Assia Benabdallah

Yves Dermenjian
email: dermenji@cmi.univ-mrs.fr

Jérôme Le Rousseau

Carleman estimates for stratified media *

Keywords: AMS 2010 subject classification: 35J15, 35K10 elliptic operators, parabolic operators, non-smooth coefficients, stratified media, Carleman estimate, observation location, Contents 1

come

that unique continuation does not hold in general for a C 0,α Hölder regularity of the coefficient with 0 < α < 1 [START_REF] Pliś | On non-uniqueness in Cauchy problem for an elliptic second order differential equation[END_REF][START_REF] Miller | Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients[END_REF], which ruins any hope to prove a Carleman estimate.

In the present article, we consider coefficients that are discontinuous across a smooth interface, yet regular on each side. This question was first addressed in [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF] for a parabolic operator P = ∂ t -∇ x (c(x)∇ x ), with a monotonicity assumption: the observation takes place in the region where the diffusion coefficient c is the 'lowest'. In the one-dimensional case, the monotonicity assumption was relaxed for general piecewise C 1 coefficients [START_REF] Benabdallah | Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications[END_REF][START_REF]Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem[END_REF], and for coefficients with bounded variations [START_REF] Rousseau | Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients[END_REF]. Simultaneously to these results, a controllability result for linear parabolic equations with c ∈ L ∞ was proven in [START_REF] Alessandrini | Null-controllability of one-dimensional parabolic equations[END_REF] in the one-dimensional case without Carleman estimate. This controllability result does not cover more general semi-linear equations. An earlier result was that of [START_REF]On the null controllability of the one-dimensional heat equation with BV coefficients[END_REF] where the controllability of a linear parabolic equations was proven in one dimension with c ∈ BV through D. Russel's method.

The case of an arbitrary dimension without any monotonicity condition in the elliptic case was solved in [START_REF] Bellassoued | Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization[END_REF][START_REF] Rousseau | Carleman estimate for elliptic operators with coefficents with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations[END_REF] and in the parabolic case in [START_REF]Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces[END_REF]. In [START_REF] Rousseau | Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions[END_REF][START_REF]Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions[END_REF] the case of a discontinuous anisotropic matrix coefficients is treated and a sharp condition on the weight function is provided for the Carleman estimate to hold.

The methods used in [START_REF] Bellassoued | Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization[END_REF][START_REF] Rousseau | Carleman estimate for elliptic operators with coefficents with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations[END_REF][START_REF]Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces[END_REF][START_REF] Rousseau | Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions[END_REF][START_REF]Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions[END_REF] focus on a neighborhood of a point at the interface where the interface can be given by {x n = 0} for an appropriate choice of coordinates x = (x ′ , x n ), x ′ ∈ R n-1 , x n ∈ R. Then, through microlocal techniques (Calderón projector or first-order factorization), a local Carleman estimate is proven. However, these methods require strong regularity for the coefficients and for the interface. Moreover, they fall short if the interface crosses the boundary. This configuration is typical in bounded stratified media such as those we consider below.

In stratified media, a controllability result for a linear parabolic equation in arbitrary dimension was obtained in [START_REF]On the controllability of linear parabolic equations with an arbitrary control location for stratified media[END_REF]. The approach was based on the 1D Carleman estimates of [START_REF] Benabdallah | Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications[END_REF][START_REF] Rousseau | Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients[END_REF] in the parabolic case and a spectral inequality as in [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF][START_REF] Lebeau | Null-controllability of a system of linear thermoelasticity[END_REF][START_REF] Jerison | Chicago Lectures in Mathematics, ch. Nodal sets of sums of eigenfunctions[END_REF] for the transverse elliptic operator, whose coefficients are smooth. The precise definition of such stratified media is given below.

The controllability result obtained in [START_REF]On the controllability of linear parabolic equations with an arbitrary control location for stratified media[END_REF] left the question of deriving a Carleman estimate open for stratified media in dimension greater than two in both the elliptic and the parabolic cases. This result is achieved here. One of the consequences of this result in the parabolic case is the null-controllability of classes of semi-linear parabolic equations. We refer to [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF] for these developments.

Remark 1.1. The following observation also provides hints that Carleman estimates can be derived for stratified media [START_REF] Poisson | Personnal communication[END_REF]. As we shall assume below interfaces cross the boundary transversely. Pick a point at the intersection of an interface and the boundary and choose local coordinates such that the interface is orthogonal to the boundary. Assume that the coefficients associated with the transverse part of the operator are flat at the boundary. Then, by reflection at the boundary, the system under consideration can be turned into a problem with a smooth interface away from any boundary which permits to use the results of [START_REF] Rousseau | Carleman estimate for elliptic operators with coefficents with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations[END_REF][START_REF]Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces[END_REF][START_REF] Rousseau | Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions[END_REF][START_REF]Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions[END_REF]. This situation is however not general.

We finish this introductory presentation by pointing out the difficulty that arises when deriving a Carleman estimate for the operator A = -∇ x (c(x)∇ x ) or P = ∂ t -∇ x (c(t, x)∇ x ) in dimension greater than two, in the presence of an interface S . In fact, the standard Carleman derivation method leads to interface terms involving 1. trace of the function u |S . Zero-and first-order operators in the tangent direction act on u |S .

2. traces of its normal derivative ∂ x n u |S ± , on both sides of S . This interface contribution can be interpreted as a quadratic form (see [START_REF] Benabdallah | Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications[END_REF]). In [START_REF] Rousseau | Carleman estimate for elliptic operators with coefficents with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations[END_REF][START_REF]Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces[END_REF] the authors show that this quadratic form is only non-negative for low (tangential) frequencies. Here we shall recover this behavior where the tangential Fourier transform is replaced by Fourier series, built on a basis of eigenfunctions of the transverse part of the elliptic operator. For high (tangential) frequencies, the tangential derivative term (i.e., the action of a the first-order operator on u |S ) yields a negative contribution, unless a monotonicity assumption on the coefficient c is made as in [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF]. In [START_REF] Bellassoued | Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization[END_REF][START_REF] Rousseau | Carleman estimate for elliptic operators with coefficents with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations[END_REF][START_REF]Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces[END_REF] the authors have used microlocal methods in the high frequency regime to solve this difficulty, and more recently in [START_REF]Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions[END_REF][START_REF] Rousseau | Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions[END_REF]. Here, because of the intersection of the interface with the boundary, and because of the little regularity required for the diffusion coefficients, such methods cannot be used directly. However, the separated-variable assumption made on the diffusion coefficients allows us to use Fourier series and similar ideas can be developed: low frequencies and high frequencies are treated differently. In the parabolic case the separation we make between the two frequency regimes is time dependent. Here, the separated-variable assumption yields explicit computations, which reveals the behavior of the solution in each frequency regime.

In the present article, a particular class of anisotropic coefficients is treated. The question of deriving Carleman estimates for more general coefficients in the neighborhood of the intersection of an interface, where the coefficients jump, with the boundary is left open.

Setting and notation

We let

Ω be an open subset in R n , with Ω = Ω ′ × (-H, H), where Ω ′ is a nonempty bounded open subset of R n-1 with C 1 boundary 2 .
We shall use the notation x = (x ′ , x n ) ∈ Ω ′ × (-H, H). We set S = Ω ′ × {0}, that will be understood as an interface where coefficients and functions may jump. For a function f = f (x) we define by [ f ] S its jump at S , i.e.,

[ f ] S (x ′ ) = f (x)| x n =0 + -f (x)| x n =0 -.
For a function u defined on both sides of S , we set

u |S ± = u |Ω ± |S , with Ω + = Ω ′ × (0, H) and Ω -= Ω ′ × (-H, 0).
Let B(t, x), t ∈ (0, T ) and x ∈ Ω, be with values in M n (R), the space of square matrices with real coefficients of order n. We make the following assumption.

Assumption 1.2. The matrix diffusion coefficient B(t, x

′ , x n ) has the following block diagonal form B(t, x ′ , x n ) = c 1 (t, x n )C 1 (x ′ ) 0 0 c 2 (t, x n )
where the functions c i , i = 1, 2, are3 in C 1 ((0, T ) × Ω ± ) with a possible jump at x n = 0. We assume

C 1 ∈ W 1,∞ (Ω ′ , M n-1 (R))
and that C 1 (x ′ ) is hermitian. We further assume uniform ellipticity

0 < c min ≤ c i (t, x n ) ≤ c max < ∞, (t, x n ) ∈ (0, T ) × (-H, H) and i = 1, 2, 0 < c min Id n-1 ≤ C 1 (x ′ ) ≤ c max Id n-1 , x ′ ∈ Ω ′ .
To lighten notation we shall often write c i-:= c i|x n =0 -and c i+ := c i|x n =0 + for i = 1, 2.

Remark 1.3. Here, the matrix coefficient B is chosen time dependent in preparation for the Carleman estimate in the parabolic case. We shall also prove such an estimate in the elliptic case: see Theorem 1.4 below and its proof in Section 3. For this theorem we shall of course use B independent of time.

For the proof of Theorem 1.4 (elliptic case) we shall further assume c 1 = c 2 . In fact, this simplification allows us to provide a fairly simple proof of the Carleman estimates that shows the different treatment of two frequency regimes. These frequency regimes are connected to the microlocal regions used in [START_REF] Rousseau | Carleman estimate for elliptic operators with coefficents with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations[END_REF] and [START_REF] Rousseau | Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions[END_REF][START_REF]Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions[END_REF]. Note however that the case c 1 c 2 can also be treated in the elliptic case. The proof is then closer to that of the parabolic case of Theorem 1.5 in Section 4. We have omitted this proof for the sake of the clarity of the exposition.

Let T > 0. For each t ∈ [0, T ], we consider the symmetric bilinear H 1 0 -coercive form [START_REF] Dautray | Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques[END_REF], p. 1211). In the elliptic case, we shall denote by • L 2 (Ω) the L 2 norm over Ω and by | • | L 2 (S ) the L 2 norm over the interface S of codimension 1.

a t (u, v) = ∫ Ω (B(t, •)∇ x u) • ∇ x vdx, with domain D(a t ) = H 1 0 (Ω). It defines a selfadjoint operator A t = -∇ x • (B(t, •)∇ x ) in L 2 (Ω) with compact resolvent and with domain D(A t ) = {u ∈ H 1 0 (Ω); ∇ x • (B(t, •)∇ x u) ∈ L 2 (Ω)} (see e.g.
We set Q T = (0, T )×Ω, S T = (0, T )×S . We shall also consider the following parabolic operator P = ∂ t + A t on Q T . In the parabolic case, we shall denote by

• L 2 (Q T ) the L 2 norm over Q T and by | • | L 2 (S T ) the L 2 norm over the interface S T of codimension 1.
In this article, when the constant C is used, it refers to a constant that is independent all the parameters. Its value may however change from one line to another. We shall use the notation a b if we have a ≤ Cb for such a constant. If we want to keep track of the value of a constant we shall use another letter.

Statements of the main results

We consider ω, a nonempty open subset of Ω. For a function β in C 0 (Ω) we set ϕ(x) = e λβ(x) , λ > 0, to be used as weight function. We consider first a matrix coefficient independent of the parameter t. A proper choice of the function β, with respect to the operator A, ω and Ω (see Assumption 2.1 and Assumption 3.2), yields the following Carleman estimate for the elliptic operator A.

Theorem 1.4 (Elliptic case).

There exist C > 0, λ 0 and s 0 > 0 such that

sλ 2 e sϕ ϕ 1 2 ∇u 2 L 2 (Ω) + s 3 λ 4 e sϕ ϕ 3 2 u 2 L 2 (Ω) + sλ |e sϕ ϕ 1 2 ∇ τ u |S | 2 L 2 (S ) + |e sϕ ϕ 1 2 ∂ n u |S | 2 L 2 (S ) + s 3 λ 3 |e sϕ ϕ 3 2 u |S | 2 L 2 (S ) ≤ C e sϕ Au 2 L 2 (Ω) + s 3 λ 4 e sϕ ϕ 3 2 u 2 L 2 (ω) ,
for all u ∈ D(A), λ ≥ λ 0 , and s ≥ s 0 .

Here, ∇ τ is the tangential gradient on the interface S . Note that belonging to the domain D(A) implies some constraints on the function u at the interface S , namely u ∈ H 1 and B∇ x u ∈ H(div, Ω). We shall first prove the result for piecewise smooth functions satisfying

u |S -= u |S + , (c∂ x n u) |S -= (c∂ x n u) |S + ,

and then use their density in D(A).

With a function β > 0 that satisfies Assumption 2.1 below, we introduce β = β + m β ∞ where m > 1. For λ > 0 we define the following weight functions ϕ(x) = e λβ(x) , η(x) = e λβ(x)e λβ , a(t) = t(Tt) -1 , with β = 2m β ∞ (see [START_REF] Fursikov | Controllability of evolution equations[END_REF][START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF]). For β satisfying some additional requirements (Assumption 4.2), that will be provided in Section 4, we prove the following Carleman estimate for the parabolic operator P.

Theorem 1.5 (Parabolic case).

There exist C > 0, λ 0 and s 0 > 0 such that s -1 e saη (aϕ)

-1 2 ∂ t u 2 L 2 (Q T ) + e saη (aϕ) -1 2 A t u 2 L 2 (Q T ) + sλ 2 e saη (aϕ) 1 2 ∇u 2 L 2 (Q T ) + s 3 λ 4 e saη (aϕ) 3 2 u 2 L 2 (Q T ) + sλ |e saη (aϕ) 1 2 ∇ τ u |S T | 2 L 2 (S T ) + |e saη (aϕ) 1 2 ∂ n u |S T | 2 L 2 (S T ) + s 3 λ 3 |e saη (aϕ) 3 2 u |S T | 2 L 2 (S T ) ≤ C e saη Pu 2 L 2 (Q T ) + s 3 λ 4 e saη (aϕ) 3 2 u 2 L 2 ((0,T )×ω) , for all u ∈ C 2 ((0, T ) × Ω ± ) such that u |S - T = u |S + T , (c 2 ∂ x n u) |S - T = (c 2 ∂ x n u) |S + T , λ ≥ λ 0 , and s ≥ s 0 (T + T 2 ).
By a density argument, we can extend this estimate to functions in

∫ ⊕ [0,T ] D(A t ) dt∩H 1 (0, T ; L 2 (Ω)).
Larger function spaces, with rougher behaviors can also be handled such as explosion at times t = 0 and t = T if they are compensated by the rapidly vanishing weight function e saη . The r.h.s. of the estimate can be used to define a norm. The larger the parameters s and λ, the bigger the associated spaces will be. Such choices can be driven by applications.

Outline

In Section 2, we provide some spectral properties of operator A, which yields a Hilbert direct decomposition of L 2 (Ω) = ⊕ k∈N * H k that reduces A. We also provide the precise assumptions made on the weight function. In Section 3, we prove the Carleman estimate for the elliptic case. In Section 4 we prove the Carleman estimate for a parabolic case. Some intermediate and technical results are collected in the appendices.

Spectral properties and weight function

Similarly to A t = -∇ x • (B(t, x)∇ x ), one can define the time independent selfadjoint transverse operator on L 2 (Ω ′ )

A ′ = -∇ x ′ • (C 1 ∇ x ′ ), D(A ′ ) = {u ∈ H 1 0 (Ω ′ ); ∇ x ′ • (C 1 ∇ x ′ u) ∈ L 2 (Ω ′ )}.
We consider an orthonormal basis of L 2 (Ω ′ ), composed of eigenfunctions (φ k ) k≥1 , associated with the eigenvalues, with finite multiplicities, 0

< µ 2 1 ≤ µ 2 2 ≤ • • • ≤ µ 2 k ≤ µ 2 k+1 ≤ • • • , with µ k → ∞. With this basis (φ k ) k≥1 , we build an unitary transform F : L 2 (Ω) → ∞ ⊕ k=1 L 2 (-H, H) defined by (F u)(k, x n ) := ∫ Ω ′ φ k (x ′ )u(x ′ , x n ) dx ′ , (2.1)
with the following properties (recall that here

∇ x ′ = ∇ τ ) v(x ′ , x n ) = ∞ k=1 (v(., x n ), φ k ) L 2 (Ω ′ ) φ k (x ′ ) = ∞ k=1 v(k, x n )φ k (x ′ ), ∇ x ′ v(x ′ , x n ) = ∞ k=1 v(k, x n )∇ x ′ φ k (x ′ ).
We shall often write vk = v(k, .).

As the family (C 1/2 1 ∇φ k ) k is orthogonal in L 2 (Ω ′ ) (C 1 is a positive definite matrix) we have

C 1/2 1 ∇ x ′ v(., x n ) 2 L 2 (Ω ′ ) = ∞ k=1 |v(k, x n )| 2 C 1/2 1 ∇ x ′ φ k 2 L 2 (Ω ′ ) = ∞ k=1 |v(k, x n )| 2 µ 2 k , which gives (c max ) -1 ∞ k=1 µ 2 k v(k, x n ) 2 ≤ ∇ τ v(., x n ) 2 L 2 (Ω ′ ) ≤ (c min ) -1 ∞ k=1 µ 2 k v(k, x n ) 2 . (2.2)
We choose a weight function β that satisfies the following properties.

Assumption 2.1. The function β ∈ C 0 (Ω), and

β |Ω ± ∈ C 2 (Ω ± ) and β ≥ C > 0, |∇ x β| ≥ C > 0 in Ω \ ω, β = Cst on Ω ′ × {-H} and β = Cst on Ω ′ × {H}. ∇ x ′ β = 0 on ∂Ω ′ × (-H, H), ∂ x n β > 0 on Ω ′ × {-H}, and ∂ x n β < 0 on Ω ′ × {H}.
There exists a neighborhood V of S in Ω of the form V = Ω ′ × (-δ, δ) in which β solely depends on x n and is a piecewise affine function of x n .

In particular β |S is constant. As the open set ω can be shrunk if necessary, we further assume that ω ∩ Ω ′ × (-δ, δ) = ∅.

Such a weight function β can be obtained by first designing a function that satisfies the proper properties at the boundaries and at the interface and then construct β by means of Morse functions following the method introduced in [START_REF] Fursikov | Controllability of evolution equations[END_REF].

Here, in addition we assume that ∂ x n β = β ′ > 0 on S + and S -, which means that the observation region ω is chosen in Ω ′ ×(0, H), i.e., where x n ≥ 0. There is no loss in generality as we can change x n into -x n to treat the case of an observation ω ⊂ Ω ′ × (-H, 0). Note that Assumption 2.1 will be completed below by Assumption 3.2 in the elliptic case and Assumption 4.2 in the parabolic case respectively.

The elliptic case: proof of Theorem 1.4

As mentioned in the introductory section, we have consider only the case c 1 = c 2 = c in this proof. The case c 1 c 2 can be treated following the lines of the proof of Theorem 1.5 in Section 4.

Local Carleman estimates can be stitched together to form a global estimate of the form presented in Theorem 1.4 (see e.g. [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators[END_REF][START_REF]Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces[END_REF]). Such local estimates are classical away from the interface (see [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF][START_REF] Fursikov | Controllability of evolution equations[END_REF][START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators[END_REF]). To prove the elliptic Carleman estimate of Theorem 1.4 it thus remains to prove such a local estimate at the interface S , for functions u ∈ D(A) with support near the interface. We shall thus assume that supp(u) ⊂ Ω ′ × (-δ, δ), where the weight function β depends only on x n and is piecewise affine.

Piecewise smooth functions that satisfy the transmission conditions

u |S -= u |S + , (c∂ x n u) |S -= (c∂ x n u) |S + , (3.1)
are dense in D(A). We may thus restrict our analysis to such functions. Because of these transmission conditions we shall write u |S and (c∂

x n u) |S in place of u |S ± and (c∂ x n u) |S ± respectively.
Applying the unitary transform of Section 2, the equation Au = f can be written

(-∂ x n c∂ x n + cµ 2 k )û k (x n ) = fk (x n ), x n ∈ (-δ, 0) ∪ (0, δ),
with supp ûk ⊂ (-δ, δ).

Our starting point is the following proposition.

Proposition 3.1. Let the weight function β satisfy Assumption 2.1. There exist C, C

′ , C ′′ > 0, λ 0 > 0, s 0 > 0 such that C sλ 2 ϕ 1 2 ∂ x n vk 2 L 2 (-δ,δ) + sλ 2 ϕ 1 2 µ k vk 2 L 2 (-δ,δ) + s 3 λ 4 ϕ 3 2 vk 2 L 2 (-δ,δ) + sλϕ |S [c 2 β ′ |∂ x n vk | 2 ] S + |sλϕv k|S | 2 [c 2 β ′3 ] S -|µ k vk|S | 2 [c 2 β ′ ] S ≤ C ′ e sϕ fk 2 L 2 (-δ,δ) + Z, (3.2) for all k ∈ N * , vk = e sϕ ûk , λ ≥ λ 0 and s ≥ s 0 , with Z = -C ′′ sλ 2 ϕ |S Re[c 2 β ′2 ∂ x n vk ] S vk|S .
We emphasize that the constants are uniform with respect to the transverse-mode index k. Such a result can be obtained by adapting the derivations in [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF] for instance. We provide a short proof in Appendix A.1. In particular we have

|Z| ≤ Csλ 2 ϕ |S |∂ x n vk|S -| + |∂ x n vk|S + | |v k|S |. (3.3)
Moreover, in addition to Assumption 2.1, we shall consider the following particular form of β Assumption 3.2. For K = c - c + and some r ≥ 0, we have

L = β ′ |S + β ′ |S - =              2 if K = 1, K if K > 1, (r + 1) -rK if K < 1. (3.4) Remark 3.3.
1. With this assumption we note that we have L > 1 and

L → 1 as K → 1, K 1.
Here we choose L = 2 if K = 1, to preserve interface terms in the Carleman estimates even for this case that corresponds to coefficients with no jump.

2. The value r = 3 is admissible in (3.4) (see Lemma 3.6 and its proof). In the spirit of what is done in [START_REF] Rousseau | Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients[END_REF] one may wish to control the jump of the slope of the weight function by choosing other values for r.

3. To construct the weight function β we first choose its slopes on both sides of the interface satisfying Assumption 3.2. Here the slopes are positive as we wish to observe the solution in {x n > 0}. We may then extend the function β on both sides of the interface. The additional requirements of Assumption 2.1 only concern the behavior of the β away from the interface. The two assumptions are compatible.

We now set

B(v) = sλϕ |S [c 2 β ′ |∂ x n v| 2 ] S + |sλϕv |S | 2 [c 2 β ′3 ] S .
Lemma 3.4. We have

B(v k ) = sλϕ |S e 2sϕ |S B 1 |γ(û k )| 2 + B 2 |sλϕû k|S | 2 , γ(û k ) = c∂ x n ûk|S + c + β ′ |S - L 2 -K L -1 (sλϕû k ) |S , with B 1 = β ′ |S -(L -1) > 0, and
B 2 = c 2 + (β ′ |S -) 3 2(L 3 -K 2 ) - L 2 -K 2 L -1 . (3.5)
For a proof see Appendix A.2. Note that L > 1 by Assumption 3.2.

We shall consider two cases: K > 1 and 0 < K ≤ 1.

Case K > 1. Then L = K and -[c 2 β ′ ] S = -c 2 + β ′ |S -(L -K 2 ) > 0, B 1 > 0, B 2 = c 2 + (β ′ |S -) 3 K 2 (K -1) > 0.
The trace terms in (3.2) thus yield a positive contribution. We have

B(v k ) -sλϕ |S |µ k vk|S | 2 [c 2 β ′ ] S sλϕ |S 3 e 2sϕ |û k|S | 2 + sλϕ |S e 2sϕ |γ(û k )| 2 + |µ k ûk|S | 2 sλϕ |S 3 e 2sϕ |û k|S | 2 + sλϕ |S e 2sϕ |∂ x n ûk|S | 2 + |µ k ûk|S | 2 .
In particular for s sufficiently large the remainder term Z estimated in (3.3) can be 'absorbed'. We thus obtain

sλ 2 ϕ 1 2 ∂ x n vk 2 L 2 (-δ,δ) + sλ 2 ϕ 1 2 µ k vk 2 L 2 (-δ,δ) + s 3 λ 4 ϕ 3 2 vk 2 L 2 (-δ,δ) + sλϕ |S 3 e 2sϕ |û k|S | 2 + sλϕ |S e 2sϕ |∂ x n ûk|S | 2 + |µ k ûk|S | 2 e sϕ fk 2 L 2 (-δ,δ) , (3.6)
for all k ∈ N * . Summing over k, using (2.2) we obtain the sought local Carleman estimate in the case K > 1

sλ 2 ϕ 1 2 ∇v 2 L 2 (Ω ′ ×(-δ,δ)) + s 3 λ 4 ϕ 3 2 v 2 L 2 (Ω ′ ×(-δ,δ)) + sλϕ |S 3 e 2sϕ |S |u |S | 2 L 2 (S ) + sλϕ |S e 2sϕ |S |∇u |S | 2 L 2 (S ) e sϕ f 2 L 2 (Ω ′ ×(-δ,δ)) . (3.7)
The Carleman estimate of Theorem 1.4 can then be deduced classically. This case, K > 1 is the case originally covered by [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF].

Case 0 < K ≤ 1. Then, either L = (r + 1) -rK > 1 or L = 2, which gives B 1 > 0. Lemma 3.6 below implies that B 2 > 0. Hence, for s sufficiently large the remainder term Z estimated in (3.3) can be 'absorbed'. We now aim to estimate the tangential term in (3.2).

Proposition 3.5.

There exists C > 0, and ε > 0 such that for all k ∈ N we have

sλϕ |S [c 2 β ′ ] S |µ k vk|S | 2 ≤ 1 1 + ε sλϕ |S B 2 sλ(ϕv k ) |S 2 (3.8) + C e sϕ fk 2 L 2 (-δ,δ) + s 2 λ 2 ϕv k 2 L 2 (-δ,δ) + ∂ x n vk 2 L 2 (-δ,δ) .
Proof. Let 0 < ε < 1. The value of ε will be determined below. We treat low and high values of µ k differently.

Low frequencies.

Set k 1 as the largest integer such that (1

+ ε) [c 2 β ′ ] S µ 2 k < B 2 (sλϕ) 2 |S , that is (1 + ε)µ 2 k < (β ′ |S -) 2 (sλϕ) 2 |S 1 |L -K 2 | 2(L 3 -K 2 ) - L 2 -K 2 L -1 . (3.9)
We then have

(1 + ε)sλϕ |S [c 2 β ′ ] S |µ k vk|S | 2 < sλϕ |S B 2 sλ(ϕv k ) |S 2 , k ≤ k 1 . (3.10)
High frequencies. Here we consider frequencies µ k that satisfy

(1 -ε)µ k ≥ s|∂ x n ϕ |S -| = sλϕ |S β ′ |S -. (3.11)
We denote by k 2 the smallest integer that satisfies (3.11).

We write

(∂ 2 x n -µ 2 k )û k = - fk c - ∂ x n c c ∂ x n ûk = -ĝ k .
As ûk (-δ) = ûk (δ) = 0, with the transmission conditions (3.1), the computations4 of Appendix A.3 yield

µ k ûk | x n =0 + = 1 (c + + c -) δ ∫ 0 sinh µ k (δ -x n ) cosh(µ k δ) c + ĝk (x n ) + c -ĝk (-x n ) dx n . (3.12)
We have

sinh µ k (δ -x n ) cosh(µ k δ) = e µ k (δ-x n ) -e -(µ k (δ-x n )) e µ k δ + e -(µ k δ) ≤ e -µ k x n . (3.13)
We note that

ϕ(0) -ϕ(-x n ) = x n 1 ∫ 0 ϕ ′ (-x n + σx n ) dσ = x n λβ ′ |S - 1 ∫ 0 ϕ(-x n + σx n ) dσ,
as the weight function β = β(x n ) is affine in (-δ, 0). Since β ′ > 0, the function ϕ increases with x n and we have ϕ(0

) ≤ ϕ(-x n ) + x n λϕ(0)β ′ |S -, if x n > 0.
As we have assumed (3.11) here we obtain

sϕ(0) -µ k x n ≤ sϕ(-x n ) -εµ k x n , x n > 0. (3.14)
We also have

sϕ(0) -µ k x n ≤ sϕ(x n ) -εµ k x n , x n > 0. (3.15)
From (3.12) we thus obtain

µ 3 2 k e sϕ| S |û k|S | ≤ 1 (c + + c -) δ ∫ 0 e sϕ(-x n ) |c -ĝk (-x n )| + e sϕ(x n ) |c + ĝk (x n )| µ 1 2 k e -εµ k x n dx n e sϕ ĝk L 2 (-δ,0) + e sϕ ĝk L 2 (0,δ) δ ∫ 0 µ k e -2εµ k x n dx n 1 2 ε -1 2 e sϕ ĝk L 2 (-δ,δ) ε -1 2 e sϕ fk L 2 (-δ,δ) + sλ ϕv k L 2 (-δ,δ) + ∂ x n vk L 2 (-δ,δ) , which leads to, for k ≥ k 2 , sλϕ |S [c 2 β ′ ] S |µ k vk|S | 2 (1 -ε)|β ′ -| -1 [c 2 β ′ ] S µ 3 k |v k|S | 2 (1 -ε)ε -1 |β ′ -| -1 [c 2 β ′ ] S e sϕ fk 2 L 2 (-δ,δ) + s 2 λ 2 ϕv k 2 L 2 (-δ,δ) + ∂ x n vk 2 L 2 (-δ,δ) .
We have thus seen that low frequencies in (3.8) are estimated by boundary terms and high frequencies are estimated by the r.h.s. of (3.2) and "absorbable" terms. It remains to prove that we cover the whole spectrum with the two estimates we have obtained. A sufficient condition is then

(1 -ε) -2 (sλϕ |S ) 2 (β ′ |S -) 2 ≤ 1 1 + ε (β ′ |S -) 2 (sλϕ) 2 |S 1 |L -K 2 | 2(L 3 -K 2 ) - L 2 -K 2 L -1 ,
that is

P(K, L) := -|L -K 2 |(L -1) + (1 -ε) 2 1 + ε 2(L 3 -K 2 )(L -1) -L 2 -K 2 ≥ 0. (3.16)
We recall that L = (r + 1) -rK if 0 < K < 1. The following lemma provides a positive answer (see Appendix 3.6 for a proof).

Lemma 3.6.

There exists ε 0 > 0 such that for 0 < ε < ε 0 ,

• P(K, L) ≥ 0 if K = 1,
• there exists r ≥ 1 such that P(K, L) ≥ 0 for K ∈ (0, 1). In particular the value r = 3 is admissible.

In particular we have B 2 > 0.

This concludes the proof of Proposition 3.5.

Arguing as we did for (3.7) in the case K > 1, we now obtain

sλ 2 ϕ 1 2 ∇v 2 L 2 (Ω ′ ×(-δ,δ)) + s 3 λ 4 ϕ 3 2 v 2 L 2 (Ω ′ ×(-δ,δ)) + sλϕ |S 3 e 2sϕ |S |u |S | 2 L 2 (S ) + sλϕ |S e 2sϕ |S |∇u |S | 2 L 2 (S ) e sϕ f 2 L 2 (Ω ′ ×(-δ,δ)) + s 2 λ 2 ϕv 2 L 2 (Ω ′ ×(-δ,δ)) + ∂ x n v 2 L 2 (Ω ′ ×(-δ,δ)) . (3.17)
The last two terms on the r.h.s. can be "absorbed" by the l.h.s. by choosing s sufficiently large. This concludes the case 0 < K ≤ 1 and the proof of Theorem 1.4. 4 The parabolic case: proof of Theorem 1.5

Here, the matrix coefficient B is assumed to be time dependent as stated in Assumption 1.2. The coefficients c 1 (t, x n ) and c 2 (t, x n ) can be different.

We choose a function β > 0 that satisfies the requirements of Assumption 2.1 and we introduce β = β + m β ∞ where m > 1. Observe that β also satisfies Assumption 2.1.

For T > 0 and λ > 0 we define the following weight functions

ϕ(x) = e λβ(x) , η(x) = e λβ(x) -e λβ , x ∈ Ω, a(t) = t(T -t) -1 , t ∈ (0, T ), (4.1)
with β = 2m β ∞ (see [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF]). Note that η < 0. As in the previous sections we choose β ′ > 0 on S + and S -, which means that the observation region ω is chosen in Ω ′ × (0, H), i.e., where x n ≥ 0. It suffices to prove a local Carleman estimate at the interface S , i.e., for functions u with support near the interface, supp(u) ⊂ [0, T ] × Ω ′ × (-δ, δ), where the weight function β depends only on x n and is piecewise affine.

We assume moreover that u satisfies the transmission conditions

u |S - T = u |S + T , (c 2 ∂ x n u) |S - T = (c 2 ∂ x n u) |S + T . (4.2)
Applying the unitary transform of Section 2, the equation ∂ t u + Au = f can be written

(∂ t -∂ x n c 2 ∂ x n + c 1 µ 2 k )û k (t, x n ) = fk (t, x n ), t ∈ (0, T ), x n ∈ (-δ, 0) ∪ (0, δ), k ≥ 1, with supp(û k ) ⊂ [0, T ] × (-δ, δ).
Setting q T,δ = (0, T ) × (-δ, δ), our starting point is the following proposition.

Proposition 4.1. Let T > 0. There exist C, C ′ > 0, λ 0 > 0, s 0 > 0 such that C s -1 (aϕ) -1 2 ∂ t vk 2 L 2 (q T,δ ) + sλ 2 (aϕ) 1 2 ∂ x n vk 2 L 2 (q T,δ ) + sλ 2 (aϕ) 1 2 µ k vk 2 L 2 (q T,δ ) + s 3 λ 4 (aϕ) 3 2 vk 2 L 2 (q T,δ ) + sλ T ∫ 0 aϕ |S [c 2 2 β ′ |∂ x n vk | 2 ] S + |sλa(ϕv k ) |S | 2 [c 2 2 β ′3 ] S dt ≤ C ′ e saη fk 2 L 2 (q T,δ ) + sλ T ∫ 0 aϕ |S |µ k vk|S T | 2 [c 1 c 2 β ′ ] S dt + Z, (4.3)
for all k ∈ N * , vk = e saη ûk , λ ≥ λ 0 and s ≥ s 0 (T + T 2 ), with

|Z| s 1 2 λT T ∫ 0 aϕ |S |∂ x n vk | 2 |S - T + |∂ x n vk | 2 |S + T dt + s(T 3 + T 4 )λ + s 3 2 T 3 λ 3 T ∫ 0 a 3 ϕ |S |v k | 2 |S T dt. ( 4.4)
We emphasize that the constants are uniform with respect to the transverse-mode index k. Such a result can be obtained by adapting the derivations in [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF] for instance. We provide a short proof in Appendix A.5.

As in Section 3, we set

B p (v k ) = sλaϕ |S [c 2 2 β ′ |∂ x n vk | 2 ] S + |sλaϕv k|S T | 2 [c 2 2 β ′3 ] S , L = β ′ |S + β ′ |S - , K i (t) = c i-(t) c i+ (t) , K i = inf t∈[0,T ] K i (t), K i = sup t∈[0,T ] K i (t), i = 1, 2. (4.5)
and

B = B(L) = inf t∈[0,T ] (c 2 2+ (t) ) (β ′ |S -) 3 K 2 2 + L 3 (L -L) L -1 , with L = max{K 2 , 2}, (4.6)
and finally

D = D(L) = sup t∈[0,T ] (c 1+ c 2+ )(t) β ′ |S -(L + K 1 K 2 ) > 0. (4.7)
We make the following assumption on the weight function in addition to Assumption 2.1.

Assumption 4.2. The weight function β is chosen such that

L ≥ L = max{K 2 , 2} and
1 2 ≥ max 2 D B , 4β ′ -|S σ D B , σ = inf t,x n c 1 (t, x n ) c 2 (t, x n ) 1 2 , ( 4.8)
The coefficients c 1 , c 2 being fixed, the forms of the coefficients D and B show that this can be achieved by first choosing the value of β ′ |S -> 0 and then picking a sufficiently large value for L.

Remark 4.3. To construct the weight function β we first choose its slopes on both sides of the interface satisfying Assumption 4.2. Here the slopes are positive as we wish to observe the solution in {x n > 0}. We may then extend the function β on both sides of the interface. The additional requirements of Assumption 2.1 only concern the behavior of the β away from the interface. The two assumptions are compatible.

Lemma 4.4. We have

B p (v k ) = sλaϕ |S e 2saϕ |S B 1 |γ(û k )| 2 + B 2 |sλa(ϕû k ) |S | 2 , with γ(û k ) = (c 2 ∂ x n ûk ) |S + c 2+ β ′ |S - L 2 -K 2
L-1 (sλaϕû k ) |S and where

B 1 = β ′ |S -(L -1), B 2 (t) = c 2 2+ (t)(β ′ |S -) 3 2(L 3 -K 2 2 (t)) - L 2 -K 2 (t) 2 L -1 .
If β Assumption 4.2 we have B 1 > 0 and B 2 ≥ B, with B defined in (4.6).

Proof. The proof of Lemma 3.4 in Appendix A.2 can be directly adapted and gives the first part of the lemma. As L ≥ 1 we have

B 1 > 0. A direct computation yields B 2 (t) = c 2 2+ (t)(β ′ |S -) 3 P p (L,K 2 (t))
L-1

with

P p (L, Y) = Y 2 (1 -2L) + 2Y L 2 + L 4 -2L 3 = L 3 (L -L) + L 3 (L -2) + 2LY(L -Y) + Y 2 .
As L ≥ 2, and L ≥ K 2 ≥ K 2 (t) ≥ K 2 > 0, we thus obtain P p (L, K 2 (t)) ≥ K 2 2 + L 3 (L -L). We now prove the following key result, providing an estimate of the tangential derivative of v, i.e., µ k vk , in the Fourier decomposition. Proposition 4.5. For a weight function β that satisfies Assumptions 2.1 and 4.2 there exists C > 0 such that for all k ∈ N * we have sλ

T ∫ 0 aϕ |S |[c 1 c 2 β ′ ] S | |µ k vk|S | 2 dt ≤ B 4 (sλ) 3 |(aϕ |S ) 3 2 vk ) |S | 2 L 2 ((0,T )) + C e saη fk 2 L 2 (q T,δ ) + s 3 λ 3 (aϕ) 3 2 vk 2 L 2 (q T,δ ) + sλ (aϕ) 1 2 ∂ x n vk 2 L 2 (q T,δ ) + sλ (aϕ) 1 2 µ k vk 2 L 2 (q T,δ ) . (4.9)
for λ and s/(T + T 2 ) both sufficiently large.

Proof. We fix k ≥ 1 and we shall keep track of the dependency of the constants on k.

We have If µ k > µ s,λ , there exists t k := t k (s, λ) ∈ (0, T/2) such that

[c 1 c 2 β ′ ] S ≤ (c 1+ c 2+ )(t) β ′ |S -(L + K 1 K 2 (t)) ≤ D,
T t 0 Φ t k T -t k T 2 µ s,λ I k I k µ k
µ k = Φ(t k ; s, λ) = Φ(T -t k ; s, λ). (4.11)
For µ k ≥ µ s,λ , we set

I k := (0, t k ) ∪ (T -t k , T ), J k := (0, T ) \ I k = (t k , T -t k ), Jk := t k 2 , T - t k 2 .
For µ k < µ s,λ , we set I k := (0, T ).

We then introduce

I(k; s, λ) := sλD ∫ I k a(t)ϕ |S µ 2 k |v k|S | 2 dt, (4.12) J(k; s, λ) := sλD ∫ J k a(t)ϕ |S µ 2 k |v k|S | 2 dt, ( 4.13)
so that the term on the l.h.s. of (4.9) is less than the sum of the two previous quantities.

The first term, I(k; s, λ), involving time t close to 0 or , will be estimated by a trace term. The second term, J(k; s, involving time t away from 0 and T , will be estimated by volume terms.

Step 1: µ k ≤ µ s,λ or t ∈ I k In the (t, µ k ) plane presented in Figure 1 this corresponds to the shaded region. We thus treat low (tangential) frequencies here. Lemma 4.6. For all k ≥ 1 we have

D|µ k vk|S | 2 ≤ B 4 sλa(t)(ϕv k ) |S 2 ,
with B as defined in (4.6), if either (1) µ k ≤ µ s,λ or (2) µ k > µ s,λ and t ∈ I k .

Proof. The first point follows from the definition of µ s,λ . The second point is a direct consequence of the definition of t k in (4.11) as the function t → a(t) decreases on (0, T/2).

For all k ∈ N * , we thus obtain

I(k; s, λ) ≤ B 4 (sλ) 3 |(aϕ |S ) 3 2 vk ) |S | 2 L 2 ((0,T )) .
Step 2: µ k > µ s,λ and t in a neighborhood of J k , preliminary result. In each open set (0, T ) × (-δ, 0) and (0, T ) × (0, δ), the function ûk satisfies the following equation We consider a cutoff function (0, T ) ∋ t → χ k (t), such that

-∂ 2 x n ûk + c 1 c 2 µ k ûk + 1 c 2 ∂ t ûk = fk c 2 + ∂ x n c 2 c 2 ∂ x n ûk . ( 4
χ k ≡ 1 on J k , 0 ≤ χ k ≤ 1, supp(χ k ) ⊂ Jk and χ ′ k ∞ ≤ C/t k ,
and we introduce

w = w(t, k, x n ; s, λ) = 1 2 χ k (t)p(t; s, λ)|û k (t, x n )| 2 . (4.16)
Note that χ k depends on the index k. Yet, as this dependency will only appear below through the estimate of χ ′ k ∞ we shall write χ in place of χ k for concision. Observe that w ≥ 0 and that it satisfies the same transmission conditions (4.2) as u. The function w satisfies

∂ 2 x n w - c 1 c 2 (2 -γ)µ 2 k - p ′ c 1 p w = -g, (4.17)
with 0 < γ < 1 and

g = -χp|∂ x n ûk | 2 - 1 c 2 ∂ t w + χ 1 c 2 p Re fk ûk - c 1 c 2 µ 2 k γ 2 χp|û k | 2 + χp ∂ x n c 2 c 2 Re ûk ∂ x n ûk + χ ′ 2c 2 p|û k | 2 .

Lemma 4.7.

There exist s 0 > 0, λ 0 > 0, depending on L and γ, such that

(2 -γ)µ 2 k - p ′ c 1 p ≥ µ 2 k if t k 2 < t < T - t k 2 , x n ∈ (-δ, δ), (4.18)
for s > s 0 (T + T 2 ) and λ > λ 0 .

See Appendix A.6 for a proof.

Step 3: µ k > µ s,λ and t in a neighborhood of J k , conclusion. For t ∈ Jk we begin by replacing the time-space dependent coefficient c 1 c 2 (2-γ)µ 2 k -p ′ c 1 p by σ 2 µ 2 k on the l.h.s. of (4.17) (the constant σ is introduced in (4.8)). This will allow us to argue as in the elliptic case, viz. solving an ordinary differential equation with constant coefficients.

We set q(t, x n ; k, s, λ)

:= -σ 2 µ 2 k + c 1 c 2 (2 -γ)µ 2 k - p ′ c 1 p .
We have

∂ 2 x n w -σ 2 µ 2 k w = -g, (4.19)
with

g := -qw -χp|∂ x n ûk | 2 - 1 c 2 ∂ t w + χ 1 c 2 p Re fk ûk - c 1 c 2 χ γ 2 pµ 2 k |û k | 2 + χp ∂ x n c 2 c 2 Re ûk ∂ x n ûk + χ ′ 2c 2 p|û k | 2 .
Observe that Lemma 4.7 gives

q(t, x n ; k, s, λ) ≥ 0, t k 2 < t < T - t k 2 , x n ∈ (-δ, δ), s > s 0 , λ > λ 0 .
From (4.19) and Appendix A.3 we obtain

J(k; s, λ) ≤ 2 ∫ Jk µ 2 k w |S dt = 2µ k σ ∫ Jk δ ∫ 0 sinh σµ k (δ -x n ) (c 2+ + c 2-) cosh(σµ k δ) c 2+ g(t, x n ) + c 2-g(t, -x n ) dx n dt. (4.20)
Note that the introduction of Jk , instead of J k , is due to the cut-off function χ. Substituting g in (4.20) we obtain seven terms. We shall provide the details for the contribution of c 2+ g(t, x n ). For the contribution c 2-g(t, -x n ) details are given if difference occurs. As in the elliptic case, we shall use that the kernel e -2sa(t)η |S sinh(σµ k (δ-x n )) cosh(σµ k δ)

be estimated by the weight e 2saη .

1. We have

δ ∫ 0 sinh(σµ k (δ -x n )) cosh(σµ k δ) -qw -χp|∂ x n ûk | 2 dx n ≤ 0. (4.21)
The negative sign is fortunate as the absolute value of this term cannot be reasonably bounded, i.e., by a term that can be "absorbed" by the l.h.s. of (4.3).

(a)

∫ Jk δ ∫ 0 sinh(σµ k (δ -x n )) (c 2+ + c 2-) cosh(σµ k δ) - c 2± c 2 ∂ t w(t, ±x n ) dx n dt = ∫ Jk δ ∫ 0 sinh(σµ k (δ -x n )) cosh(σµ k δ) ∂ t c 2± (c 2+ + c 2-)c 2 w(t, ±x n ) dx n dt,
and, by (3.13), we have

µ k ∫ Jk δ ∫ 0 sinh(σµ k (δ -x n )) (c 2+ + c 2-) cosh(σµ k δ) - c 2± c 2 ∂ t w(t, ±x n ) dx n dt sλD ∫ Jk δ ∫ 0 a(t)ϕ |S e -σµ k x n e 2sa(t)η |S µ k |û k (t, ±x n )| 2 dx n dt.
We shall thus obtain an estimate of this term by the r.h.s. of (4.9) if we prove

-σµ k x n + 2sa(t)η |S ≤ 2sa(t)η(±x n ), ∀(t, x n ) ∈ Jk × (0, δ). (4.22)
This is clear for the case

+ since η |S ≤ η(x n ).
The argument is different for the case -. Using that β is a piecewise affine, we have

η(-x n ) -η |S = ϕ(-x n ) -ϕ |S ≥ -x n λ β ′ -ϕ |S , x n ∈ (0, δ).
Therefore, (4.22) will be satisfied if

σµ k ≥ 2sa(t)λ β ′ -ϕ |S , ∀t ∈ Jk ,
which, by the definition of Φ in (4.10), can be written as

σµ k ≥ 4β ′ -|S D B Φ(t; s, λ), ∀t ∈ Jk .
As max t∈ Jk Φ(t; s, λ) = Φ t k 2 ; s, λ , it suffices to have

σµ k = σΦ(t k ; s, λ) ≥ 4β ′ -|S D B Φ t k 2 ; s, λ , ∀t ∈ Jk .
This holds if we have

a(t k ) a( t k 2 ) ≥ 4β ′ -|S σ D B . (4.23)
Observing that a(t k ) a(t k /2) ≥ 1 2 , we find that (4.23) is fulfilled by Assumption 4.2.

(b) Term χ( 1 c 2 p Re f ûk - c 1 c 2 γ 2 pµ 2 k | ûk | 2 ).
We shall prove that the associated term in (4.20) is estimated by e saη f 2 L 2 (q T,δ ) . Applying the Young inequality, we obtain

µ k p Re fk ûk c 2 ≤ D 2γ inf t∈[0,T ] (c 1 c 2 ) |e saη |S fk | 2 + c 1 c 2 γsλaϕ |S pµ 2 k |û k | 2 2 . (4.24)
Observe that

µ k ≥ sλa(t)ϕ |S , t ∈ Jk ⇔ µ k ≥ 2 D B Φ(t; s, λ), t ∈ Jk .
Arguing as above this will be fulfilled if

a(t k ) a( t k 2 ) ≥ 2 D B
, which holds by Assumption 4.2. We thus find, for t ∈ Jk ,

µ k δ ∫ 0 sinh(σµ k (δ -x n )) cosh(σµ k δ) χ( 1 c 2 p Re fk ûk - c 1 γ 2c 2 pµ 2 k |û k | 2 ) dx n ≤ δ ∫ 0 De -2( σµ k 2 x n -saη |S ) 2γ inf t∈[0,T ] (c 1 c 2 ) | fk | 2 dx n ,
and proceeding as in 2.(a) we find

µ k ∫ Jk δ ∫ 0 sinh(σµ k (δ -x n )) (c 2-+ c 2+ ) cosh(σµ k δ) c 2+ χ 1 c 2 p Re fk ûk - c 1 γ 2c 2 pµ 2 k |û k | 2 dx n dt e saη fk 2 L 2 (q T,δ ) . (c) Term χp ∂ xn c 2 c 2 Re ûk ∂ x n ûk . With the Young inequality we find µ k χp Re ûk ∂ x n ûk ≤ 1 2 p|∂ x n ûk | 2 + µ 2 k 1 2 p|û k | 2 . With (4.

22), arguing as above we obtain

µ k ∫ Jk δ ∫ 0 sinh σµ k (δ -x n ) (c 2+ + c 2-) cosh(σµ k δ) c 2+ χp ∂ x n c 2 c 2 Re ûk ∂ x n ûk dx n dt sλ (aϕ) 1 2 e saη ∂ x n ûk 2 L 2 (q T,δ ) + sλ (aϕ) 1 2 µ k vk 2 L 2 (q T,δ ) sλ (aϕ) 1 2 ∂ x n vk 2 L 2 (q T,δ ) + s 3 λ 3 (aϕ) 3 2 vk 2 L 2 (q T,δ ) + sλ (aϕ) 1 2 µ k vk 2 L 2 (q T,δ ) . (d) Term χ ′ 2c 2 p| ûk | 2 . As we have χ ′ ∞ ≤ C/t k we get χ ′ ∞ T a(t k ) T Φ(t k ; s, λ) sλϕ |S D B T µ k sλϕ |S D B .
We thus find

µ k δ ∫ 0 sinh σµ k (δ -x n ) cosh(σµ k δ) χ ′ 2c 2 p|û k | 2 dx n µ 2 k T a(t) D 3 B δ ∫ 0 sinh σµ k (δ -x n ) cosh(σµ k δ) e 2saη |S |û k | 2 dx n .
Arguing as above with (4.22) we obtain

µ k ∫ Jk δ ∫ 0 sinh σµ k (δ -x n ) (c 2+ + c 2-) cosh(σµ k δ) c 2+ χ ′ 2σc 2 p|û k | 2 dx n dt T a 1 2 µ k vk 2 L 2 (q T,δ ) sλ (aϕ) 1 2 µ k vk 2 L 2 (q T,δ ) , if s ≥ s 0 T , with s 0 > 0, and λ ≥ λ 0 > 0.
Collecting all the estimates we have obtained we conclude the proof of the Proposition 4.5.

End of the proof of Theorem 1.5.

With Proposition 4.1, estimate (4.4), Lemma 4.4 and Proposition 4.5, for λ and s/(T + T 2 ) sufficiently large, we obtain, for all k ∈ N * ,

s -1 (aϕ) -1 2 ∂ t vk 2 L 2 (q T,δ ) + sλ 2 (aϕ) 1 2 ∂ x n vk 2 L 2 (q T,δ ) + (aϕ) 1 2 µ k vk 2 L 2 (q T,δ ) + s 3 λ 4 (aϕ) 3 2 vk 2 L 2 (q T,δ ) + sλ |(aϕ |S ) 1 2 ∂ x n vk | 2 L 2 ((0,T )) + |(aϕ |S ) 1 2 µ k vk | 2 L 2 ((0,T )) + s 3 λ 3 |(aϕ |S ) 3 2 vk | 2 L 2 ((0,T )) e saη fk 2 L 2 (q T,δ ) . (4.25)
Summing over k, using (2.2) we obtain

s -1 (aϕ) -1 2 ∂ t v 2 L 2 (Q T ) + sλ 2 (aϕ) 1 2 ∂ x n v 2 L 2 (Q T ) + (aϕ) 1 2 µ k v 2 L 2 (Q T ) + s 3 λ 4 (aϕ) 3 2 v 2 L 2 (Q T ) + sλ |(aϕ |S ) 1 2 ∂ x n v| 2 L 2 (S T ) + |(aϕ |S ) 1 2 µ k v| 2 L 2 (S T ) + s 3 λ 3 |(aϕ |S ) 3 2 v| 2 L 2 (S T ) e saη f 2 L 2 (Q T ) . (4.26)
The remainder of the proof of the Carleman estimate is now classical (see e.g. [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators[END_REF]).

Remark 4.8. It is important to note that Proposition 4.5 is not a trace result, otherwise a stronger Sobolev norm would appear on the r.h.s. of (4.9). The L 2 -norm of the trace of the tangential derivative is estimated by an L 2 ((0, T ); H 1 (Ω))-norm, but this is valid only for solutions of Pu = f . This result appears to us as an expression of the parabolic regularization effect.

Observe that the estimate of Proposition 4.5 is also valid in the case where c 1 and c 2 are smooth if the weight function β is chosen with a discontinuous derivative across S according to Assumptions 2.1 and 4.2.

A Proof of some intermediate results

A.1 Proof of Proposition 3.1

For later use of this proof in Section 4 we consider the case c 1 c 2 here. The inequality we prove is uniform w.r.t. k. We shall thus remove the Fourier notation ûk and simply write (-∂ x n c 2 ∂ x n + c 1 µ 2 )u = f . We introduce v = e sϕ u and g = e sϕ f and we obtain

-∂ x n c 2 ∂ x n -c 2 (sϕ ′ ) 2 + c 1 µ 2 + 2sc 2 ϕ ′ ∂ x n + s∂ x n (c 2 ϕ ′ ) v = g,
which, following [START_REF] Fursikov | Controllability of evolution equations[END_REF], we write

M 1 v + M 2 v = g, with M 1 = -∂ x n c 2 ∂ x n -c 2 (sϕ ′ ) 2 + c 1 µ 2 , M 2 = 2sc 2 ϕ ′ ∂ x n + spc 2 ϕ ′′ , g = g + (p -1)sc 2 ϕ ′′ v -s(∂ x n c 2 )ϕ ′ v, 1 < p < 3.
The introduction of the parameter p is for instance explained in [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators[END_REF]. Following the classical method to prove Carleman estimates we compute

g 2 L 2 (R + ) = M 1 v 2 L 2 (R + ) + M 2 v 2 L 2 (R + ) + 2 Re(M 1 v, M 2 v) L 2 (R + ) , (A.1)
considering only the region {x n > 0} for now. We focus on the computation of the third term which we write as sum of 4 terms I i j , 1 ≤ i ≤ 2, 1 ≤ j ≤ 2, where I i j is the inner product of the i th term in the expression of M 1 v and the j th term in the expression of M 2 v above.

Term I 11 . With an integration by parts we have

I 11 = -2 Re ∫ x n >0 sϕ ′ ∂ x n c 2 ∂ x n v c 2 ∂ x n v dx n = -∫ x n >0 sϕ ′ ∂ x n |c 2 ∂ x n v| 2 dx n = sϕ ′ |c 2 ∂ x n v| 2 |x n =0 + + ∫ x n >0 sϕ ′′ |c 2 ∂ x n v| 2 dx n .
Term I 21 . Similarly we find

I 21 = Re ∫ x n >0 -c 2 (sϕ ′ ) 3 + sc 1 µ 2 ϕ ′ c 2 ∂ x n |v| 2 dx n = c 2 (sϕ ′ ) 3 -sc 1 µ 2 ϕ ′ c 2 |v| 2 |x n =0 + + ∫ x n >0 c 2 3s 3 c 2 (ϕ ′ ) 2 ϕ ′′ -sc 1 µ 2 ϕ ′′ |v| 2 dx n + ∫ x n >0 2c 2 ∂ x n c 2 (sϕ ′ ) 3 -(c 1 ∂ x n c 2 + c 2 ∂ x n c 1 )sµ 2 ϕ ′ |v| 2 dx n .
Term I 12 . We have

I 12 = -sp Re ∫ x n >0 (∂ x n c 2 ∂ x n v)c 2 ϕ ′′ v dx n = sp ∫ x n >0 ϕ ′′ |c 2 ∂ x n v| 2 dx n + spϕ ′′ Re(c 2 ∂ x n v)c 2 v |x n =0 + + sp Re ∫ x n >0 ∂ x n (c 2 ϕ ′′ )(c 2 ∂ x n v)v dx n .
Term I 22 . We directly find

I 22 = sp ∫ x n >0 c 2 -c 2 (sϕ ′ ) 2 + c 1 µ 2 ϕ ′′ |v| 2 dx n .
Collecting together the different terms we have obtained we find

1 2 g 2 L 2 (R + ) ≥ ∫ x n >0 α 0 |v| 2 dx n + ∫ x n >0 α 1 |c 2 ∂ x n v| 2 dx n + γ 0 |v| 2 |x n =0 + + γ 1 |c 2 ∂ x n v| 2 |x n =0 + + X + Y, with α 0 = s(p -1)c 1 c 2 µ 2 ϕ ′′ + (3 -p)s 3 (c 2 ϕ ′ ) 2 ϕ ′′ , α 1 = s(p + 1)ϕ ′′ , γ 0 = c 2 2 (sϕ ′ ) 3 |x n =0 + -c 1 c 2 sµ 2 ϕ ′ |x n =0 + , γ 1 = sϕ ′ , X = sp Re ∫ x n >0 ∂ x n (c 2 ϕ ′′ )(c 2 ∂ x n v)v dx n + ∫ x n >0 2c 2 ∂ x n c 2 (sϕ ′ ) 3 -(c 1 ∂ x n c 2 + c 2 ∂ x n c 1 )µ 2 sϕ ′ |v| 2 dx n , Y = spc 2 ϕ ′′ Re(∂ x n v)v |x n =0 + .
Because of the form of ϕ, a direct computation shows that α 0 sλ 2 µ 2 ϕ + s 3 λ 4 ϕ 3 , α 1 Csλ 2 ϕ, for λ chosen sufficiently large. Recalling that β is affine in the region we consider we find

X = sp Re ∫ x n >0 c 2 2 λ 3 β ′3 + c 2 (∂ x n c 2 )λ 2 β ′2 ϕ(∂ x n v)v dx n + ∫ x n >0 2c 2 ∂ x n c 2 (sλβ ′ ϕ) 3 -(c 1 ∂ x n c 2 + c 2 ∂ x n c 1 )µ 2 sλβ ′ ϕ |v| 2 dx n , and
g 2 L 2 (R + ) g 2 L 2 (R + ) + s 2 (λ 4 + λ 2 ) ∫ x n >0 ϕ 2 |v| 2 .
Choosing s and λ sufficiently large, with the Young inequality, we obtain

C g 2 L 2 (R + ) ≥ C ′ ∫ x n >0 sλ 2 µ 2 ϕ + s 3 λ 4 ϕ 3 |v| 2 dx n + C ′ ∫ x n >0 sλ 2 ϕ|∂ x n v| 2 dx n + γ 0 |v| 2 |x n =0 + + γ 1 |c 2 ∂ x n v| 2 |x n =0 + + Y. (A.2)
The same type of estimate can be obtained in the region {x n < 0} with opposite signs for the trace terms. The sum of (A.2) from both sides yields the result.

A.2 Proof of Lemma 3.4

Here we drop the vk notation and simply write v. It follows that

c∂ x n v = e sϕ (c∂ x n u + cs(∂ x n ϕ)u) = e sϕ c∂ x n u + cβ ′ (sλϕu) .
We set a = c∂ x n u and b = sλϕu. We then have

|c∂ x n v| 2 β ′ S = e 2sϕ [β ′ ] S |a| 2 + [c 2 (β ′ ) 3 ] S |b| 2 + 2[c(β ′ ) 2 ] Re ab .
We thus obtain

B(v) = sλϕ |S e 2sϕ Aw, w),
with w = (a, b) t and where A is the following symmetric matrix

A = [β ′ ] S [c(β ′ ) 2 ] S [c(β ′ ) 2 ] 2[c 2 (β ′ ) 3 ] S = β ′ - (L -1) c + β ′ -(L 2 -K) c + β ′ -(L 2 -K) 2(c + β ′ -) 2 (L 3 -K 2 )
.

We then see that

(Aw, w) = β ′ -(L -1) a + c + β ′ - L 2 -K L -1 b 2 + β ′ -2(c + β ′ -) 2 (L 3 -K 2 ) -(c + β ′ -) 2 L 2 -K 2 L -1 |b| 2 .
which gives the result.

A.3 Traces of the solution

We consider the following ODEs

v ′′ -µ 2 v = F, s ∈ (-δ, 0) ∪ (0, δ), (A.3) v(-δ) = v(δ) = 0, v |s=0 -= v |s=0 + , cv ′ |s=0 -= cv ′ |s=0 + . (A.4)
Here µ > 0. The solutions of (A.3) can be written as

v(s) = A ± cosh(µs) + B ± sinh(µs) + µ -1 s ∫ 0 sinh µ(s -σ) F(σ) dσ, s ∈ (-δ, 0) ∪ (0, δ).
We then have

A ± = v |s=0 ± , µB ± = v |s=0 ± and v(±δ) = µ -1 cosh(µδ) µA ± + µB ± tanh(±µδ) + ±δ ∫ 0 sinh µ(±δ -σ) cosh(µδ) F(σ) dσ .
The boundary conditions (A.4) then yield

µ tanh(µδ) µ -c + c -tanh(µδ) v + (0) v ′ + (0) =                - δ ∫ 0 sinh µ(δ-σ) cosh(µδ) F(σ) dσ - -δ ∫ 0 sinh µ(-δ-σ) cosh(µδ) F(σ) dσ                .
We observe that the determinant of this system,

D = -c -1 -tanh(µδ)µ(c + + c -), is non zero as µ > 0. It thus follows that v -(0) = v + (0) = - c + µ δ ∫ 0 sinh µ(δ -σ) (c + + c -) cosh(µδ) F(σ) dσ - c - µ -δ ∫ 0 sinh µ(-δ -σ) (c + + c -) cosh(µδ) F(σ) dσ.
A.4 Proof of Lemma 3.6

We first consider the case K = 1. Then L = 2 and P(K, L) = -(1 + ε) + 5(1ε) 2 . The result is clear for ε sufficiently small.

We now consider the case 0 < K < 1. Then L > 1; we have L -K 2 > 0 and thus

P(K, L) = -(L -K 2 )(L -1) + (1 -ε) 2 1 + ε 2(L 3 -K 2 )(L -1) -L 2 -K 2 .
For convenience we write (1ε) 2 /(1 + ε) = 1α with 0 < α < 1. We then find

Q(K) = P(K, (r + 1) -rK) = -(K -1) 2 S (K), S (K) = aK 2 + bK + c, with a = -(1 -α)r 4 < 0, b = 2(1 -α)(r 4 + r 3 -r 2 ) -(1 -2α)r, c = -(1 -α)(r 4 + 2r 3 -1) + r 2 + (3 -2α)r.
As S is a concave quadratic polynomial it suffices to prove that S (1) ≤ 0 and S ′ (1) ≥ 0. We find

S (1) = (2α -1)r 2 + 2r + 1 -α, S ′ (1) = r 2(1 -al)r 2 -2(1 -α)r -(1 -2α) .
We see that S (1) < 0 and S ′ (1) > 0 if α = 0 and r = 3. It thus remains true for α sufficiently small.

A.5 Proof of Proposition 4.1

The inequality we prove is uniform w.r.t. k. We shall thus remove the Fourier notation ûk and simply write (∂ t -∂ x n c 2 ∂ x n + c 1 µ 2 )u = f . We introduce v = e saη u and g = e saη f and we obtain

∂ t -∂ x n c 2 ∂ x n -c 2 (saη ′ ) 2 + c 1 µ 2 + 2sc 2 aη ′ ∂ x n + sa∂ x n (c 2 η ′ ) -sa ′ η v = g, which we write M 1 v + M 2 v = g, with M 1 = -∂ x n c 2 ∂ x n + [-c 2 (saη ′ ) 2 + c 1 µ 2 ] -sa ′ η, M 2 = 2sc 2 aη ′ ∂ x n + spc 2 aη ′′ + ∂ t , (A.5) g = g + (p -1)sc 2 aη ′′ v -s(∂ x n c 2 )aη ′ v, 1 < p < 3.
In preparation for what follows we observe that

1 T 2 a, |a ′ | T a 2 , |a ′′ | T 2 a 3 , |η| ϕ 2 .
We have

g 2 L 2 ((0,T )×R + ) g 2 L 2 ((0,T )×R + ) + s 2 (λ 4 + λ 2 )T 2 a 3 2 ϕ 1 2 v 2 L 2 ((0,T )×R + ) .
We compute

g 2 L 2 ((0,T )×R + ) = M 1 v 2 L 2 ((0,T )×R + ) + M 2 v 2 L 2 ((0,T )×R + ) + 2 Re(M 1 v, M 2 v) L 2 ((0,T )×R + ) , (A.6)
considering only the region {x n > 0} for now. For the computation of the last term in (A.6), we set I i j , 1 ≤ i ≤ 3, 1 ≤ j ≤ 3, where I i j is the inner product of the i th term in the expression of M 1 v and the j th term in the expression of M 2 v above. For the computations of I 11 , I 12 , I 21 and I 22 we refer to the computations performed in Appendix A.1 (simply replacing ϕ by aϕ and integrating in time).

Term I 13 . By integration by parts we find

I 13 = Re T ∫ 0 ∫ x n >0 -(∂ x n c 2 ∂ x n v)∂ t v dx n dt = 1 2 T ∫ 0 ∫ x n >0 c 2 ∂ t |∂ x n v| 2 dx n dt + Re T ∫ 0 (c 2 ∂ x n v)∂ t v |x n =0 + dt = - 1 2 T ∫ 0 ∫ x n >0 (∂ t c 2 )|∂ x n v| 2 dx n dt + Re T ∫ 0 (c 2 ∂ x n v)∂ t v |x n =0 + dt.
We have

1 2 T ∫ 0 ∫ x n >0 (∂ t c 2 )|∂ x n v| 2 dx n dt T 2 T ∫ 0 ∫ x n >0
a|∂ x n v| 2 dx n dt.

Term I 23 . By integration by parts we have

I 23 = 1 2 T ∫ 0 ∫ x n >0 -c 2 (saη ′ ) 2 + c 1 µ 2 ∂ t |v| 2 dx n dt = s 2 T ∫ 0 ∫ x n >0 c 2 aa ′ η ′2 |v| 2 dx n dt + 1 2 T ∫ 0 ∫ x n >0 [(∂ t c 2 )(saη ′ ) 2 -(∂ t c 1 )µ 2 ]|v| 2 dx n dt
We thus find

|I 23 | (T + T 2 )s 2 λ 2 T ∫ 0 ∫ x n >0 a 3 ϕ 2 |v| 2 dx n dt + T 2 T ∫ 0 ∫ x n >0
aµ 2 |v| 2 dx n dt.

Term I 33 . By integration by parts we find

I 33 = - s 2 T ∫ 0 ∫ x n >0 a ′ η∂ t |v| 2 dx n dt = s 2 T ∫ 0 ∫ x n >0
a ′′ η|v| 2 dx n dt.

This yields

|I 33 | T 2 s T ∫ 0 ∫ x n >0 a 3 ϕ 2 |v| 2 dx n dt.
Terms I 31 and I 32 . We have

I 31 = -s 2 T ∫ 0 ∫ x n >0 c 2 aa ′ ηη ′ ∂ x n |v| 2 dx n dt = s 2 T ∫ 0 ∫ x n >0 aa ′ ∂ x n (c 2 ηη ′ )|v| 2 dx n dt.
We find directly

I 32 = -ps 2 ∫ T 0 ∫ x n >0 c 2 aa ′ ηη ′′ |v| 2 dx n dt.
We then obtain

|I 31 | + |I 32 | T s 2 (λ + λ 2 ) T ∫ 0 ∫ x n >0 a 3 ϕ 3 |v| 2 dx n dt.
With the computations of Appendix A.1 we find, for λ and sa s/T 2 sufficiently large, The same type of estimate can be obtained in the region {x n < 0} with opposite signs for the trace terms. The sum of (A.7) from both sides yields C sλ 2 (aϕ) A. With the definition of Φ, this reads

- η |S ϕ 2 |S ≤ c1 32T (1 -γ) B D sλ 2
or equivalently

e λ( β-2β |S ) -e -λβ |S ≤ c1 32T (1 -γ) B D sλ 2 .
As β < 2β |S by construction of β (see the beginning of Section 4), this will hold for λ and s/T sufficiently large.

Remark 3 . 7 .

 37 Here, the weight function does not depend on x ′ . Observe that the local Carleman estimate that we obtain in Ω ′ × (-δ, δ) does not require any regularity for the boundary of the open set Ω ′ . The minimal regularity of the boundary ∂Ω to achieve a Carleman estimate remains an open question to our knowledge.

Figure 1 :

 1 Figure 1: The function Φ. The shaded region is treated in the first step of the proof.

C g 2 L 2 ( 0 ∫ x n >0 sλ 2 µ 2 0 ∫ x n >0 sλ 2 aϕ|∂ x n v| 2 dx n dt + 1 2 M 2 v 2 L 2 ( 0 γ 0 |v| 2 0 ∫[2c 2 ( 2 0 ∫ x n >0 sλ 2 µ 2 0 ∫ x n >0 sλ 2 aϕ|∂ x n v| 2 dx n dt + 1 2 M 2 v 2 L 2 ( 0 γ 0 2 0 (c 2 ∂

 220202220202202022200202 (0,T )×R + ) ≥ C ′ T ∫ aϕ + s 3 λ 4 (aϕ) 3 |v| 2 dx n dt + C ′ T ∫ (0,T )×R + ) + T ∫ |x n =0 + + γ 1 |c∂ x n v|2 |x n =0 + dt + X + Y + I 13 + I 23 + I 33 + I 31 + I 32 , withγ 0 = c 2 2 (saϕ ′ ) 3 |x n =0 +c 1 c 2 sµ 2 aϕ ′ |x n =0 + , γ 1 = saϕ ′ , X = sp Re T ∫ x n >0 a∂ x n (c 2 ϕ ′′ )(c 2 ∂ x n v)v dx n dt + ∂ x n c 2 )(saϕ ′ ) 3 -∂ x n (c 1 c 2 )µ 2 saϕ ′ ]|v| 2 dx n dt, aϕ ′′ Re(∂ x n v)v |x n =0 + dt.For λ and s/(T + T 2 ) sufficiently large, the estimations we found above yield,C g 2 L 2 ((0,T )×R + ) ≥ C ′ T ∫ aϕ + s 3 λ 4 (aϕ) 3 |v| 2 dx n dt + C ′ T ∫ (0,T )×R + ) + T ∫ |v| 2 |x n =0 + + γ 1 |c 2 ∂ x n v|2 |x n =0 + dt + Y, (A.7aϕ ′′ Re(∂ x n v)v |x n =0 + dt + Re T ∫ x n v)∂ t v |x n =0 + dt.

T ∫ 0 c 2 ∂ 3 2 λ 3 T 3 T ∫ 0 a 3 ϕa c 2 ∂ 0 ∂ 0 a 3 ϕ 2 L 2 ( 1 2 λ∂ x n v 2 L 2 ( 1 2 λ 2 v 2 L 2 (M 2 v 2 L 2 ( 1 2 ∂ x n v 2 L 2 ( 1 2 v 2 L 2 ( 1 2 ∂ x n v 2 L 2 ( 2 v 2 L 2 (

 02330320032212212222122122122222 x n v S ∂ t v |S T dt. |S |v| 2 |S T dt.As u = e -saη v we have c∂ x n u = c 2 (∂ x n vsa(η ′ )v)e -saη and thusc 2 ∂ x n v S = sa c 2 η ′ S v |S T = sλa c 2 β ′ S (vϕ) |S T .By integration by parts, we thus haveỸ2 x n β S ϕ |S ∂ t |v| 2 |S T dt =t (ac 2 )∂ x n β S ϕ |S |v| 2 |S T dt.We thus obtain| Ỹ2 | s(T 3 + T 4 )λ T ∫ |S |v| 2 |S T dt. (A.9)Finally, from the form of M 2 in (A.5), we have(saϕ) (0,T )×R * ) + (saϕ) Q T,δ ) + (saϕ) Q T,δ ) (0,T )×R * ) + sλ 2 (aϕ) Q T,δ ) + sλ 4 (aϕ) Q T,δ ) M 2 v 2 L 2 ((0,T )×R * ) + sλ 2 (aϕ) Q T,δ ) + s 3 λ 4 (aϕ) 3 Q T,δ ) ,as here sa s/T 2 ≥ s 0 , for some s 0 > 0 and ϕ ≥ 1.

Term 1 c 2 ∂ t w.

  Because of the cut-off function χ we have w |t=t k /2 = w |t=T -t k /2 = 0 and we get

  ′ |∂ x n v| 2 ] S + |sλaϕv |S T | 2 [c 2 2 β ′3 ] S dt + Ỹ ≤ C e saη f 2 L 2 (Q T,δ ) + sλ |S |µv |S T | 2 [c 1 c 2 β ′ ] S dt, (A.8) aϕ |S Re c 2 2 β ′2 ∂ x n v S v |S T dt,

	1 2 ∂ x n v	2 L 2 (Q T,δ ) + sλ 2 (aϕ)	1 2 µv	2 L 2 (Q T,δ ) + s 3 λ 4 (aϕ)	3 2 v	2 L 2 (Q T,δ )
	+ 2 β with Ỹ = Ỹ1 + Ỹ2 , where 1 2 M 2 v 2 L 2 ((0,T )×R * ) + sλ T ∫ 0 aϕ |S [c 2
		Ỹ1 = spλ 2	T ∫
				0
	and
		Ỹ2 = Re

T ∫ 0 aϕ

  6 Proof of Lemma 4.7 Computing p ′(t) = sλe 2sa(t)η |S a ′ (t)ϕ |S [1 + 2sη |S a(t)]D, we have ≤ t < T -t k2 , because of the form of η in (4.1) we find p ′ p ≤ 0, for sa(t) s/T 2 and λ both sufficiently large. This implies that inequality (4.18) holds for these values of t.We now consider the case t k 2 < t < T 2 . Note that p ′ p is nonnegative, for s/T 2 and λ large, as here a ′ (t) < 0. Setting c1 = inf t,x n c 1 (t, x n ), and using the definition of t k in (4.11), it suffices to prove

							p ′ p	(t) =	a ′ a	(t)[1 + 2sa(t)η |S ],
	If T 2 p ′ (t) c1 p(t)	≤ (1 -γ)Φ 2 (t k ; s, λ),	t k 2	< t <	T 2	.
	For all s, λ, we have
	p ′ (t) p(t) [2sη As we have = 2t -T t(T -t) 1 2 <		Φ(t k ; s, λ) Φ(t k /2; s, λ)	=	a(t k ) a(t k /2)	<	3 4	,	(A.10)
	it is sufficient to prove
	-	2T c1	sη |S a 2 (t) ≤	1 4	(1 -γ)Φ 2 t k /2; s, λ ,	t k 2	< t <	T 2	.	(A.11)
	As the function Φ decreases on (0, T/2), (A.11) holds if we have
	-	2T c1	sη |S a 2 (t) ≤	1 4	(1 -γ)Φ 2 (t; s, λ),	t k 2	< t <	T 2	.

|S a(t) + 1] ≤ 2t -T t(Tt) 2sη |S a(t) ≤ -2T sη |S a 2 (t).

a b stands for a ≤ Cb for some constant C > 0.

Note that the derivation of a Carleman estimate in the case of singular domains can be achieved (see[START_REF] Belghazi | Carleman inequalities for the heat equation in singular domains[END_REF]). Addressing the more general case of Lipschitz boundary is an open question to our knowledge.

Concerning the regularity of the coefficients c i , an inspection of the proof of the Carleman estimate in the parabolic case shows that the time derivative of the trace of c 2 at x n = 0 needs to make sense (see above (A.9) in Appendix A.[START_REF] Bellassoued | Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization[END_REF]). An alternative regularity is then W 1,∞ (0, T ; W 1,∞ (Ω)).

This is the precise point where c 1 = c 2 is used. In the case c 1 c 2 the result of Appendix A.3 cannot be used and we have to proceed as in Section 4.

The authors wish to thank an anonymous referee for his valuable corrections and suggestions that improved the readability of this article. The authors were partially supported by l'Agence Nationale de la Recherche under grant ANR-07-JCJC-0139-01.


A Alfalou
email: ayman.al-falou@isen.fr

C Brosseau
email: brosseau@uiv-brest.fr

A New Robust and Discriminating Method for Face Recognition Based on Correlation Technique and Independent Component Analysis Model

We demonstrate a novel technique for face recognition combined the independent component analysis (ICA) model with the optical correlation technique. Our approach relies on the performances of a strongly discriminating optical correlation method along with the robustness of the ICA model. Simulations were performed to illustrate how this algorithm can identify a face with images from the Pointing Head Pose Image Database (PHPID). While maintaining algorithmic simplicity, this approach based on ICA representation significantly increases the true recognition rate compared to that obtained with an all numerical ICA identity recognition method, that we recently developed, and with another based on optical correlation and a standard composite filter.

The field of face recognition has made significant progress in recent years. Computational models of face recognition are important because they can contribute to theoretical insights, e.g. image codes, as well as practical applications, i.e. biometrics, security systems, and human-computer interaction. Many different implementations are being actively explored, including eigenfaces [START_REF] Turk | Eigenfaces for recognition[END_REF], Gabor wavelets [START_REF] Shen | A Review on Gabor wavelets for face recognition[END_REF], and principal component analysis (PCA) [START_REF] Jolliffe | Principal Component Analysis[END_REF]. Noteworthy, the independent component analysis (ICA) model has sparked interest in searching for a linear transformation to express a set of random variables as linear combinations of statistically independent source variables [START_REF] Comon | Independent component analysis: a new concept?[END_REF]. ICA provides a more powerful data representation than PCA as its goal is that of providing an independent rather than uncorrelated image decomposition and representation. Interest in ICA mixture models is motivated in part by their exceptional properties, including high sensitivity to high-order relationships among pixels and enhanced face recognition performance when it is carried out in a compressed and whitened space [START_REF] Bartlett | Face recognition by independent component analysis[END_REF]. Motivated by these developments, we investigate an optical recognition technique using ICA as a preprocessing stage for face recognition. In this Letter, our two primary goals are to validate the principle of the new algorithm proposed to recognize a target face using reference facial images contained in the (supervised) learning base, and to focus on its implementation using a standard correlator which can be optically or numerically implemented. In this paper, we used a simple POF filter to validate the principle of our approach. However, other correlation filters, more efficient, and a large set of subjects will be used i.e. that is the goal of an article in progress [6]. One main drawback of the existing face recognition methods arises from the sensitivity to rotation of the target image with the reference facial images. Several studies have shown how composite filters might be designed and optimized for optical correlators [START_REF] Kumar | Tutorial survey of composite filter designs for optical correlators[END_REF]. However, it was recognized that the performance of these methods deteriorates for large values of the number of reference facial to be incorporated in the composite filter [START_REF] Alfalou | Optical implementation of segmented composite filtering[END_REF].

In this context, it should be noted that the pioneering studies which developed the theoretical basis of the above mentioned methods have highlighted their good performances for object recognition and identification purposes. However, several limitations related to the use of optical modulators, and to their sensitivity to rotation, illumination conditions, and scaling of the target object relative to the references images were emphasized. Recent works [START_REF] Diaz-Ramirez | Constrained composite filter for intraclass distortion invariant object recognition[END_REF][START_REF] Ahad | What can we learn from the shape of a correlation peak for position estimation?[END_REF] have suggested some improvements to overcome these limitations. Moreover Alsamman et al [START_REF] Alsamman | Comparative study of face recognition techniques that use joint transform correlation and principle component analysis[END_REF] have also demonstrated that the recognition performances of the PCA method are weaker than those obtained with a standard correlation technique. Since the ICA method is robust against the change of image versus rotation, lighting, background, and scale for a database of independent components [START_REF] Alfalou | All optical video-image encryption enforced security level using ICA[END_REF][START_REF] Alfalou | Independent component analysis based approach to biometric recognition[END_REF], the purpose of this Letter is to use a simple correlation approach. Our approach relies on the performances of a strongly discriminating optical correlation method along with the robustness of the ICA model.

Our algorithm is based on four steps: (1) information mining characterizing a face, (2) fabrication of a learning base modeling the target face, (3) comparison of the characteristics of the target face with the learning base, and (4) assessment of the performance of this approach by evaluating the true recognition and the false recognition rates. To illustrate the idea of this Letter in a face recognition context and to assess the sensitivity of our algorithm in terms of the dimensionality of the reduced space which is constructed by the statistically independent components of face images we used the PHPID and we calculate and compare their adapted Receiver Operating Characteristic (ROC) plots. We first describe technical details of the ICA algorithm which provides an optimal signal representation technique in the mean square error sense [START_REF] Comon | Independent component analysis: a new concept?[END_REF]. The ICA of a vector S (or a vector representing an image) searches for a linear transformation which minimizes the statistical dependence between its components [START_REF] Comon | Independent component analysis: a new concept?[END_REF].

i i n i n n C a C a C a C a S ∑ = = + + + = 1 2 2 1 1 L . ( 1
)
Wher C i are the independent components, n its number. In [START_REF] Alfalou | All optical video-image encryption enforced security level using ICA[END_REF], we used this merging property to suggest and validate an optical encryption method. As a simpler analogue of Eq. ( 1), a given face can be considered as a linear combination of several independent components arising from reference facial images contained in a learning base [START_REF] Alfalou | Independent component analysis based approach to biometric recognition[END_REF]. Here we consider a set of faces of the same subject (to be identified) differing from the head orientations, i.e. V i with 1 i n ≤ ≤ , which forms the learning base. Our starting point is to apply the FastICA method (see Ref. [START_REF] Alfalou | All optical video-image encryption enforced security level using ICA[END_REF][START_REF] Alfalou | Independent component analysis based approach to biometric recognition[END_REF] for details) for the reference facial images V i such as described by the diagram shown in Fig. 1 (a) after transforming each face into a vector V and applying a specific nonlinear function operation to ensure a good convergence of the FastICA algorithm [START_REF] Alfalou | All optical video-image encryption enforced security level using ICA[END_REF]. We have chosen the independent components that give the best result corresponding to the subject X base used in this study. In the next step, each subject V i of the learning base is correlated with different phase only filters (POF) F j =POF j [START_REF] Horner | Phase-only matched filtering[END_REF], i.e. each correlation filter is calculated from one independent component C j Fig. 1(b). Each correlation yields a reference peak-to-correlation energy (PCE) value [START_REF] Horner | Metrics for assessing pattern-recognition performance[END_REF] ij PCE ( )

j i ij pof V f PCE ⊗ = , ( 2
)
where ⊗ denotes the correlation product, f(...) defined as the energy of the correlation peak divided by the energy of the correlation plane, pof j is the inverse Fourier transform of F j and

i j ij C C δ ⊗ = , δ ij being the Kronecker symbol.
Once the components C j and the ij PCE values have been determined we follow the algorithm shown in Fig. 1(c). Basically, it consists in writing the vector V ? representing the target face as a linear mixture in terms of the n independent components (IC base)

n n C b C b V + + = L 1 1 ?
. Now, by correlating V ? with the different filters j POF one can obtain the ( )

' ' 2 ' 1 , , , _ n PCE PCE PCE Corr Val L =
vector that used to identify the target face among the reference facial images. We have

( ) ( ) ( ) ( ) ( ) ( ) ( ) noise j j n n j j j n n j j N b C C b C C b C C b f C C b C b f pof V f PCE + = ⊗ + + ⊗ + ⊗ = ⊗ + + = ⊗ = L L 2 2 1 1 1 1 ? ' , (3)
where noise

N

represents the noise in the correlation plane. We now define an error ε vector as

nn n n n n n n n n PCE PCE PCE PCE PCE PCE PCE PCE PCE PCE PCE PCE PCE PCE PCE PCE PCE PCE - + + - + - = - + + - + - = - + + - + - = ' 2 ' 2 1 ' 1 2 ' 22 ' 2 21 ' 1 2 1 ' 12 ' 2 11 ' 1 1 L M L L ε ε ε . ( 4
)
Each ε i corresponds to the difference between the Val_Corr vector and the ith row elements of the PCE matrix. At this point, it is important to recall that the PCE in elements appearing in Eq. ( 4) correspond to the face V i of the learning base, as shown in Fig. 1 (b). We then have to find the mean value of the ε i s, ( )

1 n n ε ε ε = +L
, which additionally should be less than a given acceptance threshold Th (in order to avoid a false alarm). Thus, the target face V ? is identified as the face of the subject X. The Choice of value of Th will be discussed in the following paragraph (Fig. 3(c)). What is noticeable in the above algorithm is that finding the coefficient ' j PCE can be realized by introducing the target face in a standard VLC correlator using a POF realized from a single independent component (Fig. 2). Without loss of generality a simple POF was used to validate the principle of our method; however optimized correlation filters, see e.g. [START_REF] Alfalou | Optical implementation of segmented composite filtering[END_REF][START_REF] Diaz-Ramirez | Constrained composite filter for intraclass distortion invariant object recognition[END_REF][START_REF] Ahad | What can we learn from the shape of a correlation peak for position estimation?[END_REF], can be employed to increase the robustness in the determination of the values of the ' j PCE coefficients. Mathematically, finding the ' i PCE coefficient requires two things.

On the one hand, we realize a nonlinear operation within the correlation plane ( ) k plane C , where k defines the degree of the nonlinearity (k >1). On the other hand, the PCE of the correlation plane should be determined. For the purpose of minimizing the errors in Eq. ( 2), a normalization of the ij PCE coefficients was realized in practice by considering the different responses obtained from the correlation of the reference facial images with the independent components.

Having discussed the principle of our method, we now consider a set of results obtained from images of the PHPID [START_REF] Gourier | Estimating Face Orientation from Robust Detection of Salient Facial Features[END_REF]. One of the basic questions is how to choose the value of n i.e. the number and the choice of C j depends on the number and the choice of the considered reference images. In this letter, we chosen only three references X(9) (Rightrotation) , X(27) (without-rotation) and X(45) (left-rotation). In a progress article, [6], we are studying the effect of the choice and the number of C j on the performance of our approach. However, to validate the principle of our approach, we have chosen a set of three independent components, i.e. only three faces (those which are represented by red frames in Fig. 3 (b)) were considered in the reference facial images. The horizontal axis of Fig. 3 (a) represents the face number presented at the input of our system. The vertical axis corresponds to the values of ε obtained with different positions of the subject X(i) with 1

53 i ≤ ≤
and Y(32), i.e. the subject Y without rotation which is framed in green in Fig. 3 (d). The two curves of Fig. 3 (a) show the good performance of our algorithm. Whatever the facial position X(i) of subject X which can be considered the error obtained by correlating the face Y(32) of subject Y with the independent components base (C j=1,…,3 ) is always larger than the error obtained by correlating subject X with (C j=1,..,3 ). However, to take a reliable decision on the subject identification, an acceptance threshold Th should be chosen such that subject X should be recognized for every facial position X(i) and leads to the rejection of the subject Y whatever his or her facial position. The choice of the value for Th is important since it contributes to false alarm and no detection events. To assess the robustness of our method, adapted ROCs [START_REF] Egan | Signal Detection Theory and ROC Analysis[END_REF] were graphed (Fig. 3 (c)) for a given value of the threshold Th. Every point of the ROC plot is calculated with a specific value of Th within the range [ ] 2000, , 2200 L

. The horizontal axis of the Figure 3-(c) represents the false recognition rate (when subject Y is identified as subject X) and the vertical axis corresponds to the true recognition rate (when subject X is correctly identified). The red curve presents the reference values when the true recognition is equal to the false recognition. We observe that the recognition rate is ). In summary, we have discussed an optical correlation technique using the ICA model to propose solutions to many of the problems associated with face recognition. The fact that this architecture is optically implementable makes this technique well suited for designing improved image processing systems. The face recognition performance obtained with our fast and simple algorithm is larger than that obtained from a composite filter with a significantly smaller false alarm rate. In order to test the robustness of our results, we need to consider effects of adding noises to the input images, effects of increasing the number of independent component, and compare these results with those obtained from using optimized correlation filters [6]. We believe our results can be directly tested experimentally.

V 1 V 2 V n C 1 C n C 2

Independent component

Training Base

b) V 1 V n . . . . Correlator ij PCE Target Face ? V Correlator x V V ≅ ? nn n n n n n n n n PCE PCE PCE PCE PCE PCE PCE PCE PCE PCE PCE PCE PCE PCE PCE PCE PCE PCE - + + - + - = - + + - + - = - + + - + - = ' 2 ' 2 1 ' 1 2 ' 22 ' 2 21 ' 1 2 1 ' 12 ' 2 11 ' 1 1 L M L L ε ε ε < Th           = nn n n PCE PCE PCE PCE PCE 1 1 11 O M L FastICA V 1 V 2 V n C 1 C n C 2

Independent component

Training Base

  Th arbitrarily to 2008 ≈

Figure captions FIG. 1 :

 captions1 Figure captions

FIG 3 :

 3 FIG 3: (Color online) Simulation results: (a) the curve in blue color represents the identification results obtained with different facial positions of subject X, the curve in green color corresponds to face Y(32) of subject Y. For both curves, three independent components were used. (b) Test base representing several positions of face X (the three faces of X used to compute the independent components are framed in red). (c) ROC plots using 3 independent components.

FastICA

1 PCEFIG. 2 :

 12 FIG. 1: Alfalou and Brosseau

Acknowledgments

Work in L@bISEN was supported by Groupe-ISEN. Work in Lab-STICC was supported in part by the CNRS. Lab-STICC is Unité Mixte de Recherche CNRS 3192.


A Alfalou
email: ayman.al-falou@isen.fr

C Brosseau

A Dual Encryption Scheme of Images Using Polarized Light

Motivated by recent interest in polarization encoding, we propose and analyze a dual encryption/decryption scheme. Compared to standard optical encryption methods which are based on phase and amplitude manipulation, this encryption procedure relying on Mueller-Stokes formalism provides a large flexibility in the key encryption design. The effectiveness of our algorithm is discussed thanks to a numerical simulation of the polarization encryption/decryption procedure of a 256 gray-level image. Of additional special interest is the immunity of this encryption algorithm to brute force attacks.

The development of advanced coding methods is an extremely active research area of great visibility and importance (see for a recent review [START_REF] Alfalou | Optical image compression and encryption methods[END_REF]). In particular, much effort has been devoted to searching for new types of encryption methods which can be implemented in an optical setup [START_REF] Refregier | Optical image encryption based on input plane and Fourier plane random encoding[END_REF][START_REF] Gopinathan | A polarization encoded optical encryption system using ferroelectric spatial light modulator[END_REF][START_REF] Gopinathan | Polarization encoding and multiplexing of two-dimensional signals: application to image encryption[END_REF][START_REF] Alfalou | Double random phase encryption scheme to multiplex and simultaneous encode multiple images[END_REF]. These efforts are aimed at combining the excellent control possible with spatial light modulators, with the miniaturization, parallelism, and integrability of optical devices. Despite their efficiency, many processing techniques remain uneasy to be implemented using optical techniques, can lead to complex-valued encoded images, and can be potentially insecure against attacks.

On the other hand, the interplay between encryption and polarization has piqued the interest of optical physicists for several decades [START_REF] Dollfus | Videopolarimeter and its applications in physics and telescopic observations[END_REF][START_REF] Biener | Space-variant polarization scrambling for image encryption obtained with subwavelength gratings[END_REF][START_REF] Matoba | Secure holographic memory by double-random polarization encryption[END_REF][START_REF] Tan | Secure optical memory system with polarization encryption[END_REF][START_REF] Mogesen | A phase-based optical encryption system with polarization encoding[END_REF][START_REF] Richert | Adapted polarization state contrast image[END_REF]. Since the early work of Dolfuss and co-workers [START_REF] Dollfus | Videopolarimeter and its applications in physics and telescopic observations[END_REF] dealing with polarization imaging, several polarization encryption methods were considered (see [START_REF] Gopinathan | A polarization encoded optical encryption system using ferroelectric spatial light modulator[END_REF][START_REF] Gopinathan | Polarization encoding and multiplexing of two-dimensional signals: application to image encryption[END_REF][START_REF] Matoba | Secure holographic memory by double-random polarization encryption[END_REF][START_REF] Tan | Secure optical memory system with polarization encryption[END_REF] and references therein).

In this Letter, we propose an alternative scheme. The method is based on Mueller-Stokes formalism and serves as a good starting point toward ultimate understanding of secure transmission of optical images using polarization encoding. The Letter is organized as follows: the principle of the polarization algorithm is first described. Next, we shall illustrate the effectiveness of this approach by working out an example of a 256 gray-level image. We further test the strength of this encryption algorithm against unauthorized decryption. Some preliminary notation is in order. Let us consider a narrow band stochastic field which can be represented by an ensemble of realizations, which we shall assume to be statistically stationary, at least in the wide sense. Each realization of the fluctuating electric field vector is represented by a complex analytic signal. The four Stokes parameters, S j , defined as the covariances of the analytic signal components, are the observables of the field vector at optical frequencies [START_REF] Brosseau | Fundamentals of Polarized Light[END_REF][START_REF] Wolf | Introduction to the Theory of Coherence and Polarization of Light[END_REF]. Let

              , (1)
We find for the right-hand side of Eq.

(1):

2 1 00 = pr m , ( ) ( ) ( ) ( ) ( ) ( ) θ θ ϕ θ ϕ
    + + +     + + +     = =     + + +             . ( 2
)
Analogously, an encryption wave is sent to a key image k ( )

0 1 2 3 T k k k k k S S S S = S
. A graylevel image with 256× 256 pixels randomly distributed in the range [ ] ) ( )

' ' ' ' ' 0 1 2 3 0 0 1 0 0 2 T T k k k k k k k S S S S S S = = S
. Fig. 2 shows that the two output signals are multiplexed using a beam-splitter. Accordingly, the resulting signal is ( )

' ' ' 0 0 0 0 1 0 0 2 T R I k I k I k S S S S = + = + + S S S
. Fig. 2 shows also the transformation of this image Consequently, each pixel of the encrypted image is given by ) , ( '

j i I C ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ' 0 ' ' , ' 1 ' 2 ' 3 , , , 2 , , , , , , 2 ( , ) ( ,
I i j I i j I i j i j I i j I i j ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ       = =         = M S ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 , , , 2 , 0 0 , , ( , ) ( , )
( , ) ( , ) 1 0 2 0 0 0 0 0 0 1 cos 2 cos 2 cos 2 1 ( , ) ( , ) 4 sin 2 1 cos 2 0 I k I k i j rand i j rand i j i j rand rand I k i j i j rand rand S i j S i j S i j S i j S i j S i j ϕ ϕ ϕ ϕ ϕ ϕ   +         +                     +     +   = +   +       , (3)
The measured intensity of the encrypted image, stored with a CCD camera (Fig. 2), in this case is given by

( ) ( ) [ ] ' , 0 0 0 1 ( , ) ( , ) 1 cos 2 ( , ) ( , ) 4 i j C C rand I k I i j I i j S i j S i j ϕ = = + + . ( 4
)
Two important points should be considered when interpreting the images. First, we remark that only real numbers, i.e. Eq. ( 4), are considered since we used the Mueller matrix formalism. This is at odds with the standard encoding methods, e.g. the double-random phase encryption technique, which transform the input image into a complex-amplitude stationary white noise [START_REF] Refregier | Optical image encryption based on input plane and Fourier plane random encoding[END_REF]. Second, the decryption method is a two-step process. On the one hand, the encrypted signal is passed through a linear polarizer Pol decry (4) whose polarization angle is oriented such that the term ( ) ( )

j i rand
, 0 0 _ , ' 0 _ , , ' 1_ 0 0 , , ( ) [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
I i j M S S S i j S i j ϕ ϕ ϕ ϕ ϕ ϕ   + +     =             +     +   = + =   +       ' 2 _ ' 3_ decryp decryp S S               . ( 5
)
From Eq. ( 5) it is clear that the image,

) ( ' _ 0 decryp S
, is still encrypted. The decryption process can be implemented by extracting one particular component of the relevant Stokes vector, Eq. ( 5). Here, ( The encryption procedure described above is quite general and can be implemented both optically and numerically. In order to test our method, we simulated numerically the different steps of this procedure with a 256 gray-level original image (Fig. 3(a)). The results of the simulation for the encrypted (resp. decrypted) images in the aforesaid steps are depicted in Fig. 3 (c) (resp. Fig. 3 (d)). The key image (Fig. 3(b)) is also completely depolarized. It can be seen that the target image is not recognizable. Based on the above described decryption procedure and the knowledge of the two encryption keys (key image and angles of the encrypting polarizers Pol(3)), the target image is clearly observed in Fig 3 (d).

{ } ' , , 1_ 0 0 1 cos 2 1 cos 2 ( , ) ) ( ) ( ) [ ]
The value of a general and reliable image encryption/decryption algorithm depends on a clear understanding and control of all possible attacks, i.e. plain-text, ciphertext, statistical, and brute force attacks. Here, we argue that our system show an excellent resistance against a specific type of brute force attack. A more extensive analysis will be reported elsewhere. A first example where our results are applicable occurs where the target image has a spatially uniform intensity.

From the above derivation, we Hence, it is impossible for the attacker to have access either in the key image and in the set of angles of Pol(3).

In a second case, we assume that the attacker has the ability to trick a legitimate user of the system into encrypting images ((s)he knows the key image), has a priori knowledge of the principle of the polarization algorithm, and know the nominal values of the polarization angles of Pol(1) and Pol(2) and also the phase shifts of Ret(1) and Ret [START_REF] Refregier | Optical image encryption based on input plane and Fourier plane random encoding[END_REF]. For the evaluation of the decryption quality, the mean square error (MSE) was used which can be calculated by

( ) ( ) ∑∑ - × = - N i N j I decryp j i S j i S N N MSE 2 0 ' 1 , , 1
. Here i, j label the pixels; ' 1 decryp S - characterizes the output decrypted image (Eq. ( 5)), and 0 I S denotes the intensity of the unpolarized input image, respectively. As can be seen from Figs. 3 (e)-(f) we find no significant variation in the magnitude of the MSE. This result leads to the conclusion that the attacker is unable to find the target image even after more than 5 10 5 trials. In summary, the numerical results demonstrated that the proposed encryption/decryption procedure of images based on Mueller-Stokes formalism has several interesting features. First of all, it was demonstrated that the polarization algorithm based on a dual encryption scheme is very general. Secondly, we showed that our encryption scheme remains robust under brute force attacks. All aspects of this scheme can be optically implemented using current state-of-the art technology. We note that, in a real optical implementation, the difficulty to manipulate and measure the involved polarimetric quantities adds some challenges to the attacker. Because these ideas have a broad significance they are also expected to impact related areas demanding secure data. This work was supported by Lab-STICC which is Unité Mixte de Recherche CNRS 3192.

  where T means the transpose, denote the Stokes vector and M is the Mueller matrix of a polarization element. Let the input 

  The state of polarization of the output signal is completely determined by the nominal values of θ and ϕ . This optical system functions as a polarization encoder, encoding the pixels of the signal (image) by θ and ϕ .The basis of the experiment is depicted schematically in Fig.2. The concept is first demonstrated quantitatively using the Mueller-Stokes formalism. Consider first the partially polarized wave object to be encrypted (target image I). The state of polarization is determined by the configuration of the polarization elements shown in Fig.

2 ϕ

 2 next for illustrative purpose. Then, the wave is passed through the similar set of polarization elements shown in Fig. 1. The output signal (image) is ' k S . For the current discussion we will consider only the simplest situation of unpolarized signals, i.e. denote the polarization angles of Pol(1) and Pol(2), and 1 θ and 2 θ are the phase shifts of Ret(1) and Ret(2), respectively. Hence

  to a new polarization state via a matrix of linear polarizers (Pol(3); each of them is characterized by angle the randomly chosen angle rand rand ϕ = π in the range [-π,π].

  vanishes. At the output of Pol decry (4) the Stokes vector is ( )

  to the primary image.

  of the encrypting polarizers Pol(3).

Figure captions FIG. 1 :FIG. 2 :FIG. 3 :

 captions123 Figure captions

  1. Using Eq. (1), one finds that the output Stokes vector is

			S	' I	0	00 pr m S	I	0	01 pr m S	I	1	02 pr m S	I	2	03 pr m S	I	3
	S	' I	' ' I I S S	1 2	10 20 pr pr m S m S	I I	0 0	11 21 pr pr m S m S	I I	1 1	12 22 pr pr m S m S	I I	2 2	13 23 pr pr m S m S	I I	3 3
			S	' I	3						0