R Labbas

S Maingot

D Manceau

A Thorel

On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain

Keywords: 2010 Mathematics Subject Classification. 35A09, 35B65, 35C15, 35J30, 35R20, 47D06 Population dynamics, Abstract differential equations, UMD spaces, Bounded imaginary powers, Interpolation spaces, Maximal regularity

In this paper, we consider a generalized diffusion problem arising in population dynamics. To this end, we study a fourth order operational equation of elliptic type, with various boundary conditions. We show existence, uniqueness and regularity of a classical solution on a cylindrical domain under some necessary and sufficient conditions on the data. This elliptic problem is solved in L p (a, b; X), p ∈ (1, +∞), where (a, b) ⊂ R and X is a UMD Banach space. Our techniques use essentially the functional calculus and the semigroup theory.

Introduction

Many problems in biology may be described by partial differential equations. It means that the model is constructed by averaging the density of a population and keeping only time and space variables. This leads to study the population density, so-called population dynamics, governed by reaction-diffusion equations. The model considered in general is the following:

∂ t u = α∆u + f (u), in R + × Ω, (1.1)
where u(t, x) is the density of population at time t and position x, Ω is an open set of R d , d ≥ 1, and f is a non-linear growth interaction. The diffusion operator term ∆u is obtained from Fick's law. In 1981, D.S. Cohen and J.D. Murray [START_REF] Cohen | A generalized diffusion model for growth and dispersal in population[END_REF] derived a more complete model with a biharmonic term added to the harmonic one. This model has been obtained by studying the motility of cells, for which they noticed that the classical diffusion model was not sufficient. This model is:

∂ t u = k 1 ∆u -k 2 ∆ 2 u + f (u) in R + × Ω, (1.2)
where k 1 , k 2 ∈ R \ {0}. The biharmonic term ∆ 2 u represents the long range diffusion, whereas the harmonic term ∆u represents the short range diffusion. Another derivation of this model has been obtained in 1984 by F.L. Ochoa [START_REF] Ochoa | A generalized reaction-diffusion model for spatial structures formed by motile cells[END_REF] who studied the dynamics of (1.2) when f is a cubic function.

His work brings out the importance of the biharmonic term in the diffusion of the population. The classical study of nonlinear problems like (1.1) or (1.2) needs first to consider the associated linearized steady problem. This first step is essential to deduce maximal regularity results for the linearized unsteady problem. From these maximal regularity results, one can obtain the existence and uniqueness of the nonlinear problem by a fixed point theorem. In the case where Ω is an open set with C 4 -boundary, the result is well-known. For problems (1.1), we can find a detailed study in [START_REF] Prüss | Maximal regularity for evolution equations in L p -spaces[END_REF], see section 4, p. 13 for instance. For problems (1.2), results of existence and regularity are obtained in Hilbertian case in [START_REF] Novick-Cohen | On Cahn-Hilliard type equations[END_REF] and some other results were recently obtained in [START_REF] Cherfils | On a generalized Cahn-Hilliard equation with biological applications[END_REF].

In this paper, we study problem (1.2) We follow the work of [START_REF] Favini | A simplified approach in the study of elliptic differential equations in UMD spaces and new applications[END_REF] for second order problems that we adapt to fourth order problems. For this, we set k := k 1 /k 2 and we consider the Laplace operator A 0 on the Banach space L q (ω), q ∈ (1, +∞), with homogeneous Dirichlet boundary condition on ω. Thus, equation (1.3) can be formulated as the fourth order operational differential equation u (4) (x) + (2A 0 -kI)u (x) + (A 2 0 -kA 0 )u(x) = f (x), x ∈ (a, b),

where f ∈ L p (a, b; L q (ω)), p ∈ (1, +∞), with u(x) := u(x, •) and f (x) := f (x, •). Then, we will consider a generalization of this problem, that is

u (4) (x) + (2A -kI)u (x) + (A 2 -kA)u(x) = f (x), x ∈ (a, b), (1.4)
under various boundary conditions, with (A, D(A)) a BIP operator of angle θ ∈ (0, π) on a UMD space X (see section 2 below for the definitions of BIP operator and UMD spaces) and f ∈ L p (a, b; X). More precisely, we study (1.4) under one of the following boundary conditions:

u(a) = ϕ 1 , u(b) = ϕ 2 , u (a) = ϕ 3 , u (b) = ϕ 4 , (BC1) u (a) = ϕ 1 , u (b) = ϕ 2 , u (a) + Au(a) = ϕ 3 , u (b) + Au(b) = ϕ 4 , (BC2) u(a) = ϕ 1 , u(b) = ϕ 2 , u (a) = ϕ 3 , u (b) = ϕ 4 , (BC3) or u (a) = ϕ 1 , u (b) = ϕ 2 , u (a) = ϕ 3 , u (b) = ϕ 4 , (BC4)
where ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 ∈ X. We obtain (see Theorem 2.5) the existence and uniqueness of a classical solution u of (1.4)-(BCi), i = 1, 2, 3, 4, if and only if ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 are in some real interpolation spaces.

Let B 0 , • • • , B n-1 be linear operators in X, we recall that u is a classical solution (in L p (a, b; X)) of a differential equation u (n) (x) + n-1 j=0 B j u (j) (x) = f (x), x ∈ (a, b), (1.5)
if u is a solution of (1.5) with the following regularity

u ∈ W n,p (a, b; X) and x -→ B j u (j) (x) ∈ L p (a, b; X), j = 0, • • • , n -1. (1.6)
Similarly, u is a classical solution of problem (1.5)-(BC), where (BC) are some boundary conditions if u is a classical solution of (1.5) and satisfies (BC). The paper is organised as follows. In section 2, first we detail our assumptions on space X and operator A and we describe our results. First, we state the existence and the uniqueness of the solution of (1.4)-(BC2) in Theorem 2.2. This allows us to establish in Proposition 2.3, a general representation formula for classical solutions of (1.4). Then, we state a result about the regularity of a difference of two analytic semigroups (Theorem 2.4) which is an interesting result in itself and constitutes an important tool to prove the main result (Theorem 2.5). We then present Theorem 2.5, generalizing Theorem 2.2, which gives existence and uniqueness of a classical solution u of (1.4) under one of the boundary conditions (BC1), (BC2), (BC3) or (BC4). As a consequence of this theorem, we obtain the Corollary 2.7 which states existence and uniqueness of a solution u in W 4,p (Ω) of (1.3). In section 3, we prove Theorem 2.2 and Proposition 2.3. Section 4 is devoted to the proof of Theorem 2.4. Finally, in the last section, we prove Theorem 2.5.

Assumptions and statement of results

In all the paper, (X, • ) is a complex Banach space and L(X) stands for the space of bounded linear operators in X. For any linear operator T in X, we denote by D(T ), R(T ) the domain and the range of T while

ρ(T ) := {λ ∈ C : (λ I -T ) -1 ∈ L(X)} and σ(T ) := C \ ρ(T ),
denote the resolvent set and the spectrum of T . Recall (see, for instance, [START_REF] Haase | The functional calculus for sectorial Operators[END_REF]) that a closed linear operator

T is called sectorial of angle θ ∈ (0, π) if i) σ(T ) ⊂ S θ , ii) ∀ θ ∈ (θ, π), M (T, θ ) := sup λ(λ I -T ) -1 L(X) , λ ∈ C \ S θ < ∞,
where S θ := {z ∈ C : z = 0 and |arg z| < θ} .

(2.1)

It is known that any injective sectorial operator T admits imaginary powers T is , s ∈ R, but, in general, T is is not bounded (see [START_REF] Haase | The functional calculus for sectorial Operators[END_REF], p. 70). We say that a linear operator T in X has bounded imaginary powers, denoted by T ∈ BIP(X), if T is an injective sectorial operator such that

D(T ) ∩ R(T ) = X and ∀ s ∈ R, T is ∈ L(X).
Moreover, if T ∈ BIP(X), we set

θ T := inf θ ≥ 0 : ∃ C > 0, T is L(X) ≤ Ce |s|θ , ∀ s ∈ R ,
and we write T ∈ BIP(X, θ) if T ∈ BIP(X) and θ ≥ θ T . All the results stated in this paper used the well-known Dore-Venni Theorem (see [START_REF] Dore | On the closedness of the sum of two closed operators[END_REF] and its generalization [START_REF] Prüss | On operators with bounded imaginary powers in Banach spaces[END_REF]), which needs to consider a UMD space X. Recall that a Banach space X is a UMD space if and only if for some p ∈ (1, +∞) (and thus for all p) the Hilbert transform is bounded from L p (R, X) into itself ( [START_REF] Arendt | Vector-valued Laplace transforms and Cauchy problems[END_REF][START_REF] Bourgain | Some remarks on Banach spaces in which martingale difference sequences are unconditional[END_REF][START_REF] Burkholder | A geometrical characterisation of Banach spaces in which martingale difference sequences are unconditional[END_REF]).

In all the sequel, k ∈ R \ {0}, A denotes a linear operator in X and we assume:

(H 1 ) X is a UMD space, (H 2 ) 0 ∈ ρ(A), (H 3 ) -A ∈ BIP(X, θ) for some 0 < θ < π, (H 4 ) [k, +∞) ⊂ ρ(A).
For some of our results, we need a supplementary assumption:

(H 5 ) σ(A) ⊂ (-∞, 0) and ∀ θ ∈ (0, π), sup λ∈S θ λ(λ I -A) -1 L(X) < +∞,
which means that is a sectorial operator of any angle θ ∈ (0, π). Let us give some remarks on our assumptions.

Remark 2.1.

Assumption (H 4

) is relevant only if k < 0, since for k > 0, (H 2 ) and (H 3 ) imply (H 4 ).

2. The case k = 0 has been stated (with particular boundary conditions) in [START_REF] Favini | Transmission Problem for an Abstract Fourth-order Differential Equation of Elliptic Type in UMD Spaces[END_REF]. The study is different from our and requires a particular representation formula for the solution. 3. To solve (1.4) in the scalar case (with -A > 0), it is necessary to introduce the roots ± √ -A + k, ± √ -A of the characteristic equation

r 4 + (2A -k)r 2 + (A 2 -kA)r = 0,
this is why, in our operational case, we consider the operators

L := - √ -A + k I and M := - √ -A. (2.2)
Due to (H 3 ) and (H 4 ), -A and -A + k I are sectorial operators, so the existence of L and M is ensured (see for instance [START_REF] Haase | The functional calculus for sectorial Operators[END_REF]). 4. Applying Proposition 3.19 in [START_REF] Haase | The functional calculus for sectorial Operators[END_REF], we have D(L) = D(M ). Thus, for k, l ∈ N and l k

D(L k ) = D(M k ) = D(L l M k-l ).
5. Due to (H 3 ) and (H 4 ), we get that -A + k I is sectorial. So -A + k I ∈ BIP(X, θ) (see [START_REF] Arendt | Functional calculus, variational methods and Liapunov's theorem[END_REF], Theorem 2.3, p. 69), from which we deduce -L, -M ∈ BIP(X, θ/2),

(see [START_REF] Haase | The functional calculus for sectorial Operators[END_REF], Proposition 3.2.1, e), p. 71). Since 0 < θ/2 < π/2, L and M generate analytic semigroups (e xL ) x 0 and (e xM ) x 0 (see [START_REF] Prüss | On operators with bounded imaginary powers in Banach spaces[END_REF], Theorem 2, p. 437).

6. From (H 2 ) and (H 4 ), we deduce that 0 ∈ ρ(M ) ∩ ρ(L) (see [START_REF] Haase | The functional calculus for sectorial Operators[END_REF], Proposition 3.3.1, e), p. 62). Thus, assumptions (H 1 ), (H 2 ) and (H 3 ) lead us to apply Dore-Venni Theorem [START_REF] Dore | On the closedness of the sum of two closed operators[END_REF] to obtain 0 ∈ ρ(L + M ).

Definitions (2.2) imply that

∀ ψ ∈ D(L 2 ) = D(M 2 ), (L 2 -M 2 )ψ = k ψ. (2.3) Moreover L -M = (L -M )(L + M )(L + M ) -1 , so ∀ ψ ∈ D(L), (L -M )ψ = k(L + M ) -1 ψ. (2.4)
To determine the conditions on ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 ∈ X characterizing the existence of a unique classical solution of (1.4)-(BC1), (1.4)-(BC2), (1.4)-(BC3) and (1.4)-(BC4), we need to introduce interpolation spaces. To this end, we denote (Y, X) θ,q , 0 < θ < 1, 1 ≤ q ≤ ∞, the real interpolation spaces (see, for instance, [START_REF] Grisvard | Spazi di tracce e applicazioni[END_REF][START_REF] Lions | Sur une classe d'espaces d'interpolation[END_REF]) between a subspace Y ⊂ X and X. Moreover, we will use the following notation:

(D(A), X) k+θ,q := ψ ∈ D(A k ) : A k ψ ∈ (D(A), X) θ,q .
First, we solve problem (1.4)-(BC1) whose boundary conditions are well adapted to the equation and which can be easily treated by considering two second order problems. We obtain:

Theorem 2.2. Let f ∈ L p (a, b; X) with a, b ∈ R, a < b and p ∈ (1, ∞). Assume that (H 1 ), (H 2 ), ( H 3
) and (H 4 ) hold. Then, there exists a unique classical solution of (1.4)-(BC1) if and only if

ϕ 1 , ϕ 2 ∈ (D(A), X) 1+ 1 2p ,p and ϕ 3 , ϕ 4 ∈ (D(A), X) 1 2p ,p .
This unique classical solution is denoted by F Φ,f with Φ := (ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 ) and is explicitly described by

F Φ,f (x) = e (x-a)M Zϕ 1 + e (b-x)M Zϕ 2 - 1 2 e (x-a)M ZM -1 b a e (s-a)M v 0 (s) ds - 1 2 e (b-x)M ZM -1 b a e (b-s)M v 0 (s) ds + 1 2 M -1 x a e (x-s)M v 0 (s) ds + 1 2 M -1 b x e (s-x)M v 0 (s) ds -e (b-x)M e (b-a)M ϕ 1 -e (x-a)M e (b-a)M ϕ 2 + 1 2 e (x-a)M Ze (b-a)M M -1 b a e (b-s)M v 0 (s) ds + 1 2 e (b-x)M Ze (b-a)M M -1 b a e (s-a)M v 0 (s) ds, x ∈ [a, b],
where

v 0 (x) := e (x-a)L W (ϕ 3 + Aϕ 1 ) + e (b-x)L W (ϕ 4 + Aϕ 2 ) - 1 2 e (x-a)L W L -1 b a e (s-a)L f (s) ds - 1 2 e (b-x)L W L -1 b a e (b-s)L f (s) ds + 1 2 L -1 x a e (x-s)L f (s) ds + 1 2 L -1 b x e (s-x)L f (s) ds -e (b-x)L e (b-a)L (ϕ 3 + Aϕ 1 ) -e (x-a)L e (b-a)L (ϕ 4 + Aϕ 2 ) + 1 2 e (x-a)L W e (b-a)L L -1 b a e (b-s)L f (s) ds + 1 2 e (b-x)L W e (b-a)L L -1 b a e (s-a)L f (s) ds, x ∈ [a, b], Z := I -e 2(b-a)M -1 and W := I -e 2(b-a)L -1 .
The existence of Z and W are proved in virtue of (5.9) in subsection 5.1.

Under other boundary conditions, equation (1.4) is more complicated to study. We need two results which are important by themselves. The first one is a representation formula for the classical solution of equation (1.4).

Proposition 2.3. If u is a classical solution of (1.4), then there exist K i ∈ X, i = 1, 2, 3, 4, such that for any x ∈ [a, b] u(x) = e (x-a)M K 1 + e (b-x)M K 2 + e (x-a)L K 3 + e (b-x)L K 4 + F 0,f (x).
(2.5)

where F 0,f is defined in Theorem 2.2.

The second result concerns the regularity of the difference of two analytic semigroups stated below.

Theorem 2.4. Assume that (H 1 ), (H 2 ), (H 3 ) and (H 4 ) hold. For any ψ ∈ X and x ∈ [a, b], we set u ψ (x) := e (x-a)L -e (x-a)M ψ and v ψ (x) := e (b-x)L -e (b-x)M ψ.

Then, we have

1. u ψ ∈ W 2,p (a, b; X) ∩ L p a, b; D(M 2 ) and u ψ ∈ L p (a, b; D(M )) . 2. u ψ ∈ W 4,p (a, b; X) ∩ L p a, b; D(M 4 ) if and only if ψ ∈ (D (M ) , X) 1+ 1 p ,p . Furthermore, in this case, u ψ ∈ L p a, b; D(M 2 ) .

Statements 1. and 2. hold true if we replace

u ψ by v ψ (it suffices to write v ψ (x) = u ψ (b + a -x)). Note that if ψ ∈ (D(M ), X) 3+ 1 p ,p then e (•-a)L ψ, e (•-a)M ψ ∈ W 4,p (a, b; X) ∩ L p a, b; D(M 4 ) ,
(see Remark 4.2 below), from which we deduce that

u ψ ∈ W 4,p (a, b; X) ∩ L p a, b; D(M 4 ), but our result is more precise since (D(M ), X) 3+ 1 p ,p ⊂ (D (M ) , X) 1+ 1 p ,p .
We are now in position to state our main result which generalizes Theorem 2.2: Theorem 2.5. Let f ∈ L p (a, b; X) with a < b, a, b ∈ R and p ∈ (1, ∞). Assume that (H 1 ), (H 2 ), (H 3 ) and (H 4 ) hold. Then 1. There exists a unique classical solution u of (1.4)-(BC1) if and only if

ϕ 1 , ϕ 2 ∈ (D(A), X) 1+ 1 2p ,p and ϕ 3 , ϕ 4 ∈ (D(A), X) 1 2p ,p .
2. There exists a unique classical solution u of (1.4)-(BC2) if and only if

ϕ 1 , ϕ 2 ∈ (D(A), X) 1+ 1 2 + 1 2p ,p and ϕ 3 , ϕ 4 ∈ (D(A), X) 1 2p ,p .
If in addition, we assume (H 5 ), then 3. There exists a unique classical solution u of (1.4)-(BC3) if and only if

ϕ 1 , ϕ 2 ∈ (D(A), X) 1+ 1 2p ,p and ϕ 3 , ϕ 4 ∈ (D(A), X) 1+ 1 2 + 1 2p ,p .
4. There exists a unique classical solution u of (1.4)-(BC4) if and only if

ϕ 1 , ϕ 2 ∈ (D(A), X) 1+ 1 2 + 1 2p ,p and ϕ 3 , ϕ 4 ∈ (D(A), X) 1 2p ,p .
Remark 2.6.

1. The result for the boundary conditions (BC1) can be directly generalized to the following boundary conditions:

u(a) = ϕ 1 , u(b) = ϕ 2 , u (a) + Au(a) = ϕ 3 , u (b) + Au(b) = ϕ 4 . (BC5)
2. In the same way, one can obtain similar results for mixed boundary conditions.

From Theorem 2.5, we deduce regularity results for equation (1.3) under some particular boundary conditions (other conditions can be considered). Let us explicit Theorem 2.5, statement 1 for instance, when we consider problem (1.

3)-(BC1) with

A := A 0 . Corollary 2.7. Consider a cylindrical domain Ω := (a, b) × ω of R d , where a, b ∈ R, a < b and ω is a bounded domain of R d-1 with C 2 -boundary. Let f ∈ L p (Ω), p ∈ (1, +∞); let k 1 , k 2 ∈ R \ {0} such that k 1 > k 2 C ω , where C ω > 0 is the Poincaré constant in ω. Then, there exists a unique solution u ∈ W 4,p (Ω) of                          -k 1 ∆u(x, y) + k 2 ∆ 2 u(x, y) = f (x, y), (x, y) ∈ Ω, u(x, y) = ∆ y u(x, y) = 0, (x, y) ∈ (a, b) × ∂ω, u(a, y) = ϕ 1 , y ∈ ω, u(b, y) = ϕ 2 , y ∈ ω, ∆ y u(a, y) = ϕ 3 , y ∈ ω, ∆ y u(b, y) = ϕ 4 , y ∈ ω, (2.6)
if and only if

ϕ 1 , ϕ 2 ∈ W 2,p (ω) ∩ W 1,p 0 (ω) and ∆ϕ 1 , ∆ϕ 2 , ϕ 3 , ϕ 4 ∈ W 2,p (ω) ∩ W 1,p 0 (ω), L p (ω) 1 2p ,p . (2.7)
In view to make explicit the above result, let us consider

ω := (c, d) ⊂ R. Then X := L p (c, d) and D(A 0 ) := W 2,p (c, d) ∩ W 1,p 0 (c, d), (A 0 ψ)(y) := ψ (y).
With these notations, problem (2.6) becomes (1.4) with the boundary conditions (BC1). From [START_REF] Rubio De Francia | Martingale and integral transforms of Banach space valued functions[END_REF] (Proposition 23), X satisfies (H 1 ) and from [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] (Theorem 19.15 and Lemma 9.17), A 0 satisfies (H 2 ). Theorem C of [START_REF] Prüss | Imaginary powers of elliptic second order differential operators in L p -spaces[END_REF] ensured that (H 3 ) is satisfied and, from [START_REF] Kato | Perturbation theory for linear operators[END_REF], we have 2 . Since k > C ω , we obtain [k, +∞) ⊂ ρ(A 0 ). Thus, assumption (H 4 ) is satisfied. Finally, all the assumptions of Theorem 2.5 are satisfied. We have

σ(A 0 ) = - n 2 π 2 (d -c) 2 : n ∈ N \ {0} . So σ(A 0 ) ⊂ (-∞, C ω ], where C ω := - π 2 (d -c)
(D(A 0 ), X) 1 2p ,p = W 2,p (ω) ∩ W 1,p 0 (ω), L p (ω) 1 2p ,p = ψ ∈ B 2-1 p p (c, d) : ψ(c) = ψ(d) = 0 , where B 2-1 p p (0, 1
) is a Besov space described in [START_REF] Grisvard | Spazi di tracce e applicazioni[END_REF], p. 680.

Note that since 2

-1 p > 1 p , we have C([c, d]) → B 2-1 p p (c, d) with continuous embedding and, since 2 -1 p > 1 p is never integer, B 2-1 p p (c, d) = W 2-1 p ,p (c, d). Therefore, (D(A 0 ), X) 1 2p ,p = ψ ∈ W 2-1 p ,p (c, d) : ψ(c) = ψ(d) = 0 , and
(D(A 0 ), X) 1+ 1 2p ,p = ψ ∈ D(A 0 ) : A 0 ψ ∈ (D(A 0 ), X) 1 2p ,p = ψ ∈ W 2,p (c, d) ∩ W 1,p 0 (c, d) : ψ ∈ (D(A 0 ), X) 1 2p ,p = ψ ∈ W 2,p (c, d) ∩ W 1,p 0 (c, d) : ψ (c) = ψ (d) = 0 and ψ ∈ W 2-1 p ,p (c, d) = ψ ∈ W 2,p (c, d) : ψ ∈ W 2-1 p ,p (c, d) and ψ(c) = ψ(d) = ψ (c) = ψ(d) = 0 .
Now, the classical solution u of

u (4) (x) + (2A 0 -k I)u (x) + (A 2 0 -k A 0 )u(x) = f (x), a.e. x ∈ (a, b), satisfies        u ∈ W 4,p (a, b; L p (c, d)) x -→ A 0 u (x) ∈ L p (a, b; L p (c, d)) x -→ A 2 0 u(x) ∈ L p (a, b; L p (c, d)), and                                        u(a, y) = ϕ 1 (y), y ∈ (c, d) u(b, y) = ϕ 2 (y), y ∈ (c, d) ∂ 2 u ∂y 2 (a, y) = ϕ 3 (y), y ∈ (c, d) ∂ 2 u ∂y 2 (b, y) = ϕ 4 (y), y ∈ (c, d) u(x, c) = ∂ 2 u ∂y 2 (x, c) = 0, x ∈ (a, b) u(x, d) = ∂ 2 u ∂y 2 (x, d) = 0, x ∈ (a, b). Since              u ∈ W 2,p (a, b; L p (c, d)) u ∈ L p (a, b; W 2,p (c, d) ∩ W 1,p 0 (c, d)) u(x, c) = u(x, d) = 0 ϕ 1 (c) = ϕ 1 (d) = 0,
it is possible to use the Sobolev extension Theorem to R 2 and the Mihlin Theorem (see [START_REF] Mihlin | On the multipliers of Fourier integrals[END_REF]). We deduce that

u ∈ W 2,p ((a, b) × (c, d)) = W 2,p (Ω).
By reiterating the same arguments to the other regularities, we obtain

u ∈ W 4,p ((a, b) × (c, d)) = W 4,p (Ω).
In the same way, we can obtain this result with ω in R d-1 . Equation (1.3) can be studied with other boundary conditions taking into account the results of Theorem 2.5. We can also obtain anisotropic results by considering f ∈ L p (a, b; L q (ω)), p, q ∈ (1, +∞).

Proofs of preliminary results

First, we recall some classical trace results:

Remark 3.1.
1. Let T be a closed linear operator in X and k ∈ N \ {0}. From [START_REF] Grisvard | Spazi di tracce e applicazioni[END_REF] p. 677 and [START_REF] Lions | Sur une classe d'espaces d'interpolation[END_REF] p. 39, if

u ∈ W n,p (a, b; X) ∩ L p (a, b; D(T k )),
where n ∈ N \ {0}, then for any j ∈ N satisfying the Poulsen condition 0 < 1 p + j < n and c ∈ {a, b}, we have

u (j) (c) ∈ (D(T k ), X) j n + 1 np ,p . (3.1)
Moreover, recall that, if we denote (X, D(T k )) 1-θ,p := (D(T k ), X) θ,p , where k ∈ N \ {0} and θ ∈ (0, 1) such that kθ / ∈ N, from reiteration theorem [START_REF] Grisvard | Spazi di tracce e applicazioni[END_REF][START_REF] Lions | Sur une classe d'espaces d'interpolation[END_REF], we have

(D(T k ), X) θ,p = (X, D(T k )) 1-θ,p = (X, D(T )) k-kθ,p . (3.2) 2. Let u ∈ W 4,p (a, b; X) ∩ L p (a, b; D(A 2 )
). Then, from (3.1), for any c ∈ {a, b}, we obtain

u(c) ∈ (D(A 2 ), X) 1 4p ,p , u (c) ∈ (D(A 2 ), X) 1 4 + 1 4p ,p and u (c) ∈ (D(A 2 ), X) 1 2 + 1 4p ,p .
From (3.2), we deduce, for any c ∈ {a, b},

u(c) ∈ (D(A), X) 1+ 1 2p ,p , u (c) ∈ (D(A), X) 1+ 1 2 + 1 2p ,p
and u (c) ∈ (D(A), X) 1 2p ,p . (3.3)

Proof of Theorem 2.2

Let ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 ∈ X be such that

ϕ 1 , ϕ 2 ∈ (D(A), X) 1+ 1 2p ,p and ϕ 3 , ϕ 4 ∈ (D(A), X) 1 2p ,p . (3.4)
From (3.4), there exists (see e.g. [START_REF] Favini | A simplified approach in the study of elliptic differential equations in UMD spaces and new applications[END_REF], Theorem 4, p. 200) a unique classical solution

v 0 ∈ W 2,p (a, b; X) ∩ L p (a, b; D(A)) , of    v (x) + (A -k I)v(x) = f (x), a. e. x ∈ (a, b) v(a) = ϕ 3 + Aϕ 1 , v(b) = ϕ 4 + Aϕ 2 . (3.5) Then, since ϕ 1 , ϕ 2 ∈ (D(A), X) 1+ 1 2p ,p ⊂ (D(A), X) 1 2p
,p , there exists a unique classical solution

u 0 ∈ W 2,p (a, b; X) ∩ L p (a, b; D (A)) , of    u (x) + Au(x) = v 0 (x), a. e. x ∈ (a, b) u(a) = ϕ 1 , u(b) = ϕ 2 . (3.6)
A simple computation shows that u 0 satisfies (1. Then A -1 w 0 is a classical solution of (3.6). So, by uniqueness of the solution, we obtain

u 0 = A -1 w 0 ∈ L p a, b; D(A 2 ) .
Moreover, from (3.6), we have

u 0 = v 0 -Au 0 = v 0 -w 0 ∈ W 2,p (a, b; X) ∩ L p (a, b; D(A)) .
Hence u 0 is a classical solution of (1.4)-(BC1). Uniqueness is straightforward since if u is a classical solution of (1.4)-(BC1) then v := u + Au is a classical solution of (3.5). By uniqueness, v = v 0 , so u is a classical solution of (3.6) and again, by uniqueness, u = u 0 . Conversely, if there exists a classical solution of (1.4)-(BC1), then (3.4) holds (see (3.3)).

Proof of Proposition 2.3 (Representation formula)

Let u be a classical solution of (1.4). By Theorem 2.2, we can consider the classical solution F 0,f of (1.4)-(BC1); i.e. which satisfies

F 0,f (a) = F 0,f (b) = F 0,f (a) = F 0,f (b) = 0. (3.7)
We set u hom := u -F 0,f . Then, u hom is a classical solution of

u (4) (x) + (2A -kI)u (x) + (A 2 -kA)u(x) = 0, a. e. x ∈ (a, b).
So it remains to show the existence of constants K i , for i = 1, 2, 3, 4 such that

u hom (x) = e (x-a)M K 1 + e (b-x)M K 2 + e (x-a)L K 3 + e (b-x)L K 4 , a. e. x ∈ (a, b). (3.8) It implies that u (4) (x) = (L 2 + M 2 )u (x) -L 2 M 2 u(x), a. e. x ∈ (a, b). (3.9)
Since u and F 0,f are classical solutions of (1.4), u hom is also a classical solution, so

u hom ∈ W 4,p (a, b; X) ∩ L p a, b; D(M 4 ) and u hom ∈ L p a, b; D(M 2 ) .
Thus, for almost every x ∈ (a, b), we can set

v(x) := k -1 L 2 u hom (x) -k -1 u hom (x) ∈ L p a, b; D(M 2 ) w(x) := -k -1 M 2 u hom (x) + k -1 u hom (x) ∈ L p a, b; D(L 2 ) .
(3.10) From (2.3), we obtain

v(x) + w(x) = k -1 (L 2 -M 2 )u hom (x) = u hom (x), a. e. x ∈ (a, b). (3.11)
Moreover, from (3.9), for almost every x ∈ (a, b), we deduce that

L -2 v (x) = k -1 u hom (x) -k -1 L -2 u (4) hom (x) = k -1 u hom (x) -k -1 L -2 (L 2 + M 2 )u (x) -L 2 M 2 u(x) = M 2 L -2 v(x),
and in the same way

M -2 w (x) = -k -1 u hom (x) + k -1 M -2 u (4) hom (x) = L 2 M -2 w(x).
Thus, v and w are, respectively, solutions of

v (x) -M 2 v(x) = 0 and w (x) -L 2 w(x) = 0, a. e. x ∈ (a, b).
The previous functional equalities are set in L p (a, b; X) and make sense due to (3.9). From [START_REF] Favini | A simplified approach in the study of elliptic differential equations in UMD spaces and new applications[END_REF], we obtain

v(x) = e (x-a)M K 1 + e (b-x)M K 2 and w(x) = e (x-a)L K 3 + e (b-x)L K 4 , x ∈ (a, b). (3.12)
Then, from (3.11) and (3.12), we deduce (3.8).

Proof of Theorem 2.4

The proof is essentially based on analytic semigroups results (see [START_REF] Lunardi | Analytic semigroups and optimal regularity in parabolic problems[END_REF]) and interpolation spaces (see, for instance, [START_REF] Da Prato | Somme d'opérateurs linéaires et équations différentielles opérationnelles[END_REF] p. 381-386 and [START_REF] Triebel | Interpolation theory, function Spaces, differential Operators[END_REF], Theorem p. 96) and on a corollary of the well-known Dore-Venni Theorem [START_REF] Dore | On the closedness of the sum of two closed operators[END_REF]. We recall below these results as lemmas for the readers convenience.

Lemma 4.1 ([7, 26]). Let ψ ∈ X and T be a generator of an analytic semigroup in X. Then, for any n ∈ N \ {0}, the two next properties are equivalent:

1. x → T n e (x-a)T ψ ∈ L p (a, b; X), 2. ψ ∈ (D(T n ), X) 1 np ,p . Remark 4.2.
Under the assumptions of Lemma 4.1, from (3.2) it follows that, for any n ∈ N \ {0}, the two next properties are equivalent:

1. x → e (x-a)T ψ ∈ W n,p (a, b; X) ∩ L p (a, b; D(T n )), 2. ψ ∈ (D(T ), X) n-1+ 1 p ,p .
x → e (x-a)T ∈ C ∞ ((a, b]; X) ∩ C 0 ([a, b]; X), (4.1)
and for any x > a,

e (x-a)L ψ ∈ D(L ∞ ) = D(M ∞ ) and e (x-a)M ψ ∈ D(M ∞ ), (4.2)
where

D(M ∞ ) = D(L ∞ ) := k 0 D(M k ) = k 0 D(L k ). (4.3)
From (4.1) and (4.2), we deduce

∀ x > a, u ψ ∈ C ∞ ((a, b]; X) ∩ C 0 ([a, b]; X) and u(x) ∈ D(M ∞ ).
Then, from (2.4), we obtain

∀ x > a, u ψ (x) = Le (x-a)L -M e (x-a)M ψ = M e (x-a)L -e (x-a)M ψ + (L -M ) e (x-a)L ψ = M u ψ (x) + k (L + M ) -1 e (x-a)L ψ. Thus, u ψ ∈ C 1 ((a, b]; X) ∩ C 0 ([a, b]; X) is a solution of the Cauchy problem u (x) = M u (x) + k (L + M ) -1 e (x-a)L ψ, a.e. x ∈ (a, b], u(a) = 0.
Hence, (see e.g. [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF], Corollary 2.2 p. 106), u is given by the variation of constant formula:

∀ x ∈ (a, b], u ψ (x) = x a e (x-s)M k (L + M ) -1 e (s-a)L ψ ds = k (L + M ) -1 M -1 M
x a e (x-s)M e (s-a)L ψ ds .

(4.4) Since s -→ e (s-a)L ψ ∈ C 0 ([a, b]; X) ⊂ L p (a, b; X) ,
from Lemma 4.3, we have

g ψ : x -→ M x a e (x-s)M e (s-a)L ψ ds ∈ L p (a, b; X) .
Then, since M 2 (L + M ) -1 M -1 ∈ L(X), we deduce that

x -→ M 2 u ψ (x) = kM 2 (L + M ) -1 M -1 g ψ (x) ∈ L p (a, b; X) .
We also have

u ψ ∈ W 1,p (a, b; X) because u ψ ∈ C 1 ((a, b]; X) ∩ C 0 ([a, b]; X) and u ψ (•) = M u ψ (•) + k (L + M ) -1 e (•-a)L ψ ∈ L p (a, b; X) . Furthermore, x -→ M u ψ (x) = M 2 u ψ (x) + k M (L + M ) -1 e (x-a)L ψ ∈ L p (a, b; X) . so u ψ ∈ L p (a, b; D(M )). For a. e. x ∈ (a, b], since e (x-a)L ψ ∈ D(M 2 ), from (2.
3) we obtain

u ψ (x) = L 2 e (x-a)L -M 2 e (x-a)M ψ = M 2 e (x-a)L -e (x-a)M ψ + L 2 -M 2 e (x-a)L ψ = M 2 u ψ (x) + k e (x-a)L ψ. (4.5)
Thus

u ψ ∈ L p (a, b; X), hence u ψ ∈ W 2,p (a, b; X).
2. First step. Assume that ψ ∈ (D(M ), X) 1+ 1 p ,p . We set ϕ := M ψ ∈ (D(M ), X) 1 p ,p . Then, we have M u ψ = u ϕ . Thus, equality (4.4), for a. e. x ∈ (a, b), yields

M 3 u ϕ (x) = k M (L + M ) -1 M 2 M -1 M x a e (x-s)M e (s-a)L ϕ ds = k M (L + M ) -1 M L -1 M x a e (x-s)M Le (s-a)L ϕ ds . (4.6) Since ϕ = M ψ ∈ (D(M ), X) 1 p ,p , from Lemma 4.1, we have s -→ Le (s-a)L ϕ ∈ L p (a, b; X) .
Then, from Lemma 4.3, we deduce

x -→ M

x a e (x-s)M Le (s-a)L ϕ ds ∈ L p (a, b; X) .

Hence, from (4.6) and the fact that

M (L + M ) -1 M L -1 ∈ L(X), we obtain x -→ M 4 u ψ (x) = M 3 u ϕ ∈ L p (a, b; X) . (4.7)
It follows

u ψ ∈ L p a, b; D(M 4 ) and u ϕ ∈ L p a, b; D(M 3 ) . (4.8)
Moreover, from (4.5) and (2.4), for x > a, we have

u ψ (x) = M 2 u ψ (x) + k Le (x-a)L ψ = M 2 Le (x-a)L -M e (x-a)M ψ + k Le (x-a)L ψ = M 3 e (x-a)L -e (x-a)M ψ + M 2 (L -M )e (x-a)L ψ + k Le (x-a)L ψ = M 3 u ψ (x) + k M 2 (L + M ) -1 e (x-a)L ψ + k Le (x-a)L ψ = M 3 u ψ (x) + k M 2 (L + M ) -1 + L e (x-a)L ψ = M 2 u ϕ (x) + k M 2 (L + M ) -1 + L M -1 e (x-a)L ϕ. Since M 2 (L + M ) -1 + L M -1 ∈ L(X)
, from (4.8) and (4.1), we deduce

u ψ ∈ L p (a, b; X) . (4.9)
5.1 Proof of statement 2. (Boundary conditions (BC2))

Assume (H 1 ) ∼ (H 4 ) and

ϕ 1 , ϕ 2 ∈ (D(A), X) 1+ 1 2 + 1 2p ,p and ϕ 3 , ϕ 4 ∈ (D(A), X) 1 2p ,p . (5.1)
First we prove the uniqueness of the solution of (1.4)-(BC2) by determining the constants K i , i = 1, 2, 3, 4, of the representation formula (2.5). Then, we show that the formula obtained constitutes a classical solution.

First step: Uniqueness. If u is a classical solution of (1.4)-(BC2), then from Proposition 2.3, u satisfies (2.5). In order to obtain a simplified model, we set

α 1 := K 1 -K 2 2 , α 2 := K 3 -K 4 2 , α 3 := K 1 + K 2 2 and α 4 := K 3 + K 4 2 . (5.2)
Then for a.e. x ∈ (a, b), u is given by

u(x) = e (x-a)M -e (b-x)M α 1 + e (x-a)L -e (b-x)L α 2 + e (x-a)M + e (b-x)M α 3 + e (x-a)L + e (b-x)L α 4 + F 0,f (x) .
(

Since F 0,f satisfies (3.7), the boundary conditions (BC2) applied to M -2 u(x) for a.e. x ∈ (a, b), imply the following relations

I + e (b-a)M M -1 α 1 + I + e (b-a)L LM -2 α 2 + I -e (b-a)M M -1 α 3 + I -e (b-a)L LM -2 α 4 = M -2 ϕ 1 -F 0,f (a) , (5.4
)

e (b-a)M + I M -1 α 1 + e (b-a)L + I LM -2 α 2 + e (b-a)M -I M -1 α 3 + e (b-a)L -I LM -2 α 4 = M -2 ϕ 2 -F 0,f (b) , (5.5)
I -e (b-a)L L 2 -M 2 M -2 α 2 + I + e (b-a)L L 2 -M 2 M -2 α 4 = M -2 ϕ 3 , (5.6
)

e (b-a)L -I L 2 -M 2 M -2 α 2 + e (b-a)L + I L 2 -M 2 M -2 α 4 = M -2 ϕ 4 .
(5.7)

Note that we have considered M -2 u(x) for a.e. x ∈ (a, b), because we do not know if α i ∈ D(M ), i = 1, 2, 3, 4. Using (2.3) and summing (5.4) with (5.5) and (5.6) with (5.7), we obtain the abstract system

                             I + e (b-a)M M -1 α 1 + I + e (b-a)L LM -2 α 2 + I -e (b-a)M M -1 α 3 + I -e (b-a)L LM -2 α 4 = M -2 ϕ 1 -F 0,f (a) , 2 I + e (b-a)M M -1 α 1 + 2 I + e (b-a)L LM -2 α 2 = M -2 ϕ 1 + ϕ 2 -F 0,f (a) -F 0,f (b) , k I -e (b-a)L M -2 α 2 + k I + e (b-a)L M -2 α 4 = M -2 ϕ 3 , 2k I + e (b-a)L M -2 α 4 = M -2 (ϕ 3 + ϕ 4 ) .
This leads to the system

                     M I -e (b-a)M M -2 α 3 + L I -e (b-a)L M -2 α 4 = M -2 φ2 M I + e (b-a)M M -2 α 1 + L I + e (b-a)L M -2 α 2 = M -2 φ1 , k I -e (b-a)L α 2 = ϕ 3 -ϕ 4 2 , k I + e (b-a)L α 4 = ϕ 3 + ϕ 4 2 ,
where

φ1 := ϕ 1 + ϕ 2 -F 0,f (a) -F 0,f (b) 2 and φ2 := ϕ 1 -ϕ 2 -F 0,f (a) + F 0,f (b) 2 .
(5.8)

For T invertible with bounded inverse, from [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] p. 70, for any n ∈ N, we have:

∃ δ > 0, ∃ M > 1 : e n(b-a)T L(X)
M e -n(b-a)δ .

(5.9)

Therefore, there exists n 0 ∈ N \ {0} such that e n(b-a)n0T L(X) < 1. So I ± e n(b-a)T is invertible with bounded inverse. Then, for T := L or M , we deduce

                         α 1 = I + e (b-a)M -1 M -1 φ1 -LM -1 I + e (b-a)L α 2 α 2 = k -1 I -e (b-a)L -1 ϕ 3 -ϕ 4 2 , α 3 = I -e (b-a)M -1 M -1 φ2 -LM -1 I -e (b-a)L α 4 , α 4 = k -1 I + e (b-a)L -1 ϕ 3 + ϕ 4 2 .
(5.10)

It follows that u is uniquely determined by (5.3), (5.8) and (5.10).

Second step: Existence. Consider this uniquely determined u and set

K1 := K 1 + K 3 = α 1 + α 2 + α 3 + α 4 and K2 := K 2 + K 4 = α 3 + α 4 -α 1 -α 2 ,
thus, for any x ∈ (a, b), u(x) = e (x-a)M K1 + e (b-x)M K2 + e (x-a)L -e (x-a)M K 3 + e (b-x)L -e (b-x)M K 4 + F 0,f (x).

(5.11)

Then, from Remark 4.2 and Theorem 2.4, it suffices to show that K1 , K2 ∈ (D(M ), X) 3+ 1 p ,p and K 3 , K 4 ∈ (D(M ), X) 1+ 1 p ,p . From Lemma 5.1, for T := L or M , we have

I ± e 2(b-a)T -1 = I + R ± , where R ± (X) ⊂ D(M ∞ ) and R ± T = T R ± .
Hence, from (5.10), there exists

R i ∈ D(M ∞ ), i = 1, 2, 3, 4, such that                        α 1 = 1 2 M -1 2 φ1 -k -1 L(ϕ 3 -ϕ 4 ) + R 1 α 2 = k -1 ϕ 3 -ϕ 4 2 + R 2 , α 3 = 1 2 M -1 2 φ2 -k -1 L(ϕ 3 + ϕ 4 ) + R 3 , α 4 = k -1 ϕ 3 + ϕ 4 2 + R 4 .
From (5.2) and (5.8), we obtain

             K 1 = M -1 ϕ 1 -F 0,f (a) -k -1 LM -1 ϕ 3 + R1 K 2 = -M -1 ϕ 2 -F 0,f (b) -k -1 LM -1 ϕ 4 + R2 , K 3 = k -1 ϕ 3 + R3 , K 4 = k -1 ϕ 4 + R4 ,
where Ri ∈ D(M ∞ ), i = 1, 2, 3, 4. From (3.2), we have

(D(A), X) 1+ 1 2 + 1 2p ,p = (D(M 2 ), X) 1+ 1 2 + 1 2p ,p = (D(M ), X) 2+ 1 p ,p (5.12)
and (D(A), X) 1 2p ,p = (X, D(M )) 1+1-1 p ,p = (D(M ), X) 1+ 1 p ,p .

(5.13) Since F 0,f (a) ∈ (D(A), X) 1+ 1 2 + 1 2p ,p , from (2.4), (5.1), (5.12) and (5.13), we have K1

= M -1 ϕ 1 -F 0,f (a) + k -1 (I -LM -1 )ϕ 3 + R1 + R3 = M -1 ϕ 1 -F 0,f (a) -(L + M ) -1 M -1 ϕ 3 + R1 + R3 ∈ (D(M ), X) 3+ 1
p ,p , and, in the same way, K2 ∈ (D(M ), X) 3+ 1 p ,p . Moreover,

K 3 = k -1 ϕ 3 + R3 ∈ (D(M ), X) 1+ 1 p ,p and K 4 = k -1 ϕ 4 + R4 ∈ (D(M ), X) 1+ 1
p ,p , which gives the result.

Proof of statement 3. (Boundary conditions (BC3))

In all the sequel, we assume (H 1 ) ∼ (H 5 ). For statements 1. and 2., the representation formula is easily obtained by taking into account the boundary conditions. But for statement 3. (and also statement 4.), to build the representation, we need the invertibility of some determinant operators.

More precisely, we have to prove that U and V given by

     U := I -e (b-a)(L+M ) -k -1 (L + M ) 2 e (b-a)M -e (b-a)L , V := I -e (b-a)(L+M ) + k -1 (L + M ) 2 e (b-a)M -e (b-a)L , (5.14)
are invertible with bounded inverse. This will be a consequence of a functional calculus result.

To state this result, we need some notations and technical lemmas. For θ ∈ (0, π), we denote by H(S θ ) the space of holomorphic functions on S θ (defined by (2.1)) with values in C. Moreover, we consider the following subspace of H(S θ ):

E ∞ (S θ ) := f ∈ H(S θ ) : f = O(|z| -s ) (|z| → +∞) for some s > 0 .
In other words, E ∞ (S θ ) is the space of polynomial decreasing holomorphic functions at ∞. Let T be an invertible sectorial operator of angle θ T ∈ (0, π). If f ∈ E ∞ (S θ ), with θ ∈ (θ T , π), then we can define, by functional calculus, f (T ) ∈ L(X), see [START_REF] Haase | The functional calculus for sectorial Operators[END_REF], p. 45. Set, for z ∈ C

f k (z) := 1 -e -c( √ z+ √ z+k) -k -1 ( √ z + √ z + k) 2 e -c √ z -e -c √ z+k , and g k (z) := 1 -e -c( √ z+ √ z+k) + k -1 ( √ z + √ z + k) 2 e -c √ z -e -c √ z+k ,
where c := b -a > 0. Note that, we can write formally U = f k (-A) and V = g k (-A).

Lemma 5.2. For any k > 0, f k and g k do not vanish in R + \ {0}.

Proof. First, we consider for τ > 0, ϕ(τ

) := τ 1 + e -τ
1 -e -τ , then ϕ is a strictly increasing function on

R + \ {0} since ϕ (τ ) = 2 sinh(τ ) -τ e -τ (e τ -1) 2 > 0, τ > 0. Let x > 0. Since f k (x) ≤ g k (x), it suffices to show that f k (x) > 0. Set s := c √ x and t := c √ x + k, then 0 < s < t and f k (x) = 1 -e -(s+t) - t + s t -s e -s -e -t = 1 t -s (t -s) 1 -e -(s+t) -(t + s) e -s -e -t = 1 t -s t 1 -e -s 1 + e -t -s 1 -e -t 1 + e -s = (1 -e -s ) (1 -e -t ) t -s [ϕ (t) -ϕ (s)] > 0,
which leads to the result.

Lemma 5.3. Let P be a sectorial operator in X of angle θ, for all θ ∈ (0, π). Let f ∈ H(S θ ), for some θ ∈ (0, π), be such that

(i) 1 -f ∈ E ∞ (S θ ), (ii) f = 0 on R + \ {0}.
Then, f (P ) ∈ L(X), is invertible with bounded inverse.

Proof. Since we have 1 f = 1 + h, where h := 1 -f f , it suffices to show that h ∈ E ∞ (S θ0 ), for some θ 0 ∈ (0, π), to obtain the result. Indeed, since σ(P ) ⊂ S θ0 , from [START_REF] Haase | The functional calculus for sectorial Operators[END_REF], pp. 28 and 45, in this case we can set: From (i), there exists R > 0 such that, for all z ∈ S θ with |z| > R, we have |1 -f (z)| < 1/2. Then f (z) is never equal to 0 if z ∈ S θ and |z| > R, so h ∈ H S θ ∩ C \ B(0, R) .

(5.15)

Let θ 1 ∈ (0, θ). In the compact set S θ1 ∩ B(0, R), the holomorphic function f has only a finite number of zeros, which from (ii), are not in (0, R]. So there exists θ 0 ∈ (0, θ 1 ) such that f = 0 in S θ0 ∩ B(0, R). Thus h ∈ H S θ0 ∩ B(0, R) and from (5.15), we deduce h ∈ H (S θ0 ). Proof. We first consider the case k > 0. Due to (H 5 ), P := -A satisfies the assumptions of Lemma 5.3. Moreover, it is clear that for a given θ ∈ (0, π), f k ∈ H(S θ ) and 1 -f k ∈ E ∞ (S θ ). Furthermore, from Lemma 5.2, f k does not vanish on R + \ {0}. So applying Lemma 5.3, we get that U = f k (-A) is invertible. The invertibility of V = g k (-A) is obtained in the same way. Now, assume that k < 0. The linear operator -A+k I is a sectorial operator from Remark 2.1 5. and as in the first case, setting l := -k > 0, we deduce that U = f l (-A + k I) and V = g l (-A + k I) are invertible with bounded inverse. Now, we prove Theorem 2.5 for the boundary conditions (BC3). The proof is divided in two steps. First we prove the uniqueness of the solution of (1.4)-(BC3) by determining the constants K i , i = 1, 2, 3, 4, of the representation formula (2.5). Then, we show that the formula obtained is a classical solution. (5.17)

  in the linear steady case on a bounded cylindrical open set Ω := (a, b) × ω of R d , with ω an open bounded set of R d-1 with C 2 -boundary and f ∈ L p (Ω), p ∈ (1, +∞), i.e. k 2 ∆ 2 u -k 1 ∆u = f on Ω. (1.3)

  4)-(BC1). It remains to show that u 0 satisfies (1.6). Let w 0 ∈ W 2,p (a, b; X) ∩ L p (a, b; D(A)) be the unique classical solution of    w (x) + Aw(x) = Av 0 (x), a.e. x ∈ (a, b) w(a) = Aϕ 1 , w(b) = Aϕ 2 .

Lemma 4 . 3 (eRemark 4 . 4 .

 4344 [START_REF] Dore | On the closedness of the sum of two closed operators[END_REF]). Let T ∈ BIP (X, θ) with θ ∈ (0, π/2), and g ∈ L p (a, b; X). Then, for almost every x ∈ (a, b), we have x a e (x-s)T g(s) dt ∈ D(T ) and b x e (s-x)T g(s) dt ∈ D(T ). (x-s)T g(s) dt ∈ L p (a, b; X) and x → T b x e (s-x)T g(s) dt ∈ L p (a, b; X). The assumption that X is a UMD space is not necessary in Lemma 4.1, but is essential in Lemma 4.3. Now, we state the proof of Theorem 2.4. 1. Let ψ ∈ X. Recall that ∀ x > a, u ψ (x) := e (x-a)L -e (x-a)M ψ. Since, for T := L or M , (e xT ) x≥0 is an analytic semigroup, we have (see, for instance, [17], Proposition 2.1.1, p. 35)

1 f

 1 (P ) = I + h(P ) ∈ L(X),and the invertibility of f (P ) is obtained by writing

Since 1 -

 1 f ∈ E ∞ (S θ0 ), there exists R 0 > 0 such that, if z ∈ S θ0 and |z| > R 0 , then |1 -f | < C |z| -α where α > 0, C > 0. Moreover, since for all z ∈ S θ0 with |z| > max(R, R 0 ) we have |1 -f (z)| < 1/2, we deduce that |f (z)| > 1/2. So, for all z ∈ S θ0 with |z| > max(R, R 0 ) we obtain |h(z)| ≤ 2 C |z| -α . It follows that h ∈ E ∞ (S θ0 ).Proposition 5.4. U and V , defined by (5.14), are invertible with bounded inverse.

First step: Uniqueness. Assume that ϕ 1 , ϕ 2 ∈ 16 ) 2 +

 12162 (D(A), X) 1+ 1 2p ,p and ϕ 3 , ϕ 4 ∈ (D(A), X)If u is a classical solution of (1.4)-(BC3) then, as in section 5.1, u satisfies (5.3). Since F 0,f satisfies (3.7), the boundary conditions (BC3) applied to M -1 u(•) imply the following relationsI -e (b-a)M M -1 α 1 + I -e (b-a)L M -1 α I + e (b-a)M M -1 α 3 + I + e (b-a)L M -1 α 4 = M -1 ϕ 1 ,

* This research is supported by CIFRE contract 2014/1307 with Qualiom Eco company.

Furthermore, for x > a, we have

From Lemma 4.1, since ϕ = M ψ ∈ (D(M ), X) 1 p ,p = (D(L), X) 1 p ,p , we obtain

x → Le (x-a)L ϕ ∈ L p (a, b; X).

Then, since M 2 (L + M ) -1 L -1 + M 2 (L + M ) -1 M -1 + LM -1 ∈ L(X) and from (4.8), we deduce u

ψ ∈ L p (a, b; X) . (4.10)

From statement 1. (see above), (4.8), (4.9) and (4.10), we deduce the result.

Second step. Assume that u ψ ∈ W 4,p (a, b; X) ∩ L p a, b; D(M 4 ) . From (3.1), with k = n = 4 and j = 2, we have u ψ (a) ∈ (D(M 4 ), X) 1 4p + 1 2 ,p , then from (3.2), we deduce u ψ (a) ∈ (D(M ), X) 1+ 1 p ,p . From (4.5), for x > a, we have M -2 u ψ (x) = u ψ (x) + k M -2 e (x-a)L ψ.

Passing to the limit when x tends to a, we obtain u ψ (a) = k ψ and thus

Finally, since u ψ (x) = M -1 u ϕ (x), from (4.5), we obtain

Applying Lemma 4.1 with ϕ := M ψ ∈ (D(M ), X) 1 p ,p , the fact that M L -1 ∈ L(X) and (4.8), the previous equality yields u ψ ∈ L p (a, b; D(M 2 )).

Proof of Theorem 2.5

Statement 1 has been treated in section 2.2. We give here the proofs for the others boundary conditions. Note that from Remark 3.1, in Theorem 2.5, we only have to prove the reverse implications.

For each kind of boundary conditions, the proof of Theorem 2.5 is divided in two steps. First, from the representation formula obtained in section 2.3, we show uniqueness of a classical solution. Then, we state the existence of this solution. At this end, we establish the following technical result.

Proof. We set

Then, obviously (I + V ) -1 = I -W and W (X) ⊂ V (X) ⊂ D(T ). Moreover, for ψ ∈ D(T )

)

(5.20) Summing (5.17) with (5.18) and (5. [START_REF] Novick-Cohen | On Cahn-Hilliard type equations[END_REF]) with (5.20), we obtain the system

From which we deduce the two second order systems

where

(5.23)

Then, the unique determination of the constants α i , i = 1, 2, 3, 4, is equivalent to show that the matrices Λ 1 and Λ 2 of the systems (5.21) and (5.22) have an invertible determinant. From (2.4), we have

and in the same way

where U and V are given by (5.14). Since U and V are invertible with bounded inverse (see Proposition 5.4), from (5.21) and (5.22), we deduce

(5.24)

Note that, since F 0,f ∈ W 4,p (a, b; X) ∩ L p (a, b; D(M 2 )), from Remark 3.1 we have

This combined with (5.16) implies that F 0,f (a), F 0,f (b), ϕ i ∈ D(M 2 ), i = 1, 2, 3, 4. Thus all the previous equalities are well defined. This shows that the eventual classical solution of (1.4)-(BC3) is characterized by (5.3), (5.23) and (5.24).

Second step: Existence. It suffices to prove that u, determined in the first step, satisfies (1.6). From Lemma 5.1, we have

Set T := L or M . Then, from (4.2), since T e xT = e xT T on D(T ), for any x ≥ 0, Lemma 5.1 implies

This combined with (4.2) implies that there exist

Then, from (5.2) and (5.23), we obtain

with Ri ∈ D(M ∞ ), i = 1, 2, 3, 4. We set

thus, from (5.11), for any x ∈ (a, b),

From (3.2), we have (D(A), X) 1+ 1 2p ,p = (D(M ), X) 3+ 1 p ,p .

(5.27) Thus, since ϕ 1 , ϕ 2 ∈ (D(A), X) 1+ 1 2p ,p , from (2.3), we have

and, in the same way,

Then, from Lemma 4.1 and Remark 4.2, we obtain that

(5.28) Thus, from (5.25), (5.16) and (5.27) , M ϕ 1 , ϕ 3 , F 0,f (a) ∈ (D(M ), X) 2+ 1 p ,p . From (5.26), we deduce

(5.29)

By the same arguments, v K4 : x → e (x-a)L -e (x-a)M K 4 satisfies

(5.30)

Since F 0,f satisfies (1.6), from (5.11), (5.28), (5.29) and (5.30), we deduce that u satisfies (1.6) and so is a classical solution of (1.4)-(BC3).

Proof of statement 4. (Boundary conditions (BC4))

We proceed as in the proof of the previous statement. We only point out the differences between the two proofs.

First step: Uniqueness. Assume that

and ϕ 3 , ϕ 4 ∈ (D(A), X) 1 2p ,p .

(5.31)

If u is a classical solution of (1.4)-(BC4), then u is given by (5.3). Since F 0,f satisfies (3.7), the boundary conditions (BC4) applied to M -1 u (a), M -1 u (b), M -2 u (a), M -2 u (b) and the same computations as in the proof of subsection 5.2 lead to the two second order systems

and

where where U and V are given by (5.14). Since U and V are invertible in L(X) (see Proposition 5.4), from (5.32) and (5.33), we deduce

(5.35) Then, from (5.2) and (5.35) we obtain the uniqueness of the solution of (1.4)-(BC4).

Second step: Existence. Let u, determined in the first step, then u is also given by (5.11). Then, it suffices to show that K1 , K2 ∈ (D(M ), X) 3+ 1 p ,p and K 3 , K 4 ∈ (D(M ), X) 1+ 1 p ,p . By the same arguments as in the proof of subsection 5.2, there exist

Then, from (5.2) and (5.34), we obtain

with Ri ∈ D(M ∞ ), i = 1, 2, 3, 4. Then, combining (5.12) and (5.13), since ϕ 1 , F 0,f (a) ∈ (D(A), X) 1+ 1 2 + 1 2p ,p and ϕ 3 ∈ (D(A), X) 1 2p ,p , from (2.3), we have Moreover, since (L + M )ϕ 1 , (L + M )F 0,f (a), ϕ 3 ∈ (D(M ), X) 2+ 1 p ,p , from (5.36), we deduce

and, in the same way, K 4 ∈ (D(M ), X) 1+ 1 p ,p , which leads to the result.


Marc Mezzarobba
email: marc@mezzarobba.net

Rigorous Multiple-Precision Evaluation of D-Finite Functions in SageMath

Keywords:

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Many special functions satisfy differential equations

p r (x)f (r) (x) + • • • + p 1 (x)f ′ (x) + p 0 (x)f (x) = 0 (1)
whose coefficients p i depend polynomially on the variable x. In virtually all cases, such special functions can be defined by complementing [START_REF] Apéry | Irrationalité de ζ(2) et ζ(3)[END_REF] either with simple initial values f (0), f ′ (0), . . . or with constraints on the asymptotic behavior of f (x) as x approaches a singular point. For example, the error function satisfies erf ′′ (x) + 2x erf ′ (x) = 0, erf(0) = 0, erf

′ (0) = 2 √ π , (2)
while the modified Bessel function K 0 is the solution of

xK ′′ 0 (x) + K ′ 0 (x) -xy(x) = 0 s.t. K 0 (x) = -log(x/2) -γ + O x→0 (x). (3)
This observation has led to the idea of developing algorithms that deal with these functions in a uniform way, using the ODE (1) as a data structure [START_REF] Lanczos | Applied analysis[END_REF][START_REF] Benoit | The dynamic dictionary of mathematical functions (DDMF)[END_REF]. In this context, solutions of linear ODEs with polynomial coefficients are called D-finite (or holonomic) functions. These names originate from combinatorics, where D-finite power series arise naturally as generating functions [START_REF] Stanley | Enumerative combinatorics[END_REF][START_REF] Flajolet | Analytic Combinatorics[END_REF]. While classical special functions typically satisfy ODEs of order 2 to 4 with simple polynomial coefficients, D-finite generating functions are usually given by more general and sometimes very large equations. Other notable sources of general D-finite functions include mathematical physics and number theory. Explicit examples in theoretical works on meromorphic ODEs also tend to have polynomial coefficients.

Packages for manipulating D-finite power series symbolically exist for major computer algebra systems, and include in particular gfun [START_REF] Salvy | Gfun: A Maple package for the manipulation of generating and holonomic functions in one variable[END_REF] and the rest of the algolib bundle for Maple, GeneratingFunctions [START_REF] Mallinger | Algorithmic manipulation and transformations of univariate holonomic functions and sequences[END_REF] and the RISCErgoSum suite for Mathematica, and ore_algebra [START_REF] Kauers | Ore polynomials in Sage[END_REF] for SageMath [START_REF]SageMath mathematics software[END_REF].

We extended ore_algebra with features dedicated to D-finite analytic functions such as special functions, starting with numerical evaluation algorithms. The need for such features is most evident in the context of special functions, but they turn out to be useful in a wide range of applications, as we will see from the examples in the sequel. We refer to our implementation as oa.analytic, after the name of the subpackage of ore_algebra where most of the code resides. The present paper introduces the main features of oa.analytic, with example applications. The reader is invited to consult the package's documentation for more information. We further refer to Hille [START_REF] Hille | Ordinary differential equations in the complex domain[END_REF] for more background on complex analytic ODEs, and to Kauers [START_REF] Kauers | The holonomic toolkit[END_REF] for an overview of the field of D-finite functions in computer algebra. Also note that oa.analytic is intended to supersede NumGfun [START_REF] Mezzarobba | NumGfun: a package for numerical and analytic computation with D-finite functions[END_REF], a Maple package with similar features previously developed by the author 1 .

Since the main feature of oa.analytic is to compute numerical values of solutions of differential equations, it is, at its core, a numerical ODE solver. However, it is limited to linear ODEs with polynomial coefficients, and differs from typical ODE solvers in a number of other respects. First of all, the results it computes come with rigorous error bounds, making it a "validated" ODE solver. It works in arbitrary precision, and implements specific algorithms to make evaluations at very high precisions feasible. Last but not least, it offers extensive support for connection problems between regular singular points of differential operators. These distinctive features all rely on symbolic-numeric algorithms that make use of specific properties of D-finite functions.

The development of oa.analytic is still very much a work in progress. At the time of writing, there is no formal release of ore_algebra containing the analytic subpackage yet. The latest development version of oa.analytic can be found at http://marc.mezzarobba.net/code/ore_algebra-analytic. It is distributed under the GNU General Public Licence, version 2 or later. Comments, bug reports and feature requests are welcome.

Differential Operators in ore_algebra

In ore_algebra, differential operators are represented algebraically as Ore polynomials. More specifically, the operators we are interested in are elements of the 1 Many of the examples in this article are adapted from earlier work on NumGfun [START_REF] Mezzarobba | Autour de l'évaluation numérique des fonctions D-finies[END_REF].

ring, denoted K[x] D x , of polynomials over K[x] for some number field K ⊂ C in a variable D x subject to the commutation rule D x x = xD x + 1 (which expresses the equality (xf (x)

) ′ = xf ′ (x) + f (x)). Elements of K[x] D x act on differentiable functions in a natural way, namely by x • f = (ξ → ξf (ξ)) and D x • f = f ′ ,
and can thus be identified with differential operators.

The first thing to do in order to use ore_algebra in a Sage session is to install it in the Python search path and import it. The rings K[x] and K[x] D x and their respective generators x and D x can then be created using standard Sage syntax as follows:

sage: from ore_algebra import OreAlgebra sage: Pols.<x> = PolynomialRing(QQ); Dops.<Dx> = OreAlgebra(Pols)

Note that ore_algebra supports general Ore polynomials, but basic arithmetic with differential operators in one variable is all we are going to need here.

After these commands, the Python variable Dx contains an object of type "Ore polynomial" that satisfies

sage: Dx*x x*Dx + 1
Most features of ore_algebra, including those presented in this article, are available as methods of Ore polynomial objects. Some functions of the analytic subpackage can also be called directly in advanced usage.

Numerical Analytic Continuation

Let L = p r (x)D r x + • • • + p 0 (x) ∈ K[x] D
x be a differential operator, and assume that the origin is an ordinary point of L, i.e. that p r (0) = 0. By the classical Cauchy existence theorem [START_REF] Hille | Ordinary differential equations in the complex domain[END_REF], the equation L • f = 0 then admits an r-dimensional space of complex analytic solutions defined in a neighborhood of 0 and characterized by r initial values at 0. The numerical_solution() method of differential operators computes values of D-finite functions defined in this way. Perhaps the simplest example is the computation of e = exp(1) as the value at 1 of the solution of f ′ -f = 0 such that f (0) = 1: sage: (Dx-1).numerical_solution(ini= [START_REF] Apéry | Irrationalité de ζ(2) et ζ(3)[END_REF], path=[0,1], eps=1e-40) [2.71828182845904523536028747135266249775725 +/-7.53e-42]

The last argument, eps, provides an indication of the accuracy the user is interested in. The method outputs a "mid-rad interval", or ball, consisting of a multiple-precision center and a low-precision radius. In normal circumstances, the diameter of this ball will be of the order of magnitude of eps, but this is not guaranteed to be the case. In contrast, barring any bug, the mathematical result is guaranteed to lie within the bounds of the ball.

Evaluations on the complex plane are also supported. Based on (2), we may for instance evaluate the error function as follows:

sage: (Dxˆ2 + 2*x*Dx).numerical_solution([0, 2/sqrt(pi)], [0, i]) [+/-2.91e-25] + [1.6504257587975429 +/-2.70e-17]*I
Allowed evaluation points and initial values include rational and floating-point numbers, algebraic numbers, and constant symbolic expressions.

The reader may have noticed that the argument used to specify the evaluation point is called path. The main reason why numerical_solution() does not simply take an evaluation point is that in general, solutions of differential equations with singular points are not single-valued analytic functions. What diffop.numerical_solution(ini, path) really computes is the analytic continuation along path of the solution of diffop defined in the neighborhood of the starting point by the initial conditions ini. Consider for example the operator L = xD x + D x , obtained by differentiating the inhomogeneous equation xy ′ (x) = 1. A basis of solutions of L is {1, log(x)}, corresponding respectively to the initial values (y(1), y ′ (1)) = (1, 0) and (y(1), y ′ (1)) = (0, 1). Analytic continuation of the latter to x = -1 along a path that passes above the singular point x = 0 corresponds to the standard determination of the complex logarithm, while a path that passes below the origin yields another branch: More generally, it is possible to compute the matrix that maps a vector

[f (x 0 ), f ′ (x 0 ), 1 2 f ′′ (x 0 ), . . . , 1 (r-1)! f (r-1) (x 0 )] (4)
of initial conditions2 at x 0 to the vector [f (x 1 ), f ′ (x 1 ), . . .] of "initial" conditions at some other point x 1 that define the analytic continuation of the same solution f along a given path. Following van der Hoeven [START_REF] Van Der Hoeven | Fast evaluation of holonomic functions[END_REF], we call this matrix the transition matrix along that path. The analytic continuation algorithm behind these functions is based on work by Chudnovsky and Chudnovsky [START_REF] Chudnovsky | Computer algebra in the service of mathematical physics and number theory[END_REF] with some later improvements [START_REF] Van Der Hoeven | Fast evaluation of holonomic functions[END_REF][START_REF] Mezzarobba | Autour de l'évaluation numérique des fonctions D-finies[END_REF]. In the language of numerical methods for ODEs, it is a high-order Taylor series method with a step size of the order of the local radius of convergence of solutions. The computation of the transition matrix for each step boils down to that of sums of power series whose coefficients are generated by linear recurrence relations. Two algorithms for this task are currently implemented: a naïve iterative summation and a simple binary splitting method. In both cases, the entries of a given column are actually computed all at once, by working with truncated power series in a manner similar to direct-mode automatic differentiation. Once the terms of the sum have decreased below the target accuracy, the algorithm of [START_REF] Mezzarobba | Truncation bounds for D-finite series[END_REF] is used to get a rigorous bound on the tail of the series. Both the computation of the numerical result and that of error bounds rely on Arb [START_REF] Johansson | Arb: a C library for ball arithmetic[END_REF] for controlling the round-off errors in basic arithmetic operations.

Regular Singular Connection Problems

Singular connection problems are similar to the analytic continuation problems of the previous section, except that one or both ends of the analytic continuation path are singular points of the operator (points where the leading coefficient vanishes). At a singular point, the Cauchy theorem does not apply, and solutions of the operator may diverge or display a branch point.

Our implementation supports the situation where the singularities involved are regular singular points. A singular point x 0 of L ∈ K[x] D x is regular when L admits a full basis of solutions each of the form

(x -x 0 ) α g 0 (x) + g 1 (x) log(x -x 0 ) + • • • + g p (x) (log(x -x 0 )) p p! ( 5
)
where g 0 , . . . , g p are analytic functions and α is an algebraic number.

A basic example is that of the modified Bessel equation ( 3), which possesses a regular singular point at x = 0 and whose standard solutions are defined by their asymptotic behavior there. Starting from (3), we can compute K 0 (1) by:

sage: ini = [-1, log(2)-euler_gamma] sage: (x*Dxˆ2 + Dx -x).numerical_solution(ini, [0, 1], 1e-10) [0.42102443824 +/-2.11e-12]
This command is to be understood as follows. It is classical that, when x 0 is a regular singular point of L, a solution of L is characterized by the coefficients of the monomials (x -x 0 ) ν log(x -x 0 ) k /k! in its expansion (5) about x 0 , where ν ranges over the roots of the so-called indicial polynomial of L at x 0 and k is less than the multiplicity of the root ν. The initial conditions accepted by numerical_solution() in this case are the coefficients of these monomials (ordered in a certain way, basically by asymptotic dominance as x → x 0 ). This definition generalizes the convention (4) used at ordinary points. We call the corresponding basis of solutions the canonical local basis4 at x 0 . A convenience method provides the distinguished monomials. In the case of Equation ( 3), we see from the asymptotic condition that the coefficient of x 0 is log 2 -γ and that of x 0 log x is -1.

In calls to numerical_transition_matrix(), both ends of the path can be regular singular points. The resulting matrix expresses the change of basis between the respective canonical local bases.

Example 2 (Face-Centered Cubic Lattices). Koutschan [START_REF] Koutschan | Lattice Green's functions of the higher-dimensional face-centered cubic lattices[END_REF] shows that the lattice Green's function P (x) of the four-dimensional face-centered cubic lattice is annihilated by the operator As it turns out, the space of analytic solutions at the origin has dimension 1, and all other solutions involve logarithms. It is easy to see from the definition of P that P (0) = 1, so that the function we are interested in is exactly the last element of the local canonical basis. At x = 1, there is a three-dimensional subspace of solutions that tend to zero, spanned by the last three basis elements, while the first basis function has a finite limit. Thus the value of P there is the upper right entry of the transition matrix from 0 to 1, namely: This operator has three regular singular points: 0, α = ( √ 2 + 1) This implies that the radius of convergence of a(x) is in {α, α -1 , ∞}, so that a n must be of one of the forms α -n+o(n) , α n+o(n) and e o(n) as n → ∞. The same holds for b n . A key step of Apéry's proof is to show that

a n = α n+o(n) and b n = α n+o(n) , but b n -ζ(3)a n = α -n+o(n) .
Analytic solutions at x = 0 are determined by the first two coefficients of their Taylor expansions. At x = α -1 , the canonical local basis takes the form (f 0 , f 1 , f 2 , f 3 ) where f 0 , f 2 , f 3 are analytic and f To prove that a n = α n+o (n) , it is enough to check that the coefficient c 1 in the decomposition a(x) = c i f i (x) is nonzero.

1 (x) ∼ √ x -α -
sage: mat = diffop.numerical_transition_matrix([0, sing [START_REF] Apéry | Irrationalité de ζ(2) et ζ(3)[END_REF]], 1e-40)

From the initial values a 0 = 1, a 1 = 5, we see that c 1 is equal to This result agrees with the closed form a n ∼ √ α(2π √ 2) -3/2 n -3/2 α n from [START_REF] Hirschhorn | Estimating the Apéry numbers[END_REF]. In the same way, using the initial terms b 0 = 0, b 1 = 6 of (b n ), we can compute the coefficient of f 1 in the decomposition of b(x) and check that the ratio of the two values we obtained is ζ(3). Acknowledgements. This work was prompted by discussions with Fredrik Johansson. I am also indebted to the Sage developers, in particular Vincent Delecroix, Jeroen Demeyer and Clemens Heuberger, who reviewed or helped writing a number of Sage patches vital to this project, and to Christoph Lauter for comments on a version of this text.

  sage: (x*Dxˆ2 + Dx).numerical_solution([0, 1], [1, i, -1]) [+/-5.11e-18] + [3.1415926535897932 +/-4.38e-17]*I sage: (x*Dxˆ2 + Dx).numerical_solution([0, 1], [1, -i, -1]) [+/-5.11e-18] + [-3.1415926535897932 +/-4.38e-17]*I

Example 1 .

 1 Transition matrices along loops that turn around a singular point exactly once are called monodromy matrices. The first step of van Enckevort and van Straten's experimental study of differential equations of Calabi-Yau type[START_REF] Van Enckevort | Monodromy calculations of fourth order equations of Calabi-Yau type[END_REF] consists in computing high-precision approximations of all monodromy matrices starting from a common base point. The numerical_transition_matrix() method makes this computation straightforward. In the case of the singularity at 1 of the operator studied in Example 1 from their article, we obtain 3 : sage: diffop = ((x*Dx)ˆ4 -x*(65*(x*Dx)ˆ4+130*(x*Dx)ˆ3 + 105*(x*Dx)ˆ2 + 40*x*Dx + 6) + 4*xˆ2*(4*x*Dx+3)*(x*Dx+1)ˆ2*(4*x*Dx+5)) sage: mat = diffop.numerical_transition_matrix([1/2,1-i/2,3/2,1+i/2,1/2], 1e-1000) sage: mat[0,0] # 990 digits omitted [1.000...000 +/-6.45e-998] + [0.733...6314 +/-8.99e-998]*I

  sage: (x*Dxˆ2 + Dx -x).local_basis_monomials(0) [log(x), 1]

  sage: diffop = ((-1+x)*xˆ3*(2+x)*(3+x)*(6+x)*(8+x)*(4+ 3*x)ˆ2*Dxˆ4 + 2*xˆ2*(4+3*x)*(-3456-2304*x+3676*xˆ2+4920*xˆ3+ 2079*xˆ4+356*xˆ5+21*xˆ6)*Dxˆ3 + 6*x*(-5376-5248*x+11080*xˆ2+25286*xˆ3+19898*xˆ4+ 7432*xˆ5+1286*xˆ6+81*xˆ7)*Dxˆ2 + 12*(-384+224*x+3716*xˆ2+7633*xˆ3+6734*xˆ4+2939*xˆ5 + 604*xˆ6+45*xˆ7)*Dx + 12*x*(256+632*x+702*xˆ2+382*xˆ3+98*xˆ4+9*xˆ5)) Koutschan then evaluates P (1) in order to obtain the return probability of the lattice. Both x = 0 and x = 1 are regular singular points of the operator. Let us examine the possible behaviors of solutions at these two points: sage: diffop.local_basis_monomials(0) [1/6*log(x)ˆ3, 1/2*log(x)ˆ2, log(x), 1] sage: diffop.local_basis_monomials(1) [1, (x -1)*log(x -1), x -1, (x -1)ˆ2]

Example 3 (.

 3 sage: diffop.numerical_transition_matrix([0, 1], 1e-60)[0,3] [1.1058437979212047601829954708859 +/-4.90e-32] + [+/-4.55e-41]*I Asymptotics of Apéry Numbers). In his celebrated proof that ζ(3) is irrational [1], Apéry introduces the two sequences The generating series a(x) = ∞ n=0 a n x n and b(x) = ∞ n=0 b n x n are solutions of the following operator: sage: diffop = (xˆ2*(xˆ2-34*x+1)*Dxˆ4 + 5*x*(2*xˆ2-51*x+1)*Dxˆ3 + (25*xˆ2-418*x+4)*Dxˆ2 + (15*x-117)*Dx + 1)

sage: mat[ 1 , 2 ]

 12 + 5*mat[START_REF] Apéry | Irrationalité de ζ(2) et ζ(3)[END_REF][START_REF] Chudnovsky | Computer algebra in the service of mathematical physics and number theory[END_REF] [4.546376247522844600239593024915161553303 +/-9.85e-41]*I With a little more work, using the method of singularity analysis[START_REF] Flajolet | Analytic Combinatorics[END_REF], this is actually enough to prove that a n = n -3/2 α n (λ + O(1/n)) where λ is given by sage: CBF(-i/(2*sqrt(pi)))*sqrt(sing[START_REF] Apéry | Irrationalité de ζ(2) et ζ(3)[END_REF])*(mat[1,2] + 5*mat[START_REF] Apéry | Irrationalité de ζ(2) et ζ(3)[END_REF][START_REF] Chudnovsky | Computer algebra in the service of mathematical physics and number theory[END_REF]) [0.220043767112643 +/-2.06e-16]

sage: 6 *

 6 mat[START_REF] Apéry | Irrationalité de ζ(2) et ζ(3)[END_REF][START_REF] Chudnovsky | Computer algebra in the service of mathematical physics and number theory[END_REF]/(mat[1,2] + 5*mat[1,3]) [1.2020569031595942853997381615114499907650 +/-6.08e-41]

  1 .

	sage: diffop.local_basis_monomials(0)
	[1/2*log(x)ˆ2, log(x), 1, x]
	sage: diffop.local_basis_monomials(s[1])
	[1, sqrt(x -0.02943725152285942?), x -0.02943725152285942?,
	(x -0.02943725152285942?)ˆ2]

Note the inverse factorials in our definition of initial conditions. This convention also applies to the ini argument of numerical_solution(). It is convenient in view of the generalization to regular singular points discussed in the next section.

In situations such as this one where the singular point is regular singular, it is also possible, and typically more efficient, to express the monodromy matrix using the primitives described in the next section, as suggested by van der Hoeven[START_REF] Van Der Hoeven | Fast evaluation of holonomic functions near and in regular singularities[END_REF].

Note that there is another common choice, the Frobenius basis.

Supported in part by ANR grant ANR-14-CE25-0018-01 (FastRelax).


Raphal Tackx

Jean-Loup Guillaume

Fabien Tarissan

Revealing intricate properties of communities in the bipartite structure of online social networks

Keywords: online social networks, complex networks, bipartite graphs, overlapping, communities

Many real-world networks based on human activities exhibit a bipartite structure. Although bipartite graphs seem appropriate to analyse and model their properties, it has been shown that standard metrics fail to reproduce intricate patterns observed in real networks. In particular, the overlapping of the neighbourhood of communities is difficult to capture precisely. In this work, we tackle this issue by analysing the structure of 4 real-world networks coming from online social activities. We first analyse their structure using standard metrics. Surprisingly, the clustering coefficient turns out to be less relevant than the redundancy coefficient to account for overlapping patterns. We then propose new metrics, namely the dispersion and the monopoly coefficients, and show that they help refining the study of bipartite overlaps. Finally, we compare the results obtained on real networks with the ones obtained on random bipartite models. This shows that the patterns captured by the redundancy and the dispersion coefficients are strongly related to the real nature of the observed overlaps.

I. INTRODUCTION

Many real-world networks -also referred to as complex networks -lend themselves to the use of graphs for analysing and modelling their structure. Usually, the vertices of the graph stand for the nodes of the network and the edges between vertices stand for (possible) interactions between nodes of the network. This approach have proven to be useful to identify non trivial properties of the structure of networks in very different contexts. We can cite for instance computer networks (like the Internet, peer-to-peer systems, the web) [START_REF] Le Fessant | Clustering in peer-to-peer file sharing workloads[END_REF], [START_REF] Prieur | The stength of weak cooperation: A case study on flickr[END_REF], biological networks (protein-protein interaction networks, metabolic processes) [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF], [START_REF] Vzquez | The topological relationship between the large-scale attributes and local interaction patterns of complex networks[END_REF], social networks (friendship networks, co-publication networks) [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF], [START_REF] Newman | Random graph models of social networks[END_REF], legal networks [START_REF] Tarissan | The network of the international criminal court decisions as a complex system[END_REF], linguistics [START_REF] Cancho | The small world of human language[END_REF], economy [START_REF] Battiston | Statistical properties of corporate board and director networks[END_REF], etc. However, even though this representation based on graphs is relevant, it is often too simplistic to account for the inherent complexity of most networks. Indeed, if we consider for instance an actor-movie network [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF], [START_REF] Newman | Random graphs with arbitrary degree distributions[END_REF] which relates actors to movies, or a co-publication network [START_REF] Newman | Random graphs with arbitrary degree distributions[END_REF], [START_REF] Newman | Random graph models of social networks[END_REF] which connects authors to publications, it is then natural to distinguish formally the two kind of vertices as two disjoint sets: actors on one side, movies on the other and authors on one side and publications on the other. This observation has led the scientific community to use bipartite graphs to represent such a constrained structure. Indeed in bipartite, also referred as 2mode networks, the set of nodes is composed of two disjoint parts, and ⊥ (for example movies and actors), such that edges only relate vertices of different sets. It has been shown that this fundamental object is reliable both for analysing [START_REF] Latapy | Basic notions for the analysis of large two-mode networks[END_REF], [START_REF] Tumminello | Statistically validated networks in bipartite complex systems[END_REF], [START_REF] Ahn | Flavor network and the principles of food pairing[END_REF] and modeling [START_REF] Guillaume | Bipartite graphs as models of complex networks[END_REF], [START_REF] Tarissan | Towards a bipartite graph modeling of the internet topology[END_REF] the structure of real-world networks.

In a recent paper [START_REF] Tarissan | Towards a bipartite graph modeling of the internet topology[END_REF], this approach has been exploited in order to propose for the first time a bipartite model of the Internet topology. The model is general enough to be relevant for any network presenting a bipartite structure as it only takes as parameters the degree sequence of the nodes in the two disjoint sets. The paper showed that despite the simplicity of the model, non trivial realistic properties of the Internet topology emerge naturally. But it also showed that the model fails in reproducing the overlapping observed in the bipartite structure. This overlapping arises when two ⊥ nodes are related to several nodes (two authors publishing more than one paper together for instance). The observation drawn from the study conducted on Internet topology has been extended to a wide variety of different networks in [START_REF] Tarissan | Comparing overlapping properties of real bipartite networks[END_REF], showing that overlaps are common in complex bipartite networks. Understanding the structure of the overlaps in bipartite networks is thus a key concern in the processes of improving our ability to analyse and model such networks.

The purpose of the present paper is precisely to address the question of the real nature of overlapping patterns observed in real networks. To do so, we analyse the structure of 4 different real-world networks coming from online social activities. This choice is motivated by the fact that such networks exhibit a genuine bipartite structure since online platforms usually propose to the users the ability to gather into communities. To that regard, these networks are naturally represented by bipartite graphs with ⊥ nodes defining the users and nodes depicting the communities they belong to. We then analyse the overlapping patterns through the use of both standard metrics and new metrics proposed in this paper.

The remaining of the paper is organised as follow. First, we detail the different datasets on which we conducted the study and give proper definition of the different metrics we used to analyse their structure (Section II). Then we present the results obtained and address in particular the variety of overlaps in real networks as well as in random models (Section III). We finally conclude the paper by giving some perspective on future work (Section IV).

II. DATASETS AND DEFINITIONS

In this section, we first present the different datasets we used in this study (Section II-A) before describing the existing (Section II-B) and new (Section II-C) metrics we used to analyse them.

A. Datasets

As explained before, many real-world social network exhibit an intricate structure involving two layers. In order to have an approach that allows to draw general conclusions, we relied on a wide variety of social networks involving communities on a broad sense. We focused in particular on two membership networks (FLICKR, LIVEJOURNAL) and two publications networks (CITEULIKE, WIKIPEDIA). Here below, we briefly describe them explaining what nodes and ⊥ nodes stand for in these contexts:

• LIVEJOURNAL [START_REF] Yang | Defining and evaluating network communities based on ground-truth[END_REF]: This dataset concerns a website where users are freely allowed to post information using a blogging system. It also allows people to declare friendships to each other, and to create and join groups. Here, nodes stand for groups and ⊥ nodes for users. It involves approximately 1 million users and more than 600,000 groups.

• CITEULIKE [START_REF] Emamy | CiteULike: A researcher's social bookmarking service[END_REF]: This dataset comes from a large online library which allows users to share, cite or tag scientific publications. Here we rely on about 150,000 tags and 700,000 publications, considering that nodes are the tags and ⊥ nodes are the publications.

• WIKIPEDIA [START_REF] Althoff | Analysis and forecasting of trending topics in online media streams[END_REF]: Here the ⊥ nodes are WIKIPEDIA articles and nodes are categories that group pages on similar topics. For computational reason, we did not use the whole network and rather focused on a large connected component 1 . This sub-network involves 3 millions articles and almost 500,000 subcategories.

• FLICKR [START_REF] Yang | Defining and evaluating network communities based on ground-truth[END_REF]: This dataset is composed of 100,000 groups and 400,000 users (respectively and ⊥ nodes) from the FLICKR website that allows hosting and sharing pictures. Detailed statistics on those datasets are provided extensively in Section III.

B. Metrics for bipartite graphs

In order to best account for the real structure of the datasets presented above, we rely in this study on bipartite graphs. Bipartite graphs are triplets G b = ( , ⊥, E b ), where is the set of top nodes (e.g., the groups in LIVEJOURNAL), ⊥ the set of bottom nodes (e.g., the users in LIVEJOURNAL), and E b ⊆ × ⊥ the set of links between and ⊥ (that relate the users to their groups). We denote by n (resp. n ⊥ ) the number of top nodes (resp. bottom nodes) and by m bip the number of links.

Compared to standard graphs, nodes in a bipartite graph are separated in two disjoint sets, and the links are always between a node in one set and a node in the other set (E b ⊆ ×⊥), see Fig. 1. Note that for some datasets the information is richer since we also have links between ⊥ nodes (resp. nodes), for instance links between articles (resp. a hierarchy of categories) 1 We chose to focus on the connected component containing the Network Protocol category in WIKIPEDIA. These additional links are very rich and should be used. However since we focus on the overlapping patterns we decided to discard this extra information. But for more intricate properties, it can be tedious to propose a "natural" definition. This is for instance the case of the local density in the graph, which is defined as the density of the induced subgraph by the neighbours of a node in simple graphs. In bipartite graphs, this local density usually tries to capture how the neighbourhoods of nodes tend to overlap each other. Some definitions have been already proposed to study this property in bipartite graphs [START_REF] Opsahl | Triadic closure in two-mode networks: Redefining the global and local clustering coefficients[END_REF], [START_REF] Latapy | Basic notions for the analysis of large two-mode networks[END_REF], [START_REF] Zhang | Clustering coefficient and community structure of bipartite networks[END_REF], [START_REF] Borgatti | Network analysis of 2-mode data[END_REF], [START_REF] Snijders | The statistical evaluation of social network dynamics[END_REF]. In the present work, we rely on two of those metrics to cope with the local density : the bipartite clustering coefficient and the redundancy coefficient [START_REF] Latapy | Basic notions for the analysis of large two-mode networks[END_REF]. The first one focuses on the intersection of the neighbourhood of two nodes, while the second one focuses on the impact of removing v as regard to the relations between ⊥ nodes. More formally, let N ⊥ (u) denote the set of neighbours of a node u ∈ (i.e., bottom nodes u is linked to) and N (v) the dual definition for a ⊥ node v. We can then define the bipartite clustering coefficient between two top nodes by:

cc(u, v) = |N ⊥ (u) ∩ N ⊥ (v)| |N ⊥ (u) ∪ N ⊥ (v)| . ( 1
)
This coefficient is equal to 1 if the neighbourhoods of u and v are equal (complete overlap), 0 if they have no ⊥ nodes in common. Then, it is natural to derive the clustering coefficient of u cc(u) as the average value of cc(u, v) for all v ∈ that share at least one common neighbour with u. Finally we can derive cc bip , the clustering coefficient of the graph G b , as the average value for all nodes 2As regard the redundancy coefficient of a node u, it is formally defined as:

rd(u) = |{{v, w} ∈ N ⊥ (u) 2 s.t. N (v) ∩ N (w) = {u}}| |N ⊥ (u)| 2 . ( 2
)
Intuitively a high value of the coefficient indicates that two ⊥ nodes that u relates are likely to be also related by another node (two users belonging to a group u also share another group in common), thus revealing some overlapping pattern in the structure. As for the clustering coefficient, we can derive the redundancy coefficient rd bip of the bipartite graph G b , as the average value of the former coefficient over all nodes.

Looking at the examples provided in Fig. 2, we can assess the relevance of the redundancy coefficient for identifying overlapping pattern. In Fig. 2a in which there is no overlap for node 2, one can check that rd(2) = 0 while in Fig. 2b, the value of the coefficient is 2 3 which reflects the fact that 2 over the 3 possible relations between nodes C, D and E are not affected by the removal of node 2 due to overlapping with nodes 1 and 3.

C. Two new metrics to describe the overlap

In order to refine those classical metrics, we now propose two new coefficients, related to the overlapping structure in bipartite graphs. We first define the dispersion coefficient which focus on how are distributed the links among the neighbours of u ∈ . More precisely, we compute the maximal number of different communities that could be related to the set of neighbours of u. Let N (u) denote the set of neighbours of a node u ∈ (i.e., the set of all nodes that share at least one bottom node with u). The dispersion coefficient disp(u) is then defined by:

disp(u) = |N (u)| -1 v∈N ⊥ (u) (|N (v)| -1) (3)
Intuitively, the dispersion measures whether members a group tend to share same other groups or not. If the dispersion of a node u is complete, then the size of the set of all communities related to its ⊥ neighbours (|N (u)| -1) will match exactly the sum of their degree (minus 1). Thus the value of the coefficient is equal to 1. This is the case for node 2 in Fig. 2a. This value is directly correlated to the fact that the redundancy of this node is equal to 0, no overlap implying complete dispersion of the links. On the other hand, when some overlapping pattern are present (see Fig. 2b), the dispersion tends to decrease: disp(2) = |{1,3}| 1+2+1 = 1 2 . Finally, we propose another metric which accounts for the specificity of degree-1 ⊥ nodes that are particularly numerous in real-world networks, thus impacting the analysis of overlapping patterns. We therefore define the monopoly coefficient as the proportion of degree-1 neighbours of a node u. Formally:

mon(u) = |{v ∈ N ⊥ (u) s.t. d ⊥ (v) = 1}| |N ⊥ (u)| (4)
As for the former coefficient, it is naturally possible to derive the dispersion, or the monopoly, of a bipartite graph as the average value of the corresponding coefficients over all nodes. As we will see further in the next section, these metrics are able to refine the analysis provided by the standard metrics and help understanding the true nature of the observed overlaps in bipartite networks.

III. ANALYSIS OF THE BIPARTITE STRUCTURE

The purpose of the present section is to understand the behaviour of the different metrics computed on the four datasets presented previously. We start by providing general statistics for the different networks (Section III-A). Then we detail the analysis by investigating in particular the relevance of the traditional clustering and redundancy coefficients to characterise overlapping patterns in bipartite networks (Section III-B), whether the new proposed metrics are able to refine the analysis (Section III-C) and how random models affect the structure of the networks as regards to these properties (Section III-D).

As for the former coefficient, it is naturally possible to derive the dispersion (resp. monopoly) of a bipartite graph as the average value of the dispersion (resp. monopoly) coefficients over all nodes, which we denote by disp bip (resp. mon bip ). We first focus on some global statistics of bipartite graphs as presented in the previous section. Table I presents the results for the four datasets. It shows that the networks under investigations present standard properties commonly observed in real-world networks [START_REF] Guillaume | Bipartite graphs as models of complex networks[END_REF]. In particular one can notice that the networks are globally sparse (δ b ) while local densities are order of magnitudes higher (cc bip and rd bip ). Besides, the order of magnitude between the average degrees (k and k ⊥ ) and the maximal degrees (d + and d + ⊥ ) indicates some heterogeneity in the degree distribution of both and ⊥ nodes3 .

A. Global properties

As regard the two classical metrics usually used to capture overlaps in bipartite structures (cc bip and rd bip ), one can notice that they strongly differ. By relying on the clustering coefficient, those networks seems to present few overlaps while, on the contrary, the redundancy coefficient tends to indicates the presence of overlaps. The FLICKR case is to that regard eloquent. This raises the question of understanding which one of the two coefficients is the more relevant to depict the presence of overlaps in the structure. This question will be investigated more deeply in Section III-B.

If one focus now on the new metrics (disp bip and mon bip ), one can notice that the average monopoly is very low for each network, indicating that degree-1 ⊥ nodes seem well distributed in the communities. On the contrary, the average dispersion is high. This seems to contradict the redundancy coefficient as it tends to indicate a lack of overlaps in the structure. But such an aggregated value is difficult to analyse and a more detailed discussion is provided in Section III-C.

B. Standard bipartite metrics

The global statistics presented in the previous section do not allow to grasp the diversity of the possible situations for all nodes. In order to refine this first analysis, we then compute the value of the different coefficients for all nodes, which allows to study the distribution of the values as well as some correlations between the metrics. Fig. 3 presents the inverse cumulative distribution function of the clustering (Fig. 3a) and redundancy (Fig. 3b) coefficients, showing clearly two different distributions. Whereas the distributions sharply decrease for the clustering coefficient, it seems to be quite uniformly distributed for the redundancy coefficient, except for the extreme values (0 and 1) which are highly represented.

More importantly, one can notice that the proportion of poorly clustered nodes is large. More than 75% of the nodes have a value lower than 0.2 in all datasets. The proportion of redundant nodes are in contrast particularly high. The fraction of nodes that have a redundancy of 1 is for instance extremely important for all datasets: 17% for WIKIPEDIA, 38% for CITEULIKE and FLICKR, 58% for LIVEJOURNAL.

Considering this last case, it means that for more than half of the groups created in LIVEJOURNAL, every pair of members belong to (at least) one other group in common. This is surprising, especially since the average numbers of groups a user belong to is very low (6.24, see Table I). This indicates the presence of non trivial overlapping pattern in the networks which is not captured by the clustering coefficient, thus leading to the conclusion that the notion of redundancy is more suited to identify real overlapping patterns in bipartite networks. In order to understand why such a difference is observed, we present Fig. 4 the correlation between the degree of a node and its clustering or redundancy coefficient. More precisely, we consider for each node u a point with coordinate (d ⊥ (v), disp(v)) (Fig. 4a) or (d ⊥ (v), rd(v)) (Fig. 4b). The plot shows the average value for different intervals. Fig. 4a shows a clarifying fact: the value of the clustering seems to be strongly related to the degree of the node: the higher the degree, the lower the clustering. Such a correlation is not present for the redundancy, Fig. 4b. This strengthens our former conclusion on the relevance of the redundancy to detect overlaps. Indeed, the redundancy seems to be independent of the degree, while the clustering can be derived from it and is thus more related to the number of neighbours and less to the structure of the relations.

C. Refining the analysis with new metrics

The results presented above focus on the standard metrics proposed as extensions of the notion of local density for bipartite graphs. It shows that the notion of redundancy is well suited to detect overlaps in the networks. However, such a notion does not allow to understand how those overlaps are organised around a node. The notion of dispersion, as well as the monopoly, proposed in this paper, is an attempt to precise such organisation. This section presents the analysis conducted on the datasets through the use of those two coefficients. Fig. 5 presents the inverse cumulative distribution of the dispersion (Fig. 5a) and the monopoly (Fig. 5b) coefficients. As suggested by the average values (see Table I), the dispersion is high for large fraction of nodes. In all dataset, more than 80% of the nodes have a value higher than 0.5. This is however coherent with the presence of overlaps around those nodes since 0.5 means than half of the links creates similar relations among ⊥ and nodes, thus inducing overlaps. Yet, these high values contrast with the ones of the redundancy coefficients Inverse cumulative distribution of the dispersion and monopoly coefficient.

and reduce the importance of the overlaps in the networks. In that sense, it refines the use of redundancy.

As regard monopoly coefficients, their distribution precise our former statement on the fact that it is uniformly distributed. This is obvious in all curves as they all decrease smoothly for the complete range of strictly positive values. However, the reader might notice the high fraction of nodes having no monopoly at all. Going back to the global properties of the networks (Table I), it is easily explained by the fact that the number of ⊥ nodes is way lower than the one of nodes. Thus, even if the links are well distributed over the networks, not every top nodes can be related to a degree-1 bottom node, thus leading to a monopoly coefficient of 0. As for the clustering and the redundancy, we can precise our analysis by looking at some correlations among the metrics. Fig. 6 presents the correlations of the degree of a node with its dispersion and its monopoly. Fig 6a shows a clear tendency: the higher the degree, the lower the dispersion in average. Since low value of dispersion is related to overlaps, this refines our former analysis as we can now precise which kind of top nodes are related for overlaps.

As regard the correlation between the degree and the monopoly, Fig 6b shows that on average the monopoly seems to be independent from the degree, except for FLICKR that displays a clear decrease when the degree increases. One can note that the case of FLICKR is still consistent with our previous statement : since degree-1 nodes cannot be responsible for overlaps in the bipartite structure, it is then coherent with the fact that overlaps are more related to large top nodes.

D. Impact of random model

Having analysed the overlaps in the datasets using two traditional and two new metrics, we now turn to the impact random models have on those coefficients. To do so, we use a variant of the configuration model [START_REF] Newman | The structure and function of complex networks[END_REF] which randomly assigns edges to match a given degree sequence without adding any other expected property. In other word, the generated bipartite graphs have the same number of nodes and links but the links are distributed uniformly at random among ⊥ and nodes, according to their real degree. This section shows the impact such a shuffling have on some of the metrics. Fig. 7 shows the inverse cumulative distribution function of the clustering (Fig. 7a) and redundancy (Fig. 7b) coefficients for random networks. As expected, Fig. 7a shows that the randomisation process has little impact on the clustering distribution. Indeed, as shown in Section III-B, the value of the clustering is directly related to its degree. Since the degree sequence is kept unchanged by the model, so does the clustering. This strengthens again our former conclusion to that regard.

Concerning the redundancy, the behaviour is different depending on the dataset on which we apply the model. If the model seems to have completely remove the overlaps in the WIKIPEDIA dataset, it is the complete opposite for the FLICKR network. In this last case, the model seems to have reinforced the presence of the overlaps. If no strict conclusion can be drawn here, the reader might notice that the proportion of high redundancy coefficients is related to the density of the network. The relative density of the FLICKR network (30 times higher that the WIKIPEDIA network) could thus explain the effect of the model. Finally, we show Fig. 8 the impact of the model to the dispersion coefficient. Fig. 8a presents the inverse cumulative distribution. The plot indicates that the model has increased the values in all cases. This is not surprising since the definition of the dispersion coefficient is related to the notion of expected distribution of the links. In other word, by distributing the links uniformly at random in the networks, the model tends to maximise this coefficient. Fig. 8b shows the impact the random model has on the correlation between degree and dispersion. Surprisingly, whereas Section II-C clearly identified the degree as directly related to the dispersion in all the dataset, the model shows that this correlation is due to the nature of the network since, when shuffling the real distribution of the links, this correlation is strongly affected. Yet, the tendency identifies previously is still correct: the higher the degree, the lower the dispersion.

IV. CONCLUSION

In this paper, we studied the overlaps observed in 4 different online social networks exhibiting a bipartite structure.

To do so we start by relying on two standard metrics recently proposed to address the notion of local density in bipartite graphs, namely the clustering coefficient and the redundancy coefficient. Our analysis revealed that, the clustering coefficient is surprisingly not particularly able to detect real observed overlaps in networks. To explain this result, we showed that this coefficient is more related to the number of neighbours and less to the structure of the relations.

In order to deepen the study of the overlaps, we proposed two new metrics, namely the dispersion and the monopoly, that complete the analysis. The results show that it help refining the characterisation of the overlaps made by the redundancy coefficient by giving insights on the kind of nodes affected by overlaps. Finally, we applied a random bipartite model in order to assess how the shuffling of the links can affect the properties observed in the present study. It showed in particular that such a randomisation process affect the dispersion of the link, while it keep unchanged the clustering, thus strengthening the interest of this new metric as a valid candidate for the characterisation of the overlaps.

Since several years, many researches have studied the notion of local density in order to better understand the properties of the structure of real-world networks leading especially to applications like recommendation and link prediction systems [START_REF] Liben-Nowell | The link-prediction problem for social networks[END_REF], [START_REF] Borgatti | Network analysis of 2-mode data[END_REF], [START_REF] Snijders | The statistical evaluation of social network dynamics[END_REF]. A better understanding of the structure of bipartite networks should help contributing to this area.

Fig. 1 .

 1 Fig. 1. Example of a bipartite graph.

  Fig. 2. Two bipartite graphs depicting a different structure around node 2.

Fig. 3 .

 3 Fig. 3. Inverse cumulative distribution of the clustering and redundancy coefficient.

  Fig. 4.Correlations between the degree and the clustering/redundancy coefficients.

  Fig. 5.Inverse cumulative distribution of the dispersion and monopoly coefficient.

Fig. 6 .

 6 Fig. 6.Correlations between the degree and the dispersion/monopoly coefficients.

Fig. 7 .

 7 Fig. 7. Inverse cumulative distribution of the clustering and redundancy coefficients for random bipartite graphs.

Fig. 8 .

 8 Fig. 8.Inverse cumulative distribution of the dispersion coefficient and correlation between degree and dispersion for random networks with the same and ⊥ degree distributions.

Note that for this coefficient, as well as for the next ones, the dual notion for bottom nodes can be derived, although we will not use it in the present article.

Due to space limitation, degree distributions plots are not presented here but they confirm this statement.

ACKNOWLEDGMENT

This work is supported in part by the French National Research Agency contract CODDDE ANR-13-CORD-0017-01.