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

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

Résumé

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.

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.


, , , and

1 A family of subcritical Caffarelli-Kohn-Nirenberg interpolation inequalities

With the norms

let us define as the space of all measurable functions such that is finite. Our functional framework is a space of functions such that , which is defined as the completion of the space of the smooth functions on with compact support in , with respect to the norm given by .

Now consider the family of Caffarelli-Kohn-Nirenberg interpolation inequalities given by

(1)

Here the parameters , and are subject to the restrictions

(2)

and the exponent is determined by the scaling invariance, i.e.,

These inequalities have been introduced, among others, by L. Caffarelli, R. Kohn and L. Nirenberg in [5]. We observe that if , a case which has been dealt with in [14], and we shall focus on the sub-critical case . Throughout this paper, denotes the optimal constant in (1). We shall say that a function is an extremal function for (1) if equality holds in the inequality.

Symmetry in (1) means that the equality case is achieved by Aubin-Talenti type functions

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 [4] that symmetry breaking holds in (1) if

(3)

where

For completeness, we will give a short proof of this result in Section 2. Our main result shows that, under Condition (2), symmetry holds in the complement of the set defined by (3), which means that (3) is the sharp condition for symmetry breaking. See Fig. 1.

Theorem 1.1

Assume that (2) holds and that

(4)

Then the extremal functions for (1) are radially symmetric and, up to a scaling and a multiplication by a constant, equal to .

Figure 1: In dimension , with , the grey area corresponds to the cone determined by and 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 or, equivalently . Notice that the condition induces the restriction , so that the region of symmetry is bounded. The largest possible cone is achieved as and is limited from below by the condition .

The result is slightly stronger than just characterizing the range of 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 (1), but also of all positive solutions in of the corresponding Euler-Lagrange equations, that is, up to a multiplication by a constant and a dilation, of

(5)
Theorem 1.2

Assume that (2) and (4) hold. Then all positive solutions to (5) in are radially symmetric and, up to a scaling and a multiplication by a constant, equal to .

Up to a multiplication by a constant, we know that all non-trivial extremal functions for (1) are non-negative solutions to (5). Non-negative solutions to (5) 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 , the condition (2) amounts to , , , and (1) can be written as

In this case, we deduce from Theorem 1.1 that symmetry always holds. This is consistent with a previous result ( and , close to ) obtained in [17]. A few other cases were already known. The Caffarelli-Kohn-Nirenberg inequalities that were discussed in [14] correspond to the critical case , or, equivalently . Here by critical we simply mean that scales like . The limit case and , which is an endpoint for (2), corresponds to Hardy-type inequalities: there is no extremal function, but optimality is achieved among radial functions: see [16]. The other endpoint is , in which case . The results of Theorem 1.1 also hold in that case with , up to existence issues: according to [9], either , symmetry holds and there exists a symmetric extremal function, or , 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 [4] (also see [14, 15]). Written in spherical coordinates for a function

inequality (1) becomes

where and denotes the gradient of with respect to the angular variable . Next we consider the change of variables ,

(6)

where and are two parameters such that

Our inequality can therefore be rewritten as

Using the notation

with

Inequality (1) is equivalent to a Gagliardo-Nirenberg type inequality corresponding to an artificial dimension  or, to be precise, to a Caffarelli-Kohn-Nirenberg inequality with weight in all terms. Notice that

Corollary 1.3

Assume that , and are such that

Then the inequality

(7)

holds with optimal constant as above and optimality is achieved among radial functions if and only if

(8)

When symmetry holds, optimal functions are equal, up to a scaling and a multiplication by a constant, to

We may notice that neither nor depend on and that the curve determines the same threshold for the symmetry breaking region as in the critical case . In the case , this curve was found by V. Felli and M. Schneider, who proved in [19] the linear instability of all radial critical points if . When , symmetry holds under Condition (8) as was proved in [14]. Our goal is to extend this last result to the subcritical regime .

The change of variables is an important intermediate step, because it allows to recast the problem as a more standard interpolation inequality in which the dimension is, however, not necessarily an integer. Actually 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 and sub-critical otherwise. The critical case has been studied in [14] 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 [21, 18], and make use of the corresponding fast diffusion flow. As in [14], 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 [2] for a general exposition of the method and [12, 13] for its use in presence of nonlinear flows. Also see [6] 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 (5) are established in Section 5, which indeed show that no boundary term has to be taken into account (see Proposition 5.1).

2 Symmetry breaking

For completeness, we summarize known results on symmetry breaking for (1). Details can be found in [4]. With the notations of Corollary 1.3, let us define the functional

obtained by taking the difference of the logarithm of the two terms in (7). Let us define , where

Since  as defined in Corollary 1.3 is a critical point of , a Taylor expansion at order shows that

with and

The following Hardy-Poincaré inequality has been established in [4].

Proposition 2.1

Let , , and . Then

(9)

holds for any , with , such that , with an optimal constant given by

where is the unique positive solution to

Moreover, is achieved by a non-trivial eigenfunction corresponding to the equality in (9). If , the eigenspace is generated by , with , ,… and the eigenfunctions are not radially symmetric, while in the other case the eigenspace is generated by the radially symmetric eigenfunction .

As a consequence, is a nonnegative quadratic form if and only if . Otherwise, takes negative values, and a careful analysis shows that symmetry breaking occurs in (1) if

which means

and this is equivalent to .

3 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 [11, 7, 8, 6, 2, 18] (also see additional references therein). The inequality is of the form

(10)

and

It is known through the work in [11] that the optimizers of this inequality are, up to multiplications by a constant, scalings and translations, given by

In our perspective, the idea is to use a version of the carré du champ or Bakry-Emery method introduced in [1]: 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 has been studied by M. Costa in [10] in the case of linear diffusions (see [21] and references therein for more recent papers). In [23], 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 [3, 24]. G. Savaré and G. Toscani observed in [21] that the concavity also holds in the nonlinear case, which has been used in [18] 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

which is equivalent to , and consider the flow given by

(11)

where is related to by

The inequalities imply that

(12)

For some positive constant , one easily finds that the so-called Barenblatt-Pattle functions

are self-similar solutions of (11), where and are explicit. Thus, we see that is an optimizer for (10) for all and it makes sense to rewrite (10) in terms of the function . Straightforward computations show that (10) can be brought into the form

(13)

for some constant which does not depend on , where

is a generalized Ralston-Newman entropy, also known in the literature as Tsallis entropy, and

is the corresponding generalized Fisher information. Here we have introduced the pressure variable

The Rényi entropy power is defined by

as in [21, 18]. With the above choice of , is an affine function of if . For an arbitrary solution of (11), we aim at proving that it is a concave function of and that it is affine if and only if . For further references on related issues see [11, 22]. Note that one of the motivations for choosing the variable is that it has a particular simple form for the self-similar solutions, namely

Differentiating along the flow (11) yields

so that

More complicated is the derivative for the Fisher information:

Here and are respectively the Hessian of and the identity matrix. The computation can be found in [18]. Next we compute the second derivative of the Rényi entropy power with respect to :

With , we obtain

(14)

where we have used the notation

Note that by (12), we have that and hence we find that , which also means that is a non-increasing function. In fact it is strictly decreasing unless is a polynomial function of order two in  and it is easy to see that the expression (14) vanishes precisely when is of the form , where , , are constants (but and may still depend on ).

Thus, while the left side of (13) stays constant along the flow, the right side decreases. In [18] it was shown that the right side decreases towards the value given by the self-similar solutions  and hence proves (10) in the sharp form. In our work we pursue a different tactic. The variational equation for the optimizers of (10) is given by

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

for some constants , whose precise values are explicit. This equation can also be interpreted as the variational equation for the sharp constant in (13). Hence, multiplying the above equation by and integrating yields

We recover the fact that, in the flow picture, is, up to a positive factor, the derivative of and hence vanishes. From the observations made above we conclude that must be a polynomial function of order two in . 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 [6]. 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.

4 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 (10). In order to define a new, well-adapted fast diffusion flow, we introduce the diffusion operator , which is given in spherical coordinates by

where denotes the Laplace-Betrami operator acting on the -dimensional sphere of the angular variables, and denotes here the derivative with respect to . Consider the fast diffusion equation

(15)

in the subcritical range . The exponents in (15) and in (7) are related as in Section 3 by

and is defined by

We consider the Fisher information defined as

Here is the pressure variable. Our goal is to prove that takes the form , as in Section 3. It is useful to observe that (15) can be rewritten as

and, in order to compute , we will also use the fact that solves

(16)

4.1 First step: computation of 

Let us define

and, on the boundary of the centered ball of radius , the boundary term

(17)

where by we denote the standard Hausdorff measure on .

Lemma 4.1

If solves (15) and if

(18)

then,

(19)

Proof. For , let us consider the set , so that . Using (15) and (16), we can compute

where the last line is given by an integration by parts, upon exploiting the identity :

1) Using the definition of , we get that

(20)

2) Taking advantage again of , an integration by parts gives

and, with , we find that

(21)

Summing (20) and (21), using (17) and passing to the limits as , , establishes (19).

4.2 Second step: two remarkable identities.

Let us define

and

We observe that

is independent of . We recall the result of [14, Lemma 5.1] and give its proof for completeness.

Lemma 4.2

Let , such that , and consider a function . Then,

Proof. By definition of , we have

which can be expanded as