Lorenzo Brasco

Giovanni Franzina
email: franzina@math.fau.de

CONVEXITY PROPERTIES OF DIRICHLET INTEGRALS AND PICONE-TYPE INEQUALITIES

Keywords: 2010 Mathematics Subject Classification. 47J10, 39B62, 35B50 Nonlinear eigenvalue problems, uniqueness of eigenfunctions, Hardy inequalities, maximum principle, nonlocal equations

We focus on three different convexity principles for local and nonlocal variational integrals. We prove various generalizations of them, as well as their equivalences. Some applications to nonlinear eigenvalue problems and Hardy-type inequalities are given. We also prove a measure-theoretic minimum principle for nonlocal and nonlinear positive eigenfunctions.

1. Introduction 1.1. A general overview. The aim of this paper is to study three elementary convexity principles which have found many applications in eigenvalue problems and functional inequalities. In particular, we will focus on their mutual relations and prove that they are indeed equivalent. In order to smoothly introduce the reader to the subject and clarify the scopes of the paper, we start with the three basic examples which will serve as a model for the relevant generalizations considered in the sequel:

• the first one is the convexity of the Hamiltonian function for a system of one free massive particle in classical mechanics 1 2 |φ| 2 m , which is jointly convex as a function both of the mass m > 0 and of the conjugate momentum φ ∈ R N ;

• the second one is the convexity of the quantity |∇u| 2 along curves of the type

σ t = (1 -t) u 2 + t v 2 1 2 , t ∈ [0, 1],
where u, v ≥ 0 are differentiable functions, i.e.

|∇σ t | 2 ≤ (1 -t) |∇u| 2 + t |∇v| 2 .
This is also sometimes called hidden convexity;

• the third and last one is the so-called 1 Picone identity

∇u, ∇ v 2 u ≤ |∇v| 2 ,
where again u, v ≥ 0 are differentiable functions, this time with u > 0.

As a well-known consequence of the convexity of the previous Hamiltonian, we get that

ρ → |∇ρ(x)| 2 ρ(x) ,
is convex for every x. This in turn implies convexity of the Fisher information with respect to a reference probability measure 2 ν. For every probability measure µ, this functional is given by

J (µ|ν) = |∇ log ρ| 2 ρ dν = |∇ρ| 2 ρ dν, if µ ≪ ν and ρ = dµ dν ,
and observe that the latter can also be re-written as

J (µ|ν) = 4 |∇ √ ρ | 2 dν, if µ ≪ ν and ρ = dµ dν .
From the previous we thus get for

ρ t = (1 -t) ρ 0 + t ρ 1 |∇ √ ρ t | 2 dν ≤ (1 -t) |∇ √ ρ 0 | 2 dν + (1 -t) |∇ √ ρ 1 | 2 dν.
Thus in particular if

ρ t = (1 -t) u 2 + t v 2 ,
we then obtain that the Dirichlet integral is convex along curves of the form

σ t = (1 -t) u 2 + t v 2 , t ∈ [0, 1],
which is the hidden convexity exposed above. This striking convexity property of the Dirichlet integral seems to have been first noticed by Benguria in his Ph.D. dissertation (see [START_REF] Benguria | The Thomas-Fermi-von Weizsäcker theory of atoms and molecules[END_REF] for example). Note that along the curve of functions σ t we have

σ t 2 L 2 = (1 -t) u 2 L 2 + t v 2 L 2
, then if u, v belong to the unit sphere of L 2 , the same holds true for σ t . The latter incidentally happens to be a constant speed geodesic for the metric defined by

d 2 (u, v) = u 2 -v 2 dx, u, v ∈ L 2 .
As one should expect, the geodesic convexity described above is helpful to get uniqueness results in eigenvalue problems. We recall that eigenvalues of the Dirichlet-Laplace operator -∆ on an open set Ω ⊂ R N such that |Ω| < ∞ are defined as the critical points of the Dirichlet integral on the manifold S 2 (Ω) = u ∈ W 1,2 0 (Ω) :

Ω u 2 dx = 1 .
This constraint naturally introduces Lagrange multipliers, which by homogeneity are the eigenvalues of the Laplace operator, i.e. any constrained critical point u is a weak solution of (1.1) -∆u = λ u, in Ω, u = 0, on ∂Ω.

One says that the function u is an eigenfunction corresponding to the eigenvalue λ.

Then the idea is very simple: for a minimum convex problem, critical points are indeed minimizers. This means that hidden convexity trivializes the global analysis for the Dirichlet energy on S 2 (Ω) ∩ {u ≥ 0} and there cannot be any constant sign critical point u other than its global minimizer. Since the strong minimum principle states that any constant sign eigenfunction (up to a sign) is strictly positive, this imposes any eigenfunction v ≥ 0 to be associated with the least eigenvalue

λ 1 (Ω) = min u∈W 1,2 0 (Ω) Ω |∇u| 2 dx :
Another way to prove the same result would be precisely by means of Picone inequality. Let us call u 1 a first eigenfunction of Ω, i.e. a function achieving λ 1 (Ω). If v ≥ 0 is a nontrivial eigenfunction with eigenvalue λ, then by strong minimum principle v > 0 and by Picone inequality one would get

λ = λ Ω v u 2 1 v = Ω ∇v, ∇ u 2 1 v dx ≤ Ω |∇u 1 | 2 dx = λ 1 (Ω),
and thus λ 1 (Ω) = λ since λ 1 (Ω) is the minimal eigenvalue.

Of course, in the case of the Laplace operator -∆ simplicity of λ 1 (Ω) and uniqueness of constant sign eigenfunctions are plain consequences of the Hilbertian structure and of the strong minimum principle for supersolutions of uniformly elliptic equations. Indeed, any first eigenfunction u 1 must have constant sign and can never vanish on the interior of the connected set Ω. Then any other eigenfunction has to be orthogonal in L 2 (Ω) to u 1 (i.e. it has to change sign) unless it is proportional to u 1 ... 1.2. Aim of the paper. ...nevertheless, the advantage of the hidden convexity exposed above is that it does not involve any orthogonality concept and applies to general Dirichlet energies of the form

(1.2) Ω H(∇u) dx,
where z → H(z) is convex, even and positively homogeneous of degree p > 1. Moreover, we prove in Proposition 2.6 that this remains true for the whole class of interpolating curves

(1.3) σ t = (1 -t) u q + t v q 1 q , t ∈ [0, 1].
with 1 ≤ q ≤ p. We point out that q = 1 corresponds to convexity in the usual sense and that for q > p the property ceases to be true, see Remark 2.8.

Like in the previous model case p = q = 2 and H(z) = |z| 2 , this permits to infer (see Theorem 5.1) that the only constant sign critical points of (1.2) on the manifold S q (Ω) = u ∈ W 1,p 0 (Ω) :

Ω |u| q dx = 1 , 1 < q ≤ p,
are indeed the global minimizers, which are unique up to a sign. Observe that these critical points yield the following nonlinear version of Helmoltz equation (1.1), i.e.

-div ∇H(∇u) = λ u p-q L q (Ω) |u| q-2 u, in Ω, u = 0 on ∂Ω.

We refer the reader to [START_REF] Franzina | Existence and uniqueness for a p-Laplacian nonlinear eigenvalue problem[END_REF] for a detailed account on this nonlinear eigenvalue problem in the case

H(z) = |z| p .
The previous general version of the hidden convexity can be seen again as a consequence of the joint convexity of the generalized Hamiltonian 3

(1.4) (m, φ) → H(φ) m β , for 0 ≤ β ≤ p -1,
which in turn gives the convexity of the information functional

J H,β (µ|ν) = H (∇ log ρ) ρ p-β dν, if µ ≪ ν and ρ = dµ dν ,
where the latter can also be written as

J H,β (µ|ν) = p p -β p H ∇ρ p-β p dν, if µ ≪ ν and ρ = dµ dν .
3 The parameters β and q are linked through the relation

β p + 1 q = 1.
Finally, hidden convexity is in turn equivalent (see Section 3) to the validity of the following generalized Picone inequality

(1.5) ∇H(∇u), ∇ v q u q-1 ≤ H(∇v) q p H(∇u) q-p p ,
for all differentiable functions u, v ≥ 0 with u > 0, which is proved in Proposition 2.9. Here again we consider 1 < q ≤ p.

We point out that equivalence between Picone-type inequalities and the hidden convexity property seemed to be unknown: indeed, one of the main scopes of this paper is to precise the relation between these two properties.

Up to now, we have discussed applications of these convexity principles to uniqueness issues in linear and nonlinear eigenvalue problems. But Picone-type inequalities can be used to prove a variety of different results. Without any attempt of completeness (we refer to the seminal paper of Allegretto and Huang [START_REF] Allegretto | A Picone's identity for the p-Laplacian and applications[END_REF] and to the recent paper [START_REF] Jaroš | A-harmonic Picone's identity with applications, to appear on[END_REF] for a significant account on the topic), we focus on applications to Hardy-type functional inequalities.

The idea is that when u solves a quasilinear equation with principal part given by -div∇H(∇u), by integrating (1.5) and using the equation one can get a lower bound on H(∇v) which does not depend on derivatives of u. This procedure is now well understood, see the recent paper [START_REF] Devyver | Optimal L p Hardy-type inequalities[END_REF]. In Theorem 5.4 this is applied to get a sharp anisotropic version of the Hardy inequality which reads as follows

(1.6) N + γ -p p p R N |v| p F * (x) γ-p dx ≤ R N F (∇v) p F * (x) γ dx, v ∈ C ∞ 0 (R N \ {0}),
for 1 < p < N and γ > p -N . Here F is any C 1 strictly convex norm and F * denotes the corresponding dual norm, see Section 5.2. For the case γ = 0 a different proof, based on symmetrization arguments, can be found in [START_REF] Van Schaftingen | Anisotropic symmetrization[END_REF].

The same method can be used to get, for example, the following nonlocal version of the Hardy inequality

(1.7) C R N |v| p |x| s p dx ≤ R N R N v(x) -v(y) p |x -y| N +s p dx dy , for all v ∈ W s,p 0 (R N ) \ {0} ,
by means of the following discrete version of Picone inequality

(1.8) |u(x) -u(y)| p-2 (u(x) -u(y)) v(x) p u(x) p-1 - v(y) p u(y) p-1 ≤ |v(x) -v(y)| p ,
with the choice u(x) = |x| -s p . The constant C = C(N, s, p) > 0 is sharp and for the sake of completeness we provide details about its computation in Appendix B. We point out that (1.7) was proved by this same method by Frank and Seiringer in [START_REF] Frank | Non-linear ground state representations and sharp Hardy inequalities[END_REF], which is there called ground state substitution. Other fractional Hardy inequalities have appeared in the literature, see [START_REF] Bogdan | The best constant in a fractional Hardy inequality[END_REF][START_REF] Dyda | Fractional Hardy-Sobolev-Maz'ya inequality for domains[END_REF]. In particular, in the recent paper [START_REF] Davila | Nonlocal Minimal Lawson Cones[END_REF] Davila, del Pino and Wei observed that a suitable fractional Hardy inequality on surfaces plays a role in the stability of nonlocal minimal cones, see [START_REF] Davila | Nonlocal Minimal Lawson Cones[END_REF]Corollary 11.1].

Noteworthy, not only does the Picone inequality (1.5) have its discrete counterpart (1.8), but also the hidden convexity of the Dirichlet integral has a nonlocal version. Indeed, the Gagliardo seminorm |u(x) -u(y)| p |x -y| N +s p dx dy turns out to be convex along curves of the type (1.3), whenever u, v are positive. Correspondingly, fractional Picone inequalities (or equivalently hidden convexity) are used to get uniqueness results for positive eigenfunctions of the integro-differential operator defined by the following principal value integral

(-∆ p ) s u(x) = 2 p.v. R N |u(x) -u(y)| p-2 (u(x) -u(y)) |x -y| N +s p dy.
We point out that in order to get uniqueness results for constant sign nonlocal eigenfunctions, i.e. for solutions of

(-∆ p ) s u = λ u q-1 , in Ω, u = 0, in R N \ Ω,
as in the local case, one needs to know that non-negative eigenfunctions are indeed strictly positive, at least for Ω connected. A proof of this strong minimum principle for nonlocal eigenfunctions is provided in the appendix and seems to be new. The proof is based on a logarithmic lemma recently established in [START_REF] Castro | Local behavior of fractional p-minimizers[END_REF]. After the acceptance of the present paper, we were informed about the interesting manuscript [START_REF] Castro | Nonlocal Harnack inequalities[END_REF] which contains the proof of a weak Harnack's inequality for supersolutions (as well as a proper Harnack's inequality for solutions) of the operator (-∆ p ) s . The interested reader may find in that article a more detailed account about nonlocal Harnack's inequalities. Nevertheless, those results are not used in this paper.

1.3. Plan of the paper. In Section 2 we present and prove some generalizations of the three convexity principles discussed above, then Section 3 is devoted to discuss their equivalences. Section 4 deals with the nonlocal/discrete versions of these convexities. Applications are then given in Sections 5 and 6. The paper is concluded by two Appendices: one contains a new strong minimum principle for positive nonlocal eigenfunctions (Theorem A.1), while the second contains some computations related to the determination of the sharp constant in (1.7).

Acknowledgements. We thank Agnese Di Castro, Tuomo Kuusi and Giampiero Palatucci for having kindly provided us a copy of their work [START_REF] Castro | Local behavior of fractional p-minimizers[END_REF], as well as Enea Parini for pointing out a flaw in a preliminary version of the proof of Theorem A.1. Part of this work has been done during the conferences "Linear and Nonlinear Hyperbolic Equations" and "Workshop on Partial Differential Equations and Applications", both held in Pisa and hosted by Centro De Giorgi and the Departement of Mathematics of the University of Pisa. We acknowledge the two institutions as well as the organizers for the nice atmosphere and the excellent working environment. The second author has been supported by the ERC Starting Grant No. 258685 "AnOptSetCon".

Three convexity principles

We start with the a couple of classical results, which will be useful in order to prove some of the results of the paper. We give a proof for the reader's convenience. Lemma 2.1. Let F : R N → [0, +∞) be a positively 1-homogeneous function, i.e.

F (λ z) = λ F (z), z ∈ R N , λ ≥ 0,
which is level-convex, i.e.

(2.1)

F ((1 -t) z + t w) ≤ max{F (z), F (w)}, z, w ∈ R N , t ∈ [0, 1]. Then F is convex. Proof. Let x 0 , x 1 ∈ R N , if F (x 0 ) = F (x 1 ) = 0 then by (2.1) F ((1 -t) x 0 + t x 1 ) = 0, t ∈ [0, 1].
Let us now suppose for example that F (x 0 ) > 0 and take ε > 0, we define

z = x 0 F (x 0 ) , w = x 1 F (x 1 ) + ε and t = F (x 1 ) + ε F (x 0 ) + F (x 1 ) + ε .
By using the 1-homogeneity of F , we then obtain

F ((1 -t) z + t w) = F (x 0 + x 1 ) F (x 0 ) + F (x 1 ) + ε , while max{F (z), F (w)} = max 1, F (x 1 ) F (x 1 ) + ε = 1.
Then (2.1) implies

F (x 0 + x 1 ) ≤ F (x 0 ) + F (x 1 ) + ε, x 0 , x 1 ∈ R N ,
and since ε > 0 is arbitrary, we get

F (x 0 + x 1 ) ≤ F (x 0 ) + F (x 1 ), x 0 , x 1 ∈ R N ,
i.e. F is subadditive. This in turn implies the desired result, as

F ((1 -t) x 0 + t x 1 ) ≤ F ((1 -t) x 0 ) + F (t x 1 ) = (1 -t) F (x 0 ) + t F (x 1 ), which concludes the proof. Lemma 2.2. Let 1 < p < ∞ and let H : R N → [0, +∞) be a C 1 positively p-homogeneous convex function. If H(z) = 0 then we have ∇H(z) = 0 as well.
Proof. The statement is evident if z = 0, thus let us suppose that z = 0. Assume on the contrary that ∇H(z) = 0, then there exists h ∈ R N with unit norm such that ∇H(z), h = |∇H(z)|.

The function g(t) = H(z + t h) has the following properties

g ∈ C 1 (R), g(t) ≥ 0, g ′ (0) = |∇H(z)| > 0 = g(0).
This gives a contradiction, thus ∇H(z) = 0.

2.1. Convexity of generalized kinetic energies. The first convexity principle we consider is the following.

Proposition 2.3. Let 1 < p < ∞ and let H : R N → [0+∞) be a convex positively p-homogeneous function. For every 0 < β ≤ p -1 the function (m, φ) → H(φ) m β , (m, φ) ∈ (0, +∞) × R N , is convex. Proof. For β = p -1, it is sufficient to observe that (2.2) H(φ) m p-1 = sup (t,ξ) {t m + ξ, φ : t + H * (ξ) ≤ 0} , m > 0, φ ∈ R N ,
where H * denotes the Legendre-Fenchel transform of H. This would give the desired result, since the supremum of affine functions is a convex function.

For completeness, we verify formula (2.2): since for every m > 0 the map t → t m is increasing, the maximization in (2.2) is unchanged if we replace the inequality constraint by the condition t + H * (ξ) = 0. Then the right-hand side of (2.2) is equivalent to

sup ξ∈R N ξ, φ -H * (ξ) m = m sup ξ∈R N ξ, φ m -H * (ξ) = m H * * φ m ,
which gives the desired conclusion, by using that H * * = H and the positive homogeneity of H.

For 0 < β < p -1, let us set for simplicity

Φ(m, φ) = H(φ) m p-1 and ψ(m) = m ϑ , m > 0, φ ∈ R N ,
where ϑ = β/(p -1) < 1, then we can rewrite

H(φ) m β = Φ(ψ(m), φ),
where Φ is jointly convex thanks to the first part of the proof and decreasing in its first argument, while ψ is concave, then it is standard to see that their composition is a convex function. Indeed, for every t ∈ [0, 1], m 0 , m 1 > 0 and φ 0 , φ 1 ∈ R N , we get

Φ ψ((1 -t) m 0 + t m 1 ), (1 -t) φ 0 + t φ 1 ≤ Φ (1 -t) ψ(m 0 ) + t ψ(m 1 ), (1 -t) φ 0 + t φ 1 ≤ (1 -t) Φ(m 0 , φ 0 ) + t Φ(m 1 , φ 1 ),
which gives the desired result.

A couple of comments on the previous result are in order.

Remark 2.4. As already recalled in the Introduction, the main instance of functions considered in Proposition 2.3 is the following one

(2.3) (m, φ) → |φ| 2 m , φ ∈ R N , m > 0.
If one regards the scalar quantity m as a mass and the vector quantity φ as the moment of this mass, i.e. if we decompose φ as φ = v m with v ∈ R N (the velocity of the mass particle), then we would have

|φ| 2 m = |v| 2 m.
This simple remark is the crucial ingredient of the so-called Benamou-Brenier formula for the 2-Wasserstein distance (see [START_REF] Benamou | A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem[END_REF][START_REF] Brenier | Extended Monge-Kantorovich theory[END_REF]). The latter is a distance on the space of probability measures P(Ω) over Ω, defined by

w 2 (ρ 0 , ρ 1 ) 2 := inf T Ω×Ω |x -T (x)| 2 dρ 0 : T # ρ 0 = ρ 1 ,
for every ρ 0 , ρ 1 ∈ P(Ω). Here T # ρ 0 denotes the push-forward of the measure ρ 0 . The Benamou-Brenier formula asserts that we have

w 2 (ρ 0 , ρ 1 ) 2 = inf 1 0 Ω |v| 2 µ t dx dt : ∂ t µ t + div(v t µ t ) = 0 µ 0 = ρ 0 and µ 1 = ρ 1 .
The latter consists in minimizing the integral of the total kinetic energy (the action), under a conservation of mass constraint. Thanks to the previous discussion, this dynamical problem can be transformed in a convex variational problem under linear constraint, once we introduce the variable

φ t = v t µ t .
For generalizations of this transport problem involving functions of the form H(φ) m -β the reader can consult [START_REF] Cardaliaguet | Geodesics for a class of distances in the space of probability measures[END_REF] and [START_REF] Dolbeault | A new class of transport distances between measures[END_REF].

Remark 2.5 (Sharpness of the condition on β). The previous convexity property fails to be true in general for β > p -1. Let us fix φ 0 ∈ R N \ {0} and m 0 > 0. We take φ 1 = c φ 0 and m 1 = c m 0 with c > 1, then we consider the convex combination

(m t , φ t ) = ((1 -t) m 0 + t m 1 , (1 -t) φ 0 + t φ 1 ).
For every p -1 < β < p by strict concavity of the function τ → τ p-β we have

|φ t | p m β t = (1 -t + t c) p-β |φ 0 | p m β 0 > (1 -t) |φ 0 | p m β 0 + t c p-β |φ 0 | p m β 0 = (1 -t) |φ 0 | p m β 0 + |φ 1 | p m β 1 .
2.2. Hidden convexity. The next convexity principle has been probably first identified by Benguria in his Ph.D. dissertation in the case p = q = 2 and H(z) = |z| 2 , see [START_REF] Benguria | The Thomas-Fermi-von Weizsäcker theory of atoms and molecules[END_REF]. See also [START_REF] Belloni | A direct uniqueness proof for equations involving the p-Laplace operator[END_REF] and [START_REF] Takáč | Variational problems with a p-homogeneous energy[END_REF] for some generalizations in the case q = p.

Proposition 2.6 (General hidden convexity). Let 1 < p < ∞ and 1 < q ≤ p. Let H : R N → [0+∞) be a positively p-homogeneous convex function. For every pair of differentiable functions u 0 , u 1 ≥ 0, we define

σ t (x) = (1 -t) u 0 (x) q + t u 1 (x) q 1 q t ∈ [0, 1], x ∈ Ω.
Then there holds

(2.4) H(∇σ t ) ≤ (1 -t) H(∇u 0 ) + t H(∇u 1 ), t ∈ [0, 1].
Proof. The proof for the case p = q can be found for example in [START_REF] Brasco | A note on positive eigenfunctions and hidden convexity[END_REF]. In order to consider the case q < p, we observe that by Lemma 2.1, the function F = H q/p is a positively q-homogeneous convex function. Then the first part of the proof implies

F (∇σ t ) ≤ (1 -t) F (∇u 0 ) + t F (∇u 1 ),
and by raising to the power p/q and using the convexity of τ → τ p/q , we end up with (2.4).

Remark 2.7. We remark that neither strict convexity of H nor strict positivity of the functions is needed in the previous result, unless one is interested in identification of equality cases in (2.4). Moreover, H is only required to be only positively homogeneous, i.e. it is not necessarily even.

Remark 2.8 (Sharpness of the condition on q). Again, the condition q ≤ p is vital. Indeed, by taking a non-constant u 0 = 0 and u 1 = c u 0 for c > 1, then we have

σ t (x) = ((1 -t) u q 0 + t u q 1 ) 1 q = ((1 -t) + t c q ) 1 q u 0 , and
H(∇σ t ) = ((1 -t) + t c q ) p q H(∇u 0 ) > (1 -t) H(∇u 0 ) + t H(∇u 1 )
, by strict concavity of τ → τ p/q . 2.3. Picone inequalities. We now prove a general version of the so-called Picone inequality. The usual one, i.e.

|∇u| p-2 ∇u, ∇ v p u p-1 ≤ |∇v| p , v ≥ 0, u > 0,
proved by Allegretto and Huang, see [2, Theorem 1.1] corresponds to taking p = q and H(z) = |z| p in (2.5) below.

Proposition 2.9 (General Picone inequality). Let 1 < q ≤ p and let H : R N → [0, +∞) be a C 1 positively p-homogeneous convex function. For every pair of positive differentiable functions u, v with u > 0, we have

1 p ∇H(∇u), ∇ v q u q-1 ≤ H(∇v) q p H(∇u) p-q p . (2.5)
Proof. Let us start with the case p = q. We use the convexity inequality

H(z) ≥ H(w) + ∇H(w), z -w ,
with the choices z = ∇v and w = ∇u v u .

By using the p-homogeneity of H we then get

H(∇v) ≥ v u p H(∇u) + ∇H(∇u), ∇v v u p-1 -∇H(∇u), ∇u v u p = ∇H(∇u), ∇v v u p-1 -1 - 1 p ∇H(∇u), ∇u v u p = 1 p ∇H(∇u), ∇ v p u p-1 ,
which concludes the proof of (2.5) for q = p.

We now take 1 < q < p and set

F (z) = H(z) q p , z ∈ R N ,
which is convex and positively q-homogeneous 4 , then the first part of the proof implies

1 q ∇F (∇u), ∇ v q u q-1 ≤ F (∇v).
Observe that if H(∇u) = 0, then we have ∇H(∇u) = 0 as well (see Lemma 2.2) and (2.5) holds true. Thus we can assume H(∇u) = 0. The previous inequality is equivalent to

1 p H(∇u) q-p p ∇H(∇u), ∇ v q u q-1 ≤ H(∇v) q p .
If we multiply the previous by H(∇) (p-p)/p we eventually attains the conclusion.

Remark 2.10 (Equivalent form of the Picone inequality). For p = q, as a plain consequence of Young inequality, (2.5) implies

(2.6) 1 p ∇H(∇u), ∇ v q u q-1 ≤ q p H(∇v) + p -q p H(∇u).
Observe that for q = 1, the previous inequality reduces to

H(∇u) + ∇H(∇u), ∇v -∇u ≤ H(∇v),
which just follows from the convexity of z → H(z). On the other hand, by applying (2.6) with the choices (here ε > 0)

U = ε + H(∇u) -1 p u and V = ε + H(∇v) -1 p v, we get 1 p ε + H(∇u) q-p p ε + H(∇v) -q p ∇H(∇u), ∇ v q u q-1 ≤ 1.
By multiplying the previous by (ε + H(∇u)) (p-q)/p (ε + H(∇v))

q p and then letting ε goes to 0, we get (2.5).

Remark 2.11 (Non-homogeneous functions). All the convexity principles considered in this section have been proven under the assumption that H is positively p-homogeneous. Nevertheless, the results are still true for some H violating this condition. This is the case for example of the anisotopic function

H(z) = N i=1 H i (z i ),
where

H i (t) = |t| p i , and 1 < p 1 ≤ • • • ≤ p N .
Namely, by applying (2.5) to each H i and then summing up, we get

N i=1 |u x i | p i -2 u x i v q i u q i -1 x i ≤ N i=1 |v x i | q i p i |u x i | p i -q i p i ,
for every q 1 , . . . , q N such that 1 < q i ≤ p i . In the very same way from (2.4) we get

N i=1 (σ t ) x i p i ≤ (1 -t) N i=1 |u x i | p i + t N i=1 |v x i | p i , t ∈ [0, 1],
4 Also observe that F ∈ C 1 (R N ). It is sufficient to check that F is differentiable at the origin and that its differential vanishes at z = 0. Indeed, by using homogeneity we have

F (h) -F (0) = H(h) q p = |h| q H h |h| q p = o(|h|), h ∈ R N \ {0} such that |h| ≪ 1.
where σ t = ((1 -t) u q + t v q ) 1/q and 1 < q ≤ p 1 . From Proposition 2.3 we can infer the convexity of the function

(m, φ) → N i=1 |φ| p i m β i , (m, φ) ∈ (0, +∞) × R N ,
for every β 1 , . . . , β N such that 0 < β i ≤ p i -1.

Equivalences

In this section we will show that the three convexity principles proved in the previous section are indeed equivalent. In other words, they are just three different ways to look at the same principle.

Kinetic energies and Hidden convexity. Let u be a differentiable function on an open

set Ω ⊂ R N , which is everywhere positive. We first observe that if in the generalized kinetic energy of Proposition 2.3 we make the choice m = u and φ = ∇u, then we obtain the functional u → H(∇u) u β . which is convex in the usual sense, provided that 0 < β ≤ p -1, thanks to Proposition 2.3. This convexity is indeed equivalent to (2.4), as we now show.

Indeed, let us pick two differentiable functions u, v which are everywhere positive. We observe that by setting U = u q , V = v q and γ t = (1 -t) U + t V , we get by Proposition 2.3

H(∇γ t ) γ β t ≤ (1 -t) H(∇U ) U β + t H(∇V ) V β .
By using the homogeneity of H, the previous is equivalent to

H ∇γ p-β p ≤ (1 -t) H ∇U p-β p + t H ∇V p-β p
.

If we now choose β in such a way that 5

p -β p = 1 q , i.e. β = p 1 - 1 q ,
we see that the previous inequality becomes

H(∇σ t ) ≤ (1 -t) H(u) + t H(v),
where σ t = γ 1/q t = ((1 -t) u q + t v q ) 1/q as always.

3.2. Hidden convexity and Picone. We first show that (Hidden convexity) =⇒ (Picone).

As before, given u, v positive differentiable functions with u > 0, we set

(3.1) σ t (x) = (1 -t) u(x) q + t v(x) q 1 q t ∈ [0, 1], x ∈ Ω.
Then by using the convexity of t → H(∇σ t ), we easily get

H(∇σ t ) -H(∇u) t ≤ H(∇v) -H(∇u).
5 Observe that such a choice is feasible, since

p 1 - 1 q ≤ p -1 ⇔ p p -1 ≤ q q -1 ⇔ p ≥ q.
Observe that again by convexity, the incremental ratio on the left-hand side is monotone, then there exists the limit for t monotonically converging to 0, i.e. we obtain

(3.2) d dt H(∇σ t ) |t=0 ≤ H(∇v) -H(∇u).
The previous is indeed equivalent to (2.5). To see this, it is sufficient to compute the derivative on the left-hand side. We have

∇σ t = σ 1-q t (1 -t) ∇u u q-1 + t ∇v v q-1 and d dt σ t = 1 q σ 1-q t (v q -u q ), so that we can compute d dt ∇σ t = (1 -q) σ -q t (1 -t) ∇u u q-1 + t ∇v v q-1 d dt σ t + σ 1-q t ∇v v q-1 -∇u u q-1 .
Finally, we get

d dt ∇σ t |t=0 = -(q -1) ∇u u d dt σ t t=0 + ∇v v u q-1 -∇u = - q -1 q ∇u v u q -1 + ∇v v u q-1 -∇u.
We can now compute the left-hand side of (3.2) and obtain

d dt H(∇σ t ) |t=0 = ∇H(∇σ t ), d dt ∇σ t |t=0 = ∇H(∇u), ∇v v u q-1 - p (q -1) q H(∇u) v u q - p q H(∇u).
By inserting this in (3.2), observing that

1 q ∇H(∇u), ∇ v q u q-1 = ∇H(∇u), ∇v v u q-1 - p (q -1) q H(∇u) v u q ,
and multiplying everything by q/p, we eventually get

1 p ∇H(∇u), ∇ v q u q-1 ≤ q p H(∇v) + p -q p H(∇u).
For p = q this is exactly Picone inequality (2.5), while for q < p we just have to observe that by Remark 2.10 the previous is equivalent to (2.5).

Let us now show that (Picone) =⇒ (Hidden convexity). As we said, inequality (2.5) is actually equivalent to (3.2), for every v, u and σ t curve of the form (3.1) connecting them. We now fix u, v and σ t , then by (3.2) we get

H(∇v) -H(∇σ t ) ≥ d ds H(∇ σ s ) |s=0 , and
H(∇u) -H(∇σ t ) ≥ d ds H(∇ σ s ) |s=0 ,
where s → σ s and s → σ s are the curves of the form (3.1) connecting σ t to v and σ t to u respectively. In other words, we have

σ s = (1 -t -s (1 -t)) u q + (t + s (1 -t)) v q 1 q = σ t+s (1-t) , s ∈ [0, 1],
and

σ s = (1 -t + s t) u q + (t -s t) v q 1 q = σ t-s t , s ∈ [0, 1].

The discrete case

We now prove analogous results for functions which are not necessarily differentiable. Roughly speaking, we are going to replace derivatives by finite differences. For 1 < p < ∞ and 0 < s < 1, the resulting convexity properties have applications to nonlocal integrals of the type

R N R N |u(x) -u(y)| p |x -y| N +s p dx dy, (Gagliardo seminorm), or sup 0<|h| R N |u(x + h) -u(x)| p |h| s p dx, (Nikolskii seminorm),
and more generally

∞ 0 sup 0<|h|≤t R N |u(x + h) -u(x)| p t s p dx r p dt t , p ≤ r < ∞, (Besov seminorm).
Proposition 4.1 (Discrete hidden convexity). Let 1 < p < ∞ and 1 < q ≤ p. For every u 0 , u 1 ≥ 0, we define

σ t (x) = (1 -t) u 0 (x) q + t u 1 (x) q 1 q t ∈ [0, 1], x ∈ R N .
Then we have

(4.1) |σ t (x) -σ t (y)| p ≤ (1 -t) |u 0 (x) -u 0 (y)| p + t |u 1 (x) -u 1 (y)| p , t ∈ [0, 1], x, y ∈ R N .
Proof. The proof is as in [START_REF] Franzina | Fractional p-eigenvalues[END_REF], which deals with the case p = q. We observe that

σ t = (1 -t) 1 q u 0 , t 1 q u 1 ℓ q ,
where we set

z ℓ q = (|z 1 | q + |z 2 | q ) 1/q for z ∈ R 2 .
The triangular inequality implies that

z ℓ q -w ℓ q q ≤ z -w q ℓ q , z, w ∈ R 2 ,
and by using this with the choices

z = (1 -t) 1 q u 0 (x), t 1 q u 1 (x) and w = (1 -t) 1 q u 0 (y), t 1 q u 1 (y) , we get |σ t (x) -σ t (y)| q ≤ (1 -t) |u 0 (x) -u 0 (y)| q + t |u 1 (x) -u 1 (y)| q
. By raising both sides to the power p/q and using the convexity of τ → τ p/q , we get (4.1). Proposition 4.2 (Discrete Picone inequality). Let 1 < p < ∞ and 1 < q ≤ p. Let u, v be two measurable functions with v ≥ 0 and u > 0, then

|u(x) -u(y)| p-2 (u(x) -u(y)) v(x) q u(x) q-1 - v(y) q u(y) q-1 ≤ |v(x) -v(y)| q |u(x) -u(y)| p-q . (4.2)
Proof. We notice at first that is sufficient to prove |u(x) -u(y)| q-2 (u(x) -u(y)) v(x) q u(x) q-1 -v(y) q u(y) q-1 ≤ |v(x) -v(y)| q , (4.3) since (4.2) then follows by multiplying the previous inequality by |u(x) -u(y)| p-q .

At this aim, let us start by observing that if u(x) = u(y), inequality (4.3) is trivially satisfied. We take then u(x) = u(y) and we can always suppose that u(x) < u(y), up to exchanging the role of x and y. We further observe that if v(y) = 0, inequality (4.3) is again trivially satisfied, since |u(x) -u(y)| q-2 (u(x) -u(y)) v(x) q u(x) q-1 -v(y) q u(y) q-1 ≤ 0. We can thus suppose that v(y) = 0, then we rewrite the left-hand side of (4.3) as

|u(x) -u(y)| q-2 (u(x) -u(y)) v(x) q u(x) q-1 - v(y) q u(y) q-1 = u(x) q v(y) u(y) q 1 - u(y) u(x) q-1 v(x) u(y) v(y) u(x) q - u(y) u(x)
while the right-hand side of (4.3) rewrites as

|v(x) -v(y)| q = u(x) q v(y) u(y) q v(x) u(y) v(y) u(x) - u (y) u(x) q .
Then if we set

A = v (x) u(y) v(y) u(x) and t = u(y) u(x) ,
the previous manipulations show that (4.2) is equivalent to the following

(1 -t) q-1 (A q -t) ≤ |A -t| q , for 0 ≤ t ≤ 1,
The previous elementary inequality is true (see [START_REF] Frank | Non-linear ground state representations and sharp Hardy inequalities[END_REF]Lemma 2.6]), thus we get the desired conclusion.

Some applications: local integrals

Positive eigenfunctions.

In what follows, we denote by Ω ⊂ R N an open connected set such that |Ω| < ∞. For 1 < p < ∞, as it is customary we denote by W 1,p 0 (Ω) the closure of C ∞ 0 (Ω) with respect to the L p norm of the gradient. We also take H :

R N → [0, ∞) to be a C 1 convex p-homogeneous function such that 1 C |z| p ≤ H(z) ≤ C |z| p , z ∈ R N ,
for some C ≥ 1. Then for 1 < q ≤ p, we set (5.1)

λ p,q (Ω) = min u∈W 1,p 0 (Ω) Ω H(∇u) dx : u L q (Ω) = 1 .
Theorem 5.1 (Uniqueness of positive eigenfunctions). Let 1 < p < ∞ and 1 < q ≤ p. Let λ > 0 be such that there exists a non trivial function u ∈ W 1,p 0 (Ω) verifying

- 1 p div ∇H(∇u) = λ u q-1 , u ≥ 0, in Ω.
Then we have

(5.2) λ Ω |u| q dx q-p q = λ p,q (Ω),
and v = u u -1 L q (Ω) is a minimizer of the variational problem in (5.1).

Proof. We first observe that u > 0 almost everywhere 6 in Ω, by the strong minimum principle (see for example [START_REF] Trudinger | On Harnack type inequalities and their applications to quasilinear elliptic equations[END_REF]Theorem 1.2]). We also notice that the case p = q is now well-established (see for example [START_REF] Allegretto | A Picone's identity for the p-Laplacian and applications[END_REF][START_REF] Brasco | A note on positive eigenfunctions and hidden convexity[END_REF][START_REF] Jaroš | A-harmonic Picone's identity with applications, to appear on[END_REF][START_REF] Takáč | Variational problems with a p-homogeneous energy[END_REF]), we limit ourselves to consider the case q < p.

The proof is just based on an application of Proposition 2.9. We observe that v is a solution of

- 1 p div ∇H(∇v) = λ u q-p L q (Ω) v q-1 , v > 0, in Ω.
Moreover, v is admissible for the variational problems defining λ p,q (Ω), thus by testing the previous equation with v itself and using the homogeneity of H, we get λ u q-p L q (Ω) ≥ λ p,q (Ω). Let u 1 ∈ W 1,p 0 (Ω) be a function achieving the minimum in the right-hand side of (5.1). Then we have

λ u q-p L p (Ω) = λ u q-p L p (Ω) Ω u q 1 dx = λ u q-p L p (Ω) Ω v q-1 u q 1 v q-1 dx = 1 p Ω ∇H(∇v), ∇ u q 1 v q-1 dx ≤ Ω H(∇u 1 ) q p H(∇v) p-q p dx.
If we now apply Hölder's and Young's inequality, the previous gives the desired result.

Remark 5.2. Of course, a completely equivalent proof of the previous result could use Proposition 2.6, as in [START_REF] Brasco | A note on positive eigenfunctions and hidden convexity[END_REF]. As remarked in the Introduction, we believe that a direct application of hidden convexity provides a cleaner justification of the result, while on the other hand Picone inequality offers a quicker proof. An alternative proof can be found in [START_REF] Ôtani | On the first eigenvalue of some quasilinear elliptic equations[END_REF], later refined by Kawohl and Lindqvist in [START_REF] Kawohl | Positive eigenfunctions for the p-Laplace operator revisited[END_REF].

Remark 5.3 (Sharpness of the condition q ≤ p). For a general open set Ω with finite measure, the previous result can not hold true for q > p. Indeed, let us consider an annular domain T = {x ∈ R N : 1 < |x| < r} and take H(z) = |z| p , then the problem

λ rad p,q (T ) = min W 1,p 0 (T ) T |∇u| p dx : u radial function, u L q (T ) = 1 ,
admits a minimizer u 0 ∈ W 1,p 0 (T ), which is a positive solution of

-∆ p u = λ rad p,q (T ) u q-1 , in T, with T |u| q dx = 1.
On the other hand, Nazarov in [23, Proposition 1.2] has proved that if q > p one can always take r sufficiently close to 1 such that minimizers of (5.1) are not radial. This clearly means that λ rad p,q (T ) > λ p,q (T ). 5.2. Hardy-type inequalities. As another application of the general Picone inequality (2.5), we have the following family of sharp inequalities. Theorem 5.4 (Weighted Hardy inequalities with general norms). Let F : R N → [0, +∞) be a C 1 strictly convex norm. Let 1 < p < N , for every γ > p -N we have 6 Actually, the much stronger result

(5.3) N + γ -p p p R N |v| p F * (x) γ-p dx ≤ R N F (∇v) p F * (x) γ dx, v ∈ C 1 0 (R N \ {0}),
inf K u ≥ 1 CK , for every compact K ⋐ Ω,
holds. Here we want to point out that the weaker information "u > 0 almost everywhere" suffices for this argument to work.

where F * is the dual norm defined by

F * (z) = sup x =0 x F (x) , z , z ∈ R N .
Proof. We start proving (5.3) for positive C 1 0 (R N \ {0}) functions. We take β > 0 and set

u(x) = F * (x) -β , x ∈ R N \ {0}. Observe that u is a C 1 function in R N \ {0} such that -div(F * (x) γ F (∇u) p-1 ∇F (∇u)) = β p-1 N -p -β (p -1) + γ u p-1 F * (x) γ-p , in R N \ {0}. (5.4)
Indeed, we recall the following relations between F and F * (see [START_REF] Schneider | Convex bodies: the Brunn-Minkowski theory[END_REF] for example) (5.5)

F (∇F * (x)) = 1 and ∇F (∇F * (x)) = x F * (x) , x = 0.
Of course ∇u = -β F * (x) -β-1 ∇F * (x), x = 0, by using (5.5) and the homogeneity of F we get

F (∇u) p-1 = β p-1 F * (x) -(β+1) (p-1) = β p-1 F * (x) -β p+β-p ,
and still by (5.5) and the fact that ∇F is 0-homogeneous, we also have

∇F (∇u) = -∇F (∇F * (x)) = - x F * (x) , x = 0.
Thus we get

-div F * (x) γ F (∇u) p-1 ∇F (∇u) = β p-1 div F * (x) -β p+β-p+γ x = β p-1 [N -β p + β -p + γ] F * (x) -β (p-1)+γ-p ,
as desired, where we used that ∇F * (x), x = F * (x), x = 0, again by homogeneity. Finally, by using the definition of u, we get

F * (x) -β (p-1)+γ-p = u p-1 F * (x) γ-p .
Then u verifies

C N,p R N u p-1 F * (x) γ-p ϕ dx = R N F * (x) γ F (∇u) p-1 ∇F (∇u), ∇ϕ dx, ϕ ∈ C 1 0 (R N \ {0}).
We test the previous equation with ϕ = v p u 1-p , where v ∈ C 1 0 (R N \ {0}) is positive. We get

β p-1 N -β p + β -p + γ R N v p F * (x) γ-p dx = R N F * (x) γ F (∇u) p-1 ∇F (∇u), ∇ v p u p-1 dx.
We can now use Proposition 2.9 with the choice H(z) = F (z) p , so to obtain

β p-1 N -β p + β -p + γ R N v p F * (x) γ-p dx ≤ R N F (∇v) p F * (x) γ dx
In order to conclude, it is now sufficient to observe that the function

β → β p-1 N -β p + β -p + γ ,
is maximal for β = (N + γ -p)/p. The previous argument gives (5.3) for a positive v ∈ C 1 0 (R N \{0}). Of course, the previous proof is still valid for positive Lipschitz functions supported in R N \ {0}. The result for a general v ∈ C 1 0 (R N \ {0}) then follows by writing v = v + -v -and observing that v + , v -are positive Lipschitz functions with support in R N \ {0}.

Remark 5.5. By taking γ = 0 in (5.3), we have the usual Hardy inequality on the whole space with respect to a general norm, i.e.

(5.6)

N -p p p R N |v| F * (x) p dx ≤ R N F (∇v) p dx
A different proof of (5.6) (based on symmetrization techniques) can be found in [START_REF] Van Schaftingen | Anisotropic symmetrization[END_REF]Proposition 7.5]. For γ = 0 and F being the Euclidean norm, a related inequality can be found in [START_REF] Abdellaoui | Some improved Caffarelli-Kohn-Nirenberg inequalities[END_REF].

6. Some applications: nonlocal integrals 6.1. Positive eigenfunctions. We denote by Ω ⊂ R N an open connected set, which is now supposed to be bounded. Let 1 < p < ∞ and 0 < s < 1, in what follows we denote by W s,p 0 (Ω) the completion of C ∞ 0 (Ω) with respect to the norm

u W s,p 0 (Ω) = R N R N |u(x) -u(y)| p |x -y| N +s p dx dy 1 p
.

As before, for 1 < q ≤ p we introduce the first eigenvalue (6.1) λ s p,q (Ω) = min u∈W s,p 0 (Ω)

u p W s,p 0 (Ω) : u L q (Ω) = 1 ,
the reader is referred to [START_REF] Brasco | The fractional Cheeger problem, to appear on Interfaces Free Bound[END_REF][START_REF] Franzina | Fractional p-eigenvalues[END_REF][START_REF] Lindgren | Fractional eigenvalues[END_REF] for a more detailed account about the case q = p. We notice that a minimizer u of the previous problem is a weak solution of (-∆ p ) s u = λ s p,q (Ω) |u| q-2 u, in Ω, which means that

R N R N |u(x) -u(y)| p-2 u(x) -u(y) |x -y| N +s p ϕ(x) -ϕ(y) dx dy = λ s p,q (Ω) Ω |u| q-2 u ϕ dx,
for every ϕ ∈ W s,p 0 (Ω).

Theorem 6.1 (Uniqueness of positive eigenfunctions). Let 1 < p < ∞, 0 < s < 1 and 1 < q ≤ p. Let λ > 0 be such that there exists a non trivial function u ∈ W s,p 0 (Ω) verifying

(-∆ p ) s u = λ |u| q-2 u, u ≥ 0, in Ω.
Then we have

(6.2) λ Ω |u| q dx q-p q = λ p,q (Ω),
and v = u u -1 L q (Ω) is a minimizer of the problem in (6.1).

Proof. At first, it is again crucial to observe that u > 0 almost everywhere in Ω, thanks to the minimum principle of Theorem A.1. Then the result for the case p = q follows by using [START_REF] Franzina | Fractional p-eigenvalues[END_REF]Theorem 4.1]. We now consider the case q < p.

Again, we observe that v solves

(-∆ p ) s v = λ u q-p L q (Ω) v q-1 , u > 0, in Ω.
and since v is admissible for the variational problems defining λ s p,q (Ω) we get λ u q-p L q (Ω) ≥ λ s p,q (Ω).

Let u 1 ∈ W s,p 0 (Ω) be a function achieving the minimum in the right-hand side of (6.1). Then again we have

λ u q-p L q (Ω) = λ u q-p L q (Ω) Ω v q-1 u q 1 v q-1 dx = R N R N |v(x) -v(y)| p-2 (v(x) -v(y)) |x -y| N +s p u 1 (x) q v(x) q-1 - u 1 (y) q v(y) q-1 dx dy ≤ R N R N |u 1 (x) -u 1 (y)| q |x -y| N q p +s q |v(x) -v(y)| p-q |x -y| N p-q p +s (p-q)
dx dy

where we used Proposition 4.2. If we now apply Hölder's and Young's inequalities with exponents p/q and p/(p -q), the previous gives the desired result.

) s u = λ u p-1 , in R N \ {0}.
The latter is the content of the next technical result. Lemma 6.2. Let 1 < p < ∞ and β, r > 0. Then the function

f (x) = R N |x| -β -|y| -β p-2 |x| -β -|y| -β |x -y| N +r dy, x ∈ R N , is radial and β (1 -p) -r homogeneous. Proof. Let us pick x 1 = x 2 such that |x 1 | = |x 2 |.
Let us set call R : R N → R N the linear isometry defined by the reflection in the hyperplan

π = x ∈ R N : x, x 1 -x 2 = 0 .
Then by changing variables we have

f (x 1 ) = R N |x 1 | -β -|R z| -β p-2 |x 1 | -β -|R z| -β |x 1 -R z| N +r dz = R N |x 2 | -β -|R z| -β p-2 |x 2 | -β -|R z| -β |R x 2 -R z| N +r dz = R N |x 2 | -β -|z| -β p-2 |x 2 | -β -|z| -β |x 2 -z| N +r dz = f (x 2 ),
where we used that R x 2 = x 1 , that |R z| = |z| and the linearity of R. This shows that f is radial.

For the second part, it is sufficient to observe that

f (t x) = R N t -β |x| -β -|y| -β p-2 t -β |x| -β -|y| -β |t x -y| N +r dy = t -β (p-1)-N -r R N |x| -β -|z| -β p-2 (|x| -β -|z| -β ) |x -z| N +r t N dz = t -β (p-1)-r f (x), x ∈ R N ,
for all t > 0, which gives the desired conclusion.

We then have the following sharp Hardy inequality for the fractional Sobolev space W s,p , first proved in [START_REF] Frank | Non-linear ground state representations and sharp Hardy inequalities[END_REF]. Theorem 6.3 (Fractional Hardy inequality). Let s ∈ (0, 1) and 1 < p < ∞ such that s p < N . Then there exists a constant C = C(N, s, p) > 0 (see equation (B.2) below) such that An optimization over β leads to the desired result, see Appendix B for more details.

(6.3) C R N |v| p |x| s p dx ≤ R N R N v(x) -v(y) p |x -y| N +s p dx dy, for all v ∈ C ∞ 0 (R N \ {0}).
Appendix A. A minimum principle for positive nonlocal eigenfunctions

In the following, we provide a proof of a minimum principle for positive weak supersolutions to equation (-∆ p ) s u = 0, in Ω, u ≡ 0 in R N \ Ω.

i.e. for functions u ∈ W s,p 0 (Ω) such that R N R N |u(x) -u(y)| p-2 (u(x) -u(y)) |x -y| N +sp (ϕ(x) -ϕ(y)) dx dy ≥ 0, for all ϕ ∈ C ∞ 0 (Ω), ϕ ≥ 0.

Let x 0 be any fixed point in Ω, and for every r > 0 let B r (x 0 ) denote the ball of radius r centered at x 0 . The main ingredient of our result is the following lemma, which is a consequence of a more general logarithmic estimate recently established by Di Castro, Kuusi and Palatucci in [START_REF] Castro | Local behavior of fractional p-minimizers[END_REF].

DKP Logarithmic Lemma. Let 1 < p < ∞, s ∈ (0, 1) and u ∈ W s,p 0 (Ω) be a supersolution such that u ≥ 0 in B 2 r (x 0 ) ⋐ Ω. Then for every 0 < δ < 1 there holds where u -= max{-u, 0} and C = C(N, p, s) > 0 is a constant.

We then have the following minimum principle.

Then, by the first part of the proof u > 0 almost everywhere on each K n . By letting n go to ∞, we get the conclusion.

By using this and exchanging the order of integration, we then obtain

C(β) = 2 H N -1 (S N -1 ) R N S N -1 |x| -β -|y| -β p-2 |x| -β -|y| -β |x -y| N +s p dH N -1 (x) dy = H N -2 (S N -2 ) H N -1 (S N -1 ) R N |1 -|y| -β | p-2 (1 -|y| -β ) × 1 -1
(1 -t 2 )

N -3 2 (1 -t 2 )

N -3 2

(1 -2 t ̺ + ̺ 2 )

N +s p 2 dt, then by using polar coordinates we get

C(β) = 2 ∞ 0 ̺ N -1 1 -̺ -β p-2 (1 -̺ -β ) Φ(̺) d̺ = -2 1 0 ̺ N -1 1 -̺ -β p-1 Φ(̺) d̺ + 2 ∞ 1 ̺ N -1 1 -̺ -β p-1 Φ(̺) d̺.
We now perform the change of variable ̺ = r -1 in the second integral and observe that Φ(1/r) = r N +s p Φ(r). where the function Φ is defined in (B.1). Thus this is the best constant in the Hardy inequality (6.3) (see [START_REF] Frank | Non-linear ground state representations and sharp Hardy inequalities[END_REF]Section 3.3] for more details).

  Proof. We recall that a positively α-homogeneous function u which is radially simmetric, that isu(x) = ϕ(|x|) x ∈ R N ,is uniquely determined modulo a multiplicative constant, namely u(x) = ϕ(1) |x| α . By this elementary observation, we can deduce from Lemma 6.2 that the functionu(x) = |x| -β is a solution of R N R N |u(x) -u(y)| p-2 (u(x) -u(y)) |x -y| N +s p (ϕ(x) -ϕ(y)) dx dy = C(β) R N u p-1 |x| -s p ϕ(x) dx, for all ϕ ∈ C ∞ 0 (R N \ {0}),whereC(β) = 2 R N |x| -β -|y| -β p-2 |x| -β -|y| -β |x -y| N +s p dy, x ∈ S N -1 ,and the previous integral is constant for x ∈ S N -1 , thanks to Lemma 6.2. Then, if one picks v ∈ C ∞ 0 (R N \ {0}) positive and plugs in ϕ = v p u 1-p as a test function in the previous equation, it turns out that (6.4) C(β) R N |v| p |x| -s p dx ≤ R N R N |v(x) -v(y)| p |x -y| N +sp dx dy.

  y| N +s p dx dy≤ C r N -s p δ 1-p r s p R N \B 2 r u -(y) p-1 |y -x 0 | N +sp dy + 1 , (A.1)

( 1 -2 t |y| + |y| 2 ) N +s p 2 1 - 1

 12211 dt dywhere we used that|x -y| = 1 -2 t |y| + |y| 2 ,for every x ∈ Σ t .We now set(B.1) Φ(̺) = H N -2 (S N -2 )

̺ N -1 1 - 1 0̺ -1+s p 1 --2 1 0̺ 1 p- 1 Φ(̺) d̺ + 2 1 0̺ -1+s p 1 - 1 0̺N -s p p - 1 . 1 0̺

 11111111111 ̺ -β p-1 Φ(̺) d̺ + 2 ̺ β p-1 Φ(̺) d̺ = N -1-β (p-1) ̺ β -̺ β p-1 Φ(̺) d̺ = 2 s p-1 1 -̺ N -s p-β (p-1) 1 -̺ β p-1 Φ(̺) d̺.We now observe that the term into square brackets is positive ifβ ≤Moreover, thanks to Lemma B.1 the integrand is maximal forβ = N -s p p ,and thus we get the conclusion. Remark B.3. Observe that we have (B.2) C N -s p p = 2 s p-1 1 -̺ N -s p p p Φ(̺) d̺,

  6.2. Hardy-type inequalities. As in the local case, by means of the discrete Picone inequality (4.2) we can prove a nonlocal Hardy inequality, like in[START_REF] Frank | Non-linear ground state representations and sharp Hardy inequalities[END_REF]. The idea is still to look at power-type positive solutions of (-∆ p

Theorem A.1. Let Ω ⊂ R N be an open bounded set, which is connected. Let s ∈ (0, 1), 1 < p < ∞ and u ∈ W s,p 0 (Ω) be a weak supersolution such that u ≥ 0 in Ω. Let us suppose that (A. [START_REF] Allegretto | A Picone's identity for the p-Laplacian and applications[END_REF] u ≡ 0 in Ω.

Then u > 0 almost everywhere in Ω.

Proof. We first prove that for every K ⋐ Ω compact connected, if

then u > 0 almost everywhere in K.

Let K ⋐ Ω be a connected compact set, then K ⊂ {x ∈ Ω : dist(x, ∂Ω) > 2 r} for some r > 0. We then observe that K can be covered by a finite number of balls B r/2 (x 1 ), . . . B r/2 (x k ) such that x i ∈ K and

Let us now suppose that u = 0 on a subset of K with positive measure. Then for some i ∈ {1, . . . , k -1}, we have that the set

has positive measure. We define

for δ > 0 and claim that the following Poincaré inequality holds true

Indeed, observe that F δ (x) = 0 for every x ∈ Z , hence for every x ∈ B r/2 (x i ) and y = x with y ∈ Z, we get

Now integrating with respect to y ∈ Z gives

which proves (A.5) up to an integration with respect to x ∈ B r/2 (x i ).

We now observe that

thus if we combine (A.5), (A.1) and observe that u -≡ 0, we get

with C independent of δ. By letting δ go to 0 in (A.6), we can then infer u = 0 almost everywhere in B r/2 (x i ).

By using property (A.4), we can repeat the previous argument for the balls B r/2 (x i-1 ) and B r/2 (x i+1 ) and so on, up to obtain that u = 0 almost everywhere on K. This clearly contradicts (A.3), thus u > 0 almost everywhere in K.

Let us now assume (A.2). Since Ω is connected, there exists a sequence of connected compact sets K n ⋐ Ω such that

Appendix B. Optimal constant for the fractional Hardy inequality

In Section 6 we used that u(x) = |x| -β is a solution of

In this appendix we discuss some features of the constant

and determines the best constant in (6.3), see equation (B.2) below. Computations are very much the same as in the paper [START_REF] Frank | Non-linear ground state representations and sharp Hardy inequalities[END_REF] by Frank and Seiringer, up to some simplifications. For simplicity we focus on the case N ≥ 2 and s p < N .

Lemma B.1. Let N ≥ 2, 0 < s < 1 and 1 < p < ∞. For every 0 < ̺ < 1 the function

is maximal for β = (N -s p)/p.

Proof. We just have to differentiate the function G. Indeed, we have

that is if and only if β is such that

By passing to the logarithm, we obtain the assertion.

We can then determine the maximal values of C(β).

Lemma B.2. Let N ≥ 2, 0 < s < 1 and 1 < p < ∞. For every β > 0 we have

Proof. We recall that

and the right-hand side is independent of x ∈ S N -1 . Thus we have

|x -y| N +s p dy dH N -1 (x).

We observe that for every y ∈ R N \ {0} we can write

Σ t (y), where


Zahir Belhadi

Ferhat Ménas

Alain Bérard

Herve Mohrbach

F Menas

Quantization of soluble classical constrained systems

published or not. The documents may come

Introduction

The Hamiltonian method for the quantization of a dynamical system requires to postulate canonical Poisson brackets among coordinates and momenta. Then the correspondence principle is applied to determine the different commutators of the quantum operators associated to the classical dynamical variables. However, for singular Lagrangians or constrained systems, this procedure does not work and the brackets have to be determined and not postulated. Dirac has developed a general formalism for treating such systems [START_REF] Dirac | Lectures on Quantum Mechanics[END_REF][START_REF] Anderson | Constraints in covariant field theories[END_REF]. Dirac formalism has been widely used but the determination of the Hamiltonian and the Dirac brackets replacing the Poisson brackets can be very cumbersome. More recently Faddeev and Jackiw have proposed [START_REF] Faddeev | Hamiltonian Reduction of Unconstraint and Constraint Systems[END_REF][START_REF] Liao | Non-equivalence of Faddeev-Jackiw method and Dirac-Bergmann algorithm and the modification of Faddeev-Jackiw method for keeping the equivalence[END_REF] an alternative approach (FJ approach) based on the symplectic formalism and Darboux theorem (which is equivalent to an approach proposed by Souriau [START_REF] Souriau | Structures des systèmes dynamiques[END_REF] as remarked in [START_REF] Duval | The exotic Galilei group and the "Peierls substitution[END_REF] and [START_REF] Martina | Dynamics with exotic symmetries[END_REF] ) to find the brackets and the Hamiltonian which can often avoid many of the steps of Dirac method.

Here we propose another approach for the determination of the Hamiltonian and the brackets for unconstrained and constrained systems which avoids the Dirac categorization of constraints and the Darboux theorem of the FJ approach. Our method is based on the computation of the brackets between the constants of integration of the solutions of the Euler-Lagrange equations of motion.

From them all brackets among coordinates and momenta can be directly deduced. Therefore when a system is classically soluble, i.e the analytical solution of its equations of motion are known, it is possible to quantize the system without using Dirac or FJ formalisms.

In the two next sections, the general formalism for the computation of the brackets among the constants of integration and the link with Dirac and FJ methods is presented. Then in the following sections we provide several examples which are treated with the three methods. At the end of the paper we mention possible applications to field theory.

2 Remark on the integration constants Consider a system of N degrees of freedom characterized by its generalized coordinates q = (q 1 , q 2 , ..., q N ) and conjugate momenta p = (p 1 , p 2 , ..., p N ). Now let f = f (q, p) and g = g(q, p) be two arbitrary functions. We define an antisymmetric bilinear bracket satisfying the Leibniz rule and the Jacobi identities, such that:

{f, g} = N i,
At this level this bracket is not yet specified. It can be either the Poisson, or Dirac or also a FJ bracket. Now, suppose that q and p are functions of time and of new variables R = (R 1 , R 2 , ..., R M ) with M ≤ 2N , i.e1 q = q(R, t) and p = p(R, t). Then Eq. ( 1) becomes

{f, g} = N i,j=1 M k,l=1 {q i , q j } ∂f ∂R k ∂R k ∂q i ∂g ∂R l ∂R l ∂q j + {p i , p j } ∂f ∂R k ∂R k ∂p i ∂g ∂R l ∂R l ∂p j + N i,j=1 M k,l=1 {q i , p j } ∂f ∂R k ∂R k ∂q i ∂g ∂R l ∂R l ∂p j - ∂f ∂R k ∂R k ∂q j ∂g ∂R l ∂R l ∂p i which simplifies to {f, g} = M k,l=1   N i,j=1 {q i , q j } ∂R k ∂q i ∂R l ∂q j + N i,j=1 {p i , p j } ∂R k ∂p i ∂R l ∂p j + N i,j=1 {q i , p j } ∂R k ∂q i ∂R l ∂p j - ∂R k ∂q j ∂R l ∂p i   ∂f ∂R k ∂g ∂R l
and leads to the expression of {f, g} in terms of {R k , R l }:

{f, g} = M k,l=1 {R k , R l } ∂f ∂R k ∂g ∂R l (2)
With the introduction of the new variables, the sum goes from 1 to M (M ≤ 2N ). Thus, the presence of constraints reduces the number of variables and we will only work with the free variables. Now, if f = q i or p i and g = H the Hamiltonian of the system, we obtain the following equations (using Hamilton equations for i = 1 to N )

dq i dt = {q i , H} ⇒ ∂q i ∂t + M j=1 ∂q i ∂R j dR j dt = M j,k=1 {R j , R k } ∂q i ∂R j ∂H ∂R k dp i dt = {p i , H} ⇒ ∂p i ∂t + M j=1 ∂p i ∂R j dR j dt = M j,k=1 {R j , R k } ∂p i ∂R j ∂H ∂R k (3) Now suppose that R k , k = 1...M are constants of motion (first integrals), denoted R k = C k , k = 1.
..M, then the above equations will be reduced to the form

∂q i ∂t = M j,k=1 {C j , C k } ∂q i ∂C j ∂H ∂C k ( 4
)
∂p i ∂t = M j,k=1 {C j , C k } ∂p i ∂C j ∂H ∂C k i = 1...N (5)
Therefore the dynamics of the system can be expressed in terms of the brackets among the integration constants, without ever specifying the nature of the bracket (Poisson, Dirac or FJ brackets ). Therefore the relations Eqs.( 4) are universal and common to all types of brackets. With the introduction of the notation ξ i | i=1,...2N = (q i , p i ) we can now write Eqs.(4) in a compact form :

∂ξ i ∂t = M j,k=1 {C j , C k } ∂ξ i ∂C j ∂H ∂C k i = 1...2N (6)
These relations can obtained in the same manner as we did by starting from the equations of motion

. ξ i = {ξ i , H} = 2N j=1 {ξ i , ξ j } ∂H ∂ξ j i = 1...2N
used in the FJ approach.

3 Method of Integration constants: Constrained (and unconstrained) systems

In this section we use the previous properties Eqs. [START_REF] Liao | Non-equivalence of Faddeev-Jackiw method and Dirac-Bergmann algorithm and the modification of Faddeev-Jackiw method for keeping the equivalence[END_REF] to determine the brackets of any exactly soluble system. At the end the canonical quantization allows us to build the quantum version.

Although our approach (called in the rest of the paper the CI method) is applicable to systems without constraints, it is for constrained systems that it shows its main interest. Indeed, we propose a new method which greatly facilitates the computation of the Dirac brackets, while avoiding both the complicated Dirac algorithm and the Darboux theorem in the case of the FJ approach. Consider a classical system described by a singular autonomous Lagrangian L(q, q) where q = (q 1 , ..., q N ) are the generalized coordinates and q = ( q1 , ..., qN ) the generalized velocities. Suppose we know the (general) analytical solutions q(t) = q(t, C) of the Euler-Lagrange equations and the momenta p(t) = p(t, C) p i = ∂L ∂ ẋi , where C = (C 1 , C 2 , ..., C M ) is the set of constants of integration. For constrained systems we have obviously M < 2N.

Before going further, it is necessary to distinguish between two cases: the first is when there are no arbitrary functions in the solutions (no gauge symmetry), the second is otherwise. In the later case with a gauge symmetry, we must first choose the arbitrary functions once by adding new conditions (fixing the gauge) before moving to the canonical formalism and defining any brackets.

From the analytical solutions of the equations of motion we can write the Hamiltonian 2 as H(q(t), p(t)) = H(q(t, C), p(t, C)) and from Eqs. (3) we deduce the fundamental equations

∂ ∂t qi (t, C) = M j,k=1 {C j , C k } ∂ qi ∂C j ∂H ∂C k i = 1...N ∂ ∂t pi (t, C) = M j,k=1 {C j , C k } ∂ pi ∂C j ∂H ∂C k i = 1...N (7)
These 2N equations contain M (M -1)/2 unknown brackets {C j , C k }, with j, k = 1...M . Our method consist in determining the brackets {C j , C k } from the Eqs.( 7) via a simple identification. But, with this procedure, we easily see that only the brackets containing at least one integration constant in the expression of the Hamiltonian are available. To solve this problem, we must add supplementary terms to the Lagrangian. In other words, if for example the bracket {C i , C j } is not accessible and if one of these constants appears in the expression of the generalized coordinate q i , one has to add a term of the form ηq i to the Lagrangian, redo all the calculations and put η = 0 at the end (as an illustration see example 4.2). Using the brackets {C i , C j } we can compute the brackets {q i , q j }, {p i , p j } and {q i , p j } more easily than with any other existing approaches.

If the result of the calculation depends on the integration constants, it is possible to make them disappear by inverting the solutions q(t, C) and p(t, C). At this time, to be sure of the validity of our calculations, we can just verify that q(t, C) and p(t, C) are solutions of the equations of motion obtained from the Hamilton equations using these fundamental brackets. We see that in our method we do not even talk about constraints unlike other approaches. These brackets are essential for the quantization of the system by introducing the quantum operators q and p and the principle of correspondence we have :

[f (q, p), g(q, p)] = i {f (q, p), g(q, p)}

Note, that it might look surprising to see brackets between constants and the derivative of functions with respect to those constants. Actually, these constants are first integrals and must be considered as in the Hamilton-Jacobi formalism which also treats the constants as variables. 2 We can also obtain the Hamiltonian by putting the solutions into the Legendre transformation

H = i d qi (t,C) dt pi (t, C) -L qi (t, C), d qi (t,C) dt .
In this way we do not have to inverse the momenta with respect to the velocities.

Applications

In this section we present several appealing examples. As a first application of the method we consider the isotonic oscillator which is an unconstrained system. This case shows that the method is general and thus valid for constrained and unconstrained systems. The second application deals with a singular autonomous Lagrangian which exemplifies the previous discussion about the procedure when {C i , C j } is not directly accessible (when the Hamiltonian does not contain C i or C j ). The other examples concern different constrained systems which are studied with the Dirac, FJ and our method for comparison.

Regular Lagrangian

For the first example we consider an unconstrained system namely the Lagrangian of the isotonic (anharmonic) oscillator:

L = 1 2 ẋ2 - 1 2 ω 2 x 2 - k x 2
The Euler-Lagrange equations lead to the following non-linear equation

ẍ + ω 2 x - 2k x 3 = 0 whose solution is (see ref: [8]) x(t) = 1 Aω (A 4 ω 2 -2k) sin 2 (wt + φ) + 2k
where A and φ are the constants of integration. The conjugate momentum is given by p

x (t) = ẋ(t) = (A 4 ω 2 -2k) sin(wt+φ) cos(wt+φ) A √ (A 4 ω 2 -2k) sin 2 (wt+φ)+2k
.

Expressing the Hamiltonian H = 1 2 p 2 x + 1 2 ω 2 x 2 + k
x 2 in terms of these constants we obtain

H = A 4 ω 2 + 2k 2A 2
From the Hamilton equation ẋ = {x, H} and the property Eq. ( 7) we obtain ∂x ∂t = {φ, A} ∂x ∂φ ∂H ∂A from which we obtain the bracket between the constants of integration

{φ, A} = A 3 ω A 4 ω 2 -2k
From Eq. (2) we obtain the brackets among the dynamical variables:

{x, p x } = {φ, A} - ∂x ∂A ∂p x ∂φ + ∂x ∂φ ∂p x ∂A A short computation shows that -∂x ∂A ∂px ∂φ + ∂x ∂φ ∂px ∂A = A 4 ω 2 -2k
A 3 ω so that we retrieve the canonical Poisson bracket {x, p x } = 1 as expected for an unconstrained system.

Singular autonomous Lagrangians: first example

Consider a constrained system described by the Lagrangian :

L = ẋ2 2 + x ẏ -y ż
This system gives an illustration of the discussed procedure when the Hamiltonian does not contain all constants of integration. The Euler-Lagrange equations are ẍ -ẏ = 0 ẋ + ż = 0 ẏ = 0 whose analytical solutions are :

x(t) = at + b y(t) = c z(t) = -at + d
where a, b, c and d are the constants of integration. From the conjugate momenta p x = ẋ, p y = x and p z = -y, and the Legendre transformation we obtain the Hamiltonian

H = p 2 x 2 = a 2
2 From this Hamiltonian only the brackets containing a are accessible. To solve this issue we introduce a new Lagrangian

L = ẋ2 2 + x ẏ -y ż -λx -ξy
where λ and ξ are real parameters. The new equations of motion are thus ẍ-ẏ +λ = 0, ẋ+ ż +ξ = 0 and ẏ = 0. The general solution of this system reads

x(t) = - λ 2 t 2 + at + b y(t) = c z(t) = λ 2 t 2 -at -ξt + d (9)
The Hamiltonian now reads H = p 2

x 2 + λx + ξy or in terms of the constants of integration

H = 1 2 a 2 + λb + ξc
From the Hamilton equations we can directly obtain the brackets between the different constants of integration. Indeed we have ẋ = {x, H} ⇒ -λt + a = tλ{a, b} + ξt{a, c} + a{b, a} + ξ{b, c} and

• z = {z, H} ⇒ λt -a -ξ = -tλ{a, b} -ξt{a, c} + a{d, a} + λt{d, b} + ξ{d, c}
which by identification gives:

{a, b} = {a, d} = -1 {c, d} = 1 {a, c} = {b, c} = {b, d} = 0
A direct calculation using these results and the solution Eq. ( 9) leads the following brackets among the dynamical variables

{x, p x } = {y, z} = {z, p z } = 1 {p x , p y } = {p x , z} = -1
These brackets do not depend on the parameters λ and ξ and are thus unchanged if we put λ = ξ = 0.

In other words they are also the brackets of the initial Lagrangian L = ẋ2 2 +x ẏ-y ż-λx-ξy| λ=ξ=0 = ẋ2 2 + x ẏ -y ż.

Singular autonomous Lagrangians: second example

Consider a system described by the Lagrangian :

L = ẋ2 2 + x 2 2 ẏ - x 2 2 y

CI method

The Euler-Lagrange equations are

ẍ = x ẏ -xy x ẋ = x 2 2 p x = ẋ p y = x 2 2
whose analytical solutions are :

x(t) = ae -1 2 t y(t) = be t - 1 4 p x (t) = - a 2 e -1 2 t = - x 2 p y (t) = a 2 2 e -t
where a and b are the constants of integration. Expressing the Hamiltonian in terms of these constants we have H = p 2 x + x 2 y /2 = a 2 b/2 From Eqs. [START_REF] Martina | Dynamics with exotic symmetries[END_REF] we have for the variable x,

ẋ = {x, H} ⇒ a 2 e -1 2 t = -{ae -1 2 t , a 2 b 2 } ⇒ a 2 e -1 2 t = -e -1 2 t a 2 2 {a, b}
and then by identification {a, b} = -1 a From this brackets we obtain directly all other brackets among the dynamical variables. For instance:

{x, y} = {ae -1 2 t , be t - 1 4 } = e 1 2 t {a, b} = -1 a e 1 2 t = - 1 x .
The other brackets are calculated in the same way, leading to

{x, p x } = {x, p y } = {p x , p y } = 0 {y, p x } = - 1 2x {y, p y } = 1
To check the validity of the results we determine the Hamilton equations by using the derived brackets, which are:

ẋ = {x, H} ⇒ ẋ = -x/2 ; ẏ = {y, H} ⇒ ẏ = -p x /(2x) + y ṗx = {p x , H} ⇒ ṗx = x/4 ; ṗy = {p y , H} ⇒ ṗy = -x 2 /2
These equations are equivalent to the Euler-Lagrange equations, which validates our approach for this first example.

Dirac approach

For comparison we consider the same Lagrangian with the Dirac treatment. First we deduce the momenta p x = ẋ and p y = x 2 2 and the canonical Hamiltonian

H c = p 2 x 2 + x 2 2 y.
As we have a primary constraint φ = p y - From the different elements of the inverse matrix

M -1 = {φ, φ} -1 P B {φ, ζ} -1 P B {ζ, φ} -1 P B {ζ, ζ} -1 P B = 0 1 x -1
x 0 and those of the matrix of constraints, we can obtain the Dirac bracket {f, g} D among arbitrary functions which is defined as

{f, g} D = {f, g} P B -{f, φ 1 } P B {φ 1 , φ 2 } -1 P B {φ 2 , g} P B -{f, φ 2 } P B {φ 2 , φ 1 } -1 P B {φ 1 , g} P B
Applying this formula to the coordinates and momenta we get the following brackets: which are the same as with the CI method.

Faddeev-Jackiw approach

It is instructive to compare our method to the FJ approach. The initial Lagrangian is written in the following form, that we call the FJ Lagrangian

L F J = ẋp x + ẏp y -H = z ẋ + x 2 2 ẏ - z 2 2 - x 2 2 y
using the primary constraint p y = x 2 2 and the notation z = p x . In this approach one has to treat x, y and z as independent variables. Then the Euler-Lagrange equations 3 are

-ż = -x ẏ + xy -x ẋ = +
x 2 2 0 = -ẋ + z which can be written in matrix form:

  0 x -1 -x 0 0 1 0 0   f   ẋ ẏ ż   =   xy x 2 2 z  
The matrix f is singular and antisymmetric (det(f ) = 0). The only zero mode is ( 0 1 x ), and we obtain the secondary constraint by multiplying the last equation by this zero mode from which we obtain the constraint x 2 + z = 0, which is the same secondary constraint as with Dirac approach. This constraint must be conserved in time ( ẋ 2 + ż = 0). To assure that we define a new Lagrangian by adding the term λ( ẋ 2 + ż) to the Lagrangian L F J . This term can be transformed in -λ( x 2 + z) to give the following expression

( 0 1 x )   xy x 2 2 z   = 0
L 1 F J = z ẋ + x 2 2 ẏ - z 2 2 - x 2 2 y + λ( x 2 + z)
where λ is a Lagrange multiplier which is considered as a new variable in the following. Therefore the new Euler-Lagrange equations are

-ż = -x ẏ + xy -(1/2) λ -x ẋ = x 2 2 0 = -ẋ + z - λ -( ẋ 2 + ż) = 0
which can be written in matrix form as:

    0 x -1 +1/2 -x 0 0 0 1 0 0 1 -1/2 0 -1 0     f     ẋ ẏ ż λ     = xy x 2 2
z 0 where f is now antisymmetric and invertible as det(f ) = x 2 = 0. The inverse of this matrix is :

f -1 =     {x,
    =     0 -1 x 0 0 1 x 0 -1 2x -1 x 0 1 2x 0 -1 0 1 x 1 0    
Then it follows that {x, y} = -1/x ; {y, z} = -

1 2x ; {x, z} = 0
To have the brackets with p y we use the primary constraint p y = x 2 2 . We have obtained the same result as in the case with the CI method.

Lagrangian of a particle in a constant strong magnetic field

We consider the simple problem of a non-relativistic particle of charge q and mass m, in a constant magnetic field -→ B 0 pointing in the z direction. The Lagrangian is

L = 1 2 mv 2 + q -→ A (x, y). -→ v -qV (x, y)
where -→ A (x, y) is the vector potential in the Coulomb gauge -→ A = 1 2 -→ r ∧ -→ B 0 and V (x, y) the scalar potential. In the limit of the strong magnetic field the mass term can be neglected and the Lagrangian is approximated as:

L = qB 0 2 (x ẏ -y ẋ) -qV (x, y)
In order to simplify the mathematics we consider the case V (x, y) = 1 2 k x 2 + y 2 , thus:

L = η (x ẏ -y ẋ) - 1 2 ξ x 2 + y 2
with η = qB0 2 and ξ = qk.

CI method

The Euler-Lagrange equations are

-2η ẏ + ξx = 0 ; 2η ẋ + ξy = 0 or ẍ + ξ 2 4η 2 x = 0 ; ÿ + ξ 2 4η 2 y =

Dirac approach

We deduce from the same Lagrangian the following momenta p x = -qB0 2 y, p y = qB0 2 x and the canonical Hamiltonian H c = ξ 2 x 2 + y 2 . We therefore have two primary constraints

φ 1 = p x + qB 0 2 y ≈ 0 φ 2 = p y - qB 0 2 x ≈ 0
The total Hamiltonian with the constraints is

H T = H c + λ 1 φ 1 + λ 2 φ 2 .
The consistency conditions give λ 1 ≈ -k B0 y and λ 2 ≈ k B0 x. We also see that these constraints are second class as {φ 1 , φ

2 } P B = -{φ 2 , φ 1 } P B ≈ qB 0 = 0.
The matrix of constraints is therefore

M = 0 {φ 1 , φ 2 } P B {φ 2 , φ 1 } P B 0 = qB 0 0 1 -1 0
From this matrix we get the Dirac brackets of the system:

{x, y} D = - 1 qB 0 {x, p x } D = {y, p y } D = 1 2 {p x , p y } D = 1 4 qB 0
which is, as expected, are identical to some obtained with the CI method.

Faddeev-Jackiw approach

In this case the F-J Lagrangian is the same

L F J = η (x ẏ -y ẋ) - 1 2 ξ x 2 + y 2
with p x = -ηy and p y = ηx, the Euler-Lagrange equations are then

η ẏ = -η ẏ + ξx -η ẋ = η ẋ + ξy or in matrix form 2η 0 1 -1 0 f ẋ ẏ = ξx ξy
The inverse matrix is

f -1 = 1 2η 0 -1 1 0 = {x, x} {x, y} {y, x} {y, y}
and we again find the same brackets as with the two other approaches.

Gauge invariant Lagrangian

Consider now a system described by the Lagrangian

L = 1 2 (y ẋ + x ẏ) 2 .

CI method

The general solution of the equations of motion is obtained by introducing Q = xy, and consequently Q = y ẋ + x ẏ. Then we have Q = 0 leading to the following results

x(t) = (at + b) ε(t) ; y(t) = 1 ε(t) p x (t) = a ε(t) ; p y (t) = a 2 t + ab ε(t),
where ε(t) is an arbitrary function of time. To determine the infinitesimal gauge transformation that leaves the Lagrangian invariant we consider an infinitesimal variation δε. This leads to δx = (at + b) δε and δy = -δε ε(t) 2 which can be written δx = xyδε and δy = -y 2 δε. Fixing the gauge with the condition y -1 = 0 ⇒ ε(t) = 1, the solution becomes

x(t) = at + b ; y(t) = 1 p x (t) = a ; p y (t) = a 2 t + ab.
Replacing this solution in H = ẋp x + ẏp y -L, we obtain

H = 1 2 a 2
and using Hamilton equation dx dt = {x, H} in order to have directly the bracket {a, b} = -1

From this bracket among the constants, we determine the fundamental brackets :

{x, y} = {y, p x } = {y, p y } = 0 {x, p x } = 1 {x, p y } = x {p x , p y } = -p x .

Dirac approach

From the Lagrangian we can deduce the momenta p x = y(y ẋ + x ẏ) and p y = x(y ẋ + x ẏ). It is clear that we have a primary constraint φ = xp x -yp y ≈ 0. The canonical Hamiltonian is

H c = 1 2 p 2
x y 2 , and the total Hamiltonian is defined as H T = H c + λφ. The constraint is here first class because it is the only constraint. In order to define the Dirac brackets, it is known that one must add an additional condition fixing the gauge. Here we choose the same as previously ζ = y -1 = 0. Therefore the matrix of constraints is These brackets are identical to those obtained with the CI method.

M = {φ,

Faddeev-Jackiw approach

Using the primary constraint xp x -yp y = 0, we define the FJ Lagrangian as L F J = ẋp x + ẏp y -H c which reads

L F J = p x ẋ + xp x y ẏ - 1 2 p 2
x y 2 . The independent variables of the system are x, y, z ≡ p x , then

L F J = z ẋ + xz y ẏ - 1 2 z 2 y 2
The Euler-Lagrange equations are

-ż = - z y ẏ - ẋz y + xz y 2 ẏ - x ż y = xz y 2 ẏ - z 2 y 3 0 = -ẋ - x y ẏ + z y 2
which can also be written in the matrix form

  0 z y -1 -z y 0 -x y 1 x y 0   f   ẋ ẏ ż   =    0 -z 2 y 3 z y 2    .
The matrix f is antisymmetric and singular as det(f ) = 0. The single zero mode is ( -x z y z 1 ) but it does not help as the zero mode equation does not lead to a secondary constraint equation.

The matrix f is therefore still singular and we are in the presence of a gauge symmetry. We choose the same gauge condition y = 1 and add the term λ(y -1) to the L F J Lagrangian, i.e.;

L F J = z ẋ + xz y ẏ - 1 2 z 2 y 2 + λ(y -1) .
The Euler-Lagrange equations are now

-ż = - z y ẏ - ẋz y + xz y 2 ẏ - x ż y = xz y 2 ẏ - z 2 y 3 - λ 0 = -ẋ - x y ẏ + z y 2 -ẏ = 0 or in the matrix form     0 z y -1 0 -z y 0 -x y 1 1 x y 0 0 0 -1 0 0     f     ẋ ẏ ż λ     =     0 -z 2 y 3 z y 2 0     .
The matrix f is now antisymmetric and invertible because det(f ) = 1 = 0. The inverse matrix f -1 is given by To obtain the brackets with p y one uses the primary constraint p y = xpx y and put y = 1 at the end of the computation of the brackets.

f -1 =     0 0 1 x y 0 0 0 -1 -1 0 0 -z y -x y 1 z y 0     =     {x,
We can see on this particular example that the Dirac and Faddeev-Jackiw approaches are closer to each other than the CI method.

Hojman-Urrutia Lagrangian

There are at least two physical motivations for the study of the theory associated to this Lagrangian that we would like to point out briefly. The first goes back to the publication of Hojman and Urrutia [START_REF] Hojman | On the inverse problem of the calculus of variations[END_REF] where they built a Lagrangian from two systems of second order differential equations for which in principle no second-order Lagrangian exists (following the well known Douglas classification [START_REF] Douglas | Solution of the inverse problem of the calculus of variations[END_REF]). The second motivation goes back to the work of Feynman, reported by Dyson [11] who derived the first group of Maxwell equations from the classical equations of motion and imposed brackets among coordinates and velocities. This work has been extended to the two groups of Maxwell equations [START_REF] Bérard | Magnetic Monopole in the Feynman derivation of Maxwell equations[END_REF]. Although with this "Feynman brackets" approach, there is no Lagrangian or Hamiltonian structure, Hojman and Shepley [START_REF] Hojman | No Lagrangian, no quantization ![END_REF] showed by using a Helmholtz inverse method, that under certain conditions a Lagrangian can be associated to these Feynman brackets. In particular, in order to test their method they studied the following equations of motion ẍ = -ẏ and ÿ = -y which are not derivable from a Lagrangian. By rewriting the system in first order form as ẋ = z, ẏ = ω, ż = -ω and ω = -y , they found the corresponding Lagrangian (by using integration constants in another context)

L = (y + z) . x + ω . z + 1 2 ω 2 -2yz -z 2 .
This Lagrangian is singular and therefore this is a constrained system. This theory has been studied via the Dirac approach in [START_REF] Kulshreshtha | Hamiltonian formulation of a theory with constraints[END_REF] and by the Faddeev-Jackiw approach by [START_REF] Barcelos-Neto | Symplectic analysis of a Dirac constrained theory[END_REF]. We do not reproduce their results here but we will recover them from the CI method which turns out to be much simpler.

First case: ω not constant

The Euler-Lagrange equations are ẏ + ż = 0 0 = .

x -z . ω = .

x -y -z 0 = ż + ω Again, we obtain the same results as with the other methods. The reader will notice the efficiency of our approach.

Second case: ω constant

This case with ω = k constant studied also in [START_REF] Barcelos-Neto | Symplectic analysis of a Dirac constrained theory[END_REF] is interesting as it leads to a gauge theory [START_REF] Kulshreshtha | Hamiltonian formulation of a theory with constraints[END_REF] .

In this case, the Euler-Lagrange equations are . z = 0 .

x -z = 0 .

x -z -y = 0 whose general solution is

x = at + b y = 0 z = a
The Hamiltonian is then

H = 1 2 k 2 -2yz -z 2 = 1 2 a 2
and from the Hamilton equation .

x = {x, H} we obtain the equation a = b, 1 2 a 2 which gives directly the bracket among the constants {a, b} = -1 From this we obtain the following brackets {x, z} = 1 {x, y} = {y, z} = 0 as expected.

Application to the Dirac field

Here we present a different perspective on the quantization of free fields based on the method we developed here. The idea is that, instead of applying the quantization rules (for bosons and fermions) based on the correspondence principle among the fields and their momenta [START_REF] Itzykson | Quantum Field Theory[END_REF], we simply assume the validity of the Heisenberg equation of motion of the field operators. Then we treat the creation and annihilation operators in the same manner as the constants of motion of the classical equations of motion. As an example we consider the Dirac field Ψ, with the Lagrangian density

L = iΨγ µ ∂ µ Ψ -mΨΨ
where γ µ are the Dirac matrices. The Euler-Lagrange equations are iγ µ ∂ µ Ψ -mΨ = 0 and iΨγ µ ← -∂ µ +mΨ = 0 whose general solution is

Ψ = 2 s=1 dk f k (x)b s (k)u s (k) + f * k (x)d † s (k)v s (k) (10)
where

f k (x) = m (2π) 3 k0 e -ikx , k 0 = √ k 2 + m
H = dkk 0 2 s=1 b † s (k)b s (k) -d s (k)d † s (k) (11)
From the Heisenberg equation Ψ

= 1 iℏ [Ψ, H]
We obtain the following equalities

k 0 b s ( k) = d k ′ k ′ 0 2 s ′ =1 b s ( k), N s ′ ( k ′ ) ( 12
)
k 0 d † s ( k) = -d k ′ k ′ 0 2 s ′ =1 d † s ( k), N s ′ ( k ′ ) (13)
with

N s ( k) = b † s (k)b s (k) -d s (k)d † s (k)
. From these equations we directly read off the following brackets

b s ( k), N s ′ ( k ′ ) = b s ′ ( k ′ )δ ss ′ δ( k -k ′ ) (14) d † s ( k), N s ′ ( k ′ ) = -d † s ′ ( k ′ )δ ss ′ δ( k -k ′ ). (15)
For fermions, we express these commutators by means of anticommutators

b s ( k), N s ′ ( k ′ ) = -b † s ′ ( k ′ ) b s ( k), b s ′ ( k ′ ) + b s ( k), b † s ′ ( k ′ ) b s ′ ( k ′ ) (16) +d s ′ ( k ′ ) b s ( k), d † s ′ ( k ′ ) -b s ( k), d s ′ ( k ′ ) d † s ′ ( k ′ ) (17)
and

d † s ( k), N s ′ ( k ′ ) = -b † s ′ ( k ′ ) d † s ( k), b s ′ ( k ′ ) + d † s ( k), b † s ′ ( k ′ ) b s ′ ( k ′ ) (18) +d s ′ ( k ′ ) d † s ( k), d † s ′ ( k ′ ) -d † s ( k), d s ′ ( k ′ ) d † s ′ ( k ′ ), (19)
then by identification of Eqs. ( 14) and ( 16) we obtain the anticommutation rule

b s ( k), b † s ′ ( k ′ ) = δ ss ′ δ( k -k ′ ) (20)
In the same manner, the identification between Eqs. ( 15) and [START_REF] Ghosh | Deformed Special Relativity and Deformed Symmetries in a Canonical Framework[END_REF] gives

d s ′ ( k ′ ), d † s ( k) = δ ss ′ δ( k -k ′ ) (21)
These rules of quantization are exactly identical to the results of the canonical quantization of the Dirac field. The same approach can be used for all quantum fields (Maxwell and Klein-Gordon).

Conclusion

The quantization of constrained systems is an old topic going back to Dirac and Bergmann. Their approach with the introduction of first and second class, primary and secondary constraints is essential but is very often difficult to apply. Faddeev and Jackiw proposed an alternative approach based on symplectic geometry and Darboux theorem which is very efficient and generally simpler than Dirac approach. In this paper, we developed another method that was applied on several examples and compared with the two other approaches. This method requires the knowledge of the general solutions of the equations of motion. This is for instance similar to the semiclassical quantization from the path integral approach. It is practical because it is very direct, easy and without special formalism. The novelty of this approach is based on the (simple) computation of the (Dirac) brackets among the integration constants (CI method) of the equations of motion. These constants are therefore considered as variables, a method reminiscent of the Hamilton-Jacobi calculus. We found that the CI method is consistent with the two others and we believe that it is also simpler in general. It is however based on the knowledge of the analytical solution, which is its weak point. When the classical analytical solution is not known, the method can still be applied by starting from the free systems because the solution is accessible, and then use the perturbation method to go further. Even in the others approaches, Dirac-Bergmann and Faddeev-Jackiw methods, one must find the free solution after quantization in order to use the perturbation development. In the near future we plan to extend our approach to non soluble and more complicated constrained systems like spinning particles in an electromagnetic field [START_REF] Bérard | Constrained dynamics of an anomalous (g = 2) relativistic spinning particle in electromagnetic background[END_REF] or relativistic particles in non-commutative space-time [START_REF] Ghosh | Deformed Special Relativity and Deformed Symmetries in a Canonical Framework[END_REF]. Beyond its technical aspect this method sheds light also on another point of view of the quantization process. Indeed, assuming only the validity of Hamilton (or Heisenberg) equations for the quantum fields, the quantization rules emerge naturally instead of postulating the commutation rules among the fields.

x 2 2

 2 = 0, we introduce the total Hamiltonian H T = H c +λφ where λ is a Lagrange multiplier. The consistency condition φ ≈ {φ, H T } P B ≈ {φ, H c } P B ≈ 0 (where {, } P B denotes the Poisson bracket) leads to a secondary constraint ζ = x 2 + p x ≈ 0. The associated consistency condition ζ ≈ {ζ, H c } P B +λ {ζ, φ} P B ≈ 0 leads us to the following expression λ ≈ y-px 2x . As {φ, ζ} P B ≈ -x = 0, the constraints are second class and the matrix of constraints is given by M = {φ, φ} P B {φ, ζ} P B {ζ, φ} P B {ζ, ζ} P B = 0 -x x 0 .

3

  It is useful to write the Euler-Lagrange's equations under the formhave directly the matrix f antisymmetric.

= ξ 2 a 2 + b 2 2 {p x , p y } = 1 4

 224 0. The solutions are x = a cos (ωt) + b sin (ωt) y = -b cos (ωt) + a sin (ωt) p x = η (b cos (ωt) -a sin (ωt)) p y = η (a cos (ωt) + b sin (ωt)) with ω = ξ 2η . The Hamiltonian of this system expressed in terms of the constants of integration is H and the Hamilton equations ẋ = {x, H} and ẏ = {y, H} give the equalities -aω sin (ωt) + bω cos (ωt) = {a, b} bξ cos (ωt) + {b, a} aξ sin (ωt) bω sin (ωt) + aω cos (ωt) = -{b, a} aξ cos (ωt) + {a, b} bξ sin (ωt) from which we easily read off the bracket between the constants of integration {a, b} = ω ξ . From it we deduce the brackets between the dynamical variables {x, y} = -1 qB 0 {x, p x } = {y, p y } = 1 qB 0

1 P 1 P 1 P 1 P 1 P 1 P 1 P 1 P

 11111111 φ} P B {φ, ζ} P B {ζ, φ} P B {ζ, ζ} P B Now we can calculate the Dirac brackets among the dynamical variables {x, y} D = {x, y} P B -{x, φ} P B {φ, ζ} -B {ζ, y} P B -{x, ζ} P B {ζ, φ} -B {φ, y} P B = 0 {x, p x } D = {x, p x } P B -{x, φ} P B {φ, ζ} -1 P B {ζ, p x } P B -{x, ζ} P B {ζ, φ} -1 P B {φ, p x } P B = 1 {x, p y } D = {x, p y } P B -{x, φ} P B {φ, ζ} -B {ζ, p y } P B -{x, ζ} P B {ζ, φ} -B {φ, p y } P B = x {y, p x } D = {y, p x } P B -{y, φ} P B {φ, ζ} -B {ζ, p x } P B -{y, ζ} P B {ζ, φ} -B {φ, p x } P B = 0 {y, p y } D = {y, p y } P B -{y, φ} P B {φ, ζ} -1 P B {ζ, p y } P B -{y, ζ} P B {ζ, φ} -1 P B {φ, p y } P B = 0 {p x , p y } D = {p x , p y } P B -{p x , φ} P B {φ, ζ} -B {ζ, p y } -{p x , ζ} P B {ζ, φ} -B {φ, p y } P B = -p x .

  x} {x, y} {x, z} {x, λ} {y, x} {y, y} {y, z} {y, λ} {z, x} {z, y} {z, z} {z, λ} {λ, x} {λ, y} {λ, z} {λ, λ}     and we find again {x, y} = {y, z} = 0 {x, z} = 1

- 1 2 a 2 + b 2 -c 2 From

 22 equations of Hojman and Shepley[START_REF] Hojman | No Lagrangian, no quantization ![END_REF]). The general solution isx = a cos(t) + b sin(t) + ct + d y = -b cos(t) + a sin(t) z = -a sin(t) + b cos(t) + c ω = a cos(t) + b sin(t)and the Hamiltonian is H = the Hamilton equation for x we obtain the equality -a sin(t) + b cos(t) + c = (-{a, b} b + {a, c} c) cos(t) + ({a, b} a + {b, c} c) sin(t) + ({b, c} b + {a, c} a) t + ({b, d} b + {a, d} a -{c, d}) c In the same manner for y b sin(t) + a cos(t) = (-{a, b} a + {b, c} c) cos(t) -({a, b} b + {a, c} c) sin(t) which gives directly by identification {a, b} = {c, d} = -1 {a, c} = {a, d} = {b, c} = {b, d} = 0 and finally {x, y} = {z, ω} = 1 {y, ω} = -1 {x, z} = {x, ω} = {y, z} = 0

  2 and u s (k) and v s (k) are the usual bispinors. The b s (k) and d s (k) are operators playing the role of the constants of motion in the classical case (before quantization). Then as previously we write the Hamiltonian in terms of these operators

  {x, y} D = {x, y} P B -{x, φ} P B 1 x {ζ, y} P B + {x, ζ} P B 1 x {φ, y} P B = -1 x {x, p x } D = {x, p x } P B -{x, φ} P B 1 x {ζ, p x } P B + {x, ζ} P B 1 x {φ, p x } P B = 0 {x, p y } D = {x, p y } P B -{x, φ} P B 1 x {ζ, p y } P B + {x, ζ} P B 1 x {φ, p y } P B = 0 {y, p x } D = {y, p x } P B -{y, φ} P B 1 x {ζ, p x } P B + {y, ζ} P B 1 x {φ, p x } P B = -1 2x {y, p y } D = {y, p y } P B -{y, φ} P B 1 x {ζ, p y } P B + {y, ζ} P B 1 x {φ, p y } P B = 1 {p x , p y } D = {p x , p y } P B -{p x , φ} P B 1 x {ζ, p y } P B + {p x , ζ} P B 1 x {φ, p y } P B = 0

M =

2N corresponds to an unconstrained system. For the constrained systems, each constraint eliminates one variable.

Acknowledgements. We thank M. M. Müller and Y. Grandati for stimulating discussions. Z. B. thanks A. H. Gharbi for stimulating discussions and for support.