On the isoperimetric problem in Euclidean space with density

ON THE ISOPERIMETRIC PROBLEM IN EUCLIDEAN SPACE WITH DENSITY C´ ESAR ROSALES, ANTONIO CA ˜ NETE, VINCENT BAYLE, AND FRANK MORGAN ABSTRACT. We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that for a radial log-convex density, balls about the origin are isoperimetric regions. Finally, we prove this conjecture and the uniqueness of minimizers for the density exp(|x|2)by using symmetrization techniques. 1. INTRODUCTION The isoperimetric problem inside a Riemannian manifold seeks regions of least perimeter enclosing a fixed amount of volume. This problem can also be studied in the more general setting of a manifold with density, where a given continuous positive function on the manifold is used to weight the Riemannian volume and boundary area. Such a density is not equivalent to scaling the metric conformally by a factor λ, since in that case volume and perimeter would scale by different powers of λ. We shall consider the particular case of Euclidean space with a density f=eψ. For any Borel set Ωin Rn+1, the (weighted) volume or measure of Ω, and the (weighted) perimeter relative to an open set U in Rn+1are given by vol(Ω) = ZΩ f dv,P(Ω,U) = Z∂Ω∩U f da, where dv and da are elements of Euclidean volume and area, in general provided by Lebesgue measure and n-dimensional Hausdorff measure on Rn+1. Let P(Ω) = P(Ω,Rn+1). Much of the information of the isoperimetric problem is contained in the isoperimetric profile, which is the function If:(0, vol(Rn+1)) →Rgiven by If(V) = inf {P(Ω):Ωis a smooth open set with vol(Ω) = V}. An isoperimetric region –or simply a minimizer– of volume Vis an open set Ωsuch that vol(Ω) = Vand P(Ω) = If(V). In the last years the study of isoperimetric problems in manifolds with density has increased. One of the first and most interesting examples, with applications in probability and statistics, is the Gaussian density exp(−π|x|2). About 1975 C. Borell [Bor1] and V. N. Sudakov and B. S. Tirel’son [ST] independently proved that half-spaces minimize perimeter under a volume constraint for this density. In 1982 A. Ehrhard [Eh1] gave a new proof of the isoperimetric property of half-spaces by adapting symmetrization techniques to the Date: October 29, 2018. 2000 Mathematics Subject Classification. 49Q20, 53C17. Key words and phrases. Manifolds with density, isoperimetric problem, generalized mean curvature, stability, symmetrization. First and second authors are partially supported by MCyT-Feder research project MTM2004-01387, fourth author by the National Science Foundation. 2 C. ROSALES, A. CA ˜ NETE, V. BAYLE, AND F. MORGAN Gaussian context. His proof can be simplified by following the generalization of Steiner symmetrization to product measures given by A Ros [R]. More recently, S. Bobkov and C. Houdr´e [BoH] considered “unimodal densities” with finite total measure in the real line. These authors explicitly computed the isoperimetric profile for such densities and found some of the isoperimetric solutions. M. Gromov [Gr] studied manifolds with density as “mm spaces” and mentioned the natural generalization of mean curvature obtained by the first variation of weighted area. V. Bayle [Ba] proved generalizations of the L´evy-Gromov isoperimetric inequality and other geometric comparisons depending on a lower bound on the generalized Ricci curvature of the manifold. For recent advances on manifolds with density we refer the reader to [M4], [Ba] and references therein. In this paper we first prove some existence results of isoperimetric regions for densities in Euclidean space with infinite total measure (Theorems 2.2 and 2.6) and recall regularity properties of minimizers (Theorem 2.8). In Section 3 we use a variational approach to characterize stability of balls centered at the origin for radial densities (Theorem 3.10). This result leads us to Conjecture 3.12: for radial log-convex densities in Rn+1, balls about the origin provide minimizers of any given volume. We will prove this conjecture in the one-dimensional case (Corollary 4.12) and for the radial density exp(|x|2)in any dimension (Theorem 5.2). In Section 4 we completely describe isoperimetric regions in the real line endowed with unimodal densities (Theorems 4.3 and 4.7). We use comparison arguments that provide at the same time existence and uniqueness of minimizers. As interesting consequences we solve the isoperimetric problem for log-concave and log-convex densities in the real line (Corollaries 4.8 and 4.11), improving previous results by S. Bobkov and C. Houdr´e [BoH]. We also treat the isoperimetric problem and the free boundary problem for the closed halfline [0, +∞)and for compact intervals. In Section 5 we establish, in arbitrary dimension, the isoperimetric property of round balls about the origin for the density exp(c|x|2),c>0 (Theorem 5.2). A remarkable difference with respect to the Gaussian measure is that the density exp(c|x|2)for c>0 has infinite total volume and hence the existence of minimizers is a non-trivial question. The proof of Theorem 5.2 goes as follows. First, we apply our previous results in Section 2 to ensure existence of isoperimetric regions of any given volume. Second, we use the description of minimizers for log-convex densities on the real line (Corollary 4.12) and the symmetrization in spaces with product measures given by A. Ros [R], to construct a counterpart to Steiner symmetrization for the density exp(c|x|2). Then we use this symmetrization in axis directions as employed by L. Bieberbach [Bi] to produce centrally symmetric minimizers with connected boundary. Finally we conclude by Hsiang symmetrization [H] that such a minimizer must be a round ball about the origin. As a corollary of Theorem 5.2 we deduce an eigenvalues comparison theorem for the density exp(c|x|2),c⩾0, generalizing the Faber-Krahn Inequality. Usually the uniqueness of isoperimetric regions is difficult to prove. In the case of the Gaussian density, the complete characterization of equality cases in the isoperimetric inequality is due to E. A. Carlen and C. Kerce [CK], who proved that any perimeter minimizer for fixed volume is, up to a set of measure zero, a half-space. They obtained this result as consequence of the discussion of equality in a more general functional inequality due to S. Bobkov. Previous uniqueness results in the Gaussian setting involving a Brunn-Minkowski type inequality were given by A. Ehrhard [Eh2]. In Theorem 5.2 we also show the uniqueness of round balls centered at the origin as minimizers for the density exp(c|x|2),c>0. Since round balls appear as the result of finitely many symmetrizations, it suffices to see that if an axis symmetrization of a minimizer produces a ball, then the minimizer is a ball. We deduce this fact by standard arguments [Ch2, Lemma III.2.3]. THE ISOPERIMETRIC PROBLEM IN Rn+1WITH DENSITY 3 An interesting consequence of our characterization of stable balls in Theorem 3.10 observed by K. Brakke is that any round ball about the origin is unstable in Rn+1endowed with a radial, strictly log-concave density. This fact, together with the isoperimetric property of half-spaces in the Gaussian space and our Corollary 4.9, where we proved the isoperimetric property of half-lines for densities on the real line, might suggest that halfspaces are isoperimetric regions for any radial, log-concave density on Rn+1. In Corollary 3.13 we give an example showing that this is not true in general. We believe that this is a motivation to study in more detail the isoperimetric problem for these kind of densities where unexpected shapes appear. After circulating this manuscript we heard from Franck Barthe and Michel Ledoux that the isoperimetric property of round balls about the origin for the density exp (c|x|2),c>0, was previously proved by C. Borell [Bor2, Theorem 4.1]. Borell’s proof uses a Brunn- Minkowski inequality and does not yield uniqueness of minimizers. Acknowledgements. This work began during Morgan’s lecturers on “Geometric Measure Theory and Isoperimetric Problems” at the 2004 Summer School on Minimal Surfaces and Variational Problems held in the Institut de Math´ematiques de Jussieu in Paris. We would like to thank the organizers: Pascal Romon, Marc Soret, Rabah Souam, Eric Toubiana, Fr´ed´eric H´elein, David Hoffman, Antonio Ros and Harold Rosenberg. We also thank Franck Barthe, Christer Borell and Michel Ledoux for bringing previous results on the isoperimetric problem for the density exp (|x|2)to our attention. 2. EXISTENCE AND REGULARITY RESULTS In this section we firstly deal existence of isoperimetric regions in Euclidean space with density. In general, for a Riemannian manifold with density, standard compactness arguments of Geometric Measure Theory (see [Si, 27.3 and 31.2] or [M2, 5.5 and 9.1], and [M1, 4.1] or [RiR, Thm. 2.1]) can be applied in order to provide isoperimetric regions, except that there can be loss of volume at infinity (by regularity Theorem 2.8, these are open sets with nice boundaries). In particular, if the total measure is finite, isoperimetric regions of any prescribed volume exist. We will prove some existence results for densities with infinite total volume. We begin with the following lemma: Lemma 2.1. Let f be a positive, nondecreasing function on [0, +∞)satisfying f (r)→+∞when r→+∞. If the function ψ=log(f)satisfies ψ(r)⩽Cn+1 n−εr/2 for some n ∈N, C >0and ε∈(0, 1), then the sequence ζ(m) = f(m) f(m+2)n/(n+1) tends to infinity. Conversely, if {ζ(m)} → +∞, then there is r0>0and C >0such that ψ(r)⩽Cn+1 nr/2 ,r⩾r0. Proof. We prove the first part of the statement by contradiction. Suppose that {ζ(m)}m∈N does not tend to infinity. We can assume, by passing to a subsequence if necessary, that there is K>0 such that f(m)⩽K f (m+2)n/(n+1),m∈N, 4 C. ROSALES, A. CA ˜ NETE, V. BAYLE, AND F. MORGAN and therefore ψ(m+2)⩾n+1 n(ψ(m)−log(K)),m∈N. On the other hand, as {ψ(m)} → +∞and ε>0, we can find m0∈Nsuch that ψ(m+2)⩾n+1 n−ε 2ψ(m),m⩾m0, ψ(m0+2k)⩾n+1 n−ε 2k ψ(m0). Now take r⩾m0+2 and k∈Nsuch that m0+2k⩽r⩽m0+2(k+1). By using that ψis nondecreasing we have ψ(r)⩾ψ(m0+2k)⩾n+1 n−ε 2k ψ(m0)⩾n+1 n−ε 2r/2−m0/2−1 ψ(m0). Hence, for r≫m0+2 we deduce ψ(r)>Cn+1 n−εr/2 , and we get a contradiction. Conversely, suppose that {ζ(m)} → +∞. Then, we can find m0⩾2 such that ψ(m0)>0 and ζ(m)⩾1 for m⩾m0. As a consequence ψ(m+2)⩽n+1 nψ(m),m⩾m0, ψ(m0+2k)⩽n+1 nk ψ(m0). Finally, for r⩾m0there is k∈Nsuch that m0+2(k−1)⩽r⩽m0+2k. Hence ψ(r)⩽ψ(m0+2k)⩽n+1 nk ψ(m0)⩽n+1 nr/2 ψ(m0).  Now, we can prove our first existence result. Theorem 2.2. Let f =eψbe a density on Rn+1such that f (x)→+∞when |x| → +∞. Suppose that one of the following conditions holds: (i) The sequence defined by ζ(m) = min {f(x):m⩽|x|⩽m+2} max {f(x)n/(n+1):m⩽|x|⩽m+2} tends to infinity. (ii) The density is radial, nondecreasing in |x|and satisfies ψ(x)⩽Cn+1 n−ε|x|/2 , for some constants C >0and ε∈(0, 1). Then, minimizers of any given volume exist for this density and they are bounded subsets of Rn+1. Remark 2.3.The proof of the statement shows that it suffices to suppose that the ratio min f(x)/ max f(x)n/(n+1)on lattice cubes goes to infinity. THE ISOPERIMETRIC PROBLEM IN Rn+1WITH DENSITY 5 Proof. By Lemma 2.1 we can assume that (i) holds. Denote by v(Ω)and a(∂Ω)the Euclidean volume and boundary area of a set Ω. Partition Rn+1into lattice open cubes of diameter equal to 1 and Euclidean volume v0. There is an isoperimetric constant α>0 such that any set Ωinside a cube Cas above with v(Ω)⩽v0/2 satisfies a(∂Ω∩C)⩾αv(Ω)n/(n+1). On the other hand, there is m=m(C)∈Nsuch that the cube Cis contained in the annulus {m⩽|x|⩽m+2}. Thus, the definition of weighted volume and perimeter, together with the definition of ζ(m), implies the inequality (2.1) P(Ω,C)⩾α ζ(m)vol(Ω)n/(n+1), for any Ω⊂C⊂ {m⩽|x|⩽m+2}with v(Ω)⩽v0/2. Fix V>0, and consider a sequence of smooth open sets of volume Vwith perimeters approaching If(V)and bounded from above by If(V) + 1. By using the Compactness Theorem [M3, 9.1] we may assume that this sequence converges. Fix ε>0. By hypothesis, there is m0∈Nsuch that ζ(m)⩾(1/ε)n/(n+1)for any m⩾m0. On the other hand, as f(x)→+∞ when |x| → +∞, we can suppose that v(Ω)⩽v0/2 whenever Ω⊂C⊂ {|x|⩾m0}. In particular, we can apply (2.1) to such an Ω, so that we obtain vol(Ω)⩽P(Ω,C) α ζ(m)(n+1)/n ⩽ε α′P(Ω,C)(n+1)/n. By summing the previous inequality over the collection Cmof all cubes Ccontained in {|x|⩾m}we deduce that for any set Ωof the given minimizing sequence and any m⩾m0 vol Ω∩([ C∈Cm C)⩽ε α′ ∑ C∈Cm P(Ω,C)!(n+1)/n ⩽ε α′P(Ω)(n+1)/n⩽ε α′(If(V) + 1)(n+1)/n. Hence, there is no loss of volume at infinity and the limit of our sequence is an isoperimetric region of volume V. To prove that any minimizer Ωis a bounded subset of Rn+1, we can proceed as in [M2, Lemma 13.6]. Consider any large cube Cr= [−r,r]n+1about the origin and partition almost all its complement into congruent open cubes of diameter at most 1. Denote V(r) = vol(Ω−Cr)and P(r) = P(Ω,Rn+1−Cr). As above, we have (2.2) V(r)⩽ε α′P(r)(n+1)/n,r≫0. On the other hand, there is a constant H>0 depending on ∂Ωsuch that small volume adjustments may be accomplished inside Crat a cost |∆P|⩽H|∆V|. Thus replacing Ω−Crcosts at most H|∆V|+|V′(r)|(due to the slice of ∂Cr) for almost all large r. By using that Ωis a minimizer, we get (2.3) P(r)⩽H V(r) + |V′(r)|, for almost all r≫0. Since V(r)is nonincreasing and tends to 0 when r→+∞, combining inequalities (2.2) and (2.3) yields for some c>0, c V(r)n/(n+1)⩽−V′(r), for almost all r≫0. If we suppose that Ωis unbounded, then V(r)6=0 and (n+1) (V1/(n+1))′=V−n/(n+1)V′⩽−c<0, 6 C. ROSALES, A. CA ˜ NETE, V. BAYLE, AND F. MORGAN for almost all large r>0, a contradiction since Vis positive and nonincreasing.  Our next existence result is an improvement of Theorem 2.2 in dimension two. We need the following lemma: Lemma 2.4. Let f be a planar radial density nondecreasing on [r0,+∞). Then, for any smooth, open set Ω⊂R2contained in {|x|⩾r0}and such that P(Ω)<2πr0f(r0), we have the isoperimetric inequality P(Ω)2⩾2f(r0)vol(Ω). Proof. First, we can assume that Ωis connected. Moreover, the hypothesis on the perimeter implies that Ωis bounded and the closure of Ωcannot contain a circle about the origin. Let r1and r2be the minimum and maximum distance from Ωto the origin, respectively. The intersection Ωtof Ωwith the circle of radius t∈(r1,r2)has Euclidean length strictly less than 2πt, and the boundary ∂Ωthas at least two points. Therefore, the coarea formula gives us (2.4) P(Ω)⩾Zr2 r1 f(t)card(∂Ωt)dt ⩾2f(r0) (r2−r1), where we have used that the density is nondecreasing on [r0,+∞). On the other hand, we consider the map F:(r1,r2)×∂Ω→R2given by F(t,x) = tx/|x|. It is clear that Ω⊆F(A), where Ais the open set of the pairs (t,x)where t<|x|and f(tx/|x|)<f(x). For any (t,x)∈Athe Jacobian of Fis strictly less than 1. Thus, the definition of A, together with the coarea formula and Fubini’s theorem implies (2.5) vol(Ω)⩽vol(F(A)) ⩽ZA ftx |x|d(t,x)⩽ZA f(x)d(t,x) = (r2−r1)P(Ω). Multiplying the estimates (2.4) and (2.5) we obtain the desired inequality.  Remark 2.5.We do not see how to generalize the previous lemma to Rn+1. The analog of inequality (2.5) holds, but the estimation on the Euclidean boundary area a(∂Ωt)leading to (2.4) becomes a(∂Ωt)⩾αf(t)1/nv(Ωt)(n−1)/n⩾C v(Ωt), where v(Ωt)denotes Euclidean volume. The last inequality can be integrated to deduce P(Ω)⩾Cvol(Ω), which is inadequate to obtain an analog of Lemma 2.4. Theorem 2.6. Consider the plane endowed with a nondecreasing, radial density f such that f (x)→ +∞when |x| → +∞. Then, there are minimizers for this density of any given volume. Proof. Consider a sequence of smooth open sets of volume V>0 with perimeters approaching If(V). Applying the Compactness Theorem [M3, 9.1] we can assume that this sequence converges. We can also suppose that any set Ωof this sequence satisfies P(Ω)⩽ If(V) + 1. Moreover, as the density tends to +∞, there is m0∈Nsuch that If(V) + 1< 2πm f (m)for any m⩾m0. In particular, we can apply Lemma 2.4 to the union Ω′of all connected components of Ωinside {|x|⩾m}. Hence, we get vol(Ω′)⩽P(Ω′)2 2f(m)⩽(If(V) + 1)2 2f(m),m⩾m0. As limm→+∞f(m) = +∞we conclude that there is no loss of volume at infinity and the limit of our sequence solves the isoperimetric problem for volume V. THE ISOPERIMETRIC PROBLEM IN Rn+1WITH DENSITY 7 Example 2.7. We illustrate here that Theorem 2.6 need not hold if we do not require the density to be nondecreasing. Consider in Rn+1(n>1)the density f(x) = 1+|x|2. Now, introduce bumps into the graph of fsuch that any volume Vkcorresponding to a positive rational can be enclosed with perimeter 1/k. Then, for any given volume we may find a sequence of sets enclosing this volume and with arbitrarily small perimeter, which implies that isoperimetric regions do not exist. We finish this section by recalling regularity properties of the boundary of a minimizer in Euclidean space with density. The result is also valid for any smooth Riemannian manifold with density. Theorem 2.8 ([M3, 3.10]).Consider a smooth density on Rn+1. If Ωis a minimizer, then the boundary Σ=∂Ωis a real-analytic embedded hypersurface, up to a closed set of singularities with Euclidean Hausdorff dimension less than or equal to n −7. 3. VARIATIONAL FORMULAE. STABLE BALLS FOR RADIAL DENSITIES In this section we use a variational approach to derive some properties of sets minimizing perimeter up to second order for variations preserving volume. Let f=eψbe a smooth density on Rn+1. Denote by Ω⊂Rn+1a smooth open set with boundary Σand inward unit normal vector N. We consider a one-parameter variation {φt}|t|<ε:Rn+1→Rn+1with associated infinitesimal vector field X=dφt/dt with normal component u=hX,Ni. Let Ωt=φt(Ω)and Σt=φt(Σ). The volume and perimeter functions of the variation are V(t) = vol(Ωt)and P(t) = P(Ωt), respectively. The first variation of volume and perimeter are computed in [Ba, Chapter 3]. We include here a proof for the sake of completeness. Lemma 3.1. The first variation of volume and perimeter of a smooth region Ωwith boundary Σin Rn+1endowed with smooth density f =eψfor a flow with initial normal velocity u are given by V′(0) = −ZΣ f u dv,P′(0) = −ZΣ (nH −h∇ψ,Ni)f u da, where H is the Euclidean mean curvature of Σwith respect to N (that is, the arithmetic mean of the principal curvatures of Σ) and ∇ψis the Euclidean gradient of ψ. Proof. Denote by div X(resp. divΣX) the divergence of Xin Rn+1(resp. relative to Σ). Let X(f) = h∇f,Xi. We have V′(0) = ZΩ X(f)dv +ZΩ fd dt t=0 (dvt) =ZΩ (h∇f,Xi+fdiv X)dv =ZΩ div(f X)dv =−ZΣ f u da. In the second equality we have used that (d/dt)|t=0(dvt) = (div X)dv, see [Si,§16]. In the last one, we have applied the Gauss-Green theorem. For perimeter we have P′(0) = ZΣ X(f)da +ZΣ fd dt t=0 (dat) =ZΣ (h∇f,Xi+fdivΣX)da =ZΣ (h∇f,uNi+divΣ(f X)) da =ZΣ f u h∇ψ,Nida −ZΣ nH f u da =−ZΣ (nH −h∇ψ,Ni)f u da. 8 C. ROSALES, A. CA ˜ NETE, V. BAYLE, AND F. MORGAN To obtain the fourth equality we have used that the integral over Σof the divergence of the tangent part of f X vanishes by virtue of the divergence theorem.  We define, as in [Ba, Chapter 3], the (generalized) mean curvature of Σwith respect to N as the function (3.1) Hψ=nH −h∇ψ,Ni, so that the first variation of perimeter can be written as P′(0) = −ZΣ Hψf u da. We say that a given variation {φt}tpreserves volume if V(t)is constant for any small t. We say that Ωis stationary if P′(0) = 0 for any volume-preserving variation. It is clear that any isoperimetric region is also stationary. The following characterization of stationary sets is similar to the one established by J. L. Barbosa and M. do Carmo [BdC, Proposition 2.7] for the case of Rn+1with the standard density f≡1. The proof is based on Lemma 3.1 and on the fact that any function uorthogonal to fin L2(Σ)is the normal component of a vector field associated to a volume-preserving variation of Ω, see [BdC, Lemma 2.2]. Proposition 3.2. Consider a smooth density f =eψon Rn+1. Then, for a smooth open set Ω, the following conditions are equivalent: (i) Ωis stationary. (ii) Σ=∂Ωhas (generalized)constant mean curvature H0. (iii) There is a constant H0such that (P−H0V)′(0) = 0for any variation of Ω. Example 3.3. Let f=eψbe a smooth density defined on the real line. Then, it is easy to show that a bounded interval (a,b)is stationary if and only if ψ′(a) = −ψ′(b). Now, we introduce some examples of hypersurfaces with constant mean curvature in Rn+1with a radial density. Example 3.4. Suppose f=eψ, where ψ(x) = δ(|x|)for any x∈Rn+1. The mean curvature Hψof a hypersurface Σwith respect to a unit normal vector Nis given by Hψ(p) = nH(p)−δ′(r) rhp,N(p)i,r=|p|, where His the Euclidean mean curvature of Σwith respect to N. In particular, if Σis a sphere of radius r>0, then it has constant mean curvature if and only if Σis centered at the origin. In this case, Hψ=n/r+δ′(r)with respect to the inner normal vector. On the other hand, if Σis the hyperplane defined by {x∈Rn+1:hx,ui=c}, where |u|=1, then the mean curvature of Σwith respect to N=−uis (3.2) Hψ(p) = −cδ′(r) r,r=|p|. It follows that any hyperplane passing through the origin is a minimal hypersurface of Rn+1with a radial density. In general, we cannot expect that any hyperplane has constant mean curvature for a radial density. In fact, a straightforward analysis of equation (3.2) leads us to the following: Lemma 3.5. Let f =eψbe a smooth radial density on Rn+1. Suppose that there is a hyperplane Σ which does not contain the origin and has constant mean curvature Hψ. Then, there are constants a,b∈Rand r0>0, such that ψ(x) = ea|x|2+b,whenever |x|⩾r0. THE ISOPERIMETRIC PROBLEM IN Rn+1WITH DENSITY 9 Now, we compute the second variation formula of the functional P−HψVfor any variation of a stationary set. Proposition 3.6 ([Ba, Section 3.4.6]).Consider a stationary open set Ωin Rn+1endowed with a smooth density f =eψ. Let N be the inward unit normal vector to Σ=∂Ω, and Hψthe constant mean curvature of Σwith respect to N. Consider a variation of Ωwith associated vector field X=uN on Σ. Then, we have (3.3) (P−HψV)′′(0) = Qψ(u,u):=ZΣ f(|∇Σu|2−|σ|2u2)da +ZΣ f u2(∇2ψ) (N,N)da, where ∇Σu is the gradient of u relative to Σ,|σ|2is the squared sum of the principal curvatures of Σ, and ∇2ψis the Euclidean Hessian of ψ. Proof. The first variation formula for volume and perimeter gives us (P−HψV)′(t) = −ZΣt (Hψ)tf utdat+HψZΣt f utdat, where (Hψ)tis the mean curvature of Σt. Hence (3.4) (P−HψV)′′(0) = −ZΣ H′ ψ(0)f u da, so that we have to compute the derivative of the generalized mean curvature along Σt. Denote by DUVthe Levi-Civit´a connection on Rn+1. By (3.1), we get H′ ψ(0) = nH′(0)−hDX∇ψ,Ni−h∇ψ,DXNi =nH′(0)−u(∇2ψ)(N,N) + h∇ψ,∇Σui, where in the last equality we have used that DXN=−∇Σu. On the other hand, it is well known [Ro] that (3.5) nH′(0) = ∆Σu+|σ|2u, where ∆Σis the Laplacian relative to Σ. Thus, we have obtained H′ ψ(0) = ∆Σu+|σ|2u−u(∇2ψ)(N,N) + h∇Σψ,∇Σui. By substituting this information into (3.4) we conclude that (P−HψV)′′(0) = −ZΣ f u (∆Σu+|σ|2u)da −ZΣ f u h∇Σψ,∇Σuida +ZΣ f u2(∇2ψ) (N,N)da. Finally, by using integration by parts, we deduce −ZΣ f u (∆Σu+|σ|2u)da −ZΣ f u h∇Σψ,∇Σuida =ZΣ f(|∇Σu|2−|σ|2u2)da, and the result follows.  Remark 3.7.In a smooth Riemannian manifold with density the second variation has an additional term depending on the Ricci curvature of the manifold in the normal direction N. This term comes from (3.5) and it is given by −ZΣ Ric(N,N)f u2da. 16 C. ROSALES, A. CA ˜ NETE, V. BAYLE, AND F. MORGAN (i) If f is increasing, then the unique minimizers for the free boundary problem in [0, +∞)are intervals of the form (0, x). If f is decreasing and E = +∞has finite measure, then minimizers exist and they are half-lines containing E. If E has infinite measure and a minimizer exists, then it must coincide with an interval (0, x)or a bounded interval where f equals its minimum. If a minimizer does not exist the infimum perimeter equals 2f(E). (ii) If f is increasing-decreasing and a minimizer of given volume exists, then it is an interval (0, x), a half-line containing +∞, a bounded interval where f attains its maximum or equals its right-side minimum, or the complement of one of these intervals. If it does not exist, then the infimum perimeter is approached by a bounded interval going off to +∞. (iii) If f is decreasing-increasing then minimizers of any given volume are provided by intervals (0, x)or bounded intervals in whose closure f attains its minimum. For the free boundary problem in [a,b], existence of minimizers is assured by compactness. As to the description of isoperimetric regions, we can prove the following: Theorem 4.16. Let f be a unimodal density on [a,b]. Then (i) If f is monotonic then the unique minimizers for the free boundary problem are intervals whose closure contains a boundary point. (ii) If f is increasing-decreasing, then isoperimetric regions are provided by intervals where f attains its maximum, or whose closure contains a boundary point of [a,b], or complements of these intervals. (iii) If f is decreasing-increasing, then minimizers are intervals whose closure contains a boundary point of [a,b]or a value where the minimum of f is attained, or complements of these intervals. 5. ISOPERIMETRIC INEQUALITY FOR THE DENSITY exp(|x|2) In this last section of the paper we solve the isoperimetric problem in Rn+1with the radial log-convex density f(x) = exp(c|x|2), where cis a positive constant. Precisely, we will prove that Conjecture 3.12 holds for this density: round balls about the origin provide isoperimetric regions of any given volume, like Euclidean space (c=0)and unlike Gauss space (c<0). As we pointed out in the Introduction, the proof combines Steiner symmetrization in axis directions as was employed by L. Bieberbach [Bi] together with Hsiang symmetrization [H]. We will also show uniqueness by a detailed analysis of the situation where an axis symmetrization of a minimizer produces a round ball. The use of Steiner symmetrization in our setting is natural since the ambient density can be seen as a rotationally invariant product measure. Let us recall some facts about this construction; see [R, Section 3.2] for details. Let Ωbe a compact set in Rn+1. Consider a hyperplane πin Rn+1containing the origin. The restriction of the ambient density to any straight line orthogonal to πprovides a smooth, symmetric, strictly log-convex density. We define the symmetrization of Ωwith respect to πas the set Ω∗whose intersection with any straight line Rorthogonal to πis the isoperimetric region in Rof the same weighted length as Ω∩R. By Corollary 4.12 this will be an interval centered at π∩R. It is clear that Ω∗ is symmetric with respect to π. The main property of this construction is that it preserves volume (by Fubini’s theorem) while decreasing perimeter. Lemma 5.1 ([R, Proposition 8]).For any hyperplane πthrough the origin in Rn+1, the Steiner symmetrization Ω∗of a compact set Ωsatisfies vol(Ω∗) = vol(Ω)and P(Ω∗)⩽P(Ω). Now, we will proceed to prove our main result in this section. THE ISOPERIMETRIC PROBLEM IN Rn+1WITH DENSITY 17 Theorem 5.2. In Rn+1with the density f (x) = exp(c|x|2), c >0, round balls about the origin uniquely minimize perimeter for given volume. Proof. First observe that bounded minimizers of any given volume exist for this density by Theorem 2.2. Let us prove that round balls centered at the origin are isoperimetric regions. Take a minimizer Ωof volume V>0. We apply Steiner symmetrization to Ωwith respect to any coordinate hyperplane in Rn+1so that we produce a minimizer Ω∗which is symmetric with respect to any of these hyperplanes and has connected boundary. In particular, Ω∗is centrally symmetric. Thus any hyperplane πthrough the origin divides Ω∗in two sets Ω∗ icontained in the corresponding open half-spaces πiand with the same volume. Note that the reflection with respect to πpreserves the perimeter relative to any πisince the density is radial. It follows that P(Ω1,π1) = P(Ω2,π2); otherwise, we would obtain by reflection a set with the same volume as Ω∗and strictly less perimeter. Therefore each Ω∗ i together with its reflection is a new minimizer of volume V. By the regularity properties in Theorem 2.8 and unique continuation for (real-analytic) generalized constant mean curvature surfaces, ∂Ω∗is symmetric across any hyperplane πthrough the origin. We conclude that Ω∗coincides with a ball centered at the origin. To prove uniqueness, by induction it suffices to show that if symmetrization of a minimizer Ωwith respect to a coordinate hyperplane πproduces a ball B, then Ωis a ball. We can suppose that π={xn+1=0}. Let D⊂πbe the projection of Ω. By Theorem 2.8 and Sard’s theorem, for almost all p∈Dstraight lines near porthogonal to πintersect Σ=∂Ω transversally at a fixed even number of points pi, where Σis the graph over Dp⊂Dof a smooth function hi(if we did not know that Ωis bounded, we would allow pito be ±∞). Denote by A⊆Dthe set of such points p. By the definition of Steiner symmetrization ∑ i odd Zhi+1 hi f(x)dx =2Zh∗ 0 f(x)dx on Dp, where h∗is the height function of ∂Bwith respect to π. As a consequence ∑ i odd (f(hi+1)∇hi+1−f(hi)∇hi) = 2f(h∗)∇h∗on Dp, so that we get ∑ j f(hj)|∇hj|⩾2f(h∗)|∇h∗|on Dp. On the other hand, by Corollary 4.12 we have ∑ j f(hj)⩾2f(h∗)on Dp, and equality holds if and only if the corresponding slice of Ωis an interval centered at π. Now we apply Lemma 5.3 below with αj=f(hj(p)),aj=|∇hj(p)|,α=f(h∗(p)) and a=|∇h∗(p)|. We get (5.1) ∑ j f(hj(p)) q1+|∇hj(p)|2⩾2f(h∗(p)) q1+|∇h∗(p)|2,p∈A, with equality if and only if |∇hj(p)|=|∇h∗(p)|for any j, and the slice of Ωpassing through pis an interval centered at π. 18 C. ROSALES, A. CA ˜ NETE, V. BAYLE, AND F. MORGAN Finally we use the coarea formula and inequality (5.1) to obtain P(Ω) = ZΣ f da ⩾ZA∑ j f(hj(p)) q1+|∇hj(p)|2da ⩾ZA 2f(h∗(p)) q1+|∇h∗(p)|2da =P(B), where in the last equality we have used that D−Adoes not contribute to the perimeter of B. As Ωis a minimizer we have equality above and hence in (5.1) too. It follows that for every p∈Athe slice of Ωpassing through pis a symmetric interval of the same length as the corresponding slice for B. Thus, up to a set of measure zero, Ωcoincides with a round ball about the origin.  Lemma 5.3. Suppose that we have finitely many nonnegative real numbers with ∑jαjaj⩾2αa and ∑jαj⩾2α. Then the following inequality holds ∑ j αjq1+a2 j⩾2αp1+a2, with equality if and only if aj=a for every j and ∑jαj=2α. Proof. The function g(x) = √1+x2is strictly convex with 0 <g′(x)<1 for any x>0. Let α0=∑jαj. We claim that ∑ j αj α0 g(aj)⩾g∑ j αj α0 aj⩾g2α α0 a⩾2α α0 g(a). The first inequality holds because gis convex. The second and third inequalities come from the fact that 0 <g′(x)<1 for x>0. If equality holds in the second inequality, then ∑jαjaj=2αa. If equality holds in the third inequality too, then 2α=α0=∑jαj. If equality holds in the first inequality as well, then aj=afor every j. We finish the paper with an eigenvalues comparison theorem obtained as a consequence of the isoperimetric inequality in Theorem 5.2. For a smooth bounded domain Ωin Rn+1, we consider the second order differential operator Lon C∞ 0(Ω)whose invariant measure has density f(x) = exp(c|x|2) (c⩾0), namely (5.2) (Lu)(x) = (∆u)(x)−2chx,(∇u)(x)i,u∈C∞ 0(Ω),x∈Ω, where ∆denotes the Euclidean Laplace operator on Ω. Corollary 5.4. Let Ωbe a smooth bounded domain in Rn+1. 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DEPARTAMENTO DE GEOMETR´ IA Y TOPOLOG´ IA, FACULTAD DE CIENCIAS, UNIVERSIDAD DE GRANADA, E- 18071 GRANADA, SPAIN E-mail address:[email protected] DEPARTAMENTO DE GEOMETR´ IA Y TOPOLOG´ IA, FACULTAD DE CIENCIAS, UNIVERSIDAD DE GRANADA, E- 18071 GRANADA, SPAIN E-mail address:[email protected] INSTITUTE FOURIER, BP 74, 38402 SAINT MARTIN D’HERES CEDEX, FRANCE E-mail address:[email protected] DEPARTMENT OF MATHEMATICS AND STATISTICS, WILLIAMS COLLEGE, WILLIAMSTOWN, MA 01267, U. S. A. E-mail address:[email protected]