Komla Domelevo

Stefanie Petermichl

Janine Wittwer

A linear dimensionless bound for the weighted Riesz vector

Keywords: Bellman function, Riesz transforms, weighted estimates

We show that the norm of the vector of Riesz transforms as operator in the weighted Lebesgue space L 2 ω is bounded by a constant multiple of the first power of the Poisson-A 2 characteristic of ω. The bound is free of dimension. We also show that for n > 1, the Poisson-A 2 class is properly included in the classical A 2 class.

Introduction

A weight is a positive L 1 loc function. Muckenhoupt proved in [START_REF] Muckenhoupt | Weighted norm inequalities for the Hardy maximal function[END_REF] that for 1 < p < ∞ the maximal function is bounded on L p (ω) iff the weight ω belongs to the class A p , where

ω ∈ A p iff Q p (ω) := sup B ω B ω -1/(p-1) p-1 B < ∞.
Email addresses: komla.domelevo@math.univ-toulouse.fr (Komla Domelevo), stefanie.petermichl@math.univ-toulouse.fr (Stefanie Petermichl), jwittwer@westminstercollege.edu (Janine Wittwer). 1 Research supported in part by ANR-12-BS01-0013-02. The author is a member of IUF 2 Correponding author, Tel:+33 5 61 55 76 59, Fax: +33 5 61 55 83 85 Here the notation • B denotes the average over the ball B and the supremum runs over all balls B. Hunt, Muckenhoupt, Wheeden proved in [START_REF] Hunt | Weighted norm inequalities for the conjugate function and the Hilbert transform[END_REF] that the A p condition also characterizes the boundedness of the Hilbert transform

Hf (x) = 1 π f (y)
xy dy in L p (ω). Coifman and Fefferman in [START_REF] Coifman | Weighted norm inequalities for maximal functions and singular integrals[END_REF] extended the theory to general Calderón-Zygmund operators.

The focus in this note is on the Riesz transforms in weighted spaces. The linear and optimal bound in terms of the classical A 2 characteristic has been established by one of the authors in [START_REF] Petermichl | The sharp weighted bound for the Riesz transforms[END_REF]. The same estimate holds for all Calderón-Zygmund operators by Hytonen's theorem [START_REF] Huković | Thesis: Singular integral operators in weighted spaces and Bellman functions[END_REF]. This remarkable result has been reproven by Lerner [START_REF] Lerner | A simple proof of the A 2 conjecture[END_REF]. The bound depends upon the dimension in all these proofs, as one should expect.

We are interested in a version of A 2 which is particularly well-suited for working with the Riesz transforms in R n , where we exploit the intimate connection of Riesz transforms and harmonic functions. We use the Poisson-A 2 class, which considers Poisson averages instead of box averages in the definition of A 2 and obtain a bound free of dimension for the Riesz vector. This way of measuring the characteristic of the weight arises naturally when working with Bellman functions, when convexity is replaced by harmonicity. This was also the approach in [START_REF] Nazarov | The weighted norm inequalities for Hilbert transform are now trivial[END_REF] as well as [START_REF] Petermichl | A sharp estimate for the weighted Hilbert transform via Bellman functions[END_REF] for the weighted Hilbert transform and [START_REF] Dragičević | Bellman Functions and Dimensionless Estimates Of Littlewood-Paley Type[END_REF] for unweighted Riesz transforms. From a probabilistic angle, the Poisson-A 2 characteristic also arises naturally as a consequence of the use of martingales driven by space-time Brownian motion as in Gundy-Varopoulos [START_REF] Gundy | Les transformations de Riesz et les intégrales stochastiques[END_REF].

An altered version of one-dimensional Poisson extensions of weights made a reappearance in the works concerned with the famous two-weight problem for the Hilbert transform, see [START_REF] Lacey | Two weight inequality for the Hilbert transform: A real variable characterization[END_REF] and [START_REF] Lacey | Two weight inequality for the Hilbert transform: A real variable characterization[END_REF]. It enjoys its interpretation as a 'tamed' Hilbert transform, a feature that appears to be lost in higher dimensions. In the one-dimensional case, we see a quadratic relation between the Poisson characteristic and the classical characteristic, but the classes themselves are the same. Interestingly, these different A 2 classes are not identical when the dimension is larger. We will show examples of A 2 weights whose Poisson integral diverges when the dimension is at least two. Such weights belong to A 2 but not to Poisson-A 2 . This shows that the Poisson characteristic used on a pair of weights such as for the two-weight problems, is not necessary in higher dimension. This is one of several obstacles when considering the two-weight question for the Riesz transforms, that is currently under investigation. We mention the recent advance [START_REF] Lacey | Two weight inequalities for Riesz transforms: Uniformly full dimension weights[END_REF] where the Poisson characteristic is modified.

Notation

The Riesz transforms R k in R n are the component operators of the Riesz vector R, defined on the Schwartz class by

(R k f )(ξ) = i ξ k ξ f (ξ).
We consider the space L 2 ω , where ω is a positive scalar valued L 1 loc function, called a weight. More specifically, the space L 2 ω (R n ; C) consists of all measurable functions f : R n → C so that the quantity

f ω := R n |f (x)| 2 ω(x)dx 1/2
is finite, where dx denotes the Lebesgue measure on R n . For the space of vector valued functions

L 2 ω (R n ; C n ), we replace | • | by the ℓ 2 norm • .
We are concerned with a special class of weights, called Poisson-A 2 . We say

ω ∈ Ã2 if Q2 (ω) := sup (x,t)∈R n ×R + P t (ω)(x)P t (ω -1 )(x) < ∞ (1)
where P t denotes the Poisson extension operator into the upper half space defined by

P t = e -tA
where we define A := √ -∆ and ∆ be the Laplacian in R n . The scalar Riesz transforms can be written as

R k = ∂ k • A -1 .
The Poisson kernel has the form

P t (y) = c n t (t 2 + |y| 2 ) n+1 2
where c n is its normalizing factor. The extension operator becomes

P t f (x) = c n R n f (y)P t (x -y)dy.

Main results

The main purpose of this text is to provide the dimensionless estimate:

Theorem 3.1 There exists a constant c that does not depend on the dimension n or the weight ω so that for all weights ω ∈ Ã2 the Riesz vector as an operator in weighted space

L 2 ω → L 2 ω has operator norm R L 2 ω →L 2 ω ≤ c Q2 (ω).
We also investigate the relationship between different Muckenhoupt classes. Notably, their relation changes with dimension:

Theorem 3.2 Poisson-A 2 and classical A 2 only define the same classes of weights when the dimension is one: Ã2 = A 2 if and only if n = 1.

4 The dimension-free estimate

Since R L 2 ω (R n ;C)→L 2 ω (R n ;C n ) = ω 1/2 Rω -1/2 L 2 (R n ;C)→L 2 (R n ;C n )
where the outer multiplication by ω 1/2 is a scalar multiplication. We can estimate

R L 2 ω →L 2 ω via L 2 duality. It is sufficient to estimate |( g, ω 1/2 Rω -1/2 f )| ≤ c Q2 (ω) f g
for test functions (smooth and compactly supported) f, g, where f is scalar valued and g vector valued. Or (considering ω -1/2 f instead of f and ω 1/2 g instead of g):

|( g, Rf )| ≤ c Q2 (ω) g ω -1 f ω .
To prove this estimate, we prove the following theorem:

Theorem 4.1 For test functions f, g and ω ∈ Ã2 we have the following estimate:

|( g, Rf )| ≤ c Q2 (ω)( g 2 ω -1 + f 2 ω ); (2)
here c does not depend on f, g, n, k or ω.

Considering λf and λ -1 g for appropriate λ, with the considerations above yields Theorem 3.1.

Before we turn to the proof of Theorem 4.1, let us formulate several useful lemmata.

Three useful Lemmata

The following is a well known fact. It is, for example, stated in [START_REF] Gundy | Les transformations de Riesz et les intégrales stochastiques[END_REF].

for all u ∈ R d .

We will be using this lemma for m = 2, l = 2n and k = 4.

Finally we crucially need the Lemma below, that has been proven in [START_REF] Petermichl | Heating of the Beurling operator: weakly quasiregular maps on the plane are quasiregular[END_REF]. It was deduced from the sharp weighted inequality for martingales from [START_REF] Wittwer | A sharp estimate on the norm of the martingale transform[END_REF].

Lemma 4.4 For any Q > 1 let D be a subset of R × R × C × C n × R × R D Q = {(X, Y, x, y, r, s) : |x| 2 < Xs, y 2 < Yr, 1 < rs < Q}.
For any compact K ⊂ D Q there exists an infinitely differentiable function B K,Q defined in a small neighborhood of K that still lies inside D Q so that the following estimates hold in K.

0 ≤ B K,Q ≤ 5Q(X + Y), (3)
-d 2 B K,Q ≥ 2|dx| dy . (4)
The last inequality describes an operator inequality where the left hand side is the negative Hessian of B. This function is a mollified version of the Bellman function one gets by knowing that the equivalent weighted estimate holds for a certain dyadic model operator.

Proof of the main theorem

Recall the inequality of theorem (4.1). We want to show for test functions f and g and ω ∈ Ã2 :

|( g, Rf )| ≤ c Q2 (ω)( g 2 ω -1 + f 2 ω ).
For a fixed weight ω we let Q = Q2 (ω). This gives rise to the set

D Q . We define b K,Q (x, t) = B K,Q (v(x, t)) where v(x, t) = P t (|f | 2 ω), P t ( g 2 ω -1 ), P t (f ), P t ( g), P t (ω), P t (ω -1 ) (x)
Here K is a compact subset of D Q to be chosen later.

Note that the vector v ∈ D Q for any choice of (x, t). This is ensured by Q = Q2 (ω) and several applications of Jensen's inequality. Notice also that the vector v takes compacts inside the interior of R n+1

+ to compacts K inside D Q
for fixed f, g, ω. By elementary application of the chain rule (using harmonicity of the components of v) one shows that

∆ x,t b(x, t) = n i=1 d 2 B(v) ∂ ∂x i v, ∂ ∂x i v + d 2 B(v) ∂ ∂t v, ∂ ∂t v .
Here ∆ x,t is the full Laplacian in the upper half space

∆ x,t = n i=1 ∂ 2 x i + ∂ 2 t .
Notice that condition 4 means that at any v = (X, Y, x, y, r, s) ∈ K ⊂ D Q we have the operator inequality for any

u ∈ R × R × C × C n × R × R -d 2 B K,Q (v)u, u ≥ 2(|dx| dy u, u) = 2|u 3 | u 4 .
In our situation, f, g, ω and Q are fixed, but we have varying K, x, t. So Lemma 4.3 guarantees us existence of τ x,t,K so that

-d 2 B(v) ∂ ∂x i v, ∂ ∂x i v ≥ τ x,t,K | ∂ ∂x i P t f | 2 + τ -1 x,t,K ∂ ∂x i P t g 2
for all i and

-d 2 B(v) ∂ ∂t v, ∂ ∂t v ≥ τ x,t,K | ∂ ∂t P t f | 2 + τ -1 x,t,K ∂ ∂t P t g 2 . So -∆ x,t b K,Q (x, t) ≥ τ x,t,K n i=1 | ∂ ∂x i P t f | 2 + | ∂ ∂t P t f | 2 +τ -1 x,t,K n i=1 ∂ ∂x i P t g 2 + ∂ ∂t P t g 2 ≥ 2 n i=1 | ∂ ∂x i P t f | 2 + | ∂ ∂t P t f | 2 1/2 n i=1 ∂ ∂x i P t g 2 + ∂ ∂t P t g 2 1/2 ≥ 2 n i=1 | ∂ ∂x i P t f | 2 1/2 ∂ ∂t P t g = 2 ∇P t f ∂ ∂t P t g
In what follows we first use formula 4.2, then the estimate for the Laplacian we just proved:

|( g, Rf )| ≤ 4 ∞ 0 | ∂ ∂t P t g, ∇P t f |tdt ≤ 4 ∞ 0 R n ∂ ∂t P t g ∇P t f dxtdt ≤ 2 ∞ 0 R n -∆ x,t b K,Q (x, t)dxtdt.
It remains to see that

- ∞ 0 R n ∆ x,t b K,Q (x, t)dxtdt ≤ C Q2 ( f 2 ω -1 + g 2 ω ) (5)
with C independent of n. In order to obtain this last estimate, we will apply Green's formula as well as some properties of our Bellman function. We are going to pass through values of the function b.

Recall the statement of Green's formula:

Theorem 4.5 Ω f (x)∆g(x) -g(x)∆f (x)dA(x) = ∂Ω f (t) ∂g ∂n (t) -g(t) ∂f ∂n (t) dS(t)
where n is the outward normal and dS surface measure on ∂Ω.

In order to be accurate, we are obliged to take care of a few technicalities first.

Let T R be a cylinder with square base in upper half space [-R, R] n × [0, 2R].

For R not too small, the point (0, 1) lies inside T R . Let T R,ǫ = T R + (0, ǫ). For any interior point (ξ, τ ), let G R,ǫ [(x, t), (ξ, τ )] be its Green function, meaning that

∆ x,t G R,ǫ [(x, t), (ξ, τ )] = -δ (ξ,τ ) and G R,ǫ = 0 on ∂T R,ǫ .
Notice that RT 1,0 = T R,ǫ -(0, ǫ) and the Green's functions relate as follows:

Lemma 4.6 The Green's function has the following scaling property:

R n-1 G R,ǫ [(x, t), (ξ, τ )] = G 1,0 [(R -1 (x, t -ǫ), R -1 (ξ, τ -ǫ))]. (6)
Proof. By uniqueness it suffices to see that

R -(n-1) G 1,0 [R -1 (x, t-ǫ), R -1 (ξ, τ - ǫ)
] is indeed the Green function for the region T R,ǫ at the point (ξ, τ ). It is clear that it equals zero on ∂T R,ǫ . Furthermore for test function f we have

T R,ǫ ∆ x,t R -(n-1) G 1,0 [R -1 (x, t -ǫ), R -1 (ξ, τ -ǫ)]f (x, t)dxdt = T 1,0 ∆ y,s G 1,0 [(y, s), R -1 (ξ, τ -ǫ)]f (Ry, Rs + ε)dyds = -f (ξ, τ )
We did a substitution (x, t) = (Ry, Rs+ǫ). Note that there is R -2 term arising from the switch of ∆ x,t to ∆ y,s and R n+1 from the determinant. QED

Recall that the vector v maps each T R,ǫ into a compact K = K R,ǫ ⊂ D Q . For technical reasons we have to exhaust the upper half space by compacts. We fix any compact set M in the open upper half space and consider R large enough and ǫ small enough so that M ⊂ T R,ǫ .

Let us start to use the size estimate of our Bellman function to obtain an estimate of the function value b K,Q (0, R + ǫ) from above:

b K,Q (0, R + ǫ) ≤ C Q2 (P R+ǫ |f | 2 ω -1 (0) + P R+ǫ g 2 ω(0)) = c n C Q2 R n |f | 2 (y)ω -1 (y) R + ǫ ((R + ǫ) 2 + |y| 2 ) n+1 2 dy + R n g 2 (y)ω(y) R + ǫ ((R + ǫ) 2 + |y| 2 ) n+1 2 dy ≤ c n (R + ǫ) -n C Q2 ( f 2 ω -1 + g 2 ω ).
For an estimate from below, Green's formula applied to our situation gives:

b K,Q (0, R + ǫ) = - T R,ǫ G R,ǫ ((x, t), (0, R + ǫ))∆ x,t b K,Q (x, t)dxdt - ∂T R,ǫ b K,Q (x, t) ∂G R,ǫ ((x, t), (0, R + ǫ)) ∂n dxdt + ∂T R,ǫ G R,ǫ ((x, t), (0, R + ǫ)) ∂b K,Q ((x, t)) ∂n dxdt
The first boundary term is negative because b is non-negative and the outward normal of the Green's function is negative on the boundary of T R,ǫ . The second boundary term vanishes because G R,ǫ = 0 on the boundary. So we have the following estimate:

b K,Q (0, R + ǫ) ≥ - T R,ǫ G R,ǫ ((x, t), (0, R + ǫ))∆ x,t b K,Q (x, t)dxdt. ≥ - M G R,ǫ ((x, t), (0, R + ǫ))∆ x,t b K,Q (x, t)dxdt.
since -∆b ≥ 0 and where we recall that M ⊂ T R,ǫ . We continue the estimate using the scaling properties of the Green functions [START_REF] Lacey | Two weight inequality for the Hilbert transform: A real variable characterization[END_REF].

b K,Q (0, R + ǫ) ≥ - M R -(n-1) G 1,0 (R -1 (x, t -ǫ), (0, 1))∆ x,t b(x, t)dxdt.
Since G 1,0 ((R -1 x, 0), (0, 1)) = 0 we have

b K,Q (0, R + ǫ) ≥ - M R -(n-1) G 1,0 ((R -1 x, R -1 (t -ǫ), (0, 1))- G 1,0 ((R -1 x, 0), (0, 1)) ∆ x,t b(x, t)dxdt = - M R -(n-1) ∂G 1,0 ∂t (R -1 x, τ )R -1 (t -ǫ)∆ x,t b K,Q (x, t)dxdt = - M R -n ∂G 1,0 ∂t (R -1 x, τ )∆ x,t b K,Q (x, t)dx(t -ǫ)dt, where 0 ≤ τ ≤ R -1 (t -ǫ).
Pulling this all together with the estimate from above,

- M R -n ∂G 1,0 ∂t (R -1 x, τ )∆ x,t b K,Q (x, t)dx(t -ε)dt ≤ b K,Q (0, R + ǫ) ≤ c n (R + ǫ) -n C Q2 ( f 2 ω -1 + g 2 ω ), hence - M ∂G 1,0 ∂t (R -1 x, τ )∆ x,t b K,Q (x, t)dx(t -ǫ)dt ≤ c n C Q2 ( f 2 ω -1 + g 2 ω )
uniformly with respect to R and ǫ, for all given M . When R goes to infinity, the normal derivative ∂G 1,0 ∂t (R -1 x, τ ) tends to ∂G 1,0 ∂t (0, 0) uniformly with respect to (x, t) ∈ M . But we know that the normal derivative ∂G 1,0 ∂t (0, 0) is exactly the normalizing factor c n of the Poisson kernel (see [START_REF] Dragičević | Bellman Functions and Dimensionless Estimates Of Littlewood-Paley Type[END_REF] and the references therein Couhlon-Duong). Letting R go to infinity and ε go to zero yields for all compact M of the upper half space:

- M ∆ x,t b K,Q (x, t)dxtdt ≤ C Q2 ( f 2 ω -1 + g 2 ω ).
Finally, letting M exhaust the upper half space establishes (5).

The comparison of classical and Poisson characteristic.

In this section we prove Theorem 3.2. We provide an example that demonstrates that Ã2 = A 2 if n > 1. For the case n = 1, it is known that the two classes are the same. Infact, for n = 1 the estimates Q2 (ω) Q 2 (ω) Q2 (ω) 2 are proven in [START_REF] Huković | Thesis: Singular integral operators in weighted spaces and Bellman functions[END_REF]. If n > 1 however, an easy example shows that the Poisson integral of a simple power weight diverges, although the weight belongs to classical A 2 . Consider ω α (x) = |x| α . It is well known and straightforward to check that ω α ∈ A 2 ↔ |α| < n. Also Q 2 (ω α ) ∼ 1 n 2 -α 2 . We show that the Poisson integral P t w α (0) diverges for α close to n.

P t (ω α )(0) ∼ R n t (t 2 + |x| 2 ) n+1 2 |x| α dx = |S| ∞ 0 t (t 2 + r 2 ) n+1 2 r α+n-1 dr = |S| ∞ k=1 2 k t 2 k-1 t t (t 2 + r 2 ) n+1 2 r α+n-1 dr |S| ∞ k=1 2 k-1 t t (t 2 + 2 2k t 2 ) n+1 2
(2 k-1 t) α+n-1

t α |S|2 -α-n-1 ∞ k=1 2 (α-1)k
We see that this sum converges if and only if α -1 < 0. If n ≥ 2 we have w α ∈ A 2 if and only if |α| < n so we can easily pick a valid α for which the above sum diverges.

Thus not every weight in A 2 is in Ã2 . The converse is still true, though. Let w ∈ Ã2 , and let B be a ball with center a and radius r. Then for y ∈ B, |a -y| < r, and so 1 r n ≤ 2

2 +

 2 |a -y| 2 ) n+1 2 dy ≤ C ′ P r (ω)(a),and similarly for ω -1 B . Thus Q 2 (ω) ≤ C Q2 (ω).

The proof using semigroups is very simple and concise, so we include it for the convenience of the reader. Instead of using semigroups, the same result can be obtained by the use of the Fourier transform.

Proof. Observe that F (0) = ∞ 0 F ′′ (t)tdt for sufficiently fast decaying F . So

The right hand side becomes

Now we use the fact that d dt P t = -AP t and d 2 dt 2 P t = A 2 P t and symmetry of A to see that the above equals

Using the fact that P t g is harmonic we can replace A by -∂ t in this expression. Furthermore, observe that A commutes with P t and ∂ k and that

For function f and vector function g this becomes

QED

We will need the following form of a Lemma that has been proven in [START_REF] Dragičević | Bellman Functions and Dimensionless Estimates Of Littlewood-Paley Type[END_REF] and generalised in [START_REF] Dragičević | A theorem about three quadratic forms[END_REF], the so-called 'ellipse lemma'.


Flore Nabet

Finite-Volume analysis for the Cahn-Hilliard equation with dynamic boundary conditions

This work is devoted to the convergence analysis of a finite-volume approximation of the 2D Cahn-Hilliard equation with dynamic boundary conditions. The method that we propose couples a 2d-finite-volume method in a bounded, smooth domain Ω ⊂ R 2 and a 1d-finite-volume method on ∂ Ω . We prove convergence of the sequence of approximate solutions. One of the main ingredient is a suitable space translation estimate that gives a limit in L ∞ 0, T, H 1 (Ω ) whose trace is in L ∞ 0, T, H 1 (∂ Ω ) .

Introduction

We consider a smooth, connected and bounded domain Ω ⊂ R 2 and Γ = ∂ Ω its boundary. Let T > 0 be given. We are interested here in the following phase separation model in material science (refered to as the Cahn-Hilliard equation with dynamic boundary conditions): Find the concentration of one of the two phases c : (0, T ) × Ω → R satisfying:

               ∂ t c = ∆ µ, in (0, T ) × Ω ; µ = -∆ c + f ′ b (c), in (0, T ) × Ω ; ∂ t c Γ = ∆ c Γ -f ′ s (c Γ ) -∂ n c, on (0, T ) × Γ ; ∂ n µ = 0, on (0, T ) × Γ ; c(0, .) = c 0 , in Ω ; (1)
where we have introduced an intermediate unknown: the chemical potential µ.

Flore NABET Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France e-mail: flore.nabet@univ-amu.fr

The trace of c on Γ is noted c Γ , ∆ is the Laplace-Beltrami operator on Γ and ∂ n is the normal derivative at the boundary. The Cahn-Hilliard potentials f b and f s are nonlinear and they correspond respectively to the bulk and the surface free energy densities. In fact, several physical parameters should appear in the Cahn-Hilliard equation to account for physical properties of the studied system. However, these constants affect the readability of the problem. Thus, we have chosen to write the Problem (1) without these parameters. We impose the homogeneous Neumann boundary condition for the chemical potential since no mass exchange can occur through the boundary. For many years, different authors studied the Cahn-Hilliard equation associated with the Neumann boundary condition for the order parameter c. In some cases, however, this condition is too restrictive to account for the interaction of the mixture with the walls. For this reason, physicists [START_REF] Fischer | Novel surface modes in spinodal decomposition[END_REF][START_REF] Kenzler | Phase separation in confined geometries: Solving the Cahn-Hilliard equation with generic boundary conditions[END_REF] have recently introduced the Cahn-Hilliard system with dynamic boundary conditions [START_REF] Boyer | Numerical schemes for a three component Cahn-Hilliard model[END_REF]. The associated free energy is the sum of a bulk free energy F b and a surface free energy F s :

F (c) = Ω 1 2 |∇c| 2 + f b (c) :=F b (c) + Γ 1 2 ∇ c Γ 2 + f s (c Γ ) :=F s (c) . (2)
The dynamic boundary condition on c is chosen in such a way that the total free energy decreases with respect to time:

d dt F (c(t, .)) = - Ω |∇µ(t, .)| 2 - Γ |∂ t c Γ (t, .)| 2 , t ∈ [0, T [.
f (m) b (c) ≤ C b 1 + |c| p-m , m ∈ {0, 1, 2}. A typical choice for f b is the double-well function f b (c) = c 2 (1 -c) 2 .
From a theoretical point of view, this system has already been studied (see for exemple [START_REF] Goldstein | A Cahn-Hilliard model in a domain with nonpermeable walls[END_REF] and the references therein). From a numerical point of view, we have several results. In [START_REF] Fischer | Novel surface modes in spinodal decomposition[END_REF][START_REF] Kenzler | Phase separation in confined geometries: Solving the Cahn-Hilliard equation with generic boundary conditions[END_REF], authors propose a finite-difference framework but without proof of convergence. A convergence result is proved in [START_REF] Cherfils | A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions[END_REF] with a finite element space semi-discretization, but in a slab with periodic boundary conditions in lateral directions. In this paper, we propose a convergence analysis of a finite-volume scheme for the space discretization. This method is well adapted to the coupling between the dynamics in the domain and those on the boundary by the flux term ∂ n c. Moreover, this kind of scheme preserves the mass and accounts naturally for the non-flat geometry of the boundary and the associated Laplace-Beltrami operator.

2 The discrete setting

The Finite-Volume meshes and notation

We recall that the domain Ω is not polygonal and that we have to solve an equation on Γ . Thus, our notation will be slightly different than the usual finite-volume notation (see for example [START_REF] Eymard | Finite volume methods[END_REF]). Let M be a decomposition of Ω into polygonal subsets (called control volumes and noted K ∈ M) except perhaps for those on the boundary which can have a curved edge. For each control volume K ∈ M, we associate a point x K which satisfies the orthogonality condition (see [START_REF] Eymard | Finite volume methods[END_REF]). The main differences with the usual finite-volume notation are those on the boundary mesh ∂ M. This mesh is constituted of the set of curved edges σ on the boundary Γ . With respect to the interior mesh, we keep the usual notation (Figure 1) except for control volumes K ∈ M with one edge σ , at least, belonging to the boundary. In this case, K is not polygonal (σ is curved), we note K the polygon formed by the vertices of K and by m K its Lebesgue measure. Note that K may not be included in Ω . We will use two different notations for an element of ∂ M: we note e when we consider it as a control volume belonging to ∂ M and we note σ when we consider it as the edge of an interior control volume K ∈ M. Let e ∈ ∂ M be a boundary control volume and e the corresponding chord. Their length are respectively noted m e and m e . If K ∈ M is the control volume such that e ⊂ ∂ K , we set x e as the intersection between Γ and the straight line passing through x K and orthogonal to e. Let y e be the intersection between the line (x K x e ) and the chord e. We define d K ,e as the distance between the centers x K and y e . Let V be the set of the vertices included in Γ and d e,v be the distance between the center y e and the vertex v ∈ V . For a vertex v = e|e ′ ∈ Γ which separates the control volumes e and e ′ , we note d e,e ′ the sum of d e,v and d e ′ ,v .

We can notice that all these quantities are computed by just knowing the coordinates of the vertices of the mesh in Γ . Thus, we do not need to know the equation of the boundary Γ .

Vertex v ∈ V Interior mesh M Boundary mesh ∂ M Centers xe dK ,L xL v = e|e ′ de,v y e ′ xK d e ′ ,v
dK ,e

x e ′ ye nK e nK L

Fig. 1: Finite-volume meshes

We define the mesh size by: h T = sup{diam(K ), K ∈ M}. In the results below, all the constants depend on a certain measure of regularity of the mesh. This is classical and for the sake of simplicity, we do not give here its explicit value. In short, if this quantity is bounded when the mesh size tends to 0, this amounts to assume that the control volumes do not become flat when the mesh is refined.

Discrete unknowns

With respect to the time discretization, we introduce a positive integer N. Then, we uniformly partition the temporal interval [0, T ] with the time step: ∆t = T /N. Thus, for n ∈ {0, • • • , N}, we define t n = n∆t. For each time step t n , we denote the concentration unknowns by

c n T = (c n M , c n ∂ M ) ∈ R T and the chemical potential unknowns by µ n T = (µ n M , µ n ∂ M ) ∈ R T .
Regarding the chemical potential, we have the homogeneous Neumann boundary condition; thus we can define the boundary unknown µ n ∂ M ∈ R ∂ M as follows:

µ n e = µ n K , ∀e ∈ ∂ M such that e = σ ∈ E K .
Finally, let u ∆t M (respectively u ∆t ∂ M ) be the piecewise constant function in ]0, T [×Ω (respectively ]0, T [×Γ ) such that for all t ∈ [t n ,t n+1 [:

u ∆t M (t, x) = u n+1 K if x ∈ K and u ∆t ∂ M (t, x) = u n+1 e if x ∈ e.

Inner products and norms

• Discrete L 2 inner products: For all u M , v M ∈ R M and u ∂ M , v ∂ M ∈ R ∂ M , we define: (u M , v M ) M = ∑ K ∈M m K u K v K and (u ∂ M , v ∂ M ) ∂ M = ∑ e∈∂ M m e u e v e .
The associated discrete L 2 norms are noted u M 0,M and

u ∂ M 0,∂ M . • Discrete H 1 semi-definite inner products: For all u T , v T ∈ R T and u ∂ M , v ∂ M ∈ R ∂ M : u T , v T 1,T = ∑ σ ∈E int m σ d K ,L (u K -u L )(v K -v L ) + ∑ σ ∈E ext m e d K ,e (u K -u e )(v K -v e ) and u ∂ M , v ∂ M 1,∂ M = ∑ v=e|e ′ ∈V 1 d e,e ′ (u e -u e ′ )(v e -v e ′ ).
The associated seminorms are noted

|u T | 1,T and |u ∂ M | 1,∂ M .
3 Numerical scheme and discrete energy

Finite-Volume scheme

In this section, we give the finite-volume scheme used to solve the Cahn-Hilliard equation (1). In the interior mesh M, we use the usual finite-volume approximation based on a consistent two-point flux approximation for Laplace operators. As re-gards the equation on the boundary mesh ∂ M, we use a 1d-finite-volume scheme on a curved domain and a consistent two-point flux approximation for the Laplace-Beltrami operator. We assume that c n T ∈ R T is given, the scheme is then written as follows:

Find (c n+1 T , µ n+1 T ) ∈ R T × R T such that ∀u T , v T ∈ R T :                                        c n+1 M -c n M ∆t , v M M = -µ n+1 T , v T 1,T µ n+1 M , u M M = ∑ σ ∈E int m σ d K ,L (c n+1 K -c n+1 L )(u K -u L ) + ∑ σ ∈E ext m e d K ,e (c n+1 K -c n+1 e )u K + ∑ K ∈M m K d f b (c n K , c n+1 K )u K c n+1 ∂ M -c n ∂ M ∆t , u ∂ M ∂ M = -c n+1 ∂ M , u ∂ M 1,∂ M -∑ e∈∂ M m e d f s (c n e , c n+1 e )u e -∑ σ ∈E ext m e d K ,e (c n+1 e -c n+1 K )u e (3)
With the aim of obtaining convergence result without any condition on the step time ∆t, we use a semi-implicit discretization for nonlinear terms:

d f b (x, y) = f b (y) -f b (x) y -x and d f s (x, y) = f s (y) -f s (x) y -x , ∀x, y. (4)
We can note that we mostly use in practice polynomial functions for f b and f s . Then, the term d f (x, y) can be written as a polynomial function in the variables x, y. Thus, we do not have numerical instability if x is too close to y. If we choose non polynomial functions for nonlinear terms, we have to adapt our discretization (see [START_REF] Boyer | Numerical schemes for a three component Cahn-Hilliard model[END_REF] for more details).

We remark that we can also choose an implicit discretization for nonlinear terms but in that case the same results hold only for ∆t ≤ ∆t 0 , with a small enough ∆t 0 which only depends on the parameters on the equation.

In each case, we have to use a Newton method at each time step; its convergence is achieved in a few inner iterations. We can notice that the finite-volume scheme is a low-order method. Thus, the approximation of the boundary does not influence the order of the method and it is not necessary to use curved element to improve the convergence of the scheme (3).

The boxed terms give the coupling between interior and boundary unknowns: the one in the second equation comes from the Laplacian of c in Ω and the one in the third equation stems from the normal derivative term in the dynamic boundary condition on Γ .

In order to improve the presentation and the analysis, we have written the scheme (3) in a way that looks like a variational formulation. We easily recover the usual finite-volume flux balance equations if, for each control volume, we choose the indicator function of this particular control volume as a test function in (3).

Discrete energy estimate

The discrete energy estimate is one of the key points for the proofs of existence and convergence results.

Definition 1 (Discrete free energy). The discrete free energy corresponding to the continuous definition ( 2) is defined by:

F T (c T ) = 1 2 |c T | 2 1,T + ∑ K ∈M m K f b (c K ) :=F b,T (c T ) + 1 2 |c ∂ M | 2 1,∂ M + ∑ e∈∂ M m e f s (c e ) :=F s,∂ M (c ∂ M ) , ∀c T ∈ R T .
Using the scheme (3) with

u T = c n+1 T -c n T and v T = µ n+1
T as test functions and the the discretization (4) for nonlinear terms, we obtain the following energy equality:

Proposition 1 (Discrete energy estimate). Let c n

T ∈ R T . We assume that there exists a solution (c n+1 T , µ n+1 T ) to Problem [START_REF] Eymard | Finite volume methods[END_REF]. Then, the following equality holds:

F T (c n+1 T ) -F T (c n T ) + ∆t µ n+1 T 2 1,T + 1 ∆t c n+1 ∂ M -c n ∂ M 2 0,∂ M + 1 2 c n+1 T -c n T 2 1,T + 1 2 c n+1 ∂ M -c n ∂ M 2 1,∂ M = 0. (5)
This estimate gives a L ∞ (0, T ; H 1 (Ω )) bound on the discrete solution c ∆t T and a L ∞ (0, T ; H 1 (Γ )) bound on its trace c ∆t ∂ M .

Convergence analysis

By using the topological degree theory, we can prove that if c n T ∈ R T is given, there exists at least one solution (c n+1 [START_REF] Nabet | Finite volume method for the Cahn-Hilliard equation with dynamic boundary conditions[END_REF] for more details).

T , µ n+1 T ) ∈ R T × R T to discrete Problem (3) (see
We recall the definition of a solution to Problem (1) in a weak sense:

Definition 2 (Weak formulation).

We say that a couple

(c, µ) ∈ L ∞ (0, T ; H 1 (Ω )) × L 2 (0, T ; H 1 (Ω )) such that Tr(c) ∈ L ∞ (0, T ; H 1 (Γ )) is solution to continuous Problem (1) in the weak sense if for all ψ ∈ C ∞ c [0, T [×Ω , the following identities hold: T 0 Ω (-∂ t ψc + ∇µ • ∇ψ) = Ω c 0 ψ(0, .), (6)
T 0 Ω -µψ + ∇c • ∇ψ + f ′ b (c)ψ + T 0 Γ -∂ t ψc + ∇ c • ∇ ψ + f ′ s (c)ψ (7) = Γ
Tr(c 0 )ψ(0, .).

Then, we have the following convergence result.

Theorem 1 (Convergence theorem). Assuming that Assumptions 1 hold, let us consider Problem (1) with an initial condition c 0 ∈ H 1 (Ω ) such that Tr(c 0 ) ∈ H 1 (Γ ).

Then, there exists a weak solution (c, µ) on [0, T [ (in the sense of Definition 2). Furthermore, let (c (m) , c

Γ ) m∈N and (µ (m) ) m∈N be a sequence of solutions to Problem (3) associated with a sequence of discretizations such that the space and time steps, h (m) T and ∆t (m) respectively, tend to 0. Then, up to a subsequence, the following convergence properties hold, for all q ≥ 1:

c (m) → c in L 2 (0, T ; L q (Ω )) strongly, c (m)
Γ → Tr(c) in L 2 (0, T ; L q (Γ )) strongly, and µ (m) ⇀ µ in L 2 (0, T ; L q (Ω )) weakly.

The discrete initial concentration used is the mean-value projection. The main difficulty of this proof is the passage to the limit in nonlinear terms both in Ω and on Γ . Indeed, the usual L 2 ((0, T ) × Ω ) compactness is not sufficient and we need to have an additional compactness property of the trace of c in L 2 (]0, T [×Γ ).

Theorem 2 (Estimation of space translates). There exists an extension operator φ : R T → L 2 (R 2 ) satisfying φ (u T ) = u T in Ω such that the following identity holds for all η ∈ R 2 with C > 0 independent of h T and η: For all u T ∈ R T ,

φ (u T )(. + η) -φ (u T ) 2 L 2 (R 2 ) ≤ C|η| (|η| + h T ) |u T | 2 1,T + |u ∂ M | 2 1,∂ M + u ∂ M 2 0,∂ M .
Corollary 1. Let (u T i ) i be a sequence of functions with uniform bounds on discrete H 1 -norms on Ω and Γ . We can extract a subsequence, still referred to as (u T i ) i for simplicity, which is strongly converging in L 2 (Ω ) towards a certain function u of H 1 (Ω ) whose trace belongs to H 1 (Γ ) and such that (u ∂ M i ) i is strongly converging in L 2 (Γ ) towards Tr(u).

To obtain similar results with the sequence of functions which also depends on time, we have to consider the estimation of time translates. To this end, we adapt the proof of Theorem A.2 in [START_REF] Gallouët | Analysis of a fractional-step scheme for the p1 radiative diffusion model[END_REF] and we use the particular form of the extension operator φ and the coupling between the domain Ω and its boundary Γ . Then, thanks to the a priori estimates on the solutions (see [START_REF] Nabet | Finite volume method for the Cahn-Hilliard equation with dynamic boundary conditions[END_REF]), there exists c ∈ L 2 (0, T, H 1 (Ω )) with Tr(c) ∈ L 2 (0, T, H 1 (Γ )) such that, up to a subsequence, c ∆t M strongly converges to c in L 2 (]0, T [×Ω ) and moreover, also c ∆t ∂ M strongly converges to Tr(c) in L 2 (]0, T [×Γ ). It is now more or less standard to pass to the limit in the scheme and thus to prove the convergence result.

Numerical tests

In [START_REF] Nabet | Finite volume method for the Cahn-Hilliard equation with dynamic boundary conditions[END_REF], we give numerical experiments with different choices of parameters and surface potential f s that show the different expected qualitative behavior of the solutions. In this paper, we focus on the numerical error estimates. Since no explicit non trivial solutions are known for our problem, we have to change the Problem (1). We add source term in the first equation of (1) and another one in the third equation of [START_REF] Boyer | Numerical schemes for a three component Cahn-Hilliard model[END_REF]. We notice that µ then satisfies a non homogeneous Neumann boundary condition that can be easily handled in the FV setting. We consider the manufactured solution c(t, (x, y)) = (1 + tanh(5 * (x + t)) with Ω the unit circle. We plot the error between the exact and approximate solutions at time T = 0.5 for the norm L 2 (Ω ) and L 2 (Γ ). As expected, we observe the first order convergence in time for the L 2 norm. With respect to the space convergence, as for the Laplace problem we observe a superconvergence phenomenon namely the second order convergence.

  The potentials are supposed to satisfy standard assumptions: Assumptions 1. :• Dissipativity: lim inf |c|→∞ f ′′ b (c) > 0 and lim inf |c|→∞ f ′′ s (c) > 0. • Polynomial growth for f b : there exist C b > 0 and a real p ≥ 2 such that:

  hT = 0.3 hT = 0.15 hT = 0.08 hT = 0.04