Lorenzo Brasco
email: brasco@cmi.univ-mrs.fr

Giovanni Franzina
email: franzina@science.unitn.it

AN ANISOTROPIC EIGENVALUE PROBLEM OF STEKLOFF TYPE AND WEIGHTED WULFF INEQUALITIES

Keywords: 2010 Mathematics Subject Classification. 35P30, 47A75, 34B15 Stekloff eigenvalue problem, pseudo p-Laplacian, Wulff inequality

We study the Stekloff eigenvalue problem for the so-called pseudo p-Laplacian operator. After proving the existence of an unbounded sequence of eigenvalues, we focus on the first nontrivial eigenvalue σ2,p, providing various equivalent characterizations for it. We also prove an upper bound for σ2,p, in terms of geometric quantities. The latter can be seen as the nonlinear analogue of the Brock-Weinstock inequality for the first nontrivial Stekloff eigenvalue of the (standard) Laplacian. Such an estimate is obtained by exploiting a family of sharp weighted Wulff inequalities, which are here derived and appears to be interesting in themselves.

1. Introduction 1.1. The problem. In this paper, we are concerned with some spectral properties of the so-called pseudo p-Laplacian operator (see [START_REF] Belloni | The pseudo p-Laplace eigenvalue problem and viscosity solution as p → ∞[END_REF]), defined by

∆ p u := N i=1 ∂ ∂x i ∂u ∂x i p-2 ∂u ∂x i .
For p = 2 this operator coincides with the usual Laplacian, but for p = 2 it considerably differs from the more familiar p-Laplace operator, given by

∆ p u := N i=1 ∂ ∂x i |∇u| p-2 ∂u ∂x i .
More precisely, we are concerned with investigating the Stekloff spectrum of ∆ p on a generic open bounded Lipschitz set Ω ⊂ R N , i.e. the set of those real numbers σ such that the problem (1.1)

     ∆ p u = 0, in Ω, N i=1 |u x i | p-2 u x i ν i Ω = σ|u| p-2 u,
on ∂Ω, admits nontrivial W 1,p (Ω) weak solutions, where ν Ω = (ν 1 Ω , . . . , ν N Ω ) is the outer normal versor. It is easily seen that these numbers σ can be equivalently characterized as the critical points of the restriction of the anisotropic p-Dirichlet integral

(1.2) Φ p (u) = N j=1 Ω |u x j (x)| p dx, u ∈ W 1,p (Ω),
to the manifold {u ∈ W 1,p (Ω) : u L p (∂Ω) = 1}.

1.2. One step back: the linear case. In order to smoothly present the topics and the aims of this paper, it could be useful to recall some basic facts from the linear case, i.e. we consider p = 2 in (1.1). In this case, for every open bounded Lipschitz set Ω ⊂ R N , its Stekloff eigenvalues of the Laplacian are the numbers σ such that (1.3) ∆u = 0, in Ω, ∇u, ν Ω = σ u, on ∂Ω, admits nontrivial W 1,2 (Ω) solutions. By exploiting the compactness of the embedding W 1,2 (Ω) → L 2 (∂Ω), it is easy to see that these eigenvalues form a discrete non-decreasing sequence diverging at ∞, i.e. σ 1 (Ω) ≤ σ 2 (Ω) ≤ σ 3 (Ω) ≤ . . . , with σ 1 (Ω) = 0 corresponding to constant solutions of (1.3). Again, these eigenvalues can be characterized as the critical values of the Dirichlet integral on the L 2 (∂Ω) unitary sphere. The corresponding critical points are the Stekloff eigenfunctions of ∆ on Ω, normalized by the condition on the L 2 (∂Ω) norm. Also, they turn out to form an orthonormal basis of L 2 (∂Ω). In particular, the first nontrivial eigenvalue σ 2 (Ω) coincides with the value of the best constant in the following Poincaré-Wirtinger trace inequality

c Ω ∂Ω |u(x) -u ∂Ω | 2 dH N -1 ≤ Ω |∇u(x)| 2 dx, u ∈ W 1,2 (∂Ω),
where H N -1 stands for the (N -1)-dimensional Hausdorff measure and u ∂Ω is the average of (the trace of) u on ∂Ω.

In analogy with the well-known Dirichlet and Neumann cases (see [START_REF] Henrot | Extremum problems for eigenvalues of elliptic operators[END_REF]Chapters 3 and 7]), one may be interested in the spectral optimization problem of maximizing 1 σ 2 under volume constraint. A well-known result asserts that the (unique) solutions to this problem are given by balls. This is the so-called Brock-Weinstock inequality (see [START_REF] Brock | An isoperimetric inequality for eigenvalues of the Stekloff problem[END_REF][START_REF] Weinstock | Inequalities for a classical eigenvalue problem[END_REF]). For ease of completeness, it is worth mentioning that Weinstock's result (valid only in dimension 1 On the contrary, it is not difficult to see that the problem of minimizing σ2 is always trivial. N = 2) is even stronger, since it asserts that disks are still maximizers among simply connected set of given perimeter. By observing that σ 2 scales like a length to the power -1 and that σ 2 (B R ) = R -1 for a ball of radius R, the Brock-Weinstock inequality can be written in scaling invariant form as follows (1.4)

σ 2 (Ω) ≤ ω N |Ω| 1 N
, where ω N is the measure of the N -dimensional ball of radius 1. Moreover, equality in (1.4) can hold if and only if Ω itself is a ball. Recently, a sharp quantitative stability estimate for this inequality has been given in [START_REF] Brasco | Spectral optimization for the Stekloff-Laplacian: the stability issue[END_REF]. We recall that such an "isoperimetric" property of the ball is in turn a consequence of the fact that the ball (centered at the origin) is the only solution of min

∂Ω |x| 2 dH N -1 : Ω ⊂ R N , |Ω| = c .
This result has been proved in [START_REF] Betta | A weighted isoperimetric inequality and applications to symmetrization[END_REF] (see also [START_REF] Brasco | Spectral optimization for the Stekloff-Laplacian: the stability issue[END_REF] for a different proof). In other words, we have the following weighted isoperimetric inequality

(1.5) ∂Ω |x| 2 dH N -1 ≥ N ω -1/N N |Ω| N +1 N ,
with equality if and only if Ω is a ball centered at the origin.

1.3. Results and style of the paper. Of course, problem (1.1) for p = 2 is a completely different story. We are now facing a nonlinear eigenvalue problem and no general results guarantee, for example, infiniteness and discreteness of the spectrum. Here we show that these eigenvalues forms at least a countably infinite sequence of positive numbers σ 1,p (Ω) ≤ σ 2,p (Ω) ≤ σ 3,p (Ω) ≤ . . . , diverging at infinity (Theorem 4.2). For this, we use some standard minimax methods from Nonlinear Analysis, but we will avoid to refer to the so-called Ljusternik-Schnirelmann theory and rather we will employ an alternative (and elegant) procedure introduced by Drabék and Robinson in [START_REF] Drábek | Resonance problems for the p-Laplacian[END_REF] (see the next section). Also, we will show that the first eigenvalue (again, we have σ 1,p (Ω) = 0 and it corresponds to constant eigenfunctions) is isolated in the spectrum, the latter being a close set (Proposition 3.6), and that the first nontrivial eigenvalue σ 2,p (Ω) can still be characterized as the best constant in some Poincaré-Wirtinger trace inequality (Theorem 5.6 and Remark 5.7). Namely, σ 2,p (Ω) coincides with the best constant in

c Ω min t∈R ∂Ω |u + t| p dH N -1 ≤ N i=1 Ω |u x i | p dx, u ∈ W 1,p (Ω).
We will also address the issue of generalizing the Brock-Weinstock inequality, for the first nontrivial Stekloff eigenvalue of the pseudo p-Laplacian. Indeed, by adapting Brock's method of proof, we are able to prove (Theorems 7.2 and 7.3)

(1.6) σ 2,p (Ω) ≤ |B p | |Ω| p-1 N
, where B p is the N -dimensional p unit ball, i.e.

B p = {x ∈ R N : |x 1 | p + • • • + |x N | p < 1}.
The previous inequality can be seen as a nonlinear counterpart of (1.4). Unfortunately, we are not able to detect the cases of equality in (1.6). In other words, while it is rather easy to see that equality in (1.6) implies Ω to coincide with (a translated and scaled copy of) B p , we can not guarantee that equality can really hold. This is due to the fact that we are not able to determine σ 2,p (B p ) and this in turn is intimately linked to the lack of information on the shape of the nodal line of eigenfunctions corresponding to σ 2,p (B p ) (see Remark 7.4). It is useful to recall at this point that even for the second eigenvalue of the p-Laplacian with Dirichlet boundary conditions, it is still an open problem to decide whether the nodal line in the Euclidean ball is given by a diameter (like in the linear case p = 2) or not. The proof of (1.6) is based on the anisotropic version of (1.5), which is here derived and appears to be new. Namely, we prove the following weighted Wulff inequality

(1.7) ∂Ω V ( x ) ν Ω * dH N -1 ≥ N |K| -1/N |Ω| N +1 N ,
where • and • * are norms dual to each other, while K is the unit ball of • centered at the origin and V is an increasing weight function, satisfying suitable assumptions. Here again, equality can hold if and only if Ω = K, up to a scaling factor. Inequality (1.7) is the other main contribution of this paper, the proof of (1.7) being an adaptation of the technique used in [START_REF] Brasco | Spectral optimization for the Stekloff-Laplacian: the stability issue[END_REF]. Since this result is not directly related with the study of the Stekloff spectrum, we added it in the Appendix at the end of the paper.

About the style of the paper, we point out that we tried to keep technicalities at a low level, in order to make the paper suitable for a wide audience, which can be possibly interested also in Isoperimetric Inequalities and Shape Optimization. Having this in mind, all the proofs, in despite of presenting many similarities with the vast literature on Nonlinear Spectral Theory, are self-contained. Further, where possible, our arguments are simpler and more direct. This is the case for example of the mountain pass characterization of the first nontrivial eigenvalue (Proposition 5.4), which is based on some peculiar convexity properties of the functional Φ p in (1.2) (the so-called hidden convexity, see [START_REF] Belloni | A direct uniqueness proof for equations involving the p-Laplace operator[END_REF][START_REF] Brasco | A note on positive eigenfunctions and hidden convexity[END_REF]). Our method of proof still works for the standard p-Laplacian with Dirichlet or Neumann boundary conditions, thus it can be seen as alternative to that of [START_REF] Cuesta | The beginning of the Fuĉik spectrum for the p-Laplacian[END_REF]Theorem 3.1].

1.4.

A note on the regularity of eigenfunctions. In passing, it is useful to say something about the regularity of Stekloff eigenfunctions of the pseudo p-Laplacian. We recall that in the case of the standard p-Laplacian operator, it is well-known that solutions to ∆ p u = 0 are C 1,α (see [START_REF] Dibenedetto | C 1+α local regularity of weak solutions of degenerate elliptic equations[END_REF][START_REF] Lieberman | Boundary regularity for solutions of degenerate elliptic equations[END_REF]). Then one could expect such a result to hold for ∆ p as well.

However, we point out that the by now classical results of [START_REF] Dibenedetto | C 1+α local regularity of weak solutions of degenerate elliptic equations[END_REF][START_REF] Lieberman | Boundary regularity for solutions of degenerate elliptic equations[END_REF] do not apply directly to the case of ∆ p , since the type of degeneracy is quite different. Low regularity (like L ∞ or C 0,α ) is standard routine (see [START_REF] Giusti | Direct methods in the calculus of variations[END_REF]Theorem 7.6]), due to the fact that pseudo p-harmonic functions are local minimizers of a functional having p-growth in the gradient variable. On the contrary, higher regularity is not clear.

For example, to the best of our knowledge, even the Lipschitz character of solutions seems not to be fully understood. We mention [START_REF] Uralt'seva | The boundedness of the gradients of generalized solutions of degenerate quasilinear nonuniformly elliptic equations[END_REF] where this is proved for the case p > 3 and [START_REF] Brasco | On certain anisotropic elliptic equations arising in optimal transport: local gradient bounds[END_REF], where an "almost Lipschitz" estimate is proven for p ≥ 2 (which is valid for fairly more degenerate equations).

On the other hand, in the case 1 < p < 2 the existence of second weak derivatives for pseudo p-harmonic functions can still be proved (see [START_REF] Franzina | Existence, optimization and stability for low eigenvalues of some nonlinear operators[END_REF]), thus paralleling the case of the p-Laplacian (see [START_REF] Acerbi | Regularity of minimizers of non-quadratic functionals: the case 1 < p < 2[END_REF]). 1.5. Plan of the work. We start with Section 2, where we recall some technical facts that will be useful throughout the whole paper. Then in Section 3, we introduce the Stekloff eigenvalue problem we are interested in and prove some first properties. We refine this study in Section 4, where the existence of unbounded sequence of eigenvalues is exhibited. The subsequent Section 5 is devoted to the first nontrivial Stekloff eigenvalue. We prove some equivalent characterizations for it, which imply in particular that the first eigenvalue is always isolated in the spectrum. We further analyze the first nontrivial eigenspace in Section 6, proving some nodal domains properties of eigenfunctions and giving yet another characterization of the first nontrivial eigenvalue, this time in terms of some eigenvalue problems with mixed boundary conditions. Finally, in Section 7 we provide some upper bounds for the first nontrivial eigenvalue, which can be seen as the nonlinear analogue of (1.4). As already said, a self-contained appendix devoted to weigthed Wulff inequalities complements the paper.

Technical machinery

Given 1 < p < ∞, for every x ∈ R N we will denote by x p its p norm, i.e. the quantity

x p = N i=1 |x i | p 1 p .
For an open set Ω ⊂ R N , the symbol W 1,p (Ω) will stand for the usual Sobolev space, endowed with the norm

(2.1) u W 1,p (Ω) = Ω |u(x)| p dx + Ω |∇u(x)| p 1 p .
As always, the symbol W 1,p 0 (Ω) will stand for the closure of C ∞ 0 (Ω) with respect to the norm • W 1,p (Ω) . It is useful to recall that when Ω ⊂ R N has a Lipschitz boundary, the embedding W 1,p (Ω) → L p (Ω) is compact. Moreover, each function in W 1,p (Ω) has a trace belonging to the fractional Sobolev spaces W 1-1/p,p (∂Ω). Then the embedding W 1,p (Ω) → L p (∂Ω) is compact as well.

Lemma 2.1. Let 1 < p < ∞ and Ω be an open bounded Lipschitz set in R N . Then

∇u L p (Ω) + u L p (∂Ω) , u ∈ W 1,p (Ω),
defines a norm on the Sobolev space W 1,p (Ω), which is equivalent to (2.1).

Proof. It is straightforward to check that the above quantity defines a norm. Then the thesis is a consequence of the trace inequality

u L p (∂Ω) ≤ c Ω u W 1,p (Ω) ,
and of the following Poincaré inequality

u L p (Ω) ≤ c Ω ∇u L p (Ω) + u L p (∂Ω) ,
which in turn follows by a standard contradiction argument, exploiting the compact embeddings W 1,p (Ω) → L p (Ω) and W 1,p (Ω) → L p (∂Ω).

The function t → |t| p-2 t enjoys the following monotonicity properties, which will be useful in the sequel. For the proof, the reader is referred to [28,Section 10].

Lemma 2.2. Let 2 ≤ p < ∞. Then (2.2) (|t| p-2 t -|s| p-2 s) (t -s) ≥ 2 2-p |t -s| p , t, s ∈ R.
For 1 < p < 2, we have

(2.3) (|t| p-2 t -|s| p-2 s) (t -s) ≥ (p -1) |t -s| 2 (1 + |s| 2 + |t| 2 ) 2-p 2 , t, s ∈ R.
As an application of the previous Lemma, we obtain the following fact, which we will use repeatedly.

Proposition 2.3. Let {u k } k∈N ⊂ L p (Ω) be such that (2.4) lim k→∞ Ω |u k | p-2 u k -|u| p-2 u (u k -u) dx = 0,
for some function u ∈ L p (Ω). Then {u k } k∈N converges in L p (Ω) to u.

Proof. We have two distinguish between two cases. If p ≥ 2, then by the first inequality of Lemma 2.2, it follows directly that lim k→∞ Ω |u k -u| p dx = 0.

For 1 < p < 2, we start observing that the hypothesis implies (2.5)

Ω |u k | p dx ≤ C, for every k ∈ N.
Indeed, by means of Young inequality we can infer

Ω |u k | p-2 u k -|u| p-2 u (u k -u) dx ≥ 1 - ε p p - ε q q Ω |u k | p dx + 1 - ε -p p - ε -q q Ω |u| p dx,
where we set q = p/(p -1). By taking ε small enough and using (2.4), we then obatin (2.5). We now use inequality (2.3) of Lemma 2.2, raised to the power p/2. Thus we get

Ω |u k -u| p dx ≤ C Ω (1 + |u k | 2 + |u| 2 ) (2-p) p 4 |u k | p-2 u k -|u| p-2 u (u k -u) p 2 dx.
An application of Hölder inequality with exponents 2/(2 -p) and 2/p yields

Ω |u k -u| p dx ≤ C Ω (1 + |u k | 2 + |u| 2 ) p 2 dx 2-p 2 × Ω |u k | p-2 u k -|u| p-2 u (u k -u) dx p 2 .
The conclusion is now an easy consequence of (2.4) and (2.5).

To make the paper as self-contained as possible, we also recall the definition of Palais-Smale condition. Definition 2.4. Let Φ : X → R be a C 1 functional defined on some Banach space X and let M ⊂ X be a C 1 manifold. Given x ∈ M , let us denote by T x M the tangent space at M in the point x and by DΦ(x) |TxM the differential in x of the restriction of Φ to M .

Then Φ is said to satisfy the Palais-Smale condition on M at level λ, if for every sequence

{x n } n∈N ⊂ M such that lim n→∞ Φ(x n ) = λ and lim n→∞ DΦ(x n ) |Tx n M * = 0,
we have that x n is strongly convergent, possibly up to the extraction of a subsequence. If this condition is verified for every λ, we will simply say that Φ satisfies the Palais-Smale condition on M .

In order to find critical points of functionals restricted to manifolds, in this paper we will use the following minimax procedure, introduced by Drábek and Robinson in [START_REF] Drábek | Resonance problems for the p-Laplacian[END_REF] and further studied by Cuesta in [START_REF] Cuesta | Minimax theorems on C 1 manifolds via Ekeland variational principle[END_REF]. This is alternative to the so-called Ljusternik-Schnirelmann procedure, which relies on the concept of Krasnoselskii genus (see below). Our choice permits to produce a sufficiently large set of eigenvalues, avoiding at the same time unnecessary technicalities. The general result that we need is the following, due to Cuesta (see [START_REF] Cuesta | Minimax theorems on C 1 manifolds via Ekeland variational principle[END_REF]Proposition 2.7] or [START_REF] Franzina | Existence, optimization and stability for low eigenvalues of some nonlinear operators[END_REF] for the proof).

Theorem 2.5. Let X be a uniformly convex Banach space and M ⊂ X a C 1 manifold. By indicating with S k-1 the unit sphere in R k , we set

C o (S k-1 ; M ) = {f : S k-1 → M : f is continuous and odd}.
Suppose that Φ : X → R is an even C 1 functional and define

λ k = inf f ∈Co(S k-1 ;M ) max u∈f (S k-1 ) Φ(u),
If Φ verifies the Palais-Smale condition on M at level λ k , then λ k is a critical value of Φ on M , i.e. there exists x 0 ∈ M such that

Φ(x 0 ) = λ and DΦ(x 0 ) |Tx 0 M = 0.
Remark 2.6. For reader's convenience, we recall that the Krasnoselskii genus of a compact, nonempty and symmetric subset A ⊂ X of a Banach space is defined by

γ(A) = inf k ∈ N : ∃ a continuous odd mapp f : A → S k-1 ,
with the convention that γ(A) = +∞, if no such an integer k exists. Using the Krasnoselskii genus, an infinite sequence of critical values of Φ is usually produced as follows (see [START_REF] García Azorero | Existence and nonuniqueness for the p-laplacian[END_REF][START_REF] Struwe | Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems[END_REF])

λ k = inf γ(A)≥k max u∈A Φ(u), k ∈ N.
It is an interesting open problem to decide whether or not the two previous minimax procedures give the same sets of critical values.

The Stekloff spectrum of the pseudo p-Laplacian

Let Ω be a bounded Lipschitz open set in R N and : ∂Ω → R be a measurable function satisfying

(3.1) 0 < c 1 ≤ (x) ≤ c 2 < ∞, H N -1 -a.e. on ∂Ω.
For every 1 < p < ∞, we consider the pseudo p-Laplacian, i.e. the nonlinear operator

∆ p u = N j=1 |u x j | p-2 u x j x j .
Definition 3.1. A real number σ is said to be a Stekloff eigenvalue of the pseudo p-Laplacian in Ω if the boundary value problem

(3.2)      -∆ p u = 0, in Ω, N i=1 |u x i | p-2 u x i ν i Ω = σ|u| p-2 u , on ∂Ω,
admits a nontrivial solution u. If this is the case, we say that u is a Stekloff eigenfunction corresponding to σ. We also set

S p (Ω) = {σ ∈ R : σ is a Stekloff eigenvalue},
to denote the Stekloff spectrum of the pseudo p-Laplacian on Ω.

Remark 3.2. Since the behaviour of the spectrum under varying weights is not investigated here, the notation does not account for the choice of the function : ∂Ω → R.

The solutions u of the problem (3.2) are always understood in the weak sense, i.e. u ∈ W 1,p (Ω) and

(3.3) N i=1 Ω |u x i | p-2 u x i ϕ x i dx = σ ∂Ω |u| p-2 u ϕ dH N -1 , for every ϕ ∈ W 1,p (Ω).
Observe that the integral on the right-hand side is well-defined, since the trace of a function in W 1,p (Ω) belongs to L p (∂Ω).

We start with the following basic result.

Lemma 3.3. Let 1 < p < ∞, Ω be a bounded open Lipschitz set and : ∂Ω → R be such that (3.1) holds. There exists a least eigenvalue, given by σ = 0 and corresponding to constant eigenfunctions. Moreover, any other eigenfunction whose trace does not change sign on ∂Ω, is constant in Ω.

Proof. By testing ϕ = u, equation (3.3) implies

Ω ∇u p dx = σ ∂Ω |u| p dH N -1 ,
so that every eigenvalue must be positive. Moreover, it is easily seen that σ = 0 is an eigenvalue and by the previous equality any corresponding eigenfunction is constant.

Let us now prove the second part of the statement. Let u = 0 have a constant sign on the boundary and assume, arguing by contradiction, that it corresponds to an eigenvalue σ = 0. Inserting a constant test function in (3.3) we then obtain

∂Ω |u| p-1 dH N -1 = 0,
where we also used that u does not change sign on ∂Ω. Thus, u has a null trace on ∂Ω and it solves in a weak sense the problem ∆p u = 0 in Ω, u = 0 on ∂Ω.

Solutions to the latter problem are minimizers of the strictly convex energy

v → Ω ∇v p p dx, on W 1,p 0 (Ω).
Since the unique minimizer is given by the zero constant function, there must hold u ≡ 0, a contradiction. Therefore, σ = 0 and u is a constant eigenfunction. Definition 3.4. If u is a Stekloff eigenfunction, we will call nodal domains the connected components of {x ∈ Ω : u(x) = 0}. Observe that the latter is an open set, since each pseudo p-harmonic function is locally Hölder continuous, as a local minimizer of Ω ∇v p p (see [START_REF] Giusti | Direct methods in the calculus of variations[END_REF]Theorem 7.6]). We also observe that each nodal domain is itself an open set. This follows from the fact that the connected components of an open sets are open as well.

We record the following property of eigenfunctions, which will be useful in the next section.

Lemma 3.5. Let u ∈ W 1,p (Ω) be a Stekloff eigenfunction, with eigenvalue σ > 0. Then u has at least two nodal domains, both touching the boundary.

Proof. The fact that u has to change sign follows from Lemma 3.3. Let us now take

u + (x) = max{u(x), 0} and u -(x) = max{0, -u(x)},
and let Ω 1 , Ω 2 , . . . , be the nodal domains of u. Suppose that for some j we have Ω j Ω.

Then the restriction of u to Ω j belongs to W 1,p 0 (Ω j ) and it solves

-∆ p u = 0, on Ω j ,
in the weak sense. This implies that u ≡ 0 on Ω j , hence contradicting the definition of nodal domain.

We prove that the whole collection of Stekloff eigenvalues forms a closed set.

Proposition 3.6. Let 1 < p < ∞, Ω ⊂ R N a bounded Lipschitz domain and : ∂Ω → R be a function such that (3.1) holds. Then S p (Ω) is a non empty closed subset of [0, ∞).
Proof. The fact that the collection of all the Stekloff eigenvalues is non empty and consists of nonnegative numbers is due to Lemma 3.3. In order to prove the second part of the statement, we take a sequence of eigenvalues {σ k } k∈N ⊂ S p (Ω) converging to some positive number σ and we let {u k } k∈N ⊂ W 1,p (Ω) be a sequence of corresponding eigenfunctions, normalized by the condition

∂Ω |u k | p dH N -1 (x) = 1, k ∈ N.
This implies in particular that

N i=1 Ω |u k x i | p dx = σ k , k ∈ N,
so that the sequence {u k } k∈N is bounded in W 1,p (Ω), thanks to Lemma 2.1. Thus, by the compactness of the embedding W 1,p (Ω) → L p (∂Ω), the sequence weakly converges (up to a subsequence) to some limit function u in W 1,p (Ω). Moreover, this convergence is strong in L p (∂Ω). We have to show that u is an eigenfuction with eigenvalue σ: testing the equations solved by u k with ϕ = u k -u, we then obtain

N i=1 Ω |u k x i | p-2 u k x i -|u x i | p-2 u x i (u k x i -u x i ) dx = σ k ∂Ω |u k | p-2 u k -|u| p-2 u dH N -1 - N i=1 Ω |u x i | p-2 u x i u k x i -u x i dx.
Then, by the strong convergence of {u k } k∈N ⊂ L p (∂Ω), sending k to infinity yields

lim k→∞ N i=1 Ω |u k x i | p-2 u k x i -|u x i | p-2 u x i (u k x i -u x i ) dx = 0.
Thanks to Proposition 2.3, the previous gives the strong convergence of ∇u k to ∇u in L p (Ω). Since {u k } k∈N converges to u strongly in L p (∂Ω), we also have

(3.4) lim k→∞ ∂Ω |u k -u| p dH N -1 = 0.
Thanks to these informations, we can now pass to the limit in the equation (3.3) satisfied by u k , so to obtain that u is an eigenfunction as well, with eigenvalue σ. This shows that σ ∈ S p (Ω), which is then closed.

Existence of an unbounded sequence

In this section, we will show that S p (Ω) contains an infinite sequence of eigenvalues, diverging at ∞. At this aim, we observe that the elements of S p (Ω) are the critical values of the functional

(4.1) Φ(u) = Ω ∇u(x) p p dx, u ∈ W 1,p (Ω),
on the manifold M , defined by

(4.2) M = u ∈ W 1,p (Ω) : G(u) = 1 , where G(u) = ∂Ω |u| p dH N -1 .
Then, we will use the minimax procedure of Theorem 2.5. For this, we have to check that Φ verifies the Palais-Smale condition on M . This is the content of the next result. Proof. First of all, we observe that M is a C 1 manifold. It is sufficient to verify that 1 is a regular value for G, i.e. DG(u) = 0, for every u ∈ M . This is easily verified, since for every u ∈ M , we have

DG(u)[u] = p ∂Ω |u(x)| p (x) dH N -1 (x) = p = 0.
We now verify that Φ satisfies the Palais-Smale condition on M . At this aim, we need to show that for every C > 0 and every {u n } n∈N ⊂ M , the following implication holds true

(4.3) Φ(u n ) ≤ C lim n→∞ DΦ(u n ) |T u n M * = 0    =⇒ {u n } n∈N converges in W 1,p (Ω).
We first observe that for every u ∈ M , the tangent space to M at the point u is given by

T u M = ϕ ∈ W 1,p (Ω) : ∂Ω |u| p-2 u ϕ dH N -1 = 0 .
Hence, the hypothesis in (4.3) reads

0 = lim n→∞ DΦ(u n ) |T u n M * = lim n→∞        sup ϕ∈T u n M ϕ =0 N i=1 Ω |u n x i | p-2 u n x i ϕ x i dx ϕ W 1,p (Ω)       
.

By this assumption, we can infer the existence of an infinitesimal sequence {ε n } n∈N of strictly positive numbers, such that (4.4)

N i=1 Ω |u n x i | p-2 u n x i ϕ x i dx ≤ ε n ϕ W 1,p (Ω) , for every ϕ ∈ T u n M.
We now observe that the sequence {u n } n∈N is bounded in W 1,p (Ω). Indeed, using (3.1) we have

u n p L p (∂Ω) ≤ c -1 1 G(u n ) = c -1 1 and ∇u n p L p (Ω) ≤ c Φ(u n ) ≤ C
, and the boundedness follows by Lemma 2.1.

Thus, by possibly passing to a subsequence, we can assume that {u n } n∈N converges weakly to a function u in W 1,p (Ω), and that the convergence is strong both in L p (Ω) and L p (∂Ω). In particular, such a limit function u still belongs to M . Using the strong convergence in L p (∂Ω) again, we have (4.5) lim n→∞ δ n = 0, where we set

δ n = ∂Ω |u n | p-2 u n (u n -u) dH N -1 .
Using the fact that u n ∈ M , it is easy to check that the function v n := (1 -δ n ) u n -u belongs to the tangent space T u n M . Thus, it is possible to choose ϕ = v n in (4.4), so as to obtain

N i=1 Ω |u n x i | p-2 u n x i (1 -δ n ) u n xu -u x i dx ≤ ε n (1 -δ n ) u n -u W 1,p (Ω) ≤ ε n 2 u n W 1,p (Ω) + u W 1,p (Ω) ,
where we used that |1 -δ n | ≤ 2 by (4.5), provided n is large enough. By passing to the limit as n → ∞, we then get

lim n→∞ N i=1 Ω |u n x i | p-2 u n x i (1 -δ n ) u n xu -u x i = 0,
and so

(4.6) lim n→∞ N i=1 Ω |u n x i | p-2 u n x i u n x i -u x i dx = 0.
Since u n weakly converges to u in W 1,p (Ω), we also have

(4.7) lim n→∞ N i=1 Ω |u x i | p-2 u x i u n x i -u x i dx = 0.
Subtracting (4.7) to (4.6) and using again Proposition 2.3 on each component u n x i , we then get

lim n→∞ Ω |∇(u n -u)| p dx = 0. (4.8)
Owing to Lemma 2.1, finally the strong convergence of {u n } n∈N in W 1,p (Ω) follows by (4.8). This shows the validity of the implication (4.3), thus concluding the proof.

We can then assure that S p (Ω) contains an unbounded sequence. In the following, we keep the same notation as in Section 2.

(4.9) σ k,p (Ω) = inf f ∈Co(S k-1 ;M ) max u∈f (S k-1 ) Ω ∇u p p dx.
Then each σ k,p (Ω) is a Stekloff eigenvalue of the pseudo p-Laplacian on Ω. Moreover,

(4.10) 0 = σ 1,p (Ω) < σ 2,p (Ω) ≤ . . . ≤ σ k,p (Ω) ≤ . . . and σ k,p (Ω) → ∞ as k → ∞.
Proof. For the sake of readability, we divide the proof in various steps.

Eigenvalues. We already observed that the Stekloff eigenvalues of the pseudo p-Laplacian are precisely the critical values of the functional (4.1) on the C 1 manifold M . It is then sufficient to observe that the assumptions of Theorem 2.5 are satisfied, thanks to Lemma 4.1.

Hence the first part of the statement is proved.

The sequence is non-decreasing. Let k ∈ N and f : S k → M be an odd continuous mapping. Then, we take a k-dimensional vector subspace E ⊂ R k+1 and we consider the restriction

g E of f to the intersection S k ∩ E S k-1 . Whence max u∈f (S k ) Ω ∇u p p dx ≥ max u∈f (S k ∩E) Ω ∇u p p dx = max u∈g E (S k-1 ) Ω ∇u p p dx ≥ inf g∈Co(S k-1 ;M ) max u∈g(S k-1 ) Ω ∇u p p dx = σ k,p (Ω).
Since the mapping f was arbitrarly chosen, passing to the infimum we get

σ k+1,p (Ω) ≥ σ k,p (Ω),
as desired.

The first element is zero. We now observe that the first element of the sequence (4.9) is in fact zero. To this end, we note that any continuous odd mapping from S 0 = {1, -1} to M can be identified with the choice of an antipodal pair u, -u on the symmetric manifold M . This and the fact that the functional is even imply that if k = 1 formula (4.9) gives the minimum of (4.1) on M . The latter is of course zero, corresponding to constant functions.

Existence of a gap. We prove that there is a gap between zero and the second term of the sequence, i.e. We now observe that

c = ∂Ω (x) dH N -1 (x) -1 p
defines the unique (modulo the choice of the sign) constant function belonging to M . Let 0 < ε < 1/2 and consider the two neighborhoods

B + ε = {u ∈ M : u -c L p (∂Ω, ) < ε}, B - ε = {u ∈ M : u -(-c) L p (∂Ω, )
< ε}, which are disjoint, by construction. Here we set for brevity

ϕ L p (∂Ω; ) = ∂Ω |ϕ(x)| p (x) dH N -1 (x) 1 p .
Since the mapping f n is odd and continuous, for every n ∈ N the image f n (S 1 ) is symmetric and connected, then it can not be contained in B + ε ∪ B - ε , the latter being symmetric and disconnected. So we can pick an element (4.12)

u n ∈ f n (S 1 ) \ B + ε ∪ B - ε .
This yields a sequence {u n } n∈N ⊂ M , which is bounded in W 1,p (Ω) by (4.11). Hence, there exists a function v ∈ M such that {u n } n∈N converges to v weakly in W 1,p (Ω) and strongly in L p (∂Ω), possibly by passing to a subsequence. By the weak convergence it follows that

Ω ∇v p p dx ≤ lim inf n→∞ Ω ∇u n p p dx ≤ lim n→∞ 1 n = 0.
This in turn shows that v ∈ M is constant, so that either v = c or v = -c. Let us assume that v = c, for example: using the strong convergence in L p (∂Ω), we get

0 = v -c L p (∂Ω, ) = lim n→∞ u n -c L p (∂Ω, ) ≥ ε,
where in the last inequality we used (4.12). This gives a contradiction and thus σ 2,p (Ω) > 0.

Unboundedness. It remains to prove that this sequence of eigenvalues is unbounded. To this aim, we suitably modified the argument of [START_REF] García Azorero | Existence and nonuniqueness for the p-laplacian[END_REF]Proposition 5.4] (for a different proof, avoiding the use of Schauder bases, one could adapt the argument of [START_REF] Franzina | Existence and uniqueness for a p-Laplacian nonlinear eigenvalue problem[END_REF]Theorem 5.2]). We recall that the Sobolev space W 1,p (Ω) admits a Schauder basis (see [START_REF] Fučík | On the existence of Schauer bases in Sobolev spaces[END_REF][START_REF] Matveev | Bases in Sobolev spaces on bounded domains with Lipschitz boundary[END_REF]). Hence, there exists an ordered countable set of elements {e n } n∈N ⊂ W 1,p (Ω) with the property that for all u ∈ W 1,p (Ω), we have

u = ∞ n=1 α n e n ,
for a (uniquely determined) sequence of scalars {α n } n∈N . Here the converge of the previous series has to be understood in the sense of the norm topology. Clearly, if we now denote by E n = Vect({e 1 , . . . , e n }), the linear envelope of the first n elements of the basis, then the union n∈N E n is dense in W 1,p (Ω). Let us also set F k = Vect({e n } n>k ), which is the topological supplement of the finite-dimensional vector space E k , and define the new sequence

τ k,p (Ω) = inf f ∈Co(S k-1 ;M ) max u∈f (S k-1 )∩F k-1 Ω ∇u(x) p p dx, k ∈ N.
At first, we verify that such a sequence is actually well defined. Indeed, let f be an odd and continuous map from the unit sphere S k-1 to M and assume that the intersection f (S k-1 ) ∩ F k-1 is empty: this implies that for every ω ∈ S k-1 , the element f (ω) always has at least a nontrivial component on E k-1 . By composing f with the continuous odd operator

P k-1 : W 1,p (Ω) → E k-1 ,
given by the natural projection on the linear space E k-1 , the map P k-1 •f is odd, continuous and P k-1 • f (ω) = 0, for every ω ∈ S k-1 . That is, we constructed an odd continuous map from

S k-1 to E k-1 \ {0} R k-1 \ {0}
, which is impossible thanks to the Borsuk-Ulam Theorem 2 . Hence the image of any f ∈ C o (S k-1 ; M ) has to intersect F k-1 , for every k ∈ N.

Since obviously τ k,p (Ω) ≤ σ k,p (Ω), it sufficies now to show that lim k→∞ τ k,p (Ω) = +∞.

At this aim, assume by contradiction that τ k,p (Ω) < τ , for all k ∈ N. Then, for every k ∈ N, we can take a mapping f ∈ C o (S k ; M ) and

u k ∈ f S k-1 ∩ F k-1 such that (4.13) Ω ∇u k p p dx < τ.
Since u k ∈ M for all k ∈ N, equation (4.13) implies that the sequence {u k } k∈N is bounded in W 1,p (Ω) and weakly converges (up to a subsequence) to some limit function u ∈ M . On the other hand, by definition of Schauder basis, the functionals defined on W 1,p (Ω) by

φ k (u) = α k , if u = +∞ n=1 α n e n ∈ W 1,p (Ω), k ∈ N,
are linear and they turn out to be continuous, thanks to [3, page 83]. By the weak convergence of the sequence {u k } k∈N to u, it follows that lim k→∞ φ n (u k ) = φ n (u), for all n ∈ N. Since u k ∈ F k-1 , we have that φ n (u k ) = 0, for every n ≤ k -1, thus φ n (u) = 0 for all n ∈ N. This means that u = 0, contradicting the fact that u ∈ M .

Remark 4.3. If Ω has m connected components Ω 1 , . . . , Ω m , equation (4.9) still defines an infinite sequence of eigenvalues, diverging at ∞, but in this case we have

σ 1,p (Ω) = • • • = σ m,p (Ω) = 0,
corresponding to the (normalized) piecewise constant eigenfunctions

c i = ∂Ω i (x) dH N -1 (x) -1 p • 1 Ω i (x).

The first nontrivial eigenvalue

We are going to show that σ 2,p (Ω) is actually the first nontrivial Stekloff eigenvalue of -∆ p . In other words, the first eigenvalue σ = 0 is always isolated in the spectrum and any other eigenvalue has to be greater than σ 2,p (Ω). Then the quantity σ 2,p (Ω) can also be seen as the fundamental gap of the pseudo p-Laplacian, with Stekloff boundary conditions. Theorem 5.1. Let u ∈ W 1,p (Ω) be a Stekloff eigenfunction, with eigenvalue σ > 0. Then we have σ ≥ σ 2,p (Ω). 2 We recall that the Borsuk-Ulam states the following: "for every continuous map f : S k → R k , there exists x0 ∈ S k such that f (x0) = f (-x0)".

Since our function P

k-1 • f is odd, this would give that 0 ∈ Im(P k-1 • f ), that is a contradiction.
Proof. The proof is inspired to [START_REF] Juutinen | On the higher eigenvalues for ∞-eigenvalue problem[END_REF]Theorem 3.4]. At first, we observe that the positive and negative parts u + and u -of u are both not identically zero, thanks to Lemma 3.5. Also, they belong to W 1,p (Ω), hence they have a trace on the boundary ∂Ω. Moreover, we have trace |∂Ω (u + ) = (trace |∂Ω u) + and trace |∂Ω (u -) = (trace |∂Ω u) -, so that using u + and u -as test functions in (3.3), we obtain

Ω ∇u + (x) p p dx = σ ∂Ω |u + (x)| p (x) dH N -1 (x),
and

Ω ∇u -(x) p p dx = σ ∂Ω |u -(x)| p (x) dH N -1 (x).
Let us now consider the odd and continuous mapping f : S 1 → M defined by

f ω (x) = ω 1 u + (x) -ω 2 u -(x) |ω 1 | p ∂Ω |u + | p dH N -1 + |ω 2 | p ∂Ω |u -| p dH N -1 , ω = (ω 1 , ω 2 ) ∈ S 1 .
If we choose f = f in the definition of σ 2,p (Ω), we then get

σ 2,p (Ω) ≤ max ω∈S k-1 |ω 1 | p Ω ∇u + p p dx + |ω 2 | p Ω ∇u - p p dx |ω 1 | p ∂Ω |u + | p dH N -1 + |ω 2 | p ∂Ω |u -| p dH N -1 = σ,
concluding thus the proof.

We devote the rest of this section to give some alternative characterizations of σ 2,p (Ω). The first one is a mountain pass characterization. The main tool we need is the following Lemma, which guarantees that we can always join functions on M , by a curve on which the energy Ω ∇u p p is controlled by the values at the endpoints. For this, some assumptions on the sign of functions involved are necessary. Lemma 5.2. Let u, v ∈ M , with v ≥ 0 on Ω and u satisfying one of the following assumptions:

(i) u ≥ 0 on Ω; (iii) the positive and negative parts of u are both not identically zero and

(5.1) u + ≡ 0 on ∂Ω and

Ω ∇u + p p dx ∂Ω u p + dH N -1 ≤ Ω ∇u - p p dx ∂Ω u p -dH N -1
, with the convention that (5.1) is satisfied if u -≡ 0 on ∂Ω. Then there exists a continuous curve γ : [0, 1] → M , such that

Ω ∇γ t (x) p p dx ≤ max Ω ∇u(x) p p dx, Ω ∇v(x) p p dx , t ∈ [0, 1].
Proof. If u is positive on Ω, in order to conclude it sufficies to observe that the anisotropic Dirichlet integral is convex on the curve defined by (5.2) Then the first nontrivial Stekloff eigenvalue has the following Mountain Pass characterization

γ t (x) = (1 -t) u(x) p + t v(x) p 1 p , x ∈ Ω, t ∈ [0, 1].
(5.3) σ 2,p (Ω) = inf γ∈Γ Ω (c,-c) max u∈γ Ω ∇u p p dx.
Proof. For every γ ∈ Γ Ω (c, -c), the closed path on M obtained by gluing γ and -γ is in fact the image of some odd continuous mapping f γ from S 1 to M . Hence, by definition of σ 2,p (Ω) we get

σ 2,p (Ω) ≤ max u∈fγ (S 1 ) Ω ∇u(x) p p dx = max u∈γ Ω ∇u(x) p p dx.
By taking the infimum among all admissible paths γ, we easily obtain

σ 2,p (Ω) ≤ inf γ∈Γ Ω (-c,c) max u∈γ Ω ∇u(x) p p dx.
To prove the reverse inequality, we proceed as follows: for every n ∈ N, let us take

{f n } n∈N ⊂ C o (S 1 ; M ) such that max u∈fn(S 1 ) Ω ∇u(x) p p dx ≤ σ 2,p (Ω) + 1 n .
Let us then pick up u n ∈ f n (S 1 ) such that one of the hypotheses of Lemma 5.2 is satisfied 4 . Then we can assure the existence of a continuous curve σ n on M , connecting the constant function c to u n and such that

Ω ∇σ n t (x) p p dx ≤ Ω ∇u n (x) p p dx ≤ σ 2,p (Ω) + 1 n .
Symmetrically, the path -σ n connects -c to -u n ∈ f n (S k-1 ) and the previous estimate still holds true on this path, since the functional is even. Then gluing together the three paths σ n , -σ n and f n , we get a continuous curve Σ n ∈ Γ Ω (c, -c) such that max

u∈Σ n Ω ∇u(x) p p dx ≤ σ 2,p (Ω) + 1 n .
Passing to the infimum over Γ Ω (c, -c), we then get

inf γ∈Γ Ω (c,-c) max u∈γ Ω ∇u p p dx ≤ σ 2,p (Ω) + 1 n .
By letting n tends to ∞, we finally get the desired result. In what follows, we will use the shortcut notation

(5.4) R Ω (u) = Ω ∇u p p dx ∂Ω |u| p dH N -1 , u ∈ W 1,p (Ω) \ {0},
where it is understood that R(u) = +∞ whenever u has zero trace on the boundary. The following is the main result of this section. It gives a simpler variational description of σ 2,p (Ω) just in terms of a minimization, rather than through a minimax procedure.

Theorem 5.6. Let Ω ⊂ R N be an open bounded Lipschitz set. Then the infimum

(5.5) inf u∈W 1,p (Ω)\{0}
R Ω (u) :

∂Ω |u| p-2 u dH N -1 = 0 ,
is attained and coincides with σ 2,p (Ω). Moreover, every minimizer of (5.5) is a Stekloff eigenfunction.

Proof. We first observe that if Ω is not connected, then the infimum in (5.5) is 0. Since in this case we have σ 2,p (Ω) = 0 as well (see Remark 4.3), then the result is proven.

Let us now suppose that Ω is connected. We start by proving that the infimum (5.5) is attained. A standard contradiction argument exploiting the compactness of the embedding W 1,p (Ω) → L p (∂Ω) leads to the existence of a constant C p,Ω such that

∂Ω |v| p dH N -1 ≤ C p,Ω Ω |∇v| p dx, for all v ∈ W 1,p (Ω) verifying (5.6) ∂Ω |v| p-2 v dH N -1 = 0.
Then, by the equivalence of all norms in R N , it is not difficult to deduce that R Ω (u) ≥ C p,Ω > 0, for all ϕ ∈ W 1,p (Ω) satisfying (5.6), possibly for a different constant C p,Ω . This shows that the infimum (5.5) is strictly positive. The existence of a minimizer is again a straightforward consequence of Lemma 2.1 and of the compact embedding W 1,p (Ω) → L p (∂Ω).

We now denote by σ * the minimum value (5.5) and take a function u ∈ W 1,p (Ω) realizing it. Then u minimizes the functional

v → Ω ∇v p p dx -σ * ∂Ω |v| p dH N -1 ,
as well, among functions v ∈ W 1,p (Ω) satisfying the zero-mean condition (5.6). The Euler-Lagrange equation corresponding to this problem is precisely given by (3.3), with σ = σ * , since the Lagrange multiplier corresponding to (5.6) is zero (some care is needed for the case 1 < p < 2, see Lemma 5.8 below). This in turn implies that σ * is a Stekloff eigenvalue.

Let us now suppose that u ∈ W 1,p (Ω) is an eigenfuction for some eigenvalue σ = 0, then by testing equation (3.3) with ϕ = u, we necessarily have R Ω (u) = σ. Similarly, by taking a constant test function in ϕ in (3.3), we get that u verifies (5.6). This implies that each nontrivial Stekloff eigenfunction u is admissible for problem (5.5) and thus

σ * ≤ σ 2,p (Ω).
By Theorem 5.1, the reverse inequality holds as well, since σ * > 0.

Remark 5.7. The value (5.5) coincides with the best constant in the following Poincaré-Wirtinger trace inequality

c Ω min t∈R ∂Ω |u + t| p dH N -1 ≤ Ω ∇u p p dx, u ∈ W 1,p (Ω).
It is sufficient to observe that for every u ∈ W 1,p (Ω), the function t → u + t p L p (∂Ω; ) is C 1 strictly convex and coercive (see below), then the value

min t∈R ∂Ω |u + t| p dH N -1 ,
is uniquely realized and we have

t minimizes ∂Ω |u + t| p dH N -1 ⇐⇒ u + t is admissible in (5.5).
We conclude this section with the following technical result, which we used to deduce the characterization of σ 2,p given by Theorem 5.6.

Lemma 5.8 (Euler-Lagrange equation).

Let Ω ⊂ R N be an open bounded set, having Lipschitz boundary. Let u ∈ W 1,p (Ω) be a minimizer of the functional

F p (v) = 1 p Ω ∇v p p dx - σ p ∂Ω |v| p d H N -1 , v ∈ W 1,p (Ω),
on the set of admissible functions A = v ∈ W 1,p (Ω) : ∂Ω |v| p-2 v dH N -1 = 0 . Then u is a Stekloff eigenfunction with eigenvalue σ.

Proof. For p ≥ 2, observe that A is a C 1 manifold, thus the thesis is a plain consequence of the Lagrange Multipliers Theorem. Indeed, in this case u has to satisfy

N i=1 Ω |u x i | p-2 u x i ϕ x i dx -σ ∂Ω |u| p-2 u ϕ dH N -1 + µ Ω |u| p-2 ϕ dH N -1 = 0, for every ϕ ∈ W 1,p (Ω),
for some µ ∈ R. By choosing as ϕ any constant function and by using that u ∈ A, we can then easily conclude that µ = 0, i.e. u satisfies (3.3).

For 1 < p < 2, some care is needed, since the constraint A is no more a C 1 manifold and we can not directly conclude as before. In this case, we modify the argument in [START_REF] Dacorogna | Sur une généralisation de l'inégalité de Wirtinger[END_REF], the only difference being the fact that we are not assuming u to be in L ∞ (Ω). Let ϕ ∈ Lip(Ω) and n ∈ N \ {0}, then the C 1 convex function

h n (c) = ∂Ω u + 1 n ϕ + c p dH N -1 , c ∈ R,
is coercive, since we have

h n (c) ≥ 2 1-p |c| p ∂Ω dH N -1 - ∂Ω u + 1 n ϕ p dH N -1 .
In particular, for every n ∈ N \ {0}, h n admits a minimum point c n , which thus satisfies

h n (c n ) = 0, that is ∂Ω u + 1 n ϕ + c n p-2 u + 1 n ϕ + c n dH N -1 = 0,
i.e. u + 1/n ϕ + c n ∈ A. Moreover, as n goes to ∞, we can guarantee that the quantity n c n stays uniformly bounded. More precisely, for every n ∈ N, there must exist x n ∈ ∂Ω such that

(5.7) ϕ(x n ) + n c n = 0.
Indeed, if this would not be true, then either ϕ(x) + n c n > 0 for every x ∈ ∂Ω or ϕ(x) + n c n < 0, thanks to the continuity of ϕ on ∂Ω. Since the function τ → |u + τ | p-2 τ is strictly increasing, we would obtain

0 = ∂Ω u + 1 n ϕ + c n p-2 u + 1 n ϕ + c n dH N -1 > ∂Ω |u| p-2 u dH N -1 = 0 or 0 = ∂Ω u + 1 n ϕ + c n p-2 u + 1 n ϕ + c n dH N -1 < ∂Ω |u| p-2 u dH N -1 = 0.
In both cases, we would get a contradiction, so (5.7) must be true. This in turn implies that, possibly passing to a subsequence, the sequence {n c n } n∈N converges to some real number C, as n goes to ∞. Using the minimality of u and the fact that u + 1/n ϕ + c n is admissible, we then get

0 ≤ lim n→∞ F p u + 1 n (ϕ + n c n ) -F p (u) 1 n = N i=1 Ω |u x i | p-2 u x i ϕ x i dx -σ ∂Ω |u| p-2 u (ϕ + C) dH N -1 . Since u ∈ A, the previous is equivalent to 0 ≤ N i=1 Ω |u x i | p-2 u x i ϕ x i dx -σ ∂Ω |u| p-2 u ϕ dH N -1 , ϕ ∈ Lip(Ω).
The same argument with -ϕ in place of ϕ shows that u satisfies equation (3.3), for every Lipschitz test function. The conclusion then follows by exploiting the density of Lipschitz functions in W 1,p (Ω), which is true since Ω has Lipschitz boundary (see [START_REF] Giusti | Direct methods in the calculus of variations[END_REF]Theorem 3.6]).

Further properties

The next result concerns some nodal properties of the first nontrivial eigenvalue. The proof is inspired to the linear case (see [START_REF] Alessandrini | Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions[END_REF][START_REF] Kuttler | An inequality for a Stekloff eigenvalue by the method of defect[END_REF]). Proposition 6.1. Let Ω ⊂ R N be an open and connected bounded set, having Lipschitz. There exists a first nontrivial Stekloff eigenfunction w ∈ W 1,p (Ω) with exactly two nodal domains.

Proof. Let us take u ∈ W 1,p (Ω) a first nontrivial eigenfuction, thanks to Lemma 3.5 we have that u has at least two nodal domains.

Let us now suppose that u has n ≥ 3 nodal domains, Ω 1 , . . . , Ω n ⊂ Ω. We then take the functions

v k = u • 1 Ω k , k = 1, 2, i.
e. the restrictions of u to Ω 1 and Ω 2 , respectively and we define

w = α v 1 + β v 2 .
This is a function in W 1,p (Ω) and observe that we can always choose α, β ∈ R such that

∂Ω |w| p-2 w dH N -1 (x) = 0.
By construction w is admissible for the variational problem (5.5) which gives σ 2,p (Ω). Moreover, we can infer

Ω ∇w p p dx = α p Ω 1 ∇v 1 p dx + β p Ω 2 ∇v 2 p dx = σ 2,p (Ω) α p ∂Ω 1 ∩∂Ω |v 1 | p dH N -1 + β p ∂Ω 2 ∩∂Ω |v 2 | p dH N -1 = σ 2,p (Ω) ∂Ω |w| p dH N -1 .
Owing to the characterization of Theorem 5.6 for σ 2,p (Ω), we then get that w is a first nontrivial Stekloff eigenfuction of Ω, having exactly two nodal domains.

Remark 6.2. Very likely, the previous property is verified by every Stekloff eigenfunction corresponding to σ 2,p (Ω), i.e. every first nontrivial eigenfunction should have exactly two nodal domains. The main obstruction to the proof is the lack of a unique continuation principle for pseudo p-harmonics functions. Indeed, observe that in the previous proof we constructed a function w which satisfies ∆w = 0 and identically vanishes on a open subset of Ω, but we can not get a contradiction from this. We also like to point out that Harnack's inequality is of not use here, since we can not guarantee that ∂Ω 1 ∩ Ω does not coincide with ∂Ω 2 ∩ Ω. This is linked to the existence of the so-called Lakes of Wada, i.e. triples of open connected sets in the plane, which share the same boundaries.

In the case of the second Dirichlet eigenvalue of the p-Laplacian, the use of the unique continuation property can be avoided, as proved in [START_REF] Cuesta | A nodal domain property for the p-Laplacian[END_REF]. However, also this proof can not be applied here, since our eigenfunctions are not known to be in C 1 , as required by the argument in [START_REF] Cuesta | A nodal domain property for the p-Laplacian[END_REF].

|Ω 1 ∪ Ω 2 | ≤ |Ω|, Ω 1 ∩ Ω 2 = ∅ and H N -1 (∂Ω i ∩ ∂Ω) > 0, i = 1, 2.
We also set Hal(Ω) = {(Ω 1 , Ω 2 ) halving pair of Ω}.

If Σ ⊂ Ω is such that Γ := ∂Σ ∩ Ω = ∅ and this is a Lipschitz surface, we also introduce the following quantity (6.2) Λ p (Σ; Ω) = min

u∈W 1,p (Σ)\{0} R Σ (u) : u = 0 on Γ .
An optimal function in (6.2) is a weak solution of the following mixed Dirichlet-Stekloff eigenvalue problem

(6.3)              -∆ p u = 0, in Σ u = 0, on Γ, N i=1 |u x i | p-2 u x i ν i Ω = λ |u| p-2 u , on ∂Ω ∩ ∂Σ,
with λ = Λ p (Σ; Ω), i.e. a minimizer of (6.2) satisfies

N i=1 Ω |u x i | p-2 u x i ϕ x i dx = Λ p (Σ; Ω) ∂Ω∩∂Σ |u| p-2 u ϕ dH N -1 ,
for every ϕ ∈ W 1,p (Σ) with ϕ = 0 on Γ. Lemma 6.4. With the previous notation, for every p ∈ (1, ∞) problem (6.2) admits a unique positive solution u ∈ W 1,p (Σ) satisfying the normalization condition

∂Ω∩∂Σ |u(x)| p dH N -1 (x) = 1.
Moreover, the boundary value problem (6.3) admits a positive (weak) solution if and only if λ = Λ p .

Proof. Existence of a solution for this problem is straightforward. Positivity follows as always by observing that for every admissible u, the function |u| is still admissible and

R Σ (|u|) = R Σ (u).
Uniqueness can be proved using the device of Belloni and Kawohl, that we already used in Lemma 5.2. Suppose to have two distinct strictly positive 5 solutions u 0 and u 1 such that (6.4)

∂Ω∩∂Σ |u i (x)| p dH N -1 (x) = 1, i = 0, 1.
As in Lemma 5.2, we set γ t (x) = [(1 -t) u 0 (x) p + t u 1 (x) p ] 1/p , for a given 0 < t < 1. This still satisfies the normalization condition (6.4) and (6.5)

t → R Σ (γ t ) is strictly convex on [0, 1].
Then γ t is still a solution and we must have

R Σ (γ t ) = R Σ (u 0 ) = R Σ (u 1 ), t ∈ [0, 1].
This can hold if and only if u 0 = µ u 1 for some µ > 0 (see [START_REF] Belloni | A direct uniqueness proof for equations involving the p-Laplace operator[END_REF] for more details). By using (6.4), we get µ = 1 and thus we obtain a contradiction.

The second part of the statement can be proved along the same lines of [9, Theorem 3.1], still using property (6.5). One just needs to observe that every λ such that (6.3) has a solution is a crititical value of Ω ∇u p p on the manifold v ∈ W 1,p (Ω) : v = 0 on Γ and

∂Ω |v| p dH N -1 = 1 .
This concludes the proof. 5 Strict positivity is a consequence of Harnack's inequality. Indeed, as already observed, a pseudo p-harmonic function is a local minimizer of the Dirichlet energy Ω ∇u p p dx. Then Harnack's inequality for these functions is a consequence of [START_REF] Giusti | Direct methods in the calculus of variations[END_REF]Theorem 7.11].

Using problem (6.3), we have yet another minimax characterization of σ 2,p (Ω), this time in terms of the eigenvalues Λ p . For this, we assume some smoothness on the nodal domains. Proposition 6.5. Let Ω ⊂ R N be an open bounded connected set, having Lipschitz boundary. Suppose that the nodal domains Ω + and Ω -of a first nontrivial eigenfunction u belongs to Hal(Ω). Then there holds (6.6) σ 2,p (Ω) = min max{Λ p (Ω 1 ; Ω), Λ p (Ω 2 ; Ω)} : (Ω 1 , Ω 2 ) ∈ Hal(Ω) .

The minimum above is realized by the pair (Ω + , Ω -) and

(6.7) Λ p (Ω + ; Ω) = R Ω + (u) = R Ω -(u) = Λ p (Ω -; Ω).
Proof. Let us take a halving pair (Ω 1 , Ω 2 ) and u i ∈ W 1,p (Ω i ) such that u i = 0 on ∂Ω i ∩ Ω, with

Ω i ∇u i (x) p p dx = Λ p (Ω i ; Ω) and ∂Ω i ∩∂Ω |u i (x)| p dH N -1 (x) = 1, i = 1, 2.
Then we can choose two parameters α 1 , α 2 ∈ R in such a way that

v(x) = 2 i=1 α i u i (x) • 1 Ω i (x), x ∈ Ω,
satisfies the zero-mean condition (5.6). Thus, we can infer

σ 2,p (Ω) ≤ α p 1 Ω 1 ∇u 1 (x) p p dx + α p 2 Ω 2 ∇u 2 (x) p p dx α p 1 + α p 2 = α p 1 Λ p (Ω 1 ; Ω) + α p 2 Λ p (Ω 2 ; Ω) α p 1 + α p 2 ≤ max{Λ p (Ω 1 ; Ω), Λ p (Ω 2 ; Ω)},
and since this is true for every halving pair (Ω 1 , Ω 2 ), this remains true taking the infimum over Hal(Ω).

Let us now take an eigenfunction u ∈ W 1,p (Ω) relative to σ 2,p (Ω), i.e. a minimizer of (5.5). By Proposition 6.1, we can choose it in such a way that it has two nodal domains Ω + and Ω -, both touching the boundary of Ω. Using the equation, we then have

R Ω + (u) = R Ω -(u) = σ 2,p (Ω).
By definition of Λ p and the hypothesis on Ω + , Ω -, we then get

Λ p (Ω + ; Ω) ≤ R Ω + (u) and Λ p (Ω -; Ω) ≤ R Ω -(u), so that max{Λ p (Ω + ; Ω), Λ p (Ω -; Ω)} ≤ σ 2,p (Ω).
This concludes the proof of (6.6) and shows that the minimum is realized by the pair (Ω + , Ω -). In order to prove (6.7), it is sufficient to observe that u restricted to Ω + is a positive solution of (6.3), with λ = σ 2,p (Ω). By the second part of Lemma 6.4, we can infer that Λ p (Ω + ; Ω) = R Ω + (u). The same observation applies to Ω -, thus leading to (6.7).

7. An upper bound for σ 2,p

In this section we prove an upper bound for σ 2,p (Ω), in terms of geometric quantities. For this, we need the following simple result. It guarantees that the coordinate functions ϕ j (x) = x j , j = 1, . . . , N are always admissible in (5.5), modulo a translation.

∂Ω |x i | p-2 x i (x + z) dH N -1 (x) = 0, i = 1, . . . , N.
Proof. Let us consider the following function

g(y) = N i=1 1 p ∂Ω |x i -y i | p (x) dH N -1 (x), y = (y 1 , . . . , y N ) ∈ Ω.
It is not difficult to see that this function is C 1 and admits a global minimum point, thus there exists z such that

∂Ω |x i -z i | p-2 (x i -z i ) (x) dH N -1 (x) = 0, i = 1, . . . , N.
Let us now make the change of variable y = x -z. By defining Ω = Ω -z, the previous turns out to be equal to

∂Ω |y i | p-2 y i (y + z) dH N -1 (y) = 0, i = 1, . . . , N,
which gives the thesis.

The following is the main result of this section, dealing with the case of a general weight . This is the nonlinear counterpart of Brock's inequality for the first nontrivial Stekloff eigenvalue of the Laplacian (compare with [START_REF] Brock | An isoperimetric inequality for eigenvalues of the Stekloff problem[END_REF]Theorem 1]). Its proof crucially exploits the weighted Wulff inequality derived in Theorem A.4 and Corollary A.5 of the Appendix. Theorem 7.2. Let 1 < p < ∞ and p = p/(p -1). Let Ω ⊂ R N be an open bounded set, having Lipschitz boundary and a function satifying (3.1). Then there holds

(7.2) σ 2,p (Ω) ≤     ∂Ω (x) -1 p-1 ν Ω (x) p p dH N -1 (x) N |Ω|     p-1 .
Proof. Let z ∈ R N be as in Lemma 7.1 and let us set Ω = Ω -z. By the characterization (5.5) of σ 2,p (Ω), we obtain

σ 2,p (Ω) ≤ Ω ∇ϕ i (x) p p dx ∂Ω |ϕ i (x)| p (x + z) dH N -1 (x) = |Ω| ∂Ω |x i | p (x + z) dH N -1 (x) , i = 1, . . . , N,
where ϕ i (x) = x u , as before. Taking the sum over i = 1, . . . , N , we obtain

σ 2,p (Ω) ≤ N |Ω| ∂Ω x p p (x + z) dH N -1 (x)
, then we observe that by Hölder inequality, we have

∂Ω x p p (x + z) dH N -1 (x) ≥ ∂Ω x p ν Ω (x) p dH N -1 (x) p ∂Ω (x + z) -1 p-1 ν Ω (x) p p dH N -1 (x) p-1 ,
and the numerator in the right-hand side is the weigthed anisotropic perimeter P p,1 (Ω ) of Ω , with the notation (A.12). Also observe that ν Ω (x) = ν Ω (x + z), then using the weigthed Wulff inequality of Corollary A.5 with β = 1, we get

σ 2,p (Ω) ≤ N |Ω| ∂Ω x p ν Ω (x) p dH N -1 (x) p ∂Ω (x) -1 p-1 ν Ω (x) p p dH N -1 (x) p-1 ≤ N |Ω| N p |Ω| p ∂Ω (x) -1 p-1 ν Ω (x) p p dH N -1 (x) p-1
, which gives the desired estimate.

A significant and intrinsic instance of weight function verifying (3.1) is given by

(x) = ν Ω (x) p , x ∈ ∂Ω.
In this case, a more elegant and simpler bound is possible, that should be compared with the Brock-Weinstock inequality (1.4). , where B p = {x ∈ R N : x p < 1}.

Proof. Again, we take ϕ i (x) = x i , i = 1, . . . , N , then up to a translation of Ω (which does not affect σ 2,p (Ω)), we can suppose that (7.1) is satisfied. We again obtain

σ 2,p (Ω) ≤ Ω ∇ϕ i (x) p p dx ∂Ω |ϕ i (x)| p ν Ω (x) p dH N -1 (x) = |Ω| ∂Ω |x i | p ν Ω (x) p dH N -1 (x) , i = 1, . . . , N.
that is, summing up over i = 1, . . . , N , we have

σ 2,p (Ω) ≤ N |Ω| ∂Ω x p p ν Ω (x) p dH N -1 (x)
.

Using the isoperimetric property of B p given by Corollary A.5, this time with β = p, we eventually obtain the thesis.

Remark 7.4. We conjecture the bounds (7.2) and (7.3) to be "isoperimetric", like in the linear case corresponding to the Brock-Weinstock inequality (1.4). In other words, we conjecture that equality holds in (7.3) if and only if Ω = B p , up to dilations and translations. For (7.2) one also needs to require

(x) = c ν Ω (x) p , x ∈ ∂Ω.
To prove this conjecture, one would need to show that σ 2,p (B p ) = 1, i.e. the coordinate functions ϕ i (x) = x i , i = 1, . . . , N are first nontrivial eigenfuctions of -∆ p on B p . It is easily seen that x 1 , . . . , x N are indeed Stekloff eigenfunctions on B p , with corresponding eigenvalue 1. Of course, it could happen that σ 2,p (B p ) < 1. To conclude, it would be sufficient to prove the existence of a first nontrivial eigenfunction having {x j = 0} as nodal line, for some j = 1, . . . , N . The thesis would then follows from Lemma 6.4 and Proposition 6.5.

Appendix A. Weighted Wulff Inequalities

We start this self-contained Appendix with some facts about the theory of convex bodies. For more details, the reader should consult [START_REF] Schneider | Convex bodies: the Brunn-Minkowski theory[END_REF].

Let K ⊂ R N be a bounded convex set with non empty interior, containing the origin. We define the 1-positively homogeneous function x = inf{λ > 0 : x ∈ λ K}, and observe that K turns out to be the unit ball for this "norm" (actually, this is not a true norm since -x = x ). We also define the dual norm

ξ * = max x∈K x, ξ ,
which is sometimes called support function of K. Then the polar set K * is usually defined as the unit ball for • * , i.e.

K * = {ξ ∈ R N : ξ * ≤ 1}.
By definition, we have the following general version of the Cauchy-Schwarz inequality

(A.1) | x, ξ | ≤ x ξ * , x, ξ ∈ R N ,
with equality if and only if ξ belongs to the normal cone N K (x/ x ) to K at the point x/ x . In particular, if K is C 1 , equality holds if and only if ξ = t ν K (x/ x ), for some t ≥ 0.

Given Ω ⊂ R N a bounded open Lipschitz set, if we define its anisotropic perimeter by

P K (Ω) = ∂Ω ν Ω (x) * dH N -1 ,
we have the classical Wulff inequality (A.2)

P K (Ω) ≥ N |K| 1 N |Ω| N -1
N . Recalling that P K (K) = N |K|, the previous is equivalent to say that K minimizes P K , among sets with given measure. Moreover, strict equality holds in (A.2), if Ω is not a scaled and translated copy of K. See for example [START_REF] Figalli | A mass transportation approach to quantitative isoperimetric inequalities[END_REF] for a detailed study of Wulff inequality. Definition A.1. Let V : [0, +∞) → [0, +∞) a Borel function such that V (0) = 0. For every Ω ⊂ R N open bounded Lipschitz set, we define its weigthed anisotropic perimeter by

P V,K (Ω) = ∂Ω V ( x ) ν Ω (x) * dH N -1 (x).
Remark A.2. When K coincides with the unit ball of the Euclidean norm | • |, it easily seen that x = x * = |x| and P V,K coincides with the weighted perimeter ∂Ω V (|x|) dH N -1 (x), already studied in [START_REF] Betta | A weighted isoperimetric inequality and applications to symmetrization[END_REF][START_REF] Brasco | Spectral optimization for the Stekloff-Laplacian: the stability issue[END_REF].

Let us now further suppose that V ∈ C 1 ([0, ∞)), V (t) > 0 for t > 0 and it satisfies the following condition

(A.3) v(t) := V (t) + (N -1) V (t) t ,
is non decreasing on (0, +∞).

We consider the vector field

W (x) = V ( x ) x x , x ∈ R N ,
with the convention that W (0) = 0. The crucial property of W is expressed by the following Lemma, which extends to the anisotropic case a straightforward calculation of the Euclidean one.

Lemma A.3. With the previous notations, there holds

(A.4) div W (x) = v( x ), x ∈ R N \ {0}.
In particular, div W is a non decreasing function of • .

Proof. First of all, we observe that x → x is convex and thus differentiable almost everywhere. We also have

(A.5) ∇ x = ν K x x ν K x x * and ν K x x , x = ν K x x * x ,
where these relations hold almost everywhere. Observe that (A.4) is a simple consequence of (A.5). Indeed, using these we get div W (x) = V ( x ) ∇ x , Observe that the second relation in (A.5) comes again from the cases of equality in the Cauchy-Schwarz inequality, by simply noticing that

x x + N V ( x ) x -V ( x ) ∇ x , x x 2 = V ( x ) x + (N -1) V ( x ) x = v( x
ν K x x , x = ν K x x , x x x .
This prove (A.5) and thus the thesis.

We are now ready for the main result of this appendix.

Theorem A.4 (Weighted Wulff inequality). Let Ω ⊂ R N be an open bounded Lipschitz set. Then we have

(A.7) P V,K (Ω) ≥ N |K| 1 N |Ω| 1-1 N V |Ω| |K| 1 N
, with equality if and only if Ω coincides with K, up to dilations. In other words, K is the only minimizer of P K,V , under measure constraint, i.e. then by using the monotone behaviour of v, we can infer I 2 (Ω) ≥ 0. Thus (A.9) shows that K minimizes P V,K among sets with given measure.

Cases of equality. Let us suppose that P V,K (Ω) = P V,K (K). Again by (A.9) we can infer

I 1 (Ω) = 0 = I 2 (Ω).
If the function v is strictly increasing, then the previous and (A.10) easily imply that |Ω∆K| = 0, i.e. Ω has to coincide with K. On the contrary, if v is simply a non decreasing functions, the proof is a bit more complicated. In this case, the information I 2 (Ω) = 0 is useless and we need to exploit the first one i.e. I 1 (Ω) = 0. Keeping into account that V (t) > 0 for t > 0, from the latter we can infer that (A.11) ν Ω (x) * = x x , ν Ω (x) , for H N -1 -a.e. x ∈ ∂Ω.

This implies that the standard anisotropic perimeter of Ω can be written as

P K (Ω) = ∂Ω ν Ω (x) * dH N -1 (x) = Ω div x x dx = Ω N -1 x dx,
where we used the computations of Lemma A.3, with V ≡ 1. We now observe that the last integrand is a strictly decreasing function of • . Then using that K = {x : x ≤ 1} and that |Ω| = |K|, we have with strict inequality if |Ω∆K| = 0. This implies that P K (Ω) ≤ P K (K) and P K (Ω) < P K (K) as soon as |Ω∆K| = 0. Appealing to the Wulff inequality (A.2), we then conclude that |Ω∆K| = 0, that is Ω coincides with K also in this case. Some significant instances of functions V satisfying our hypothesis (A.3) are given by convex powers, i.e.

V (t) = t β , t ≥ 0, for every β ≥ 1. In particular, choosing as K the unit ball B p of the p norm centered at the origin, i.e.

B p = x ∈ R N : x p < 1 and using the distinguished notation (A.12) P p,β (Ω) =

∂Ω

x β p ν Ω (x) p dH N -1 (x),

we have the following particular case of Theorem A.4, that we enunciate as a separate result.

Lemma 4 . 1 .

 41 Let Ω ⊂ R N be an open bounded Lipschitz set and Ω : ∂Ω → R be a function such that (3.1) holds. Then Φ satisfies the Palais-Smale condition on the manifold M .

Theorem 4 . 2 .

 42 Given 1 < p < ∞, let Ω ⊂ R N be an open bounded connected set, having Lipschitz boundary. Let also : ∂Ω → R be a function such that (3.1) holds. For every k ∈ N, we define

  σ 2,p (Ω) > 0. At this aim, let us argue by contradiction and suppose that inf f ∈Co(S 1 ;M ) max u∈f (S 1 ) Ω ∇u p p dx = 0, so that, for all n ∈ N, there exists an odd continuous mapping f n from S 1 to M such that (4.11) max u∈fn(S 1 ) Ω ∇u p p dx ≤ 1 n .

Proposition 5 . 4 .

 54 Let Ω ⊂ R N be an open bouned connected Lipschitz set. Let us define the constant function c = ∂Ω (x) dH N -1 (x)

Remark 5 . 5 .

 55 The previous argument can be easily adapted to give a shorter proof of the mountain pass characterization of the second Dirichlet eigenvalue of the p-Laplacian (see[START_REF] Cuesta | The beginning of the Fuĉik spectrum for the p-Laplacian[END_REF] Theorem 3.1]).

Definition 6 . 3 .

 63 Let Ω ⊂ R N be an open bounded connected set, with Lipschitz boundary. Let us consider two open connected Lipschitz subsets Ω 1 , Ω 2 ⊂ Ω, then (Ω 1 , Ω 2 ) is said a halving pair for Ω if the following conditions are satisfied: (6.1)

Lemma 7 . 1 .

 71 Let Ω ⊂ R N be a bounded open set, having Lipschitz boundary. Let : ∂Ω → R be a function satisfying (3.1). Then there exists z ∈ R N such that the translated set Ω = Ω -z satisfies (7.1)

Theorem 7 . 3 .

 73 Let Ω ⊂ R N be an open bounded set, having Lipschitz boundary. Then there holds

  ), for a.e. x ∈ R N , which gives the desired result.So, let us now prove (A.5): we first recall some basic facts of convex analysis. If F : R N → R ∪ {+∞} is a convex lower semicontinuous function, we have ξ ∈ ∂F (x) if and only ifF (x) + F * (ξ) = x, ξ ,where F * denotes the Legendre-Fenchel conjugate of F and ∂F (x) is the subdifferential of F at the point x.Choosing F (x) = x , it is easy to see that its Legendre-Fenchel conjugate function is given by F * (ξ) = δ K * (ξ), i.e. the indicator function of the polar set K * . This yields ξ ∈ ∂ x if and only if ξ * ≤ 1 and ξ, x = x .In particular, if x = 0 and ξ ∈ ∂ x , by (A.1) we getx = ξ, x ≤ x ξ * ≤ x ,i.e. ξ * = 1 and equality holds in (A.1). This implies that if x = 0, the subdifferential of ∂ x is characterized by(A.6) ξ ∈ ∂ x if and only if ξ * = 1 and ξ ∈ N K x x .Since for almost every x ∈ R N we have∂ x = {∇ x } and N K x x = z ∈ R N : z = t ν K x xfor some t ≥ 0 , the characterization (A.6) gives the first relation in (A.5).

(A. 8 )

 8 P V,K (K) = min{P V,K (Ω) : |Ω| = |K|}.Proof. It is easily seen that (A.7) and (A.8) are equivalent, so let us suppose that |Ω| = |K|. We divide the proof in two steps: first we prove the inequality, then we detect the cases of equality.Inequality. By using the Divergence Theorem and Lemma A.3 we getΩ v( x ) dx = Ω div W (x) = ∂Ω V ( x ) x x , ν Ω (x) dH N -1 (x) = ∂Ω V ( x ) x x , ν Ω (x) -ν Ω (x) * dH N -1 (x) + P V,K (Ω), while integrating v over K yields K v( x ) dx = K div W (x) = ∂K V ( x ) x, ν K (x) dH N -1 (x) = ∂K V ( x ) ν Ω (x) * dH N -1 (x) = P V,K (K),since by definition x = 1 on ∂K. Subtracting the two equalities, we get (A.9)P V,K (Ω) -P V,K (K) = I 1 (Ω) + I 2 (Ω)where we setI 1 (Ω) = ∂Ω V ( x ) ν Ω (x) * -x x , ν Ω (x) dH N -1 (x),andI 2 (Ω) = Ω v( x ) dx -K v( x ) dx.It is not difficult to see that both quantities are positive. For the first, this is a simple consequence of the Cauchy-Schwarz inequality (A.1); for the second, we just observe that (A.10)I 2 (Ω) = Ω\K [v( x ) -v(1)] dx + K\Ω [v(1) -v( x )] dxthanks to the fact that |K \ Ω| = |Ω \ K|, since K and Ω have the same measure. On the other hand, there holds Ω \ K ⊂ {x : x ≥ 1} and K \ Ω ⊂ {x : x ≤ 1},

Observe that it is always possible to make such a choice, since fn(S k-1 ) is symmetric, i.e. if u ∈ fn(S k-1 ), then -u ∈ fn(S k-1 ) as well.

Acknowledgements. We wish to thank Guido De Philippis, Luigi De Pascale and Carlo Nitsch for some useful conversations. Part of this work has been written during a visit of the second author to Marseille. The FRUMAM and LATP institutions and their facilities are gratefully acknowledged. G. F. has been partially supported by the Ph.D. program of the University of Trento.

Namely, inequality

holds for all t ∈ [0, 1], owing to [START_REF] Brasco | A note on positive eigenfunctions and hidden convexity[END_REF]Lemma 2.1]. We point out that such a curve γ lives on M and is indeed continuous in the norm topology of W 1,p (Ω) (see [START_REF] Franzina | Existence, optimization and stability for low eigenvalues of some nonlinear operators[END_REF]).

Thus, let us now suppose that u + and u -are both non identically zero on Ω and that (5.1) holds. Set

Then σ t is a continuous curve on M , connecting u to its (renormalized) positive part. Since u + and u -have disjoint supports, we get

for all t ∈ 0, 1 2 , by (5.1) and basic calculus 3 . In order to conclude, it is now sufficient to connect the (renormalized) positive part of u to v: for this, we can simply use a curve like (5.2), suitably reparametrized, i.e.

and exploits once again the convexity of the functional along this curve. Finally, gluing together the two curves, i.e. defining

we get the desired conclusion. Observe that the same construction above is still feasible if u -≡ 0 on ∂Ω.

Remark 5.3. Of course, the positivity of the function v in the previous Lemma can be dropped and replaced by condition (5.1). We kept it just for ease of exposition.

Given a pair of functions u, v ∈ M , we denote by Γ Ω (u, v) the set of all continuous paths in M , parametrized on [0, 1] and connecting u to v, i.e.

Γ Ω (u, v) = {γ : [0, 1] → M : γ is continuous and γ(0) = u, γ(1) = v} , where continuity is understood in the norm topology of W 1,p . Then we have the following alternative characterization for σ 2,p (Ω). 3 We use the simple fact that


Jiangbo Yuan

Guillaume Gravier

Sébastien Campion

Xiuwen Liu

Hervé Jégou

Efficient Mining of Repetitions in Large-Scale TV Streams with Product Quantization Hashing

Duplicates or near-duplicates mining in video sequences is of broad interest to many multimedia applications. How to design an effective and scalable system, however, is still a challenge to the community. In this paper, we present a method to detect recurrent sequences in large-scale TV streams in an unsupervised manner and with little a priori knowledge on the content. The method relies on a product k-means quantizer that efficiently produces hash keys adapted to the data distribution for frame descriptors. This hashing technique combined with a temporal consistency check allows the detection of meaningful repetitions in TV streams. When considering all frames (about 47 millions) of a 22-day long TV broadcast, our system detects all repetitions in 15 minutes, excluding the computation of the frame descriptors. Experimental results show that our approach is a promising way to deal with very large video databases.

Introduction

Mining repetitions in video data consists in finding occurrences of repeating segments, also referred to as motifs or repetitions, within the video. In the most general setup, no prior knowledge of the relevant motifs is available. A motif is solely characterized by the fact that it is repeated with very limited variability. Finding out such repeated segments in an automatic manner is of interest for several applications, in particular for TV stream structuring where repeating inter-programs-e.g., advertisements, jingles, credits-are key elements [START_REF] Naturel | Detecting repeats for video structuring[END_REF][START_REF] Manson | Automatic tv broadcast structuring[END_REF][START_REF] Ibrahim | Tv stream structuring[END_REF] to be identified. Motif discovery in video is also used for commercial detection [START_REF] Gauch | Finding and identifying unknown commercials using repeated video sequence detection[END_REF] or to analyze news video [START_REF] Yang | Efficient short video repeat identification with application to news video structure analysis[END_REF]. However, most methods proposed in the literature do not scale up and can hardly deal with more than a few million frames.

In this paper, we focus on efficiently discovering unknown and repeating sequences in very long TV streams, where the targeted repetitions are either short repeating clips or more complex sequences such as commercials, trailers, or longer programs. A naive approach to the problem consists in using an offthe-shelf video retrieval system to index all the frames and retrieve repeated ones by submitting each frame in turn as a query. The efficiency of such querybased approaches is improved by using approximate nearest neighbor search techniques [START_REF] Yuan | Mining repetitive clips through finding continuous paths[END_REF][START_REF] Yang | Efficient short video repeat identification with application to news video structure analysis[END_REF], yet this approach is tractable only on a limited scale as the complexity is quadratic in the amount of data. Frame sub-sampling is usually performed to skirt the scale issue, with typically one frame extracted per shot in a repeatable manner, however at the cost of a reduced temporal precision. A better way to find all cross-neighbors consists in directly constructing a knearest neighbor graph. But even state-of-the-art algorithms [START_REF] Dong | Efficient k-nearest neighbor graph construction for generic similarity measures[END_REF][START_REF] Wang | Scalable k-nn graph construction for visual descriptors[END_REF] are limited to a few million vectors and cannot deal with week-long TV streams, in particular because of memory requirements for such methods where a temporary version of the graph must be stored in memory during the construction phase.

As an alternative to indexing, clustering [START_REF] Goh | Audio-visual event detection based on mining of semantic audio-visual labels[END_REF][START_REF] Berrani | A non-supervised approach for repeated sequence detection in tv broadcast streams[END_REF] or hashing [START_REF] Pua | Real time repeated video sequence identification[END_REF][START_REF] Gauch | Finding and identifying unknown commercials using repeated video sequence detection[END_REF][START_REF] Döhring | Mining tv broadcasts for recurring video sequences[END_REF][START_REF] Ibrahim | Tv stream structuring[END_REF] techniques are used to discover repeating segments by grouping similar frames. Clustering and hashing methods are close in spirit as they share the same underlying goal of mapping frame-level features to integers, based on a partition of the feature space [START_REF] Paulevé | Locality sensitive hashing: a comparison of hash function types and querying mechanisms[END_REF]. Clustering approaches better fit the data distribution, but traditional clustering methods produce a relatively coarse partition, which is detrimental for the precision of the matching and for the complexity of the subsequent stages. In contrast, hashing-based methods enable fast projection with a reasonable large hash table. It is however difficult, if possible, to balance the recall and precision of the projection, in particular because the lack of data adaptation leads to unevenly group the vectors [START_REF] Paulevé | Locality sensitive hashing: a comparison of hash function types and querying mechanisms[END_REF].

In this work, we propose a method for efficient discovery of repeated segments in large TV streams, bridging the gap between quantization techniques and hashing techniques. In particular, we show the interest of a hashing scheme based on product quantization hashing to combine the efficiency of hashing techniques to produce hash keys with the possibility to better fit the data distribution offered by clustering. The scheme is inspired by previous work on the use of product quantization (PQ) as a descriptor encoding approach for approximate nearest neighbor search in the compressed domain [START_REF] Jégou | Product quantization for nearest neighbor search[END_REF]. In our case, product quantization is used for hashing purposes to overcome the efficiency and scalability limitations of traditional clustering approaches. Compared to k-means and hierarchical k-means, product quantization is able to implicitly generate a large number of cells (as large as 2 64 or even larger) with a very fast learning stage, and, as importantly, with a very compact representation of the centroids. As a consequence, the resulting partition is of significantly better quality than in traditional hashing-methods with comparable hash-table size.

In the context of repetition mining, product quantization hashing is used to assign each frame in a video stream to some bucket based on the resulting code with a typical size of 32 bits. As a result of the hashing step, each bucket contains visually similar frames-as defined by the representation selected for frames, GIST features in our case-from which repeated sequences are searched.

The paper is organized as follows. After describing the frame representation used, Section 2 describes the proposed product quantization hashing scheme. Section 3 describes the discovery procedure used to extract maximal-length repeated patterns from the hash table. Experimental results on three weeks of annotated TV data (47 million frames) are reported in Section 4. Finally, concluding remarks are given in Section 5.

Product quantization hashing

As previously mentioned, we want to consider all video frames in the problem of discovering repeated patterns so as to maintain good temporal localization. Clearly, this choice is possible only if one defines a representation of a frame that is both compact, discriminative and exhibit an efficient indexing structure. To this end, we use global descriptors for each frame. Product quantization is then used to efficiently map global descriptors to a very large number of clusters. Finally, a hash code is derived from the product quantization. We describe in turn each step of the process.

Frame representation

Similar to previous work on TV stream description [START_REF] Döhring | Mining tv broadcasts for recurring video sequences[END_REF][START_REF] Ibrahim | Tv stream structuring[END_REF], frames are represented by color GIST descriptors [START_REF] Oliva | Modeling the shape of the scene: A holistic representation of the spatial envelope[END_REF] which are especially effective to cope with compression artifacts if an image has undergone no or limited geometrical distortion [START_REF] Douze | Evaluation of GIST descriptors for web-scale image search[END_REF], as is typically the case in video. A GIST feature is a low-dimensional representation that captures the global layout of a scene, and is specifically successful at recognizing different categories of scenes that avoids the recognition of individual objects or regions. The components of the description aim at reflecting the so-called spatial envelope of the images, which is related to a set of perceptual dimensions (naturalness, openness, roughness, expansion, ruggedness) representing the dominant spatial structure of a scene.

In this work, the image is divided into a 4x4 spatial resolution grid at 3 different scales, and orientation histograms are computed in each of the scale/spatial positions. Given that the description is computed for the three RGB channels, a frame is finally described by a 960-dimensional vector representation. Finally, PCA is applied, reducing the final dimension from 960 to 64.

Product quantization

Product quantization refers to the process of separately quantizing parts of a vector. It is a compromise between a component-wise scalar quantization and a vector quantizer applied to the full vector. By encoding the subspaces independently, a product quantizer can efficiently map the input vectors to a very large number of clusters while keeping, to some extent, the advantage of a vector quantizer. Recently, a k-means version of product quantization was proposed in an approximate nearest neighbor search algorithm [START_REF] Jégou | Product quantization for nearest neighbor search[END_REF]. The structure of the quantizer allows the estimation of the Euclidean distance between two vectors directly from the quantized indexes at query time.

Here, we use the product quantization step to assign GIST-PCA vectors representing similar visual content to a unique cluster from which a compact hash code will be derived. Given a collection of D-dimensional descriptors, the product k-means clustering step works as follows:

• Divide the space into m subspaces of dimension d each ;

• Quantize each subspace with k-means assuming the same number of clusters k for each subspace ; • Encode each vector as an index of m integers ∈ [1, k], known as a PQ code of length m × ⌈log 2 k⌉ bits.

In the context of repetition discovery, product quantization offers several advantages compared to traditional clustering algorithms (e.g., flat k-means, BIRCH). In particular, the quantizers operate on low dimensional subspaces and are trained with limited number of training vectors. In our case, 20, 000 vectors randomly selected from the first day of the TV stream were used to learn the codebooks. Most importantly, PQ generates a very large number of clusters with low assignment complexity. In this paper, the regular k-means was trained on one million vectors to produce K cells, resulting in a K × D assignment complexity. In comparison, PQ was trained on 20,000 vectors for K = k m clusters with an assignment complexity of k × D.

In this work, product quantization is used so that frames in the same cluster are visually similar and are thus expected to be from sequentially close segments or from repeating segments. In other words, each frame in a cluster should belong to one of the occurrences of the same repeated pattern. While increasing the number of bits per code provides smaller quantization error leading in better approximate nearest neighbor search, this is not a desirable property here. Indeed, in the discovery problem as we formulate it, having all visually similar frames as part of the same cluster is a necessity. Within each cell, contrary to nearest neighbor search, we therefore target high recall rather than high precision. We therefore chose m = 8 and k = 8 or 32, i.e., 24 bits or 40 bits codes, instead of the 64 bits typically used in [START_REF] Jégou | Product quantization for nearest neighbor search[END_REF].

Hashing PQ codes

Even though the PQ codes are compact, the space of possible codes remain sparsely populated, even for real-world video sizes. In practice, they are further hashed to gain additional memory and computational efficiency. For instance, PQ-codes with m = 8 and k = 32 partitions the space into 32 8 distinct cells. This number is significantly larger than the total number of frames in a 24-hour long video stream, which comprises approximately 2×10 6 frames. The m-dimensional PQ-code is therefore subsequently hashed (with any good hashing function such as a universal hash function) and the resulting key is used within a regular hashing scheme. As a result, each frame is represented by a 32 bits signature, with visually similar frames sharing with high probability the same signature and therefore assigned to the same bucket. Obviously, as with any non reversible hashing functions, collisions might appear.

Discovering repetitions using temporal consistency

We exploit the structure of the hash table resulting from product quantization hashing to discover a set of repeated patterns (a.k.a. repetitions) in each bucket.

Algorithm 1 Find initial small repeated segments As discussed above, high recall was preferred to high precision. Filtering out noise from the bucket is therefore a necessity before identifying small repeated sequences. As a final stage, boundary refinement is applied before merging the small sequences to form maximal ones.

Extracting small repeated sequences from the hash table

The first step in identifying maximal length repeated sequences is to find in each bucket short repeated sequences. As the hashing scheme was designed to ensure high recall within each bucket to the expense of precision, finding such sequences in each bucket must cope in some way with outliers. This is performed by sorting frames within the bucket according to their time stamp in order to identify sequences of contiguous frames, allowing for small gaps between consecutive frames to deal with noise and outliers. As all frames in a bucket are visually similar, the short repeated sequences identified are deemed to be occurrences of the pattern.

The procedure is given in Algorithm 1 where the output, RepSets, of the algorithm is the list of all sets of repeated segments. The threshold b min is used to avoid searching in very small buckets while l min filters out very short repetitions. Finally, g max is the maximum temporal gap tolerated for two frames to be considered contiguous. The use of g max mostly reflects that the recall in each bucket is not perfect even if good.

The size of the RepSets list usually is considerably large: In our experiments, with b min = 8, l min = 4 and g max = 25 (i.e., 1 s), between 20,000 to 40,000 sets where found for a 24 hours long TV stream. This number is reduced to a few thousands after removing all singletons. However, these short repeated segments detected by the initial stage of the algorithm are far from the maximal length repeated sequences one wants to discover. The next steps therefore aim at extending each individual segment and refining their boundaries before merging related sets to form longer repeated sequences.

Algorithm 2 Segment boundary optimization

for each segment Si ∈ RepSets do → find the monochrome segment M k ∈ right before Si if start ID(Si) -end ID(M k ) < bmax then → start ID(Si) ← end ID(M k ) end if → find the monochrome segment M l ∈ right after Si if start ID(M l ) -end ID( Si) < bmax then → end ID(Si) ← start ID(M l ) end if → label Si with <M k , M l > end for

Segment boundary optimization

In the particular case of advertisements in TV streams, one can take advantage of specific markers such as inter-commercial monochrome segments as in [START_REF] Gauch | Finding and identifying unknown commercials using repeated video sequence detection[END_REF][START_REF] Naturel | Fast structuring of large television streams using program guides[END_REF]. Inter-commercial markers are usually very short sequences, i.e., less than 1 s, which consist of monochrome frames with no audio signal. A large majority of the repeated sequences in TV streams corresponds to advertisements which intuitively provide an appealing feature for TV stream structuring [START_REF] Berrani | A non-supervised approach for repeated sequence detection in tv broadcast streams[END_REF]. Moreover, taking benefit from product quantization hashing, inter-commercial markers can be detected at almost no extra cost. These facts justify that specific processing be devoted to inter-commercial monochrome frames to adjust boundaries of short repeating segments.

As inter-commercial monochrome short segments are numerous in TV streams and are by definition visually very similar, they all fall into the same bucket after product quantization hashing and therefore generate an entry in the RepSets list easily identifiable by its many occurrences. The set of monochrome repeated segments is therefore characterized and identified by its abnormally large population.

Based on the location of monochrome segments, one can refine the boundaries of the repeated segments: Whenever a monochrome segment appears near the beginning (respectively, the end) of an occurrence of a repeated segment, the starting (respectively, ending) frame ID of the latter is modified to match the corresponding boundary in the monochrome segment. This procedure is formally described by Algorithm 2, where b max is set to 200. The use of a threshold reflects the fact that monochrome segments appear with commercials, thus reducing the probability of optimizing the boundaries of a non commercial repeated segment.

Finding out maximal length repeated segments

The previous steps enable to identify short repeated segments which are then merged to yield longer repeated segments. For sake of comparison with other methods, the fusion stage was limited to basic operations to avoid over-fitting problems and application specific solutions. The fusion successively takes place at two levels. An inner-set fusion is first performed independently within each entry of the RepSets list before merging across all entries.

Inner-set fusion indicates that, in a list of short repeated segments, several may belong to a continuous program but somehow are cut by the detection. For each entry in the RepSets list, two segments with a gap (say, S i S i+1 ) less than a threshold g ′ max = 1, 000 are merged. When we take advantage of monochrome segments, we increase g ′ max = 10, 000 if the two segments have the same labels (say, <M k , M k+1 >, as defined in Algorithm 2).

Inter-set fusion is expensive since it has to exhaustively check all identical sets. The basic idea is the following: Given two entries r 1 and r 2 in RepSets (assume the size of r 1 is larger than the size of r 2 ), we mix and merge their segments together by using a gap threshold g ′′ max = 500, then we accept the returned set if its size is equal or smaller than r 2 . All the accepted new repSets form the final RepSets that will be evaluated.

Experimental results

Dataset and ground-truth

Experiments are carried out on a 22-day long TV stream. The dataset was manually annotated using the individual TV programs as basic units. Each unit is identified by its start and end times, a title and a genre (e.g., program, jingle). A ground truth reference is built from these annotations by considering all units with the same title as repeated segments/events. Note that such repetitions vary in genre, in length and in the number of occurrences.

Experiments are carried out either on the entire dataset or on small portions identified as TV-DAY1 and TV-DAY2 (see Tab. 1), corresponding to respectively the first and second day of the stream. The former is used to train the product quantizers and to tune parameters while the latter is used for sake of comparison to existing approaches which cannot scale to the 22 days.

All the algorithms have been developed in C and Matlab (for and only for the detection part). Experiments have been performed on a single core machine under Unix.

Evaluation measures

Performance measures were designed to assess the detection of repeated segments both in terms of purity and temporal localization. Given the detected list of repeated segments-i.e., RepSets-and the ground truth list, there is often a many-to-one mapping between the lists because of the difficult sets fusion. We therefore map each entry in the ground truth list to the best corresponding entry in RepSets. Each detected segment is then mapped to at most one occurrence of the corresponding event in the ground truth, assuming two segments coincide if they overlap by more than θ × l where l is the length of the longer segment and θ a user defined parameter. Relying on the mapping above, the detection is evaluated by the F1-score for a particular repeating segment, i.e., the harmonic mean of precision and recall defined as

• recall=(# correctly detected reference segments )/(# of reference segments) • precision=(# correctly detected segments)/(# of detected segments)

The measurements are averaged over all repeating events to yield global measures denoted as AF (for a single-day data) and mAF (for the 22-day data).

Results

Experiments are performed with different hashing and clustering techniques, namely: standard k-means (KM), hashing (STDHASH), locality sensitive hashing (LSH), and our product quantization hashing (PQH) with 24 or 40 bits keys. Comparative results are reported in Fig. 1 for the TV-DAY2 subset for varying values of the overlap threshold θ. Final results on the entire corpus are given in Fig. 2 for PQ hashing and LSH which are the only scalable methods. In both cases, results are reported with and without boundary optimization using monochrome frames.

Standard k-means and hashing are special cases of our method: Setting m = 1 results in a standard k-means while defining m = 64 and k = 2 (or a small value) yields an approach very similar to standard hashing schemes as used in [START_REF] Pua | Real time repeated video sequence identification[END_REF][START_REF] Gauch | Finding and identifying unknown commercials using repeated video sequence detection[END_REF][START_REF] Döhring | Mining tv broadcasts for recurring video sequences[END_REF][START_REF] Ibrahim | Tv stream structuring[END_REF]. LSH use random projections with a hash function defined as

h r (x) = 1, if r T x 0 0, otherwise (1)
where r is a random vector subject to a N (0, I ) distribution. See [START_REF] Charikar | Similarity estimation techniques from rounding algorithms[END_REF] for more details. The length of binary codes (b) depends on how many hash functions are used. From the results in Fig. 1, we observe that longer codes (e.g., LSH-128) are less adapted as they allow very limited frame distortion. Fig. 1 shows that the standard hashing method, though the fastest, provides poor results. On the opposite, the methods based on regular k-means clustering are effective, especially for large values of k. However, the learning and assignment steps are too expensive for real-time purpose, and even infeasible for the very large values of k needed for the purpose of hashing because of the lack of sufficient amount of training data. Therefore, these two methods are not adapted to process large-scale streams. In contrast, PQH-24, PQH-40, and LSH-64 provide overall competitive results with reasonable processing time. PQH-40 achieved the best timing performance (about 20 s for encoding and hashing, and 17 s for detection). The accuracy of PQH-24 is slightly better however with a slower processing time, in particular in the detection step (9 s for encoding and hashing, and 42 s for detection). LSH-64 is faster than PQH-24 on encoding but takes about 230 s for detection. Note that the timing performance highly depends on the size of the candidate RepSets.

Performance measures on the 22 days long TV stream, as reported in Fig. 2, demonstrate the effectiveness of our approach to accurately discover repeating segments in very large data streams.

Conclusion

In this paper, we have proposed a new method for efficiently and effectively detecting repeated sequences in large TV streams, without requiring prior knowledge. By hashing PQ-codes to produce compact but effective frame-signatures, our method is scalable in terms of both storage and complexity. Several directions can be envisioned to further refine the method. Boundary refinement using shot boundaries might be considered when monochrome frames are not available. Using key-frames is an option to drastically speed-up the process that needs to be explored. Finally, the PQ hashing scheme proposed meets the needs of other applications. For instance, we plan to extend the work for web-based near duplicate video detection by encoding entire video clips in order to fetch a higher-level signature for video retrieval/mining.

Fig. 1 .

 1 Fig. 1. Detection performance on TV-DAY2 without (left) and with (right) monochrome segments for boundary refinement.

Fig. 2 .

 2 Fig. 2. Detection performance on TV-22 without (left) and with (right) monochrome segments for boundary refinement.

→

  RepSets ← {} for each bucket do if bucket size > bmin then → append a new entry to RepSets → sort frames by their temporal indexes in ascending order for each segment of contiguous frames with length > lmin, where contiguous means that the distance between two neighboring frames < gmax do → append the start and end frame of the segment to the current RepSets

	entry
	end for
	end if
	end for
	→ remove all entries in RepSets within only one segment
	→ return RepSets

Table 1 .

 1 Datasets and statistics of the reference repeated segments

	Name	#frames	length	#rep seg
	TV-DAY1	2,160,026	24 h	133
	TV-DAY2	2,159,920	24 h	148
	TV-22	47,517,004	528 h	1,502

Acknowledgments. This project was partially supported by the Quaero project, funded by OSEO, french agency for innovation.