Critical domains for certain Dirichlet integrals in weighted manifolds
Levi Lopes de Lima




Universidade Federal do Ceará (UFC),
	Departamento de Matemática, Campus do Pici, Av. Humberto Monte, s/n, Bloco 914, 60455-760,
	Fortaleza, CE, Brazil.
levi@mat.ufc.br



§ ABSTRACT 
We start by revisiting the derivation of the variational formulae for the functional assigning to  a bounded regular domain in a Riemannian manifold its first Dirichlet eigenvalue 

and
extend it to (not necessarily bounded) domains in certain weighted manifolds. 
This is further extended to other functionals defined by certain Dirichlet energy integrals, with a Morse index formula for the corresponding critical domains being established. We complement these infinitesimal results by proving a couple of global rigidity theorems for (possibly critical) domains in Gaussian half-space, including an Alexandrov-type soap bubble theorem. Although we provide direct proofs of these latter results, we find it worthwhile to point out that the main tools employed (specifically, certain Pohozhaev and Reilly identities) can be formally understood as limits (when the dimension goes to infinity) of tools previously established by Ciarolo-Vezzoni  <cit.> and Qiu-Xia <cit.> to handle similar problems in round hemispheres, with the notion of “convergence” of weighted manifolds being loosely inspired by the celebrated Poincaré's limit theorem in the theory of Gaussian random vectors.
	


Critical domains for certain Dirichlet integrals in weighted manifolds
    Levi Lopes de Lima
    September 9, 2024
======================================================================






§ INTRODUCTION



We consider a complete Riemannian manifold (M,g) of dimension n≥ 2 and a smooth function ϕ:M→ℝ and then form the weighted manifold (M,g,e^-ϕd vol_g), 
where d vol_g is the Riemannian volume element associated to g. 
We fix a connected, proper
domain Ω⊂ M with a smooth boundary Σ=∂Ω. In the purely Riemannian case, where ϕ equals a constant, we always assume that Ω is compact. Otherwise, we impose that Ω has finite ϕ-volume (in the sense that vol_ϕ(Ω)<+∞,
where for  any Borel subset B⊂ M we set vol_ϕ(B)=∫_B d vol_ϕ, with  d vol_ϕ=e^-ϕd vol_g).  In particular, if M itself has finite ϕ-volume, which in fact is the only case of interest here, then if needed we can assume that d vol_ϕ defines a probability measure on M and with this normalization our assumption on Ω boils down to vol_ϕ(Ω)<1. 


	A notable example of a weighted manifold is the Gaussian space
		
    (ℝ^n,δ, e^-ϕ_nd vol_δ),  ϕ_n(x)=e^-|x|^2/2-n/2log(2π),

	where δ is the standard flat metric, which satisfies vol_ϕ_n(ℝ^ n)=1. A somewhat informal interpretation of Poincaré's limit theorem (see <cit.> and Section  <ref> below) allows us to view this space as the appropriate limit of orthogonal projections onto ℝ^n of (volume re-scaled) round spheres of dimension k-1 and radius √(k) as k→+∞, a viewpoint that, at least on heuristic grounds,  we will find useful to employ here; see Remarks <ref> and <ref> and Section <ref> for more on this.  
	
	
 
Our aim here is to study, initially from a variational perspective, the most elementary properties of optimal configurations for Dirichlet functionals of the type

    (Ω,g,e^-ϕd vol_ϕ)↦ℰ(Ω):=inf_u∫_Ω(α|∇ u|^2+β u^2-fu)d vol_ϕ,

where α,β∈ℝ, α>0, f is an auxiliary function and u:Ω→ℝ is  assumed to vary in a suitable Sobolev space of functions determined by appropriate boundary conditions (usually of Dirichlet type) and possibly satisfying an extra integral constraint. Whenever the functional in question displays a geometric flavor it is natural to assume further that Ω varies in the set of domains with a fixed ϕ-volume, a viewpoint we will always adopt here. Although a minimizer (henceforth, an optimal domain) to this variational problem is known to exist under mild conditions on the data (see for instance <cit.> and the references therein), a complete classification is far from being available in general. However, and this is our initial motivation here, we may use variational methods to have a perception on the properties an optimal domain should eventually satisfy. Instead of delving into the intricacies of a general theory, we prefer here to illustrate the methods and results by considering a couple of important examples.    

 (The first Dirichlet eigenvalue)
  Let ℰ(Ω,g,d vol_ϕ)=λ(Ω,g,d vol_ϕ) be the functional ascribing to any weighted domain the first eigenvalue λ(Ω,g,d vol_ϕ) associated to the Dirichlet problem

    {[ Δ_ϕ u+λ u=0           Ω;         u=0           Σ ].

where 

    Δ_ϕ= div_ϕ∇=Δ-⟨∇ϕ,·⟩

is the weighted Laplacian. Here, 

    div_ϕ=e^ϕ div_ge^-ϕ= div -⟨∇ϕ,·⟩

 is the weighted divergence, ⟨ ,⟩=g(·,·), Δ is the metric Laplacian and ∇ is the metric gradient; we refer to Section <ref> for a discussion on the technical assumptions needed to make sure that the existence problem for (<ref>) (in particular, the existence of a first eigenvalue with the expected properties) is well-posed for appropriate choices of Ω. Granted those conditions, it turns out that 

    λ(Ω,g,d vol_ϕ)=∫_Ω|∇ u|^2d vol_ϕ,

where, besides solving (<ref>), u is further constrained to satisfy 

    ∫_Ω u^2d vol_ϕ=1.

Due to the obvious geometric meaning of λ, it is natural to investigate its variational properties under a ϕ-volume-preserving assumption. More precisely, if 0<V< vol_ϕ(M),

    𝒟_V={Ω⊂ M; vol_ϕ(Ω)=V 
    }
 
and 

    λ_V=inf_Ω∈𝒟_Vλ(Ω),

the question is to determine those Ω∈𝒟_V such that λ_V=λ(Ω).	


 (A Dirichlet energy)
For β∈ℝ and f∈ L^2_ϕ(Ω) the Dirichlet (β,f)-energy of Ω⊂ M is 
	
    ℰ_β,f(Ω)=inf_v
    		J_β,f(v),
	
	where 
	
    J_β,f(v)=∫_Ω(1/2|∇ v|^2+β/2v^2-fv)d vol_ϕ

	which, under suitable conditions (cf. Remark <ref> below), is well defined whenever β>-λ(Ω) in the sense that there exists w:Ω→ℝ such that ℰ_β,f(Ω)=J_β,f(w)>-∞ with w satisfying 
	
    {[ -Δ_ϕ w+β w=f            Ω;          w=0            Σ ].
 
	An interesting problem here is to classify the domains which minimize the energy (<ref>) in the class of all domains with a given ϕ-volume. Thus, 
if

    ℰ_V,β,f=inf_Ω∈𝒟_Vℰ_β,f(Ω),

the question is to determine those Ω∈𝒟_V such that ℰ_V,β,f=ℰ_β,f(Ω).	
	

A recurrent theme here is to try to understand how ℰ(Ω) varies with Ω at an  infinitesimal level (and of course restricting ourselves to the examples above). For this we consider variations of Ω induced by one-parameter families of diffeomorphisms of the type t∈(-ϵ,ϵ)↦φ_t:M→ M with φ_0= Id, which we assume to be ϕ-volume-preserving in the sense that vol_ϕ(Ω_t)= vol_ϕ(Ω) for t∈(-ϵ,ϵ). In the context of Example <ref>, and 
following the pioneering works by Garabedian and Schiffer <cit.> and Shimakura <cit.>, where the Euclidean case (ℝ^n,δ,d vol_δ) has been treated in detail, our first goal is to compute the variational formulae for 

    d/dtλ(Ω,g,d vol_ϕ)|_t=0   and  d^2/dt^2λ(Ω,g,d vol_ϕ)|_t=0,   Ω_t=φ_t(Ω),

in terms of the 
variational function ξ=⟨ v,ν⟩, where 
v:=dφ_t/dt|_t=0 is the variational field and ν is the outward unit normal vector field to Σ (Propositions <ref> and <ref>). 

Needless to say, we also perform similar calculations in the setting of Example <ref> (see the proof of Proposition <ref> culminating in (<ref>) and Proposition <ref>).




Our calculation for (<ref>) should be compared with the more general approaches in <cit.> (in the purely Riemannian case) and in <cit.> (for weighted manifolds but restricted to the first variation), where the authors handle more general classes of deformations (with the time varying metrics not being necessarily induced by embeddings of Ω into a fixed Riemannian manifold). Besides its simplicity, the advantage of the (more pedestrian) approach adopted here, which aligns with the computations in <cit.>, lies in the fact that it can be easily extended to other naturals energy functionals, as illustrated here for the examples above in the category of weighted manifolds. 



A common (and certainly well-known) feature that emerges from our computations for the first variational formula is that optimal domains support functions (in our examples, the functions u and w satisfying (<ref>) and (<ref>), respectively) meeting an extra boundary condition involving its normal derivative. In other words, these functions satisfy an over-determined elliptic boundary value problem, which means that optimal domains tend to be quite rigid and hence in principle amenable to a classification; see Remark <ref>.  



With the variational  formulae at hand, we then proceed to extend the Morse index formula first established in <cit.> (for λ and in the Euclidean setting) to the more general framework considered here ( Theorems <ref> and <ref>), with some interesting applications briefly indicated (Remark <ref> and <ref>).  

Although variational methods certainly may be explored to access the most elementary properties of 
optimal domains for Dirichlet integrals, they do not seem to be of much use in the rather challenging problem of their explicit characterization. Thus, it is natural to seek for alternate routes by resorting to global methods relying on suitable variations of certain classical differential and integral identities (usually associated to Pohozhaev and Reilly, respectively). Our first result in this direction is Theorem <ref>, where we classify solutions to an over-determined system associated to a certain Dirichlet energy (ℰ_-1,1 in the notation of Example <ref>) defined in a domain contained in Gaussian half-space. This constitutes the exact analogue of a result in  <cit.> for domains in a round hemisphere and similarly relies on a certain Pohozhaev identity (Proposition <ref>). We stress that, as explained in Remark <ref>, both our over-determined system and the associated Pohozhaev identity may be viewed as “Poincaré's limits” of the corresponding  entities in <cit.>, which points toward the existence of an underlying, although informal, principle which may be useful in other contexts. We confirm this by means of Theorem <ref>, a version of Alexandrov's soap bubble theorem for embedded hypersurfaces in Gaussian half-space. As in the previous example, this is the exact analogue of the classical spherical case, as approached in <cit.>, thus similarly hinging on a Reilly-type identity (Proposition <ref>) which, as explained in Remark <ref>, may be viewed as the “Poincaré's limit” of the corresponding identity in <cit.>. Hopefully, this kind of “Poincaré's convergence”, which is used here  
as an informal device to transfer tools and results from spheres to Gaussian space, may be successfully employed in other classes of problems. 


	
Acknowledgments. The author would like to thank S. Almaraz, C. Barroso and J.F. Montenegro for conversations. 	
 



§ SOME PRELIMINARY FACTS


We collect here the preliminary technical ingredients needed to carry out the proofs of our main results.
We insist that the smooth boundary Σ=∂Ω will always be oriented by its outward pointing unit normal vector ν and is endowed with the weighted area element d area=e^-ϕd area_g, where d area_g is the area element induced by g. 
As usual, we let L^2_ϕ(Ω) denote the space of all measurable functions f:Ω→ℝ such that ∫_Ω f^2d vol_ϕ<+∞. More generally, if s∈ℕ_0={0,1,2,…} we denote by W^s_ϕ(Ω) the  Sobolev space of all such functions such that its weak derivatives {∇^jf}_j=0^s up to order s lie in 
L_ϕ^2(Ω)=W_ϕ^0(Ω)
with the Hilbertian norm 

    f_W^s_ϕ(Ω)^2=∑_j=1^s∇^jf^2_L^2_ϕ(Ω).

Note that we may extend this definition to s∈ℤ (by duality) and to s∈ℝ (by interpolation) Finally, if Ω is proper we denote by W^s_ϕ[0](Ω) the closure of C^∞_0(Ω) in W^s_ϕ(Ω). 


	Since ϕ is smooth,  whenever Ω is bounded 
	the Sobolev norm (<ref>) is equivalent to the standard metric Sobolev norm (which is obtained by taking ϕ≡ 0 and gives rise to the standard Sobolev spaces W^s(Ω), etc.). In this case we have identifications W^s_ϕ(Ω)=W^s(Ω), etc.  
	

We impose a few working assumptions  (which are automatically satisfied when Ω is bounded) on our otherwise general setup. 


	
  * A.1
A Poincaré inequality holds for elements of W^1_ϕ[0](Ω): there exists C=C_Ω,ϕ>0 such that 

    ∫_Ω |∇ f|^2d vol_ϕ≥ C∫_Ω f^2 d vol_ϕ,    f∈ 
    W^1_ϕ[0](Ω),

which implies that the square root of the integral in the left-hand side defines a norm in W^1_ϕ[0](Ω) which is equivalent to _W^1_ϕ(Ω). 

  * A.2
The embedding W^1_ϕ(M)↪ L^2_ϕ(M) is compact. 


Taken together, these assumptions imply that the embedding W^1_ϕ[0](Ω)↪ L^2_ϕ(Ω) is compact for any Ω as above. Thus, by standard spectral theory, the (weak) formulation of the eigenvalue problem in (<ref>) admits a non-trivial solution (an eigenfunction) for λ varying in a discrete set {λ_l}_l=1^+∞ with λ_l→+∞ as l→+∞. Also, under these conditions it is known that the first eigenvalue λ=λ_1>0 is such that the corresponding eigenfunction u does not change sign throughout Ω, so that the corresponding eigenspace is simple (that is, one-dimensional). Hence, λ depends smoothly on smooth variations of Ω (for instance, variations of the type Ω_t=φ_t(Ω) as in Section <ref>). Finally, since both Σ and ϕ are assumed to be smooth, standard elliptic theory implies that u is smooth up to the boundary. Hence, if we normalize u such that (<ref>) is satisfied then λ is given by (<ref>);
we refer to the discussion in <cit.> and the references therein for details; see also Remark <ref> below.


The existence of a smallest eigenvalue λ for Δ_ϕ implies that the operator 

    -Δ_ϕ+β:W^1_ϕ[0](Ω)→ L^2_ϕ(Ω)
 is invertible if β>-λ, which justifies the existence of a minimizer for J_β,f in (<ref>) via a solution of (<ref>).


The assumptions above suffice to justify the computations leading to the variational formulae in Sections <ref> and <ref>. However, in order to establish finer properties of the underlying functional (such as the Morse index formula in Theorems <ref> and <ref>) one more requirement is needed.  


	
  * A.3 The standard “elliptic package” for non-homogeneous boundary problems of the type
	 
    {[ Δ_ϕ u=f       Ω;     u=ξ       Σ ].

	 holds true, including suitable trace/extensions theorems, the fact that the map
	 
    u∈ W^s+2_ϕ(Ω)↦ (Δ_ϕ u,u|_Σ)∈ W_ϕ^s(Ω)× W_ϕ^s+3/2(Σ),    s≥ 0,

	 is an isomorphism and the corresponding regularity theory (up to the boundary); see Remark <ref>.
 




	Poincaré inequality (<ref>) holds in the Gaussian setting of Example <ref> 
	with Ω being any proper domain with vol_ϕ_nΩ)<1. 
	Proofs may be found in <cit.> and the method of proof, which is based on Gaussian rearrangements, may be extended to weighted manifolds for which the validity of a certain relative isoperimetric inequality is taken for granted <cit.>.
	On the other hand,  the compactness of the embedding W^1_ϕ(M)↪ L^2_ϕ(M) seems to hold in a more general setting (if we further assume that vol_ϕ(M)<+∞). 
	For instance, it holds 
	for (ℝ^n,δ,e^-ϕd vol_δ), with ϕ(x)∼ |x|^θ, θ>1, as |x|→+∞ <cit.>. More generally (that is, in the presence of a Riemannian background (M,g)) it is well-known that this property follows from the validity of a certain logarithmic Sobolev inequality, which by its turn holds true whenever the corresponding weighted manifold satisfies Ric_ϕ≥ρ g, ρ>0,  where 
	
    Ric_ϕ= Ric_g+∇^2ϕ

		 is the Bakry-Émery Ricci tensor; see <cit.>, where this is discussed in the abstract framework of diffusion Markov triples satisfying the curvature-dimension condition CD(ρ,∞), ρ>0. 
	 



	If Ω is compact then the elliptic package mentioned above is classically known <cit.>. In the case (ℝ^n,δ, e^-ϕ), it is established in <cit.> under the assumption that Σ is smooth with a positive reach and under mild assumptions under ϕ which are too complicated to exactly reproduce here but which essentially boil down to assuming that ϕ(x) grows at least as |x|^2 as |x|→ +∞). In particular, this covers the Gaussian case in (<ref>). Keeping the same growth conditions on ϕ, we further mention that the methods in <cit.> may be adapted without much difficulty to the general case (M,g,e^-ϕ) if there exist a compact K⊂ M, r>0 and a diffeomorphism ψ:(M\ K,g)→(ℝ^n\ B_r(0⃗),δ) with uniformly bounded distortion (up to a sufficiently high order).  
	


The assumptions A.1, A.2 and A.3 above entail the fact that both Ω and Σ are sufficiently well behaved at infinity (both topologically and metrically) in order that the validity of a couple of technical ingredients further needed below is ensured. 
First, the following integration by parts formula holds:

    ∫_Ω fΔ_ϕ h  d vol_ϕ+∫_Ω⟨∇ f,∇ h⟩  d vol_ϕ=
    	∫_Σ f∂ h/∂νd area_ϕ,

	where f∈ W_ϕ^1(Ω)∩ C^∞(Ω) and h∈ W_ϕ^2(Ω)∩ C^∞(Ω); this follows from the corresponding divergence theorem
	
    ∫_Ω div_ϕ Y d vol_ϕ=∫_Σ⟨ Y,ν⟩ d area_ϕ,

where now Y is vector field on Ω lying in a suitably defined Sobolev space. 
We remark that this latter formula follows 	
	from the standard one (for bounded domains) by a simple approximation.
	In the same vein, Σ is is assumed to support its own weighted Sobolev scale W_ϕ^s(Σ), so that 
		the intrinsic analogue of (<ref>) holds:
		
    ∫_Σ fΔ_Σ,ϕ h  d area_ϕ=-∫_Σ⟨ D f, D h⟩  d area_ϕ,

			where now f and h are functions on Σ lying in a suitable Sobolev space (say, W^2_ϕ(Σ)),  D is the covariant derivative of g|_Σ and we use self-explanatory notation stemming from the fact that Σ itself is a weighted manifold with the induced structures. As above, this should follow from the analogue of (<ref>),
		
    ∫_Σ div_Σ,ϕ Y d vol_ϕ=0,

			where Y is tangent to Σ (and also lies in a suitable Sobolev space). Finally, we note that at some points of this paper, specifically in the proofs of Theorem <ref> and <ref> (which concern properties of domains in Gaussian space) we assume that Σ is such that certain test functions, which depend polynomially on |x|, lie in W^2_ϕ(Σ), so that the integration by parts formulae above may be applied. In this regard, polynomial volume growth at infinity is more than enough.  

	
Standing assumptions: Henceforth we will always take it for granted that (Ω,g,e^-ϕ) and Σ=∂Ω meet the assumptions A.1, A.2 and A.3 above; see also Remarks <ref>, <ref> and <ref>. 	
	

	
	
Another ingredient we will use is 
a simple adaptation to the weighted setting of certain variational formulae for bulk and surface integrals which are well-known in the area of shape optimization <cit.>. To state it we recall that the  weighted mean curvature of Σ is 

    H_ϕ= div_ϕν=H-⟨∇ϕ,ν⟩,
 	
	where H= div ν is the usual mean curvature. 


	For any variation t∈ (-ϵ, ϵ)↦Ω_t as above (not necessarily preserving volume) and any smooth family of smooth maps t∈ (-ϵ, ϵ)↦Ψ(t, ·): Ω_t→ℝ such that
	Ψ(t,·) and ∂Ψ/∂ t(t,·) lie in W^1_ϕ(Ω_t)∩ C^∞(Ω_t)  there hold
	
    d/dt∫_Ω_tΨ  d vol_ϕ=∫_Ω_t∂Ψ/∂ td vol_ϕ+
    		∫_Σ_tΨξ d area_ϕ,   Σ_t=∂Ω_t,

		and
		
    d/dt∫_Σ_tΨ  d area_ϕ=∫_Σ_t(
    				∂Ψ/∂ t+(⟨∇Ψ,ν⟩+H_ϕΨ)ξ)d area_ϕ,

			where ∇ here means the metric gradient with respect to the x variable. 
	
	
	
		The simple proof <cit.> in the Euclidean case (with Ω bounded) applies verbatim to the Riemannian setting. From this the weighted versions above (again with Ω bounded) may be easily derived by straightforward manipulations. The general case, in which Ω is only assume to have  a finite ϕ-volume, follows by a simple approximation (based on our standing assumptions).  
	
	
	
	If Ψ=1 in (<ref>) we obtain the first variation formula for the ϕ-volume:
	
    d/dt_|_t=0 vol_ϕ(Ω_t)=∫_Σξ d area_ϕ.

	In particular, if the variation is ϕ-volume-preserving then the variational function ξ satisfies  
	
    ∫_Σξ d area_ϕ=0.

	Conversely, it is well-known that for any compactly supported, smooth function ξ:Σ→ℝ satisfying (<ref>) there exists a ϕ-volume-preserving variation whose variational vector is ξν. For any such variation we may use (<ref>) with Ψ=ξ to check that 
	
    0=	d^2/dt^2_|_t=0 vol_ϕ(Ω_t)=
    			∫_Σ(ξ̇+(∂ξ/∂ν+H_ϕξ)ξ)d area_ϕ,
 
	where here and in the following a dot means d/dt|_t=0. 
	




§ THE VARIATIONAL FORMULAE FOR Λ
	
 
 We start by stating our version of the classical Hadamard formula for the first variation of λ; see also <cit.> and the references therein for other versions of this result in the weighted setting.
 
 
 If 
 Ω_t=φ_t(Ω) is a (not necessarily ϕ-volume-preserving) variation of domains as above and λ_t is the corresponding first eigenvalue then
 
    λ̇=-∫_Σ(∂ u/∂ν)^2ξ d area_ϕ.

 
 
 
	The corresponding first eigenfunction u_t satisfies
	
    {[ Δ_ϕ u_t+λ_t u_t=0               Ω_t;             u_t=0               Σ_t ].

so upon derivation we see that u̇ satisfies 
  
    {[ Δu̇+λ̇ u+λu̇=0              Ω;    u̇=-∂ u/∂νξ              Σ ].

We now observe that, due to our normalization (<ref>),

    1=∫_Ω_tu_t^2d vol_ϕ,

so if we use (<ref>) with Ψ=u_t^2 we get 

    0=2∫_Ω uu̇ d vol_ϕ+∫_Σ u^2ξ d area_ϕ,

 and since u=0 on Σ,
 
    ∫_Ω uu̇ d vol_ϕ=0.

We also have, this time using the normalization (<ref>),  

    λ_t=∫_Ω_t|∇ u_t|^2d vol_ϕ,

which by (<ref>) with Ψ=|∇ u_t|^2 gives   

    λ̇
     = 2
    ∫_Ω⟨∇ u,∇u̇⟩ d vol_ϕ+
    ∫_Σ|∇ u|^2ξ d area_ϕ.

By means of (<ref>) we can handle the first integral in the right-hand side above in two slightly different ways. We first have 

    ∫_Ω⟨∇ u,∇u̇⟩ d vol_ϕ    =     
     -∫_Ω uΔ_ϕu̇ d vol_ϕ+∫_Σ u∂u̇/∂νd area_ϕ
       (<ref>)=   ∫_Ω u(λ̇ u+λu̇) d vol_ϕ,

so that (<ref>) gives

    ∫_Ω⟨∇ u,∇u̇⟩ d vol_ϕ=λ̇.

On the other hand, 	

    ∫_Ω⟨∇ u,∇u̇⟩ d vol_ϕ    =     
    	-∫_Ωu̇Δ_ϕ u  d vol_ϕ+∫_Σu̇∂ u/∂ν d area_ϕ
       (<ref>)+(<ref>)=     
    	-∫_Ωu̇(-λ u) d vol_ϕ+
    	∫_Σ(-∂ u/∂νξ)∂ u/∂νd area_ϕ,

so that, again using (<ref>),  

    ∫_Ω⟨∇ u,∇u̇⟩ d vol_ϕ=-∫_Σ(∂ u/∂ν)^2ξ  d area_ϕ,

and the result follows from this and 
(<ref>).


We now explore (<ref>) by recalling a definition first put forward in <cit.>.
	
	
	Let Σ'⊂Σ be an open subset. We say that (Ω,g) is Σ'-critical (for λ)  if λ̇=0 for any ϕ-volume-preserving deformation such that supp ξ⊂Σ'.
	
	





	
	
		That Ω is Σ'-critical is equivalent to the first eigenfunction u satisfying
	
    {[ Δ_ϕ u+λ u= 0            Ω;       s_ξ= ξ            Σ;      |∇ u|=κ           Σ' ].
	
	for some constant κ>0. In particular, if Σ'=Σ and Ω lies in a simply connected space form 𝒮^n_K with sectional curvature K∈ℝ, results in <cit.> imply that Ω is a geodesic ball (where Ω is require to lie in a hemisphere in the spherical case K>0). On the other hand, it is clear that any annular domain Ω⊂𝒮^n_K enclosed by geodesic spheres (so that ∂Ω=𝕊^n-1_r_1(x_1)∪𝕊^n-1_r_2(x_1), the r_i>0 and x_i∈𝒮^n_K, i=1,2) is 𝕊^n-1_r_i(x_i)-critical for i=1,2. Thus, the localization in Definition <ref> allows for some more flexibility in the somewhat arduous task of finding non-trivial, explicit solutions to the over-determined system (<ref>).
		
		
	
		If Ω is optimal for λ then it is Σ-critical and (<ref>) means that the “optimal” eigenfunction u satisfies an extra condition along Σ (its normal derivative is constant). Of course, this aligns with the comments in Remark <ref>. 
			

	We now proceed to compute λ̈ assuming that (Ω,g) is Σ'-critical. 
	We first notice that under Σ'-criticality any such ξ admits a natural extension towards Ω. Indeed, observing that u̇=0 outside supp ξ, it follows from (<ref>) that s_ξ:=-(∂ u/∂ν)^-1u̇ satisfies

    {[    Δ_ϕ s_ξ+λ s_ξ= 0                   Ω;              s_ξ= ξ                   Σ; ∫_Ω s_ξ u d vol_ϕ=0 ].
  
In particular, this allows us to define ∂ξ/∂ν:=∂ s_ξ/∂ν along Σ. 
We may now state our version of the second variation formula for λ in the weighted setting; see Remark <ref> for other approaches to this calculation in the purely Riemannian setting. 


	If Ω is Σ'-critical (for λ) then 

    λ̈ = 2κ^2Q_ϕ(ξ),

where the quadratic form Q_ϕ is given by 

    Q_ϕ(ξ)=∫_Σ'ξℒ_ϕξ  d area_ϕ,   ℒ_ϕξ=∂ξ/∂ν+H_ϕξ.
	
	


An useful information here is obtained by first noticing that, along Σ', where |∇ u|=|∂ u/∂ν|=κ,

    0  =  -λ u =Δ_ϕ u=Δ_gu-∂ u/∂ν⟨∇ϕ,ν⟩,

where ∇^2 is the metric Hessian operator.
Since

    Δ_gu=Δ_Σ u+∂ u/∂ν H+(∇^2u)(ν,ν)=∂ u/∂ν H+(∇^2u)(ν,ν)

and Δ_Σ u=0, where Δ_Σ is the intrinsic Laplacian of Σ, we conclude that 

    (∇^2u)(ν,ν)=-∂ u/∂νH_ϕ.

	
With these preliminaries at hand we  note that, by (<ref>),

    dλ/dt=-∫_Σ'_t⟨∇ u_t,ν_t⟩^2ξ_t d area_ϕ,    Σ'_t=φ(Σ'),

so that (<ref>) gives 

    -λ̈=∫_Σ'(Ψ̇+(⟨∇Ψ,ν⟩+Ψ H_ϕ)ξ)d area_ϕ,   Ψ=⟨∇ u_t,ν_t⟩^2ξ_t.

	We now compute the various terms in the right-hand side above. First, 
	
    Ψ̇= 
    		2 ⟨∇ u,ν⟩(⟨∇u̇,ν⟩+⟨∇ u ,ν̇⟩)ξ
    		+⟨∇ u,ν⟩^2ξ̇.

		Now recall that u̇=-∂ u/∂ν  s_ξ and 
 |∇ u|=κ along Σ'. Also, without loss of generality we may assume that v is normal (v=ξν), which gives  ν̇=-Dξ <cit.>.
Hence,

    Ψ̇=-2κ^2ξ∂ξ/∂ν+κ^2ξ̇.
	
On the other hand, again using Σ'-criticality,

    ⟨∇Ψ,ν⟩+Ψ H_ϕ=ν(⟨∇ u,ν⟩^2)ξ+
    κ^2(∂ξ/∂ν+H_ϕξ).

But

    ν(⟨∇ u,ν⟩^2)=2∂ u/∂ν(⟨∇_ν∇ u,ν⟩+⟨∇ u,∇_νν⟩)

and since 

    ⟨∇ u,∇_νν⟩=∂ u/∂ν⟨ν,∇_νν⟩=0

we find that 

    ν(⟨∇ u,ν⟩^2)=2∂ u/∂ν⟨∇_ν∇ u,ν⟩(<ref>)=-2κ^2H_ϕ.

Thus, if we put together the pieces of our computation we get

    λ̈ =2κ^2∫_Σ'ξ(∂ξ/∂ν+H_ϕξ)d area_ϕ -κ^2∫_Σ(ξ̇+(∂ξ/∂ν+H_ϕξ)ξ)d area_ϕ,

so the proof is completed using (<ref>).








§ A MORSE INDEX FORMULA


The computations leading to (<ref>)-(<ref>) justify the following definitions, which are direct extensions of concepts already appearing in <cit.>. Given Σ⊂Σ, we denote by W^1_ϕ[0](Σ) the space of functions ξ∈W^1_ϕ[0](Σ) such that ∫_Σξ d area_ϕ=0.


	If Ω⊂ M is Σ'-critical (for λ) with Σ'⊂Σ, then we say that it is  Σ'-stable if Q_ϕ(ξ)≥ 0 for any ξ∈W^1/2_ϕ[0](Σ'). In case Q_ϕ|_W^1/2_ϕ[0](Σ') is positive definite then we say that Ω it is strictly Σ'-stable. If Σ'=Σ then we simply say that Ω is stable (or strictly stable). 
 


If Ω is Σ'-critical (with Σ'⊂Σ) and Σ”⊂Σ', we define  the nullity and the index  of the pair (Ω,Σ”) by 

    null(Ω,Σ”)={ξ∈W^1/2_ϕ[0](Σ”)); Q_ϕ(ξ)=0 },
	
and 

    ind(Ω,Σ”)=sup{ V; V⊂W^1/2_ϕ[0](Σ”)   and  Q_ϕ|_V   is  positive  definite},

respectively. 




	It  has been proved in <cit.> (in the classical Euclidean setting) that any Σ'-critical domain is locally strictly stable in the sense that for any x_0∈Σ' there exists a small neighborhood U⊂Σ' with x_0∈ U such that Ω is strictly U-stable. In <cit.> this result has been reformulated in a more conceptual framework (again in the classical case) by means of the establishment of a Morse index formula for the quadratic form Q_ϕ. We will check below that this latter result may be extended to our more general setting  with essentially the same proof (Theorem <ref>).



	The stability (or lack thereof) of (purely) Riemannian domains has been recently studied from a global viewpoint (i.e with Σ'=Σ) in <cit.>, notably in the two-dimensional case (n=2).  In this regard, 
	it follows from Faber-Krahn inequality <cit.> that a geodesic ball B_r(x)⊂𝒮^n_K in a simply connected space form is the only (volume-preserving) minimizer for λ. In particular, it is not only 𝕊^n-1_r(x)-critical but also strictly 𝕊^n-1_r(x)-stable, where 𝕊^n-1_r(x)=∂ B_r(x) is the associated geodesic sphere. An interesting problem, first put forward in  <cit.>, is to check whether in the spherical case K>0 the converse of this statement (namely, if Ω⊂ℍ_K^n is Σ-stable then Ω is a geodesic ball) holds true. Despite an affirmative answer in <cit.> for n=2, this problem remains wide open if n≥ 3, except for a further contribution in <cit.>, where it is shown that the domain is a hemisphere in case its boundary is minimal. 
	
	
	
		A Faber-Krahn inequality also holds in the Gaussian setting, with the optimal domains being the Gaussian half-spaces
		
    ℋ^n_ u,d={
    			x∈ℝ^n;⟨ x, u⟩_δ>d
    			}.

		where u∈ℝ^n, u=1, and d∈ℝ  <cit.>. In particular, one may ask whether these are the only stable domains for λ in Gaussian space. In fact, Poincaré's limit theorem (cf. Section <ref>) suggests that this should hold true if and only if the corresponding conjectured property for 𝒮^n_K mentioned in Remark <ref> holds as well (at least for all n large enough). 
			

We now turn to the numerical invariants appearing in Definition <ref>. At first sight, it not clear how the index/nullity relate to an algebraic count of negative/null “eigenvalues” of the quadratic form Q_ϕ, which is defined in terms of the manifestly non-local operator ℒ_ϕ. In particular, it is not even clear that these invariants are finite if Σ' is compact. However, as already mentioned in Remark <ref>, standard results in elliptic theory ensured by our standing assumption (specially A.3) may be used to settle this problem. A first ingredient here is the following Gärding inequality satisfied by Q_ϕ. 


	There exist constants C_1,C_2>0 (depending only on Ω) such that, 	for any ξ∈ W^1/2(Σ), 
	
    Q_ϕ(ξ)≥ C_1ξ^2_W^1/2(Σ)-C_2ξ^2_W^0(Σ),

where W^s(Σ), s∈ℝ, is the standard Sobolev scale of Σ. 
	  
	
	
	We follow the proof of <cit.> closely (see also <cit.> for a slightly different argument). Let us 
	consider the isomorphism 
		
    ξ∈ W^s_ϕ(Σ)↦ h_ξ∈ W_ϕ^s+1/2(Ω)

	that to each ξ associates its ϕ-harmonic extension (that is, Δ_ϕ h_ξ=0 on Ω); cf. (<ref>). 
	Also,
	let us set w_ξ=s_ξ-h_ξ, where s_ξ∈ W_ϕ^s+1/2(Ω) is given by (<ref>). Since Δ_ϕ w_ξ=-λs_ξ, elliptic regularity implies that w_ξ∈  W_ϕ^s+5/2(Ω). Hence, standard trace theory gives that ∂ξ/∂ν=∂ w_ξ/∂ν∈ W_ϕ^s+1(Γ), so that the operator ξ∈ W_ϕ^s(Σ)↦ℒ_ϕξ∈ W_ϕ^s+1(Σ) has order -1. Thus, if 
	
    𝒬_ϕ(ξ)=∫_Σξ∂𝔰_ξ/∂νd vol_ϕ

	for ξ∈ W_ϕ^1/2(Σ)⊂ W_ϕ^0(Σ),
	we may apply this with s=0 to obtain
	
    |Q_ϕ(ξ)-𝒬_ϕ(ξ)|≤ C_3f^2_W_ϕ^0(Σ),    C_2>0,

	from which we see that 
	
    Q_ϕ(ξ)≥𝒬_ϕ(ξ)-C_3ξ^2_W^0(Σ).

	On the other hand, by (<ref>) and (<ref>),
	
    𝒬_ϕ(ξ)
    		    =    ∫_Ω|∇ h_ξ|^2 d vol_ϕ
        =    h_ξ^2_W^1_ϕ(Ω)- h_ξ^2_W^0_ϕ(Ω)
       ≥    C_1ξ^2_W_ϕ^1/2(Σ) - C_4ξ^2_W_ϕ^-1/2(Σ),

	where C_1, C_4>0 and we used (<ref>) with s=1/2 and s=0 in the last step. Now, the Sobolev embedding W_ϕ^0(Σ)↪ W_ϕ^-1/2(Σ) gives 
	
    ξ_W_ϕ^-1/2(Σ)≤ C_5ξ_W_ϕ^0(Σ),    C_5>0,

	so we obtain (<ref>) with C_2=C_3+C_4C_5^2. 
	
	
	We may now state the weighted version of the Morse index formula proved in <cit.>.
	
	
		Let Σ_1⊂Σ be such that
		Ω is Σ_1-critical with Σ_1 compact. Then ind(Σ_1)<+∞. Moreover, if Σ_0⊂Σ_1 
		 and there exists a smooth deformation Σ_t⊂Σ, 0≤ t≤ 1, connecting Σ_0 to Σ_1
		 then there holds 
		 
    ind(Ω,Σ_1)- ind(Ω,Σ_0)=∑_0<t<1 null(Ω,Σ_t).

		 		
		 		
	 <cit.> If Ω is Σ'-critical and x∈Σ' then there exits U⊂Σ' with x∈ U such that Ω is strictly U-stable.		
		
		
		 (of Theorem <ref>) We resort to the abstract strategy in <cit.>, so we only need to check the validity of two assertions regarding ℒ_ϕ and Q_ϕ:
			
				
  * Q_ϕ satisfies a Gärding inequality;
				
  * ℒ_ϕ satisfies the unique continuation property (UCP) : if ξ meets ℒ_ϕξ=λξ, λ∈ℝ, and ξ≡ 0 in some neighborhood U⊂Σ_1 then ξ≡ 0 on Σ. 
			
			Now, the appropriate Gärding inequality has been established in Proposition <ref>. As for UCP, notice that ∂ξ/∂ν= on U. Since ℳ_ϕ=Δ_ϕ+λ is elliptic, its principal symbol never vanishes and hence U is non-characteristic for ℳ_ϕ. Thus, by  Hölmgren's uniqueness, s_ξ vanishes in a neighborhood of U (inside Ω). But ℳ_ϕ is known to satisfy its own version of the UCP, which implies that s_ξ≡ 0 and hence ξ≡ 0  on Σ, as desired. This completes the proof. 
			 
			
			
				 Among the many interesting (potential) applications of Theorem <ref> we mention:
				
					
  * the annular domains in Remark <ref> are strictly U-stable if U⊂𝕊^n-1_r_i(x_i) is small enough;
					
  * any of the many critical domains constructed more recently by perturbative methods (see <cit.> and the references therein) are locally strictly stable;
					
  * very likely, the perturbative results mentioned in the previous item should admit counterparts in the weighted setting, so that local stability of critical domains should extend to this case as well;
					
  * the nullity/index can now be identified to an algebraic count of negative/null eigenvalues of the associated quadratic form, which opens up the possibility of effective computations of these invariants in examples.  
				 
					





§ THE VARIATONAL FORMULAE FOR J_Β,Γ


We now turn to the Dirichlet energy ℰ_β,f in Example <ref>. We start by exhibiting an important example where minimizing the corresponding Dirichlet integral J_β,f makes sense. 


	In the Gaussian setting (<ref>), 
	the half-spaces ℋ^n_ u,0 defined in (<ref>) satisfy  
 λ(ℋ^n_ u,0)=1 (with the corresponding eigenspace being generated by u(x)=⟨ x, u⟩_δ). Thus, 
	if Ωℋ^n_ u,0 then λ(Ω)> 1 by eigenvalue  monotonicity, so that (<ref>) has a solution for each β≥ -1 by Remark <ref>, and under this condition it makes sense to minimize ℰ_β,f(Ω), where Ω∈𝒟_V, V= vol_ϕ_n(ℋ^n_ u,d), d>0. 
		
	
As already observed, the general problem of characterizing optimal domains for ℰ_β,f, even in the rather special case of Example <ref>, seem to lie beyond the current technology.	
Nevertheless, we may try to use the variational methods discussed above to have a first idea on the properties an optimal domain should satisfy, at least in the case f≡γ, a real constant. 
In fact, as a consequence of the first variation formula (<ref>) for J_β,γ, we next check that optimal domains for ℰ_β,γ satisfy the over-determined elliptic system (<ref>) below. 



	If Ω is optimal in the sense ℰ_V,β,γ=ℰ_β,γ(Ω) then the corresponding “optimal” function w (that is, ℰ_β,γ(Ω)=J_c,γ(w)) satisfies the over-determined system
	
    {[ -Δ_ϕ w+β w=γ            Ω;          w=0            Σ;     |∇ w|= c            Σ ].
 
	for some c> 0. 
	 
	
	
		If V= vol_ϕ(Ω_0) and t∈(-ϵ,ϵ)↦Ω_t∈𝒟_V is a smooth ϕ-volume-preserving variation of Ω with Ω_0=Ω then optimality of Ω implies that J_β,γ(w)≤ J_β,γ(w_t), where w_t is the solution of (<ref>) with Ω=Ω_t (and f=γ). Hence, J̇_̇β̇,̇γ̇=0, where as always the dot means derivative at t=0. We now compute this derivative using the formalism stemming from Proposition <ref>. From (<ref>),
		
    J_β,γ(w_t)=∫_Ω_tΨ_t d vol_ϕ,   Ψ_t=1/2|∇ w_t|^2+β/2w_t^2-γ w_t,
 
		so that, by (<ref>),
		
    J̇_̇β̇,̇γ̇
    			 =  ∫_Ω(⟨∇ w_0,∇ẇ⟩+β wẇ-γẇ)d vol_ϕ+1/2∫_Σ |∇ w|^2η  d area_η,

		where η is the variational function and ẇ satisfies 
		
    {[ -Δ_ϕẇ+βẇ=0          Ω; ẇ=-∂ w/∂νη          Σ ].

		Now, by (<ref>), 
		
    ∫_Ω⟨∇ w,∇ẇ⟩ d vol_ϕ    =     
    			-∫_ΩẇΔ_ϕ w  d vol_ϕ+∫_Σẇ∂ w_0/∂ν d area_ϕ
       =     
    			-∫_Ωẇ(β w-γ) d vol_ϕ+
    			∫_Σ(-∂ w/∂νη)∂ w/∂νd area_ϕ,

		and leading this to (<ref>) we see that 
		
    J̇_β,γ=-1/2∫_Σ(∂ w/∂ν)^2η  d area_ϕ.

		The result then follows (with c=|∂ w/∂ν|) because we already know that this derivative vanishes for any η such that ∫_Ση  d area_ϕ=0. 
	  
	
	As in Section <ref>, this proposition justifies a notion of Σ-criticality for J_β,γ, so that optimal domains for ℰ_β,γ are automatically Σ-critical and hence satisfy (<ref>). Naturally, the question remains of classifying such domains in each specific case. The inherent difficulty in approaching this problem is best illustrated by certain examples in Gaussian space taken from <cit.>.  
	
	
		In Gaussian space of Example <ref>, the following domains are easily verified to be Σ-critical for J_-1,1 (equivalently, they support a function w satisfying (<ref>) for suitable values of β):
		
			
  * round balls centered at the origin;
			
  * the complements of the balls in the previous item;
			
  * the slabs {x∈ℝ^n;|⟨ x, u⟩|<ε}, where ε>0 and | u|=1; 
			
  * the half-spaces ℋ^n_ u,d in (<ref>); in this case the explicity solution to (<ref>) for d≥ 0 is w(x)=β x_1-1 with β d=1.
		
		
	
	Note the the set of all domains in each class of examples above exhausts the possible values for the ϕ_n-volume of a proper domain in Gaussian space, which indicates that the problem of deciding which  Σ-critical domains are optimal is far from trivial given that at least four candidates concur for each value of the volume. As a first step toward approaching this difficulty we may look at how J_β,γ varies to second order around such a domain. Thus, we proceed by computing J̈_β,γ along variations passing through the given Σ-critical domain.      

We first observe that, as in Section <ref>, criticality here also implies that any η∈ H^1/2(Σ) admits a natural extension to Ω. Indeed, it follows from (<ref>) that 𝔯_η:=-(∂ w/∂ν)^-1ẇ satisfies 
	
    {[ -Δ_ϕ𝔯_η +β𝔯_η=0               Ω;          𝔯_η= η               Σ ].
  	
which allows us to set ∂η/∂ν:=∂𝔯_η/∂ν along Σ.


	If Ω is Σ-critical for J_β,γ with β≠ 0 then 
		
    J̈_β,γ=c^2
    		𝒬_β,λ(η),    c=|∂ w/∂ν|,

	where the quadratic form 𝒬_β,γ is given by
	
    𝒬_β,γ(η)=∫_Σηℒ_β,γη d area_ϕ,   ℒ_β,γη=∂η/∂ν+H_ϕη.

	 	
	
	
		This uses essentially the same argument as in the previous computation of λ̈ leading to (<ref>). Indeed, 
	it follows from (<ref>) and (<ref>) that 
	
    -J̈_β,γ=1/2∫_Σ(Ψ̇+(⟨∇Ψ,ν⟩+Ψ H_ϕ)η)d area_ϕ,   Ψ=⟨∇ w_t,ν_t⟩^2η_t.

	As before, we may assume that the variational vector field is normal along Σ, which immediately gives  
	
    Ψ̇=-2c^2ξ∂ξ/∂ν+c^2ξ̇.
	
	On the other hand, 
	the only novelty in the computation of the remaining term inside the integral in (<ref>) is that instead of (<ref>) we now have 
	
    (∇^2 w)(ν,ν)=-∂ w/∂νH_ϕ-γ,
 
	so we end up with 
	
    J̈_β,γ    =     
    	c^2∫_Ση(∂η/∂ν+H_ϕη)d area_ϕ
    	-γ/2∂ w/∂ν∫_Σ_0η d area_ϕ
             
    	 -c^2/2∫_Σ(η̇+(∂η/∂ν+H_ϕη)η)d area_ϕ,

	which reduces to (<ref>) if we recall that the variation is ϕ-volume-preserving and use (<ref>) and (<ref>) with ξ=η. 
	

We stress the formal similarity between (<ref>)-(<ref>) and (<ref>)-(<ref>), the only essential difference being in the way the variational functions extend to Ω in each case. In particular, we can define here the notions corresponding  to those in Definitions <ref> and <ref> above, so that the same argument as in the proof of Theorem <ref> yields the following result. 


	The appropriate Morse index formula, similar to (<ref>), holds in the present setting. In particular, any Σ'-critical domain is locally strictly stable. 
  


As a consequence of Theorem <ref>, all the Σ-critical domains (for J_-1,1) in Example <ref> are locally strictly stable, thus being variationally indistinguishable from this viewpoint.  




§ A POHOZHAEV IDENTITY AND OPTIMAL DOMAINS FOR ℰ_-1,1 IN GAUSSIAN HALF-SPACE


As already observed in the Introduction and confirmed by Remark <ref>, in general (infinitesimal) variational methods by themselves do not seem to provide effective tools for the classification of  
optimal domains for Dirichlet integrals. Thus, we henceforth seek for alternate routes by resorting to global methods relying on suitable differential/integral identities.    
We first illustrate this approach by means of the next result, which completely classifies optimal domains for ℰ_-1,1 in the Gaussian half-space ℋ^n_ u,0. It constitutes the exact analogue of a result in the spherical case <cit.> and we refer to Remark <ref> for details on the heuristics behind this analogy, which relies on a naive application of Poincaré's limit theorem (Section <ref>).


	If Ωℋ^n_ u,0 be a Σ-critical domain for J_-1,1 in Gaussian space then Ω=ℋ^n_ u,d, d>0. 
	
	
	
		If Ωℋ^n_ u,0 is optimal for ℰ_-1,1 then Ω=ℋ^n_ u,d, d>0.
			

A key ingredient in our proof of Theorem <ref> is the following Pohozhaev-type identity in Gaussian space. 

  
	If w:ℝ^n→ℝ is a C^2 function and X is parallel (in the sense that ∇ X≡ 0, where ∇ is the covariant derivative induced by the flat metric δ) then 
	
    div_ϕ_n(|∇ w|^2/2X-⟨ X,∇ w⟩∇ w)=|∇ w|^2/2 div_ϕ_n X-⟨ X,∇ w⟩Δ_ϕ_n w.

	


	We have 
	
    div_ϕ_n(|∇ w|^2/2X)=|∇ w|^2/2 div_ϕ_n X +
    	⟨∇(|∇ w|^2/2),X⟩

	and 
	
    div_ϕ_n(⟨ X,∇ w⟩∇ w)=⟨ X,∇ w⟩ div_ϕ_n∇ w+
    		⟨∇⟨ X,∇ w⟩,∇ w⟩,

	and since 
	
    ⟨∇(|∇ w|^2/2),X⟩=
    		⟨∇⟨ X,∇ w⟩,∇ w⟩

	because X is parallel, the result follows. 


We now observe that, by Proposition <ref>, any Ω satisfying the conditions of Theorem <ref> supports a function w such that 
	
    {[ -Δ_ϕ_n w-w=1            Ω;          w=0            Σ;     |∇ w| =c            Σ ].
 
for some c>0. This leads to our next preparatory result.


	If w:Ω→ℝ satisfies (<ref>) then 
	
    Δ_ϕ_n(|∇ w|^2)≥ 0,

	with the equality occurring if and only if ∇^2w≡ 0. 
	


	In the Gaussian case, the weighted Bochner-Weitzenböck identity says that 
	
    1/2Δ_ϕ_n(|∇ w|^2)=|∇^2w|^2+⟨∇ w,∇(Δ_ϕ_n w)⟩+|∇ w|^2;

	see <cit.>. 
	It follows that 
	
    1/2Δ_ϕ_n(|∇ w|^2)
    		   ≥   ⟨∇ w,∇(-w-1)⟩ +|∇ w|^2
        =     0,

	with the equality occurring if and only if |∇^2w|=0.


The final ingredient in the proof of Theorem <ref> is the next rigidity result for solutions of (<ref>) in Ωℋ^n_ u,0, in whose proof the Pohozhaev in Proposition <ref> plays a key role. 


	If w satisfies (<ref>) and Ωℋ^n_ u,0 then |∇ w|=c everywhere on Ω. 
	
	
	
		By rotational invariance we may assume that u=e_1=(1,0,⋯,0) so that x_1≥ 0 on Ω. By (<ref>), the maximum principle and the fact that |∇ w|=c along Σ we see that |∇ w|≤ c on Ω, which immediately gives 
		
    c^2∫_Ω x_1 d vol_ϕ_n > ∫_Ω x_1|∇ w|^2 d vol_ϕ_n

		unless there already holds |∇ w|=c everywhere on Ω. 
		However, 
		
    x_1|∇ w|^2
    		    =     div_ϕ_n (x_1 w|∇ w|)-w⟨∇ x_1,∇ w⟩-x_1wΔ_ϕ w
        =     div_ϕ_n (x_1 w|∇ w|)-w∂ w/∂ x_1+x_1w(w+1),

		so if we take this to (<ref>) we obtain
		
    c^2∫_Ω x_1 d vol_ϕ_n > 	
    		-∫_Ω w∂ w/∂ x_1d vol_ϕ_n+
    		∫_Ω x_1w(w+1)d vol_ϕ_n.

		We now check that this contradicts the Pohozhaev-type identity (<ref>) for X=-e_1, for which there holds div_ϕ_n X=x_1. Indeed, integrating the right-hand side of (<ref>) over Ω with this choice and using the divergence theorem twice we get 
		
    ∫_Σ(c^2/2⟨ X,ν⟩-⟨ X,cν⟩⟨ cν,ν⟩)d area_ϕ_n    =     
    		-c^2/2∫_Σ⟨ X,ν⟩ d area_ϕ_n
        =     -c^2/2∫_Ω div_ϕ_n X d vol_ϕ_n
        =     -c^2/2∫_Ω x_1 d vol_ϕ_n.
	
		On the other hand, the right-hand side of (<ref>) gives 
		
    1/2∫_Ω x_ 1|∇ w|^2 d vol_ϕ_n-∫_Ω⟨-e_1,∇ w⟩ (-w-1)d vol_ϕ_n    =    1/2∫_Ω x_ 1|∇ w|^2 d vol_ϕ_n
             -∫_Ω⟨ (w+1)e_1,∇ w⟩ d vol_ϕ_n.

		Now, the last integral may be manipulated as follows. We have
		
    ⟨ (w+1)e_1,∇ w⟩    =     div_ϕ_n (w(w+1)e_1)-w div_ϕ_n((w+1)e_1) 	
        =     div_ϕ_n (w(w+1)e_1)-w(w+1) div_ϕ_n e_1-w⟨∇ w,e_1⟩
        =     div_ϕ_n (w(w+1)e_1)+x_1w(w+1)-w∂ w/∂ x_1,

		so that 
		
    ∫_Ω⟨ (w+1)e_1,∇ w⟩ d vol_ϕ_n=∫_Ω x_1w(w+1) d vol_ϕ_n-
    		∫_Ω w∂ w/∂ x_1  d vol_ϕ_n,

		and we see 
		that altogether (<ref>) gives
		
    c^2∫_Σ x_1 d area_ϕ_n=
    		-∫_Ω x_ 1|∇ w|^2 d vol_ϕ_n+2∫_Ω x_1w(w+1) d vol_ϕ_n-2∫_Ω w∂ w/∂ x_1  d vol_ϕ_n.

		Comparing this with (<ref>) we conclude that 
		
    ∫_Ω x_1w(w+1) d vol_ϕ_n-∫_Ω w∂ w/∂ x_1  d vol_ϕ_n
    		-∫_Ω x_ 1|∇ w|^2 d vol_ϕ_n>0.

		However, 
		
    ∫_Ω x_1w(w+1) d vol_ϕ_n    =     -∫_Ω x_1wΔ_ϕ_n w d vol_ϕ_n
        =    ∫_Ω⟨∇ (x_1w),∇ w⟩ d vol_ϕ_n
        =    ∫_Ω(w∂ w/∂ x_1+x_1|∇ w|^2) d vol_ϕ_n,

		so we end up with 0>0, a contradiction which completes the proof. 
	
	
	 (of Theorem <ref>) By Proposition <ref> there exists w:Ω→ℝ satisfying (<ref>), so Proposition <ref> applies to ensure that |∇ w|=c everywhere on Ω. In particular, Δ_ϕ_n(|∇ w|^2)=0 and the equality holds in (<ref>). Thus, by Proposition <ref>, ∇^2w≡ 0 and w is an affine function. It follows that Σ=∂Ω lies in a hyperplane (because w vanishes there) and since 
	Ωℋ^n_ u,0, the result follows.
	
	
	
		Our proof of Theorem <ref> is modeled on <cit.>, where solutions of the over-determined system 
			
    {[ -Δ w-(k-1)Kw=1              Ω;            w=0              Σ;       |∇ w| =c              Σ ].

		are shown to be quite rigid.
		Here, Ω⊂𝒮_K,+^k-1, where 𝒮_K,+^k-1 is a round hemisphere of dimensional k-1 and radius 1/√(K), K>0. In particular,  it is proved that Ω is a geodesic ball. Now, if we take K=1/k, so that 𝒮_1/k,+^k-1 is a hemisphere of radius √(k), and take the limit as k→ +∞ then (<ref>) formally converges to the problem 
			
    {[ -Δ_∞ w-w=1          Ω;        w=0          Σ;   |∇ w| =c          Σ ].

			Besides precisely determining the nature of the elusive “Laplacian” Δ_∞, the question remains of realizing in which space this limiting problem should be treated. From an entirely heuristic viewpoint, we may approach this by making use of the Poincaré's limit theorem (see Section  <ref> below), which may be loosely interpreted as saying that the push-forward of the weighted manifold (𝒮_1/k^k-1,δ_ sph_k,e^-ϕ_(k)d vol_δ_ sph_k) under the orthogonal projection Π_k,n:ℝ^k→ℝ^n converges in a suitable sense to the Gaussian space (ℝ^n,δ,e^-ϕ_n) as k→+∞. Here, δ_ sph_k is the standard spherical metric in 𝒮_1/k^k-1 and ϕ_(k)>0 is chosen such that vol_δ_ sph_k(𝒮_1/k^k-1)=e^ϕ_(k), so that ℙ_k:=e^-ϕ_(k)d vol_δ_ sph_k-1 defines a probability measure. This clearly suggests that, as we did above, the limiting problem (<ref>) should be treated in Gaussian half-space ℋ^n_ u,0=(ℝ^n_+,δ,e^-ϕ_n), with the proviso that  Δ_∞ must be replaced by the corresponding weighted Laplacian Δ_ϕ_n. With the right over-determined system at hand, it remains to transplant to our setting the methods from <cit.>, which rely on a Pohozhaev-type identity verified in <cit.>. As it is well-known, this kind of identity is a manifestation of the existence of certain conformal fields on spheres which may be intrinsically characterized as gradient vector fields of functions lying in the first eigenspace of the metric Laplacian. Since spectral data behave quite well under Poincaré's limit <cit.>, we were naturally led to formulate Proposition <ref> in terms of parallel vector fields in Gaussian space, since they can be written as ∇ψ with ψ satisfying Δ_ϕ_nψ+ψ=0.    
		

	
	
	 
	

§ A REILLY FORMULA AND AN ALEXANDROV-TYPE THEOREM IN GAUSSIAN HALF-SPACE

	
	
	Inspired by Theorem <ref>, and taking into account the heuristics behind it explained in Remark <ref>, we are led to speculate on possible extensions of other classical results to the Gaussian setting by similar methods. The next theorem confirms this expectation and aligns itself with the well-known fact that Serrin's uniqueness result (or rather its reformulation by Weinberger <cit.>) may be used to provide an alternate proof of the classical Alexandrov's soap bubble theorem (cf. the comments in <cit.>).     
	
	
		Let Σ⊂ℋ^n_ u,0 be a smooth embedded hypersurface whose weighted mean curvature is (a non-zero) constant. Then Σ=∂ℋ^n_ u,d for some d≥ 0. 
		
		
		
			There are no compact embedded hypersurfaces with (non-zero) constant weighted mean curvature in ℋ^n_ u,0.
		

	We start the proof  of this result by recalling that if B is a bi-linear symmetric form on vectors on Ω, its ϕ-divergence is given by
	
    div_ϕ B=e^ϕ∘ div  B∘ e^-ϕ= div B- i_∇ϕB,

	where div B_j=∇^iB_ij is the metric divergence and  i_Y means  contraction by a vector field  Y. 
	
	
	Under the conditions above, if h is a smooth function on Ω then
	
    div_ϕ(h  i_YB) 
    		 =  B(∇ h,Y) +h i_Y div_ϕ B+h/2⟨ B,ℒ_Yg⟩,

	where ℒ denotes Lie derivative. 
	
	
	
		We have
		
    div(h i_YB)
    			   =    ∇^i(hB_ijY^j)
        =     B_ij∇^i h X^j+h∇^iB_ij Y^j+hB_ij∇^iY^j
        =     B(∇ h,Y) + h i_Y div B + 
    			h/2⟨ℒ_Yg,B⟩,

		and using (<ref>) twice,
		
    div_ϕ(h i_YB)
    		   =     
    		B(∇ h,Y) + h i_Y divB + 
    		h/2⟨ℒ_Yg,B⟩ -h i_∇ϕ( i_YB)⟩
        =     
    			B(∇ h,Y)+h i_Y( div_ϕ B+ i_∇ϕB)
    			+ 
    			h/2⟨ℒ_Yg,B⟩ -h i_∇ϕ( i_YB)⟩,

		so the result follows.
	
	
	It turns out that (<ref>), for appropriates choices of B and Y, is the starting point in establishing a Reilly-type formula holding for any weighted domain (Ω,g,d vol_ϕ) for which our standing assumptions hold.
	
	
	
		Under our standing assumptions, if h and f are functions on Ω then there holds 
			
    ∫_Ω h((Δ_ϕ f+f)^2-|∇^2f|^2)
    			    =    ∫_Σ h(2f_νΔ_Σ,ϕ f+H_ϕ f_ν^2+A_Σ(Df,Df)+2f_ν f)
           
    			+∫_Σ h_ν(|Df|^2-f^2)
           +∫_Ω(
    			(∇^2h)(∇ f,∇ f)-Δ_ϕ h· |∇ f|^2 -2h|∇ f|^2+h Ric_ϕ(∇ f,∇ f)	)
           +∫_Ω(Δ_ϕ h+h)f^2,

		where A_Σ is the second fundamental form of Σ and f_ν=⟨∇ f,ν⟩, etc.
		
	
	
	
	We take B=∇^2 f and Y=∇ f  in (<ref>) and integrate over Ω to obtain 
	
    ∫_Ω h|∇^2f|^2=∫_Σ h(∇^2f)(∇ f,ν)-∫_Ω (∇^2f)(∇ h,∇ f)-∫_Ω h( div_ϕ∇^2f)(∇ f),

	where for convenience we omit the weighted area and volume elements in the integrals. 
	Now,
	
    -∫_Ω (∇^2f)(∇ h,∇ f) 
    	    =     
    	 -1/2∫_Ω⟨∇ h,∇(|∇ f|^2)⟩
        =    1/2∫_ΩΔ_ϕ h·|∇ f|^2-1/2∫_Σ |∇ f|^2h_ν

	and the Ricci identity easily implies that
	
    -∫_Ω h( div_ϕ∇^2f)(∇ f)=-∫_Ω h⟨∇(Δ_ϕ f),∇ f⟩ -∫_Ω h  Ric_ϕ(∇ f,∇ f),

	so that 
	
    ∫_Ω h|∇^2f|^2 
    		    =    ∫_Σ h(∇^2f)(∇ f,ν) + 1/2∫_ΩΔ_ϕ h·|∇ f|^2-1/2∫_Σ |∇ f|^2h_ν
           -∫_Ω h⟨∇(Δ_ϕ f),∇ f⟩ -∫_Ω h Ric_ϕ(∇ f,∇ f).

	On the other hand, the next to the last term in the right-hand side above may be treated as
	
    -∫_Ω h⟨∇(Δ_ϕ f),∇ f⟩    =     -∫_Ω⟨∇(hΔ_ϕ f),∇ f⟩ +∫_ΩΔ_ϕ f⟨∇ h,∇ f⟩
        =    ∫_Ω  h(Δ_ϕ f)^2-∫_Σ hΔ_ϕ f· f_ν
    		 +∫_ΩΔ_ϕ f⟨∇ h,∇ f⟩,

	so we see that
	
    ∫_Ω h|∇^2f|^2
    	    =    ∫_Σ h(∇^2f)(∇ f,ν) -1/2∫_Σ |∇ f|^2h_ν
    	+ 1/2∫_ΩΔ_ϕ h·|∇ f|^2
           -∫_Σ hΔ_ϕ f· f_ν +
    	∫_Ω  h(Δ_ϕ f)^2+∫_ΩΔ_ϕ f⟨∇ h,∇ f⟩
           -∫_Ω h Ric_ϕ(∇ f,∇ f).

	
	We now observe that 
	
    ∫_Ω h((Δ_ϕ f+f)^2-|∇^2f|^2)
    	    =    ∫_Ω h(Δ_ϕ f)^2+ 2∫_Ω hΔ_ϕ f· f 
           +∫_Ω hf^2-∫_Ω h|∇^2 f|^2

	and 
	
    ∫_Ω hΔ_ϕ f· f
    		    =    ∫_Σ h ff_ν-∫_Ω⟨∇(hf),∇ f⟩
        =    ∫_Σ h ff_ν-∫_Ω h|∇ f|^2 -∫_Ω f⟨∇ h,∇ f⟩,

	so we obtain 
	
    ∫_Ω h((Δ_ϕ f+f)^2-|∇^2f|^2)
    	    =    ∫_Ω h(Δ_ϕ f)^2 +
    	2∫_Σ h ff_ν-2∫_Ω h|∇ f|^2 -2∫_Ω f⟨∇ h,∇ f⟩
            +∫_Ω h f^2 
            -∫_Σ h(∇^2f)(∇ f,ν) +1/2∫_Σ |∇ f|^2h_ν
    	- 1/2∫_ΩΔ_ϕ h·|∇ f|^2
           +∫_Σ hΔ_ϕ f· f_ν -
    	∫_Ω  h(Δ_ϕ f)^2-∫_ΩΔ_ϕ f⟨∇ h,∇ f⟩
           +∫_Ω h Ric_ϕ(∇ f,∇ f)
        =    ∫_Σ hΔ_ϕ f· f_ν +1/2∫_Σ |∇ f|^2h_ν
    	-∫_Σ h(∇^2f)(∇ f,ν) +	2∫_Σ h ff_ν
           - 1/2∫_ΩΔ_ϕ h·|∇ f|^2 
    	-∫_ΩΔ_ϕ f⟨∇ h,∇ f⟩  +∫_Ω h  Ric_ϕ(∇ f,∇ f)
           -2∫_Ω h|∇ f|^2 -2∫_Ω f⟨∇ h,∇ f⟩ +∫_Ω h f^2.

	Since we can handle the sixth and ninth terms in the right-hand side above as 
	
    -∫_ΩΔ_ϕ f⟨∇ h,∇ f⟩    =     -∫_Σ f_ν⟨∇ h,∇ f⟩  + ∫_Ω⟨∇ f,∇⟨∇ h,∇ f⟩⟩
        =     -∫_Σ f_ν⟨∇ h,∇ f⟩ +∫_Ω(∇^2h)(∇ f,∇ f)+1/2∫_Ω⟨∇ h,∇ |∇ f|^2⟩
        =     -∫_Σ f_ν⟨∇ h,∇ f⟩ +∫_Ω(∇^2h)(∇ f,∇ f)
           +1/2∫_Σ |∇ f|^2h_ν-1/2∫_ΩΔ_ϕ h· |∇ f|^2

	and 
	
    -2∫_Ω f⟨∇ h,∇ f⟩    =     -∫_Ω⟨∇ h,∇ f^2⟩
        =     -∫_Σ f^2 h_ν +∫_Ω f^2Δ_ϕ h,

	we end up with 
	
    ∫_Ω h((Δ_ϕ f+f)^2-|∇^2f|^2)
    		    =    ∫_Σ h(Δ_ϕ f · f_ν-(∇^2f)(∇ f,ν)) 
           +∫_Σ(|∇ f|^2h_ν-f_ν⟨∇ h,∇ f⟩)
           + 2∫_Σ hff_ν -∫_Σ h_ν f^2
           +∫_Ω(
    		(∇^2h)(∇ f,∇ f)-Δ_ϕ h· |∇ f|^2 -2h|∇ f|^2+h Ric_ϕ(∇ f,∇ f)	)
           +∫_Ω(Δ_ϕ h+h)f^2.

	
	We now handle the first two boundary terms above by computing in an adapted frame along Σ. First,
	
    ∫_Σ h(Δ_ϕ f · f_ν-(∇^2f)(∇ f,ν))
    	    =     -∫_Σ h⟨∇ϕ,∇ f⟩	f_ν + 
    	∫_Σ h(Δ f · f_ν-(∇^2f)(∇ f,ν))
        =     -∫_Σ h⟨∇ϕ,∇ f⟩	f_ν
           +∫_Σ h(f_νΔ_Σ f+H f_ν^2-⟨ D f_ν,Df⟩+A_Σ(Df,Df))
        =     -∫_Σ h⟨∇ϕ,∇ f-f_νν⟩	f_ν
    	 +∫_Σ hf_ν⟨ Dϕ,Df⟩
            +∫_Σ h(f_νΔ_Σ,ϕ f+H_ϕ f_ν^2-⟨ D f_ν,Df⟩+A_Σ(Df,Df))
        =    ∫_Σ h(f_νΔ_Σ,ϕ f+H_ϕ f_ν^2
    	-⟨ D f_ν,Df⟩+A_Σ(Df,Df)),

	where we used Gauss-Weingarten in the second step.
	Also, 
	
    ∫_Σ(|∇ f|^2h_ν-f_ν⟨∇ h,∇ f⟩)
    		    =    ∫_Σ(|Df|^2 h_ν-f_ν⟨ Dh,Df⟩)
        =    ∫_Σ(|D f|^2h_ν -⟨ D(f_ν h),Df⟩
    		+h⟨ Df_ν,Df⟩)
        =    ∫_Σ(
    		|D f|^2h_ν +h⟨ Df_ν,Df⟩
    		+f_ν hΔ_Σ,ϕf
    		),

	so if we substitute these identities back in (<ref>) the result follows after a few manipulations. 
	
	
	 (of Theorem <ref>) By rotational invariance we may assume that u=e_1. We apply our Reilly formula in Proposition <ref> with h=x_1, so that Δ_ϕ_n h+h=0 and choose f satisfying the Dirichlet problem
	
    {[ Δ_ϕ_n f+f=1           Ω;         f=0           Σ ].

	where we may assume that Σ≠∂ℋ^n_e_1,0, hence bounding a domain Ωℋ^n_e_1,0; cf. Remark <ref>. 

	Since Ric_ϕ_n=δ, the formula reduces to 
	
    ∫_Ω x_1-∫_Ω|∇^2f|^2=∫_Σ H_ϕ_n x_1f_ν^2= H_ϕ_n∫_Σ x_1f_ν^2,
	
	and using Cauchy-Schwarz, 
	
    ∫_Ω x_1-∫_Ω|∇^2f|^2≥ H_ϕ_n(∫_Σ x_1 f_ν)^2/∫_Σ x_1.

	Now, as a consequence of (<ref>),
	
    ∫_Σ x_1 f_ν =∫_Ω x_1Δ_ϕ_n f - ∫_Ω fΔ_ϕ_n x_1=∫_Ω x_1.

	On the other hand, the Minkowski formula in Gaussian space <cit.> gives
	
    ∫_Σ x_1=-H_ϕ_n∫_Σ⟨ e_1,ν⟩,

	and since 
	
    ∫_Σ⟨ e_1,ν⟩=∫_Ω div_ϕ_n e_1=-∫_Ω x_1,

	we see from (<ref>) that ∇^2f≡ 0 and therefore f is an affine function. Thus, 
	Σ lies in a hyperplane and since Ω⊂ℋ^n_e_1,0, the proof is completed. 
		
	
	
	Proposition <ref> implies that an optimal domain Ωℋ^n_e_1,0 for ℰ_-1,-1 supports a function f satisfying the over-determined sustem
	
    {[ Δ_ϕ_n f+f=1           Ω;         f=0           Σ;     |∇ f|=c           Σ ].

	for some c>0. 
	Although the methods of Section <ref>, which are appropriate to handle the slightly different problem in (<ref>), do not seem adequate to provide a rigidity result for such domains, the proof above  
	shows that if we replace the third condition in (<ref>) by the assumption that the weighted mean curvature of Σ=∂Ω is constant then rigidity is recovered.
		
	
	
		Our argument leading to Theorem <ref> is modeled on the proof of <cit.>, where Alexandrov's soap bubble theorem in simply connected space forms is retrieved as a consequence of a generalized Reilly formula <cit.>. It turns out that if we apply their formula to (a compact domain in) the round hemisphere 𝒮^k-1_1/k,+ in Remark <ref>, take the limit as k→ +∞ and argue (always on heuristic grounds!) that not only the metric Laplacian but also the standard mean and Ricci curvatures should give rise to their weighted counterparts along Poincaré's convergence, we obtain our Reilly-type formula in Proposition <ref> (as applied to Gaussian space). Of course, the long computation in the proof of this latter proposition provides a rigorous justification of this rather informal argument based on Poincaré's limit (Section <ref>). Finally, we mention that Proposition <ref> (again, as applied to Gaussian space) also follows formally as the appropriate Poincaré's limit of a Reilly-type formula in <cit.>, which by its turn generalizes to weighted domains the one in <cit.> referred to above.
		
		
		
			Hypersurfaces in Gaussian space with constant weighted mean curvature are usually referred to as λ-hypersurfaces and there is a huge literature on trying to classify them under a most varied set of assumptions. We refer to <cit.> for a recent survey on the subject. 
			



§ POINCARÉ'S LIMIT THEOREM


	For the reader's convenience, we provide here the precise formulation of Poincaré's limit theorem <cit.>, which has been used in a rather informal way in the bulk of the text. Using the notation of Remark <ref>, let  
	Π_k,n:ℝ^k→ℝ^n be the orthogonal projection associated to the natural embedding ℝ^n↪ℝ^k and let (𝒮^k-1_1/k,ℙ_k) be the 
	probability space defined by ℙ_k=e^-ϕ_(k)d vol_δ_ sph_k.
 With this terminology, and using standard probabilistic jargon, Poincaré's limit theorem says that the  
	random vectors 
	
    Π_k,l=Π_k,n|_𝒮^k-1_1/k:(𝒮^k-1_1/k,ℙ_k)→ℝ^n

	 converge weakly to a ℝ^n-valued random vector Z whose distribution is the Gaussian density d vol_ϕ_n=e^-ϕ_nd vol_δ,
	which means that 𝔼(ψ(Π_k,l))→𝔼(ψ(Z))) for any ψ:ℝ^n→ℝ uniformly bounded and continuous. The standard probabilistic proof of this remarkable result combines the Law of Large Numbers and the well-known fact that a 
	random vector Z^[k] uniformly distributed over 𝒮^k-1_1/k may be expressed as 
	
    Z^[k]=√(k)X^[k]/X^[k],

	where X^[k] is a ℝ^k-valued independent standard Gaussian vector <cit.>. 
	With a bit more of effort it may be checked that 
	this statement actually holds true with ψ= 1_A, the indicator function 
	of an arbitrary Borel set A⊂ℝ^n. Precisely, 
	
    lim_k→ +∞ℙ_k(Π_k,n^-1(A))
    		= vol_ϕ_n(A);

	see <cit.> or <cit.>, where it is also explained how this sharpened version may be used to transfer the solution of the isoperimetric problem from (𝒮^k-1_1/k,ℙ_k) to Gaussian space as k→ +∞, a celebrated result first proved by Borell <cit.> and Sudakov-Tsirel'son <cit.> and which lies at the heart of the “concentration of measure phenomenon” <cit.>. 
	Inspired by this circle of ideas we are led to loosely interpret (<ref>)  as saying that the “projections” of the weighted manifolds (𝒮^k-1_1/k,δ_ sph_k,ℙ_k) onto ℝ^k converge (in a sense whose actual meaning is not relevant here) to the Gaussian space (ℝ^n,δ,d vol_ϕ_n), a perspective we have naively explored above in order to transplant problems and techniques from (high dimensional) spheres to Gaussian space. Finally, we refer to 
	<cit.> 
	for a detailed guide to the proofs of both versions of this foundational result mentioned above.    
	

alpha




