω ), we can associate a vector of signed first 1 k eigenfunctions. To fix ideas, from now on we always consider nonnegative eigenfunctions. The secondpartofouranalysisconcernsthepropertiessatisfiedbyanyminimizerofproblems(A)and (B). Theorem 1.3. Let u = (u ,...,u ) be either any minimizer of J in H , or a vector of first 1 k ∞ ∞ eigenfunctions associated to an optimal partition (ω ,...,ω ) of (1.2). Then u is a vector of 1 k nonnegative functions in Ω, and denoting by S the positivity set {x ∈ Ω : u > 0}, for every i i i=1,...,k, we have: (1) Subsolution in Ω: We have that −∆u (cid:54)0 in distributional sense in Ω, if u is a solution to problem (A), i −∆u (cid:54)λ (ω )u in distributional sense in Ω, if u is a solution to problem (B). i 1 1 i (2) Solution in S : We have that i −∆u =0 in int(S ), if u is a solution to problem (A), i i −∆u =λ (ω ) in int(S ), if u is a solution to problem (B). i 1 i i 2WedenotebyBr(x)theballofcenterxandradiusr inRN. Incasex=0,wesimplywriteBr. 4 N.SOAVE,H.TAVARES,S.TERRACINI,ANDA.ZILIO (3) Exterior sphere condition for the positivity sets: S satisfies the 1-uniform exterior sphere i condition in Ω, in the following sense: for every x ∈ ∂S ∩Ω there exists a ball B with 0 i radius 1 which is exterior to S and tangent to S at x , i.e. i i 0 S ∩B =∅ and x ∈S ∩B. i 0 i Moreover, in B∩B (x ) we have u ≡0 for every j =1,...,k (including j =i). 1 0 j (4) Lipschitz continuity: u is Lipschitz continuous in Ω, and in particular S is an open set, i i for every i. (5) Lebesgue measure of the free-boundary: the free-boundary ∂{u > 0} has zero Lebesgue i measure, and its Hausdorff dimension is strictly smaller than N. (6) Exact distance between the supports: for every x ∈∂S ∩Ω there exists j (cid:54)=i such that 0 i B (x )∩∂suppu (cid:54)=∅. 1 0 j Notice that, if y ∈∂S is such that |x −y |=1, then B (y ) is an exterior sphere to S at x . 0 j 0 0 1 0 i 0 Moreover, by the Hopf lemma, the interior Lipschitz regularity is optimal. Regardingtheregularityofavectorofeigenfunctionsuofproblem(B),ifweaskthatΩsatisfies the exterior sphere condition, then we have actually a stronger statement. Theorem1.4. Letubeavectoroffirsteigenfunctionsassociatedtoanoptimalpartition(ω ,...,ω ) 1 k of(1.2). AssumethatΩsatisfiestheexteriorsphereconditionwithradiusr >0. Thenuisglobally Lipschitz continuous in Ω. Next, we establish a relation involving the normal derivatives of two “adjacent components” on the regular part of the free boundary. In what follows, for each i, ν (x) will denote the exterior normal at a point x ∈ ∂S (at points i i where such a normal vector does exist). Assumptions. Let x ∈ ∂S ∩ Ω, and let us assume that ΓR := ∂S ∩ B (x ) is a smooth 0 i i i R 0 hypersurface, for some R > 0. By the 1-uniform exterior sphere condition, we know that the principal curvatures of ∂S in x , denoted by χi(x ), h=1,...,N −1, are smaller than or equal i 0 h 0 to 1 (where we agree that outward is the positive direction). We further suppose that the strict inequality holds, that is there exists δ >0 such that (1.4) χi(x ),...,χi (x )(cid:54)1−δ. 1 0 N−1 0 We know that there exists j (cid:54)=i and y ∈∂suppu such that |x −y |=1. 0 j 0 0 Theorem 1.5. Let u = (u ,...,u ) be either any minimizer of J in H , or a vector of first 1 k ∞ ∞ eigenfunctionsassociatedtoanoptimalpartition(ω ,...,ω )of(1.2). Underthepreviousassump- 1 k (cid:83) tions and notations, we have that y =x +ν (x ) is the unique point in ∂suppu at distance 0 0 i 0 k(cid:54)=i k 1 from x . If y ∈∂suppu ∩Ω, then ∂suppu is also smooth around y , and 0 0 j j 0  (cid:12) (cid:12) (1.5) (∂νui(x0))2 = Nh(cid:89)=−11 (cid:12)(cid:12)(cid:12)(cid:12)χχihjh((xy00))(cid:12)(cid:12)(cid:12)(cid:12) if χih(x0)(cid:54)=0 for some h, (∂ u (y ))2 ν j 0  χih(x0)(cid:54)=01 if χi(x )=0 for all h=1,...,N −1. h 0 We stress that, since the sets S and S are at distance 1 from each other and (1.4) holds, i j χi(x ) (cid:54)= 0 if and only if χj(y ) (cid:54)= 0, and hence the term on the right hand side is always well h 0 h 0 defined. The proof of Theorem 1.5 is based on the introduction of a family of domain variations for the minimizeru. Asweshallsee,thepossibilityofproducingadmissibledomainvariations,preserving VARIATIONAL PROBLEMS WITH LONG-RANGE INTERACTION 5 the constraint on the distance of the supports in H , presents major difficulties. At the moment, ∞ wecanonlyovercomesuchobstructionsandproducemoreorlessexplicitvariationssupposingthat ∂S is locally regular. This is the main problem when trying to study the regularity of the free i boundary. Regarding this point, we mention that the proofs of all our results (and also of those in [7], in a nonvariational case) are completely different with respect to the analogue counterpart in problems with point-wise interaction. Indeed, all the local techniques, such as blow-up analysis and monotonicity formulae, cannot be straightforwardly adapted when dealing with long-range interaction; thereasonisthattheinterfacebetweendifferentpositivitysets{u >0}and{u >0} i j with i(cid:54)=j is now a strip of width at least 1, and hence with a standard blow-up one cannot catch the interaction on the free-boundary at the limit. Wealsomentionthatthevalidityofauniformexteriorsphereconditiondoesnotdirectlyimply any extra regularity for ∂S : if we could show that ∂S is a set with positive reach (see [14]), i i then we could argue as in [7, Corollary 6.3] and prove at least that the Hausdorff dimension of ∂S is N −1 (see also [8, Theorem 4.2] for a different proof of this fact), but on the other hand i sets enjoying the uniform exterior sphere condition are not necessarily of positive reach, as shown in [19, Section 2]. Remark 1.6. A very interesting feature of Theorem 1.5 stays in the fact that it reveals a deep differencebetweensegregationmodelswithpoint-wiseinteraction,andwithlong-rangeinteraction. To explain this difference, let us consider a sequence {u } of solutions to (1.3), with p = 1 and β β →+∞. Thisisthesettingstudiedin[7]. In[7,Theorem9.1],theauthorsderiveafree-boundary condition analogous to (1.5) for the limit configurations in case p = 1, but in their situation, the left hand side is replaced by the ratio between the normal derivatives, ∂ u (x )/∂ u (y ). This ν i 0 ν j 0 differenceisincontrastwithrespecttosegregationphenomenawithpoint-wiseinteraction,where, as proved in [25], limit configurations associated with (cid:88) (cid:88) −∆u =−βu u or −∆u =−βu u2 i i j i i j j(cid:54)=i j(cid:54)=i belongtothesamefunctionalclass[13,25], andhenceinparticularsatisfythesamefree-boundary condition, that is |∂ u (x )| = |∂ u (x )| on the regular part of the free boundary. A similar ν i 0 ν j 0 difference has been observed in [27,28,29] in the case of fractional operators, that is when the non-locality is in the differential operator. Finally, in comparison with the free boundary condition derived in [7], it is worthwhile noticing that the analogue of (1.5) there involves the plain quotient of the normal derivatives, while here we find the squared one. Remark1.7. Thepreviousresultmayfailiftherighthandsidein(1.4)isreplacedbytheconstant 1. Indeed, if ∂S ∩B (x )=∂B (0)∩B (x ) for some x ∈∂B (0) and R>0, and the set S is i R 0 1 R 0 0 1 i contained in the exterior of B (0), then y =0 is a cusp for ∂S . 1 0 j 1.2. Structure of the paper. We first treat problem (A). In Section 2 we prove Theorem 1.1 forthisproblem,relatingthissegregationproblemwithavariationalcompetition–diffusionoftype (1.3). Then some qualitative properties of any possible minimizer of problem (A) are shown in Section 3, where we prove Theorem 1.3 for this problem. Section 4 contains the proof of the free boundary condition contained in the statement of Theorem 1.5 for problem (A). The analogous statements for problem (B) – existence and properties of minimizers, and free boundary condition – are proved in Section 5. Finally, in Appendix A we state and prove an Hadamard’s type formula which we need along this paper. 6 N.SOAVE,H.TAVARES,S.TERRACINI,ANDA.ZILIO 2. Existence of a minimizer for Problem (A) In this section we prove Theorem 1.1. To this purpose, we introduce a competition parameter β >0 which allows us to remove the segregation constraint. To be precise, let H ={u∈H1(Ω ,Rk): u =f a.e. in Ω \Ω}⊃H , 1 i i 1 ∞ and let β >0. We consider the minimization of the functional ˆ ¨ k (cid:88) (cid:88) J (u)= |∇u |2+ β1 (x−y)u2(x)u2(y)dxdy β i B1 i j i=1 Ω 1(cid:54)i<j(cid:54)k Ω1×Ω1 in the set H. With respect to the search of a minimizer for inf J , the advantage stays in the H∞ ∞ fact that we can get rid of the infinite dimensional constraint dist(suppu ,suppu )(cid:62)1 for i(cid:54)=j, i j and we can easily show that a minimizer for J in H does exists, and satisfies an Euler-Lagrange β equation of type (1.3) with p = 2. This allows us to obtain Theorem 1.1 as a direct corollary of the following statement: Theorem 2.1. For every β >0, there exists a minimizer u =(u ,...,u ) for inf J , which β 1,β k,β H β is a solution of −∆u =−βu (cid:80) (1 (cid:63)u2) in Ω  i i j(cid:54)=i B1 j (2.1) u >0 in Ω i u =f in Ω \Ω. i i 1 The family {u : β > 0} is uniformly bounded in H1(Ω ,Rk)∩L∞(Ω ), and there exists u = β 1 1 (u ,...,u )∈H such that: 1 k (1) u →u strongly in H1(Ω) as β →+∞, up to a subsequence; β (2) dist(suppu ,suppu )(cid:62)1 for every i(cid:54)=j, so that u∈H ; i j ∞ (3) for every i(cid:54)=j, ¨ lim 1 (x−y)u2 (x)u2 (y)dxdy =0 β→+∞ Ω1×Ω1 B1 i,β j,β (4) u is a minimizer for inf J . In particular, u is a solution to problem (A). H∞ ∞ Remark 2.2. Without any additional complication, we can replace in the previous theorem the indicator function 1 with a more general function V ∈ L∞(RN) satisfying V > 0 a.e. in B , B1 1 V =0 a.e. on RN \B . 1 The proof of Theorem 2.1 is the object of the rest of the section. Before proceeding, we observe that, by the definition of support given in [3, Proposition 4.17], the set H can be defined in the ∞ following equivalent way: ¨ (cid:26) (cid:27) H = u∈H : 1 (x−y)u2(x)u2(y)dxdy =0 ∀i(cid:54)=j ∞ B1 i j Ω1×Ω1 (see the proof of Lemma 3.1 below for more details). Remark 2.3. Here it is worth to stress that we consider the functions u as defined in Ω , and i 1 hence the supports have to be considered in this set (and not only in Ω). Proof of Theorem 2.1. Theexistenceofaminimizeru followsbythedirectmethodofthecalculus β ofvariations,andthefactthatminimizerssolve(2.1)isstraightforward. Observethatf (cid:62)0,hence i the minimizers are positive in Ω, by the strong maximum principle. VARIATIONAL PROBLEMS WITH LONG-RANGE INTERACTION 7 For the uniform L∞ estimate, since u >0 is subharmonic in Ω for every i=1,...,k, by the i,β maximum principle we have (cid:107)u (cid:107) (cid:54)(cid:107)f (cid:107) . Let us set i,β L∞(Ω) i L∞(∂Ω) c :=infJ and c := inf J . β β ∞ ∞ H H∞ We observe that, since J (v) = J (v) for every v ∈ H , we have c (cid:54) c . Then, by the β ∞ ∞ β ∞ minimality of u , for every β > 0 we have J (u ) (cid:54) c . Since moreover u ≡ f in Ω \Ω, the β β β ∞ i,β i 1 uniform H1(Ω ,Rk) boundedness of {u } follows. Hence, up to a subsequence, u (cid:42)u weakly in 1 β β H1(Ω ,Rk) and a.e. in Ω. Moreover 1 ¨ lim 1 (x−y)u2(x)u2(y)dxdy =0 ∀i(cid:54)=j β→+∞ Ω1×Ω1 B1 i j and by the Fatou lemma we have ¨ ¨ 0(cid:54) 1 (x−y)u2(x)u2(y)dxdy (cid:54)liminf 1 (x−y)u2 (x)u2 (y)dxdy =0 Ω1×Ω1 B1 i j β→+∞ Ω1×Ω1 B1 i,β j,β for every i(cid:54)=j. This in particular proves point (2) in the thesis and implies that u∈H , defined ∞ in (1.1). On the other hand, by the the minimality of u and weak convergence, β ˆ ˆ k k (cid:88) (cid:88) c (cid:54)J (u)= |∇u |2 (cid:54)liminf |∇u |2 ∞ ∞ i i,β i=1 Ω β→∞ i=1 Ω ˆ k (cid:88) (cid:54)limsup |∇u |2 (cid:54)limsupJ (u )=limsupc (cid:54)c . i,β β β β ∞ β→∞ i=1 Ω β→∞ β→∞ This means that all the previous inequalities are indeed equalities, and in particular: • we have convergence (cid:107)∇u (cid:107) → (cid:107)∇u (cid:107) , which together with the weak conver- i,β L2(Ω) i L2(Ω) gence ensures that u →u strongly in H1(Ω,Rk) (recall that Ω is bounded); β • point (3) of the thesis holds; • we have c =J (u), which proves the minimality of u∈H . (cid:3) ∞ ∞ ∞ 3. Properties of minimizers for problem (A) ThissectionisdevotedtotheproofofTheorem1.3forthesolutionsofproblem(A).Letthenu be a minimizer for inf J . Theorem 1.1 (see also Theorem 2.1) does not give any information H∞ ∞ about the continuity of u , and in particular we do not know if the sets S = {x ∈ Ω : u (x) > 0} i i i are open. On the other hand it is reasonable to work at a first stage with the functions ˆ Φ :Ω→R, Φ (x):= u2(y)dy, i i i B1(x) which are clearly continuous due to the Lebesgue dominated convergence theorem. Let us consider the open sets   (cid:91) Ci =Ω∩ B1(y), Di :=int(Ω\Ci), y∈{Φi=0} for i=1,...,k, so that Ω=C ∪D ∪(∂D ∩Ω), and ∂D ∩Ω=∂C ∩Ω. i i i i i 8 N.SOAVE,H.TAVARES,S.TERRACINI,ANDA.ZILIO Observe that, by the definition of Φ , we have u =0 a.e. in C . Moreover i i i D ={x∈Ω:dist(x,{Φ =0})>1}⊂{Φ >0}. i i i The strategy of the proof of Theorem 1.3 can be summarized as follows: • At first, we prove some simple properties of the set D and of the restriction of u on D . i i • In particular, we show that S is the union of connected components of D , so that the i i regularity of u in Ω is reduced to the regularity of u on ∂D . i i i • Using the basic properties of D , we show that u is locally Lipschitz continuous across i i ∂D , andhenceinΩ. ItfollowsinparticularthatS isopen, anddirectlyinheritsfrom D i i i properties (3) and (5) in Theorem 1.3. Moreover, points (1) and (2) holds. • As a last step, we prove point (6) by using the minimality of u. Lemma 3.1. The function u is harmonic in D . In particular, if D˜ is any connected component i i i of D , then either u ≡0 or u >0 in D˜ . i i i i Proof. The set D is open. If we know that dist(D ,suppu ) (cid:62) 1, then we can consider any i i j φ∈C∞(D ) and observe that, by the minimality of u for J on the set H , the function c i ∞ ∞ f(ε):=J (u ,...,u ,u +εφ,u ,...,u ) ∞ 1 i−1 i i+1 k has a minimum at ε = 0. This implies that u is harmonic in D , and all the other conclusions i i follow immediately. Therefore, in what follows we have to show that (3.1) dist(D ,suppu )(cid:62)1 ∀j (cid:54)=i. i j By definition of H we have u2(x)u2(y)1 (x−y)=0 for a.e. x,y ∈Ω , that is ∞ i j B1 1 u2(x)u2(y)=0 for a.e. x,y ∈Ω , |x−y|<1. i j 1 As a consequence, u (x)Φ (x)=0 for a.e. x∈Ω and every j (cid:54)=i. In particular, this implies that j i (3.2) {Φ >0}⊂(Ω\suppu ). i j Let x ∈D . Then by definition of D , dist(x ,{Φ =0})>1, and hence B (x )⊂{Φ >0}. But 0 i i 0 i 1 0 i then, due to (3.2), and since x has been arbitrarily chosen, we deduce that (3.1) holds. (cid:3) 0 Let A be the union of the connected components of D on which u > 0, and let N be the i i i i union of those on which u ≡0, so that D =A ∪N . We know that u is positive and harmonic i i i i i in A , while u = 0 a.e. in N ∪C . Since A , N and C are open, this means that (if necessary i i i i i i i replacingu withadifferentrepresentativeinitssameequivalenceclass)u iscontinuousinA ,N , i i i i and C . To discuss the continuity of u in Ω, we have to derive some properties of the boundary i i ∂D ∩Ω = (∂A ∪∂N )∩Ω = ∂C ∩Ω. In the next lemma we show that D satisfies a uniform i i i i i exterior sphere condition. Lemma 3.2. For each i, the set D satisfies the 1-uniform exterior sphere condition in Ω, in the i following sense: for every x ∈∂D ∩Ω there exists a ball B of radius 1 such that 0 i D ∩B =∅ and x ∈D ∩B. i 0 i Moreover, in B we have u ≡0. i Proof. This comes directly from the definitions: we have ∂D ∩Ω=∂C ∩Ω={x:dist(x,{Φ =0})=1}∩Ω. i i i Thus, given x ∈ ∂D ∩Ω, there exists y ∈ ∂B (x) with Φ (y) = 0. The ball B (y) is the desired i 1 i 1 exterior tangent ball, since B (y)⊂C , and hence B (y)∩D =∅. (cid:3) 1 i 1 i VARIATIONAL PROBLEMS WITH LONG-RANGE INTERACTION 9 The exterior sphere condition permits to deduce that ∂D has zero Lebesgue measure. i Lemma 3.3. The boundary ∂D is a porous set, and in particular it has 0 Lebesgue measure and i dim (∂D )<N. H i For the definition of “porosity”, we refer to [20, Section 3.2], while here and in what follows dim denotes the Hausdorff dimension. H Proof. Since∂D ⊂Ωisbounded,toproveitsporosityitissufficienttoshowthatthereexistsδ >0 i suchthat: foreveryballB (x )withx ∈∂D ,thereexistsy ∈B (x )withB (y)⊂B (x )\∂D r 0 0 i r 0 δr r 0 i (see [20, Exercise 3.4]). The existence of such δ = 1/2 follows immediately by the exterior sphere condition: given x ∈ ∂D , there exists z ∈ Ω such that B (z) is exterior to D . Let then y be the point on 0 i 1 1 i the segment x z at distance r/2 from D . The ball B (y) is contained both in Ω \∂D and in 0 i r/2 1 i B (x ), and this proves that ∂D is porous. The rest of the proof follows by [20, Page 62]. (cid:3) r 0 i It is not difficult now to deduce that u is continuous at every point of ∂N . Indeed, notice that i i ∂N ⊂∂C , and in both N and C we have u ≡0. Since ∂N ⊂∂D has 0 Lebesgue measure, we i i i i i i i deduce that u =0 a.e. in N ∪C =Ω\A . That is, up to the choice of a different representative, i i i i u ≡ 0 in Ω\A , and hence it is real analytic therein. At this stage, it remains to discuss the i i continuity of u on ∂A . This is the content of the forthcoming Corollary 3.6, where we show that i i actually u is locally Lipschitz continuous in Ω. We postpone the proof, proceeding here with the conclusion of Theorem 1.3. The continuity of u implies in particular that {u > 0} is open for i i every i, so that {u > 0} = A . Thus, Lemmas 3.1-3.3 establish the validity of points (2) and (5) i i in Theorem 1.3. The subharmonicity of u , point (1), follows from (2). i Regardingpoint(3),theexistenceofanexteriorsphereB ofradius1for{u >0}atanybound- i ary point x comes directly from Lemma 3.2. We also know that u ≡ 0 in B, and furthermore, 0 i by (3.1), B (x )∩suppu =∅ for every j (cid:54)=i. This proves the validity of (3). 1 0 j It remains only to show that also point (6) holds. Proof of Theorem 1.3-(6). Thisisaconsequenceoftheminimality. Takex ∈∂S ∩Ωandassume, 0 i in view of a contradiction, that dist(x ,suppu )>1 for some x ∈∂S ∩Ω, for every j (cid:54)=i. Then 0 j 0 i there exists ρ>0 such that B (x )⊂Ω and ρ 0 (3.3) dist(B (x ),suppu )>1 ∀j (cid:54)=i. ρ 0 j Let v be the harmonic extension of u in B (x ): i ρ 0 (cid:40) ∆v =0 in B (x ) ρ 0 v =u on ∂B (x ). i ρ 0 Since u (cid:54)≡ 0 on ∂B (x ), we infer that v > 0 in B (x ), and in particular v (cid:54)≡ u in B (x ). Let i ρ 0 ρ 0 i ρ 0 now u˜ be defined by (cid:40) u in Ω\B (x ) u˜ = i ρ 0 , u˜ =u ∀j (cid:54)=i. i j j v in B (x ) ρ 0 Due to(3.3), it belongs to H , so that byminimality J (u)(cid:54)J (u˜). On theother hand, by the ∞ ∞ ∞ definition of harmonic extension we have also J (u˜) < J (u) (the strict inequality comes from ∞ ∞ the fact that v (cid:54)≡u in B (x )), a contradiction. (cid:3) i ρ 0 10 N.SOAVE,H.TAVARES,S.TERRACINI,ANDA.ZILIO Remark 3.4. In [7], the authors proved harmonicity, local Lipschitz continuity, and exterior sphereconditionforlimitsofanysequenceofsolutionsto(2.1). Nevertheless,theresulthereisnot containedin[7],sinceweestablishharmonicity,Lipschitzcontinuity,andexteriorspherecondition for any minimizer of inf J , independently on wether it can be approximated with a sequence H∞ ∞ ofsolutionsto(2.1)ornot. Also,itisworthtopointoutthattheapproachiscompletelydifferent: while in [7] the authors proceed with careful uniform estimates for viscosity solution of (1.3), here we use the variational structure of the limit problem. 3.1. Lipschitz continuity of the minimizers. In this subsection we show that the solutions of problem (A) are Lipschitz continuous inside Ω, which is the highest regularity one can expect for the minimizers of J (by the Hopf lemma). This is a consequence of the following general ∞ statement. Theorem3.5. LetΛbeadomainofRN,andletA⊂Λbeanopensubset,satisfyingther-uniform exterior sphere condition in Λ: for any x ∈ ∂A∩Λ there exists a ball B with radius r which is 0 exterior to A and tangent to ∂A at x , i.e. 0 A∩B =∅ and x ∈A∩B. 0 Let f ∈L∞(Λ), and let u∈H1(Λ)∩L∞(Λ) satisfy (cid:40) −∆u=f in A u=0 a.e. in Λ\A Then u is locally Lipschitz continuous in Λ, and for every compact set K (cid:98) Λ there exists a constant C =C(r,N,K)>0 such that (cid:107)∇u(cid:107) (cid:54)C(cid:0)(cid:107)u(cid:107) +(cid:107)f(cid:107) (cid:1). L∞(K) L∞(Λ) L∞(Λ) For the sake of generality, we required no sign condition on the function u, even though we will apply the result only to nonnegative solutions. Corollary 3.6. Let u be any minimizer of J in H . Then u is locally Lipschitz continuous in ∞ ∞ Ω. Proof. WeapplyTheorem3.5totheharmonicfunctionsu inA:=A ,withΛ:=Ωandr =1. (cid:3) i i TheproofofTheorem3.5isbaseduponasimplebarrierargument. ForanyR>0,letusdefine (cid:40) 1 −∆w =1 in B w (x):= (R2−|x|2)+ =⇒ R R R 2N w =0 in RN \B , R R and let (3.4) w∗(x):=(cid:18)R (cid:19)N−2w (cid:18)R2 x(cid:19)= RN (cid:0)|x|2−R2(cid:1)+ R |x| R |x|2 2N|x|N be its Kelvin transform with respect to the sphere of radius R. It is not difficult to check that (cid:18)R (cid:19)N+2 (cid:18)R2 (cid:19) (cid:18)R (cid:19)N+2 (3.5) −∆w∗(x)=− ∆w x = . R |x| R |x|2 |x| With this preliminary observation, we can easily prove the following estimate: