domain: info stat

text:
Matěj Kubička
email: matej.kubicka@lss.supelec.fr

Jan Klusáček

Antonio Sciarretta
email: antonio.sciarretta@ifpen.fr

Arben Cela
email: arben.cela@esiee.fr

Hugues Mounier
email: hugues.mounier@lss.supelec.fr

Laurent Thibault
email: laurent.thibault@ifpen.fr

S I Niculescu
email: silviu.niculescu@lss.supelec.fr.

Performance of current eco-routing methods

Eco-routing is a vehicle navigation method that aims to minimize fuel or energy consumption for a given trip. It is based on a hypothesis that we can trade extra travel time for lower consumption. While the hypothesis was experimentally verified the design of a method that would fully exploit its potential proves challenging. Current solutions hinge on assumption that energy spent on any given road does not change in time. We challenge validity of this assumption by studying performance of such methods in detailed second-bysecond simulation that pronounces the time-dependencies. This allows us to quantify the real savings attainable with current eco-routing.

I. INTRODUCTION

. The idea is to minimize energy (or fuel) consumption by route selection: given some origin and destination, eco-routing plots a route such that energy (fuel) needed to finish the trip is minimal.

The routing is usually done on a graph where nodes represent junctions, edges represent roads and costs are estimated energies needed to travel between two junctions the road connects. Minimal path routing 2 can then be used to find the route that minimizes total energy for the trip. Authors typically reduce the complex time-variant functions that describe the costs. They must be time-invariant and nonnegative in order to use Dijkstra's routing algorithm, which is a common choice between authors.

Validation is often done using the same assumptions. Full experimental validation would require a host of identical vehicles to depart from spatially and temporally identical place in order to measure consumptions on different paths to destination reliably. This is difficult to realize in practice. Consequently, authors estimate performance of their methods

by comparing estimated consumption of the eco-routes to estimated consumption of the shortest routes. In other words, the same model with the same assumptions is used for both routing and validation. We validate eco-routing methods without the time-invariancy assumption and with consumption models experimentally validated on real vehicles. This allows us to reason about ability of eco-routing to lower the consumption. Three eco-routing methods are studied in this way. Their performance is evaluated for both electric and conventional vehicles. We chose to make distinction between the two because the electric vehicle can recuperate braking energy. This has fundamental effect on consumption.

The original study that considered the idea of eco-routing is by Ericsson et al. [START_REF] Ericsson | Optimizing route choice for lowest fuel consumption-potential effects of a new driver support tool[END_REF]. Authors tried to estimate the potential of eco-routing based on route choices of real drivers. They used a collection of 15437 recorded commutes to compute typical consumptions on streets of Lund, Sweden. Then, they looked at 109 real journeys and estimated how many of them could be optimized and what is the expected gain for those that could be. They conclude that fuelefficiency could be enhanced for 46% of the trips and that fuel savings would be 8.2% on the average.

A study similar to ours is by Richter et al. [START_REF] Richter | Comparison of eco and time efficient routing of ICEVs, BEVs and PHEVs in inner city traffic[END_REF]. Authors study the potential of eco-routing in a simulation of inner city traffic. They used simulation tool ULTraSim [START_REF] Richter | ULTraSim, a traffic simulator incorporating submicroscopic BEV, HEV, ICEV models[END_REF] to compute average consumption on every road in their map for electric, plugin hybrid and conventional vehicles. Then, they chose thirteen origin-destination pairs and plotted the shortest route and the eco-route for each. Reported fuel savings were 8.5% on the average for eco-route against the shortest route. This is similar to the findings of Ericsson et al..

A field study of the impact of route choice on the consumption and emissions was conducted by Ahn and Rakha [START_REF] Ahn | The effects of route choice decisions on vehicle energy consumption and emissions[END_REF]. Authors recorded thirty-nine trips over two routes during morning commute. These trips shared the same origin and destination, but eighteen of them took the arterial route and twenty-one the highway. These two roads are reported to have different speed limits and different commute patterns. Authors observe that fuel consumption is lower by 4 -8% when taking the slower arterial route.

Another field study similar to the one by Ahn and Rakha was conducted by Minett et al. [START_REF] Minett | Eco-routing: comparing the fuel consumption of different routes between an origin and destination using field test speed profiles and synthetic speed profiles[END_REF]. Authors recorded forty trips between the cities Delft and Zoetermeer taking either motorway, local or provincial road. They report consumption on local road lower by 17% with respect to the motorway.

Our study is most similar to the approach of Richter et al. [START_REF] Richter | Comparison of eco and time efficient routing of ICEVs, BEVs and PHEVs in inner city traffic[END_REF] and Ericsson et al. [START_REF] Ericsson | Optimizing route choice for lowest fuel consumption-potential effects of a new driver support tool[END_REF]. Our contribution with respect to these works is multifold:

• We compare eco-routes to the shortest and to the fastest routes. The idea of eco-routing is to trade extra travel time for lower consumption, hence it makes more sense to draw conclusions with respect to the fastest routes rather than the shortest ones. • We simulate every route (eco-route, shortest route, fastest route) separately and ensure that the initial conditions are identical for each (same departure time, same traffic state) to sustain comparability. • We ran our simulations on a large set of randomized trip origins and destinations. This allows us to sample the probability distribution of energy savings. The paper is organized as follows: section II below defines eco-routing, section III reviews tested eco-routing methods and section IV presents the design of our experiment. Then, our findings are shown in section V and discussed in section VI. Conclusions are presented in section VII.

II. PROBLEM STATEMENT

Let G = (V, E) be a road network graph, where V = {n k : k = 1 . . . M } is a set of nodes with cardinality M and E = {e k : k = 1 . . . N } is a set of arcs with cardinality N . The nodes represent roads between two junctions, arcs describe the connections between them. We refer to G simply as map.

Lets assume we know cost function f : V → R (abbrev. f k for k-th node in V ) that describe how much energy is needed to travel between the two junctions the road connects. Let p be a path (also called a route) defined as a contiguous sequence of nodes p = {n o , . . . , n d }, where n o and n d are the origin and destination nodes. We distinguish three types of paths: the shortest path p s , the fastest path p f , and the eco-route p e . The energy consumption of a path is a sum of energies lost at each node, hence

E(p) = k∈p f k (1)
We distinguish the energy of the eco-route, E e = E(p e ), the energy of the shortest route, E s = E(p s ), and the energy of the fastest route, E f = E(p f ). The eco-routing is defined as optimization problem

p e (o, d) = argmin p∈P E(p) s.t. n 1 = o n |p| = d ( 2
)
where P is the set of all simple paths in the map G, n i is i-th node in the path p and |p| is cardinality of the path p.

The shortest path and the fastest path routing can be defined analogously such that the shortest path routing minimizes the distance and the fastest path routing minimizes the travel time.

III. METHODS UNDER TEST

This study was conducted on three well-known eco-routing methods, all of them conform to the model [START_REF] Ahn | The effects of route choice decisions on vehicle energy consumption and emissions[END_REF]. First method is by Barth et al., originally proposed in [START_REF] Barth | Environmentally-friendly navigation[END_REF] and further extended in [START_REF] Boriboonsomsin | Eco-routing navigation system based on multisource historical and real-time traffic information[END_REF]. This is the first published eco-routing method we know of. Authors use a model called CMEM to estimate both fuel consumption and emissions. The CMEM is microscopic emissions and fuel intake model validated in [START_REF] Barth | Recent validation efforts for a comprehensive modal emissions model[END_REF]. The consumption on each road is approximated using

ln(f k ) = β 0 + β 1 v k + β 2 v 2 k + β 3 v 3 k + β 4 v 4 k + β 5 s k (3)
where v k is the average traffic speed and s k is the road grade. Authors obtained the β-coefficients using multivariate nonlinear regression on previously collected speed profiles. Average speed is sourced in real-time from a traffic information system.

Second method is by Andersen et al. [START_REF] Andersen | Ecotour: reducing the environmental footprint of vehicles using ecoroutes[END_REF]. Authors posses large amount of recorded trips in Denmark. Their positioning data originate from the GPS receivers, consumption is read from the vehicle using an on-board diagnostics interface. The routing is done on a road network of Denmark. Their function f k assigns each road the average consumption observed there. This method is interesting due to its simplicity. However, we believe authors are mixing different vehicle types in the same average which must lead to confounding since some vehicles show notoriously higher consumption than the others.

Third method is by Juřík et al. [START_REF] Juřík | Energy optimal real-time navigation system[END_REF]. This method is based on physical consumption modeling: authors adapt the longitudal vehicle model [START_REF] Guzzella | Vehicle propulsion systems: introduction to modeling and optimization[END_REF] to estimate the consumption. They consider losses incurred by altitude changes, frictious losses and aerodynamic drag. Powertrain efficiency and heating in friction brakes is neglected. Their model reads

f k = E r + (α -1)E p if E p ≤ 0 E r if E p > 0 (4)
where E r represents rolling friction and aerodynamic losses, E p is a potential energy and α is a constant representing recuperation capability of the vehicle. The E r and E p is given as

E r = ρ 2 A f C d v 2 k l k + M gµ r l k cos(θ k ) (5)
E p = M gµ r l k sin(θ k ) (6)
where v k is the average speed and l k is the road length on the road k. Constants ρ, A f , C d , M , µ r are vehicle-specific and g is the gravitational constant.

IV. METHODOLOGY

Our experiments were conducted within a traffic simulation of real European city. The simulator allows us to introduce a vehicle, to set its route and to observe its progress in time. We ran simulations with the vehicles on different routes to the same destination, each time under identical initial conditions (same departure time, same origin, same traffic state). This allows us to record vehicle speed profiles which is essential when computing realistic estimates of consumption.

The departure time was set to midnight. The traffic is minimal at this time, hence perturbances due to the traffic are minimal. While it would be interesting to study eco-routing in congested urban settings this should come only after it was shown that eco-routing can perform in less perturbed environment.

We used two vehicle models: a model of FAM F-City electric vehicle and a model of conventional Renault Scénic. They are both standard consumption models based on onedimensional longitudal dynamics and powertrain efficiency maps with experimentally identified coefficients. The F-City model was validated in [START_REF] Dib | Evaluation of the energy efficiency of a fleet of electric vehicle for ecodriving application[END_REF]. Authors observed 1.7% consumption error during the test drive. We define the vehicle model as a function ρ(v, a, s) : R n × R n × R n → R that maps speed, acceleration and slope profiles of length n to energy consumption. The speed profiles were extracted from the simulations, accelerations are based on difference quotients of the speed and the slope profiles come from digital elevation model of Europe [START_REF]EU Digital Elevation Model[END_REF]. It covers European continent with grid cell size of 25 × 25 meters. Bicubic interpolation was used to interpolate the terrain.

The simulation was conducted in a tool called SUMO [START_REF] Daniel | Recent development and applications of SUMO -Simulation of Urban MObility[END_REF] and on a scenario of Luxembourg developed and validated by Codecà et al. [START_REF] Codecà | Luxembourg SUMO traffic (LuST) scenario: 24 hours of mobility for vehicular networking research[END_REF]. The scenario is based upon a map of Luxembourg, with synthetic traffic matched to Luxembourg's own traffic patterns. It features 286, 884 vehicles over twenty-four hours on 931.12 kilometers of residential, arterial and highway roads. It has roundabouts, traffic lights, multi-lane roads and other features characteristic for urban traffic.

The toolchain used to conduct our experiment is on Figure 1. First we built a set of origin-destination pairs. For that we reused scenario's own trip origins and destinations as they are generated from real traffic demand in the Luxembourg city. This yielded 108, 356 origin-destinations pairs. We plotted eight routes between each origin and destination: three ecoroutes according to Barth The shortest routes were routed using the cost function f k = l k , the fastest routes were routed using the cost function f k = l k v k . Note that the "fastest" route is an approximation of the real fastest route based on the average travel speeds. We resolved to this solution as computing the real fastest routes is a hard problem. The routing produced 866, 848 routes in the map of Luxembourg. We simulated each route in the SUMO simulator, collected their speed profiles and computed their consumption using the reference models of Renault Scénic and FAM F-City. This required an equivalent of 559 hours of single core computing time on 96-core computing cluster. We have excluded data from 4781 origindestination pairs as they failed validity checks. While it is likely that the samples were in fact correct we prefer to remove samples for which we cannot be certain of their validity.

Methods

(i) k .
The speed profile is a time series made of speeds measured at different time instants. For every vehicle, the consumption f

(i) k = ρ(v (i) k , v(i)
k , s k ) and the average speed denoted vk (i) is stored for each pass on each road in a quadruple (f

(i) k , v(i)
k , s k , l k ) together with the road slope and road length. We have collected a total of 22, 592, 081 of these quadruples, 316 were removed due to failed sanity checks.

A. Eco-routing implementation details

Consumption estimates according to Barth et al. [START_REF] Barth | Environmentally-friendly navigation[END_REF] are based on equation [START_REF] Andersen | Ecotour: reducing the environmental footprint of vehicles using ecoroutes[END_REF]. We needed to compute the βcoefficients specific to our vehicles. For that reason we replaced the original vehicle model with our own and collected the quadruples (f

(i) k , v(i) k , s k , l k )
as discussed above. This provided necessary data to fit the β-coefficients using regression. Note that we had convergence problems when performing nonlinear regression on (3), instead we obtained the β-coefficients by regression on linearized model. See Table I for the results.

The method by Andersen et al. can suffer with a lack of observations in back alleys and other scarcely used roads. We used f k = ∞ on roads on which we had no historical data (that is, when m = 0). On the roads where we had the data we took their average. Hence, compound from different vehicles with varying consumption characteristics. This necessarily leads to more uncertainty in estimated consumptions. We don't mix vehicle types in this study since we have only two vehicles. Instead, we use costs f k specific either for the electric vehicle or for the conventional vehicle. This is idealized with respect to the original.

f k =    1 m m i=1 f (i) k if m > 0 ∞ if m = 0 (7)
We also adapted the consumption model of Juřík et al. as our vehicle description did not match the one used by the authors. Equation ( 5) was changed to

E r = (cv 2 k + bv k + a)l k (8)
where a, b and c are coast-down parameters: coefficients of polynomial fitted to experimentally measured relation between the force acting against vehicle movement and the speed. From modeling point of view this has comparable meaning to the original approach, however, it neglects dependence of rolling losses on road grade. For the values see Table I.

V. RESULTS

The results presented in this section are based on a collection of 828, 600 simulated routes between 103, 575 unique origin-destination pairs. Raw experiment data is available online [START_REF] Matěj | Performance of current eco-routing methods -experiment data[END_REF], our findings are summarized below. II.

Probability distributions of relative consumptions are shown on figures 3 and 4. Former shows consumption of the eco-routes relative to the fastest routes (ratio E e /E f ), latter relative to the shortest routes (ratio E e /E s ). There are three curves in each plot, one for each method. We observed exceptionally high probability of the eco-route being identical to the shortest (resp. fastest) route. Since the distributions have discontinuity at this point we removed it from the plots and report probabilities at this particular point in Table III where we summarize the results. The Table III has following sections 1) "Average savings" and "Estimated savings" list the average of actually saved energy and the average of a priori estimates of saved energy. Former tells how much was actually saved while latter tells how much was expected to be saved. 2) "Probability of failure" gives sample probability that the eco-route is less economic than the fastest (resp. shortest) route and that it requires longer travel time than the fastest (resp. shortest) route. 3) "Eco-route same as reference" gives sample probability that eco-route is identical to the fastest (resp. shortest) route. 4) "Median of time-delay" and "95-th percentile time delay" are statistics of travel time delays. They have heavy-tailed distributions so we report the median rather than the mean to give a robust estimate of central tendency. Sampled eco-route consumptions do not come from the same population as the shortest and fastest routes. We reject the hypothesis that any of the six measured eco-route consumption sample sets (three eco-routing methods for two distinct vehicle types) come from the same population as the samples of the shortest or the fastest route consumption. We tested the hypothesis using Mann-Whitney U test. In the total of twelve tests the largest two-sided p-value was smaller than 10 -7 , hence the differences are statistically significant on 99% level. We used the U test rather than the t-test because the t-test assumes the population distribution is normal. The d'Agostino's normality test on the sampled distributions returns high χ 2 statistic for all sample sets. Corresponding p-values are smaller than 10 -10 for the hypothesis that the consumption distributions are normal, hence we adapted the alternative hypothesis.

VI. DISCUSSION

The method by Andersen et al. shows highest correlation of consumption estimates with the reference and the standard estimation error of 96 Wh for the electric vehicle is lowest between the competing methods. However, the estimation can't be expected to work so well in practice. As discussed in section IV-A we used vehicle-specific consumption averages while the original method mixes consumptions from different vehicle types into a single average. Quality of the consumption estimates reflects on eco-routing performance: Andersen et al. achieves the highest average savings (up to 12.5%) and low probability of failure (ranging from 6% to 10%).

The consumption estimates by Barth et al. show more spread than estimates by Andersen et al. (see Figure 2) but still retain high correlation (0.95 for both vehicles). Standard errors are roughly tripled with respect to Andersen et al.. Routing performance shows average savings close to 2.5% for both vehicles when compared to the fastest route. When compared to the shortest route the average savings drop to negative values. According to this result we are more likely to save energy if we drive along the shortest route, rather than along the eco-route.

The method by Juřík et al. underestimates consumption severely, as can be seen on Figure 2. This is by design as au- thors neglect powertrain efficiency. They observed that scale error will not affect correlation, nor eco-routing performance. Indeed, structure of the shortest paths in a graph where all the costs are scaled by the same positive amount will not be altered. Consumption correlation is 0.92 for the electric vehicle and 0.83 for the conventional vehicle. Standard errors are roughly six times higher than those of Andersen et al. Routing performance is best for the electric vehicle, showing 3.5% savings with respect to the fastest route. However, ecoroute still performs worse than the shortest route on average. For the conventional vehicle we see worst performance among all: -8.8% savings with respect to the shortest route. This is not surprising as They must be, by definition, positive. However, we have seen that the real savings are often negative on the average. This cannot be accounted for as long as we assume that the consumption does not change in time. Unfortunately, this is also the assumption that makes the current eco-routing feasible.

Results are generally better for the electric vehicle. Even when correlation score is high for both electric and conventional vehicles the savings are higher for the electric vehicle. For example in the case of Andersen et al. the correlation is 0.99 for both, but the electric vehicle enjoys average savings of 12.5% while conventional vehicle only 8.4%.

VII. CONCLUSION

We have recreated three eco-routing methods published in the literature and tested them in simulated urban traffic to quantify their ability to lower the energy or fuel consumption needed to finish a trip. The three methods were chosen such that they make use of different ideas to achieve their task: Andersen et al. [START_REF] Andersen | Ecotour: reducing the environmental footprint of vehicles using ecoroutes[END_REF] simply averages observed energy consumptions, Barth et al. [START_REF] Barth | Environmentally-friendly navigation[END_REF] uses regression model to estimate the consumption and Juřík et al. [START_REF] Juřík | Energy optimal real-time navigation system[END_REF] uses a physical model of a vehicle to do the same. We cannot guarantee that our results are representative of reality. However, the simulations are matched to mobility patterns of a real city and highly detailed both in terms of temporal resolution and size. Moreover, our referential vehicle models are based upon real vehicles and standard modeling methodology.

The results support the eco-routing hypothesis: eco-routes can have a lower consumption than the fastest routes on the average as shown by methods of Barth et al. and Andersen et al. Among the three competing methods Andersen et al. have shown the best results (12.5% average savings), but we conjecture that this is not attainable in practice as we idealized the method. When we compare the eco-routes to the shortest routes we find that the methods by Juřík et al. and Barth et al. offered eco-routes that were less economic than the shortest routes. Eco-routes by Andersen et al. performed better, however, since the method was idealized it remains unanswered whether this will hold in practice. However, it shows that eco-routing has the potential to deliver interesting savings. In order to attain it we need better consumption estimates.

We observed that the eco-routes often fail to save energy. This is particularly problematic if the eco-route also requires longer time to destination. Then, the eco-routing advises us to take less economic route for a time. We formulated occurrences of this as "probability of failure" and observed that it happens between 5% and 55% of the time, depending on the method and used vehicle. This is to our best knowledge the first time the likelihood of failure was observed and further study to its causes is necessary. Indeed, no sensible user would accept 50% chance that eco-routing will navigate him on a path that is less economic and incurs travel time delays.

We provide open access to the dataset with experiment data [START_REF] Matěj | Performance of current eco-routing methods -experiment data[END_REF]. It contains an XML file with all the routes, simulation results and consumptions. A script written in Python that generates the plots and tables shown here is attached to the dataset.

  et al., Juřík et al. and Andersen et al. for both electric and conventional vehicles plus shortest and fastest routes for reference. The routing was done upon a map G = (V, E) extracted from the Luxembourg scenario.

Fig. 1 :

 1 Fig. 1: Experiment toolchain (simplified)

  Correlations of consumption estimates according to Barth et al., Juřík et al. and Andersen et al. with the reference consumptions are shown on Figure 2. Results for the electric vehicle are on Figure 2a and for the conventional vehicle on Figure 2b. Each sample represents a single trip. Perfect estimation would be achieved along the black line. Pearson's product-moment correlation coefficients and standard errors are listed in Table

Fig. 2 :

 2 Fig. 2: Correlations of estimated and reference consumptions

Fig. 3 :Fig. 4 :

 34 Fig. 3: Sample distribution of energy savings relative to fastest routes

TABLE I :

 I

	COEFFICIENTS FOR ECO-ROUTING
			F-City	Renault Scénic
		a	113.5 N	110.45 N
		b	0.774 N/(m/s)	1.5175 N/(m/s)
	Juřík et al. [12]	c	0.4212 N/(m/s) 2	0.5119 N/(m/s) 2
		M	1, 190 kg	1, 588 kg
		α	0.85	0.0
		β 0	-0.7123	0.9580
		β 1	-2.2703 • 10 -2	-6.3418 • 10 -2
	Barth et al. [4]	β 2 β 3	-3.2109 • 10 -3 1.0617 • 10 -4	-1.1529 • 10 -3 7.0035 • 10 -5
		β 4	-5.8308 • 10 -7	-4.1114 • 10 -7
		β 5	2.3320	1.6148

  by Juřík et al. and Barth et al. require real-time information about the average speeds in the network. Furthermore, Barth et al. and Andersen et al. require historically observed consumptions on the roads. We extracted necessary data beforehand from scenario's native traffic. Lets denote speed profile of i-th vehicle when traveling on road k as v

TABLE II

 II

			:
	CONSUMPTION ESTIMATION RESULTS
			F-City	Scénic
		Andersen	0.987	0.989
	Correlation	Barth	0.956	0.957
		Juřík	0.916	0.826
		Andersen	96 Wh	339 Wh
	Std. error	Barth	234 Wh	792 Wh
		Juřík	484 Wh 2,493 Wh

TABLE III

 III

				:
		RESULTS SUMMARY
			fastest	shortest
			F-City Scénic F-City Scénic
	Average savings	Andersen Barth Juřík	12.5% 2.6% 3.5%	8.4% 2.8% -3.9%	7.5% -3.6% -1.8%	4% -2.1% -8.8%
	Estimated savings	Andersen Barth Juřík	14.7% 1.4% 17.8%	9.9% 1.5% 18.5%	9.2% 2.4% 4.4%	5.7% 2.5% 6.8%
	Maximum savings	Andersen Barth Juřík	60.7% 58.1% 60.9%	58.6% 54.6% 56.5%	51.4% 40.9% 51.4%	40.2% 35.6% 39.1%
	Probability of failure	Andersen Barth Juřík	6.9% 5.1% 31.7%	8.4% 4.5% 48.1%	8.4% 13.3% 37.1%	9.6% 14.1% 54.7%
	Eco-route	Andersen	8.7%	13.3%	12%	17.5%
	same as	Barth	60.6%	63.0%	30.8%	30.2%
	reference	Juřík	6.6%	6.2%	16.7%	11.8%
	Median of time delay	Andersen Barth Juřík	4.7% 0% 18.0%	0.2% 0% 16.5%
	95 th perc. time delay	Andersen Barth Juřík	56% 36.8% 78%	47.6% 38.2% 85.2%

  Juřík et al. designed their method for electric and hybrid vehicles. The method by Juřík et al. is most practical: it is based on a standard vehicle consumption model and the physical laws it entails. Hence, there is no need to utilize historical data, nor to fit any coefficients. Method by Barth et al. requires a set of prerecorded trips and Andersen et al. requires specialized hardware. Juřík et al. and Andersen et al. overestimate the expected savings.

Note that the term "eco-routing" is also used in a context of emissions minimization; we do not consider emissions in this study.

Commonly also known as shortest path routing; we don't use this term to avoid confusion with routing that minimizes travel distance.


Jean Dolbeault
email: dolbeaul@ceremade.dauphine.fr

Maria J Esteban
email: esteban@ceremade.dauphine.fr

Michael Loss
email: loss@math.gatech.edu

Matteo Muratori
email: matteo.muratori@unipv.it

Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities

Keywords: Functional inequalities, interpolation, Caffarelli-Kohn-Nirenberg inequalities, weights, optimal functions, best constants, symmetry, symmetry breaking, semilinear elliptic equations, rigidity results, uniqueness, flows, fast diffusion equation, carré du champ, Emden-Fowler transformation Mathematics Subject Classification (2010) : Primary : 26D10, 35B06, 35J20. Secondary : 49K30, 35J60, 35K55

We use the formalism of the Rényi entropies to establish the symmetry range of extremal functions in a family of subcritical Caffarelli-Kohn-Nirenberg inequalities. By extremal functions we mean functions which realize the equality case in the inequalities, written with optimal constants. The method extends recent results on critical Caffarelli-Kohn-Nirenberg inequalities. Using heuristics given by a nonlinear diffusion equation, we give a variational proof of a symmetry result, by establishing a rigidity theorem: in the symmetry region, all positive critical points have radial symmetry and are therefore equal to the unique positive, radial critical point, up to scalings and multiplications. This result is sharp. The condition on the parameters is indeed complementary of the condition which determines the region in which symmetry breaking holds as a consequence of the linear instability of radial optimal functions. Compared to the critical case, the subcritical range requires new tools. The Fisher information has to be replaced by Rényi entropy powers, and since some invariances are lost, the estimates based on the Emden-Fowler transformation have to be modified.

Symmétrie des fonctions extrémales pour des inégalités de Caffarelli-Kohn-Nirenberg sous-critiques

Nous utilisons le formalisme des entropies de Rényi pour établir le domaine de symétrie des fonctions extrémales dans une famille d'inégalités de Caffarelli-Kohn-Nirenberg sous-critiques. Par fonctions extrémales, il faut comprendre des fonctions qui réalisent le cas d'égalité dans les inégalités écrites avec des constantes optimales. La méthode étend des résultats récents sur les inégalités de Caffarelli-Kohn-Nirenberg critiques. En utilisant une heuristique donnée par une équation de diffusion non-linéaire, nous donnons une preuve variationnelle d'un résultat de symétrie, grâce à un théorème de rigidité : dans la région de symétrie, tous les points critiques positifs sont à symétrie radiale et sont par conséquent égaux à l'unique point critique radial, positif, à une multiplication par une constante et à un changement d'échelle près. Ce résultat est optimal. La condition sur les paramètres est en effet complémentaire de celle qui définit la région dans laquelle il y a brisure de symétrie du fait de l'instabilité linéaire des fonctions radiales optimales. Comparé au cas critique, le domaine sous-critique nécessite de nouveaux outils. L'information de Fisher doit être remplacée par l'entropie de Rényi, et comme certaines invariances sont perdues, les estimations basées sur la transformation d'Emden-Fowler doivent être modifiées.

A family of subcritical Caffarelli-Kohn-Nirenberg interpolation inequalities

With the norms

w L q,γ (R d ) := R d |w| q |x| -γ d x 1/q , w L q (R d ) := w L q,0 (R d ) ,
let us define L q,γ (R d ) as the space of all measurable functions w such that w L q,γ (R d ) is finite. Our functional framework is a space H p β,γ (R d ) of functions w ∈ L p+1,γ (R d ) such that ∇w ∈ L 2,β (R d ), which is defined as the completion of the space D(R d \{0}) of the smooth functions on R d with compact support in R d \{0}, with respect to the norm given by w 2 := (pp) w 2 L p+1,γ (R d )

+ ∇w 2 L 2,β (R d )
. Now consider the family of Caffarelli-Kohn-Nirenberg interpolation inequalities given by

w L 2p,γ (R d ) ≤ C β,γ,p ∇w ϑ L 2,β (R d ) w 1-ϑ L p+1,γ (R d ) ∀ w ∈ H p β,γ (R d ) . (1)
Here the parameters β, γ and p are subject to the restrictions

d ≥ 2 , γ -2 < β < d -2 d γ , γ ∈ (-∞, d ) , p ∈ 1, p with p := d -γ d -β -2 (2)
and the exponent ϑ is determined by the scaling invariance, i.e., ϑ = (d -γ) (p -1)

p d + β + 2 -2 γ -p (d -β -2) .
These inequalities have been introduced, among others, by L. Caffarelli, R. Kohn and L. Nirenberg in [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF]. We observe that ϑ = 1 if p = p , a case which has been dealt with in [START_REF]Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF], and we shall focus on the sub-critical case p < p . Throughout this paper, C β,γ,p denotes the optimal constant in [START_REF] Muratori | Diffusions hypercontractives[END_REF]. We shall say that a function

w ∈ H p β,γ (R d
) is an extremal function for (1) if equality holds in the inequality.

Symmetry in [START_REF] Muratori | Diffusions hypercontractives[END_REF] means that the equality case is achieved by Aubin-Talenti type functions

w (x) = 1 + |x| 2+β-γ -1/(p-1) ∀ x ∈ R d .
On the contrary, there is symmetry breaking if this is not the case, because the equality case is then achieved by a non-radial extremal function. It has been proved in [START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF] that symmetry breaking holds in [START_REF] Muratori | Diffusions hypercontractives[END_REF] if

γ < 0 and β FS (γ) < β < d -2 d γ (3)
where β FS (γ) := d -2 -(γd ) 2 -4 (d -1) .

For completeness, we will give a short proof of this result in Section 2. Our main result shows that, under Condition [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF], symmetry holds in the complement of the set defined by [START_REF] Blachman | The convolution inequality for entropy powers[END_REF], which means that (3) is the sharp condition for symmetry breaking. See Fig. 1.

Theorem 1.1 Assume that (2) holds and that

β ≤ β FS (γ) if γ < 0 . ( 4
)
Then the extremal functions for (1) are radially symmetric and, up to a scaling and a multiplication by a constant, equal to w .

The result is slightly stronger than just characterizing the range of (β, γ) for which equality in (1) is achieved by radial functions. Actually our method of proof allows us to analyze the symmetry properties not only of extremal functions of [START_REF] Muratori | Diffusions hypercontractives[END_REF], but also of all positive solutions in H p β,γ (R d ) of the corresponding Euler-Lagrange equations, that is, up to a multiplication by a constant and a dilation, of div |x| -β ∇w = |x| -γ w 2p-1w p in R d \ {0} .

(

) 5
Theorem 1.2 Assume that (2) and (4) hold. Then all positive solutions to (5) in H p β,γ (R d ) are radially symmetric and, up to a scaling and a multiplication by a constant, equal to w .

Up to a multiplication by a constant, we know that all non-trivial extremal functions for (1) are non-negative solutions to [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF]. Non-negative solutions to [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF] are actually positive by the standard Strong Maximum principle. Theorem 1.1 is therefore a consequence of Theorem 1.2. In the particular case when β = 0, the condition (2) amounts to d ≥ 2, γ ∈ (0, 2), p ∈ 1, (d -γ)/(d -2) , and (1) can be written as

w L 2p,γ (R d ) ≤ C 0,γ,p ∇w ϑ L 2 (R d ) w 1-ϑ L p+1,γ (R d ) ∀ w ∈ H p 0,γ (R d ) .
In this case, we deduce from Theorem 1.1 that symmetry always holds. This is consistent with a previous result (β = 0 and γ > 0, close to 0) obtained in [START_REF] Dolbeault | Weighted interpolation inequalities: a perturbation approach[END_REF]. A few other cases were already known. The Caffarelli-Kohn-Nirenberg inequalities that were discussed in [START_REF]Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF] correspond to the critical case θ = 1, p = p or, equivalently

β = d -2 + (γ -d )/p.
Here by critical we simply mean that w L 2p,γ (R d ) scales like ∇w L 2,β (R d ) . The limit case β = γ -2 and p = 1, which is an endpoint for (2), corresponds to Hardy-type inequalities: there is no extremal function, but optimality is achieved among radial functions: see [START_REF] Dolbeault | On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities[END_REF]. The other endpoint is

β = (d -2) γ/d , in which case p = d /(d -2)
. The results of Theorem 1.1 also hold in that case with p = p = d /(d -2), up to existence issues: according to [START_REF] Wang | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF], either γ ≥ 0, symmetry holds and there exists a symmetric extremal function, or γ < 0, and then symmetry is broken but there is no optimal function.

Inequality (1) can be rewritten as an interpolation inequality with same weights on both sides using a change of variables. Here we follow the computations in [START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF] (also see [START_REF]Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF][START_REF]Symmetry of optimizers of the Caffarelli-Kohn-Nirenberg inequalities[END_REF]). Written in spherical coordinates for a function w(r, ω) = w(x) , with r = |x| and ω

= x |x| , inequality (1) becomes ∞ 0 S d -1 | w| 2p r d -γ-1 d r d ω 1 2p ≤ C β,γ,p ∞ 0 S d -1 |∇ w| 2 r d -β-1 d r d ω ϑ 2 ∞ 0 S d -1 | w| p+1 r d -γ-1 d r d ω 1-ϑ p+1
where |∇ w| 2 = ∂ w ∂r 2 + 1 r 2 |∇ ω w| 2 and ∇ ω w denotes the gradient of w with respect to the angular variable ω ∈ S d -1 . Next we consider the change of variables r → s = r α ,

w(r, ω) = v(s, ω) ∀ (r, ω) ∈ R + × S d -1 (6)
where α and n are two parameters such that

n = d -β -2 α + 2 = d -γ α .
Our inequality can therefore be rewritten as

∞ 0 S d -1 |v| 2p s n-1 d s d ω 1 2p ≤ K α,n,p ∞ 0 S d -1 α 2 ∂v ∂s 2 + 1 s 2 |∇ ω v| 2 s n-1 d s d ω ϑ 2 ∞ 0 S d -1 |v| p+1 s n-1 d s d ω 1-ϑ p+1 , with C β,γ,p = α ζ K α,n,p and ζ := ϑ 2 + 1 -ϑ p + 1 - 1 2 p = (β + 2 -γ) (p -1) 2 p d + β + 2 -2 γ -p (d -β -2) .
Using the notation Then the inequality

D α v = α ∂v ∂s , 1 s ∇ ω v , with α = 1 + β -γ 2 and n = 2 d -γ β + 2 -γ , Inequality (1)
v L 2p,d -n (R d ) ≤ K α,n,p D α v ϑ L 2,d -n (R d ) v 1-ϑ L p+1,d -n (R d ) ∀ v ∈ H p d -n,d -n (R d ) , (7)
holds with optimal constant K α,n,p = α -ζ C β,γ,p as above and optimality is achieved among radial functions if and only if

α ≤ α FS with α FS := d -1 n -1 . ( 8
)
When symmetry holds, optimal functions are equal, up to a scaling and a multiplication by a constant, to

v (x) := 1 + |x| 2 -1/(p-1) ∀ x ∈ R d .
We may notice that neither α FS nor β FS depend on p and that the curve α = α FS determines the same threshold for the symmetry breaking region as in the critical case p = p . In the case p = p , this curve was found by V. Felli and M. Schneider, who proved in [START_REF] Felli | Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type[END_REF] the linear instability of all radial critical points if α > α FS . When p = p , symmetry holds under Condition (8) as was proved in [START_REF]Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF]. Our goal is to extend this last result to the subcritical regime p ∈ (1, p ).

The change of variables s = r α is an important intermediate step, because it allows to recast the problem as a more standard interpolation inequality in which the dimension n is, however, not necessarily an integer. Actually n plays the role of a dimension in view of the scaling properties of the inequalities and, with respect to this dimension, they are critical if p = p and sub-critical otherwise. The critical case p = p has been studied in [START_REF]Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF] using tools of entropy methods, a critical fast diffusion flow and, in particular, a reformulation in terms of a generalized Fisher information. In the subcritical range, we shall replace the entropy by a Rényi entropy power as in [START_REF] Savaré | The concavity of Rényi entropy power[END_REF][START_REF] Dolbeault | Nonlinear diffusions: Extremal properties of Barenblatt profiles, best matching and delays[END_REF], and make use of the corresponding fast diffusion flow. As in [START_REF]Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF], the flow is used only at heuristic level in order to produce a well-adapted test function. The core of the method is based on the Bakry-Emery computation, also known as the carré du champ method, which is well adapted to optimal interpolation inequalities: see for instance [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF] for a general exposition of the method and [START_REF] Dolbeault | Nonlinear flows and rigidity results on compact manifolds[END_REF][START_REF]Interpolation inequalities on the sphere: linear vs. nonlinear flows[END_REF] for its use in presence of nonlinear flows. Also see [START_REF] Carrillo | Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities[END_REF] for earlier considerations on the Bakry-Emery method applied to nonlinear flows and related functional inequalities in unbounded domains. However, in non-compact manifolds and in presence of weights, integrations by parts have to be justified. In the critical case, one can rely on an additional invariance to use an Emden-Fowler transformation and rewrite the problem as an autonomous equation on a cylinder, which simplifies the estimates a lot. In the subcritical regime, estimates have to be adapted since after the Emden-Fowler transformation, the problem in the cylinder is no longer autonomous. This paper is organized as follows. We recall the computations which characterize the linear instability of radially symmetric minimizers in Section 2. In Section 3, we expose the strategy for proving symmetry in the subcritical regime when there are no weights. Section 4 is devoted to the Bakry-Emery computation applied to Rényi entropy powers, in presence of weights. This provides a proof of our main results, if we admit that no boundary term appears in the integrations by parts in Section 4. To prove this last result, regularity and decay estimates of positive solutions to [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF] are established in Section 5, which indeed show that no boundary term has to be taken into account (see Proposition 5.1).

Symmetry breaking

For completeness, we summarize known results on symmetry breaking for [START_REF] Muratori | Diffusions hypercontractives[END_REF]. Details can be found in [START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF]. With the notations of Corollary 1.3, let us define the functional

J [v] := ϑ log D α v L 2,d -n (R d ) + (1 -ϑ) log v L p+1,d -n (R d ) + log K α,n,p -log v L 2p,d -n (R d )
obtained by taking the difference of the logarithm of the two terms in [START_REF] Carrillo | Asymptotic L 1 -decay of solutions of the porous medium equation to self-similarity[END_REF]. Let us define d µ δ := µ δ (x) d x, where µ δ (x) := 1

(1 + |x| 2 ) δ .
Since v as defined in Corollary 1.3 is a critical point of J , a Taylor expansion at order ε 2 shows that

D α v 2 L 2,d -n (R d ) J v + ε µ δ/2 f = 1 2 ε 2 ϑ Q[ f ] + o(ε 2 ) with δ = 2 p p-1 and Q[ f ] = R d |D α f | 2 |x| n-d d µ δ - 4 p α 2 p -1 R d | f | 2 |x| n-d d µ δ+1 .
The following Hardy-Poincaré inequality has been established in [START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF].

Proposition 2.1 Let d ≥ 2, α ∈ (0, +∞), n > d and δ ≥ n. Then R d |D α f | 2 |x| n-d d µ δ ≥ Λ R d | f | 2 |x| n-d d µ δ+1 (9) holds for any f ∈ L 2 (R d , |x| n-d d µ δ+1 ), with D α f ∈ L 2 (R d , |x| n-d d µ δ ), such that R d f |x| n-d d µ δ+1 = 0, with an optimal constant Λ given by Λ =          2 α 2 (2 δ -n) if 0 < α 2 ≤ (d -1) δ 2 n (2 δ -n) (δ -1) , 2 α 2 δ η if α 2 > (d -1) δ 2 n (2 δ -n) (δ -1) ,
where η is the unique positive solution to

η (η + n -2) = d -1 α 2 .
Moreover, Λ is achieved by a non-trivial eigenfunction corresponding to the equality in [START_REF] Wang | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF].

If α 2 > (d -1) δ 2 n (2 δ-n) (δ-1)
, the eigenspace is generated by ϕ i (s, ω) = s η ω i , with i = 1, 2,. . . d and the eigenfunctions are not radially symmetric, while in the other case the eigenspace is generated by the radially symmetric eigenfunction ϕ 0 (s,

ω) = s 2 -n 2 δ-n .
As a consequence, Q is a nonnegative quadratic form if and only if

4 p α 2
p-1 ≤ Λ. Otherwise, Q takes negative values, and a careful analysis shows that symmetry breaking occurs in (1) if

2 α 2 δ η < 4 p α 2 p -1 ⇐⇒ η < 1 , which means d -1 α 2 = η (η + n -2) < n -1
, and this is equivalent to α > α FS .

The strategy for proving symmetry without weights

Before going into the details of the proof we explain the strategy for the case of the Gagliardo-Nirenberg inequalities without weights. There are several ways to compute the optimizers, and the relevant papers are [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF][START_REF] Carrillo | Asymptotic L 1 -decay of solutions of the porous medium equation to self-similarity[END_REF][START_REF] Carrillo | Fine asymptotics for fast diffusion equations[END_REF][START_REF] Carrillo | Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities[END_REF][START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF][START_REF] Dolbeault | Nonlinear diffusions: Extremal properties of Barenblatt profiles, best matching and delays[END_REF] (also see additional references therein). The inequality is of the form

w L 2p (R d ) ≤ C 0,0,p ∇w ϑ L 2 (R d ) w 1-ϑ L p+1 (R d ) with 1 < p < d d -2 (10)
and

ϑ = d (p -1) p d + 2 -p (d -2) .
It is known through the work in [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF] that the optimizers of this inequality are, up to multiplications by a constant, scalings and translations, given by

w (x) = 1 + |x| 2 -1 p-1 ∀ x ∈ R d .
In our perspective, the idea is to use a version of the carré du champ or Bakry-Emery method introduced in [START_REF] Muratori | Diffusions hypercontractives[END_REF]: by differentiating a relevant quantity along the flow, we recover the inequality in a form which turns out to be sharp. The version of the carré du champ we shall use is based on the Rényi entropy powers whose concavity as a function of t has been studied by M. Costa in [START_REF] Costa | A new entropy power inequality[END_REF] in the case of linear diffusions (see [START_REF] Savaré | The concavity of Rényi entropy power[END_REF] and references therein for more recent papers). In [START_REF] Villani | A short proof of the "concavity of entropy power[END_REF], C. Villani observed that the carré du champ method gives a proof of the logarithmic Sobolev inequality in the Blachman-Stam form, also known as the Weissler form: see [START_REF] Blachman | The convolution inequality for entropy powers[END_REF][START_REF] Weissler | Logarithmic Sobolev inequalities for the heat-diffusion semigroup[END_REF]. G. Savaré and G. Toscani observed in [START_REF] Savaré | The concavity of Rényi entropy power[END_REF] that the concavity also holds in the nonlinear case, which has been used in [START_REF] Dolbeault | Nonlinear diffusions: Extremal properties of Barenblatt profiles, best matching and delays[END_REF] to give an alternative proof of the Gagliardo-Nirenberg inequalities, that we are now going to sketch.

The first step consists in reformulating the inequality in new variables. We set

u = w 2p ,
which is equivalent to w = u m-1/2 , and consider the flow given by

∂u ∂t = ∆u m , ( 11
)
where m is related to p by

p = 1 2 m -1 . The inequalities 1 < p < d d -2 imply that 1 - 1 d < m < 1 . ( 12
)
For some positive constant κ > 0, one easily finds that the so-called Barenblatt-Pattle functions

u (t , x) = κ d t -d d m-d +2 w 2p κ t -1 d m-d +2 x = a + b |x| 2 -1 1-m
are self-similar solutions of [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF], where a = a(t ) and b = b(t ) are explicit. Thus, we see that w = u m-1/2 is an optimizer for [START_REF] Costa | A new entropy power inequality[END_REF] for all t and it makes sense to rewrite [START_REF] Costa | A new entropy power inequality[END_REF] in terms of the function u. Straightforward computations show that ( 10) can be brought into the form

R d u d x (σ+1) m-1 ≤ C E σ-1 I where σ = 2 d (1 -m) -1 (13)
for some constant C which does not depend on u, where

E := R d u m d x
is a generalized Ralston-Newman entropy, also known in the literature as Tsallis entropy, and

I := R d u |∇P| 2 d x
is the corresponding generalized Fisher information. Here we have introduced the pressure variable

P = m 1 -m u m-1 .
The Rényi entropy power is defined by

F := E σ
as in [START_REF] Savaré | The concavity of Rényi entropy power[END_REF][START_REF] Dolbeault | Nonlinear diffusions: Extremal properties of Barenblatt profiles, best matching and delays[END_REF]. With the above choice of σ, F is an affine function of t if u = u . For an arbitrary solution of [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF], we aim at proving that it is a concave function of t and that it is affine if and only if u = u . For further references on related issues see [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF][START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations[END_REF]. Note that one of the motivations for choosing the variable P is that it has a particular simple form for the self-similar solutions, namely

P = m 1 -m a + b |x| 2 .
Differentiating E along the flow ( 11) yields

E = (1 -m) I , so that F = σ (1 -m) G with G := E σ-1 I .
More complicated is the derivative for the Fisher information:

I = -2 R d u m Tr Hess P -1 d ∆P Id 2 + m -1 + 1 d (∆P) 2 d x .
Here Hess P and Id are respectively the Hessian of P and the (d × d ) identity matrix. The computation can be found in [START_REF] Dolbeault | Nonlinear diffusions: Extremal properties of Barenblatt profiles, best matching and delays[END_REF]. Next we compute the second derivative of the Rényi entropy power F with respect to t :

(F ) σ E σ = (σ -1) E 2 E 2 + E E = (σ -1) (1 -m) 2 I 2 E 2 + (1 -m) I E =: (1 -m) H . With σ = 2 d 1 1-m -1, we obtain H = -2 Tr Hess P -1 d ∆P Id 2 + (1 -m) (1 -σ) (∆P -〈∆P〉) 2 (14)
where we have used the notation

〈A〉 := R d u m A d x R d u m d x .
Note that by ( 12), we have that σ > 1 and hence we find that F = (E σ ) ≤ 0, which also means that G = E σ-1 I is a non-increasing function. In fact it is strictly decreasing unless P is a polynomial function of order two in x and it is easy to see that the expression ( 14) vanishes precisely when P is of the form a + b |xx 0 | 2 , where a, b ∈ R, x 0 ∈ R d are constants (but a and b may still depend on t ). Thus, while the left side of ( 13) stays constant along the flow, the right side decreases. In [START_REF] Dolbeault | Nonlinear diffusions: Extremal properties of Barenblatt profiles, best matching and delays[END_REF] it was shown that the right side decreases towards the value given by the self-similar solutions u and hence proves [START_REF] Costa | A new entropy power inequality[END_REF] in the sharp form. In our work we pursue a different tactic. The variational equation for the optimizers of ( 10) is given by -∆w = a w 2 p-1b w p .

A straightforward computation shows that this can be written in the form

2 m u m-2 div u ∇P + |∇P| 2 + c 1 u m-1 = c 2
for some constants c 1 , c 2 whose precise values are explicit. This equation can also be interpreted as the variational equation for the sharp constant in [START_REF]Interpolation inequalities on the sphere: linear vs. nonlinear flows[END_REF]. Hence, multiplying the above equation by ∆u m and integrating yields

R d 2 m u m-2 div u ∇P + |∇P| 2 ∆u m d x + c 1 R d u m-1 ∆u m d x = c 2 R d ∆u m d x = 0 .
We recover the fact that, in the flow picture, H is, up to a positive factor, the derivative of G and hence vanishes.

From the observations made above we conclude that P must be a polynomial function of order two in x. In this fashion one obtains more than just the optimizers, namely a classification of all positive solutions of the variational equation. The main technical problem with this method is the justification of the integrations by parts, which in the case at hand, without any weight, does not offer great difficulties: see for instance [START_REF] Carrillo | Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities[END_REF]. This strategy can also be used to treat the problem with weights, which will be explained next. Dealing with weights, however, requires some special care as we shall see.

The Bakry-Emery computation and Rényi entropy powers in the weighted case

Let us adapt the above strategy to the case where there are weights in all integrals entering into the inequality, that is, let us deal with inequality (7) instead of inequality [START_REF] Costa | A new entropy power inequality[END_REF]. In order to define a new, well-adapted fast diffusion flow, we introduce the diffusion operator L α = -D * α D α , which is given in spherical coordinates by

L α u = α 2 u + n -1 s u + 1 s 2 ∆ ω u
where ∆ ω denotes the Laplace-Betrami operator acting on the (d -1)-dimensional sphere S d -1 of the angular variables, and denotes here the derivative with respect to s. Consider the fast diffusion equation

∂u ∂t = L α u m ( 15
)
in the subcritical range 1 -

1 n < m = 1 -1 ν < 1.
The exponents m in [START_REF]Symmetry of optimizers of the Caffarelli-Kohn-Nirenberg inequalities[END_REF] and p in [START_REF] Carrillo | Asymptotic L 1 -decay of solutions of the porous medium equation to self-similarity[END_REF] are related as in Section 3 by

p = 1 2 m -1 ⇐⇒ m = p + 1 2 p
and ν is defined by

ν := 1 1 -m .
We consider the Fisher information defined as

I [P] := R d u |D α P| 2 d µ with P = m 1 -m u m-1 and d µ = s n-1 d s d ω = s n-d d x .
Here P is the pressure variable. Our goal is to prove that P takes the form a + b s 2 , as in Section 3. It is useful to observe that (15) can be rewritten as ∂u ∂t

= D * α (u D α P)
and, in order to compute d I dt , we will also use the fact that P solves

∂P ∂t = (1 -m) PL α P -|D α P| 2 . (16) 4
m m-1 |D α P| 2 -2 (1 -m) P m m-1 P L α P d ς = s n-1 S d -1 ∂ ∂s P m m-1 |D α P| 2 -2 (1 -m) P m m-1 P L α P d ω (s) , (17)
where by d ς = s n-1 d ω we denote the standard Hausdorff measure on ∂B s .

Lemma 4.1 If u solves (15) and if lim s→0 + b(s) = lim S→+∞ b(S) = 0 , (18)
then,

d dt I [P] = -2 R d K [P] u m d µ . ( 19
)
Proof. For 0 < s < S < +∞, let us consider the set

A (s,S) := x ∈ R d : s < |x| < S , so that ∂A (s,S) = ∂B s ∪ ∂B S .
Using [START_REF]Symmetry of optimizers of the Caffarelli-Kohn-Nirenberg inequalities[END_REF] and ( 16), we can compute

d dt A (s,S) u |D α P| 2 d µ = A (s,S)
∂u ∂t

|D α P| 2 d µ + 2 A (s,S) u D α P • D α ∂P ∂t d µ = A (s,S) L α (u m ) |D α P| 2 d µ + 2 A (s,S) u D α P • D α (1 -m) PL α P -|D α P| 2 d µ = A (s,S) u m L α |D α P| 2 d µ + 2 (1 -m) A (s,S) u PD α P • D α L α Pdµ + 2 (1 -m) A (s,S) u D α P • D α PL α Pdµ -2 A (s,S) u D α P • D α |D α P| 2 d µ + α 2 ∂B S (u m ) |D α P| 2 -u m ∂ ∂s (|D α P| 2 ) d ς -α 2 ∂B s (u m ) |D α P| 2 -u m ∂ ∂s (|D α P| 2 ) d ς = - A (s,S) u m L α |D α P| 2 d µ + 2 (1 -m) A (s,S) u PD α P • D α L α Pdµ + 2 (1 -m) A (s,S) u D α P • D α PL α Pdµ + α 2 ∂B S (u m ) |D α P| 2 + u m ∂ ∂s (|D α P| 2 ) d ς -α 2 ∂B s (u m ) |D α P| 2 + u m ∂ ∂s (|D α P| 2 ) d ς ,
where the last line is given by an integration by parts, upon exploiting the identity u D α P = -D α (u m ):

A (s,S) u D α P • D α |D α P| 2 d µ = - A (s,S) D α (u m ) • D α |D α P| 2 d µ = A (s,S) u m L α |D α P| 2 d µ -α 2 ∂B S u m ∂ ∂s (|D α P| 2 ) d ς + α 2 ∂B s u m ∂ ∂s (|D α P| 2 ) d ς .
1) Using the definition of A [P], we get that

- A (s,S) u m L α |D α P| 2 d µ = -2 A (s,S) u m A [P] d µ -2 A (s,S) u m D α P • D α L α Pdµ. (20)
2) Taking advantage again of u D α P = -D α (u m ), an integration by parts gives

A (s,S) u D α P • D α PL α Pdµ = - A (s,S) D α (u m ) • D α PL α Pdµ = A (s,S) u m (L α P) 2 d µ + A (s,S) u m D α P • D α L α Pdµ -α 2 ∂B S u m P L α Pd ς + α 2 ∂B s u m P L α Pd ς .
and, with u P = m 1-m u m , we find that 10

2 (1 -m) A (s,S) u PD α P • D α L α Pdµ + 2 (1 -m) A (s,S) u D α P • D α PL α Pdµ = 2 (1 -m) A (s,S) u m (L α P) 2 d µ + 2 A (s,S) u m D α P • D α L α Pdµ -2 (1 -m) α 2 ∂B S u m P L α Pd ς + 2 (1 -m) α 2 ∂B s u m P L α Pd ς . ( 21
)
Summing ( 20) and ( 21), using [START_REF] Dolbeault | Weighted interpolation inequalities: a perturbation approach[END_REF] and passing to the limits as s → 0 + , S → +∞, establishes [START_REF] Felli | Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type[END_REF].

4.2. Second step: two remarkable identities.

Let us define

k[P] := 1 2 ∆ ω |∇ ω P| 2 -∇ ω P • ∇ ω ∆ ω P -1 n-1 (∆ ω P) 2 -(n -2) α 2 |∇ ω P| 2
and

R[P] := K [P] - 1 n -(1 -m) (L α P) 2 .
We observe that

R[P] = 1 2 L α |D α P| 2 -D α P • D α L α P - 1 n (L α P) 2
is independent of m. We recall the result of [14, Lemma 5.1] and give its proof for completeness.

Lemma 4.2 Let d ∈ N, n ∈ R such that n > d ≥ 2
, and consider a function

P ∈ C 3 (R d \ {0}). Then, R[P] = α 4 1 - 1 n P - P s - ∆ ω P α 2 (n -1) s 2 2 + 2 α 2 s 2 ∇ ω P - ∇ ω P s 2 + k[P] s 4 .
Proof. By definition of R[P], we have

R[P] = α 2 2 α 2 P 2 + |∇ ω P| 2 s 2 + α 2 2 n -1 s α 2 P 2 + |∇ ω P| 2 s 2 + 1 2 s 2 ∆ ω α 2 P 2 + |∇ ω P| 2 s 2 -α 2 P α 2 P + α 2 n -1 s P + ∆ ω P s 2 - 1 s 2 ∇ ω P • ∇ ω α 2 P + α 2 n -1 s P + ∆ ω P s 2 - 1 n α 2 P + α 2 n -1 s P + ∆ ω P s 2 2 ,
which can be expanded as

R[P] = α 2 2 2 α 2 P 2 + 2 α 2 P P + 2 |∇ ω P | 2 + ∇ ω P • ∇ ω P s 2 - 8
∇ ω P • ∇ ω P s 3 + 6 |∇ ω P| 2 s 4 + α 2 n -1 s α 2 P P + ∇ ω P • ∇ ω P s 2 - |∇ ω P| 2 s 3 + 1 s 2 α 2 P ∆ ω P + α 2 |∇ ω P | 2 + ∆ ω |∇ ω P| 2 2 s 2 -α 2 P α 2 P + α 2 n -1 s P -α 2 n -1 s 2 P - 2
∆ ω P s 3 + ∆ ω P s 2 - 1 s 2 α 2 ∇ ω P • ∇ ω P + α 2 n -1 s ∇ ω P • ∇ ω P + ∇ ω P • ∇ ω ∆ ω P s 2 - 1 n α 4 P 2 + α 4 (n -1) 2 s 2 P 2 + (∆ ω P) 2 s 4 + 2 α 4 n -1 s P P + 2 α 2 P ∆ ω P s 2 + 2 α 2 n - 1
s 3 P ∆ ω P .
Collecting terms proves the result. Now let us study the quantity k[P] which appears in the statement of Lemma 4.2. The following computations are adapted from [START_REF] Dolbeault | Nonlinear flows and rigidity results on compact manifolds[END_REF] and [14, Section 5]. For completeness, we give a simplified proof in the special case of the sphere (S d -1 , g ) considered as a Riemannian manifold with standard metric g . We denote by H f the Hessian of f , which is seen as (d -1) × (d -1) matrix, identify its trace with the Laplace-Beltrami operator on S d -1 and use the notation A 2 := A : A for the sum of the squares of the coefficients of the matrix A. It is convenient to define the trace free Hessian, the tensor Z f and its trace free counterpart respectively by

L f := H f - 1 d -1 (∆ ω f ) g , Z f := ∇ ω f ⊗ ∇ ω f f and M f := Z f - 1 d -1 |∇ ω f | 2 f g whenever f = 0. Elementary computations show that L f 2 = H f 2 - 1 d -1 (∆ ω f ) 2 and M f 2 = Z f 2 - 1 d -1 |∇ ω f | 4 f 2 = d -2 d -1 |∇ ω f | 4 f 2 . ( 22
)
The Bochner-Lichnerowicz-Weitzenböck formula on S d -1 takes the simple form

1 2 ∆ ω (|∇ ω f | 2 ) = H f 2 + ∇ ω (∆ ω f ) • ∇ ω f + (d -2) |∇ ω f | 2 (23)
where the last term, i.e., Ric(∇ ω f ,

∇ ω f ) = (d -2) |∇ ω f | 2
, accounts for the Ricci curvature tensor contracted with

∇ ω f ⊗ ∇ ω f .
We recall that α FS := d -1 n-1 and ν = 1/(1m). Let us introduce the notations

δ := 1 d -1 - 1 n -1 and B[P] := S d -1 1 2 ∆ ω (|∇ ω P| 2 ) -∇ ω (∆ ω P) • ∇ ω P -1 n-1 (∆ ω P) 2 P 1-ν d ω , so that S d -1 k[P]P 1-ν d ω = B[P] -(n -2) α 2 S d -1
|∇ ω P| 2 P 1-ν d ω . ),

S d -1 k[P]P 1-ν d ω ≥ (n -2) α 2 FS -α 2 S d -1 |∇ ω P| 2 P 1-ν d ω + c(n, m, d ) S d -1
|∇ ω P| 4 P 2 P 1-ν d ω .

Proof. If d = 2, we identify S 1 with [0, 2π) θ and denote by P θ and P θθ the first and second derivatives of P with respect to θ. As in [14, Lemma 5.3], a direct computation shows that

k[P] = n -2 n -1 |P θθ | 2 -(n -2) α 2 |P θ | 2 = (n -2) α 2 FS |P θθ | 2 -α 2 |P θ | 2 .
By the Poincaré inequality, we have

S 1 ∂ ∂θ P 1-ν 2 P θ 2 d θ ≥ S 1 P 1-ν 2 P θ 2 d θ .
On the other hand, an integration by parts shows that

S 1 P θθ |P θ | 2 P P 1-ν d θ = 1 3 S 1 ∂ ∂θ |P θ | 2 P θ P -ν d θ = ν 3 S 1 |P θ | 4 P 2 P 1-ν d θ
and, as a consequence, by expanding the square, we obtain

S 1 ∂ ∂θ P 1-ν 2 P θ 2 d θ = S 1 P θθ + 1 -ν 2 |P θ | 2 P 2 P 1-ν d θ = S 1 |P θθ | 2 P 1-ν d θ - (ν -1) (ν + 3) 12 S 1 |P θ | 4 P 2 P 1-ν d θ .
The result follows with c(n, m, 2) = n-2 n-1

1 12 (ν -1) (ν + 3) = n-2 n-1 m (4-3 m) 12 (1-m) 2 from S 1 |P θθ | 2 P 1-ν d θ ≥ S 1 |P θ | 2 P 1-ν d θ + (ν -1) (ν + 3) 12 S 1 |P θ | 4 P 2 P 1-ν d θ .
Assume next that d ≥ 3. We follow the method of [14, Lemma 5.2]. Applying [START_REF] Villani | A short proof of the "concavity of entropy power[END_REF] with f = P and multiplying by P 1-ν yields, after an integration on S d -1 , that B[P] can also be written as

B[P] = S d -1 HP 2 + (d -2) |∇ ω P| 2 -1 n-1 (∆ ω P) 2 P 1-ν d ω .
We recall that n > d ≥ 3 and set

P = f β with β = 2 3-ν . A straightforward computation shows that H f β = β f β-1 H f + (β -1) Z f and hence B[P] = β 2 S d -1 H f + (β -1) Z f 2 + (d -2) |∇ ω f | 2 -1 n-1 Tr (H f + (β -1) Z f ) 2 d ω = β 2 S d -1 L f + (β -1) M f 2 + (d -2) |∇ ω f | 2 + δ Tr (H f + (β -1) Z f ) 2 d ω .
Using [START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations[END_REF], we deduce from

S d -1 ∆ ω f |∇ ω f | 2 f d ω = S d -1 |∇ ω f | 4 f 2 d ω -2 S d -1 H f : Z f d ω = d -1 d -2 S d -1 M f 2 d ω -2 S d -1 L f : Z f d ω - 2 d -1 S d -1 ∆ ω f |∇ ω f | 2 f d ω that S d -1 ∆ ω f |∇ ω f | 2 f d ω = d -1 d + 1 S d -1 d -1 d -2 M f 2 d ω -2 S d -1 L f : Z f d ω = d -1 d + 1 S d -1 d -1 d -2 M f 2 d ω -2 S d -1
L f : M f d ω on the one hand, and from (23) integrated on S d -1 that

S d -1 (∆ ω f ) 2 d ω = d -1 d -2 S d -1 L f 2 d ω + (d -1) S d -1 |∇ ω f | 2 d ω
on the other hand. Hence we find that

S d -1 Tr (H f + (β -1) Z f ) 2 d ω = S d -1 (∆ ω f ) 2 + 2 (β -1) ∆ ω f |∇ ω f | 2 f + (β -1) 2 |∇ ω f | 4 f 2 d ω = d -1 d -2 S d -1 L f 2 d ω + (d -1) S d -1 |∇ ω f | 2 d ω + 2 (β -1) d -1 d + 1 S d -1 d -1 d -2 M f 2 d ω -2 S d -1 L f : M f d ω + (β -1) 2 d -1 d -2 S d -1 M f 2 d ω .
Altogether, we obtain

B[P] = β 2 S d -1 a L f 2 + 2 bL f : M f + c M f 2 d ω + β 2 d -2 + δ (d -1) S d -1 |∇ ω f | 2 d ω where a = 1 + δ d -1 d -2 , b = (β -1) 1 -2 δ d -1 d + 1 and c = (β -1) 2 1 + δ d -1 d -2 + 2 (β -1) δ (d -1) 2 (d + 1) (d -2) .
A tedious but elementary computation shows that

B[P] = aβ 2 S d -1 L f + b a M f 2 d ω + c -b 2 a β 2 S d -1 M f 2 d ω + β 2 (n -2) α 2 FS S d -1
|∇ ω f | 2 d ω can be written in terms of P as

B[P] = S d -1 Q[P] P 1-ν d ω + (n -2) α 2 FS S d -1
|∇ ω P| 2 P 1-ν d ω

where

Q[P] := α 2 FS n -2 d -2 LP + 3 (ν-1) (n-d ) (d +1) (n-2) (ν-3) MP 2 + (d -1) (ν-1) (n-d ) [((4 d -5) n+d -8) ν+9 (n-d ))] (d -2) (d +1) 2 (ν-3) 2 (n-2) (n-1)
MP 2 is positive definite. This concludes the proof in the case

d ≥ 3 with c(n, m, d ) = m (n-d ) [4 (d +1) (n-2)-9 m (n-d )] (d +1) 2 (3 m-2) 2 (n-2) (n-1)
. Let us recall that

K [P] = R[P] + 1 n -(1 -m) (L α P) 2 .
We can collect the two results of Lemmas 4.2 and 4.3 as follows. 1-m u m-1 for some m ∈ (1 -1 n , 1), then there exists a positive constant c(n, m, d ) such that

R d R[P] u m d µ ≥ α 4 1 - 1 n R d P - P s - ∆ ω P α 2 (n -1) s 2 2 u m d µ + 2 α 2 R d 1 s 2 ∇ ω P - ∇ ω P s 2 u m d µ + (n -2) α 2 FS -α 2 R d 1 s 4 |∇ ω P| 2 u m d µ + c(n, m, d ) R d 1 s 4
|∇ ω P| 4 P 2 u m d µ .

Third step: concavity of the Rényi entropy powers and consequences

We keep investigating the properties of the flow defined by [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF]. Let us define the entropy as

E := R d u m d
F := E σ
for some exponent σ to be chosen later, and find that

F = σ (1 -m) G where G := E σ-1 I . With H := E -σ G , by using Lemma 4.1, we also find that E -σ F = σ (1 -m) H where E 2 H = E 2-σ G = 1 σ (1 -m) E 2-σ F = (1 -m) (σ -1) R d u |D α P| 2 d µ 2 -2 R d u m d µ R d K [P] u m d µ = (1 -m) (σ -1) R d u |D α P| 2 d µ 2 -2 1 n -(1 -m) R d u m d µ R d (L α P) 2 u m d µ -2 R d u m d µ R d R[P] u m d µ if lim s→0 + b(s) = lim S→+∞ b(S) = 0. Using u D α P = -D α (u m ), we know that R d u |D α P| 2 d µ = - R d D α (u m ) • D α Pdµ = R d u m L α Pdµ
and so, with the choice

σ = 2 n 1 1 -m -1 ,
we may argue as in Section 3 and get that

E 2 H + (1 -m) (σ -1) E R d u m L α P -R d u |D α P| 2 d µ R d u m d µ 2 d µ + 2 E R d R[P] u m d µ = 0 if lim s→0 + b(s) = lim S→+∞ b(S) = 0. So, if α ≤ α FS and P is of class C 3 , by Corollary 4.4, as a function of t , F is con- cave, that is, G = E σ-1 I is non-increasing in t .
Formally, G converges towards a minimum, for which necessarily L α P is a constant and R[P] = 0, which proves that P(x) = a+b|x| 2 for some real constants a and b, according [START_REF] Muratori | Diffusions hypercontractives[END_REF], since for some constant κ,

to Corollary 4.4. Since 2 (1-ϑ) ϑ (p+1) = σ -1, the minimization of G under the mass constraint R d u d µ = R d v 2p d µ is equivalent to the Caffarelli-Kohn-Nirenberg interpolation inequalities
G = E σ-1 I = κ R d v p+1 d µ σ-1 R d |D α v| 2 d µ with v = u m-1/2 .
We emphasize that [START_REF]Symmetry of optimizers of the Caffarelli-Kohn-Nirenberg inequalities[END_REF] preserves mass, that is,

d dt R d v 2p d µ = d dt R d u d µ = R d L α u m d µ = 0
because, as we shall see in Proposition 5.1, no boundary terms appear when integrating by parts if v is an extremal function associated with [START_REF] Carrillo | Asymptotic L 1 -decay of solutions of the porous medium equation to self-similarity[END_REF]. In particular, for mass conservation we need lim

s→0 + ∂B s u P d ς = lim S→+∞ ∂B S u P d ς = 0 . ( 25
)
The above remarks on the monotonicity of G and the symmetry properties of its minimizers can in fact be extended to the analysis of the symmetry properties of all critical points of G . This is actually the contents of Theorem 1.2.

Proof of Theorem 1.2. Let w be a positive solution of equation [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF]. As pointed out above, by choosing

w(x) = u m-1/2 (r α , ω) ,
we know that u is a critical point of G under a mass constraint on R d u d x, so that we can write the corresponding Euler-Lagrange equation as dG (25). Using L α u m as a test function amounts to apply the flow of ( 15) to G with initial datum u and compute the derivative with respect to t at t = 0. This means [START_REF] Weissler | Logarithmic Sobolev inequalities for the heat-diffusion semigroup[END_REF] holds. Here we have used Lemma 4.1. We emphasize that this proof is purely variational and does not rely on the properties of the solutions to [START_REF]Symmetry of optimizers of the Caffarelli-Kohn-Nirenberg inequalities[END_REF], although using the flow was very useful to explain our strategy. All we need is that no boundary term appears in the integrations by parts.

[u] = C , for some constant C . That is, R d dG [u] • L α u m d µ = C R d L α u m d µ = 0 thanks to
0 = R d dG [u] • L α u m d µ = E σ H = -(1 -m) (σ -1) E σ-1 R d u m L α P -R d u |D α P| 2 d µ R d u m d µ 2 d µ -2 E σ-1 R d R[P] u m d µ if lim s→0 + b(s) = lim S→+∞ b(S) = 0 and
Hence, in order to obtain a complete proof, we have to prove that ( 18), ( 24) and ( 25) hold with b defined by [START_REF] Dolbeault | Weighted interpolation inequalities: a perturbation approach[END_REF], whenever u is a critical point of G under mass constraint. This will be done in Proposition 5.1. Using Corollary 4.4, we know that R[P] = 0, ∇ ω P = 0 a.e. in R d and

L α P = R d u |D α P| 2 d µ R d u m d µ
a.e. in R d , with P = m 1-m u m-1 . We conclude as in [START_REF]Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF]Corollary 5.5] that P is an affine function of s 2 . 2

Regularity and decay estimates

In this last section we prove the regularity and decay estimates on w (or on P or u) that are necessary to establish the absence of boundary terms in the integrations by parts of Section 4.

Proposition 5.1 Under Condition (2), if w is a positive solution in H p β,γ (R d ) of (5), then (18)
, ( 24) and (25) hold with b as defined by [START_REF] Dolbeault | Weighted interpolation inequalities: a perturbation approach[END_REF], u = v 2p and v given by (6).

To prove this result, we split the proof in several steps: we will first establish a uniform bound and a decay rate for w inspired by [START_REF] Dolbeault | Weighted interpolation inequalities: a perturbation approach[END_REF] in Lemmas 5.2, 5.3, and then follow the methodology of [START_REF]Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF]Appendix] in the subsequent Lemma 5.4. Lemma 5.2 Let β, γ and p satisfy the relations (2). Any positive solution w of (5) such that

w L 2p,γ (R d ) + ∇w L 2,β (R d ) + w 1-ϑ L p+1,γ (R d ) < +∞ . ( 26
)
is uniformly bounded and tends to 0 at infinity, uniformly in |x|.

Proof. The strategy of the first part of the proof is similar to the one in [17, Lemma 3.1], which was restricted to the case β = 0. Let us set δ 0 := 2 (p -p). For any A > 0, we multiply ( 5) by (w ∧ A) 1+δ 0 and integrate by parts (or, equivalently, plug it in the weak formulation of ( 5)): we point out that the latter is indeed an admissible test function since w ∈ H p β,γ (R d ). In that way, by letting A → +∞, we obtain the identity

4 (1 + δ 0 ) (2 + δ 0 ) 2 R d ∇w 1+δ 0 /2 2 |x| -β d x + R d w p+1+δ 0 |x| -γ d x = R d w 2p+δ 0 |x| -γ d x .
By applying (1) with p = p (so that ϑ = 1) to the function w = w 1+δ 0 /2 , we deduce that

w 2+δ 0 L 2p+δ 1 ,γ (R d ) ≤ (2 + δ 0 ) 2 4 (1 + δ 0 ) C 2 β,γ,p w 2p+δ 0 L 2p+δ 0 ,γ (R d )
with 2 p + δ 1 = p (2 + δ 0 ). Let us define the sequence {δ n } by the induction relation

δ n+1 := p (2 + δ n ) -2 p for any n ∈ N, that is, δ n = 2 p -p p -1 p n+1 -1 ∀ n ∈ N
, and take q n = 2 p + δ n . If we repeat the above estimates with δ 0 replaced by δ n and δ 1 replaced by δ n+1 , we get

w 2+δ n L q n+1 ,γ (R d ) ≤ (2 + δ n ) 2 4 (1 + δ n ) C 2 β,γ,p w q n L qn ,γ (R d )
.

By iterating this estimate, we obtain the estimate

w L qn ,γ (R d ) ≤ C n w ζ n L 2p ,γ (R d ) with ζ n = (p -1) p n p -1 + (p -p) p n
where the sequence {C n } is defined by C 0 = 1 and

C 2+δ n n+1 = (2 + δ n ) 2 4 (1 + δ n ) C 2 β,γ,p C q n n ∀ n ∈ N .
The sequence {C n } converges to a finite limit C ∞ . Letting n → ∞ we obtain the uniform bound

w L ∞ (R d ) ≤ C ∞ w ζ ∞ L 2p ,γ (R d ) ≤ C ∞ C β,γ,p ∇w L 2,β (R d ) ζ ∞ ≤ C ∞ C β,γ,p w p L 2p,γ (R d ) ζ ∞ where ζ ∞ = p -1 p -p = lim n→∞ ζ n .
In order to prove that lim |x|→+∞ w(x) = 0, we can suitably adapt the above strategy. We shall do it as follows: we truncate the solution so that the truncated function is supported outside of a ball of radius R 0 and apply the iteration scheme. Up to an enlargement of the ball, that is, outside of a ball of radius R ∞ = a R 0 for some fixed numerical constant a > 1, we get that w L ∞ (B c R∞ ) is bounded by the energy localized in B c R 0 . The conclusion will hold by letting R 0 → +∞. Let us give some details.

Let ξ ∈ C ∞ (R + ) be a cut-off function such that 0 ≤ ξ ≤ 1, ξ ≡ 0 in [0, 1) and ξ ≡ 1 in (2, +∞). Given R 0 ≥ 1, consider the sequence of radii defined by

R n+1 = 1 + 1 n 2 R n ∀ n ∈ N .
By taking logarithms, it is immediate to deduce that {R n } is monotone increasing and that there exists a > 1 such that

R ∞ := lim n→∞ R n = a R 0 .
Let us then define the sequence of radial cut-off functions {ξ n } by

ξ n (x) := ξ 2 |x| -R n R n+1 -R n + 1 ∀ x ∈ R d .
Direct computations show that there exists some constant c > 0, which is independent of n and R 0 , such that

|∇ξ n (x)| ≤ c n 2 R n χ B R n+1 \B Rn , ∇ξ 1/2 n (x) ≤ c n 2 R n χ B R n+1 \B Rn , |∆ξ n (x)| ≤ c n 4 R 2 n χ B R n+1 \B Rn ∀ x ∈ R d . ( 27
)
From here on we denote by c, c , etc. positive constants which are all independent of n and R 0 . We now introduce the analogue of the sequence {δ n } above, which we relabel {σ n } to avoid confusion. Namely, we set σ 0 := 2 p -2 and σ n+1 = p (2 + σ n ) -2, so that σ n = 2 (p p n -1). If we multiply (5) by ξ n w 1+σ n and integrate by parts, we obtain:

R d ∇ ξ n w 1+σ n • ∇w |x| -β d x + R d ξ n w p+1+σ n |x| -γ d x = R d ξ n w 2p+σ n |x| -γ d x , whence 4 (1 + σ n ) (2 + σ n ) 2 R d ξ n ∇w 1+σ n /2 2 |x| -β d x + 1 2 + σ n R d ∇ξ n • ∇w 2+σ n |x| -β d x ≤ B c Rn w 2p+σ n |x| -γ d x .
By integrating by parts the second term in the l.h.s. and combining this estimate with

R d ∇ ξ 1/2 n w 1+σ n /2 2 |x| -β d x ≤ 2 R d ξ n ∇w 1+σ n /2 2 |x| -β d x + 2 R d ∇ξ 1/2 n 2 w 2+σ n |x| -β d x ,
we end up with

2 (1 + σ n ) (2 + σ n ) 2 R d ∇ ξ 1/2 n w 1+σ n /2 2 |x| -β d x - 4 (1 + σ n ) (2 + σ n ) 2 R d ∇ξ 1/2 n 2 w 2+σ n |x| -β d x - 1 2 + σ n R d |x| -β ∆ξ n -β |x| -β-2 x • ∇ξ n w 2+σ n d x ≤ B c Rn w 2p+σ n |x| -γ d x .
Thanks to (27), we can deduce that

R d ∇ ξ 1/2 n w 1+σ n /2 2 |x| -β d x ≤ B R n+1 \B Rn 2 c 2 + c R 2 n n 4 + β c R n n 2 |x| -1 w 2+σ n |x| -β d x + (2 + σ n ) 2 2 (1 + σ n ) B c Rn w 2p+σ n |x| -γ d x .
In particular,

R d ∇ ξ 1/2 n w 1+σ n /2 2 |x| -β d x ≤ c n 4 B c Rn w 2+σ n |x| -β-2 d x + (2 + σ n ) 2 2 (1 + σ n ) w 2p-2 ∞ B c Rn w 2+σ n |x| -γ d x .
Since [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF] implies that β + 2 > γ, by exploiting the explicit expression of σ n and applying (1) with p = p (and ϑ = 1) to the function ξ 1/2 n w 1+σ n /2 , we can rewrite our estimate as w

2+σ n L 2+σ n+1 ,γ (B c R n+1 ) c p n w 2+σ n L 2+σn ,γ (B c
Rn ) . After iterating the scheme and letting n → ∞ we end up with

w L ∞ (B c R∞ ) ≤ c w L 2p,γ (B c R 0
) .

Since w is bounded in L 2p,γ (R d ), in order to prove the claim it is enough to let R 0 → +∞. 1) .

C 1 |x| (γ-2-β)/(p-1) ≤ w(x) ≤ C 2 |x| (γ-2-β)/(p-
Proof. By Lemma 5.2 and elliptic bootstrapping methods we know that w ∈ C ∞ (R d \ {0}). Let us now consider the function h(x) := C |x| (γ-2-β)/(p-1) , which satisfies the differential inequality

-div |x| -β ∇h + (1 -ε) |x| -γ h p ≥ 0 ∀ x ∈ R d \ {0}
for any ε ∈ (0, 1) and

C such that C p-1 > 2-γ+β 1-ε d -γ-p (d -2-β) (p-1) 2
. On the other hand, by Lemma 5.2, w 2p-1 is negligible compared to w p as |x| → ∞ and, as a consequence, for any ε > 0 small enough, there is an

R ε > 0 such that -div |x| -β ∇w + (1 -ε) |x| -γ w p ≤ 0 if |x| ≥ R ε . With q := (1 -ε) |x| -γ h p -w p h-w ≥ 0, it follows that -div |x| -β ∇(h -w) + q (h -w) ≥ 0 if |x| ≥ R ε .
Hence, for C large enough, we know that h(x) ≥ w(x) for any x ∈ R d such that |x| = R ε , and we also have that lim |x|→+∞ h(x)w(x) = 0. Using the Maximum Principle, we conclude that 0 ≤ w(x) ≤ h(x) for any x ∈ R d such that |x| ≥ R ε . The lower bound uses a similar comparison argument. Indeed, since

-div |x| -β ∇w + |x| -γ w p ≥ 0 ∀ x ∈ R d \ {0}
and

-div |x| -β ∇h + |x| -γ h p ≤ 0 ∀ x ∈ R d \ {0} if we choose C such that C p-1 ≤ (2 -γ + β) d -γ-p (d -2-β) (p-1) 2
, we easily see that

w(x) ≥ min |x|=1 w(x) ∧C |x| (γ-2-β)/(p-1) ∀x ∈ R d \ B 1 .
This concludes the proof.

Our next goal is to obtain growth and decay estimates, respectively, on the functions P and u as they appear in the proof of Theorem 1.2 in Section 4, in order to prove Proposition 5.1. We also need to estimate their derivatives near the origin and at infinity. Let us start by reminding the change of variables [START_REF] Carrillo | Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities[END_REF], which in particular, by Lemma 5.3, implies that for some positive constants C 1 and C 2 ,

C 1 s 2/(1-p) ≤ v(s, ω) ≤ C 2 s 2/(1-p) as s → +∞ .
Then we perform the Emden-Fowler transformation

v(s, ω) = s a ϕ(z, ω) with z = -log s , a = 2 -n 2 , ( 28
)
and see that ϕ satisfies the equation

-α 2 ϕ -∆ ω ϕ + a 2 α 2 ϕ = e ((n-2) p-n) z ϕ 2p-1 -e ((n-2) p-n-2) z/2 ϕ p =: h in C := R × S d -1 (z, ω) . ( 29
)
From here on we shall denote by the derivative with respect either to z or to s, depending on the argument. By definition of ϕ and using Lemma 5.3, we obtain that

ϕ(z, ω) ∼ e 2-n 2 + 2 p-1 z as z → -∞ ,
where we say that f (z, ω) ∼ g (z, ω) as z → +∞ (resp. z → -∞) if the ratio f /g is bounded from above and from below by positive constants, independently of ω, and for z (resp. -z) large enough. Concerning z → +∞, we first note that Lemma 5.

(z, ω)| , |ϕ (z, ω)| , |∇ ω ϕ(z, ω)| , |∇ ω ϕ (z, ω)| , |∇ ω ϕ (z, ω)| , |∆ ω ϕ(z, ω)| ≤ O(ϕ(z, ω)) , (30)
uniformly in ω. Here we denote by ∇ ω the differentiation with respect to ω. As a consequence, we have, uniformly in ω, and for ∈ {0, 1, 2}, t ∈ {0, 1},

|∂ z ∇ t ω h(z, ω)| ≤ O e -n+2 2 z as z → +∞ , |∂ z ∇ t ω h(z, ω)| ≤ O(e -n+2 2 + 2 p p-1 z ) as z → -∞ , ( 31
) |∆ ω h(z, ω)| ≤ O e -n+2 2 z as z → +∞ , |∆ ω h(z, ω)| ≤ O(e -n+2 2 + 2 p p-1 z ) as z → -∞ . ( 32
)
Lemma 5.4 Let β, γ and p satisfy the relations (2) and assume α ≤ α FS . For any positive solution w of (5) satisfying (26), the pressure function P = m 1-m u m-1 is such that P , P /s, P/s 2 , ∇ ω P /s, ∇ ω P/s 2 and L α P are of class C ∞ and bounded as s → +∞. On the other hand, as s → 0 + we have

(i) S d -1 |P (s, ω)| 2 d ω ≤ O(1), (ii) S d -1 |∇ ω P(s,ω)| 2 d ω ≤ O(s 2 ), (iii) S d -1 |P (s, ω)| 2 d ω ≤ O(1/s 2 ), (iv) S d -1 ∇ ω P (s, ω) -1 s ∇ ω P(s,ω) 2 d ω ≤ O(1), (v) S d -1 1
s 2 ∆ ω P(s,ω) 2 d ω ≤ O(1/s 2 ).
Proof. By using the change of variables (28), we see that

P(s,ω) = p+1 p-1 e -1 2 (n-2) (p-1) z ϕ 1-p (z, ω) , z = -log s .
From (30) we easily deduce that uniformly in ω, P , P /s, P/s 2 , ∇ ω P /s, ∇ ω P/s 2 and L α P are of class C ∞ and bounded as s → +∞. Moreover, as s → 0 + , we obtain that

P (s, ω) ≤ O 1 s ϕ (z, ω) ϕ(z, ω) -a and 1 s ∇ ω P(s,ω) ≤ O 1 s ∇ ω ϕ(z, ω) ϕ(z, ω)
are of order at most 1/s uniformly in ω. Similarly we obtain that

|P (s, ω)| ≤ O 1 s 2 ϕ (z, ω) ϕ(z, ω) -p |ϕ (z, ω)| 2 |ϕ(z, ω)| 2 + 1 -2 a (1 -p) ϕ (z, ω) ϕ(z, ω) + a 2 (1 -p) -a , ∇ ω P (s, ω) s - a(1 -p) s 2 ∇ ω P(s,ω) ≤ O 1 s 2 ∇ ω ϕ (z, ω) ϕ(z, ω) - p ϕ (z, ω) ∇ ω ϕ(z, ω) |ϕ(z, ω)| 2 , 1 s 2 |∆ ω P(s,ω)| ≤ O 1 s 2 ∆ ω ϕ(z, ω) ϕ(z, ω) -p |∇ ω ϕ(z, ω)| 2 |ϕ(z, ω)| 2 ,
are at most of order 1/s 2 uniformly in ω. This shows that |b(s)| ≤ O(s n-4 ) as s → 0 + and concludes the proof if 4 ≤ d < n. When d = 2 or 3 and n ≤ 4, more detailed estimates are needed. Properties (i)-(v) amount to prove that

(i) S d -1 ϕ (z,ω) ϕ(z,ω) -a 2 d ω ≤ O(e -2 z ), (ii) S d -1 ∇ ω ϕ(z,ω) ϕ(z,ω) 2 d ω ≤ O(e -2 z ), (iii) S d -1 ϕ (z,ω) ϕ(z,ω) -p |ϕ (z,ω)| 2 |ϕ(z,ω)| 2 + 1 -2 a (1 -p) ϕ (z,ω) ϕ(z,ω) + a 2 (1 -p) -a 2 d ω ≤ O(e -2 z ), (iv) S d -1 ∇ ω ϕ (z,ω) ϕ(z,ω) - p ϕ (z,ω) ∇ ω ϕ(z,ω) |ϕ(z,ω)| 2 2 d ω ≤ O(e -2 z ), (v) S d -1 ∆ ω ϕ(z,ω) ϕ(z,ω) -p |∇ ω ϕ(z,ω)| 2 |ϕ(z,ω)| 2 2 d ω ≤ O(e -2 z ),
as z → +∞.

Step 1: Proof of (ii) and (iv). If w is a positive solution of (5), then ϕ is a positive solution to (29). With ∈ {0, 1, 2}, applying the operator ∇ ω ∂ z to the equation (29) we obtain

-α 2 (∇ ω ∂ z ϕ) -∇ ω ∆ ω ∂ z ϕ + a 2 α 2 ∇ ω ∂ z ϕ = ∇ ω ∂ z h(z, ω) in C . Define χ (z) := 1 2 S d -1 |∇ ω ∂ z ϕ| 2 d ω ,
which by (30) converges to 0 as z → ±∞. Assume first that χ is a positive function. After multiplying the above equation by ∇ ω ∂ z ϕ, integrating over S d -1 , integrating by parts and using

χ = S d -1 ∇ ω ∂ z ϕ ∇ ω ∂ z ϕ d ω and χ = S d -1 ∇ ω ∂ z ϕ ∇ ω ∂ z ϕ d ω + S d -1 |∇ ω ∂ z ϕ | 2 d ω ,
we see that χ satisfies

-χ + S d -1 |∇ ω ∂ z ϕ | 2 d ω + 1 α 2 S d -1 |∆ ω ∂ z ϕ| 2 d ω -λ 1 S d -1 |∇ ω ∂ z ϕ| 2 d ω + 2 a 2 + λ 1 α 2 χ = h α 2 , with h := S d -1 ∇ ω ∂ z h ∇ ω ∂ z ϕ d ω. Then, using S d -1 ∇ ω ∂ z ϕ d ω = 0, by the Poincaré inequality we deduce S d -1 |∆ ω ∂ z ϕ| 2 d ω ≥ λ 1 S d -1 |∇ ω ∂ z ϕ| 2 d
In the case where χ is equal to 0 at some points of R, it is enough to do the above comparison argument on maximal positivity intervals of χ to deduce the same asymptotic estimate. Finally we observe that ϕ(z, ω) ∼ e a z as z → +∞, which ends the proof of (ii) considering the above estimate for χ when = 0. Moreover, the same estimate for = 1 together with (ii) and (30) proves (iv).

Step 2: Proof of (v). By applying the operator ∆ ω to (29), we obtain

-α 2 (∆ ω ϕ) -∆ 2 ω ϕ + a 2 α 2 ∆ ω ϕ = ∆ ω h in C .
We proceed as in Step 1. With similar notations, by defining This ends the proof of (v), using (ii), (30) and noticing again that ϕ(z, ω) ∼ e a z as z → +∞.

Step 3: Proof of (i) and (iii). Let us consider a positive solution ϕ to (29) and define on R the function ϕ 0 (z) := 1

S d -1 S d -1 ϕ(z, ω) d ω .
By integrating (29) on S d -1 , we know that ϕ 0 solves based on the estimate (33) with = 2. This ends the proof of (iii).

-ϕ 0 + a 2 ϕ 0 = 1 α 2 S d -1 S d -1 h(z, ω) d ω =: h 0 (z) α 2 ∀ z ∈ R , with |h 0 (z)| ≤ O e -n+2
Proof of Proposition 5.1 It is straightforward to verify that the boundedness of P , P /s, P/s 2 , ∇ ω P /s, ∇ ω P/s 2 , L α P as s → +∞ and the integral estimates (i)-(v) as s → 0 + from Lemma 5.4 are enough in order to estab- lish ( 18), ( 24) and (25).

2

Figure 1 .

 1 Figure 1. In dimension d = 4, with p = 1.2, the grey area corresponds to the cone determined by d -2 + (γd )/p ≤ β < (d -2) γ/d and γ ∈ (-∞, d ) in (2). The light grey area is the region of symmetry, while the dark grey area is the region of symmetry breaking. The threshold is determined by the hyperbola (d -γ) 2 -(βd + 2) 2 -4 (d -1) = 0 or, equivalently β = β FS (γ). Notice that the condition p ≤ p induces the restriction β ≥ d -2 + (γd )/p, so that the region of symmetry is bounded. The largest possible cone is achieved as p → 1 and is limited from below by the condition β > γ -2.

- 2 . 1 . 3

 213 is equivalent to a Gagliardo-Nirenberg type inequality corresponding to an artificial dimension n or, to be precise, to a Caffarelli-Kohn-Nirenberg inequality with weight |x| n-d in all terms. Notice that p = n n Corollary Assume that α, n and p are such that d ≥ 2 , α > 0 , n > d and p ∈ 1, p .

. 1 . 2 L

 12 First step: computation of dI dt Let us define K [P] := A [P] -(1m) (L α P) 2 where A [P] := 1 α |D α P| 2 -D α P • D α L α P and, on the boundary of the centered ball B s of radius s, the boundary term b(s) := ∂B s ∂ ∂s P

Lemma 4 . 3

 43 Assume that d ≥ 2 and 1/(1m) = ν > n > d . There exists a positive constant c(n, m, d ) such that, for any positive function P ∈ C 3 (S d -1

Corollary 4 . 4

 44 Let d ∈ N, n ∈ R be such that n > d ≥ 2, and consider a positive function P ∈ C 3 (R d \ {0}). If u is related to P by P = m

Lemma 5 . 3

 53 Let β, γ and p satisfy the relations (2). Any positive solution w of (5) satisfying (26) is such that w ∈ C ∞ (R d \ {0}) and there exists two positive constants, C 1 and C 2 with C 1 < C 2 , such that for |x| large enough,

1 z

 1 2 and (28) show that ϕ(z, ω) ≤ O(e a z ). The lower bound can be established by a comparison argument as in [14, Proposition A.1], after noticing that |h(z, ω)| ≤ O(e (a-2)z ). Hence we obtain that ϕ(z, ω) ∼ e a z = e 2-n 2 z as z → +∞ . Moreover, uniformly in ω, we have that |h(z, ω)| ≤ O e -n+2 2 z as z → +∞ , |h(z, ω)| ∼ e as z → -∞ , which in particular implies |h(z, ω)| = o ϕ(z, ω) as z → +∞ and |h(z, ω)| ∼ ϕ(z, ω) as z → -∞ . Finally, using [20, Theorem 8.32, p. 210] on local C

  ω as e.g. in [12, Lemma 7], where λ 1 := d -1. A Cauchy-Schwarz inequality implies that --Schwarz inequality and (31) we infer that |h /ζ | = O e (a-2) z for z → +∞, and |h /ζ | = O e (a+2/(p-1)) z for z → -∞. By a simple comparison argument based on the Maximum Principle, and using the convergence of χ to 0 at ±∞, we infer that ζ (z) ≤ -e t |h (t )| ζ (z) dtif ν := a 2 + λ 1 /α 2 . This is enough to deduce that ζ (z) ≤ O e (a-1)z as z → +∞ after observing that the condition-ν =a 2 + λ 1 /α 2 ≤ a -1is equivalent to the inequality α ≤ α FS . Hence we have shown that if χ is a positive function, then for α ≤ α FS , χ (z) ≤ O e 2 (a-1) z as z → +∞ .

2 SS d - 1 ∆ ω ϕ ∆ 2 ω ϕ d ω = S d - 1 |∇ ω ∆ ω ϕ| 2 d ω ≥ λ 1 S d - 1 |∆ ω ϕ| 2 d ω , we obtain -χ 5 + |χ 5 | 2 2 χ + 2 a 2 + λ 1 α 2 χ 5 ≤ |h 5 | α 2 with h 5 :

 211115525 d -1 |∆ ω ϕ| 2 d ω , after multiplying the equation by ∆ ω ϕ and using the fact that-= S d -1 ∆ ω h ∆ ω ϕ d ω.Again using the same arguments as above, together with (32), we deduce that χ 5 (z) ≤ O e 2 (a-1) z as z → +∞ .

2 ze-a z 2 a α 2 ∞- 1 |ψ| 2 d ω = e - 2az S d - 1 |ϕ -ϕ 0 | 2 d ω ≤ e -2az λ 1 - 1 |ψ| 2 d ω + S d - 1 |ψ | 2 -p |ϕ | 2 |ϕ| 2 + 1 -

 2212az1111221 as z → +∞ , |h 0 (z)| ∼ e -at h 0 (t ) dt -e z e at h 0 (t ) dt , we deduce that as z → +∞, ϕ 0 (z) ∼ e a z and ϕ 0 (z)a ϕ 0 (z)ϕ(z, ω) ∼ e -2a z ∞ z e at h 0 (t ) dt = O(e -2 z) .If we define the function ψ(z, ω) := e -a z ϕ(z, ω) -ϕ 0 (z) , we may observe that it is bounded for z positive and moreoverϕ (z, ω) ϕ(z, ω) a = O(e -2 z ) + ψ (z, ω) e -a z ϕ(z, ω) as z → +∞ .We recall that e -a z ϕ(z, ω) is bounded away from 0 by a positive constant as z → +∞. Hence we know thatϕ (z, ω) ϕ(z, ω) a ≤ O |ψ (z, ω) + O(e -2 z ) . (34)By the Poincaré inequality and estimate (33) with = 0, we haveS d S d -1 |∇ ω ϕ| 2 d ω ≤ O(e -2z ) .Moreover, by the estimate (33) with = 1, we also obtaine -2az S d -1 |ϕ -ϕ 0 | 2 d ω ≤ e -2az λ 1 S d -1 |∇ ω ϕ | 2 d ω ≤ O(e -2z ) .22Hence, since ψ =a ψ + e -az (ϕ -ϕ 0 ), the above estimates imply thatS d d ω ≤ O(e -2z ) ,which together with (34) ends the proof of (i).To prove (iii), we first check thatϕ ϕ 2 a (1p) ϕ ϕ + a 2 (1p)a = O(|ψ | + |ψ | 2 + |ψ |) + O(e -2 z ) ,and so it remains to prove that S d -1 |ψ | 2 d ω is of order O(e -2 z ). Since ψ = a 2 ψ -2 a e -az (ϕ -ϕ 0 ) + e -az (ϕ -ϕ 0 ) , using the above estimates, we have only to estimate the term with the second derivatives. This can be done as above by the Poincaré inequality, e -2azS d -1 |ϕ -ϕ 0 | 2 d ω ≤ e -2az λ 1 S d -1 |∇ ω ϕ | 2 d ω ≤ O(e -2z) ,

Acknowlegments. This research has been partially supported by the projects STAB (J.D.) and Kibord (J.D.) of the French National Research Agency (ANR), and by the NSF grant DMS-1301555 (M.L.). J.D. thanks the University of Pavia for support. M.L. thanks the Humboldt Foundation for support. M.M. has been partially funded by the National Research Project "Calculus of Variations" (PRIN 2010-11, Italy). © 2016 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.