K Gittins
email: katie.gittins@unine.ch.

C Léna

Upper bounds for Courant-sharp Neumann and Robin eigenvalues

Keywords: MSC classification (2010): 35P15, 49R05, 35P05 Courant's Nodal Domain theorem, Neumann eigenvalues, Robin eigenvalues, Euclidean domains

We consider the eigenvalues of the Laplacian on an open, bounded, connected set in R n with C 2 boundary, with a Neumann boundary condition or a Robin boundary condition. We obtain upper bounds for those eigenvalues that have a corresponding eigenfunction which achieves equality in Courant's Nodal Domain theorem. In the case where the set is also assumed to be convex, we obtain explicit upper bounds in terms of some of the geometric quantities of the set.

Introduction 1.Statement of the problem

Let Ω be an open, bounded, connected set in R n , n ≥ 2, with Lipschitz boundary ∂Ω. Consider the Neumann Laplacian acting on L 2 (Ω) and note that it has discrete spectrum since Ω is bounded. The Neumann eigenvalues of Ω can hence be written in a non-decreasing sequence, counted with multiplicity,

0 = µ 1 (Ω) < µ 2 (Ω) ≤ • • • ≤ µ k (Ω) ≤ . . . ,
where the only accumulation point is +∞. By Courant's Nodal Domain theorem, any eigenfunction corresponding to µ k (Ω) has at most k nodal domains. If u k is an eigenfunction corresponding to µ k (Ω) with k nodal domains, then we call it a Courant-sharp eigenfunction. In this case, we also call µ k (Ω) a Courant-sharp eigenvalue of Ω.

The Courant-sharp property was first considered by Pleijel [START_REF] Pleijel | Remarks on Courant's nodal line theorem[END_REF] in 1956 for the Dirichlet Laplacian. In particular, Pleijel proved that there are only finitely 1 many Courant-sharp Dirichlet eigenvalues of a bounded, planar domain with sufficiently regular boundary. See [START_REF] Bérard | Inégalités isopérimétriques et applications[END_REF][START_REF] Peetre | A generalization of Courant's nodal domain theorem[END_REF] for generalisations of Pleijel's theorem to higher dimensions and other geometric settings. Following from Pleijel's result, natural questions are, for a given domain, how many such eigenvalues are there and how large are they?

The recent articles [START_REF] Bérard | The weak Pleijel theorem with geometric control[END_REF][START_REF] Van Den Berg | On the number of Courant-sharp Dirichlet eigenvalues[END_REF] consider these questions and give upper bounds for the largest Courant-sharp Dirichlet eigenvalue and the number of such eigenvalues in terms of some of the geometric quantities of the underlying domain. In Theorem 1.3 of [START_REF] Bérard | The weak Pleijel theorem with geometric control[END_REF], the authors bound this number using the area, perimeter, maximal curvature and minimal cut-distance to the boundary, for a set in R 2 which is sufficiently regular but not necessarily convex (the cut-distance will be defined in Section 3). In [START_REF] Van Den Berg | On the number of Courant-sharp Dirichlet eigenvalues[END_REF], such geometric upper bounds are obtained for an open set in R n with finite Lebesgue measure. In the case where the domain is convex, the upper bound given in Example 1 of [START_REF] Van Den Berg | On the number of Courant-sharp Dirichlet eigenvalues[END_REF] could be expressed in terms of the isoperimetric ratio of the domain. From this, one can deduce that if the domain has a large number of Courant-sharp Dirichlet eigenvalues then its isoperimetric ratio is also large.

It was shown recently in [START_REF] Léna | Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions[END_REF] that if Ω is an open, bounded, connected set in R n with C 1,1 boundary, then the Neumann Laplacian acting in L 2 (Ω) has finitely many Courant-sharp eigenvalues (we refer to [START_REF] Léna | Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions[END_REF] for a description of prior results). As mentioned in [START_REF] Bérard | The weak Pleijel theorem with geometric control[END_REF], the aforementioned questions are also interesting for the Courant-sharp eigenvalues of the Neumann Laplacian.

Goal of the article

The aim of the present article is to obtain upper bounds for the Courant-sharp Neumann eigenvalues in the case where Ω ⊂ R n is open, bounded, connected with C 2 boundary. In the case where Ω is also convex, we obtain explicit upper bounds for the Courant-sharp Neumann eigenvalues of Ω, and for the number of such eigenvalues, in terms of some of the geometric quantities of Ω. These results correspond to some of those mentioned above for the Dirichlet case, with some additional hypotheses due to the difficulties in handling the Neumann boundary condition.

We follow the same strategy that was used in [START_REF] Léna | Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions[END_REF]. This involves distinguishing between the nodal domains of a Courant-sharp eigenfunction u for which the majority of the L 2 norm of u comes from the interior (bulk domains) and those for which the majority of the L 2 norm of u comes from near the boundary (boundary domains), and then obtaining upper bounds for the number of each type of nodal domain. In the first case, the argument used by Pleijel [START_REF] Pleijel | Remarks on Courant's nodal line theorem[END_REF], which rests upon the Faber-Krahn inequality, can be used as the eigenfunction in a bulk domain almost satisfies a Dirichlet boundary condition. For the boundary domains, it is not possible to employ the same argument as Pleijel as these nodal domains have mixed Dirichlet-Neumann boundary conditions so the Faber-Krahn inequality cannot be employed. The strategy of [START_REF] Léna | Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions[END_REF] to deal with the boundary domains is to locally straighten the boundary of the domain Ω and then to reflect the nodal domain in order to obtain a new domain that almost satisfies a Dirichlet boundary condition. One then has to compare the L 2 norm of the gradient of an eigenfunction corresponding to a Courant-sharp eigenvalue on the boundary domain to the L 2 norm of the gradient of the reflected eigenfunction on the reflected domain. See Section 5.

We restrict our attention to Euclidean domains with C 2 boundary. We can then make use of tubular coordinates in order to set up and describe the reflection procedure explicitly. This allows us to keep explicit control of the constants appearing in the aforementioned estimates in order to obtain estimates for the Courant-sharp Neumann eigenvalues.

In Proposition 7.1, we obtain an upper bound for the Courant-sharp Neumann eigenvalues of Ω in terms of some of its geometric quantities. More specifically, it depends on |Ω| the area of Ω, ρ(Ω) the isoperimetric ratio to the power 1/2, t + (Ω) the smallest radius of curvature of the boundary, and the cut distance to the boundary (see Section 3 for precise definitions of the latter quantities).

A simpler presentation of this upper bound is possible in the case where Ω is convex, since one of the additional conditions in the general case is no longer required (see Section 8). In addition, we obtain an upper bound for the number of such eigenvalues by using the upper bound for the Neumann counting function which is proved in Appendix A. In particular, we have the following proposition.

Proposition 1.1. Let Ω be an open, bounded, convex set in R 2 with C 2 boundary. There exist constants C > 0 and C > 0, that do not depend on Ω, such that for any Courant-sharp eigenvalue µ k (Ω),

µ k (Ω) ≤ C |Ω| t + (Ω) 4 + ρ(Ω) 8 |Ω| (1)
and

k ≤ C |Ω| 2 t + (Ω) 4 + ρ(Ω) 8 .
(

) 2
We note that the left-hand side and right-hand side of Inequality (1) have the same homogeneity with respect to scaling, and that Inequality (2) is scaling invariant. In addition, in Section 8, we obtain an upper bound for the Courantsharp Neumann eigenvalues which also depends upon the diameter of Ω.

By Proposition 1.1, we then observe that if Ω is a sufficiently regular convex set with a large number of Courant-sharp eigenvalues, it has a large isoperimetric ratio or a large curvature at some point of its boundary (or both). If we additionally assume that µ k (Ω) is large compared with |Ω|t + (Ω) -4 , we can conclude that the isoperimetric ratio is large. We note that a large isoperimetric ratio is enough to generate a large number of Courant-sharp Neumann eigenvalues. Indeed, this is the case for a rectangle (0, 1) × (0, L) with L large. By contrast, to the best of the authors' knowledge, it is not known whether a boundary point with large curvature alone can generate many Courant-sharp eigenvalues. It could be interesting to investigate this further.

By -∆ β Ω , we denote the Laplacian on Ω with the following Robin boundary condition ∂u ∂ν

+ βu = 0 on ∂Ω,
where ∂u ∂ν is the exterior normal derivative and β : ∂Ω → R is a non-negative, Lipschitz continuous function. We denote the corresponding eigenvalues by (µ k (Ω, β)) k≥1 . It was shown in [START_REF] Léna | Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions[END_REF] that there are finitely many Courant-sharp eigenvalues of -∆ β Ω . By monotonicity of the Robin eigenvalues with respect to β, we obtain the same results for the Courant-sharp Robin eigenvalues (see Subsection 2.2).

In addition, we obtain analogous results to those mentioned above for any dimension n ≥ 3, namely Propositions 9.2, 9.3 and 9.4 in Section 9.

Organisation of the article

In Section 2, we show that in order to obtain upper bounds for the largest Courant-sharp eigenvalue µ, it is sufficient to obtain upper bounds for the number of nodal domains and the remainder of the Dirichlet counting function. Estimates for the latter are obtained in Section 6. To deal with the former, we first consider the 2-dimensional case and set up tubular coordinates in Section 3. Following [START_REF] Léna | Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions[END_REF], we then define cut-off functions in Section 4 that allow us to distinguish between bulk and boundary domains. In Subsection 5.1 we perform the straightening of the boundary procedure and obtain the desired estimates. We then use these estimates in Subsection 5.2 to obtain an explicit upper bound for the number of Courant-sharp eigenvalues. In Subsection 5.3 and Subsection 5.4, by taking the geometry of the domain into account, we improve the estimates from Subsection 5.1 in special cases. We then combine all of the preceding results in Section 7 to obtain an upper bound for the largest Courant-sharp eigenvalue. In Section 8, we obtain explicit upper bounds for the largest Courant-sharp eigenvalue and the number of Courant-sharp eigenvalues of an open and convex planar domain with C 2 boundary that involve some of its geometric quantities. In particular, we prove Proposition 1.1. In Section 9, we obtain analogous results in arbitrary dimension n ≥ 3. In Appendix A, we prove an upper bound for the Neumann counting function of a convex set, which is used in the two preceding sections to control the number of Courant-sharp eigenvalues.

Preliminaries

Strategy for Courant-sharp Neumann eigenvalues

For µ > 0, we define the Neumann counting function as follows:

N N Ω (µ) := {k ∈ N : µ k (Ω) < µ}.
Let (λ k (Ω)) k≥1 denote the Dirichlet eigenvalues of the Laplacian on Ω. By the min-max characterisations of the Neumann and Dirichlet eigenvalues, we have, for k ∈ N, that

µ k (Ω) ≤ λ k (Ω). (3)
For µ > 0, we define the Dirichlet counting function:

N D Ω (µ) := {k ∈ N : λ k (Ω) < µ},
and the corresponding remainder R D Ω (µ) such that

N D Ω (µ) = ω n |Ω| (2π) n µ n/2 -R D Ω (µ), (4)
where ω n denotes the Lebesgue measure of the ball of radius 1 in R n , and the first term in the right-hand side of Equation 4 corresponds to Weyl's law. By (3), we have N N Ω (µ) ≥ N D Ω (Ω), and therefore

N N Ω (µ) ≥ ω n |Ω| (2π) n µ n/2 -R D Ω (µ).
Consider an eigenpair (µ, u) for the Neumann Laplacian, and denote by ν(u) the number of its nodal domains. If u is a Courant-sharp eigenfunction associated with µ > 0, µ = µ k (Ω) with ν(u) = k. On the other hand, Courant's Nodal Domain theorem implies that µ k-1 (Ω) < µ k (Ω), so that N N Ω (µ) = k -1. We therefore have

N N Ω (µ) -ν(u) < 0. (5)
Hence, in order to obtain upper bounds for µ, we require upper bounds for ν(u) and R D Ω (µ). These will be obtained in Sections 5, 6 respectively. To obtain an upper bound for ν(u), we follow the strategy of [START_REF] Léna | Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions[END_REF] in which an important step is to straighten the boundary locally. By restricting our attention to domains with C 2 boundary, we can make use of tubular coordinates to straighten the boundary which allow us to obtain the desired explicit estimates in Subsection 5.1.

Application to Courant-sharp Robin eigenvalues

Analogous arguments to the above hold for the Robin eigenvalues (µ k (Ω, β)) k≥1 . For µ > 0, we define the Robin counting function:

N β Ω (µ) := {k ∈ N : µ k (Ω, β) < µ}.
For k ∈ N, the Robin eigenvalues satisfy the following monotonicity property with respect to β ≥ 0:

µ k (Ω) ≤ µ k (Ω, β) ≤ λ k (Ω),
where β ≡ 0 gives rise to the Neumann eigenvalues and β → +∞ corresponds to the Dirichlet eigenvalues. We then have

N β Ω (µ) ≥ N D Ω (Ω),
and hence

N β Ω (µ) ≥ ω n |Ω| (2π) n µ n/2 -R D Ω (µ).
(See, for example, [10, Section 4]).

Tubular coordinates in 2D

For any r > 0, we define the inner tubular neighbourhood of ∂Ω with radius r:

∂Ω + r := {x ∈ Ω ; dist(x, ∂Ω) < r},
and its volume τ (r) := |∂Ω + r |. Let us now assume that Ω is simply connected (we consider multiply connected domains at the end of Section 5). Let γ : [0, L] → R 2 be a closed, simple and C 2 curve parametrised by arc length, such that ∂Ω = γ([0, L]). In particular, L is the total length of ∂Ω. For each s ∈ [0, L], we write t(s) := γ (s), the unit tangent vector at γ(s), and denote by n(s) the unit vector such that (t(s), n(s)) is a direct orthonormal basis. Up to reversing the orientation of the curve γ, we can assume that n(s) points towards the interior of Ω. The signed curvature of γ at the point γ(s) (relative to our choice of orientation), denoted by κ(s), is then defined by t (s) = κ(s) n(s).

We define the mapping

F : [0, L] × R → R 2 q := (s, t) → x := γ(s) + t n(s). (6)
The function F is of class C 1 , and its differential at q := (s, t), expressed from the base ((1, 0), (0, 1)) to the base (t(s), n(s)), is

J(q) = 1 -t κ(s) 0 0 1 . (7)
Following Subsection 3.1 of [START_REF] Bérard | The weak Pleijel theorem with geometric control[END_REF], we define

t + (Ω) := sup s∈[0,L] |κ(s)| -1 , (8)
and, for s ∈ [0, L], the (internal) cut-distance to ∂Ω at γ(s):

δ + (s) := sup{δ > 0 ; dist(F (s, t), ∂Ω) = t for all t ∈ [0, δ]}. (9)
We set

δ + (Ω) := inf s∈[0,L] δ + (s) (10)
and

δ 0 (Ω) := min{t + (Ω), δ + (Ω)}. (11)
By construction, F is a diffeomorphism of class C 1 from (0, L) × (0, δ 0 (Ω)) to ∂Ω + δ0(Ω) \ 0 , where 0 is the segment F ({0} × (0, δ 0 (Ω))). Given f : R → R continuous and piecewise C 1 and δ ≤ δ 0 (Ω), we define the function ϕ :

∂Ω + δ \ 0 → R by ϕ(x) = f (t)
, where x = F (s, t). By definition of δ 0 (Ω), this can alternatively be written as ϕ(x) = f (dist (x, ∂Ω)).

Proposition 3.1. The function ϕ is continuous, and is of class C 1 except on the regular arcs

Γ i = {x ∈ Ω ; dist(x, ∂Ω) = t i },
where {t i ; 1 ≤ i ≤ N } are the points of discontinuity of f in (0, δ 0 (Ω)]. Fur- thermore, ∇ϕ(x) = f (t)n(s), ( 12
)
with x = F (s, t), if x / ∈ N i=1 Γ i .
Proof. This follows from the chain rule, using the expression for the Jacobian matrix given in Equation 7.

Cut-off functions

The purpose of this section is to define cut-off functions ϕ δ 0 , ϕ δ 1 in order to characterise the nodal domains as bulk domains or boundary domains, as in Subsection 2.2 of [START_REF] Léna | Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions[END_REF]. The key point here is to obtain explicit estimates.

Step 1: We construct two functions χ 0 , χ 1 : R → R which are continuous, piecewise C 1 , and satisfy i. for t ≤ 1 4 , χ 0 (t) = 0 and χ

1 (t) = 1; ii. for t ∈ 1 4 , 3 4 , 0 ≤ χ 0 (t), χ 1 (t) ≤ 1; iii. for t ≥ 3 4 , χ 0 (t) = 1 and χ 1 (t) = 0; iv. χ 2 0 + χ 2 1 = 1;
We write

B := max( χ 0 L ∞ , χ 1 L ∞ ).
We first construct ψ : R → R, continuous, piecewise C 1 and non-decreasing, such that ψ(0) = 0 and ψ(1) = 1 and satisfying 0 ≤ ψ ≤ 1. We ask for the additional condition

ψ(t) 2 + ψ(1 -t) 2 = 1
for all t ∈ R. One possible choice for ψ is given at the end of this section.

We then set

χ 0 (t) := ψ 2 t - 1 4
;

χ 1 (t) := ψ 2 3 4 -t .
The functions χ 0 and χ 1 have the desired properties with B = 2 ψ L ∞ .

Step 2: For each δ ∈ (0, δ 0 (Ω)], we construct two functions ϕ δ 0 , ϕ δ 1 : Ω → R which are continuous and piecewise C 1 (the gradient is continuous except for finite jumps on regular arcs), and satisfy, for some positive constant C independent of δ,

i. for dist (x, ∂Ω) ≤ δ 4 , ϕ δ 0 (x) = 0 and ϕ δ 1 (x) = 1; ii. for dist (x, ∂Ω) ∈ δ 4 , 3 δ 4 , 0 ≤ ϕ δ 0 (x), ϕ δ 1 (x) ≤ 1; iii. for dist (x, ∂Ω) ≥ 3 δ 4 , ϕ δ 0 (x) = 1 and ϕ δ 1 (x) = 0; iv. ϕ δ 0 2 + ϕ δ 1 2 = 1; v. ∇ϕ δ 0 L ∞ , ∇ϕ δ 1 L ∞ ≤ Cδ -1 .
The functions χ 0 and χ 1 of Step 1 being given, we define, for x ∈ ∂Ω + δ ,

ϕ δ 0 (x) := χ 0 t δ ; ϕ δ 1 (x) := χ 1 t δ .
We make the obvious extensions of ϕ δ 0 and ϕ δ 1 so that both functions are continuous. From the enumerated properties of χ 0 and χ 1 and Equation ( 12), it follows that ϕ δ 0 and ϕ δ 1 have the desired properties with C = B.

Explicit constants: In order to obtain explicit estimates in what follows, it is necessary to specify ψ. One possible choice for ψ is given by

ψ(t) 2 := 1 1 0 s(1 -s) ds t 0 s(1 -s) ds, for t ∈ [0, 1], extended by 0 for t / ∈ [0, 1]. More explicitly, ψ(t) = 3t 2 -2t 3 for t ∈ [0, 1]. In that case ψ L ∞ = √ 3.
The functions χ 0 and χ 1 that we construct satisfy the listed properties with B = 2 √ 3, and the functions ϕ δ 0 and

ϕ δ 1 with C = 2 √ 3.

Estimates of the nodal count

We wish to count the number of each type of nodal domain. For the bulk domains, one considers a Pleijel-type argument via the Faber-Krahn inequality.

For the boundary domains, one reflects them in the boundary (after straightening it) and then applies the Faber-Krahn inequality to the reflected domains. See Section 2 of [START_REF] Léna | Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions[END_REF] for the full details and Subsection 5.2 below.

Straightening of the boundary

We start by giving explicit versions of some estimates in Subsection 2.4 of Reference [START_REF] Léna | Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions[END_REF], in order to control some quantities of interest when we straighten the boundary using tubular coordinates.

Lemma 5.1. Let V be an open set V ⊂ (0, L)×(0, 3 δ 0 (Ω)/4) and let U = F (V ).

There exist constants 0 < m -≤ m + such that, for any measurable and nonnegative function g :

V → R, m - V g dq ≤ U f dx ≤ m + V g dq,
where f = g • F -1 . Furthermore, we can choose m -= 1/4 and m + = 7/4.

Proof. This follows directly from the change of variable formula

U f (x) dx = V g(q) (1 -t κ(s)) dq,
and the fact that V ⊂ (0, L) × (0, 3 δ 0 (Ω)/4), so that t|κ(s)| ≤ 3/4.

By taking g = 1, we deduce the following from Lemma 5.1.

Corollary 5.2. If V is an open set V ⊂ (0, L) × (0, 3 δ 0 (Ω)/4) and U = F (V ), we have m -|V | ≤ |U | ≤ m + |V |, (13)
where m + , m -are the constants in Lemma 5.1.

Corollary 5.3. There exists a positive constant M such that, for all r ∈ (0, 3δ 0 (Ω)/4], τ (r) ≤ M Lr.

Furthermore, we can choose M = m + , where m + is the constant in Lemma 5.1.

Proposition 5.4. There exists a positive constant K such that the following holds

: if V is an open set V ⊂ (0, L) × (0, 3 δ 0 (Ω)/4), U = F (V ), u ∈ H 1 (U ) and v := u • F , we have V |∇v| 2 dq V v 2 dq ≤ K U |∇u| 2 dx U u 2 dx . ( 15
)
Furthermore, we can choose K = 4m + , where m + is the constant in Lemma 5.1.

Proof. From Lemma 5.1, we have directly

U u 2 dx ≤ m + V v 2 dq.
On the other hand, for any q ∈ V ,

∇v(q) = J(q) T (∇u) • F (q), so that |∇u| 2 • F (q) = J(q) T -1 ∇v(q) 2 .
From this we deduce, using the change of variable x = F (q), that

U |∇u| 2 dx = V 1 1 -t κ(s) (∂ s v(q)) 2 + (1 -t κ(s)) (∂ t v(q)) 2 dq ≥ 1 4 V |∇v| 2 dq.
Putting together the two previous estimates, we obtain Inequality (15).

Number of nodal domains

To ease the notation in the computations that follow, we set A := |Ω| in this section. Let us fix ε 0 ∈ (0, 1). We consider an eigenpair (µ, u) for the Neumann Laplacian (as in Subsection 2.2 of [START_REF] Léna | Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions[END_REF]) or for -∆ β Ω (as in Section 4 of [START_REF] Léna | Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions[END_REF])). We define u 0 := ϕ δ 0 u and u 1 := ϕ δ 1 u, so that

u 2 = u 2 0 + u 2 1 .
Following Subsection 2.1 of [START_REF] Léna | Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions[END_REF], we say that a nodal domain D of u is a bulk domain when

D u 2 0 dx ≥ (1 -ε 0 ) D u 2 dx
and a boundary domain when

D u 2 1 dx > ε 0 D u 2 dx.
We denote by ν 0 (ε 0 , u) and ν 1 (ε 0 , u) the number of bulk and boundary domains respectively, so that ν(u) = ν 0 (ε 0 , u) + ν 1 (ε 0 , u).

We now need to bound ν 0 (ε 0 , u) and ν 1 (ε 0 , u) from above. Reproducing the argument in Subsection 2.3 of [START_REF] Léna | Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions[END_REF] with n = 2, we obtain

ν 0 (ε 0 , u) ≤ A Λ 1 + ε 0 1 -ε 0 µ + 1 + 1 ε0 1 -ε 0 C 2 δ 2 , ( 16
)
where Λ is the first eigenvalue of the Dirichlet Laplacian on the disc of unit area and C is the constant given in the construction of ϕ δ 0 and ϕ δ 1 in Section 4. Let us now consider a boundary domain

D 1 j (1 ≤ j ≤ ν 1 (ε 0 , u)).
In a similar way to Subsection 2.4 of [START_REF] Léna | Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions[END_REF], we define

U := {x ∈ D 1 j ; u 1 (x) = 0} ⊂ D 1 j ∩ ∂Ω + 3δ 4
.

To simplify notation, we set ũ := u 1 . We consider V = F -1 (U ) and ṽ = ũ • F . We define σ(s, t) := (s, -t) and V R as the interior of the closure of V ∪ σ(V ). We extend ṽ to a function ṽR on V R such that ṽR • σ = ṽR (i.e. ṽR is even with respect to the variable t). By the Faber-Krahn inequality,

Λ |V R | ≤ V R ∇ṽ R 2 dq V R |ṽ R | 2 dq , and therefore Λ 2 |V | ≤ V |∇ṽ| 2 dq V |ṽ| 2 dq .
Applying Inequalities ( 13) and ( 15), we obtain

m -Λ 2 |U | ≤ K U |∇ũ| 2 dx U |ũ| 2 dx . ( 17
)
Let us note that in the computation leading to Inequality [START_REF] Schneider | Encyclopedia of Mathematics and its Applications[END_REF], we implicitly assume that the boundary of V is regular enough, so that V R is a connected open set and ṽR belongs to H 1 0 (V R ). In order to avoid such an assumption, we can proceed as in Subsection 2.5 of [START_REF] Léna | Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions[END_REF]: we perform the steps indicated above, in a super-level set V α = {ṽ R > α} for some α > 0, and afterwards let α go to 0. By Sard's theorem, we can find a sequence of α's tending to 0 so that V α is regular enough (this method was introduced in [START_REF] Bérard | Inégalités isopérimétriques et applications[END_REF]). Furthermore, as shown in Section 4 of [START_REF] Léna | Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions[END_REF], Inequality (17) also holds for an eigenfunction u of -∆ β Ω . Since ũ = u 1 = ϕ δ 1 u, we find, after applying the Leibniz formula and Young's inequality,

U |∇ũ| 2 dx ≤ 2 D 1 j |∇u| 2 dx + 2 C 2 δ 2 D 1 j u 2 dx, 10
and we have, by definition of a boundary domain,

U ũ2 dx = D 1 j u 2 1 dx > ε 0 D 1 j u 2 dx,
for our choice of ε 0 . Substituting this into Inequality [START_REF] Schneider | Encyclopedia of Mathematics and its Applications[END_REF], we get

1 ≤ 4K |U | m -Λ ε 0 µ + C 2 δ 2 .
Summing over all boundary domains, we find

ν 1 (ε 0 , u) ≤ 4K m -Λ ε 0 ∂Ω + 3δ 4 µ + C 2 δ 2 .
By Inequality ( 14) and since δ ∈ (0, δ 0 (Ω)], we have

∂Ω + 3δ 4 ≤ M L 3δ 4 ,
and therefore

ν 1 (ε 0 , u) ≤ 3KM Lδ m -Λ ε 0 µ + C 2 δ 2 . ( 18
)
In order to deduce from Inequalities ( 16) and ( 18) a bound on ν(u) which is invariant by scaling, we set

δ := A µ 1/4
.

We note that this choice of δ assumes that µ is large enough. That is, µ ≥ Aδ 0 (Ω) -4 . For our choice of ε 0 ∈ (0, 1), we therefore obtain that

ν(u) ≤ 1 Λ 1 + ε 0 1 -ε 0 (Aµ) + 3KM ε 0 m - ρ 2 (Aµ) 3/4 + C 2 1 + 1/ε 0 1 -ε 0 (Aµ) 1/2 + 3KM C 2 ε 0 m - ρ 2 (Aµ) 1/4 , (19)
where the parameter ρ is the isoperimetric ratio for the domain Ω to the power 1/2:

ρ := L 1/2 A 1/4 .

Geometry of the domain

Let us first sketch how to extend the above results to a multiply connected domain. We now assume only that Ω is an open, bounded and connected set in R 2 , with a C 2 boundary ∂Ω. The set R 2 \ Ω has an unbounded connected component, which we denote by D 0 . We define b ∈ N ∪ {0} as the number of bounded connected components of R 2 \ Ω, which we denote by D 1 , . . . , D b . For h ∈ {0, 1, . . . , b}, ∂D h is a C 2 -regular connected curve, and we denote its length by L h . We set

s h := h i=0 L i
for h ∈ {0, 1, . . . , b} and s -1 := 0. As in Section 3, we choose an arc-length parametrisation

γ h : [s h-1 , s h ] → R 2 of ∂D h
, oriented in such a way that for all s ∈ [s h-1 , s h ], the normal vector n h (s) to γ h at γ h (s) points towards the inside of Ω. Furthermore, we denote by κ h (s) the signed curvature of γ h at γ h (s). We keep the notation L for the total length of the boundary ∂Ω, so that

L = L 0 + L 1 + • • • + L b .
We define

E = b h=0 (s h-1 , s h ).
We give a natural generalisation of the definition of the function F in Equation ( 6):

F : E × R → R 2 is now defined by F (s, t) := γ h (s) + tn h (s) if s ∈ (s h-1 , s h ).
Extending the definition given in Equation ( 8), we write

t + := max h∈{0,...,b} sup s∈[s h-1 ,s h ] |κ h (s)| -1
, while we continue to define δ + (s), δ + and δ 0 (Ω) by Equations ( 9), ( 10) and [START_REF] Matheron | La formule de Steiner pour les érosions[END_REF]. By reasoning separately on each of the connected components of ∂Ω, we can show that

F is a C 1 diffeomorphism from E × (0, δ 0 (Ω)) to ∂Ω + δ0(Ω) \ b h=0 h
, where the segment h is defined by

h := γ h (s h-1 ) + (0, δ 0 (Ω))n h (s h-1 ).
Furthermore, the constructions of Section 4 and the estimates of Section 5 still hold.

Let us now discuss how to improve the estimates of Subsection 5.1 for domains with particular geometric properties. We have the following improvement of Corollary 5.3. Proposition 5.5. If Ω is either simply connected or homeomorphic to an annulus, then, for all δ ∈ (0, δ 0 (Ω)],

τ (δ) ≤ Lδ.
Proof. Let us first consider the simply connected case. From the change of variable x = F (q), we obtain

τ (δ) = L 0 δ 0 (1 -t κ(s)) dt ds = L δ - δ 2 2 L 0 κ(s) ds.
Since γ is a simple, closed, positively-oriented curve,

L 0 κ(s) ds = 2 π,
giving the desired inequality (see for instance Corollary 9.5.2 of [START_REF] Berger | Differential geometry: manifolds, curves, and surfaces[END_REF]). In the case where Ω is homeomorphic to an annulus, ∂Ω has two connected components. After a similar computation for each of them, we find

τ (δ) = L δ - δ 2 2 s0 0 κ 0 (s) ds + s1 s0 κ 1 (s) ds .
Since the normal vector has been chosen to point inwards, γ 0 and γ 1 have opposite orientations and the second term in the right-hand side vanishes. The desired inequality is an equality in this case.

Remark 5.6. Suppose Ω is a multiply-connected domain that is homeomorphic to a disc with b ∈ N ∪ {0} discs removed. By the Gauss-Bonnet theorem, we then have

L 0 κ(s) ds = ∂Ω k g (s) ds = 2πχ(Ω) = 2π(1 -b),
where χ(Ω) is the Euler Characteristic of Ω and k g denotes the geodesic curvature. If the width of the interior tubular neighbourhood is sufficiently small, that is δ ≤ δ 0 (Ω), then we obtain

τ (δ) = Lδ -π(1 -b)δ 2 .
We recover the cases b = 0, 1 as discussed above. If b ≥ 2, then we have that

τ (δ) = Lδ + π(b -1)δ 2 > Lδ,
for all δ ∈ (0, δ 0 (Ω)]. The quantity Lδ is no longer an upper bound of τ (δ), but is the leading order term for δ small. A similar upper bound was obtained in [START_REF] Sz | Über Parallelmengen nichtkonvexer ebener Bereiche[END_REF], but there the width of the interior tubular neighbourhood need only be smaller than the in-radius of the domain (see also [START_REF] Bandle | Isoperimetric Inequalities[END_REF]).

Application to convex domains

In this subsection, we assume that Ω is a convex domain. We show that Lemma 5.1 and Proposition 5.4 hold for convex domains, and give explicit constants in the corresponding bounds. We first make the following remark.

Remark 5.7. Let us assume that Ω is convex. Then δ 0 (Ω) = t + (Ω). Indeed, this is equivalent to saying that a ball of radius smaller than t + (Ω) can roll freely inside the set Ω. This follows from a more general result that was established in [START_REF] Brooks | Blaschke's rolling theorem in R n[END_REF], where this particular case is discussed on page 53, in answer to a question of J.A. Delgado [START_REF] Delgado | Blaschke's theorem for convex hypersurfaces[END_REF]. Furthermore, Inequality (14) holds with M = 1 for any r > 0. Indeed, let us define the inner parallel set at distance r > 0 by Ω r := Ω \ ∂Ω + r . The set Ω r is convex. We write m(r) := |Ω r | and we have, by definition, τ (r) := |Ω| -m(r) for all r > 0. According to [START_REF] Matheron | La formule de Steiner pour les érosions[END_REF], the function r → m(r) is differentiable and m (r) = -|∂Ω r |. Therefore, for r > 0,

τ (r) = r 0 |∂Ω t | dt.
The desired inequality then follows from the fact that the perimeter is nondecreasing with respect to inclusion among convex sets. We note that both results in the present remark hold in arbitrary dimension.

+ = 1. That is, if V is an open set V ⊂ (0, L) × (0, 3 t + (Ω)/4
) and U = F (V ) with F as defined in (6), then for any measurable and non-negative function g : V → R,

1 4 V g dq ≤ U f dx ≤ V g dq,
where f = g • F -1 .

Proposition 5.4 holds with

K = 4. That is, if V is an open set V ⊂ (0, L) × (0, 3 t + (Ω)/4), U = F (V ), u ∈ H 1 (U ) and v := u • F , we have V |∇v| 2 dq V v 2 dq ≤ 4 U |∇u| 2 dx U u 2 dx .
Proof. Since Ω is convex, we have κ(s) ≥ 0 for all s ∈ [0, L], so that

1/4 ≤ 1 -t κ(s) ≤ 1
for all t ∈ (0, 3t + (Ω)/4). Using these inequalities in the change of variable formula, we obtain the desired results.

6 An upper bound for R D Ω (µ)

As discussed in Section 2, in order to obtain an upper bound for the Courantsharp Neumann (or Robin) eigenvalues, we require an upper bound for the remainder of the Dirichlet counting function R D Ω (µ). In [START_REF] Van Den Berg | On the number of Courant-sharp Dirichlet eigenvalues[END_REF], the authors obtain an upper bound for R D Ω (µ) by using an inner partition of Ω into squares of uniform side-length and Dirichlet bracketing. Suppose these squares have side-length . Then, in the notation of the present article, Inequality (13) of [START_REF] Van Den Berg | On the number of Courant-sharp Dirichlet eigenvalues[END_REF] reads:

R D Ω (µ) ≤ τ ( √ 2 ) 4π µ + |Ω| √ 2 µ 1/2 .
Corollary 5.3 gives an upper bound for the volume of the inner tubular neighbourhood provided that the width is sufficiently small. Choosing > 0 so that √ 2 ≤ 3δ 0 (Ω)/4, and applying Corollary 5.3, we obtain

R D Ω (µ) ≤ √ 2M L 4π|Ω| (|Ω|µ) + |Ω| 1/2 √ 2 (|Ω|µ) 1/2 .
We then use the freedom in the choice of to minimise the right-hand side, meaning that we set

:= 2π|Ω| M L µ -1/4
to get

R D Ω (µ) ≤ M L|Ω| π µ 3/4 = M π ρ(Ω)(|Ω|µ) 3/4 , (20)
as soon as µ is large enough for √ 2 ≤ 3δ 0 (Ω)/4 to be satisfied. Since we used Corollary 5.3 in this section, Inequality (20) holds with M = 7/4, for all Ω ⊂ R 2 with a C 2 boundary, as soon as µ is large enough for the condition √ 2 ≤ 3δ 0 (Ω)/4 to be satisfied. If we additionally assume Ω to be simply connected or homeomorphic to an annulus, Proposition 5.5 tells us that Inequality (20) holds with M = 1 as soon as µ is large enough so that √ 2 ≤ δ 0 (Ω). Finally, if we assume Ω to be convex, Inequality (20) holds with M = 1 for all µ > 0.

Upper bound for Courant-sharp eigenvalues

We again set A := |Ω| to ease the notation. Let us consider an eigenpair (µ, u) such that u is a Courant-sharp eigenfunction. We assume that µ is large enough that

δ = A µ 1/4 ≤ δ 0 (Ω) (21) and √ 2 = 4πA M L µ -1/4 ≤ 3 4 δ 0 (Ω). ( 22
)
We recall that ε 0 ∈ (0, 1) is fixed as in Subsection 5.2. By substituting Inequality (19), Equation ( 4) and Inequality (20) into Inequality [START_REF] Berger | Differential geometry: manifolds, curves, and surfaces[END_REF], we obtain the necessary condition

1 4π - 1 + ε 0 (1 -ε 0 )Λ (Aµ) - 3KM ε 0 m -Λ ρ 2 + M π ρ (Aµ) 3/4 -C 2 1 + 1/ε 0 (1 -ε 0 )Λ (Aµ) 1/2 - 3KM C 2 ε 0 m -Λ ρ 2 (Aµ) 1/4 < 0. ( 23
)
We can reformulate Inequality (23) by saying that if µ is Courant-sharp and satisfies (21) and ( 22),

f ρ (ξ) := a 0 ξ 3 + a 1 (ρ)ξ 2 + a 2 ξ + a 3 (ρ) < 0, with a 0 := 1 4π - 1 + ε 0 (1 -ε 0 )Λ ; a 1 (ρ) := - 3KM ε 0 m -Λ ρ 2 + M π ρ ; a 2 := -C 2 1 + 1/ε 0 (1 -ε 0 )Λ ; a 3 (ρ) := - 3KM C 2 ε 0 m -Λ ρ 2 ; ξ := (Aµ) 1/4 .
Therefore, we have that Aµ ≤ ξ * (ρ) 4 , where ξ * (ρ) is the largest real zero of the function ξ → f ρ (ξ). We summarise the above discussion in the following proposition. 4 |Ω| .

µ ≤ max 4096π 2 |Ω| 81M 2 ρ(Ω) 4 δ 0 (Ω) 4 , |Ω| δ 0 (Ω) 4 , ξ * (ρ(Ω))
We note that the function f ρ , and therefore also ξ * (ρ), only depend on the geometric properties of the domain Ω and the choice of ε 0 . Sections 4 and 5 allow us to specify several sets of possible choices of constants in the definition of f ρ , according to the nature of the domain Ω, which we always assume to be open, bounded and connected with a C 2 boundary. We summarise them in Table 1, where j 0,1 denotes the smallest positive zero of the Bessel function of the first kind J 0 .

8 Geometric upper bounds for Courant-sharp eigenvalues of convex planar domains

Geometric upper bounds

In this subsection, we suppose that Ω is an open, bounded, convex planar domain with C 2 boundary. To simplify the notation, we set A := |Ω|, t + := t + (Ω) and ρ := ρ(Ω). Let µ be a Courant-sharp eigenvalue of the Neumann Laplacian on Ω, or of -∆ β Ω . We assume that µ is large enough so that Inequality (21) is satisfied. We recall that ε 0 ∈ (0, 1) is fixed as in Subsection 5.2.

In order to obtain upper bounds for a Courant-sharp eigenvalue µ in terms of some of the geometric quantities of the domain Ω, we rewrite Inequality (23), with the constants from the last row of Table 1, as follows.

1 4π - 1 + ε 0 (1 -ε 0 )Λ (Aµ) ≤ 3KM ε 0 m -Λ ρ 2 + M π ρ + C 2 (Aµ) 1/4 1 + 1/ε 0 (1 -ε 0 )Λ + 3KM C 2 ε 0 m -Λ ρ 2 (Aµ) 1/2 (Aµ) 3/4 , which implies that Aµ ≤ 1 4π - 1 + ε 0 (1 -ε 0 )Λ -4 × 3KM ε 0 m -Λ ρ 2 + M π ρ + C 2 (Aµ) 1/4 1 + 1/ε 0 (1 -ε 0 )Λ + 3KM C 2 ε 0 m -Λ ρ 2 (Aµ) 1/2
As Ω is convex, δ 0 (Ω) = t + (see Remark 5.7). In particular, there exists a ball of radius t + which is contained in Ω, that we denote by B t+ . Then

A = |Ω| ≥ |B t+ | = πt 2
+ , so At -2 + ≥ π. Together with Inequality (21), we then have

Aµ ≥ A t 2 + 2 ≥ π 2 .
Substituting this into Inequality (24), we obtain

Aµ ≤ D -4 2 288ρ 2 Λ + ρ π 1/2 + 504 5π 1/2 Λ + 3456ρ 2 πΛ 4 , (25)
which implies

µ ≤ D -4 2 A -1 288ρ 2 Λ + ρ π 1/2 + 504 5π 1/2 Λ + 3456ρ 2 πΛ 4 =: L 1 (ρ, A), ( 26
)
where D 2 = 1 4π -7 5Λ . With Proposition 7.1 and the preceding discussion in mind, we have that, if

µ is Courant-sharp, then µ ≤ max A t 4 + , L 1 (ρ, A) . (27)
Note that since Ω is convex, Inequality ( 14) holds for all r > 0 (see Remark 5.7), so Inequality ( 22) is not necessary in the convex case.

In what follows, we obtain an upper bound for µ that involves some of the other geometric quantities of Ω. We consider Inequality (24) and use the fact that µ = µ k (Ω) ≥ µ 2 (Ω) for k ≥ 2. We can then make use of known geometric estimates for the first positive Neumann eigenvalue of Ω. This is sufficient for our purposes due to the monotonicity of the Robin eigenvalues with respect to the function β. That is, for k ∈ N, µ k (Ω) ≤ µ k (Ω, β).

From Inequality (24), we have

Aµ ≤ D -4 2 288ρ 2 Λ + ρ π 1/2 + 504 5Λ(Aµ 2 ) 1/4 + 3456ρ 2 Λ(Aµ 2 ) 1/2 4 , (28)
which implies that

µ ≤ D -4 2 288ρ 2 ΛA 1/4 + ρ π 1/2 A 1/4 + 504 5ΛA 1/2 µ 1/4 2 + 3456ρ 2 ΛA 3/4 µ 1/2 2 4 . ( 29
)
Since Ω is convex, we invoke the 2-dimensional version of the classical inequality due to Payne and Weinberger, [START_REF] Payne | An Optimal Poincaré Inequality for Convex Domains[END_REF], for the first positive Neumann eigenvalue of Ω,

µ 2 (Ω) ≥ π 2 diam(Ω) 2 , (30)
where diam(Ω) denotes the diameter of Ω, to obtain

µ ≤ D -4 2 288ρ 2 ΛA 1/4 + ρ π 1/2 A 1/4 + 504diam(Ω) 1/2 5Λπ 1/2 A 1/2 + 3456ρ 2 diam(Ω) πΛA 3/4 4 =: L 2 (ρ, A, diam(Ω)). (31)
By Proposition 7.1 and the fact that Ω is convex, we have the following inequality:

µ ≤ max A t 4 + , L 2 (ρ, A, diam(Ω)) .
Remark 8.1. It is also possible to obtain an upper bound for the largest Courantsharp eigenvalue of Ω involving only the geometric quantities A, ρ and t + via the above approach by making use of Lemma 5.2 of [START_REF] Milman | On the role of convexity in isoperimetry, spectral gap and concentration[END_REF] with K = Ω and L a ball of radius t + that is contained in Ω.

We collect the preceding upper bounds in the following proposition.

µ ≤ max |Ω| t + (Ω) 4 , L 1 (ρ(Ω), |Ω|) , and µ ≤ max |Ω| t + (Ω) 4 , L 2 (ρ(Ω), |Ω|, diam(Ω)) ,
where the functions L 1 and L 2 are defined in (26) and (31) respectively.

Suppose that µ is large enough so that (21) holds. Note that since Ω is convex, diam(Ω) ≤ L 2 and so µ 2 (Ω) ≥ 4π 2 L 2 , by the Payne-Weinberger inequality. Substituting this into (28), we obtain

µ ≤ ρ 4 AD 4 2 288ρ Λ + 1 π 1/2 + 504 5(2π) 1/2 Λ + 3456ρ 3 2πΛ 4 =: A -1 D -4 2 ρ 4 G(ρ) 4 ,
where ρ → G(ρ) is an increasing function for ρ ≥ 0. In fact, in this convex case, we can deduce a simplified upper bound for µ and obtain a geometric upper bound for the number of Courant-sharp eigenvalues. Proposition 8.3. Let Ω be an open, bounded and convex set with a C 2 boundary. Let µ = µ k (Ω) or µ = µ k (Ω, β) where β is a non-negative Lipschitz continuous function on ∂Ω. There exist positive constants C and C that do not depend on Ω such that, if µ is Courant-sharp, then 8 |Ω| , and

µ ≤ C |Ω| t + (Ω) 4 + ρ(Ω)
k ≤ C |Ω| 2 t + (Ω) 4 + ρ(Ω) 8 .
The first inequality of Proposition 8.3 follows from Inequalities (25) and (27), by using the Isoperimetric inequality ρ ≥ √ 2π 1 4 . The second inequality follows from the first one and Corollary A.2, together with Remark A.3, after applying Young's inequality repeatedly. Indeed, by following this procedure, we obtain

C = D -4 2 288 Λ + 1 √ 2π 3/4 + 35064 10πΛ 4 ≈ 8.98 × 10 17 ,
and

C ≤ C + 1.
We remark that this proves Proposition 1.1 since the latter is Proposition 8.3 restricted to the Neumann case.

Application to the disc of unit area

For the disc D ⊂ R 2 of unit area, we have ρ(D) = (4π) 1/4 and δ 0 (D) = t + (D) = π -1/2 . We wish to evaluate Proposition 7.1, so we must first calculate ξ * (ρ). We use the last line of Table 1. The value of ξ * (ρ) can be computed either numerically or with Cardano's formula. We find, for any Courant-sharp Neumann By comparing the two preceding upper bounds, we remark that we do not lose a great deal by appealing to the bound of Payne and Weinberger, even though it is not the natural choice for our situation. Indeed, equality is achieved in (30) for a rectangle of fixed area with one side shrinking to 0. Such shrinking behaviour is not possible in our situation since Ω contains a ball of radius t + (Ω) > 0.

We now evaluate the first inequality of Proposition 8. For comparison with the above upper bounds, we recall that the third positive Neumann eigenvalue is the largest Courant-sharp Neumann eigenvalue as proved in [START_REF] Helffer | On nodal domains in Euclidean balls[END_REF]. In addition, µ 4 (D) = π(j 2,1 ) 2 ≈ 3.054 2 π < 30 (where j 2,1 is the first positive zero of J 2 , the derivative of the Bessel function J 2 ). The Courantsharp Robin eigenvalues of D are not known (to date).

Generalisation to arbitrary dimension 9.1 Tubular neighbourhoods of the boundary

To simplify the notation, we write Γ for the submanifold ∂Ω. As before, for r > 0, we define the inner tubular neighbourhood with radius r:

Γ + r := {x ∈ Ω ; dist(x, Γ) < r}.
For all x ∈ Γ, we denote by n(x ) the outward unit normal vector at x . In order to parametrise the inner tubular neighbourhoods, we introduce the normal bundle of Γ. Furthermore, for all x ∈ Γ, we identify the one-dimensional vector space spanned by n(x ) with the real line R. We therefore write the normal bundle as a trivial product:

N (Γ) := Γ × R
and we define a C 1 mapping F :

N (Γ) → R n by F (x , t) = x -tn(x ). ( 32
)
Let us recall that the mapping x → n(x ) from Γ to S n-1 is known as the Gauss map (see for instance Subsection 2.5 of [START_REF] Schneider | Encyclopedia of Mathematics and its Applications[END_REF]). Its differential is a symmetric linear endomorphism of T x Γ, the tangent space to Γ at x , seen as a subspace of R n . It is called the Weingarten endomorphism, and we denote it by W x . Its eigenvalues are called the principal curvatures of Γ at x . Let (e 1 , . . . , e n-1 ) be an orthonormal basis of T x Γ consisting of eigenvectors of W x and let (κ 1 (x ), . . . , κ n-1 (x )) be the associated principal curvatures. Then (e 1 , . . . , e n-1 , 1) and (e 1 , . . . , e n-1 , n(x )) are orthonormal bases of T (x ,t) N (Γ), identified with T x Γ ⊕ R, and R n respectively, and the matrix of the differential D (x ,t) F in those bases is

     1 -tκ 1 (x ) • • • 0 0 . . . . . . . . . . . . 0 • • • 1 -tκ n-1 (x ) 0 0 • • • 0 -1     
.

We now define

t + (Ω) := sup x ∈Γ max i∈{1,...,n-1} |κ i (x )| -1
, and, for any x ∈ Γ, the (internal) cut-distance to Γ at x :

δ + (x ) := sup{δ > 0 ; dist(F (x , t), Γ) = t for all t ∈ [0, δ]}.
We set δ + (Ω) := inf

x ∈Γ δ + (x ) and δ 0 (Ω) := min{t + (Ω), δ + (Ω)}.
By construction, F is a diffeomorphism of class C 1 from Γ × (0, δ 0 (Ω)) to Γ + δ0(Ω) . Lemma 9.1. Let V be an open set in Γ × (0, δ 0 (Ω)) and let U = F (V ). For any measurable, non-negative function g :

V → R, U f (x) dx = Γ +∞ 0 1 V (x , t)g(x , t) n-1 i=1 (1 -tκ i (x )) dt dx
where f = g • F -1 , 1 V is the characteristic function of V and dx is the surface measure in Γ induced by the Lebesgue measure in R n .

Proof. The proof is similar to the proof of Weyl's formula for the volume of tubes given in Chapter 6 of [START_REF] Berger | Differential geometry: manifolds, curves, and surfaces[END_REF]. First, we use Proposition 3.3.16 of [START_REF] Berger | Differential geometry: manifolds, curves, and surfaces[END_REF] to get

U f (x) dx = V g F * (dx),
where dx denotes the Lebesgue measure in R n and F * (dx) the pullback of dx by the diffeomorphism F .

Then, we note that, up to a change of sign, the mapping F defined in Equation ( 32) is the canonical map defined in Equation 2.7.5 of [START_REF] Berger | Differential geometry: manifolds, curves, and surfaces[END_REF]. According to Equation 6.8.2 of [START_REF] Berger | Differential geometry: manifolds, curves, and surfaces[END_REF], there exists a function G defined on Γ × (0, δ 0 (Ω)), to be specified below, such that

F * (dx) = G(x , t) dx ⊗ dt,
where dx ⊗ dt denotes the product measure in N (Γ) = Γ × R, with dt the Lebesgue measure in R. From Equation 6.7.16 of [START_REF] Berger | Differential geometry: manifolds, curves, and surfaces[END_REF], we obtain

V g F * (dx) = Γ×R 1 V (x , t)g(x , t)G(x , t) dx ⊗ dt = Γ R 1 V (x , t)g(x , t)G(x , t) dt dx .
It remains to give an explicit formula for G(x , t). We fix u → x(u) a local parametrisation of Γ such that x(0) = x and, for i ∈ {1, . . . , n -1}, ∂x ∂u i (0) = e i .

From Equation 6.8.9 and Proposition 6.6.2 of [START_REF] Berger | Differential geometry: manifolds, curves, and surfaces[END_REF], we obtain

G(x , t) = det Π ∂x ∂u1 (0) -t ∂n•x ∂u1 (0) , . . . , Π ∂x ∂un-1 (0) -t ∂n•x ∂un-1 (0) det Π ∂x ∂u1 (0) , . . . , Π ∂x ∂un-1 (0)
, where Π is the orthogonal projection from R n to T x Γ. It follows from the chain rule that, for all i ∈ {1, . . . , n -1},

∂n • x ∂u i (0) = W x [e i ] = κ i (x )e i .
This finally implies

G(x , t) = n-1 i=1
(1 -tκ i (x )).

Lemma 9.1 allows us to extend the estimates of Section 5 to higher dimensions in a straightforward manner. Indeed, Lemma 5.1, Corollaries 5.2 and 5.3, and Proposition 5.4 hold with the constants given in Table 2. then we have the following upper bound for the nodal count

ν(u) ≤ 1 Λ(n) n 2   1 + ε 0 1 -ε 0 ξ + 1 + 1 ε0 1 -ε 0 C 2 ξ 1/2 n 2 + 3 • 2 n 2 -1 M K n 2 ρ 2 m -ε n 2 0 ξ -1 4 ξ + C 2 ξ 1/2 n 2 , (35)
where ξ := V 2/n µ and ρ

:= S 1/2 /V 1/2-1/2n .
An upper bound for the remainder R D Ω (µ) is given in Section 2 of [START_REF] Van Den Berg | On the number of Courant-sharp Dirichlet eigenvalues[END_REF] for arbitrary dimension. Repeating the argument in Section 6, we obtain, for any

∈ (0, 3(4 √ n) -1 δ 0 (Ω)), R D Ω (µ) ≤ ω n M S √ n (2π) n µ n 2 + πn 3 2 ω n V (2π) n µ n-1 2 . Setting = πnV M S µ -1 4 , we find that, if µ is large enough so that πnV M S µ -1 4 ≤ 3δ 0 (Ω) 4 √ n , (36)
we have the following upper bound for the remainder

R D Ω (µ) ≤ 2nω n (2π) n √ πM ρξ n 2 -1 4 . (37)
As before, if the eigenfunction u is Courant-sharp,

ν(u) = N Ω (µ) + 1,
and therefore, if Inequalities (34) and (36) are satisfied, from Inequalities (35) and (37), we obtain

f ρ (ξ) := ω n (2π) n ξ n 2 - 2nω n √ πM (2π) n ρξ n 2 -1 4 - 1 Λ(n) n 2 1 + ε 0 1 -ε 0 ξ + 1 + 1 ε0 1 -ε 0 C 2 ξ 1 2 n 2 - 3 • 2 n 2 -1 M K n 2 Λ(n) n 2 m -ε n 2 0 ρ 2 ξ -1 4 ξ + C 2 ξ 1 2 n 2 < 0. ( 38
)
A straightforward computation shows that, since n . The second follows from the first and Corollary A.2, together with Remark A.3, after applying Young's inequality repeatedly. Both C n and C n could be computed explicitly.

0 < ε 0 < 1 -γ(n) 2 n 1 + γ(n)
As described in the introduction for the case of dimension 2, the previous inequality implies that a sufficiently regular convex set with a large number of Courant-sharp eigenvalues has a large isoperimetric ratio or a point in its boundary where the curvature is large (or both). Additionally, if there exists a Courant-sharp eigenvalue which is large with respect to |Ω| 2 n t + (Ω) -4 , the set has a large isoperimetric ratio.

Acknowledgments: KG was supported by the Swiss National Science Foundation grant no. 200021 163228 entitled Geometric Spectral Theory during the academic year 2016 -2017. KG also acknowledges support from the Max Planck Institute for Mathematics, Bonn, from 1 October 2017 to 31 July 2018. CL acknowledges support from the ERC (project COMPAT, ERC-2013-ADG no. 339958), the Portuguese FCT (Project OPTFORMA, IF/00177/2013) and the Swedish Research Council (Grant D0497301). Both authors wish to thank the referee for her/his insightful comments and suggestions. (i + 1)t + (Ω) i µ

A An upper bound for the Neumann counting function of a convex domain

n-i-1 2 , ( 47
)
where t + (Ω) is the smallest radius of curvature, as defined by Equation (8). The eigenvalue µ K(a)+1 (a) is the smallest among the non-trivial eigenvalues of the sets Ω α,a with α ∈ I a , and therefore, according to the Payne-Weinberger inequality, π 2 na 2 ≤ µ K(a)+1 (a). It follows from the variational characterisation of the eigenvalues that µ k (a) ≤ µ k (Ω) for any integer k (see, for example, Proposition 4(c) of [START_REF] Reed | Methods of Modern Mathematical Physics IV, Analysis of Operators[END_REF]), so that π 2 na 2 ≤ µ K(a)+1 (Ω).

Equivalently, the counting function N N Ω (•) satisfies

N N Ω π 2 na 2 ≤ K(a). (48)
To finish the proof, let us give an upper bound for K(a). We define D a as the interior of α∈Ia C α,a , and note that |D a | = K(a)a n . Furthermore, we have

D a ⊂ Ω + √ naB,
where B is the ball of radius 1 in R n and the right-hand side is understood as a Minkowski sum. We obtain

K(a) ≤ a -n Ω + √ naB . ( 49
)
Using normal coordinates in the exterior of Ω (similarly to Subsection 9.1 but for the outer parallel sets), we find, for any δ > 0,

|Ω + δB| = |Ω| + δ 0 ∂Ω n-1 i=1
(1 + tκ i (x )) dx dt.

Expanding and using κ i (x ) ≤ t + (Ω) -1 for all x ∈ ∂Ω and i ∈ {1, . . . , n -1}, we find

|Ω + δB| ≤ |Ω| + |∂Ω| n-1 i=0
n -1 i δ i+1

(i + 1)t i + (Ω)

.

We now choose δ := √ na := π/ √ µ. We obtain the desired result by substituting this value into the previous inequality and by using Inequalities (49) and (48) successively.

Let us denote by F n (|Ω|, |∂Ω|, t + (Ω), µ) the right-hand side of Inequality (47). It is a continuous and increasing function of µ. (50)

Proof. Let us fix ε > 0. We have N N Ω ((1 + ε)µ k (Ω)) ≥ k. We obtain Inequality (50) by applying Inequality (47) and letting ε go to 0.

Remark A.3. We recall that µ k (Ω) ≤ µ k (Ω, β) for every non-negative Lipschitz continuous function β on ∂Ω, and every k ≥ 1. Since F n (|Ω|, |∂Ω|, t + (Ω), µ) is increasing in µ, Inequalities (47) and (50) hold when substituting N β Ω (µ) for N N Ω (µ) and µ k (Ω, β) for µ k (Ω), respectively.

Proposition 5 . 8 .

 58 If Ω ⊂ R 2 is convex, then 1. Lemma 5.1 holds with m -= 1/4 and m

Proposition 8 . 2 .

 82 Let Ω be an open, bounded and convex set with a C 2 boundary. If µ is a Courant-sharp eigenvalue of the Neumann Laplacian on Ω or of -∆ β Ω , then we have that

  (or Robin) eigenvalue µ(D) of D, µ(D) ≤ 2.67 × 10 17 . It is well-known that the first positive Neumann eigenvalue of D is π(j 1,1 ) 2 where j 1,1 is the first positive zero of J 1 , the derivative of the Bessel function J 1 . By substituting this into Inequality (29), we obtain that any Courant-sharp Neumann eigenvalue µ(D) of D satisfies µ(D) ≤ 1.26 × 10 20 . We also have diam(D) = 2π -1/2 . By Inequality (31), we have that any Courant-sharp Neumann (or Robin) eigenvalue µ(D) of D satisfies µ(D) ≤ 2.09 × 10 20 .

  2 for the disc of unit area. We obtain that any Courant-sharp Neumann (or Robin) eigenvalue µ(D) of D satisfies µ(D) ≤ 1.42 × 10 20 , which is better than the second inequality of Proposition 8.2 for the disc of unit area, but worse than using Inequality (29) directly (see above). In addition, Proposition 8.3 applied to the disc of unit area gives that any Courant-sharp Neumann (or Robin) eigenvalue µ(D) of D satisfies µ(D) ≤ 1.51 × 10 20 , and the number of such eigenvalues k ≤ 1.51 × 10 20 .

Proposition A. 1 .

 1 Let Ω be an open, bounded and convex set in R n with a C 2 boundary. Then, for all µ > 0, we have

Proof.

  We combine the Payne-Weinberger lower bound (see Inequality (44)) with Neumann bracketing. Reference[START_REF] Van Den Berg | On the number of Courant-sharp Dirichlet eigenvalues[END_REF] uses a similar approach with Dirichlet bracketing.For α ∈ Z n , we define the open hypercube C α := α + (0, 1) n . Let us fix a > 0. We defineI a := {α ∈ Z n ; aC α ∩ Ω = ∅}and denote by K(a) the cardinality of I a . For all α ∈ I a , we define Ω α,a := Ω ∩ aC α . By construction, Ω α,a is a convex, open set with diameter bounded from above by √ na. We defineΩ a := α∈Ia Ω α,a ,which is open and bounded, but not connected. We consider the spectrum of the Neumann Laplacian on Ω a , which we denote by (µ k (a)) k≥1 . Since Ω a has K(a) connected components, 0 is an eigenvalue of multiplicity K(a), that is0 = µ 1 (a) = µ 2 (a) = • • • = µ K(a) (a).

Corollary A. 2 .

 2 Let Ω be an open, bounded and convex set in R n with a C 2 boundary. For any integer k ≥ 1,k ≤ F n (|Ω|, |∂Ω|, t + (Ω), µ k (Ω)).

Table 1 :

 1 Possible choices for the constants

	Nature of Ω General case Simply or doubly connected πj 2 Λ πj 2 0,1 0,1 Convex πj 2 0,1	C √ 2 2 √ 2 √	m -3 1/4 1/6 7/4 7 ε 0 M K 3 1/4 1/6 1 7 3 1/4 1/6 1 4
	Proposition 7.1. If µ is a Courant-sharp eigenvalue of the Neumann Laplacian
	on Ω or of -∆ β Ω , we have

  Proposition 9.4. Let Ω be an open, bounded and convex set with a C 2 boundary. Let µ = µ k (Ω) or µ = µ k (Ω, β), with β a non-negative Lipschitz continuous function on ∂Ω. There exist positive constants C n and C n that do not depend on Ω such that, if µ is Courant-sharp,

	and
		µ ≤ max	2 n t + (Ω) 4 , M 2 (n, ρ(Ω), |Ω|) , |Ω|
	with M 1 and M 2 defined in Formulas (43) and (45) respectively.
	Similarly to the end of Section 8, we can find simpler, although less explicit,
	upper bounds.
		µ ≤ C n	|Ω| t + (Ω) 4 + 2 n	ρ(Ω) 8 2 |Ω| n
	and	k ≤ C n	  |Ω| t + (Ω) 1 n	2n	+ ρ(Ω) 4n	  .
	The first inequality in Proposition 9.4 follows from Inequalities (42) and (46), taking into account ρ(Ω) ≥ 1 √ nω 2n
							2	,
							n
	we have
			lim ξ→+∞	f ρ (ξ) = +∞.
	Taking into account the conditions (34), (36) and (38), we can summarise our
	result in the following proposition.

We now assume that Ω is a bounded, open and connected set in R n , with n ≥ 3, such that ∂Ω is a C 2 submanifold of R n . We extend the methods used in the previous sections to this situation.

Upper bounds for Courant-sharp eigenvalues

To simplify the formulas, we write V := |Ω| and S := |∂Ω| (the total (n -1)dimensional surface measure of ∂Ω) in the rest of this section. We denote by N Ω either N N Ω or N β Ω , since the lower bound that we derive holds for both types of counting function.

Following [START_REF] Helffer | On nodal domains in Euclidean balls[END_REF], we define

, and we recall that γ(n) < 1 for all integers n ≥ 2. For the rest of this section, we set

This choice of ε 0 will be explained below. We consider (µ, u), an eigenpair of the Neumann Laplacian on Ω or of -∆ β Ω , with µ large enough, in a sense to be made precise.

Following the steps of Subsection 5.2, we obtain an upper bound for the number of bulk domains

, where Λ(n) is the first eigenvalue of the Dirichlet Laplacian on a ball of volume

, where ω n denotes the volume of a ball of radius 1 in R n and j n 2 -1,1 the smallest positive zero of the Bessel function of the first kind J n 2 -1 . We also obtain the following upper bound for the number of boundary domains

.

Both bounds hold assuming δ ∈ (0, δ 0 (Ω)]. Let us now set

.

It follows from the two previous upper bounds that, if µ is large enough so that

being defined in Formula (38).

We note that the function f ρ , and therefore also ξ * (ρ), only depend on the dimension n, the geometric properties of the domain Ω and the choice of ε 0 .

Geometric upper bounds for Courant-sharp eigenvalues of convex bodies

We now assume that the set Ω is convex. In that case, t + (Ω) = δ + (Ω), as discussed in Remark 5.7, and therefore δ 0 (Ω) = t + (Ω). To simplify the formulas, we write t + := t + (Ω). Again ε 0 is fixed as in (33). Additionally, the constants appearing on the left-hand side of Inequality (38) can be chosen as in the second row of Table 2, which implies a smaller value of ξ * (ρ). Furthermore, again from Remark 5.7, Corollary 5.3 holds, with M = 1 and for any r > 0. The estimate (37) on the remainder therefore holds without Condition (36). We conclude that, if µ is Courant-sharp, then

We now obtain an explicit upper bound for ξ * (ρ).

From Inequality (38) and the constants from the second row of Table 2, we have

In what follows, we use the fact that, by (34),

We also invoke the Mean Value theorem which gives that for N ≥ 1, there exists z ∈ (0, x) such that

and (ξ + 12ξ

)

Substituting these into (39), we obtain

We note that the coefficients of

only depend on the dimension, while the coefficient of ξ n 2 -1 4 depends on the dimension and the isoperimetric ratio. Inequality (40) implies that

where

25

(41), we obtain

which implies that

If instead we consider the bound due to Payne and Weinberger, [START_REF] Payne | An Optimal Poincaré Inequality for Convex Domains[END_REF],

√ πV 1 2n

.

The following bound due to Gritzmann, Wills and Wrase, [START_REF] Gritzmann | A new isoperimetric inequality[END_REF],

.

By substituting the latter inequality into Inequality (41), we have

We can summarise both versions of the upper bound in the following proposition.


Valentin Vielzeuf

Alexis Lechervy

Stéphane Pateux

Frédéric Jurie

Frederic Jurie

Towards a General Model of Knowledge for Facial Analysis by Multi-Source Transfer Learning

whether they are published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

An increasing number of deep neural networks has been implemented and trained during the last few years. These existing models can be seen as incredibly rich and compressed sources of knowledge about diverse domains, which can be reused to tackle novel tasks by transferring this knowledge. In this context, the standard way for knowledge transfer consists in selecting a single source, generally under the form of the parameters of a statistical model (e.g., a pre-trained network), and to re-use it on a new task by fine-tuning the parameters. The knowledge source is often empirically chosen, typically by selecting the closest task according to human judgment or a complex rich task such as the ImageNet classification task.

To automate this selection process, recent works [START_REF] Li | Task relation networks[END_REF][START_REF] Achille | Task2vec: Task embedding for meta-learning[END_REF][START_REF] Zamir | Taskonomy: Disentangling task transfer learning[END_REF] have shown that a relational space between diverse ba-sic tasks/models can be exploited, allowing to combine few potential candidate models and make the transfer more informative than using a single model.

However, the models discarded by this selection process may still contain useful knowledge. By analogy with multimodal problems, one may consider each independent model (specifically the last hidden representation of the neural networks) as a modality for the new task. Some modalities taken in isolation can yield bad results while they carry useful information complementary to other modalities. On the contrary, two well-performing modalities can be redundant and combining them brings less improvement.

Extending this reasoning to M modalities can be done by learning a common representation embracing all the modalities. However, this is not a trivial problem. For example, naively concatenating the local embeddings provided by each model does indeed produce a common embedding, but does not work well when dealing with many tasks/modalities, as observed by Zamir et al. [START_REF] Zamir | Taskonomy: Disentangling task transfer learning[END_REF].

We can formulate this multi-source transfer learning as follows. We define knowledge items as (domain, task) pairs, which can be associated with a model trained on this domain to fulfill this task. Supposing having M of such source knowledge, how to group them into a unique and general model of knowledge, performing well both on source knowledge and on novel target knowledge?

By addressing such a goal, this paper makes three main contributions: (a) the definition of a carefully designed ensemble of source and target knowledge related to facial analysis. Facial analysis is a topic of broad interest having received a lot of attention from the community, and for which large sets of knowledge (i.e. pre-trained models) are available. Moreover, a general knowledge in such a field would be of practical interest, as the number of new face-related tasks and domains grows exponentially. (b) a simple yet efficient methodology to project the source embeddings into a unique one, which is accurate both on the source and the target knowledge. (c) A distillation process is introduced to transfer the learned general knowledge into one lightweight convolutional neural network. This simple model outperformed its teacher, being on par with state-ofthe-art models specifically built for solving specific tasks on specific domains. Moreover, it fits real-world application requirements, with 2 million parameters.

Related Work

The proposed work is related to several fields including information compression, transfer learning or distillation.

Information Fusion and Multi-Task Learning Combining the knowledge from several existing models into a single representation can be seen as a multimodal fusion, where model features are the input modalities. Classical methods exist such as Principal Components Analysis [START_REF] Joliffe | Principal component analysis and exploratory factor analysis[END_REF] or Canonical Correlation Analysis [START_REF] Hardoon | Canonical correlation analysis: An overview with application to learning methods[END_REF]. The recent trend is often focusing on equivalent neuronal methods such as the one Ngiam et al. [START_REF] Ngiam | Multimodal deep learning[END_REF] proposed, using multimodal autoencoders to learn a representation able to reconstruct all the modalities. The literature on deep multimodal fusion can be divided in i) multimodal architectures [START_REF] Snoek | Early versus late fusion in semantic video analysis[END_REF][START_REF] Neverova | Moddrop: adaptive multi-modal gesture recognition[END_REF][START_REF] Pérez-Rúa | Mfas: Multimodal fusion architecture search[END_REF] focusing on where to fuse the information in the network, ii) representations based on constraints [START_REF] Andrew | Deep canonical correlation analysis[END_REF][START_REF] Chandar | Correlational neural networks[END_REF][START_REF] Shahroudy | Deep multimodal feature analysis for action recognition in rgb+ d videos[END_REF] building on the relations between the modalities (e.g. correlations). More details about this field can be found in these two surveys [START_REF] Atrey | Multimodal fusion for multimedia analysis: a survey[END_REF][START_REF] Baltrušaitis | Multimodal machine learning: A survey and taxonomy[END_REF].

Information fusion and multitask learning approaches are often related, as in the multitask autoencoder proposed by Ghifary et al. [START_REF] Ghifary | Domain generalization for object recognition with multi-task autoencoders[END_REF] allowing to improve domain generalization. The recent work by Ruder et al. [START_REF] Ruder | Latent multi-task architecture learning[END_REF] on multitasking architecture search allows to learn latent architectures for multitasking problems. Multitask approaches for face analysis have been proven to be as efficient or even better than single-task learning [START_REF] Chang | Deep, landmark-free fame: Face alignment, modeling, and expression estimation[END_REF]. In the context of this paper, we can also see the fusion of diverse existing models both as a multimodal problem (each model is a modality) and as a multitask problem (the final student model has to be efficient on all tasks).

Transfer Learning As the goal of the paper is to transfer knowledge to novel tasks, it can be related to transfer learning. We share the same motivations as Taskonomy proposed by Zamir et al. [START_REF] Zamir | Taskonomy: Disentangling task transfer learning[END_REF], helping to select which combination of existing models to use when tackling a new task. Nevertheless in our case, because the existing models are already partially related to the target task, there is a benefit to select them all and keep what is useful in each model. Geyer et al. [START_REF] Geyer | Transfer learning by adaptive merging of multiple models[END_REF] propose to merge two pre-trained models before transferring them, based on incremental moment matching [START_REF] Lee | Overcoming catastrophic forgetting by incremental moment matching[END_REF]. Chen et al. [17] designed a coupled end-toend transfer learning, distilling the knowledge of one source model into the target model, while selective adversarial networks are proposed in [START_REF] Cao | Partial transfer learning with selective adversarial networks[END_REF] to select positive transfers and discard negative ones in the particular case where the target label space is a subspace of the source label space. Finally, regarding domain generalization, Mancini et al. [START_REF] Mancini | Best sources forward: domain generalization through source-specific nets[END_REF] propose to fuse the outputs of domain-specific neural networks, after predicting the domain of target samples.

Self-supervised Learning Our work uses a large dataset to learn an autoencoder in an unsupervised fashion, and, by this means, exhibits the relations between the given models. Therefore, training the Teacher can be seen as a selfsupervised learning process, which is a widely explored topic [START_REF] Zhang | Colorful image colorization[END_REF][START_REF] Zhang | Split-brain autoencoders: Unsupervised learning by cross-channel prediction[END_REF][START_REF] Noroozi | Representation learning by learning to count[END_REF]. The goal of self-supervised learning is to design an efficient and cost-less proxy task helping to solve the target task for which we don't have enough annotations. Doersch et al. [START_REF] Doersch | Multi-task selfsupervised visual learning[END_REF] showed the benefit of using several proxy tasks in self-supervised learning, which is basically what we are doing with the six different embeddings to reconstruct. The difference lies in the very definition of the proxy task. Ours are coming from previously learned knowledge. In traditional self-supervised, the tasks are low-level objectives such as solving a jigsaw puzzle [START_REF] Noroozi | Unsupervised learning of visual representations by solving jigsaw puzzles[END_REF] or evaluating the rotation of an image [27]. Radenovic et al. [START_REF] Radenovic | Deep shape matching[END_REF] used a proxy task closer to ours, using state-of-theart models to extract edges of images as labels for their visual model.

Other methods falling in this unsupervised learning category can also be linked to ours, such as the deep clustering approach [START_REF] Caron | Deep clustering for unsupervised learning of visual features[END_REF] where a convolutional neural network if trained using the output of a k-means clustering algorithm.

Model Compression and Knowledge Distillation

A last important aspect of our method is to distill [START_REF] Hinton | Distilling the knowledge in a neural network[END_REF] With a goal close to ours, Chebotar et al. [START_REF] Chebotar | Distilling knowledge from ensembles of neural networks for speech recognition[END_REF] used as a teacher a weighted average prediction of an ensemble of neural networks. More recently the idea of data distillation was developed by [START_REF] Radosavovic | Data distillation: Towards omnisupervised learning[END_REF], applying the same teacher model to diverse transformations of the input image and taking the average prediction as a label for the student. Li et al. [START_REF] Li | Delta: Deep learning transfer using feature map with attention for convolutional networks[END_REF] use feature map attention to regularize the learning of the student. Finally, multimodal distillation bears similarities with our approach, using some well-known modalities of a given input to master other views with fewer annotations. For instance, the SoundNet model [START_REF] Aytar | Soundnet: Learning sound representations from unlabeled video[END_REF] learns an audio model from the labels yield by a visual model, while Xu et al. [START_REF] Xu | Pad-net: multi-tasks guided prediction-and-distillation network for simultaneous depth estimation and scene parsing[END_REF] proposed to fuse the predictions coming from diverse modalities to improve the quality of the final main task of the student.

Methodology

To the best of our knowledge, transferring multi-source knowledge across both tasks and domains has not been addressed yet in the literature. Therefore, we will first study how to formally and practically formalize such a problem. We will then detail the two main steps composing our approach: dimensionality reduction and distillation.

Multi-source Multi-domain Transfer Learning

General formulation Let's first define the concept of knowledge as the abstract ability to perfectly solve a given task t on a given domain d. We limit ourselves to the family of tasks including classification / regression with machine learning techniques. The knowledge extracted when solving this problem (e.g. through the training of a deep neural network) can be denoted as The general problem we tackle is the one of adapting transfer learning and domain adaptation to the case where we have not only one already solved problem, but a set S of solved problems.

K (t,d) = (E (t,d) , C (t,d) ), where • E (t,
Following transfer learning and domain transfer, our approach is based on defining an operator E G = G(E (ti,di) , i ∈ S) where G is a combination function using all the h i embedding (coming from the K S (t S i ,d S i ) ). It aims at gathering all knowledge information regarding the t i and the d i into one embedding h G .

One possible solution is to concatenate all the outputs of E (ti,di) , i ∈ S to generate h G . But as observed in Taskonomy [START_REF] Zamir | Taskonomy: Disentangling task transfer learning[END_REF], it generally suffers from lack of generalization.

Recent approaches such as Taskonomy [START_REF] Zamir | Taskonomy: Disentangling task transfer learning[END_REF][START_REF] Achille | Task2vec: Task embedding for meta-learning[END_REF] have addressed this question by selecting the K (ti,dj ) where t i and t j are most correlated.

In our approach (see Section 3.2), G is a neuronal encoder allowing to estimate a representation h G of reduced dimensionality, still approximating well the h i and potentially leading to better generalizations by removing biases due to over-complexity.

Nevertheless, E G is then composed of all E (t S i ,d S i ) and of G and thus mapping a given x to h G has a high computational cost. In Section 3.3 we propose the use of distillation to transform E G into a unique and lightweight model E G unique , directly projecting x to ĥG and therefore allowing easier transfer when using it to get a new knowledge.

Source and target knowledge for facial analysis

As building a general knowledge on all tasks and domains existing would not be feasible, we propose to validate our approach experimentally by carefully designing a set S of Source-Knowledge and a set T of Target-Knowledge to cover a facial analysis general knowledge, including the 15 different knowledge described in Table 1.

S gathers M=6 source knowledge, which leads to a domain i d S i of around 22 million faces with M=6 different tasks contains in around 106 million parameters. To train G (in an unsupervised fashion), instead of using i d S i , we will use a domain D unsup of 4.12 millions of faces (discarding annotations), extracted from VGGFace2 [START_REF] Cao | Vggface2: A dataset for recognising faces across pose and age[END_REF] (3.14 million), EmotioNet [START_REF] Benitez-Quiroz | Emotionet: An accurate, real-time algorithm for the automatic annotation of a million facial expressions in the wild[END_REF] (0.72 million) and IMDb-WIKI [START_REF] Rothe | Dex: Deep expectation of apparent age from a single image[END_REF] (0.26 million).

T contains 9 various knowledge on facial analysis. Note that two tasks with the same name (for instance Expression Classification) may still be different (for instance facial expression is subjective and depends on its annotators [START_REF] Zeng | Facial expression recognition with inconsistently annotated datasets[END_REF]). 14M [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF] Table 1. The M=6 source and 9 target knowledge (source is in bold). The task, as well as the used pre-trained model for source knowledge, are described in the associated papers. Information about the used domain can be found in the second citation on each row. Moreover, more details about all the selected knowledge will be provided in supplementary materials.

K (ti,di) t i d i size Expr-AffectNet Emotion Classif. [

Dimensionality Reduction for a More General Knowledge

We explain in the previous subsection that we dispose of M source knowledge encoders E (t S i ,d S i ) , allowing to extract M h i embedding from a given face x. We study here the operator G allowing to combine these embedding into a compact and general embedding h G .

Motivations Defining G as a basic concatenation function implies a very large and redundant h G . Therefore there is a meaning in reducing the dimension of h G , leading to discard some redundancies and exploit the complementarity between the h j . Dimensionality reduction is a wellexplored topic, from linear projections such as PCA [START_REF] Joliffe | Principal component analysis and exploratory factor analysis[END_REF] to more complex non-linear approaches such as manifold learning [START_REF] Huo | A survey of manifold-based learning methods[END_REF]. The major drawback of a linear approach is that the representation is projected into a plane and therefore can miss the real shape of the data. Thus, learning the manifold represented by h G with a (non-linear) neural network [START_REF] Li | Measure, manifold, learning, and optimization: A theory of neural networks[END_REF] makes sense.

Adopted Approach As we want to reduce the dimensionality without supervision, we propose to train G with an auto-encoding objective L Rec , as shown in Step 1 of Figure 1. In other words, we are building h G as an answer to all source tasks (represented by the h j ) but on the D unsup domain. For that, we optimize the parameters of both G (as an encoder) and of G (as a decoder reconstructing the h i ):

h G = G((h i ) i=1..M ) (1)
( hi ) i=1..M = G(h G ) (2)
L Rec = Σ M m=1 || ĥi -h i || 2 (3)
Note that we choose G to have a symmetric architecture to G. We will further discuss the architecture choice of an autoencoder in the experiments section, by also experimenting with PCA, regular autoencoders [START_REF] Ballard | Modular learning in neural networks[END_REF], variational autoencoders [START_REF] Kingma | Auto-encoding variational bayes[END_REF] and denoising autoencoders [START_REF] Vincent | Stacked denoising autoencoders: Learning useful representations in a deep network with a local denoising criterion[END_REF].

Another important choice is the choice of the dimensionality of h G , as it drives the knowledge compression process. As we are aiming at creating a unique general knowledge, we consider as a rule of thumb to fix it as the average dimensionality of the h i . We will conduct a empirical study in subsection 4.4 on the impact of this dimension on the quality of h G and check that there is no optimal dimension, only extreme values (very low-dimensional or high-dimensional) clearly degrading our approach.

Real-world Transfer Learning by Distillation

As mentioned in the problem definition, obtaining h G by step 1 implies using E G , composed of M=6 different pretrained models combined to G. It leads to a huge number of parameters (130M) and greatly limits the possibility of domain adaptation when dealing with the target knowledge, as we can't adapt such a large number of parameters on a new domain. A natural way to solve this problem would then to compress the so-obtained model. Model compression can be addressed with several approaches as shown in the recent literature [START_REF] Cheng | A survey of model compression and acceleration for deep neural networks[END_REF], mainly gathered into four categories: parameters pruning, low-rank factorization, compact convolutional filters and knowledge distillation.

Our current model is composed of M=6 different specific branches and we do not only want to reduce the number of parameters: we want to achieve a unique encoder. It is the promise of the distillation approach, allowing to train a new Student model, supervised by the previous big Teacher model [START_REF] Hinton | Distilling the knowledge in a neural network[END_REF]. Thus, we will not only make E G lighter but also transform it to a conventional estimator E G unique , such as a classic convolutional neural network architecture, on which well-known methods such as data augmentation can be easily applied. Moreover previous works [19] have shown that fine-tuning all the parameters of a model may lead to better domain adaptation and improve the quality of the transfer.

Distillation To achieve distillation, we still are training our new E G unique on D unsup , as illustrated in Figure 1. We consider E G unique as a neural network of arbitrary architecture (we choose a ResNet-18 [31] for all experiments), taking the face image x and directly projecting it into a representation ĥG .

The training is down by minimizing L Distillation

L Distillation = D c (h G , ĥG ) (4)
where

D c is the cosine metric (i.e. D c (a, b) = a b a • b ).
Using E G unique to estimate knowledge from T Finally when E G unique has been trained, the last step is straight-forward and only consists in adding a Multi-Layer-Perceptron on top of it and train all parameters on to estimate the new target knowledge.

About Using 2 Step Training

The two previous steps may be seen as training two parts of the same model and may be done at the same time, by training E G M T = G • E G unique to directly fit the h i . This approach may be considered as a weakly supervised multitask (MT) learning. In practice we observe that it does not converge as well as our two-step approach, which is disentangling the processing relative to the tasks (first step) and to the domains (second step). It also is in line with several progressive approaches observed in the literature [START_REF] Karras | Progressive growing of gans for improved quality, stability, and variation[END_REF], achieving better model convergence by progressively training different parts of the model.

Experiments

This section validates our approach, first by detailing results of E G unique on the different source and target knowledge, compared to state-of-the-art dedicated approaches, then by running an ablation study on the different steps of our method and on the benefit to decompose the learning process into these steps.

Domain

Implementation details

We pre-process all faces from all domains with same operations. We use a private face detector to first detect and loosely crop the detected faces. If more than one face is detected, we select the closest to the image center. A landmark-based aligner is then applied on the detected faces, which are finally resized to 300 × 300 × 3 pixels. Table 2 reports the number of images where a face was detected for each domain (first column). When a face is not detected (second column), we will consider during evaluation of our models (a) that the prediction is not correct when addressing classification problems (b) and that the prediction is the average between minimum and maximum values when dealing with regression problems. Note that the rest of the paper is following this rule.

As we use different tasks and domains, the evaluation protocols are changing for each knowledge. We follow the exact evaluation protocol used by the state-of-the-art dedicated approaches we are comparing to.

Experimental Validation

We first propose to evaluate our final approach ĥ(t,d) G on target knowledge. Looking at the two last columns of Table 3, we observe that our approach adapt well to all target knowledge, even outperforming dedicated state-ofthe-art approaches on AgeC-UTK, Expr-SFEW, Expr-RAF, Ethnic-UTK, AgeR-UTK. Note that the model for ĥ(t,d) G counts only 2.2 millions parameters, which is almost always far less than the parameters used by other approaches, helping to better generalize and better fits real-world applications constraints.

Target Knowledge Moreover, this level of performance with a lightweight model is difficult to achieve, as shown by the results of the CNN baseline, which has the same architecture than KG unique but is trained from scratch on the domain of the target knowledge. Indeed, this CNN is not always converging (e.g. on AgeR-FG with only 1002 images) and there is a huge gap between its scores and our approach, highlighting the clear benefit of transferring knowledge. A second stronger baseline called P-CNN is basically the clas- sic way for transfer learning: it consists in the use of a pretrained CNN, chosen among the source encoders as the one transferring the best for the target task and then fine-tuned on the target task for all its parameters. See also Table 6 for the scores obtained by selecting the best source encoder but without fine-tuning all parameters. Note that fine-tuning all parameters does not lead always to an improvement.

Source Knowledge

We also analyze the performance of ĥ(t,d) G on the source knowledge and compare it to the original pre-trained models used as source knowledge estimators in Table 4. For Expression-AffectNet, Attrib-CelebA and Gender-UTK we observe a clear improvement, while for Age-FG and for Identity-LFW the performance is slightly degraded. For Age-FG it may be explained by performance saturation, as the domain is very small and the human performance is around 4.6 in MAE [START_REF] Han | Age estimation from face images: Human vs. machine performance[END_REF]. For Identity-LFW, the loss can come from the difference in representation size: the original model representation has 2048 features dedicated to Identity Matching, while our model counts only 1024 features. Augmenting the dimension of ĥg may lead to more comparable performance and will be discussed in subsection 4.4. Finally, to avoid a long training time we do not compare the models on ImageNet but on the smaller dataset TinyImageNet [START_REF] Yao | Tiny imagenet classification with convolutional neural networks[END_REF]. The obtained accuracy (on the validation set) clearly underlines the importance in the choice of D unsup : the unsupervised training has been done only on face images and thus is not beneficial for tasks such as Ima-geNet classification. Despite this loss in performance, note that when training a ResNet-18 from scratch on TinyIma-geNet, we achieve only 52% accuracy. task (of size 5488 in our case). Then, the goal is to generate a compact representation h G (which size is fixed to 1024 in all the experiments) of the M=6 embeddings (h i ).

Ablation study

As this dimension reduction step is a crucial operation of our approach, we propose to study the impact of different variations on this step. Therefore, we first evaluate the ability of G to reconstruct the original h i features from h G . We report in Table 5 the normalized Root Mean Squared Error between each h i and hi with diverse variations on G. PCA stands for Principal Components Analysis, AE for standard Autoencoder, VAE for Variational Autoencoder and DAE for Denoising Autoencoder. Note that AE, VAE and DAE have the same number of parameters (the encoder G has 3 fully connected layers: 5488×3136, 3136×1792 and 1792×1024 and the decoder G is the symmetrical). The linear PCA baseline is easily outperformed by other approaches. The two best performing are the AE and the DAE. Surprisingly, the DAE does not generalize better than the AE, and is performing better only for the reconstruction of the ImageNet embedding, which is the less relevant in a context of face inputs. Still note that the gap of performance is not very important and all non-linear methods allows a fairly decent reconstruction of the embeddings.

h i PCA AE VAE DAE E G M T E G

Contribution of the distillation: E

G M T versus E G unique
The last two columns of the and E G unique , by their ability to reconstruct the h i from the embedding they produced. To achieve such a reconstruction, we extracted the ĥMT and ĥG from all element of D unsup and then trained a decoder (with a similar architecture to G) to reconstruct the h i .

We can observe that the reconstruction error of E G M T is sometimes very high (e.g. for the identity embedding, explaining the low results of ĥMT when use for Identity-LFW in Table 3). Note that simply generating random embeddings (according to a uniform distribution in the range of the target embeddings) allows to achieve a normalized RMSE of 1.4. Therefore a score of 1.22 is almost random.

Without surprise, the reconstruction error of E G unique is higher than the one obtained by G, which can be explained by both the distillation approximation between h G and ĥG and the lower capacity of our small model. Nevertheless, a low reconstruction error is not a guarantee of better performance when trying to apply the h G representation in other tasks and domains. Thus, we propose to evaluate the very transfer learning operation using E G instead of E G unique . The first column of Table 3 reports the results obtained by E G on the target knowledge, while Table 4 provides the results on the source knowledge. On almost all knowledge, E G is outperformed by E G unique by a significant margin. Yet, if we try to understand from where the distillation improvement is coming, we can observe that only using a MLP on top of ĥG (without fine-tuning all parameters) does not bring improvements compared with using a similar MLP on top of h G , some small changes in term of performance being observed. Thus, the distillation error has a limited impact when dealing with transfer. The bigger improvement is observed when using ĥ(t,d) G , meaning that we fine-tune all parameters of E G unique on the new knowledge. It illustrates that the main contribution of the distillation lies in the reduced size of the encoder, giving it the ability to more easily adapt to new tasks and domains. 3).

Concatenation, reduction with G or selection ? If we come back to the operator G, we discussed in section 3 the good reasons to reduce dimensionality of the h i . We propose to empirically compare this choice to other methods and validate the intuition that selection may discard useful information. Thus, for each target knowledge, we propose to evaluate G and several alternatives summarized in Table 6. We first evaluate a Multi Layer Perceptron taking as input the concatenation (Concat of all h i ) and having G architecture. The MLP parameters are trained from scratch on the target knowledge. Our G is then evaluated by training a small MLP on top of h G (extracted from all images of the target domain). We also compare the benefits brought by using a non-linear G by also reporting the Princpal Components Analysis (PCA) results. Finally, another concurrent approach consists in selecting the embedding (resp. the combination of embedding) yielding the Best Transfer (BT) (resp. the Best Combination Transfer (BCT)). Several works [START_REF] Zamir | Taskonomy: Disentangling task transfer learning[END_REF][START_REF] Li | Task relation networks[END_REF] proposed a method to automatically select such a best combination of embeddings. We choose here to reproduce this approach in a naive way, by brute force testing all the possible single transfers (BT) or combinations (by concatenation) of transfers (BCT) and reporting the best found results on each target kowledge in the two last columns of Table 6.

Table 6 shows that the Concat baseline is outperformed by all other approaches on almost all target knowledge. It is in line with what Zamir et al. [START_REF] Zamir | Taskonomy: Disentangling task transfer learning[END_REF] observed and it might be explained by the large dimensionality of the input of the Concat method and the limited size of some of the target domains. Then, it is interesting to see that the simple PCA method is nevertheless several times on par with the BT results, while G performances are better than BT results and often better than the BCT. Thus, it validates the choice of a reduction of dimensionality applied on all embedding h i , moreover illustrating the intuition developed in introduc- tion: when the embeddings from which we want to transfer knowledge are correlated, there is a benefit to exploit these redundancies instead of discarding some information by block, as done by the selection methods. Note that in contrast to the direct Concat baseline approach, G is able to extract a meaningful h G from the h i because of the large number of observed samples of D unsup .

Impact of h G dimension

During the presentation of the method, we propose to choose the size of h G as the average size of h i . In Figure 2 we discuss this choice, by showing that there is no optimal representation size for all knowledge. Nevertheless, a too small representation conduct to low results and a large representation implies a high-computational cost and lower performance.

Effect of Learning in 2 Stages

During the construction of our model, we argue that there was a benefit to adopt a 2 stage approach, disentangling the compression and the distillation steps. We already have observed the low ability of h M T to reconstruct the source embedding h i . Yet, what is the real impact in term of transfer learning of this insufficient convergence.

Let's study in details the columns dedicated to E G M T and E G unique in Table 3. We observe that if we are not fine-tuning the whole parameters of the model and only using the h M T and ĥG as features, the gap is significant on most of the source and target knowledge. Moreover, these results are correlated task by task to the reconstruction error observed on each h i . Nevertheless h M T allows to achieve far better performance than the CNN baseline, still underlining the benefit of a pre-training, even when it is a noisy one. We can also project ourselves in the case where we had much more knowledge to master. For instance, multiplying the number of target knowledge by 2 will imply a factor of time of 8 between E G unique and BCT.

Conclusions

This paper has introduced a novel approach to the problem of multi-source transfer learning, validated in the context of facial analysis. A unique and general model E G unique , obtained by merging six different source knowledge, can be transferred on 9 different target knowledge. Building this model is done through two successive training steps. First, an autoencoder G if trained to combine the hidden representations of the existing models into one single unifying embedding h G . Then, distilling this model to a light-weight student CNN allows to reduce the number of parameters and improve the adaptation ability of the model. The approach was experimentally validated by an exhaustive ablation study and performances on par with state-of-the-art methods on the 15 different knowledge, with a single simple model. On overall, the approach provides an efficient way to obtain universal models compressing the knowledge included in several existing models, without loss in performance, allowing an easy exploitation in real-world applications.

Figure 1 .

 1 Figure 1. Overview of the method. Step 1 is the training of G on Dunsup, to obtain an encoder E G of x in a compact embedding hG, gathering all sources of knowledge (represented by the extracted hj). Step 2 aims at reducing the number of parameters used to get hG by distilling the knowledge of 130 millions parameters into a much simpler encoder E G unique , leading to ĥG. In step 3 this lightweight encoder E G unique can then be plugged with a MLP C (tn,dn) and all parameters adapted to learn new knowledge K (tn,dn) .

Figure 2 .

 2 Figure 2. Impact of the size of hG on the reconstruction error of G and on the transfer for some of the target knowledge. Black cross is the optimal representation size for a given knowledge.

Figure 3 .

 3 Figure 3. Total training time in hours (logscale) with one P100 GPU for different methods, on Dunsup and on all target domains.

4. 6 .

 6 Training TimeFinally, we report from Figure 3 the training time spent on D unsup and on the tartget domains for different approaches. Even if the training on D unsup is relatively long (almost 70 hours), the transfer learning step is then really faster than training a model from scratch. Moreover compared to the brute-force baseline of selecting all possible combinations (BCT), the total training time of the E G unique (taking the training time of G into account) is divided by 4.

the learned knowledge into a single lightweight model. In this spirit, Romero et al. proposed a deep and thin student named Fit- net [56]. It learns from both last layer and hidden layers of the teacher and outperforms it. Aiming to ensure privacy of the learning dataset, Papernot et al. [52] proposed to use several teachers in a semi-supervised fashion.

Table 2 .

 2 Number of detected/undetected faces for each domain.

		# Detected Faces # Undetected Faces
	CelebA	202442	177
	UTKFace	24018	98
	FG-NET	1002	0
	SFEW	1732	37
	RAF	15330	9
	ShoulderPain 48391	0
	LFW	13233	0
	D unsup	4.12 M	0.02M

Table 3 .

 3 Performance of diverse end-to-end approach evaluated on the target knowledge. All approaches are described in the methods or detailed in the text. Note that for the specific case of LFW, two types of results are reported. For result between parenthesis, before evaluation on LFW, the network is first trained (or fine-tuned) on a subset of 100,000 faces from VGGFace2 to predict Identity. The other LFW results are obtained directly from embedding h. Acc. is Accuracy and ER is the Average Error Rate.

Table 4 .

 4 Contribution of G As described in Section 3, G takes as input the concatenation of the 6 embeddings (h i ), one per Performance of the original models E S

	Knowledge	Metric	Original E G	E G unique
	Expr-AffectNet	Accuracy 63.5	64.0 64.4
	Identity-LFW	Accuracy 99.65	99.1 99.42
	Gender-UTK	Accuracy 96.5	97.2 97.9
	Object-TinyImageNet Accuracy 76.2	71.8 56.5
	AgeR-FG	MAE	2.85	2.85 3.05
	Attrib-CelebA	Error rate 8.03	7.7	7.67
			(t S j ,d S

j ) , of E G and of E G unique on the original tasks.

Table 5 .

 5 Reconstruction normalized RMSE obtained on Dunsup test set by considered methods for each source knowledge embedding hi and on average.

	unique

Table 5

 5

	allow to evaluate E G M T

Table 6 .

 6 Performance of different variations of G and of selection approaches on the 9 targets knowledge (performance metrics are the same used in Table

	Knowledge		G		Selection
		Concat PCA	AE	BT	BCT
	AgeC-UTK	65.50	65.00 68.80 63.2	67.20
	Gender-UTK	96.92	96.7	97.20 96.50 97.10
	Expr-SFEW	45.70	32	52.10 52.20 53.1
	Expr-RAF	84.89	81.9	86.51 85.48 85.74
	Ethnic-UTK	86.65	62.5	89.20 83.40 86.20
	Identity-LFW 88.10	96.8	99.10 99.65 99.7
	AgeR-UTK	4.45	4.68	4.39	4.70	4.54
	Pain-UNBC	0.69	0.6	0.54	0.53	0.51
	AgeR-FG	3.22	3.18	2.85	2.85	2.85
	Attrib-CelebA 8.07	8.03	7.70	8.03	7.85