Isoperimetric and stable sets for log-concave perturbations of Gaussian measures

Let $\Omega$ be an open half-space or slab in $\mathbb{R}^{n+1}$ endowed with a perturbation of the Gaussian measure of the form $f(p):=\exp(\omega(p)-c|p|^2)$, where $c>0$ and $\omega$ is a smooth concave function depending only on the signed distance from the linear hyperplane parallel to $\partial\Omega$. In this work we follow a variational approach to show that half-spaces perpendicular to $\partial\Omega$ uniquely minimize the weighted perimeter in $\Omega$ among sets enclosing the same weighted volume. The main ingredient of the proof is the characterization of half-spaces parallel or perpendicular to $\partial\Omega$ as the unique stable sets with small singular set and null weighted capacity. Our methods also apply for $\Omega=\mathbb{R}^{n+1}$, which produces in particular the classification of stable sets in Gauss space and a new proof of the Gaussian isoperimetric inequality. Finally, we use optimal transport to study the weighted minimizers when the perturbation term $\omega$ is concave and possibly non-smooth.


Introduction
The Gaussian isoperimetric inequality states that in Euclidean space R n+1 endowed with the weight γ c (p) := exp(−c|p| 2 ), c > 0, any half-space is an isoperimetric region, i.e., it has the least weighted perimeter among sets enclosing the same weighted volume. This result was independently proved in the mid-seventies by Sudakov and Tirel'son [72], and Borell [19], by means of an approximation argument which is sometimes attributed to Poincaré. As it is explained in [61, Thm. 18.2] and [67,Prop. 6], the Gaussian probability measure in R n+1 is the limit of orthogonal projections into R n+1 of high dimensional spheres S m−1 ( √ m) with uniform probability densities. Hence, the isoperimetric inequality in the Gauss space can be deduced as the limit of the spherical isoperimetric inequality. The complete characterization of equality cases does not follow from the previous approach and was subsequently studied by Carlen and Kerce [26]. Indeed, they discussed when equality holds in a more general functional inequality of Bobkov, and showed in particular that any isoperimetric set in the Gauss space coincides, up to a set of volume zero, with a half-space. Other proofs of the Gaussian isoperimetric inequality were found by Ehrhard [32], who adapted Steiner symmetrization to produce a Brunn-Minkowski inequality in the Gaussian context, by Bakry and Ledoux [6], see also [46], who gave a semigroup proof involving Ornstein-Uhlenbeck operators, and by Bobkov [16], as a consequence of sharp two point functional inequalities and the central limit theorem. Bobkov's inequalities were later extended to the sphere and used to deduce isoperimetric estimates for the unit cube by Barthe and Maurey [11]. An interesting result of Bobkov and Udre [18] shows that a symmetric probability measure µ on R such that all coordinate half-spaces in R n+1 are isoperimetric sets for the product measure µ n+1 is necessarily of Gaussian type. Recently, Cianchi, Fusco, Maggi and Pratelli [29] have obtained a sharp quantitative estimate for the Gaussian isoperimetric inequality.
The mathematical interest in the Gaussian isoperimetric question comes also from its wide range of applications in Probability Theory and Functional Analysis. The surveys of Ledoux [45], [47] show its relation to the concentration of measures phenomenon, the theory of spectral gaps for diffusion generators, and the logarithmic Sobolev inequalities. In Morgan's book [61, Ch. 18] two different variational proofs of the Gaussian isoperimetric inequality are included, together with a brief description of how it was used by Perelman in his proof of the Poincaré conjecture. Applications to Brownian motion appear in Borell [19].
Some of the methods employed to solve the isoperimetric problem in Gauss space imply isoperimetric comparisons for certain perturbations of the Gaussian measure. For example, Bakry and Ledoux [6], see also Bobkov [17], obtained a Lévy-Gromov type inequality by showing that, for probability measures on R n+1 having a log-concave density with respect to γ c , i.e., those of the form (1.1) exp(δ(p) − c|p| 2 ), for some concave function δ(p), the weighted perimeter of a set is greater than or equal to the perimeter of a half-space with the same volume for the measure γ c scaled to have unit volume. Indeed, this comparison is still valid for weighted Riemannian manifolds where the associated Bakry-Émery-Ricci curvature has a positive lower bound, see also Bayle [12,Ch. 3] and Morgan [61,Ch. 18]. A recent result of Milman [54] contains more general isoperimetric inequalities for weighted Riemannian manifolds depending on a curvature-dimension-diameter condition. Other modifications of Gaussian weights were considered by Fusco, Maggi and Pratelli [35], who classified the isoperimetric sets for the measures exp(−|x| 2 /2) dx dy in R n × R k .
1.1. The partitioning problem for Euclidean densities.
Inspired by the aforementioned works, in the present paper we study the isoperimetric question in Euclidean open sets for some log-concave perturbations of the Gaussian measure. In order to motivate and describe our results in detail we need to introduce some notation and definitions.
Let Ω be an open subset of R n+1 . By a density on Ω we mean a continuous positive function f = e ψ on Ω which is used to weight the Hausdorff measures in R n+1 . In particular, for any open set E ⊂ Ω with smooth boundary ∂E ∩Ω, the weighted volume and the weighted perimeter in Ω of E are given by where dv and da denote the Euclidean elements of volume and area, respectively. Observe that only the area of ∂E inside Ω is taken into account, so that ∂E ∩ ∂Ω has no contribution to P f (E, Ω). By following a relaxation procedure as in Ambrosio [2] and Miranda [58], we can define the weighted perimeter P f (E, Ω) of any Borel set E ⊆ Ω, see (2.17). The resulting perimeter functional satisfies some of the basic properties of Euclidean perimeter, see Section 2.5. The reader interested in the theory of finite perimeter sets and functions of bounded variation in Euclidean domains with density or in more general metric measure spaces is referred to the papers of Bellettini, Bouchitté and Fragalà [15], Baldi [8], Ambrosio [2], and Miranda [58].
Once we have suitable notions of volume and perimeter, we can investigate the isoperimetric or partitioning problem in Ω, which seeks sets E ⊂ Ω with the least possible weighted perimeter in Ω among those enclosing the same weighted volume. When such a set exists then it is called a weighted isoperimetric set or a weighted minimizer in Ω.
The study of isoperimetric problems in Euclidean open sets with density has increased in the last years. However, in spite of the last advances, the characterization of the solutions has been achieved only for some densities having a special form or a nice behaviour with respect to a certain subgroup of diffeomorphisms. In particular, radial and homogeneous densities are being a focus of attention, see the related works [62], [43], [64], [34], [28], [23], [25], [57], and the references therein.
For the Gaussian measure γ c the partitioning problem has been considered inside domains with simple geometric properties. On the one hand, Lee [48] employed approximation as in the first proofs of the Gaussian isoperimetric inequality to state that, in a half-space Ω ⊂ R n+1 , the intersections with Ω of half-spaces perpendicular to ∂Ω are weighted minimizers for any given volume. As we pointed out before, this approach does not provide uniqueness of weighted minimizers. Later on, Adams, Corwin, Davis, Lee and Visocchi [1] derived some properties of weighted isoperimetric sets inside planar sectors with vertex at the origin, and showed that the half-spaces perpendicular to ∂Ω are the unique weighted minimizers for a half-space Ω ⊂ R n+1 with linear boundary. Since the Gaussian density is invariant under linear isometries of R n+1 the proof of this fact follows by reflection across ∂Ω and the characterization of equality cases in the Gaussian isoperimetric inequality [26]. Unfortunately, this argument does not hold if 0 / ∈ ∂Ω, and so the uniqueness of weighted minimizers for an arbitrary half-space Ω remains open.
On the other hand, Lee [48] studied the partitioning problem when Ω is a horizontal strip symmetric with respect to the x-axis in the Gauss plane. In particular, some partial results led to the conjecture [48,Conj. 5]: the weighted minimizers in Ω must be bounded by curves different from line segments and meeting the two components of ∂Ω. However, we must point out that the perimeter decreasing rearrangement used in [48,Prop. 5.3] to discard isoperimetric curves meeting a vertical segment two or more times leaves invariant a region in Ω bounded by such segments. In particular, this kind of region cannot be excluded as a weighted minimizer in Ω. Note also that the previously mentioned reflection argument across ∂Ω does not apply in this case. To the best of our knowledge, the solution to the isoperimetric problem in Gaussian slabs of any dimension is still an open question.
The partitioning problem in a Euclidean half-space endowed with a modification of the Gaussian density was considered by Brock, Chiacchio and Mercaldo in [21]. These authors employed the Gaussian isoperimetric inequality and a transport map to establish that, in the half-space Ω := {(x 1 , . . . , x n+1 ) ∈ R n+1 ; x n+1 > 0} with density f (p) := x m n+1 exp(−|p| 2 /2), m 0, the intersections with Ω of half-spaces perpendicular to ∂Ω uniquely minimize the weighted perimeter for any weighted volume. Note that f can be expressed as exp(m log(x n+1 ) − |p| 2 /2), and so it is a log-concave perturbation of the Gaussian measure as in (1.1), with perturbation term depending only on the coordinate function x n+1 . We must remark that the isoperimetric question for the density x m n+1 exp(|p| 2 /2), m 0, has a completely different answer, since the same authors showed in [22] that half-balls centered at the origin are always weighted minimizers.

Results and proofs.
Motivated by the previous works, in this paper we investigate the partitioning problem in a more general setting that we now describe. Let S be a linear hyperplane of R n+1 . We denote by d S (p) := p, ξ the signed distance from S associated to a unit vector ξ normal to S. Note that d S (p) represents the ξ-coordinate of a point p ∈ R n+1 with respect to any orthonormal basis {e 1 , . . . , e n , ξ} of R n+1 . For an open set Ω ⊆ R n+1 , we consider the perturbation of the Gaussian density γ c given by (1.2) f (p) := exp ω(d S (p)) − c|p| 2 , where ω is a continuous function on J Ω := d S (Ω), and c is a positive constant. Equation (1.2) defines a very large family of Euclidean densities, which includes the Gaussian measure γ c and weights of the form exp ω(x i ) − c|p| 2 , where x i is a coordinate function in R n+1 . In particular, for the linear hyperplane S of equation x n+1 = 0, the half-space x n+1 > 0, the function ω(s) := m log(s), m 0, and the constant c := 1/2, we get the situation studied in the work of Brock, Chiacchio and Mercaldo [21].
In Section 2.2 we obtain some basic properties for a density f as in (1.2). A straightforward computation shows that the perturbation term ω(d S (p)) is concave as a function of p ∈ Ω if and only if ω(s) is a concave function of s ∈ J Ω . Thus, the concavity of ω implies that f is a log-concave perturbation as in (1.1) of the Gaussian measure γ c . In particular, the Lévy-Gromov inequality of Bakry and Ledoux [6] provides an isoperimetric comparison, possibly non-optimal, for the density f . Moreover, as we see in Lemma 2.2, the concavity of ω entails finite weighted volume for Ω and finite weighted perimeter for any half-space intersected with Ω. Hence, from a well-known result that we recall in Theorem 3.1, we deduce existence of weighted isoperimetric sets in Ω for any weighted volume. By these reasons (and some more that we will explain later), we are naturally lead to assume concavity of ω for densities as in (1.2). Indeed the main result of the paper, that we prove in Theorems 5.5 and 5.12, establishes the following: Let Ω ⊂ R n+1 be an open half-space or slab with boundary parallel to S. Consider a density in Ω as in (1.2) with ω a concave function. Then, the intersections with Ω of half-spaces perpendicular to ∂Ω are weighted isoperimetric sets. Moreover, if ω is smooth on J Ω , then these are the unique weighted minimizers in Ω.
For the Gaussian measure γ c , our theorem gives uniqueness of weighted minimizers in arbitrary half-spaces, thus improving the aforementioned results in [48] and [1]. It also contains the complete solution to the partitioning problem inside Gaussian slabs, for which only partial results in the planar case [48] were known. Moreover, for the half-space Ω := {x n+1 > 0} with density f (p) := exp(m log(x n+1 ) − |p| 2 /2), m 0, we deduce the minimality of half-spaces perpendicular to ∂Ω already proved in [21]. In fact, our theorem extends this isoperimetric comparison to any half-space or slab Ω, and any concave (possibly non-smooth) function ω. Unfortunately, we do not provide uniqueness of weighted minimizers when ω is non-smooth.
Unlike previous approaches, our methods do not heavily rely on the Gaussian isoperimetric inequality (indeed, it follows from our techniques, as we explain at the end of this Introduction). The proof of the main theorem is divided into two parts. We first assume that ω is smooth on J Ω . This allows us to invoke some known results from Geometric Measure Theory to obtain in Section 3 existence of weighted isoperimetric regions with nice interior boundaries. In Section 4 we employ variational tools to study stable sets, i.e., second order minima of the weighted perimeter for deformations preserving the weighted volume. We are able to classify these sets under some regularity conditions, by showing that they are all half-spaces parallel or perpendicular to ∂Ω. Finally, in Section 5 we utilize integration of second order differential inequalities to show that perpendicular half-spaces are isoperimetrically better than the parallel ones. Finally, the case where ω is non-smooth is deduced from the smooth case by means of an approximation argument, which makes use of a recent theorem of Azagra [4]. Now, we shall explain in more detail the different ingredients of the proof and other interesting results for general densities on Euclidean open sets.
Let Ω be an open subset of R n+1 with density f = e ψ . In general, the existence of weighted minimizers in Ω is a non-trivial question, see for instance [68], [64] and [24]. However, if the weighted volume of Ω is finite, then we can ensure that there are weighted isoperimetric sets in Ω of any weighted volume (Theorem 3.1). Moreover, by the regularity results in [37], [41] and [59], if Ω has smooth boundary and f ∈ C ∞ (Ω) then, for any weighted minimizer E, the interior boundary ∂E ∩ Ω is a disjoint union Σ ∪ Σ 0 , where Σ is a smooth embedded hypersurface, and Σ 0 is a closed set of singularities with Hausdorff dimension less than or equal to n − 7 (Theorem 3.2). As we pointed out before, for a density f as in (1.2) with ω smooth and concave, the weighted volume of any smooth open set Ω is finite (Lemma 2.2), and so weighted minimizers with sufficiently regular interior boundaries exist in Ω.
On the other hand, the Gaussian measure shows that the interior boundary of a minimizer need not be bounded, see some related results in [68], [64] and [30]. This lack of compactness leads us to study the more general condition of null weighted capacity. The capacity of a compact set in a Riemannian manifold was introduced by Choquet and it physically represents the total electrical charge that the set can hold while it maintains a certain potential energy. The reader interested in the theory of capacities and its wide range of applications in isoperimetric problems, harmonic analysis and Brownian motion is referred to the surveys of Grigor'yan [38] and Troyanov [74]. In a very recent work, Grigor'yan and Masamune [39] have extended the notion of capacity and obtained parabolicity results for weighted Riemannian manifolds.
In our setting, a hypersurface Σ has null weighted capacity if the quantity Cap f (K) defined in (2.12) vanishes for any compact set K ⊆ Σ. It is clear that a compact hypersurface Σ satisfies Cap f (Σ) = 0, but the reverse statement is not true. This is illustrated in Example 2.8, where it is shown that Cap f (Σ) = 0 for any complete hypersurface Σ of finite weighted area. Hypersurfaces of null weighted capacity play a crucial role in our classification of stable sets in Section 4 due to an integral stability inequality valid for functions that need not vanish in Σ, see Proposition 4.2. As we aim at studying the isoperimetric problem by means of the stability condition, it is then important to establish a relation between weighted minimizers and hypersurfaces of null weighted capacity. This is done in Theorem 3.6, where we prove that, for an arbitrary smooth bounded density f on a smooth open set Ω ⊂ R n+1 , the regular part Σ of the interior boundary ∂E ∩ Ω of a weighted minimizer E is a hypersurface of null weighted capacity. The main difficulty in proving this theorem is the possible presence in high dimensions of a noncompact singular set Σ 0 . However, we have been able to adapt the arguments given for constant densities and compact interior boundaries by Sternberg and Zumbrun [71]. This requires uniform estimates for the perimeter of a weighted minimizer inside Euclidean balls (Proposition 3.4) and some technical effort to construct a sequence of functions with compact support and weighted energies tending to zero. More details and references are found in Remark 3.7.
We now turn to describe more precisely our stability results in Euclidean domains with density. Let Ω ⊆ R n+1 be a smooth open set with a density f = e ψ smooth on Ω. Motivated by the regularity properties of a weighted minimizer in Theorem 3.2 and the null weighted capacity of interior boundaries in Theorem 3.6, we consider open sets E ⊂ Ω with finite weighted perimeter and such that ∂E ∩ Ω = Σ ∪ Σ 0 , where Σ is a smooth embedded hypersurface with Cap f (Σ) = 0, and Σ 0 is a closed singular set with vanishing weighted area. If Σ has non-empty boundary then we assume ∂Σ = Σ ∩ ∂Ω. We say that E is weighted stable if it is a critical point with non-negative second derivative for the weighted perimeter functional under compactly supported variations moving ∂Ω along ∂Ω and preserving weighted volume. Thus, if E is weighted stable, then Σ is an f -stable free boundary hypersurface as defined by Castro and the author in [27] (for constant densities this coincides with the classical notion of stable constant mean curvature hypersurfaces with free boundary). As a consequence, we can apply the formulas in [27] for the first and second derivatives of weighted area and volume in order to deduce some variational properties of a weighted stable set.
On the one hand, we show that the hypersurface Σ has constant f -mean curvature and meets ∂Ω orthogonally along ∂Σ. The f -mean curvature of Σ is the function H f in (2.8) previously introduced by Gromov [40] when computing the first derivative of the weighted area. On the other hand, the associated f -index form of Σ defined in (4.1) is nonnegative for smooth functions having compact support on Σ and mean zero with respect to the weighted element of area. Note that the f -index form involves the extrinsic geometry of Σ, the second fundamental form of ∂Ω and the Bakry-Émery-Ricci curvature Ric f in (2.3). The 2-tensor Ric f was first introduced by Lichnerowicz [50], [51], and later generalized by Bakry andÉmery [5] in the framework of diffusion generators. In particular, it is easy to observe that the stability inequality becomes more restrictive provided the ambient set Ω is convex and Ric f 0. By assuming both hypotheses we deduce in Proposition 4.2 more general stability inequalities for mean zero functions satisfying certain integrability conditions. This allows us to deform a given stable set by means of infinitesimal variations that could move the singular set Σ 0 . The proof of Proposition 4.2 relies on the null weighted capacity property, which permits to extend the approximation arguments employed by Ritoré and the author for constant densities in convex solid cones, see [66] Coming back to the case where Ω is a half-space or slab in R n+1 endowed with a density f as in (1.2) with ω smooth on J Ω , we analyze in Lemma 4.7 when a half-space E intersected with Ω is a weighted stable set. We get that, if ∂Ω is parallel to the hyperplane S, then E can be weighted stable if and only if ∂E is either parallel or perpendicular to ∂Ω. Furthermore, the spectral gap inequality in Gauss space yields that half-spaces parallel to ∂Ω are weighted stable if and only if ω is convex. The weighted stability of half-spaces for Euclidean product measures was also studied by Barthe, Bianchini and Colesanti [10], and Doan [31], see Remark 4.8 for a more detailed description of their results.
From the previous discussion we conclude that half-spaces parallel to ∂Ω can be discarded as weighted minimizers if ω is strictly concave. Moreover, the concavity of ω also implies Ric f 2c > 0, which allows to apply the generalized stability inequalities in Proposition 4.2 and the connectivity of stable boundaries in Corollary 4.5. Motivated by all this, we are led to study in more detail the stability condition when ω is a concave function. As a main consequence of our analysis, in Theorem 4.11 we establish the following classification result: Let Ω ⊂ R n+1 be an open half-space or slab with boundary parallel to a given hyperplane S. Consider a density in Ω as in (1.2) with ω smooth and concave on J Ω . Then, a weighted stable set of finite weighted perimeter and such that the regular part of the interior boundary has null weighted capacity is the intersection with Ω of a half-space parallel or perpendicular to ∂Ω.
The proof of this theorem employs the generalized stability inequalities in Proposition 4.2 with suitable deformations of E. More precisely, the fact that ∂Ω is totally geodesic leads us to consider equidistant sets translated along a fixed direction η of ∂Ω to keep the weighted volume constant. From an analytical point of view we show in Lemma 4.9 that these deformations have associated test functions of the form u := α + h, where α is a real constant and h is the normal component of η. After evaluating the f -index form over u, it turns out that this kind of variation decreases the weighted perimeter unless the interior boundary of E is a hyperplane intersected with Ω. From here, it is not difficult to conclude that this hyperplane must be parallel or perpendicular to ∂Ω, and the proof is complete. We must remark that similar test functions already appeared in previous stability results, see for example Sternberg and Zumbrun [70], and Barbosa, do Carmo and Eschenburg [9]. More recently, McGonagle and Ross [53] have used the same functions to describe smooth, complete, orientable hypersurfaces of constant f -mean curvature and finite index in R n+1 with Gaussian density γ c .
Our stability theorem contains the classification of weighted stable sets in Gaussian halfspaces and slabs allowing the presence of singularities, see Corollary 4.13. Indeed, from the techniques of the proof we can also deduce characterization results for smooth, complete, orientable f -stable hypersurfaces with finite weighted area and free boundary in a half-space or slab, see Corollary 4.16. We must emphasize that the finite area hypothesis is very restrictive in R n+1 with constant density since it enforces a complete constant mean curvature hypersurface to be compact. However, for general densities this is no longer true, as it is shown for example by a Gaussian hyperplane. More recent results for free boundary f -stable hypersurfaces are found in [25] and [27]. On the other hand, complete f -minimal hypersurfaces with non-negative second derivative of the weighted area for any variation have been intensively studied, see the Introduction of [27] for a very complete list of references.
Once we have ensured existence of isoperimetric minimizers and described the stable candidates, we only have to compare their weighted perimeter for fixed weighted volume. This is done in Proposition 5.1, where we finally show that half-spaces perpendicular to ∂Ω are better than half-spaces parallel to ∂Ω. The key ingredient of the proof is a second order differential inequality, which is satisfied by the relative profile associated to the family of half-spaces parallel to ∂Ω. Since this inequality becomes an equality for half-spaces perpendicular to ∂Ω the desired comparison follows from a classical integration argument. On the other hand, in Remark 5.6 we indicate that any two half-spaces perpendicular to ∂Ω of fixed weighted volume have the same weighted perimeter. Thus, our uniqueness result for weighted minimizers is the best that can be expected in this setting. We must also mention that the integration of differential inequalities was already employed in several isoperimetric comparisons, see for instance Morgan and Johnson [63], Bayle [12,Ch. 3], [13], and Bayle and the author [14].
The previous arguments complete the proof of Theorem 5.5, which assumes smoothness and concavity of the function ω on J Ω . Indeed, the smoothness assumption has been a key ingredient for the regularity of weighted minimizers in Theorem 3.2, and for the variational approach followed to characterize weighted stable sets in Theorem 4.11. Now, we turn to the proof of Theorem 5.12, where we extend the isoperimetric property of perpendicular half-spaces to any concave function ω. As in previous works (see for example Milman [55] and Cabré, Ros-Oton and Serra [23]) this will be accomplished from the smooth case by means of a approximation argument. We carry out this one in two steps. In the first step, we consider interior parallel half-spaces or slabs to get the claim when ω is smooth only in J Ω . In the second one, we invoke a result of Azagra [4] to construct a suitable sequence of smooth concave functions in J Ω converging uniformly to ω in J Ω . We think that this second step may be also obtained from standard convolution techniques and approximation of finite perimeter sets by smooth bounded sets. However, by making use of Azagra's result we have provided a more direct and less technical proof.

Some applications.
The isoperimetric theorem established in this paper has some interesting consequences involving Poincaré type inequalities and analytical properties of the weighted isoperimetric profile. We explain them now in more detail.
Let Ω be an open half-space or slab of R n+1 endowed with a log-concave perturbation f of the Gaussian measure γ c as in (1.2). If ω is smooth and concave on J Ω , then the minimality property of a half-space E perpendicular to ∂Ω implies that E is a weighted stable set. In particular, E satisfies a generalized stability inequality which is equivalent, see Corollary 5.10, to the estimate Here λ f (Σ) denotes the spectral gap of Σ := ∂E ∩ Ω defined in Remark 5.11. It is well known (see for instance Ledoux [47]) that λ f (Σ) = 2c for a complete hyperplane Σ in R n+1 with Gaussian density f = γ c . For a smooth convex domain D of a Gaussian hyperplane, the inequality λ f (D) 2c was obtained by Bakry and Qian when D is bounded [7], and by Brandolini, Chiacchio, Henrot and Trombetti when D is unbounded [20]. A more general and unified approach for spectral gap inequalities in compact weighted Riemannian manifolds can be found in [7] and the recent paper of Kolesnikov and Milman [44].
On the other hand, from our isoperimetric theorem and the arguments in the proof of Proposition 5.1, we show in Corollary 5.14 that the weighted isoperimetric profile I Ω,f defined in (3.1) is a smooth function satisfying the differential equation I ′′ Ω,f = −2c I −1 Ω,f . This equality was well-known to hold in the Gauss space, see for instance Bakry and Ledoux [6], and Bayle [12,Ch. 3]. From this identity, which is valid even if ω is non-smooth on J Ω , we can deduce that I Ω,f is a concave function. Indeed, the concavity of the isoperimetric profile in arbitrary weighted Riemannian manifolds of non-negative Bakry-Émery-Ricci curvature was obtained by Milman, see The methods employed in this paper give also characterization results for weighted stable and isoperimetric sets in R n+1 with a density f as in (1.2). More precisely, in Theorem 4.14 we establish that, if ω is smooth and concave, then any weighted stable set of finite weighted perimeter, null weighted capacity and small singular set is a half-space. Moreover, if ω is not an affine function, then the half-space must be parallel or perpendicular to the hyperplane S. From this result, we prove in Theorems 5.16 and 5.19 that, if ω is concave but not affine, then half-spaces perpendicular to S are weighted minimizers (with uniqueness if we further assume ω smooth). On the other hand, when ω is affine, it follows that any half-space in R n+1 is a weighted isoperimetric set.
In the particular case of the Gaussian density γ c on R n+1 our stability theorem implies that any weighted stable set E in the conditions defined above must be a half-space. When the set E has smooth boundary this is a particular case of a recent result of McGonagle and Ross [53] showing that the hyperplanes are the unique smooth, complete and orientable f -stable hypersurface in the Gauss space. Unfortunately, the solution to the isoperimetric problem for γ c cannot be deduced from their theorem, since weighted minimizers need not be smooth in high dimensions. However, our stability result allows the presence of a singular set in such a way that it can be applied to any weighted isoperimetric set. As a consequence, our techniques provide a new proof of the Gaussian isoperimetric inequality which includes also the characterization of equality cases.

Organization.
The paper is organized as follows. In Section 2 we introduce some preliminary material about Euclidean densities, hypersurfaces of null weighted capacity and weighted perimeter. In Section 3 we recall existence and regularity results for weighted minimizers, and show that the regular part of the boundary of a weighted minimizer is a hypersurface of null weighted capacity. Section 4 is devoted to variational properties and characterization results for weighted stable sets. Finally, Section 5 contains our isoperimetric comparisons and their consequences.
Acknowledgements. The author thanks F. Morgan for some comments and suggestions.

Preliminaries
In this section we introduce the notation and list some basic results that will be used throughout the paper.

Euclidean densities.
We consider Euclidean space R n+1 endowed with its standard Riemannian flat metric · , · . Given an open set Ω ⊆ R n+1 , by a density on Ω we mean a continuous positive function f = e ψ defined on Ω. This function is used to weight the Euclidean Hausdorff measures. In particular, for a Borel set E ⊆ Ω, the weighted volume and weighted area of E are respectively defined by where dv and da are the Lebesgue measure and the n-dimensional Hausdorff measure in R n+1 . We shall employ the notation dl f := f dl for the (n − 1)-dimensional weighted Hausdorff measure. We will say that f has finite weighted volume if V f (Ω) < +∞.
For a smooth density f = e ψ on Ω, the associated Bakry-Émery-Ricci tensor is the 2-tensor where ∇ 2 is the Euclidean Hessian operator. Clearly, if the density is constant, then Ric f = 0. The Bakry-Émery-Ricci curvature at a point p ∈ Ω in the direction of a unit vector w ∈ R n+1 is given by (Ric f ) p (w, w). If this curvature is always greater than or equal to a constant 2c, then we write Ric f 2c.

Perturbations of the Gaussian density.
For any c > 0, the Gaussian density of curvature 2c in R n+1 is defined by γ c (p) := e −c|p| 2 for any p ∈ R n+1 . Observe that γ c is bounded and radial, i.e., it is constant over round spheres centered at the origin. Hence, γ c is invariant under Euclidean linear isometries, which implies by (2.1), (2.2), and the change of variables theorem, that the weighted volume and area of E and φ(E) coincide for any Borel set E and any linear isometry φ. It is also known that γ c has finite weighted volume, and that any hyperplane of R n+1 has finite weighted area. Moreover, by using (2.3) we get that γ c has constant Bakry-Émery-Ricci curvature 2c.
In this paper we are interested in certain Euclidean measures having a log-concave density with respect to γ c . Let S be a linear hyperplane of R n+1 . For a fixed unit vector ξ normal to S, the signed distance from S is defined by d S (p) := p, ξ . Note that d S (p) is the ξ-coordinate of p with respect to any orthonormal basis {e 1 , . . . , e n , ξ} of R n+1 . Given an open set Ω ⊆ R n+1 we denote J Ω := d S (Ω). For any continuous function ω : J Ω → R and any c > 0, we consider the following density on Ω Clearly ω(d S (p)) is constant over hyperplanes parallel to S. It is also clear that f coincides with the restriction of γ c to Ω when ω = 0. If we suppose that ω is smooth on J Ω , and denote by ω ′ and ω ′′ the first and second derivatives of ω(s) with respect to s, then a straightforward computation from (2.4) and (2.3) shows that for any point p ∈ Ω and any vector w ∈ R n+1 .
From the second equality in (2.5), it follows that the perturbation term ω(d S (p)) in (2.4) is a concave function on Ω if and only if ω(s) is a concave function of s ∈ J Ω . In such a case, the density f is a particular case of (1.1), and this allows for instance to apply the Lévy-Gromov type inequality mentioned in the Introduction. In this sense, the concavity of ω seems a natural hypothesis when we consider densities as in (2.4). Furthermore, the concavity property implies other consequences that will be important in proving our main results. Some of them are stated in the next lemma.
Let Ω be an open set of R n+1 endowed with a density f = e ψ as in (2.4). If ω is concave, then f is bounded and Ω has finite weighted volume. Moreover, the intersection with Ω of any hyperplane in R n+1 has finite weighted area.
Proof. As ω is concave then it is differentiable almost everywhere in J Ω , and bounded from above by the tangent line at any point where ω ′ exists. Thus we have ψ(p) −c|p| 2 + α p, ξ + β in Ω for some constants α, β ∈ R. Hence f is bounded from above by a Euclidean density of Gaussian type and the proof finishes. Remark 2.3. In R n+1 with a smooth log-concave perturbation of γ c as in (2.4), we can apply the Heintze-Karcher inequality obtained by Morgan in [60, Th. 2] for densities with Ric f 2c > 0. Hence, we deduce that f has finite weighted volume. In fact, since any hyperplane Σ is totally geodesic in R n+1 , then the same result ensures that A f (Σ) < +∞.
Example 2.4. The previous lemma fails if ω is not assumed to be concave. For example, if S denotes the vertical axis in R 2 and ω(s) := 2cs 2 , then we get the planar density f (x, y) = e c(x 2 −y 2 ) , for which the weighted area of R 2 is not finite. Note also that any horizontal line have infinite weighted length.

Weighted divergence theorems.
Let Ω be an open set of R n+1 endowed with a density f = e ψ smooth on Ω. For any smooth vector field X on Ω, the f -divergence of X is the function where div and ∇ denote the Euclidean divergence of vector fields and the gradient of smooth functions, respectively. The f -divergence div f is the adjoint operator of −∇ for the L 2 norm associated to the weighted volume dv f . By using the Gauss-Green theorem together with equality div f X dv f = div(f X) dv, we obtain for any open set E ⊂ Ω such that ∂E ∩ Ω is a smooth hypersurface, and any smooth vector field X with compact support in Ω. In the previous formula N denotes the inner unit normal along ∂E ∩ Ω. As happens for constant densities, the previous formula can be extended to more general sets. In this paper, we will only need the following Gauss-Green theorem for open sets with almost smooth interior boundaries. Consider an open set E ⊂ Ω such that ∂E ∩ Ω is a disjoint union Σ∪Σ 0 , where Σ is a smooth hypersurface (possibly with boundary), and Σ 0 is a closed set with A f (Σ 0 ) = 0. Then, for any smooth vector field X with compact support in Ω, we have where N is the inner unit normal on Σ.
Proof. An approximation argument as in the proof of [52,Thm. 9.6] shows that the lemma holds for the constant density f = 1. For general densities the formula follows from the constant case by using that div f X dv f = div(f X) dv.
Let Σ be a smooth oriented hypersurface in Ω, possibly with boundary ∂Σ. For any smooth vector field X along Σ, we define the f-divergence relative to Σ of X by where div Σ is the Euclidean divergence relative to Σ. If N is a unit normal vector along Σ, then the f-mean curvature of Σ with respect to N is the function where H := (−1/n) div Σ N is the Euclidean mean curvature of Σ. By using the Riemannian divergence theorem it is not difficult to get for any smooth vector field X with compact support on Σ, see [25, Lem. 2.2] for details.
Here we denote by ν the conormal vector, i.e., the inner unit normal to ∂Σ in Σ. From now on, we understand that the integrals over ∂Σ are all equal to zero provided ∂Σ = ∅.
Given a function u ∈ C ∞ (Σ), the f-Laplacian relative to Σ of u is defined by where ∇ Σ is the gradient relative to Σ. For this operator we have the following integration by parts formula, which is an immediate consequence of (2.9) where u 1 , u 2 ∈ C ∞ 0 (Σ) and ∂u 2 /∂ν is the directional derivative of u 2 with respect to ν. As usual, we denote by C ∞ 0 (Σ) the set of smooth functions with compact support on Σ. Note that, when ∂Σ = ∅, a function in C ∞ 0 (Σ) need not vanish on ∂Σ.

Hypersurfaces of null weighted capacity.
Let Ω be an open set of R n+1 endowed with a density f = e ψ smooth on Ω. Given a smooth oriented hypersurface Σ ⊂ Ω, possibly with boundary, and a number q 1, we denote by L q (Σ, da f ) and L q (∂Σ, dl f ) the corresponding spaces of integrable functions with respect to the weighted measures da f and dl f . The weighted Sobolev space . We will use the notation H 1 0 (Σ, da f ) for the closure of C ∞ 0 (Σ) with respect to this norm. Following Grigor'yan and Masamune [39], we define the weighted capacity of a compact subset K ⊆ Σ by means of equality A standard approximation argument shows that we can replace H 1 0 (Σ, da f ) with C ∞ 0 (Σ) in the previous definition. Note also that the monotonicity property Cap f (K 1 ) Cap f (K 2 ) holds for two compact sets K 1 ⊆ K 2 . Thus, it is natural to define for any open set D ⊆ Σ. We say that Σ has null weighted capacity if Cap f (Σ) = 0, i.e., Cap f (K) = 0 for any compact set K ⊆ Σ. Our main interest in such a hypersurface comes from the next result, which follows from (2.12) by taking a countable exhaustion of Σ by precompact open subsets.
From the previous lemma we can generalize to hypersurfaces of null weighted capacity some properties and results valid for compact hypersurfaces. For example, we can extend the divergence theorem in (2.9) and the integration by parts formula in (2.11) to vector fields and functions satisfying certain integrability conditions. Lemma 2.7 (Generalized divergence theorem and integration by parts formula). Let Ω be an open set of R n+1 endowed with a density f = e ψ smooth on Ω. Consider a smooth oriented hypersurface Σ, possibly with boundary, and such that Cap f (Σ) = 0. Then, for any smooth vector field X on Σ satisfying As a consequence, for any two functions u 1 , u 2 ∈ C ∞ (Σ) such that we have the integration by parts formula Proof. We follow the proof given by Ritoré and the author for constant densities and vector fields tangent to Σ, see [66,Lem. 4.4]. Consider a sequence {ϕ k } k∈N as in Lemma 2.6. By applying (2.9) to the vector field ϕ k X and using that div By letting k → ∞ we obtain the first part of the statement from the dominated convergence theorem and the Cauchy-Schwarz inequality. The second part is a direct consequence of the first one by taking X : For a complete hypersurface, the null capacity property can be deduced from a suitable behaviour of the volume growth associated to metric balls centered at a fixed point, see Grigor'yan [38]. A very particular case of this situation is shown in the next example.
which tends to zero when k → ∞. Finally, given a compact set K ⊆ Σ, there is k 0 ∈ N such that ϕ k = 1 in K for any k k 0 . Hence Cap f (K) = 0 by (2.12). We remark that this conclusion also follows if Σ is a complete hypersurface with null quadratic weighted area growth at infinity.

Sets of finite weighted perimeter.
Let Ω be an open set of R n+1 endowed with a density f = e ψ smooth on Ω. The notion of f -divergence in (2.6) allows us to introduce the weighted perimeter of sets by following the classical approach by Caccioppoli and De Giorgi. More precisely, for any Borel set E ⊆ Ω, the weighted perimeter of E in Ω is given by where X ranges over smooth vector fields with compact support in Ω. Clearly P f (E, Ω) does not change by sets of volume zero. Thus, we can always assume that E satisfies for any open set U ⊆ Ω. This is an immediate consequence of (2.13) and the Gauss-Green formula in (2.7). Indeed, the previous equality also holds for sets with almost smooth interior boundary.
This shows that ∂E ∩ ∂Ω does not contribute to P f (E, Ω).
We say that E has finite weighted perimeter in Ω if P f (E, Ω) < +∞. We say that E is a of Ω. Moreover, by using the Riesz representation theorem as in Thm. 1 of [33, Sect. 5.1], we can find a generalized inner unit normal N f : Ω → R n+1 such that the Gauss-Green formula holds for any smooth vector field X with compact support in Ω. Of course, for almost smooth open sets as in Example 2.9, the inner unit normal N along the hypersurface Σ coincides with N f . Remark 2.10 (Unweighted Caccioppoli sets coincide with weighted Caccioppoli sets). If E is a finite perimeter set in Ω (in Euclidean sense) and f is bounded in Ω, then E has finite weighted perimeter in Ω. In fact, for any smooth vector field X with compact support in Ω, we have where in the second equality we have applied (2.14) for a constant density. If we denote α := sup{f (p) ; p ∈ Ω} then (2.13) implies that P f (E, Ω) α P (E, Ω), where P (E, Ω) is the Euclidean perimeter of E in Ω. Suppose now that E has finite weighted perimeter in Ω. As before, we get In general, most of the basic properties of finite perimeter sets in R n+1 can be extended to sets of finite weighted perimeter. For example, the next lemma is similar to the localization result stated in [3,Prop. 3.56] and [33, p. 196] for constant densities, with the difference that our estimates involve the weighted perimeter off of a Euclidean ball.
Lemma 2.11. Let Ω be an open set of R n+1 endowed with a density f = e ψ smooth on Ω. Given a Borel set E ⊆ Ω of finite weighted perimeter, and a point p 0 ∈ R n+1 , we have for every r > 0. Here B(p 0 , r) denotes the Euclidean open ball of radius r centered at p 0 .
Proof. We fix a smooth vector field X with compact support in Ω. Given a smooth function ϕ : R n+1 → R, an application of (2.14) to ϕX yields By approximation, the previous formula also holds when ϕ is piecewise smooth. Now, we fix p 0 ∈ R n+1 and r > 0. For any ε ∈ (0, r), we consider the function ϕ ε (p) By applying (2.15) with ϕ = ϕ ε we obtain For any r > 0, we define where in the second equality we have used the coarea formula. Then, for every r > 0, the second term at the left hand side of (2.16) tends to g ′ (r) = E∩∂B(p0,r) p−p0 |p−p0| , X da f when ε → 0 + . On the other hand, note that 0 ϕ ε 1 and {ϕ ε } ε>0 pointwise converges, as ε → 0 + , to the characteristic function of R n+1 − B(p 0 , r). From (2.16) and the dominated convergence theorem, we deduce for every r > 0. Finally, if we suppose |X| 1, then we conclude Suppose that E ⊆ Ω is a Borel set of finite weighted perimeter in Ω such that the generalized inner unit normal N f of E exists and it is continuous on ∂E ∩ Ω. Then ∂E ∩ Ω is a C 1 hypersurface.
Remark 2.13. Let Ω be an open set of R n+1 endowed with a density f = e ψ smooth on Ω. For any function u ∈ L 1 loc (Ω, dv f ), we can define the total weighted variation of u by where X ranges over smooth vector fields with compact support in Ω. It is then clear from (2.13) that P f (E, Ω) = Dχ E f (Ω) for any Borel set E ⊆ Ω. Here χ E is the characteristic function of E. Thus, a set E has finite weighted perimeter in Ω if and only if χ E ∈ BV f (Ω), where BV f (Ω) denotes the space of functions with bounded weighted variation, i.e., those functions u ∈ L 1 loc (Ω, dv f ) such that Du f (Ω) < +∞. These spaces have been studied by means of different approaches; the interested reader is referred to Bellettini, Bouchitté and Fragalà [15], see also Baldi [8].
Observe that we have defined in (2.13) the weighted perimeter functional for densities f = e ψ which are smooth on Ω. In the case where f is merely continuous on Ω we can use a relaxation procedure as in the papers of Ambrosio [2] and Miranda [58] to define the weighted perimeter of a Borel set E ⊆ Ω by where {E k } k∈N ranges over sequences of open sets in Ω such that ∂E k ∩ Ω is smooth.
In the next result we gather the basic properties of the above perimeter functional that we need in this paper.
Lemma 2.14. Let Ω be an open set of R n+1 endowed with a continuous density f = e ψ . Then, the following facts hold where X ranges over smooth vector fields with compact support in Ω. Note also that the functional defined at the right hand side of the previous equality is lower semicontinuous with respect to the convergence of sets in Thus we obtain A f (∂E ∩ Ω) P f (E, Ω), which proves (ii). Finally, statement (iii) is a direct consequence of (ii) and (2.17). In this section we first recall some known results about the existence and regularity of solutions for the weighted isoperimetric problem inside Euclidean open sets with density. Then, we will use technical arguments to show that, for bounded densities, the regular part of the interior boundary of such solutions has null weighted capacity.
Let Ω be an open subset of R n+1 endowed with a continuous density f = e ψ . The weighted isoperimetric profile of Ω is the function I Ω,f : (0, V f (Ω)) → R + 0 given by (3.1) I where V f (E) is the weighted volume in (2.1) and P f (E, Ω) the weighted perimeter in (2.17).
Obviously a weighted minimizer in Ω has finite weighted perimeter in Ω.
The existence of weighted minimizers is a non-trivial question. In the works of Bayle, Cañete, Morgan and the author [68], and Morgan and Pratelli [64], we can find some elementary examples showing that minimizers need not exist if Ω is unbounded. These papers also provide sufficient conditions for existence involving the growth of the density at infinity, see [68,Sect. 2] and [64,Sects. 3 and 7]. Here we will only use the following result, whose proof relies on the lower semicontinuity of the weighted perimeter and standard compactness arguments, see Morgan   As pointed out by Morgan [59,Sect. 3.10] the regularity of weighted isoperimetric regions inside a smooth open set Ω with smooth density f is the same as for the classical setting of constant density f = 1. The latter was studied by Gonzalez, Massari and Tamanini, who obtained interior regularity [37], and by Grüter, who proved regularity at the free boundary [41]. We gather their results in the next theorem, see also [54,Sect. 2]. . If E is a weighted isoperimetric region in Ω, then the interior boundary ∂E ∩ Ω is a disjoint union Σ ∪ Σ 0 , where Σ is a smooth embedded hypersurface with (possibly empty) boundary ∂Σ = Σ ∩ ∂Ω, and Σ 0 is a closed set of singularities with Hausdorff dimension less than or equal to n − 7.
Remark 3.3. From the previous theorem, and taking into account that the weighted perimeter in (2.13) does not change by sets of volume zero, we can always assume that a weighted isoperimetric region E in Ω is an open set. Note that E has almost smooth interior boundary as in Lemma 2.5 and Example 2.9. Therefore, we get P f (E, Ω) = A f (Σ) and, more generally, for any open set U ⊆ Ω. It is also worth mentioning that the condition ∂Σ = Σ ∩ ∂Ω prevents the existence of interior points of Σ contained in ∂Ω.
Weighted minimizers and their interior boundaries need not be bounded. This is illustrated by the Gaussian density γ c (p) := e −c|p| 2 , c > 0, for which any weighted isoperimetric region is, up to a set of volume zero, a Euclidean half-space [72], [19], [26]. Some criteria ensuring boundedness have been found by Bayle, Cañete, Morgan and the author [68, Sect. 2], Morgan and Pratelli [64,Sect. 5], and Cinti and Pratelli [30]. In spite of the possible lack of compactness we will be able to show that, for smooth bounded densities, the regular part of the interior boundary is a hypersurface of null weighted capacity. To prove this important fact we first establish a uniform upper estimate for the perimeter of a weighted minimizer inside open balls of R n+1 .
where ρ n is the Euclidean volume of B(0, 1).
Fix a point p ∈ R n+1 and take an open ball B(p, r) of radius r R. Note that by (2.1) and the definition of R. Therefore, there is a unique r ′ p R 0 such that Consider the set E ′ := (E − B(p, r)) ∪ B(p 0 , r ′ p ). It is clear that E ′ ⊂ Ω and V f (E ′ ) = V f (E). By using that E is a weighted isoperimetric region we get P f (E, Ω) P f (E ′ , Ω). This fact together with Example 2.9 and Lemma 2.11 gives us the following ∂B(p, r)) + A f (∂B(p 0 , r ′ p )), for every r R. On the other hand, we have ·) is a finite Borel measure in Ω. By combining the two previous inequalities and taking into account the definition of weighted area in (2.2), we obtain ∂B(p, r)) + A f (∂B(p 0 , r ′ p )) α λ n (r n + (r ′ p ) n ), for every r R, where λ n is the Euclidean area of the unit sphere in R n+1 .
Finally, let m 0 := inf{f (q) ; q ∈ B(p 0 , R 0 )}. As r ′ p R 0 then f (q) m 0 for any q ∈ B(p 0 , r ′ p ). From (2.1) and (3.3) we deduce The previous inequality yields r ′ p β r, for any r R, where β is a positive constant which does not depend on p and r. Plugging this information into (3.4), we conclude that for every r R. This completes the proof.   (3.2) in Ω = R n+1 endowed with a lower semicontinuous density f bounded from above and below, see the proof of [30,Thm. 5.7]. Now, we are ready to state and prove our main result in this section. Theorem 3.6 (Null capacity property for weighted minimizers). Let Ω be a smooth open set of R n+1 endowed with a bounded density f = e ψ smooth on Ω. If E is a weighted isoperimetric region in Ω, then the regular part Σ of the interior boundary ∂E ∩ Ω if a hypersurface of null weighted capacity. Remark 3.7 (Idea of the proof). Recall that Σ has null weighted capacity if Cap f (K) = 0 for any compact subset K ⊆ Σ, see (2.12). To prove the theorem it is then enough to construct a sequence {ϕ k } k∈N ⊂ H 1 0 (Σ, da f ) satisfying: (i) 0 ϕ k 1, for any k ∈ N, (ii) lim k→∞ Σ |∇ Σ ϕ k | 2 da f = 0, (iii) for any compact K ⊆ Σ there is k 0 ∈ N such that ϕ k = 1 on K, for any k k 0 .  Proof of Theorem 3.6. We will obtain a sequence {ϕ k } k∈N satisfying properties (i), (ii) and (iii) in Remark 3.7. Let Σ 0 be the singular set of ∂E ∩ Ω. We know from Theorem 3.2 that Σ 0 is a closed set with H q (Σ 0 ) = 0, for any q > n − 7. Here H q denotes the q-dimensional Hausdorff measure in R n+1 . By Remark 3.3 we get P f (E, U ) = A f (Σ ∩ U ) for any open set U ⊆ Ω. If Σ 0 = ∅, then Σ = ∂E ∩ Ω, which is a complete hypersurface of finite weighted area. In this case Σ has null weighted capacity by Example 2.8. So, we can suppose Σ 0 = ∅. This implies, in particular, that n 7 and H n−2 (Σ 0 ) = 0.
Let C > 0 and R > 0 be the constants in Proposition 3.4. Thus, we have Moreover, we may assume r i R, 2r i 1/k, and Σ 0 ∩ B(p i , r i /2) = ∅ for any i ∈ N. In particular, inequality (3.5) holds for any B(p i , r i ). From the fact that Σ 0 ∩ B(p i , r i /2) = ∅ for any i ∈ N, it follows that Moreover, we have the gradient estimate Let J ⊂ N be a finite set. We define ζ J k : R n+1 → [0, 1] by ζ J k (p) := min{ζ i (p) ; i ∈ J}. Note that ζ J k is a piecewise smooth function. Having in mind (3.8), (3.5) and (3.6), we obtain So, we have proved Now, we fix a point p 0 ∈ Σ 0 , and we denote δ k (p) : Finally, we define the function ϕ k : R n+1 → [0, 1] by ϕ k := δ k ζ J(k) k , where J(k) ⊂ N is a finite set such that the compact ball B(p 0 , k) is contained in the union of the balls {B(p i , r i /2) ; i ∈ J(k)} with the open set R n+1 − Σ 0 . Clearly, if p ∈ Σ and ϕ k (p) = 0, then p ∈ ∂E ∩ Ω ∩ B(p 0 , k) ∩ R n+1 − ∪ i∈J(k) B(p i , r i /2) . From here, it is easy to deduce that the support of ϕ k in Σ is a compact subset of ∂E ∩ Ω − Σ 0 = Σ. Note also that ϕ k ∈ H 1 0 (Σ, da f ). Moreover, by the Cauchy-Schwarz inequality we get , which tends to zero as k → ∞ by (3.10) and (3.9). On the other hand, it is clear that So, for a given compact set K ⊂ Σ, we can find by (3.7) a number k 0 ∈ N such that ϕ k = 1 on K, for any k k 0 . Therefore, the sequence {ϕ k } k∈N satisfies the desired properties and the proof is completed.

Characterization of weighted stable sets
In this section we consider the stability condition in Euclidean smooth open sets with smooth densities. By a weighted stable set we mean a second order minimum of the weighted perimeter functional for compactly supported variations preserving the weighted volume. Clearly a weighted isoperimetric region is a weighted stable set. As we aim to study the weighted minimizers by means of the stability condition, we can restrict ourselves, by Theorems 3.2 and 3.6, to stable sets of finite weighted perimeter and whose interior boundary coincides, up to a closed set of vanishing area, with a smooth hypersurface of null weighted capacity. Under these conditions we will be able to obtain a stability inequality, that we use to classify weighted stable sets in half-spaces and slabs of R n+1 endowed with a log-concave perturbation of the Gaussian density as in (2.4).

Generalized stability inequalities.
Let us first introduce some notation and recall the basic variational properties of stable sets. Consider a smooth open set Ω in R n+1 endowed with a density f = e ψ smooth on Ω. Take an open set E ⊂ Ω whose interior boundary ∂E ∩ Ω is the disjoint union of a smooth embedded hypersurface Σ and a closed singular set Σ 0 with A f (Σ 0 ) = 0. Note that Σ is a closed hypersurface of R n+1 when Σ 0 = ∅. If Σ has non-empty boundary ∂Σ, then we assume ∂Σ = Σ ∩ ∂Ω, which prevents the existence in ∂Ω of interior points of Σ. If ∂Σ = ∅ then we adopt the convention that all the integrals along ∂Σ vanish. We denote by N the inner unit normal of Σ, and by ν the conormal vector of ∂Σ, i.e., the inner unit normal along ∂Σ in Σ.
Let X be a smooth vector field on R n+1 with compact support on Ω and tangent along ∂Ω. The one-parameter group of diffeomorphisms {φ s } s∈R of X allows to define a variation of E by E s := φ s (E). The associated weighted volume and perimeter functionals are given by V f (s) := V f (E s ) and P f (s) := P f (E s , Ω), respectively. The variation is said to be volume-preserving if V f (s) = V f (E) for any s small enough. We say that E is weighted stationary if P ′ f (0) = 0 for any volume-preserving variation. We say that E is weighted stable if it is weighted stationary and P ′′ f (0) 0 for any volume-preserving variation. If we denote Σ s := φ s (Σ), then ∂Σ s = Σ s ∩ ∂Ω, and we know from Example 2.9 that P f (s) = A f (Σ s ). Thus, if E is weighted stable, then the hypersurface Σ is free boundary f -stable in the sense defined by Castro   with respect to the inner unit normal N , and Σ meets ∂Ω orthogonally along ∂Σ.
The f -index form of Σ is the symmetric bilinear form on C ∞ 0 (Σ) given by where Ric f denotes the Bakry-Émery-Ricci tensor defined in (2.3), σ is the second fundamental form of Σ with respect to N , and II is the second fundamental form of ∂Ω with respect to the inner unit normal. From the integration by parts formula in (2.11), we get In the previous equation, L f is the f -Jacobi operator of Σ, i.e., the second order linear operator given by where ∆ Σ,f is the f -Laplacian relative to Σ in (2.10). It is important to recall that L f (v) coincides with the derivative of the f -mean curvature along a variation whose velocity vector X p := (d/ds)| s=0 φ s (p) satisfies X, N = v along Σ, see [27,Eq. (3.5)]. This means that where (H f ) s denotes the f -mean curvature along the hypersurface Σ s .
Observe that the stability inequality in Lemma 4.1 (ii) is valid for mean zero functions with compact support on Σ. Geometrically these functions come from volume-preserving variations of E fixing a neighborhood of the singular set Σ 0 . Note also that the stability inequality becomes more restrictive provided the Bakry-Émery-Ricci curvature Ric f is nonnegative and Ω is convex. By assuming these hypotheses we obtain below more general inequalities for mean zero functions satisfying certain integrability conditions. Their proofs rely on approximation arguments, that can be carried out when Σ has null weighted capacity.
Proof. We follow the proof given by Ritoré and the author for bounded stable sets in convex solid cones with constant densities, see [66,Lems. 4.5 and 4.7].
Remark 4.3. If Σ 0 = ∅, then we can derive the stability inequality in Proposition 4.2 (i) even if u is not supposed to be bounded. Thus, we obtain I f (u, u) 0 for any u ∈ H 1 (Σ, da f ) with Σ u da f = 0. This improvement is due to the completeness of Σ, which allows to construct from Example 2.8 a sequence {ϕ k } k∈N ⊂ C ∞ 0 (Σ) satisfying 0 ϕ k 1, lim k→∞ ϕ k (p) = 1 for any p ∈ Σ, and |∇ Σ ϕ k | α/k in Σ for some constant α > 0. By using this gradient estimate we can reproduce the arguments in the proof of Proposition 4.2 (i) without assuming that u is bounded.
Remark 4.4. Let Σ ⊂ Ω be a smooth oriented hypersurface of null weighted capacity. From the integration by parts formula in Lemma 2.7 and the symmetry of the f -index form I f , we can prove the following: for any two bounded functions u i ∈ C ∞ (Σ) ∩ H 1 (Σ, da f ) such that ∆ Σ,f u i ∈ L 1 (Σ, da f ) and ∂u i /∂ν ∈ L 1 (∂Σ, dl f ), we have Q f (u 1 , u 2 ) = Q f (u 2 , u 1 ). This is equivalent to the identity that will be used in the proof of Theorem 4.11.
In order to obtain some consequences from the stability condition we must insert suitable test functions in the inequalities of Proposition 4.2. For example, by using constant functions, we can generalize a previous result of Sternberg and Zumbrun for isoperimetric regions with compact interior boundaries in Euclidean convex domains with constant densities, see [70,Thm. 3.1].  To finish this section we apply Lemma 4.1 and Proposition 4.2 to analyze when Euclidean half-spaces are weighted stationary or stable for the perturbations of the Gaussian density defined in (2.4).
Lemma 4.7. Let S be a linear hyperplane of R n+1 and d S (p) := p, ξ the signed distance function from S associated to a unit normal vector ξ. Consider an open half-space or slab Ω with boundary parallel to S, and endowed with the density f = e ψ , where ψ(p) := ω(d S (p)) − c|p| 2 for some ω ∈ C ∞ (J Ω ) and c > 0 (recall that J Ω := d S (Ω)). Let E be the intersection with Ω of an open half-space in R n+1 with interior boundary Σ := ∂E ∩ Ω.
(i) If Σ is either parallel or perpendicular to S, then Σ has constant f -mean curvature. Otherwise, Σ has constant f -mean curvature if and only if there are constants α, β ∈ R such that ω(d S (p)) = α p, ξ + β, for any p ∈ Ω. (ii) E is weighted stationary in Ω if and only if Σ is parallel or perpendicular to S.
Proof. Let N be the inner unit normal to Σ. By taking into account (2.8) and the first equality in (2.5), we get from which we deduce (i). Statement (ii) follows from (i) and the orthogonality condition in Lemma 4.1 (i). To prove (iii) we use the stability inequality in Lemma 4.1 (ii). Take a function u ∈ C ∞ 0 (Σ) with 0 = Σ u da f = e ω(s0) Σ u da c , where da c denotes the weighted area measure associated to the Gaussian density γ c (p) := e −c|p| 2 . From the definition of f -index form in (4.1), the second equality in (2.5), and the fact that d S (p) = s 0 in Σ, we obtain On the other hand, the Gaussian isoperimetric inequality [72], [19] implies that Σ bounds a Gaussian minimizer and, in particular, a weighted stable set for γ c . Therefore, we have the Poincaré type inequality Σ |∇ Σ u| 2 − 2cu 2 da c 0, which shows that I f (u, u) 0 if ω ′′ (s 0 ) 0. Conversely, suppose that E is weighted stable, and take a coordinate function u over Σ with respect to a fixed orthonormal basis in S. This function satisfies u ∈ H 1 (Σ, da f ) and Σ u da f = 0. Moreover, u gives equality in the previous Poincaré inequality. From Remark 4.3 we get I f (u, u) 0, and so ω ′′ (s 0 ) 0, as we claimed.
Remarks 4.8. 1. The previous result is also valid when Ω = R n+1 . In this case ∂Ω = ∅ and E is weighted stationary if and only if Σ has constant f -mean curvature. We stress that weighted stationary half-spaces which are neither parallel nor perpendicular to S appear if and only if ω : R → R is an affine function.
2. From Lemma 4.7 (iii) we deduce that, if ω is strictly concave, then half-spaces parallel to S are weighted unstable sets. In particular, they cannot bound isoperimetric minimizers.
3. Note that the weighted stability of half-spaces perpendicular to S is not discussed in Lemma 4.7. When ω is concave we will show, as an immediate consequence of Theorem 5.5, that they are all weighted stable. 4. Barthe, Bianchini and Colesanti have employed Poincaré type inequalities to study the weighted stability of half-spaces in R n+1 for a Euclidean measure µ n+1 , where µ is a probability measure on R, see [10,Sect. 3]. In particular, they obtain stability of half-spaces in R n+1 with Gaussian density. Indeed, it is proved in [10,Thm. 3.5] that weighted stability of coordinate half-spaces characterizes Gaussian type measures. By using similar arguments, Doan provides in [31, Thm. 5.1] a stability criterion for horizontal half-spaces in R n+1 = R n × R with product density f (p, t) = e ψ(p)+δ(t) . In fact, a calibration argument shows that, if δ is constant or strictly convex, then any horizontal half-space is weighted area-minimizing, see [24,Thm. 2.9] and [31, Cor. 5.3].

Characterization results.
We now turn to the classification of weighted stable sets in a half-space or slab Ω ⊂ R n+1 endowed with a smooth log-concave perturbation of the Gaussian density as in (2.4). For that, we will use the stability inequality in Proposition 4.2 (ii) with a suitable test function. Since ∂Ω is totally geodesic it is natural to produce such a function by using translations along ∂Ω. However, as a difference with respect to the case of constant density, such translations need not preserve the enclosed volume. To solve this difficulty we consider variations of a given stable set E by equidistant sets translated along ∂Ω to keep the weighted volume constant. The associated test functions along the interior boundary of E are computed in the next lemma. Proof of Lemma 4.9. The associated velocity vector of {φ s } s∈R is X p := (d/ds)| s=0 φ s (p) = N p + λ ′ (0)η, which is tangent to ∂Ω in the points of ∂Σ. Thus we have v(p) := X p , N p = 1 + λ ′ (0) η, N p , for any p ∈ Σ. Let E s ⊂ Ω be the bounded open set such that ∂E s ∩ Ω = Σ s . By using that the function V f (s) := V f (E s ) is constant for s small enough, and the first variation of weighted volume computed in [27,Sect. 3], we deduce Therefore λ ′ (0) = α −1 and the proof finishes.
For a half-space or slab Ω ⊂ R n+1 with a smooth density f as in (2.4) we discussed in Lemma 4.7 when half-spaces intersected with Ω are weighted stationary or stable. In particular, we showed that the stability of half-spaces parallel to ∂Ω becomes more restrictive when the function ω is concave. On the other hand, the concavity of ω is also equivalent by (2.5) to the inequality Ric f 2c, which allows to deduce the stability inequalities in Proposition 4.2 and the connectivity of stable boundaries in Corollary 4.5. In the main result of this section we assume concavity of ω to characterize stable sets in Ω of finite weighted perimeter and interior boundary of null weighted capacity. Theorem 4.11 (Stable sets in haf-spaces and slabs). Let S ⊂ R n+1 be a linear hyperplane and d S (p) := p, ξ the signed distance function from S associated to a unit normal vector ξ. Consider an open half-space or slab Ω with boundary parallel to S, and endowed with the density f = e ψ , where ψ(p) := ω(d S (p)) − c|p| 2 for some concave function ω ∈ C ∞ (J Ω ) and c > 0 (recall that J Ω := d S (Ω)). Let E ⊂ Ω be an open set of finite weighted perimeter in Ω such that ∂E ∩ Ω = Σ ∪ Σ 0 , where Σ is a smooth hypersurface of null weighted capacity with boundary ∂Σ = Σ ∩ ∂Ω, and Σ 0 is a closed singular set with A f (Σ 0 ) = 0. If E is weighted stable, then E is the intersection with Ω of a half-space with boundary parallel or perpendicular to ∂Ω. Moreover, if ω is strictly concave, only the latter case is possible.
Proof. Let N be the inner unit normal to Σ. From Lemma 4.1 (i) we know that Σ has constant f -mean curvature and meets ∂Ω orthogonally in the points of ∂Σ. Hence, along ∂Σ, the conormal vector ν coincides with the inner unit normal to ∂Ω. On the other hand, the weighted stability of E together with inequality Ric f 2c in (2.5) implies that Σ is connected by Corollary 4.5.
For a fixed unit vector η ∈ S, we define the vector field X p := η with associated oneparameter group of translations τ s (p) := p + sη. Let Σ s := τ s (Σ), and denote by (H f ) s the f -mean curvature of Σ s . By using (2.8), (2.5), equality d S (τ s (p)) = d S (p), and that the unit normal N s to Σ s and the Euclidean mean curvature H s of Σ s are invariant under translations, we get Hence, if we define h : Σ → R by h(p) := X p , N p = η, N p , then equations (4.3) and (4.4) give us (4.6) ∆ where ∆ Σ,f is the f -Laplacian relative to Σ and σ is the second fundamental form of Σ. Formula (4.6) shows that h is an eigenfunction for the f -Jacobi operator on Σ.
Observe that h ∈ L 1 (Σ, da f ) since h is bounded and A f (Σ) = P f (E, Ω) < +∞ by Example 2.9. Thus, we can define the test function Σ h da f and v is the function in Lemma 4.9. Note that u is a bounded smooth function on Σ with Σ u da f = 0. Let us see that u satisfies the integrability hypotheses in Proposition 4.2 (ii). From the definition of h and equation (4.6), we obtain where e i is a principal direction of Σ at p with principal curvature k i (p). Note that |∇ Σ u| 2 |σ| 2 Ric f (N, N )+|σ| 2 , which is an integrable function by Proposition 4.2. Hence, it follows that u ∈ H 1 (Σ, da f ) and ∆ Σ,f u ∈ L 1 (Σ, da f ). Now, we compute ∂u/∂ν along ∂Σ. Let D be the Levi-Cività connection in R n+1 . Then, for any vector T tangent to ∂Σ, we have since ∂Ω is totally geodesic and ν coincides with the inner unit normal to ∂Ω. The previous computation shows that D ν N is proportional to ν in the points of ∂Σ. By taking an orthonormal basis {e 1 , . . . , e n } of principal directions at p ∈ ∂Σ with e n = ν p , and having in mind (4.7), we deduce since η is tangent to ∂Ω and ν p is normal to ∂Ω at p.
At this point, we can apply Proposition 4.2 (ii) to ensure that Q f (u, u) 0, where Q f is the f -index form of Σ defined in (4.2). Observe that the boundary term in Q f (u, u) vanishes since ∂u/∂ν = II(N, N ) = 0 along ∂Σ. By using (4.6), the second equality in (2.5), and that On the other hand, equalities (4.6), (4.5) and (4.8) give us Plugging this into (4.9), and taking into account that by the concavity of ω and the Cauchy-Schwarz inequality in L 2 (Σ, da f ). As a consequence, we can ensure that α 2 (|σ| 2 − ω ′′ (d S (p)) ξ, N p 2 ) = 0 on Σ and η, N p = h(p) = −α, for any p ∈ Σ and any unit vector η ∈ S. If α = 0 for any unit vector η ∈ S, then Σ is contained in a hyperplane parallel to ∂Ω. If α = 0 for some unit vector η ∈ S, then |σ| 2 = 0, and Σ is contained in a hyperplane transversal to ∂Ω. In both cases, the unit normal N to Σ extends continuously to Σ. This implies by Lemma 2.12 that ∂E ∩ Ω is a C 1 hypersurface since the generalized unit normal N f in (2.14) equals N along Σ. In particular Σ 0 = ∅. As Σ is closed and connected, we deduce that Σ is a hyperplane intersected with Ω. From the orthogonality condition between Σ and ∂Ω in Lemma 4.1 (i) we conclude that Σ is perpendicular to ∂Ω if ∂Σ = ∅. Otherwise, Σ is parallel to ∂Ω. Moreover, if ω is strictly concave, then Σ must be perpendicular to ∂Ω by Lemma 4.7 (iii). This completes the proof of the theorem.  As a particular case of Theorem 4.11, when S is the linear hyperplane parallel to ∂Ω and ω = 0, we obtain the following result for the Gaussian density, which is interesting in itself. Consider an open set E ⊂ Ω of finite weighted perimeter in Ω such that ∂E ∩ Ω = Σ ∪ Σ 0 , where Σ is a smooth hypersurface of null weighted capacity with boundary ∂Σ = Σ ∩ ∂Ω, and Σ 0 is a closed singular set with A f (Σ 0 ) = 0. If E is weighted stable, then E is the intersection with Ω of a half-space with boundary parallel or perpendicular to ∂Ω.
The arguments in the proof of Theorem 4.11 also apply to show that a stable set of finite weighted perimeter and null weighted capacity in R n+1 with a smooth log-concave perturbation of the Gaussian density as in (2.4) must be a half-space. In this case the computations are even easier since ∂Σ = ∅. By combining this fact with Lemma 4.7, we get Theorem 4.14 (Stable sets in R n+1 ). Let S ⊂ R n+1 be a linear hyperplane and d S (p) := p, ξ the signed distance function from S associated to a unit normal vector ξ. Consider the density f = e ψ , where ψ(p) := ω(d S (p)) − c|p| 2 for some concave function ω ∈ C ∞ (R) and c > 0. Let E ⊂ R n+1 be an open set of finite weighted perimeter such that ∂E = Σ ∪ Σ 0 , where Σ is a smooth hypersurface of null weighted capacity, and Σ 0 is a closed singular set with A f (Σ 0 ) = 0. If E is weighted stable, then E is a half-space. Moreover: (i) if ω is not an affine function, then ∂E is parallel or perpendicular to S, (ii) if ω is strictly concave, then ∂E is perpendicular to S.
As the particular case of Theorem 4.14 when ω = 0 we deduce a characterization result for weighted stable sets in the Gauss space. This will be used in Section 5 to provide a new proof of the Gaussian isoperimetric inequality. where Σ is a smooth hypersurface of null weighted capacity and Σ 0 is a closed singular set with A f (Σ 0 ) = 0. If E is weighted stable, then E is a half-space.
As we showed in Example 2.8, any complete hypersurface of finite weighted area has null weighted capacity. Hence, by using the same technique as in Theorem 4.11, we obtain the following classification result for free boundary f -stable hypersurfaces. Consider an open half-space or slab Ω with boundary parallel to S, and endowed with the density f = e ψ , where ψ(p) := ω(d S (p)) − c|p| 2 for some concave function ω ∈ C ∞ (J Ω ) and c > 0. Let Σ ⊂ Ω be a smooth, complete, orientable hypersurface with boundary ∂Σ = Σ ∩ ∂Ω. If Σ is f -stable and has finite weighted area, then Σ is either a hyperplane parallel to ∂Ω or the intersection with Ω of a hyperplane perpendicular to ∂Ω. Moreover, if ω is strictly concave, then Σ is a hyperplane perpendicular to ∂Ω.
Remark 4.17. The previous corollary is valid for Gaussian half-spaces and slabs. The result also holds when Ω = R n+1 . In particular, we deduce that any smooth, complete, orientable, f -stable hypersurface of finite weighted area in the Gauss space must be a hyperplane. This was previously established by McGonagle and Ross, see [53,Cor. 2].

Characterization of weighted isoperimetric regions
Let Ω be an open half-space or slab of R n+1 endowed with a perturbation f of the Gaussian density γ c as in (2.4). Recall that the concavity of the function ω is equivalent, by the second equality in (2.5), to that f is a log-concave perturbation of γ c as in (1.1). In particular, when f is also a probability density on Ω we can apply the Lévy-Gromov isoperimetric inequality first established by Bakry and Ledoux [6,Sect. 2] (see also Morgan [61,Thm 18.7]), and later extended by Bayle [12,Sect. 3] and Milman [54,Thm. 1.2], to provide a lower bound, possibly non-optimal, for the isoperimetric profile function I Ω,f in (3.1) with isoperimetric model the Gaussian probability measure. In this section we will be able to compute explicitly I Ω,f by showing that half-spaces perpendicular to ∂Ω are always weighted isoperimetric regions in Ω. Moreover, when f is smooth on Ω we will prove that these are the unique isoperimetric regions, up to sets of volume zero. We also derive interesting consequences of these results and obtain similar statements when Ω = R n+1 .

Half-spaces and slabs.
In order to characterize the weighted minimizers in a half-space or a slab Ω we will use the stability property as a main tool. Indeed, since any weighted isoperimetric region is also a weighted stable set, we deduce from Theorem 4.11 that the only candidates to be weighted minimizers are intersections with Ω of half-spaces parallel or perpendicular to ∂Ω. When the perturbation of γ c is strictly log-concave, then we know from Lemma 4.7 (iii) that parallel half-spaces do not minimize, since they are not weighted stable. In the general case, we can show that half-spaces perpendicular to ∂Ω are always isoperimetrically better than the parallel ones.
Proposition 5.1 (Parallel half-spaces vs. perpendicular half-spaces). Let S ⊂ R n+1 be a linear hyperplane and d S (p) := p, ξ the signed distance function from S associated to a unit normal vector ξ. Consider an open half-space or slab Ω with boundary parallel to S, and endowed with the density f = e ψ , where ψ(p) := ω(d S (p)) − c|p| 2 for some concave function ω ∈ C ∞ (J Ω ) and c > 0 (recall that J Ω := d S (Ω)). Then, a half-space perpendicular to ∂Ω has strictly less weighted perimeter in Ω than a half-space parallel to ∂Ω with the same weighted volume in Ω.
Remark 5.2. The proof of the proposition relies on the fact that the profile function associated to a parallel one-parameter family of f -stationary hyperplanes satisfies a second order differential inequality, which becomes an equality when the family is perpendicular to ∂Ω. The integration of differential inequalities has been used to obtain isoperimetric comparisons in several contexts, see for example Morgan  Proof of Proposition 5.1. Let T ⊂ R n+1 be a linear hyperplane. We take an orthonormal basis {e 1 , . . . , e n , ξ T }, where {e 1 , . . . , e n } ⊂ T and ξ T is chosen so that ξ T = ξ when T = S. We identify a point p ∈ R n+1 with its coordinates (z, t) = (z 1 , . . . , z n , t) in the previous basis. Given a domain U ⊆ T we consider the family Σ s := U × {s}, where −∞ a < s < b +∞. Let C U := U × (a, b), and denote by E s the cylinder U × (a, s) with s ∈ (a, b).
By Lemma 2.2 we know that f is a bounded density of finite weighted volume, and that hyperplanes intersected with Ω has finite weighted area. Thus, we can define the weighted volume and area functions V f (s) := V f (E s ) and A f (s) := A f (Σ s ) = P f (E s , C U ). By using the change of variables formula and Fubini's theorem, we get Now, we apply differentiation under the integral sign to obtain for any s ∈ (a, b). In particular, we deduce ). This function is continuous on [0, V f (C U )] and C 2 on (0, V f (C U )). From the computations above and the concavity of ω it is straightforward to check that, when T = S, then F ′′ −2cF −1 in (0, V f (C U )) with equality if and only if ω is an affine function. Moreover, The previous arguments can be applied when Σ s is a family of hyperplanes in Ω which are either parallel or perpendicular to ∂Ω. If F and G denote the associated profile functions, then F ′′ −2cF −1 in (0, V f (Ω)) and G ′′ = −2c G −1 in (0, V f (Ω)). By taking into account that F (0) > 0, F (V f (Ω)) 0 and G(0) = G(V f (Ω)) = 0, we conclude that F > G in (0, V f (Ω)) by Lemma 5.3 below. This proves the claim. Proof. Suppose that there is t ∈ (a, b) where F (t) < G(t). Then, the minimum of the function F − G is achieved at some t 1 ∈ (a, b) for which (F − G)(t 1 ) < 0. Thus, we get This shows that F G in [a, b]. Suppose now that F (t 0 ) = G(t 0 ) for some t 0 ∈ (a, b), which in particular implies F ′ (t 0 ) = G ′ (t 0 ). As the function x → −2c/x is increasing for x > 0, we can apply a classical comparison result for ordinary differential inequalities [73,Thm. 19.1] to deduce that F G in [t 0 , b]. By using the same result with the functions F 1 (t) : . This fact together with inequality F G in [a, b] proves the claim.
Remark 5.4. It is important to observe that, for an arbitrary concave function ω : J Ω → R (possibly non-smooth on J Ω ), and a given family {Σ s } s∈R of parallel hyperplanes perpendicular to ∂Ω, the associated profile function G : ) is continuous on [0, V f (Ω)] and satisfies the differential equation G ′′ = −2c G −1 in (0, V f (Ω)). To see this note that, since any Σ s is perpendicular to ∂Ω, then the function ω(d S (z, s)) does not depend on s. Hence, the smoothness of ω is not necessary to differentiate under the integral sign in (5.2), and we can proceed as in the proof of Proposition 5.1 to get the claim.
We can now combine our main previous results to prove the following statement.
Theorem 5.5 (Weighted minimizers in half-spaces and slabs). Let S ⊂ R n+1 be a linear hyperplane and d S (p) := p, ξ the signed distance from S associated to a unit normal vector ξ. Consider an open half-space or slab Ω with boundary parallel to S, and endowed with the density f = e ψ , where ψ(p) := ω(d S (p)) − c|p| 2 for some concave function ω ∈ C ∞ (J Ω ) and c > 0 (recall that J Ω := d S (Ω)). Then, weighted isoperimetric regions of any given volume exist in Ω and, up to sets of volume zero, they are all intersections with Ω of half-spaces perpendicular to ∂Ω.
Proof. The concavity of ω implies by Lemma 2.2 that f is bounded and V f (Ω) < +∞. Hence, the existence of weighted minimizers is a consequence of Theorem 3.1. Let E ⊂ Ω be a weighted isoperimetric region with V f (E) = v ∈ (0, V f (Ω)). From the regularity results in Theorem 3.2 we know that the interior boundary ∂E ∩ Ω is a disjoint union Σ ∪ Σ 0 , where Σ is a smooth embedded hypersurface with (possibly empty) boundary ∂Σ = Σ ∩ ∂Ω, and Σ 0 is a closed singular set of Hausdorff dimension less than or equal to n − 7. By Remark 3.3 we can assume that E is, up to a set of volume zero, an open set. From Theorem 3.6, the hypersurface Σ has null weighted capacity. By the classification of weighted stable sets in Theorem 4.11 we deduce that E is the intersection with Ω of a half-space with boundary parallel or perpendicular to ∂Ω. Finally, we apply the comparison in Proposition 5.1 to conclude that half-spaces perpendicular to ∂Ω are isoperimetrically better.
Remark 5.6 (Uniqueness). The weighted minimizers in Ω of a given volume are not unique. If L denotes the straight line perpendicular to S passing through the origin, then the density f = e ψ with ψ(p) := ω(d S (p)) − c|p| 2 is invariant under any Euclidean rotation φ about L. Hence, a set E ⊂ Ω is a weighted minimizer of volume v if and only if the same holds for φ(E). In fact, for any two half-spaces E and E ′ perpendicular to ∂Ω and with the same weighted volume in Ω, there is an isometry φ as above such that φ(E) = E ′ .
An interesting particular case of Theorem 5.5 is obtained when S is the linear hyperplane parallel to ∂Ω and ω vanishes identically.
Corollary 5.7 (Weighted minimizers in Gaussian half-spaces and slabs). Let Ω be an open half-space or slab in R n+1 with Gaussian density γ c (p) := e −c|p| 2 , c > 0. Then, weighted isoperimetric regions of any given volume exist in Ω and they are all obtained, up to sets of volume zero, by intersecting Ω with half-spaces perpendicular to ∂Ω.
Remark 5.8. The isoperimetric problem in a Euclidean half-space Ω with Gaussian density has been studied in previous works. On the one hand, Lee [48,Prop. 5.1] used the same approximation argument as in [72] (see a sketch in [61, Thm. 18.2]) to prove that half-spaces perpendicular to ∂Ω are weighted minimizers. Later on, Adams, Corwin, Davis, Lee and Visocchi [1, Thm. 3.1] obtained uniqueness of minimizers when 0 ∈ ∂Ω by means of reflection across ∂Ω and the characterization of equality cases in the Gaussian isoperimetric inequality [26]. Some basic properties for weighted isoperimetric regions in symmetric planar strips with Gaussian density were established by Lee [48,Sect. 5].
Remark 5.9 (Stability vs. isoperimetry). As we announced in Remarks 4.8, the isoperimetric result in Theorem 5.5 implies that half-spaces perpendicular to ∂Ω are weighted stable sets. However, not every weighted stable set is a weighted minimizer. For example, inside an open half-space or slab Ω with Gaussian density, a half-space E parallel to ∂Ω provides a weighted stable set by Lemma 4.7 (iii). However, E is not isoperimetric since a half-space perpendicular to ∂Ω of the same weighted volume is isoperimetrically better.
The weighted stability of half-spaces perpendicular to ∂Ω allows us to apply the stability inequality in Remark 4.3 to deduce the following result. Consider an open half-space or slab Ω with boundary parallel to S, and endowed with the density f = e ψ , where ψ(p) := ω(d S (p)) − c|p| 2 for some concave function ω ∈ C ∞ (J Ω ) and c > 0. Let E be the intersection with Ω of a Euclidean half-space perpendicular to ∂Ω. If we denote Σ := ∂E ∩ Ω, then we have the inequality Remark 5.11. A Poincaré type inequality provides a lower bound on the spectral gap or Poincaré constant of Σ, which is the non-negative number defined as It is well known that λ f (Σ) = 2c for a hyperplane Σ of R n+1 with Gaussian density γ c . For a smooth and convex domain Σ of a Gaussian hyperplane, the lower bound λ f (Σ) 2c was obtained by Bakry and Qian when Σ is bounded [7,Thm. 14], and by Brandolini, Chiacchio, Henrot and Trombetti when Σ is unbounded [20, Thm. 1.1]. Our Corollary 5.10 shows that the spectral gap inequality λ f (Σ) 2c also holds when Σ is an n-dimensional strip perpendicular to ∂Ω endowed with a log-concave perturbation of γ c as in (2.4).
We now turn to establish the isoperimetric property of half-spaces perpendicular to ∂Ω for a density f as in (2.4), where the function ω : J Ω → R is concave and possibly non-smooth on J Ω . An important previous result in this direction was obtained by Brock, Chiacchio and Mercaldo [21, Thm. 2.1], who employed the Gaussian isoperimetric inequality and a transport map to establish that, in the half-space Ω := {(x 1 , . . . , x n+1 ) ∈ R n+1 ; x n+1 > 0} with density f (p) := x m n+1 e −|p| 2 /2 , m 0, the intersections with Ω of half-spaces perpendicular to ∂Ω uniquely minimize the weighted perimeter for fixed weighted volume. Note that this density is a particular case of (2.4) with c = 1/2 and ω(s) = m log(s), for any s > 0. In the general case we can prove the following statement.
Theorem 5.12 (The non-smooth case). Let S ⊂ R n+1 be a linear hyperplane and d S (p) := p, ξ the signed distance from S associated to a unit normal vector ξ. Consider an open half-space or slab Ω with boundary parallel to S, and endowed with the density f = e ψ , where ψ(p) := ω(d S (p)) − c|p| 2 for some concave function ω : J Ω → R and c > 0 (recall that J Ω := d S (Ω)). Then, the intersections with Ω of half-spaces perpendicular to ∂Ω are weighted isoperimetric regions.
The proof of this theorem relies on two approximation arguments. In the first step, we consider parallel interior domains to Ω in order to prove the claim when ω : J Ω → R is smooth. In the second step, we use a recent result of Azagra [4, Thm. 1.1] to approximate ω by a sequence of smooth concave functions ω k for which the first step can be applied.
Proof of Theorem 5.12. As in the proof of Lemma 2.2, the concavity of ω gives us the existence of constants α, β ∈ R such that (5.3) ω(d S (p)) α p, ξ + β, for any p ∈ Ω.
This inequality implies that f is bounded from above by a density of Gaussian type. In particular, f is bounded, V f (Ω) < +∞, and Euclidean hyperplanes intersected with Ω have finite weighted area.
Let E ⊂ Ω be a Borel set with V f (E) = v ∈ (0, V f (Ω)) and P f (E, Ω) < +∞. To prove the claim we must show where E * denotes the intersection with Ω of a half-space perpendicular to ∂Ω and such that V f (E * ) = v. By using the approximation property in Lemma 2.14 (iii) and the continuity of the profile function associated to the family of hyperplanes parallel to ∂E * , see Remark 5.4, it suffices to prove (5.4) when E is an open set with smooth boundary ∂E ∩ Ω. This will be accomplished in two steps.
Step 1. The isoperimetric comparison in (5.4) holds when ω is smooth on J Ω .
Let k 0 := 1/ sup{dist(p, ∂Ω) ; p ∈ Ω} (we understand that k 0 = 0 when Ω is a half-space). For any k > k 0 , we define Ω k := {p ∈ Ω ; dist(p, ∂Ω) > 1/k}. This provides an increasing family of open half-spaces or slabs {Ω k } k>k0 such that Ω k ⊂ Ω and ∪ k>k0 Ω k = Ω. Note that ω is smooth on J k , where J k := d S (Ω k ). For any k > k 0 , we denote E k := E ∩ Ω k and v k := V f (E k ). Let E * k be the intersection with Ω k of a half-space parallel to E * and such that V f (E * k ) = v k . By using Theorem 5.5 and Remark 5.6, we deduce that any E * k is a weighted isoperimetric region in Ω k for the restriction of f to Ω k . Thus, we have (5.5) P f (E k , Ω k ) P f (E * k , Ω k ), for any k > k 0 . Now, we take limits in (5.5) when k → ∞. Note that E k is an open set in Ω k with ∂E k ∩ Ω k = ∂E ∩ Ω k . As the boundary ∂E ∩ Ω is smooth, then we can apply Lemma 2.14 (ii) to get P f (E k , Ω k ) = A f (∂E ∩ Ω k ) = ∂E∩Ω k f da for any k > k 0 . As a consequence lim k→∞ P f (E k , Ω k ) = A f (∂E ∩ Ω) = P f (E, Ω), by the monotone convergence theorem. To finish the proof of Step 1 we have to show that (5.6) lim k→∞ P f (E * k , Ω k ) = P f (E * , Ω).
Let T be the linear hyperplane in R n+1 parallel to ∂E * . We shall employ the same notation as in the proof of Proposition 5.1. We identify a point p ∈ Ω with its coordinates (z, t) respect to an orthonormal basis {e 1 , . . . , e n , ξ T }, where {e 1 , . . . , e n } ⊂ T and ξ T is a unit vector normal to T . As ξ T , ξ = 0, then the function d S (z, t) does not depend on t. For any k > k 0 , let s k ∈ R be the unique number such that E * k = U k × (−∞, s k ), where U k := T ∩ Ω k . From equations (5.1) and (5.2), we easily get where U := T ∩ Ω. On the other hand, inequality (5.3) implies Ψ ∞ < +∞ since the integrand is bounded from above by a Gaussian density on T . By using this fact and that Φ : R → Φ(R) is a diffeomorphism, we deduce from (5.7) that lim k→∞ s k = s ∞ , where s ∞ is the unique number satisfying v = Φ(s ∞ ) Ψ ∞ . This shows that E * = U × (−∞, s ∞ ). Now, from equation (5.2) we have P f (E * k , Ω k ) = A f (U k × {s k }) = e −cs 2 k Ψ k , and so lim k→∞ P f (E * k , Ω k ) = e −cs 2 ∞ Ψ ∞ = A f (U × {s ∞ }) = P f (E * , Ω).
This shows that equation (5.6) holds and finishes the proof of Step 1.
Step 2. The isoperimetric comparison in (5.4) holds for any function ω concave on J Ω .
By the approximation result of Azagra [4, Thm. 1.1], for any k ∈ N, there is a smooth concave function ω k : J Ω → R such that ω ω k ω + 1/k on J Ω . We define ψ k : Ω → R by ψ k (p) := ω k (d S (p)) − c|p| 2 . Hence, if we denote f k := e ψ k , then we produce a sequence of smooth densities on Ω which pointwise converges to f when k → ∞, and satisfies f k ef in Ω for any k ∈ N. Let v k := V f k (E). Thus we have lim k→∞ v k = v by the dominated convergence theorem. Consider the intersection E * k of a half-space parallel to E * with Ω and such that V f k (E * k ) = v k . By using Step 1, we deduce that E * k is a weighted isoperimetric region in Ω for the density f k . Therefore, we get P f k (E, Ω) P f k (E * k , Ω), for any k ∈ N. Now, we take limits when k → ∞. As the boundary ∂E ∩ Ω is smooth, then Lemma 2.14 (ii) yields P f k (E, Ω) = A f k (∂E ∩ Ω) = ∂E∩Ω f k da for any k ∈ N. As a consequence lim k→∞ P f k (E, Ω) = A f (∂E ∩ Ω) = P f (E, Ω), by the dominated convergence theorem. To finish the proof we must see that (5.8) lim k→∞ P f k (E * k , Ω) = P f (E * , Ω).
We follow the same notation as in Step 1. For any k ∈ N, let s k ∈ R be the unique number such that E * k is identified with the cylinder U × (−∞, s k ) enclosing weighted volume v k with respect to f k . From equations (5.1) and (5.2), this means that Observe that we can apply (5.3) and dominated convergence to obtain lim k→∞ Ψ k = Ψ ∞ := U e ω(dS(z))−c|z| 2 dz < +∞.
From (5.9) we deduce that lim k→∞ s k = s ∞ , where s ∞ is the unique number satisfying v = Φ(s ∞ ) Ψ ∞ . This implies that E * is identified with the cylinder U × (−∞, s ∞ ) with weighted volume v with respect to f . Now, from equation (5.2) we get P f k (E * k , Ω) = A f k (U × {s k }) = e −cs 2 k Ψ k , and so lim Hence equation (5.8) holds. This finishes the proof of the theorem.
Remarks 5.13. 1. Unfortunately, our method of proof in Theorem 5.12 does not provide the uniqueness of minimizers that we obtained in Theorem 5.5 when ω is smooth on J Ω .

We think that
Step 2 can be also carried out by means of standard convolution arguments and approximation of finite weighted perimeter sets by bounded open sets with smooth boundary inside Ω. We have employed Azagra's theorem instead since it provides a more direct and less technical proof at this stage.
As an immediate consequence of Theorem 5.12 and Remark 5.4, we deduce the following analytical result. Corollary 5.14 (A differential equation for the isoperimetric profile). Let S ⊂ R n+1 be a linear hyperplane and d S (p) := p, ξ the signed distance function from S associated to a unit normal vector ξ. Consider an open half-space or slab Ω with boundary parallel to S, and endowed with the density f = e ψ , where ψ(p) := ω(d S (p)) − c|p| 2 for some concave function ω : J Ω → R and c > 0. Then, the weighted isoperimetric profile I Ω,f in (3.1) is continuous on [0, V f (Ω)], smooth on (0, V f (Ω)), and satisfies the second order differential equation I Ω,f on (0, V f (Ω)). The arguments in the proof of Theorem 5.5 can be adapted to solve the isoperimetric problem in R n+1 with a smooth log-concave perturbation of the Gaussian density as in (2.4).
Theorem 5.16 (Weighted minimizers in R n+1 ). Let S ⊂ R n+1 be a linear hyperplane and denote d S (p) := p, ξ the signed distance function from S associated to a unit normal vector ξ. Consider the density f = e ψ , where ψ(p) := ω(d S (p)) − c|p| 2 for some concave function ω ∈ C ∞ (R) and c > 0. Then, weighted isoperimetric regions of any given volume exist for this density, and we have the following: (i) if ω is not an affine function, then any weighted minimizer is, up to a set of volume zero, a half-space perpendicular to S, (ii) if ω is an affine function, then any weighted minimizer is, up to a set of volume zero, a half-space. Moreover, any half-space in R n+1 is a weighted minimizer.
Proof. Following the proof of Theorem 5.5 and using the classification of weighted stable sets in Theorem 4.14, we obtain that weighted minimizers of any volume exist, and they are all Euclidean half-spaces, up to sets of volume zero. Now we distinguish two cases.

Case 1.
If ω is not an affine function, then the boundary of any minimizer is either parallel or perpendicular to S. Let F, G : [0, V f (R n+1 )] be the profile functions for these two families of candidates. From the computations in the proof of Proposition 5.1, we get F ′′ −2cF −1 and G ′′ = −2c G −1 in (0, V f (R n+1 )). Since F and G coincides at the extremes, an application of Lemma 5.3 yields F G in [0, V f (R n+1 )]. In fact, if F (v 0 ) = G(v 0 ) for some v 0 ∈ (0, V f (R n+1 )), then we would get F = G in [0, V f (R n+1 )], and so the function F would satisfy F ′′ = −2cF −1 . From here, we would deduce that ω ′′ = 0 in R, a contradiction. So, we conclude that F > G in (0, V f (R n+1 )), which proves the claim.

Case 2.
If ω is an affine function, then ω ′′ = 0 and ω ′ is constant in R. By following the computations in Proposition 5.1, it is easy to check that the profile function F associated to an arbitrary family {Σ s } s∈R of parallel hyperplanes satisfies F ′′ = −2cF −1 in (0, V f (R n+1 )) and vanishes at the extremes. From Lemma 5.3, we infer that F does not depend on the family {Σ s } s∈R . As a consequence, any half-space in R n+1 is a weighted minimizer.
Example 5.17. The theorem can be applied for densities like f (p) = e ω(xi)−c|p| 2 , where x i is a coordinate function, c is positive, and ω : R → R is smooth and concave.
Remark 5.18. From the previous theorem we can deduce, as in Corollary 5.10 and Remark 5.11, the corresponding Poincaré type inequality and the associated lower bound on the spectral gap for hyperplanes with a log-concave perturbation of the Gaussian density as in (2.4). We must remark that these comparisons also hold for any density f in R n+1 such that Ric f 2c > 0, see for example Ledoux  By using the same approximation argument employed in Step 2 of Theorem 5.12, we can establish the following result in which ω is not assumed to be smooth.
Theorem 5.19 (The non-smooth case in R n+1 ). Let S ⊂ R n+1 be a linear hyperplane and denote d S (p) := p, ξ the signed distance function from S associated to a unit normal vector ξ. Consider the density f = e ψ , where ψ(p) := ω(d S (p)) − c|p| 2 for some concave function ω : R → R and c > 0. If ω is not an affine function, then half-spaces perpendicular to S are weighted isoperimetric regions.
As a particular case of Theorem 5.16 we provide a new proof of the isoperimetric property of Euclidean half-spaces in the Gauss space. The reader is referred to the beginning of the Introduction for an account of different proofs and applications of this important result.