Assia Benabdallah

Yves Dermenjian

Laetitia Thevenet

Carleman estimates for some non smooth anisotropic media

Keywords: AMS 2010 subject classification: 35B37, 35J15, 35J60 anisotropic elliptic operators, approximation, non-smooth coefficients, stratified media, Carleman estimate, observation location

teaching and research institutions in France or abroad, or from public or private research centers.

Lipschitz. In fact, Carleman estimates imply the unique continuation property which does not hold in general for a C 0,α Hölder regularity of the coefficients with 0 < α < 1 [START_REF] Miller | Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients[END_REF][START_REF] Pliś | On non-uniqueness in Cauchy problem for an elliptic second order differential equation[END_REF].

Here we are interested in coefficients that are non continuous across an interface S . When the observation takes place in the region where the diffusion coefficient c is the 'lowest', this question was solved 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 ). In the one dimensional case, and without assumption on the localization of the observation, the question was solved 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]. The work [START_REF]Carleman estimates for stratified media[END_REF] generalizes [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] to some stratified media with dimension n ≥ 1. Without Carleman estimate, the controllability for a one dimensional parabolic operator was proved in [START_REF] Alessandrini | Null-controllability of one-dimensional parabolic equations[END_REF] for c ∈ L ∞ but this approach does not authorize semilinear operators.

Recently, Carleman estimates for an arbitrary dimension without any condition on the localization of the observation were obtained 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], in the elliptic case, and in the parabolic case in [START_REF]Carleman estimates for stratified media[END_REF][START_REF]Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces[END_REF], but 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 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][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] require strong regularity for the coefficients and for the interface. Moreover, they fall short if the interface crosses the boundary whereas this configuration is typical in bounded stratified media, examples falling into the framework considered here and in [START_REF]Carleman estimates for stratified media[END_REF]. In [START_REF]Carleman estimates for stratified media[END_REF] the authors assumed that the diffusion coefficients have a 'stratified' structure. More precisely, they have considered operators of the form A = -∇ • (B(•)∇) in which the matrix diffusion coefficient B(x) has the following block diagonal form

B(x) = B(x , x n ) = c 1 (x n )C τ (x ) 0 0 c 2 (x n )
where Ω = Ω × (-H, H), x = (x , x n ), C τ is a smooth hermitian matrix and the coefficients c 1 , c 2 have a possible jump at x n = 0. The object of the present work is to obtain a Carleman estimate for more general diffusion coefficients without a stratified structure that separates variables. We shall consider a matrix diffusion coefficient of the form

B(x) = C τ (x) 0 0 c(x)
with C τ and c having possible jump at x n = 0.

Here, to understand the difficulties that we face, the reader can observe that attempting to prove the Carleman estimate by extending the proof as is done in the one dimensional case, leads in fact to tangential terms at the interface S that cannot be controlled. These terms existed also in [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF] where B is a scalar function c which led the authors to add conditions on the jump of diffusion coefficient c at the interface. Not to mention our approach, the main contribution of our paper is to derive an estimate of these tangential terms allowing to conclude the proof of the Carleman estimate.

In [START_REF]Carleman estimates for stratified media[END_REF], these tangential terms at the interface are controlled by using Fourier series in the tangential direction. By a suitable choice of the weight function, the low frequencies lead to a positive quadratic form. The treatment of the high frequencies needs more computations. It uses the ideas developed 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] where the normal part of the elliptic operator can be inverted. In [START_REF]Carleman estimates for stratified media[END_REF] this argument uses the assumption of the separation of the tangential and normal variables in the diffusion matrix B.

In the case we consider here, the diffusion coefficients depend on x = (x , x n ) and, contrary to [START_REF]Carleman estimates for stratified media[END_REF], one cannot decompose the operator A as ∂ x n c 2 (x n )∂ x n + A τ with A τ a tangential elliptic operator on Ω . Our method consists in the introduction of a suitable decomposition, (Ω j,δ ), of a neighborhood of the interface and, on each Ω j,δ , an approximation of the diffusion coefficient c by a function depending only on the normal variable, x n , for which the result of [START_REF]Carleman estimates for stratified media[END_REF] can be used. As these approximations depend on the Carleman large parameters s and λ (see (1.1)), we shall need a refined estimate of the tangential derivative (more precisely, for the high frequencies, see Lemma 3.1).

The question of the derivation of Carleman estimates in the case where the diffusion coefficients are totally anisotropic in the neighborhood of a point where the interface S meets the boundary ∂Ω is left open. Note also that deriving Carleman estimates for the parabolic operator associated to the elliptic operator we consider here, is also an open question. In fact, if we follow the same idea as for the elliptic case we present here, and if we use singular weight functions as introduced in [START_REF] Fursikov | Controllability of evolution equations[END_REF], we then have to consider approximations of order

1 √ t(T -t)sλϕ |S
(connected to the Carleman parameters). These approximations blow up near t = 0 and t = T .

For each pair (s, λ) of Carleman parameters, we introduce several meshes that seem to indicate a connection to numerical methods. We believe that this connection should be further investigated.

Setting and notation

Let Ω be an open subset in R n , with Ω = Ω × (-H, H) 1 , where Ω is a nonempty bounded open subset of R n-1 with C 2 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(x), 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. We further assume uniform ellipticity

0 < c min ≤ c(x) ≤ c max < ∞, x ∈ Ω, 0 < c min Id n-1 ≤ C τ (x) ≤ c max Id n-1 , x ∈ Ω.
1 As a matter of fact, we only ask that Ω is a cylinder in a neighborhood of the interface S . See the end of section 2. 2 For some particular geometries we can suppose that Ω is piecewise smooth. Nevertheless the technics used for building our approximation in a neighborhood of the interface seems to require better than C 1 . We shall take C 2 for the readability.

We consider the symmetric bilinear

H 1 0 -coercive form a(u, v) = ∫ Ω (B(•)∇u) • ∇vdx, with domain D(a) = H 1 0 (Ω). It defines a selfadjoint operator A = -∇ • (B(•)∇) in L 2 (Ω) with compact resolvent and its domain is D(A) = {u ∈ H 1 0 (Ω); ∇ • (B(•)∇u) ∈ L 2 (Ω)}.
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.

In this article, when the constant C is used, it refers to a constant that is independent of all the parameters. Its value may however change from one line to another. If we want to keep track of the value of a constant we shall use another letter or add a subscript.

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. A proper choice of the function β, with respect to the operator A, ω and Ω (see Assumption 2.4 and Assumption 4.1), yields the following Carleman estimate for the elliptic operator A.

Theorem 1.2. 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 ∂ x 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, i.e. parallel to the interface S . Note that the membership of the domain D(A) implies some constraints on the function u at the interface S , namely u ∈ H 1 0 (Ω) and B∇ x u ∈ H(div, Ω) := {v ∈ L 2 (Ω) n ; div v ∈ L 2 (Ω)}. We shall first prove the result for piecewise

C 2 functions satisfying u |S -= u |S + , (c∂ x n u) |S -= (c∂ x n u) |S + ,
and then use their density in D(A) (see Appendix C).

Outline

Choosing 0 < δ < H, our starting point is the following Carleman estimate in the open set

Ω δ := Ω × (-δ, δ).
There exist a weight function β and C, C > 0, λ 0 > 0, s 0 > 0 such that

C sλ 2 ϕ 1 2 e sϕ ∇u 2 L 2 (Ω δ ) + s 3 λ 4 ϕ 3 2 e sϕ u 2 L 2 (Ω δ ) + sλϕ |S ∫ S [c 2 β |e sϕ ∂ x n u| 2 ] S dσ + ∫ S |sλϕe sϕ u |S | 2 [c 2 β 3 ] S dσ -∫ S |e sϕ ∇ τ u| 2 [β cC τ ] S dσ ≤ C e sϕ Au 2 L 2 (Ω δ ) (1.2)
for all u ∈ D(A), λ ≥ λ 0 , s ≥ s 0 and supp u ⊂ Ω × (-δ, δ).

We have to understand [β cC τ ] S as the matrix of jumps of each term of the matrix and [β cC τ ] S is its norm that we can take in L ∞ (S ). Such an inequality 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 and a suitable choice of the weight function β. Some easy handlings (see [START_REF]Carleman estimates for stratified media[END_REF]) show that only the sign of the term ∫ S |e sϕ ∇ τ u| 2 [β cC τ ] S dσ arises a problem since we cannot exclude to have a negative quantity. In other words, the main difficulty is to estimate the tangential derivative of u at the interface S .

In Section 2, we introduce a covering (Ω j ) j of a neighborhood of Ω related to the Carleman's parameters s, λ and precise (c j ) j , the approximation of the diffusion coefficient c. Of course, we build an adapted partition of unity (χ j ) j subordinated to (Ω j ) j and we define, for each x n ∈ (-δ, δ) and each j, the tangential part of A, i.e.

A τ (x n ) = -∇ τ • C τ (•, x n )∇ τ with D j (A τ (x n )) = {u ∈ H 1 0 (Ω j ); ∇ τ • C τ (•, x n )∇ τ u ∈ L 2 (Ω j )}. So, for u ∈ D(A) that solves Au = f with f ∈ L 2 (Ω), u j := χ j u will solve                A τ (x n )u j -c j (x n )∂ 2 x n u j = f j + g j + h j on Ω ± j,δ = {x = (x , x n ); x ∈ Ω j , 0 < ±x n < δ}, u j = 0 on ∂Ω j , [u j ] S j = 0 and [c j ∂ x n u j ] S j = [(c j -c)∂ x n u j ] S j := θ j (1.3) with S j = Ω j ∩ S , f j = χ j f, g j = (c -c j )∂ 2
x n u j and

h j = [A τ , χ j ]u + (∂ x n c)(∂ x n u j ), (1.4)
where [A τ , χ j ] denotes the commutator of A τ and χ j .

It will be sufficient to estimate the tangential derivative of u j defined below. We cannot directly apply the results of [START_REF]Carleman estimates for stratified media[END_REF] for two main reasons:

1. the dependence on x n of A τ , 2. the presence of θ j and g j involving the normal derivative of u on S j and the second derivative of u on Ω j , 3. the presence of h j which depends on s, λ.

To take into account the first constraint, we consider (µ 2 j,k (x n )) k≥1 , the family of eigenvalues of (A τ (x n ), D j (A τ (x n ))), and denote by u j,k , f j,k , g j,k , h j,k and θ j,k the respective Fourier coefficients of u j , f j , g j , h j and θ j in an orthonormal basis associated to the previous eigenvalues.

To overcome the two other constraints, we prove a refined estimate of the high frequencies (i.e. coefficients associated to the large eigenvalues in the decomposition on the eigenfunctions) of the tangential derivatives of u j (see Section 3): there exist a constant C independent of s, λ, µ j,k , a constant µ 0 := µ 0 (s, λ) > 0 such that, for all µ j,k (0 + ) ≥ µ 0 , one has

sλϕ | S |µ j,k (0 + )e sϕ | S u j,k | 2 S j ≤ C |e sϕ f j,k | 2 L 2 (-δ,δ) + ϕ | S |e sϕ ϕ -1/2 g j,k | 2 L 2 (-δ,δ) + ϕ -1 | S |e sϕ ϕ 1/2 h j,k | 2 L 2 (-δ,δ) + s 3/2 λϕ | S e 2sϕ| S |θ j,k | 2 (1.5)
for s, λ sufficiently large (as we allow C τ (x) to be discontinuous through S , µ j,k (0 + ) denotes lim x n ↓0 + µ j,k (x n )). Note the peculiar weights in the right hand side (in brief r.h.s.).

The low frequencies are treated as in [START_REF]Carleman estimates for stratified media[END_REF] and, still as in [START_REF]Carleman estimates for stratified media[END_REF], we conclude by verifying that there exists a weight function β such that one can recover the spectrum of (A τ (0 + ), D j (A τ (0 + ))) (see Section 4). It remains to eliminate the three last terms of (1.5) (the additional terms with respect to [START_REF]Carleman estimates for stratified media[END_REF]). The properties of the functions χ j and the definition of c j shall be used in this step:

∇χ j ∞ ≤ C sλϕ | S , ∇ • (B∇χ j ) ∞ ≤ Csλϕ | S , c j -c ∞ ≤ C 1 sλϕ | S ,
as well as the weights of the second members obtained through the refined estimate. Collecting all the previous results we shall have proved Theorem 1.2 and this will conclude Section 4.

In order to point out the main ideas of this work, we have put almost all technical results in the appendix.

Preparation of data

The partition

Theorem 2.1. For each pair (s, λ), s > 0, λ > 0, there exist a finite family (Ω j ) j∈J of open sets such that Ω ⊂ ∪ j Ω j and a partition of unity (χ j ) j∈J subordinated to this open covering with

∇ τ χ j ∞ ≤ C sλϕ | S and ∇ τ • (C τ ∇ τ χ j ) ∞ ≤ Csλϕ | S , (2.1)
where the constant C is independant on s, λ and j. Moreover, only N functions χ j are non equal to 0 in each point of Ω with N only depending on Ω .

As a matter of fact, the proof is tricky when Ω is not a cube. In this case, we begin to define the Ω j := Ω j (s, λ) such that Ω j ⊂ Ω . They are cubes of which the length of the edges is h = h(s, λ). Next, we define the open sets Ω j that intersect ∂Ω . They are no more exactly cubes. The complete proof is given in Appendix A. We recover Ω δ by the family of cylindrical subdomains

Ω j,δ (s, λ) = {(x , x n ) ∈ R n ; x ∈ Ω j (s, λ), -δ < x n < δ}.
In the sequel, we will denote Ω j,δ := Ω j,δ (s, λ) = Ω j ×(-δ, δ) and we recall that

Ω ± j,δ = {(x , x n ); x ∈ Ω j , 0 < ±x n < δ} (already quoted for (1.3)).

Partition and transverse operators

On each subdomain Ω j,δ we define the following approximation of the diffusion coefficient c(x , x n ):

c j (x n ) =            c j + (x n ) = 1 |Ω j | ∫ Ω j c(x , x n ) dx , ∀ x n ∈ (0, δ), c j -(x n ) = 1 |Ω j | ∫ Ω j c(x , x n ) dx , ∀ x n ∈ (-δ, 0). (2.2)
So, for each x n ∈ (-δ, 0) ∪ (0, δ), we have given sense to (1.3) where the operators A τ (x n ) act in a section of Ω ± j,δ parallel to the interface S j . Similarly, we have c j (0 ± ). If Ω is a cube, we can go straight to Lemma 2.2 taking into account the construction of the partition (see Step 2 in Annexe A) since we do not need to extend the coefficients. Otherwise, a problem remains with the open cylinders Ω j,δ (s, λ) intersecting ∂Ω, which needs modifications in a neighborhood of ∂Ω . The idea is to extend the coefficients c(x , x n ), outside Ω and independently of (s, λ), in such a way that we control the behavior of the extended solutions u j that are associated to these cylinders. The reader may refer to Appendix A for more explanations. Now we mention the following result which will be useful in the next section.

Lemma 2.2. With Assumption 1.1, we have inf j∈J,k≥1 inf x n ∈(-δ,δ) µ 2 j,k (x n ) µ 2 j,k (0 + ) > 0, and sup j∈J,k≥1 sup x n ∈(-δ,δ) µ 2 j,k (x n ) µ 2 j,k (0 + ) < +∞. (2.3)
Proof. We shall easily deduce these inequalities from the variational presentation of the Min-Max Principle since all the symmetric bilinear H 1 0 -coercive forms a τ,x n ,Ω j of the operators A τ (x n ) have same domain up to a translation of variables, i.e.

H 1 0 (Ω j ). If V k denotes the generic k-dimensional linear space of L 2 (Ω j ) and if V ⊥
k is its orthogonal space in L 2 (Ω j ) for the scalar product (u, v) = ∫ uv dx (specific notation to this Lemma, as well as u 2 = (u, u)), we know that

µ 2 j,k (x n ) = max V k-1 ⊂L 2 (Ω j )        min u∈V ⊥ k ∩H 1 0 (Ω j ), u =1 a τ,x n ,Ω j (u, u)        ,
which implies, by Assumption 1.1,

c min max V k-1 ⊂L 2 (Ω j )        min u∈V ⊥ k ∩H 1 0 (Ω j ), u =1 ∇u 2        ≤ µ 2 j,k (x n ) ≤ c max max V k-1 ⊂L 2 (Ω j )        min u∈V ⊥ k ∩H 1 0 (Ω j ), u =1 ∇u 2        , from which one may conclude c min c max ≤ µ 2 j,k (x n ) µ 2 j,k (0 + ) ≤ c max c min . (2.4) Remark 2.3.
When Ω j ∩ ∂Ω ∅, we only have to modify, without repercussions, the values of c min and c max that appear in (2.4). We shall find again this situation throughout the proofs of this work.

In order to evaluate the awkward term ∫ S |e sϕ ∇ τ u| 2 [β cC τ ] S dσ that occurs in (1.2) , we have to estimate ∇ τ u on the interface S . In fact, we need this estimate for u j := χ j u. We write u j (x) = k u j,k (x n )ϕ k (x , x n ) where the family (ϕ k (•, x n )) k≥1 is an orthonormal basis associated to the eigenvalues of A τ (x n ). So, the first line of (1.3) becomes

µ 2 j,k (x n )u j,k -c j (x n )∂ 2 x n u j,k = f j,k + g j,k + h j,k , 0 < |x n | < δ. (2.5)
For x n = 0, the same relation is valid on condition to distinguish the cases x n = 0 + and x n = 0 - for the coefficients µ 2 j,k and c j . Finally, reasoning as in [START_REF]Carleman estimates for stratified media[END_REF], section 2, we find

(c max ) -1 ∞ k=1 µ 2 j,k (x n )|u j,k (x n )| 2 ≤ ∇ τ u j (•, x n ) 2 L 2 (Ω j ) ≤ (c min ) -1 ∞ k=1 µ 2 j,k (x n )|u j,k (x n )| 2 .
(2.6)

The weight function β

The open set ω having been fixed in section 1.2, we choose a weight function β that satisfies the following properties.

Assumption 2.4. 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 .

We draw reader's attention on two points: firstly, the trace β |S is constant on the interface S and, secondly, we can 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].

In the remainder of this paper 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. As we can change x n into -x n to treat the case of an observation ω ⊂ Ω × (-H, 0), we lose nothing. Note that Assumption 2.4 will be completed below by Assumption 4.1.

A refined estimation for the high frequencies of the tangential derivative

We recall that this section is a first step to achieve inequality (1.5). Taking into account (2.5) we fix, for the moment, j ∈ J, k ∈ N * and consider w solution of

               µ 2 j,k (x n )w -c j (x n )∂ 2 x n w = F on (-δ, 0) ∪ (0, δ), w(±δ) = 0 [w] S = 0 and [c j ∂ x n w] S = θ,
with, here, S = {0}, F ∈ L 2 (-δ, δ), θ ∈ R. One has Lemma 3.1. Let F belong to L 2 (-δ, δ). There exist a constant C independent of s, λ, j, k, a constant µ 0 := µ 0 (s, λ) > 0 such that for all µ j,k (0 + ) ≥ µ 0 , the following estimates are satisfied for s, λ sufficiently large and w solution of (3)

sλϕ | S µ j,k (0 + ) 2 |e sϕ w| 2 S ≤ C |e sϕ F| 2 L 2 (-δ,δ) + s 3/2 λϕ | S e 2sϕ| S |θ| 2 , (3.1) sλϕ | S µ j,k (0 + ) 2 |e sϕ w| 2 S ≤ C ϕ | S |e sϕ ϕ -1/2 F| 2 L 2 (-δ,δ) + s 3/2 λϕ | S e 2sϕ| S |θ| 2 , (3.2) sλϕ | S µ j,k (0 + ) 2 |e sϕ w| 2 S ≤ C ϕ -1 | S |e sϕ ϕ 1/2 F| 2 L 2 (-δ,δ) + s 3/2 λϕ | S e 2sϕ| S |θ| 2 . (3.3) Remark 3.2.
Even if Lemma 3.1 seems similar to Proposition 3.5 of [START_REF]Carleman estimates for stratified media[END_REF] (for elliptic operator), there are two important differences: the weights for the sources F and the presence of θ which is 0 in this Proposition.

The difference among these three inequalities is the weight of the source terms. It should be noted that there is no comparison relation between them. The source terms resulting from the approximation of the coefficient c fall into three terms (see (1.4)). The first is just the localization of the initial source term, the second is the difference between the elliptic operator and its approximation and the third comes from the action of the cut-off function χ j on the elliptic operator. As we shall see later, they should be treated differently to be absorbed by the r.h.s. of the Carleman estimate (1.2).

Proof. We begin to set

σ 2 := inf j∈J,k≥1 inf x n ∈(-δ,δ) µ 2 j,k (x n ) c j (x n )µ 2 j,k (0 + ) and µ 0 := µ 0 (s, λ) = 2sλϕ | S β |S -+ λβ |S - σ , (3.4)
and next we introduce

W(x n ) = 1 2 sλϕ | S e 2sϕ | S |w(x n )| 2 .
On the one hand it derives from (2.3) that we can choose σ > 0, on the other hand we observe that W ≥ 0 and it verifies

           ∂ 2 x n W -σ 2 µ j,k (0 + ) 2 W = -σ 2 µ 2 j,k (0 + ) - µ 2 j,k (x n ) c j (x n ) (2 -γ) W := -d, 0 < |x n | < δ, W(-δ) = W(δ) = 0, W(0 -) = W(0 + ), c j + W (0 + ) = c j -W (0 -) + θw | S sλϕ | S e 2sϕ | S with = 1 c j -sλϕ | S e 2sϕ | S F w + sλϕ | S e 2sϕ | S c j (∂ x n w) 2 + γµ 2 j,k W , (3.5)
where the real number γ will be precised later and we have omitted the subscript j in S since the function β, and therefore ϕ, depends only on x n if -δ < x n < δ. Applying Lemma B.2, one gets (c j ± := c j (0 ± ) to lighten the writing)

sλϕ | S |µ j,k (0 + )e sϕ | S w(0)| 2 = 2µ j,k (0 + ) (c j + + c j -) δ ∫ 0 sinh(σµ j,k (0 + )(δ -x n )) σ cosh(σµ j,k (0 + )δ) c j + d(x n ) + c j -d(-x n ) dx n - 2µ j,k (0 + ) tanh(σµ j,k (0 + )δ) θw | S σ(c j + + c j -) sλϕ | S e 2sϕ | S (3.6)
that is exactly the left-hand side of estimates of Lemma 3.1. Setting r(x n ) := σ 2 µ 2 j,k (0 + )-

µ 2 j,k (x n ) c j (x n ) (2- γ), we have d(x n ) = r(x n )W(x n ) -(x n
) and, immediately, we note that the definition of σ implies that r(x n ) ≤ 0 as soon as γ ≤ 1. So, we emphasize a non positive contribution that we can eliminate, namely

δ ∫ 0 sinh(σµ j,k (0 + )(δ -x n ))
σ cosh(σµ j,k (0

+ )δ) c j + r(x n )W(x n ) + c j -r(-x n )W(-x n ) dx n ≤ 0. (3.7)
Now, we consider the contribution coming fromand, similarly to the previous result, the second term of (3.5) brings a non positive contribution since

- δ ∫ 0 sinh(σµ j,k (0 + )(δ -x n )) cosh(σµ j,k (0 + )δ) sλϕ | S e 2sϕ | S c j + (∂ x n w) 2 (x n ) + c j -(∂ x n w) 2 (-x n ) dx n ≤ 0. (3.8)
The estimate of the other terms of (3.6) needs more computations. Temporarily we forget the coefficient

2 σ(c j + +c j -)
that we shall reinstate later. Let us begin by

I ± := -µ j,k (0 + ) δ ∫ 0 sinh(σµ j,k (0 + )(δ -x n )) cosh(σµ j,k (0 + )δ) c j ± c j -sλϕ | S e 2sϕ | S F(±x n )w(±x n ) + γµ 2 j,k (±x n )W(±x n ) dx n .
On one hand, applying the Young inequality we have, for any α > 0,

µ j,k (0 + )|sλϕ | S e 2sϕ | S F(±x n )w(±x n )| ≤ 1 2γα e 2sϕ | S |F(±x n )| 2 + γ 2 αs 2 λ 2 ϕ 2 | S e 2sϕ | S µ 2 j,k (0 + )|w(±x n )| 2 ,
and, on the other hand, observing that sinh(σµ j,k (0 + )(δx n )) cosh(σµ j,k (0 + )δ) = e σµ j,k (0 + )(δ-x n )e -σµ j,k (0 + )(δ-x n ) e σµ j,k (0 + )δ + e -σµ j,k (0 + )δ ≤ e -σµ j,k (0 + )x n , we obtain

I ± = µ j,k (0 + ) δ ∫ 0 sinh(σµ j,k (0 + )(δ -x n )) cosh(σµ j,k (0 + )δ) c j ± c j sλϕ | S e 2sϕ | S F(±x n )w(±x n ) -γ(µ 2 j,k W)(±x n ) dx n ≤ δ ∫ 0 e -σµ j,k (0 + )x n +2sϕ | S 2γα c j ± c j |F(±x n )| 2 dx n + δ ∫ 0 sinh(σµ j,k (0 + )(δ -x n )) cosh(σµ j,k (0 + )δ) c j ± c j γ 2 µ 2 j,k (0 + )sλϕ | S |e sϕ | S w(±x n )| 2         αsλϕ | S - µ 2 j,k (±x n ) µ j,k (0 + )         dx n . Assuming α ≤ β | S - σ c min c max , we see that (αsλϕ | S - µ 2 j,k (±x n )
µ j,k (0 + ) ) ≤ 0 since we suppose µ j,k (0 ± ) ≥ µ 0 . That allows us to omit the corresponding term and gives

I ± ≤ ∫ δ 0 e -σµ j,k (0 + )xn+2sϕ | S 2γα c j ± c j |F(±x n )| 2 dx n .
This term will be estimated by three different and non-comparable ways that will lead to the three estimates of Lemma 3.1.

Case 1. We follow exactly the way described in [START_REF]Carleman estimates for stratified media[END_REF] (point 2(a) of section 4): replacing ∂ t w by F and forgetting the dependance in t, we arrive to the condition σµ j,k (0

+ ) ≥ 2sλβ ϕ | S .
As it is verified for µ j,k (0 + ) ≥ µ 0 , we have

I ± ≤ ∫ δ 0 e 2sϕ 2γα c j ± c j |F ± x n )| 2 dx n .
This will lead to (3.1), the first estimate of Lemma 3.1.

Case 2. Since we suppose µ j,k (0 + ) ≥ µ 0 with µ 0 defined in (3.4), the estimate (B.3) of Lemma B.1 leads to

I ± ≤ δ ∫ 0 e 2sϕ(±x n ) ϕ | S 1 2γα c j ± c j |ϕ -1/2 (±x n )F(±x n )| 2 dx n ,
which will give (3.2), the second estimate of Lemma 3.1.

Case 3. Always with the same hypothesis on µ j,k (0 + ), the estimate (B.2) of Lemma B.1 leads to

I ± ≤ δ ∫ 0 e 2sϕ(±x n ) ϕ -1 | S 1 2γα c j ± c j |ϕ 1/2 (±x n )F(±x n )| 2 dx n ,
which will give (3.3), the third estimate of Lemma 3.1.

Moreover the last term in (3.6) verifies 2µ j,k (0 + ) tanh(σµ j,k (0

+ )δ) θw | S σ(c j + + c j -) sλϕ | S e 2sϕ | S ≤ C s 3/2 λϕ | S e 2sϕ| S |θ| 2 + s 1/2 λϕ | S |µ j,k (0 + )e sϕ | S w | S | 2 ,
which permits to conclude for s large enough if we collect these results with (3.7) and (3.8).

Proof of the Theorem 1.2

We start off the inequality (1.2) and, as we do not know the sign of the second parenthesis of the left-hand side, we have to improve our knowledge of the integrant which leads us to introduce the quadratic form

B(u) = sλϕ |S e 2sϕ [c 2 β |∂ x n u| 2 ] S + |sλϕu |S | 2 [c 2 β 3 ] S
(we can extract the function e 2sϕ since it is continuous) and

c ± (x ) = c(x , 0 ± ), L = β |S + β |S - , K c (x ) = c -(x ) c + (x ) , K τ = C τ (x , 0 -)(C τ (x , 0 + )) -1 L ∞ (S ) K c = inf x ∈Ω K c (x ), K c = sup x ∈Ω K c (x ).
Finally, we set

D = D(L) = β |S -sup x ∈Ω c + (x ) C τ (x , 0 + ) (L + K c K τ )
and we make the following assumption on the weight function in addition to Assumption 2.4.

Assumption 4.1. The weight function β is chosen such that L ≥ L = max{K c , 2} and

K 2 c + L 3 (L -L) (L + K c K τ )(L -1) ≥ 36N sup x ∈Ω c + (x ) C τ (x , 0 + ) σ 2 inf x ∈Ω c 2 + (x ) , 2δβ |S -≤ β(0).
The integer N is the one of Theorem 2.1. The functions c, C τ being fixed, it is the same for σ 2 , K c , K c , K τ and L, which shows this inequality can be achieved by first choosing the value of β |S -> 0 and then picking a sufficiently large value for L. The assumption 2δβ |S -≤ β(0) can easily be fulfilled since β is defined up to a constant.

Lemma 4.2. We have

B(u) = sλϕ |S e 2sϕ |S B 1 |γ(u)| 2 + B 2 |sλ(ϕu) |S | 2 , with γ(u) = (c∂ x n u) |S + c + β |S - L 2 -K c
L-1 (sλϕu) |S and where

B 1 = β |S -(L -1), B 2 (x ) = c 2 + (x )(β |S -) 3 2(L 3 -K 2 c (x )) - L 2 -K c (x ) 2 L -1 . If β satisfies Assumption 4.1 we have B 1 > 0 and B 2 (x ) ≥ B, with B B = B(L) = inf x ∈Ω c 2 + (x ) (β |S -) 3 K 2 c + L 3 (L -L) L -1 . Remark 4.3. From the inequality (a + αb) 2 + b 2 ≥ (1 + α 2 ) -1 a 2 , we deduce that s 1/2 e 2sϕ | S |c∂ x n u| 2 is less than the product of s 1/2 λϕ | S inf(B 1 , B) -1
B(u) and of a constant C depending on L but not on (s, λ), which we shall use to remove the last term in (4.4), absorbing it by the l.h.s. of (1.2) for (s, λ) large enough.

The idea of the proof of Lemma 4.2 is similar to both proofs of Appendix A.2 and Lemma 4.4 that are in [START_REF]Carleman estimates for stratified media[END_REF].

It remains to estimate the tangential derivative of u at the interface S (the third integral of the second parenthesis in the l.h.s. of (1.2)). The wording of Theorem 2.1 points out an integer N and its existence implies that we have a constant C n independent on our mesh such that

I := sλϕ |S ∫ S [cβ C τ ] S |e sϕ ∇ τ u| 2 dσ ≤ C n sλϕ |S Σ j∈J ∫ S j [cβ C τ ] S |e sϕ ∇ τ u j | 2 dσ. Using [cβ C τ ] S = c + β |S -L Id R n-1 -K c C τ-(C τ+ ) -1 C τ+ , we obtain [cβ C τ ] S ≤ β |S -max(c + C τ+ ) L + K c C τ-(C τ+ ) -1
and, from (2.6) , it suffices to estimate D sλϕ |S Σ k≥1 µ j,k (0 + ) 2 |e sϕ u j,k | 2 on S j uniformly with respect to j. In order to solve this point, we shall distinguish two cases: the small values of k and the others.

Proposition 4.4. There exists C > 0 such that, for all j ∈ J, k ∈ N * , we have

Dsλϕ | S |µ j,k (0 + )e sϕ u j,k | 2 S j ≤ B 4N (sλϕ | S ) 3 e 2sϕ | S |u j,k | 2 S j + C |e sϕ f j,k | 2 L 2 (-δ,δ) + ϕ | S |e sϕ ϕ -1/2 g j,k | 2 L 2 (-δ,δ) +ϕ -1 | S |e sϕ ϕ 1/2 h j,k | 2 L 2 (-δ,δ) + s 3/2 λϕ | S e 2sϕ| S |θ j,k | 2 (4.1)
for s, λ and L large enough.

Proof. We shall keep track of the dependency of the constants on j and k. For low frequencies, direct computations lead to

Dsλϕ | S |µ j,k (0 + )e sϕ | S u j,k | 2 S j ≤ B 4N (sλϕ | S ) 3 e 2sϕ | S |u j,k | 2 S j , (4.2)
as soon as µ j,k (0

+ ) ≤ 1 2 √ N B D sλϕ | S
. Now, we look at the high frequencies with the definition (3.4) of µ 0 and we use Lemma 3.1 for F = f j,k + g j,k + h j,k and θ = θ j,k : there exists a constant C > 0 such that we have

Dsλϕ | S |µ j,k (0 + )e sϕ | S u j,k | 2 S j ≤ C |e sϕ f j,k | 2 L 2 (-δ,δ) + ϕ | S |e sϕ ϕ -1/2 g j,k | 2 L 2 (-δ,δ) + ϕ -1 | S |e sϕ ϕ 1/2 h j,k | 2 L 2 (-δ,δ) + s 3/2 λϕ | S e 2sϕ| S |θ j,k | 2 , ∀µ j,k (0 + ) ≥ µ 0 . (4.3)
Collecting (4.2) and ( 4.3), we have (4.1) but it remains to verify that one can recover all the spectrum of A τ (0 + ). This will be true as soon as there exists a weight function β such that, in addition of Assumption 2.4, it verifies

1 2 √ N B D sλϕ | S ≥ 2sλϕ | S β |S -+ λβ |S - σ ,
which will be achieved if, for sϕ |S ≥ 1/2, one has

B D ≥ 36N(β |S -) 2 σ 2
. This inequality is equivalent to

inf x ∈Ω (c 2 + (x ) )(β |S -) 3 K 2 c +L 3 (L-L) L-1 β |S -sup x ∈Ω c + (x ) C τ (x , 0 + ) (L + K c K τ ) ≥ 36N(β |S -) 2 σ 2 .
As β verifies Assumption 4.1 the previous estimate occurs and the proof is complete.

We now prove the following key result, providing an estimate of the tangential derivative of u. Proof. The proof is a consequence of two Lemmas:

Lemma 4.6. There exists C > 0 such that for all j ∈ J, k ∈ N * , we have

ϕ | S e sϕ ϕ -1/2 g j 2 +ϕ -1 | S e sϕ ϕ 1/2 h j 2 ≤ C sλ ϕ -1/2 e sϕ f j 2 + (sλ) 2 e sϕ ϕ 1/2 ∇u 2 +s 3 λ 3 ϕ 3/2 e sϕ u 2 .
for s and λ large enough.

Proof. We recall that

h j = [A τ , χ j ]u + (∂ x n c)∂ x n u j . Since [A τ , χ j ] + (∂ x n c)∂
x n is an operator of order 1 of which the coefficients depend on ∇ τ χ j , ∇ τ • C τ ∇ τ χ j and ∂ x n c, and as e sϕ ϕ ±1/2 ≥ 1, we can apply Theorem 2.1 to obtain

ϕ -1 | S e sϕ ϕ 1/2 h j 2 ≤ C sλ e sϕ ϕ 1/2 ∇u 2 L 2 (Ω j,δ ) + s 2 λ 2 ϕ | S e sϕ ϕ 1/2 u 2 L 2 (Ω j,δ ) , e sϕ ϕ -1/2 h j 2 ≤ C sλϕ |S e sϕ ϕ -1/2 ∇u 2 L 2 (Ω j,δ ) + s 2 λ 2 ϕ 2 | S e sϕ ϕ -1/2 u 2 L 2 (Ω j,δ ) .
Moreover, because of (2.2) we have

c j -c ∞ ≤ C 1 √ sλϕ | S
where the constant C is independent on j ∈ J, and, as g j = -(c jc)∂ 2

x n u j , Lemma B.3 implies that for λ > λ 1 , defined just after (B.7), one has

ϕ | S e sϕ ϕ -1/2 g j 2 ≤ C sλ ϕ -1/2 e sϕ f j 2 + sλϕ | S ϕ -1/2 e sϕ ∇u j 2 + (sλϕ | S ) 2 ϕ -1/2 e sϕ u j 2 +(sλ) 2 ϕ 1/2 e sϕ ∂ x n u j 2 .
Gathering together the previous results we obtain

ϕ | S e sϕ ϕ -1/2 g j 2 +ϕ -1 | S e sϕ ϕ 1/2 h j 2 ≤ C sλ ϕ -1/2 e sϕ f j 2 +(sλ) 2 e sϕ ϕ 1/2 ∇u j 2 +(sλ) 2 e sϕ ϕ 1/2 ∇u 2 + sλϕ | S e sϕ ϕ -1/2 ∇u j 2 + s 2 λ 2 ϕ 2 | S e sϕ ϕ -1/2 u j 2 + s 3 λ 3 ϕ | S e sϕ ϕ 1/2 u 2 .
As soon as λ > λ 1 , sλ ≥ 1, the proof of the lemma is ended by noticing that 2δβ |S -≤ β(0) by Assumption 4.1 and, therefore, one has

β(0) ≤ 2β(x n ) on (-δ, δ) which implies ϕ |S ≤ ϕ 2 .
Lemma 4.7. There exists a constant C > 0 such that, for all j ∈ J, one has

s 3/2 λϕ | S e 2sϕ | S |θ j | 2 L 2 (S j ) ≤ Cs 1/2 e 2sϕ | S ∫ S j |c∂ x n u j | 2 dσ.
Proof. The definition

θ j = [(c j -c)∂ x n u j ] S j and (2.2) imply |θ j | 2 L 2 (S j ) ≤ C sλϕ | S |c∂ x n u j | 2 L 2 (S j ) .
Then the conclusion follows.

We recall that, for each x ∈ Ω , there are no more N (N is the integer of Theorem 2.1) elements of J such that χ j (x ) 0. So the proof of Theorem 4.5 is ended by noticing that for any function f , we have

f 2 = ( j∈J f χ j ) 2 ≤ N j∈J ( f χ j ) 2 ≤ N 2 f 2
since 0 ≤ χ j ≤ 1 and also since the convexity of the numerical function z → z 2 implies that

( 1≤ j≤N 1 N a j ) 2 ≤ 1 N 1≤ j≤N a 2 j .
Finally we obtain Theorem 1.2 since the three terms involving u in the r.h.s. of (4.4) are absorbed by the l.h.s. of (1.2) : use Remark 4.3 for the integral on S while the two others are absorbed by the l.h.s. of (1.2) by choosing s, λ large enough.

A Partition of unity

In this appendix, in order to prove Theorem 2.1 and the following results, we construct a suitable lattice in a neighborhood of

Ω . The open set Ω is a bounded set of R n-1 and its closure is included in a cube [a, b] n-1 , -∞ < a < b < ∞.
For the simplicity of the proof, we shall suppose that [a, b] = [0, 1] since only the value of the constant C in (2.1) will change in the general case. We shall proceed gradually: in a first time (Step 1), we shall suppose that Ω = (0, 1), then, in Step 2, we shall consider the case Ω = (0, 1) n-1 and, at last, in Step 3, we use as far as possible some elementary arguments to obtain the generalization to any Ω with our required regularity.

Step 1. We begin to work with n = 2 which means that the open cube is the interval (0, 1).

Let us denote

h(s, λ) := Θ √ sλϕ | S
where 1 2 < Θ ≤ 1 in such a way that

√ sλϕ | S Θ
is an even natural integer p = p(sλ). For each sλ > 1, there is a finite number of admissible values Θ. We shall choose the nearest value of 1.

In order to simplify our explanation, we work on S := (-h, 1 + h). On S we consider the lattice

x j (s, λ) = x j-1 (s, λ) + h(s, λ) where j = 0, ..., p(s, λ), x -1 (s, λ) = -h and x p+1 (s, λ) = 1 + h,
and we set

I j (s, λ) = (x j-1 (s, λ), x j+1 (s, λ)), 0 ≤ j ≤ p.
Two successive subintervals I j (s, λ) are overlaped each other and the intersection has a measure

p I p x +1 p x O 1 1+h 1 I 2 I 0 I -h {1 x 0 x 1 x 2h h
ψ(x) =                1, for -1 3 < x < 1 3 , 0, for x (-2 3 , 2
3 ), > 0, for -2 3 < x < 2 3 . Let us consider now the functions ψ j and χ j defined by

ψ j (x) = ψ x -x j h , and for -h ≤ x ≤ 1 + h, 0 ≤ j ≤ p, χ j (x) := ψ j (x) 0≤ j≤p ψ j (x) . (A.1)
Remark A.1. The previous sum contains less than two terms not vanishing for each x ∈ [-h, 1+h], and the family (χ j ) j is a partition of unity on (0, 1) if we only consider the restrictions on (0, 1) of each χ j .

Lemma A.2. There exists a constant C, independant of s, λ > 0 and j, 0 ≤ j ≤ p, such that

(χ j ) ∞ ≤ C sλϕ | S , and (χ j ) ∞ ≤ Csλϕ | S . (A.2)
Proof. Deriving the function ψ j , we have

|ψ j (x)| ≤ ψ ∞ h and |ψ j (x)| ≤ ψ ∞ h 2 .
From the definition of ψ j , there exists a constant m > 0 (m = 2 for an appropriate choice of ψ) such that 0≤ j≤n+1 ψ j (x) > 1/m. Then, using χ j (x) =

ψ j 0≤ j≤n+1 ψ j (x) - 0≤ j≤n+1 ψ j (x) ψ j ( ψ j ) 2 (x), we have |∇χ j (x)| ≤ 3m ψ ∞ h .
Similarly, Remark A.1 and the expression of χ j (x) give us the last estimation

|χ j (x)| ≤ 2m h 2 ψ ∞ + 4m ψ 2 ∞ .
Step 2. Now, we come back to the cube [0, 1] n-1 , n > 2. The pair (s, λ) being fixed, we use p = p(s, λ) defined in Step 1, and we build on each axis of coordinate a lattice similar to this one of Step 1. The product gives a lattice indexed by j := ( j 1 , j 2 , • • • , j n-1 ), 0 ≤ j i ≤ p(s, λ), which defines the small cubes Ω j = I j 1 × I j 2 × • • • × I j n-1 where each I j i has the form introduced in the previous step. The measure of each cube is (2h) n-1 and there is overlaping. We associate to each cube Ω j the function χ j (x ) := χ j 1 (x 1 )χ j 2 (x 2 ) • • • χ j n-1 (x n-1 ). So, in each point x ∈ Ω , there are q(x ) functions of the partition not vanishing with 1 ≤ q ≤ 2 n-1 . The proof of Theorem 2.1 is complete in this case. Moreover, if Ω = (0, 1) n-1 we can directly continue the coefficients c on each side of the boundary as we write it in item 2 of Step 3. The case Ω (0, 1) n-1 needs a little more work.

Step 3. In the following, the open set Ω is no more a cube but, in order to give an explanation very visual, we shall suppose n = 3 which means that Ω is a bounded open set in [0, 1] 2 and ∂Ω is C 2 . The reader will see that the extension to n > 3 is easy once this approach will be understood. In subsection 1.3, we have reduced our problem to the family of problems (1.3) posed in slices indexed by x n of the cylinder Ω × (-δ, δ), slices with sizes of same order. As the lattice of Step 2 fails now on this point, we precise our modified approach in the 6 following items. We build in items 1 and 2 a finite covering (R κ ) of the boundary ∂Ω using the C 2 regularity of ∂Ω , covering that does not depend on Carleman parameters (s, λ) and that permits to extend the coefficients in a neighborhood of Ω × (-δ, δ). Each R κ corresponds to a true cube Rκ by a diffeomophism. Then, once the pair (s, λ) is chosen, we distinguish the cubes far from the boundary (item 3) which come from the mesh of [0, 1] 2 ⊃ Ω and the 'cubes' close to the boundary (item 4) which come from a meshing of Rκ . Choosing h small enough, we show in items 5 and 6 that the covering is complete and the partition of the unity is adapted.

1. We cover the boundary of Ω by a finite family of open sets (R κ ) 1≤κ≤N b in such a way that each R κ is diffeomorphic by ϕ κ to a rectangle Rκ := (-α κ1 , α κ1 ) × (-α κ2 , α κ2 ) and R κ ∩ ∂Ω is the image by ϕ -1 κ of the straight line {0}×(-α κ2 , α κ2 ). Moreover, we can suppose that J ϕ κ , the Jacobian matrix of ϕ κ , has a determinant equal to 1. So, ∪ κ=N b κ=1 R κ is an open neighborhood of ∂Ω that contains the tube T δ := {x ∈ R 2 ; d(x, ∂Ω ) < δ } for a some δ > 0 which is now fixed. We set R± κ := {y = (y 1 , y 2 ) ∈ Rκ ; ±y 1 > 0} and

R ± κ := ϕ -1 κ ( R± κ ).
2. To each R κ and Rκ we associate the cylinders R κ,δ := R κ × (-δ, δ), Rκ,δ := R j × (-δ, δ) as well as the cylinders R ± κ,δ and R± κ,δ . We extend the functions w, defined on Rκ,δ , by antisymmetry relatively to the plane {y 1 = 0} and the coefficients crl (•, x 3 ) := c rl (ϕ -1 κ (•), x 3 ) by symmetry relatively to the same plane. As a matter of fact, crl (•, x 3 ) is continuous on Rκ and crl is C 1 on the four open subsets R± κ,δ ∩ {±x 3 > 0} of Rκ,δ . As it is usual, from A τ (x n ) acting in L 2 (R + κ ) we can define the extension Ãτ (x n ) in Rκ with the tangential matrix Cτ,κ (•,

x 3 ) = (J ϕ κ • ϕ -1 κ ) C(ϕ -1 κ (•), x 3 ) (J ϕ κ • ϕ -1 κ ) t
, and come back to R κ which gives the extended operator A τ (x n ) acting in L 2 (R κ ). The new tangential matrix will be C 1 since the map ϕ κ is C 2 .

3. Now, we take into account the parameters (s, λ). Firstly, we use Step 2 to obtain a mesh of [0, 1] 2 by the family (Ω j ) = (I j 1 × I j 2 ). We keep the cubes Ω j such that Ω j ⊂ Ω . Let J int be the set of the corresponding subscripts. Since Ω is not a cube, it is clear that ∪ j∈J int Ω j does not fill Ω . Therefore, we have to complete this family.

4. Using a process similar to Step 1 and 2, we build a covering for each rectangle Rκ with small cubes Rκ,l overlaping each other and having a size equivalent to 2h(s, λ). Let J b be this family of subscripts j = (κ, l). We deduce functions χκ,l adapted to these cubes and take theirs images χ κ,l using the functions ϕ -1 κ , 1 ≤ κ ≤ N b to come back to Ω . In the same way we obtain R κ,l using ϕ -1 κ , functions that do not depend on (s, λ). This last remark is important to claim that all the R κ,l have sizes of the same order and that the functions χ κ,l verify (2.1).

5.

We have to show that the union of (∪ j∈J int Ω j ) and (∪ j∈J b R j ) covers Ω . This will be true only if h(s, λ) is small enough, i.e. if s and λ are large enough. Once this point will be verified, it will be sufficient to normalize the functions of the partition in a similar spirit to (A.1). However, the overlap must be important enough in order that all the functions of the partition do not vanish in a same point (in Step 1, the size of I 2k ∩ I 2k+1 is half the length of each interval I k , for example). Note that ∪ J b R j ⊃ T δ and that the partition has all the required properties for each

x ∈ T δ /2 . Let us consider x ∈ Ω such that d(x , ∂Ω ) > δ 2 which implies that the open disc D(x , δ 2 ) ⊂ Ω , while the cube C(x , δ √ 2
) of center x with sides of length L = δ

). To conclude, it will be sufficient to prove Claim: If h(s, λ) ≤ L 8 , the point x is recovered by cubes Ω j such that at least one of the associated function χ j verifies χ j (x ) 0. In order to prove the claim we suppose that h(s, λ) = L 8 (the other situations will come easily). The lattice used in Step 2 does not depend on Ω . We project its points on each axis of coordinate. Let us consider the first axis of coordinate, i.e. the interval (x 1 -L 2 , x 1 + L 2 ). The projection of the lattice give 7 points (y

' ± T ) 2 | ' ± ; ' x ( D ) 2 | ' ± ; ' x ( D ' ± 2 | ' ± 2 | L { 1 ' x 1 1 0 ' @ ' ' x . ' x 2 | L + 1 ' x 4 y 3 y 1 ' x ' ) s;( h
1 , • • • , y 7 ) or 8 points (y 1 , • • • , y 8 ) included in (x 1 -L 2 , x 1 + L 2 )
. In the first case, x 1 = y 4 and in the second case, y 3 < x 1 < y 4 . We repeat this process for the second coordinate x 2 . So, there are enough points of the lattice that surround x in each direction to imply the result.

B Proof of some intermediate results

Lemma

B.1. Let σµ j,k (0 + ) ≥ 2sλϕ | S β |S -+ λβ |S -. (B.1)
Then, for all function g ∈ L 2 (-δ, δ), we have For r ∈ (0, 1), ϕ(-y + ry) ≤ ϕ(0) since β > 0 and, so, we have ϕ(0) -ϕ(-y) ≤ yλβ |S -ϕ(0), which gives

-σµ j,k (0 + )y + 2s(ϕ | S -ϕ(-y)) + λ(β | S -β(-y)) ≤ -σµ j,k (0 + )y + 2syλβ |S -ϕ(0) + λβ |S -y, y ∈ (0, δ).
Using (B.1), the first estimate (B.2) is proved.

Step 2. To prove (B.3), we will prove that ϕ -1 | S e -σµ j,k (0 + )y+2sϕ | S ≤ ϕ(±y) -1 e 2sϕ(±y) , for all y ∈ (0, δ). This is equivalent to prove that -σµ j,k (0

+ )y + 2s(ϕ | S -ϕ(±y)) -λ(β | S -β(±y)) ≤ 0.
Let us begin by the case with -y. As β | S -β(-y) ≥ 0 ( β > 0), it will be sufficient to prove that -σµ j,k (0 + )y + 2s(ϕ | S -ϕ(-y)) ≤ 0. In the first step we have seen that ϕ(0) -ϕ(-y) ≤ yλβ |S -ϕ(0), whence -σµ j,k (0

+ )y + 2s(ϕ | S -ϕ(-y)) ≤ -σµ j,k (0 + )y + 2syλβ |S -ϕ(0) ≤ 0
with the assumption on µ j,k .

Lemma B.3. There exist C > 0 and λ 1 > 0 such that

ϕ -1/2 e sϕ ∂ 2 x n u j 2 ≤ C ϕ -1/2 e sϕ f j 2 + sλϕ | S ϕ -1/2 e sϕ ∇u j 2 + (sλϕ | S ) 2 ϕ -1/2 e sϕ u j 2 +(sλ) 2 ϕ 1/2 e sϕ ∂ x n u j 2 , (B.5)
for all s ≥ 1, λ ≥ λ 1 and j ∈ J.

Proof.

In Ω ± , one has ϕ -1/2 e sϕ c∂ 2

x n u j = ϕ -1/2 e sϕ ∂ x n (c∂ x n u j ) -ϕ -

∂ x n (c∂ x n u j ) = (∇ • B∇(χ j u) -∇ τ • C τ ∇ τ u j = -f χ j + 2(C τ ∇ τ u) • ∇ τ χ j + (∇ τ • C τ ∇ τ χ j )u -∇ τ • C τ ∇ τ u j .
Let η = ϕ -1/2 e sϕ . Using (2.1), there exists C > 0 independent of s, λ, j such that where g j = -∇ • B∇(χ j u), which also leads to g j = (-∇ • B∇u)χ j -u∇ τ • ( Cτ ∇ τ χ j ) -2 B∇u • ∇χ j .

As -∇ • ( B∇u) = η 2 f -(∂ x n η 2 )c∂ x n u, we obtain

g j = η 2 f j -2B∇u • ∇χ j -u∇ τ • (C τ ∇ τ χ j ) -χ j (∂ x n η 2 )c∂ x n u.
For h 0, parallel to the interface S , we set D h ρ(x) = ρ(x+h)-ρ(x) |h| , and take an any function ψ ∈ H 1 0 (Ω j,δ ) such that ψ = D -h (D h w) where supp w Ω j,δ (we suppose h small enough). On one side, we can write since [η 2 c∂ x n u j ] S = 0. There is no problem when Ω j,δ ⊂ Ω δ , otherwise, for Ω j,δ ∩ ∂Ω ∅, the way we proceed to continue functions and coefficients does not introduce any new problem (see Appendix A). As g j = -∇ • B∇u j , we deduce from the two previous expressions the following relation ∫ We carry these two previous estimates on (B.9), which gives

∫ Ω η 2 (B(x + h)∇D h u j ) • ∇D h w ≤ C 1 (B)|η∇u j | L 2 + |η -1 g j | L 2 |η∇(D h w)| L 2 .
This inequality is, in particular, true if we choose w equal to u j : Letting h go to zero we obtain ∫ Ω (∂ x l u j )(∂ x k ψ) ≤ C 3 C 2 (B) |η -1 ψ| L 2 which leads to

|η∂ x k ∂ x l u j | L 2 ≤ C 1 (B)|η∇u j | L 2 + |η -1 g j | L 2 C 2 (B) for 1 ≤ k, l ≤ n -

Assumption 1 . 1 .

 11 The matrix diffusion coefficient B(x , x n ) has the following block diagonal formB(x) = C τ (x) 0 0 c(x)where 1. the functions C τ , c, are C 1 (Ω ± ) with a possible jump at x n = 0, 2. the two restrictions to the interface S of the function c : x → c(x , 0 ± ) are C 2 , 3. C τ (x) is an hermitian matrix of order n -1.

Theorem 4 . 5 .

 45 There exist C, λ 0 > 0 and s 0 > 0 such that, for s ≥ s 0 and λ ≥ λ 0 , we havesλϕ | S ∫ S e 2sϕ |∇ τ u| 2 S [β cC τ ] S dσ ≤ B 4 (sλϕ | S ) 3 ∫ S e 2sϕ |u| 2 S dσ+ C e sϕ f 2 + sλ ϕ 1/2 e sϕ ∇u 2 + s 2 λ 2 ϕ 3/2 e sϕ u 2 + s 1/2 ∫ S e 2sϕ |c∂ x n u| 2 dσ . (4.4)

Figure 1 :

 1 Figure 1: Covering in dimension 1 equal to h, half of the measure of each interval. Let us choose a function ψ ∈ C ∞ 0 (R) such that 0 ≤ ψ(x) ≤ 1 and

  coordinates is inscribed in D(x , δ

)

  Figure 2: Claim

δ ∫ 0 ϕ 1 | 1 ∫ 0 ϕ 1 ∫ 0 ϕ

 011010 | S e -σµ j,k (0 + )y+2sϕ | S g 2 (±y)dy ≤ S e -σµ j,k (0 + )y+2sϕ | S g 2 (±y)dy ≤ δ ∫ 0 ϕ(±y) -1 e 2sϕ(±y) g 2 (±y)dy. (B.3) Proof. Step 1. Let us prove the first estimate. Such kind of estimate is true if ϕ| S e -σµ j,k (0 + )y+2sϕ | S ≤ ϕ(±y)e 2sϕ(±y) , for all y ∈ (0, δ), which we write-σµ j,k (0 + )y + 2s(ϕ | S -ϕ(±y)) + λ(β | S -β(±y)) ≤ 0. (B.4) Since β > 0, ϕ is an increasing function on (-δ, δ) and we have ϕ(0) -ϕ(y) ≤ 0 and (β | S -β(y)) ≤ 0, y ∈ (0, δ), which gives the estimate (B.4) for +y. Let us prove that it is still valid for -y. Since the function β is affine on (-δ, 0), we have β | S -β(-y) = β |S -y for y ∈ (0, δ) whence ϕ(0) -ϕ(-y) = y (-y + ry)dr = yλβ |S - (-y + ry)dr.

η∂ x n (c∂ x n u j ) 2 ≤ C η f j 2 +

 22 sλϕ | S η∇ τ u 2 + (sλϕ | S ) 2 ηu 2 + η∇ τ • C τ ∇ τ u j 2 , (B.8)where the norms are taken in L 2 (Ω j,δ ). It remains to estimate the last term of the r.h.s. of (B.8).Let us denote B := η 2 B, C := η 2 C and let ψ ∈ H 1 0 (Ω j,δ ) be a test function. One has∫ Ω B∇(χ j u) • ∇ψ = ∫ Ω {(-∇ • B∇u)χ j -u∇ τ • ( Cτ ∇ τ χ j ) -2 B∇u • ∇χ j }ψ = ∫ Ω g j ψ

∫Ω

  B∇u j • ∇D -h (D h w) = ∫ Ω B∇u j • D -h (∇D h w) = ∫ Ω D h ( B∇u j ) • (∇D h w) = ∫ Ω [ B(x + h)D h (∇u j ) + D h ( B)∇u j ] • ∇D h wand, on the other side, we have∫ Ω B∇u j • ∇D -h (D h w) = -∫ Ω (∇ • B∇u j )D -h (D h w)

ΩB 2 .

 2 (x + h)D h (∇u j ) • ∇D h w = ∫ Ω g j D -h (D h w) -(D h ( B)∇u j ) • ∇D h w . (B.9)As η does not depend on the tangential variable, D h and η commute. This gives∫ Ω (D h ( B)∇u j ) • ∇D h w ≤ C 1 (B)|η∇u j | L 2 |η∇D h w| L 2 , ∫ Ω g j D -h (D h w) ≤ |η -1 g j | L 2 |D -h (ηD h w)| LIt is wellknown that |D -h w| L 2 ≤ |∇ w| L 2 for w ∈ H 1 0 with supp w ⊂ Ω j,δ and h small enough. Similarly, we have|D -h (ηD h w)| L 2 ≤ |∇ τ (ηD h w)| L 2 , from which one can conclude that ∫ Ω g j D -h (D h w) ≤ |η -1 g j | L 2 |η∇(D h w)| L 2 .

C 2 (

 2 B) ∫ Ω (η∇D h u j ) • η∇D h u j ≤ ∫ Ω η 2 (B(x + h)∇D h u j ) • ∇D h u j ≤ C 3 |η∇(D h u j )| L 2where the constants C 1 (B) and C 2 (B) depend on B but not on η and whereC 3 := C 1 (B)|η∇u j | L 2 + |η -1 g j | L 2 . Dividing the two extremities by |η∇(D h u j )| L 2 we obtain C 2 (B)|η∇D h u j | L 2 ≤ C 3 . For 1 ≤ l ≤ n -1, and ψ ∈ H 1 0 (Ω j,δ ), one obtains ∫ Ω (∂ x l u j )(D -h ψ) = ∫ Ω (D h u j )(∂ x l ψ) = ∫ Ω η(∂ x l D h u j )η -1 ψ ≤ C 3 C 2 (B) |η -1 ψ| L 2 .

  1/2 (∂ x n c)e sϕ ∂ x n u j . (B.6)We see straightaway thatϕ -1/2 (∂ x n c)e sϕ ∂ x n u j | ≤ c max (sλ) 2 ϕ 1/2 e sϕ ∂ x n u j 2 (B.7) if s ≥ 1 and λ > λ 1 where e λ 1 β L ∞ (Ω) = 1 λ 1 .The other term in (B.6) verifies

  1.Now, we can estimate η∇τ • C τ ∇ τ u j 2 , i.e. η∇ τ • C τ ∇ τ u j 2 ≤ C η∇u j 2 + η -1 g j 2 ≤ C η∇u j 2 + η f j 2 + sλϕ | S η∇u 2 + (sλ) 2 η -1 (∂ x n η 2 )∂ x n u 2 + (sλϕ | S ) 2 ηu 2 ≤ C η f j 2 + sλϕ | S η∇u 2 + (sλϕ | S ) 2 ηu 2 + (sλ) 2 η -1 (∂ x n η 2 )∂ x n u 2 ≤ C ϕ -1/2 e sϕ f j 2 + sλϕ | S ϕ -1/2 e sϕ ∇u 2 + (sλϕ | S ) 2 ϕ -1/2 e sϕ u 2 + (sλ) 2 ϕ 1/2 e sϕ ∂ x n u 2 .

Bringing this result in (B.8), we deduce (B.5) with the help of (B.6) and (B.7).

assia@cmi.univ-mrs.fr,

dermenji@cmi.univ-mrs.fr,

laetitiathevenet@gmail.com. L. Thevenet was partially supported by l'Agence Nationale de la Recherche under grant ANR-07-JCJC-0139-01. a LATP -39, rue F. Joliot-Curie, 13453 Marseille cedex 13, France.

Acknowledgement

The authors are grateful to F. Boyer (Université d'Aix-Marseille) for fruitful discussions on Appendix C.

Now, let us consider the case with +y. As β | S -β(y) = -β S + y, we have to prove that -σµ j,k (0 + )y+ 2s(ϕ | S -ϕ(y)) + λβ S + y ≤ 0. Using (B.1) this will be true whether 2s(ϕ | S -ϕ(y)) + λβ S + y ≤ (2sλϕ | S β |S -+ λβ |S -)y. As ϕ(y) -ϕ | S ≥ λϕ | S β S + y, we conclude by noticing that, for s ≥ 1/2, one has

Proof. The solution of this system is of the form

For s = 0 + , we obtain A + = ν(0 + ). Similarly, s = 0 -gives us A -= ν(0 -). Taking the derivative of the expression and considering s = 0 ± , we also obtain µB + = ν (0 + ) et µB -= ν (0 -). Thus, for s ∈ (-δ, 0), we have

For s ∈ (0, δ), we obtain ν(s) = ν(0) cosh(µs)

This can be written Finally, we have

C About the density in D(A)

The main ideas are in [START_REF] Agelas | The G method for heterogeneous anisotropic diffusion on general meshes[END_REF] where the authors proved the density in

Au. The reader will see here a small extension resulting from a chat with J. Droniou that we thank warmly: the convergence is in norm and not only weak. In other words, D(A) is a core of the operator A. We start knowing that the restriction to Ω ± of each u ∈ D(A) belongs to H 2 (Ω ± ) (this claim is a consequence of Appendix A and the technics developed in the proof of Lemma B.3). We begin by the two following lemmas that are easy:

1. We set c ± (x ) = lim ±x n ↓0 c(x , x n ) and we call n ± the interior normal to the interface S relatively to Ω ± which enables us to introduce a change of variables C 2 with the function

In our case, n ± (x ) = (0, • • • , 0, ±1) with the initial coordinates. We are seeing to appear the ask in point 2 of Assumption 1.1: the traces c ± : x → c ± (x ) are C 2 . So, with the new coordinates, a function belonging to H 2 (Ω + ) will keep this property in U + := Ψ(Ω + ) and if this function is 0 on ∂Ω, it will be the same with the change. Idem for Ω -and U -. As a matter of fact, we have replaced c by the value 1 on S .

2. Let u ∈ D(A) be a function of which the support is included in Ω δ and such that u ∈ H 2 (Ω ± ). Setting v(x , t) := u(Ψ(x , t)), we verify that v ∈ H 1 (U), v ∈ H 2 (U ± ) and

So, the jump of ∂v at the interface is 0 which implies that ∂ t v ∈ H 1 (U). The tangential derivatives do not give any problem when we are going through the interface. In conclusion, the function v is belonging to H 2 (U) ∩ H 1 0 (U). 3. There exists a sequence (v l ) ⊂ C 2 (U) that tends to v for the topology of H 2 (U) and we set u l = v l • Ψ -1 . From the properties of Ψ, it comes that u l ∈ H 1 (Ω) ∩ C 2 (Ω ± ) and u l → u just as well for H 1 (Ω δ ) as for H 2 (Ω ± δ ). The relation (C.1) implies the continuity of the trace of c∂ x l and from Lemma C.1 one has u l ∈ D(A) while Lemma C.2 gives the convergence of Au l to Au in L 2 (Ω δ ).


R Possamai Bastos

F Sill Torres

G Di Natale

M Flottes

B Rouzeyre

Novel transient-fault detection circuit featuring enhanced bulk built-in current sensor with low-power sleep-mode

This work presents a novel circuit for detecting transient faults in combinational and sequential logic. The detection mechanism features a built-in current sensor connected to the bulks of the monitored logic. The proposed circuit was optimized in terms of power consumption and enhanced with low-power sleep-mode. In addition, a calibration method for bulk built-in current sensors is presented. Overhead results indicate an increase of only 15% in power consumption which represents an improvement of factor 7 compared to similar existing sensors.

.

Introduction

Error resilience is a drastically rising demand on integrated systems while, at the same time, the application of ultra-deep submicron technologies requires higher robustness to natural aging processes or environment sources like radiations from cosmic origin or every day material [START_REF] Karnik | Characterization of Soft Errors Caused by Single Event Upsets in CMOS Processes[END_REF]. In addition to these natural phenomena, malicious fault-based attacks can be used to bypass security mechanisms of secure systems and extracting information on confidential data [START_REF] Leveugle | Early Analysis of Fault-Based Attack Effects in Secure Circuits[END_REF]. Both these natural or malicious phenomena on integrated circuits can induce transient effects that provoke bit-flips of stored results during the system lifetime.

Until the early 2000's, researches on transient faults focused essentially on memory elements, which were considered the system's most vulnerable circuits. Many concurrent error detection and/or correction mechanisms were proposed to mitigate soft errors induced by transient faults in memory cells. In the last decade, however, more sensitive deep-submicron technologies as well as the increasing demand for secure systems have also pushed for the development of countermeasures against transient faults in combinational parts of the circuits. These faults indeed can propagate up to storage elements and thus cause soft errors as well. On the other hand, if the transient fault does not induce any error due to an electrical, logical or latching-window masking effect, its detection is crucial all the same in secure applications since the fault itself reveals an attempt of attack.

The today's trend in efficient protections against transient faults is applying mitigation techniques at different abstraction levels of the design [START_REF] Lisboa | Using Built-in Sensors to Cope with Long Duration Transient Faults in Future Technologies[END_REF] [START_REF] Shazli | Transient Error Detection and Recovery in Processor Pipelines[END_REF]. The idea behind is the avoidance of costly fault-tolerance mechanisms like triple modular redundancy, taking advantage of cheaper mitigation techniques that ensure satisfactory soft error coverage for the system's most recurrent operations. This strategy is exemplified through system's recovery schemes fired in function of the indication of concurrent error detection (CED) circuitries.

CED mechanisms implemented at transistor or gate level guarantee an early detection, as soon as the faults happen, preventing more critical failure scenarios such as the induction and propagation of multiple errors to other clock cycles, stages, or parts of the system. In case of a fault, the generated error flag is able to activate, for instance, recovery machines already implemented in modern systems for dealing with branch misprediction 2 [3][4]. Thereby, faulty operation can be repeated in faultfree conditions, adapting the system to perform again its normal computational sequence.

This paper proposes a new low-cost CED scheme that efficiently identifies transient faults. The proposed circuit monitors transistor bulks of system's blocks such as similar existing Bulk Built-In Current Sensor (BBICS) [START_REF] Neto | Using Bulk Built-in Current Sensors to Detect Soft Errors[END_REF][6], which we further discuss in section 2. Our solution, though, is optimized to satisfy today's need for low-power transient-fault robust systems. More precisely, to the best of our knowledge, the innovative contributions of this paper are:

 An optimization of the original BBICS's circuitry [START_REF] Neto | Using Bulk Built-in Current Sensors to Detect Soft Errors[END_REF][6] to achieve reasonable overheads in power consumption (section 3);  The introduction of the sleep-mode for BBICS that allows additionally energy savings when the system is on standby (section 3);  A design method for calibrating BBICS with the capability of detecting a minimum transient-fault profile (section 4);  An extensive transistor-level simulation-based exploration of the BBICS size limits in function of their response times and power consumptions (section 5).

Fundamentals

This section presents preliminary information regarding soft errors, built-In current sensors and leakage currents in nanometer technologies.

Transient faults in integrated circuits

Transient voltage variations during the lifetime of combinational or sequential circuits are defined as transient faults. The first harmful effects of transient faults are soft errors by inverting stored results of system operations (i.e. bit-flips of storage elements).

Due to the transistor shrinking and the growing communication of confidential data, soft errors can happen today even at ground level by means of perturbation events arisen from environmental or intentional sources. Examples of environment events are alpha particles released by radioactive impurities and more importantly neutrons from cosmic rays [START_REF] Karnik | Characterization of Soft Errors Caused by Single Event Upsets in CMOS Processes[END_REF]. On the other hand, intentional perturbation events are usually produced by optical sources such as flashlights or laser beams [START_REF] Leveugle | Early Analysis of Fault-Based Attack Effects in Secure Circuits[END_REF] that can maliciously induce transient effects on secure circuits like smartcards to retrieve their secret information.

Soft errors and transient faults are also known as single event upsets (SEU) and single event transients (SET) in fault-tolerance-related fields. If provoked by malicious fault-based attacks, such circuit misbehaviors provide fundamental information for cryptanalysis methods that break security applications.

The modeling of transient faults in CMOS circuits is done by injecting a double exponential current pulse I fault at the sensitive node. Thereby, the shape of the pulse can be approximated by following equation [START_REF] Srinivasan | Accurate, predictive modeling of soft error rate due to cosmic rays and chip alpha radiation[END_REF]:

       f r t t f t t fault f r Q I ee tt (1)
with Q f is the charge collected due to the particle strike, t f means the decay time of the current pulse, and t r labels the time constant for initially establishing the ion track

Built-in current sensors detecting transient faults

Built-In Current Sensors (BICS) were initially proposed as a mechanism for detecting large increases in the current I DDQ consumed by a CMOS circuit during its quiescent state, i.e. when the circuit is not switching. The mechanism allows thus testing CMOS circuits against permanent faults [START_REF] Athan | A Novel Built-in Current Sensor for I DDQ Testing of Deep Submicron CMOS ICs[END_REF]. Further, BICS were also adapted for detecting transient faults in memory cells (i.e. bit-flips) [START_REF] Lo | Design of Static CMOS Self-checking Circuits using Built-In Current Sensing[END_REF][10] [11][12]. Recently, efforts were made for monitoring transient faults in combinational logic as well [START_REF] Narsale | Variation-tolerant hierarchical voltage monitoring circuit for soft error detection[END_REF]. All these techniques connect BICS to the power lines (V DD and GND) of the monitored circuit to distinguish anomalous transient currents from normal currents. The today's problem is that the amplitude of transient currents induced by radiation effects or fault attacks can have the same order of currents normally generated by switching activities in combinational logic circuits. Hence, schemes monitoring power lines are very limited for detecting just small range of transient faults.

On the other hand, BICS connected to the bulks of the monitored circuit's transistors are able to detect a wide range of transient faults [START_REF] Neto | Using Bulk Built-in Current Sensors to Detect Soft Errors[END_REF][6] [START_REF] Wirth | Bulk built in current sensors for single event transient detection in deep-submicron technologies[END_REF]. As Fig. 1 (a) and (b) illustrate, Bulk-BICS (BBICS) identify anomalous transient currents I fault flowing through the junction between a bulk and a reversely-biased drain of a perturbed transistor (MOSFETs "off" in Fig. 1). BBICS indeed take advantage of two facts:

(1) In fault-free scenarios (i.e. I fault = 0) the bulk-todrain (or drain-to-bulk) current I B is negligible even if the MOSFET is switching in function of new input stimuli;

(2) During transient-fault scenarios, I fault is much higher than the leakage current flowing through the junction.

The range of detectable transient faults is easily adjustable by calibrating the size of the transistors that constitute the BBICS. Hence, schemes based on BBICS can be designed to latch a flag of fault indication for abnormal currents within a defined range that represents a risk of resulting in soft errors. Note that in such a strategy the connection between the monitored circuit (e.g. system's block 1) and the BBICS circuitry (e.g. NMOS-BBICS 1) is done via metal from the body-ties of each monitored transistor (e.g. T11, T12, and T13) up to the input of the BBICS circuitry. Thereby, the peak of the anomalous transient current (i.e. the transient fault) is almost not attenuated, ensuring thus an efficient detection [START_REF] Wirth | Bulk built in current sensors for single event transient detection in deep-submicron technologies[END_REF]. In fact, this very small attenuation is a function of the local distance between the struck zone of the monitored transistor and its body-tie.

The work in [START_REF] Lisboa | Using Built-in Sensors to Cope with Long Duration Transient Faults in Future Technologies[END_REF] shows that area overheads imposed by BBICS's mechanisms for protecting adder circuits can be up to 13.4 % without impact on the system's operating frequency. The costs therefore are considerably smaller than the ones due to classic CED schemes [START_REF] Bastos | How to Sample Results of Concurrent Error Detection Schemes in Transient Fault Scenarios?[END_REF]. Moreover, BBICS approach is much more efficient for dealing with transient faults of long duration and multiple faults. Nevertheless, the negative issue of existing BBICS is the elevated power consumption to provide high detection sensitivity in ultra-deep submicron technologies [START_REF] Neto | Using Bulk Built-in Current Sensors to Detect Soft Errors[END_REF][6] [START_REF] Zhang | A new bulk bult-in current sensing circuit for single-event transient detection[END_REF].

Leakage in nanometer technologies

Ideally, CMOS cells draw no current or rather dissipate no power when idle. Unfortunately, this is not true for real cells realized in nanometer technologies. A major impact originates from sub-threshold leakage current I sub which is the current between source and drain of a transistor when the device should in fact be cut off. A commonly used approximation of I sub is [START_REF] Hu | Berkeley short channel igfet model[END_REF]:

1             gs th B q Vv nk T sub W I a e L (2)
with a 1 is a technology factor, W means the gate width, L labels the gate length, q corresponds to the charge of an electron, n means the sub-threshold swing coefficient, k B is Boltzmann's constant, T labels the operating temperature, V gs is the gate-source voltage, and v th labels the threshold voltage. Further, the threshold voltage can be modeled with [START_REF] Hu | Berkeley short channel igfet model[END_REF]:

2 3      th th0 sb ds v v a V a V ( 3
)
with th0 is the zero-bias threshold voltage, γ is the bodybias coefficient, a 2 and a 3 label technology constants, η corresponds to the Drain Induced Barrier Lowering (DIBL) coefficient, V sb labels the source-bulk voltage, and V ds is the drain-source voltage.

Sleep-mode improved bulk built-in current sensor

The circuit of our Sleep-mode Improved Bulk Built-In Current Sensor (SIBBICS) is presented in Fig. 2 (a). If the mode of operation is identical to the original BBICS circuit [START_REF] Neto | Tbulk-BICS: A Built-In Current Sensor Robust to Process and Temperature Variations for Soft Error Detection[END_REF] shown in Fig. 2 (b), our structure is optimized in such a way that the power consumption could be largely reduced. The following subsections detail the basic structure and the low-power features of our proposed scheme.

Basic structure

BBICS circuits in Fig. 2 are designed for monitoring NMOS bulks in pull-down networks, and thus they represent the blocks NMOS-BBICS in Fig. 1 (c). W min represents the minimum diffusion width of the transistors, L min is the minimum channel length, and design factors X and Y are used for calibrating the sensors (see section 4).

The PMOS-BBICS structures for monitoring pull-up networks are the complement counterparts of the schemes in Fig. 2. For the sake of simplicity, the similar illustrations and behaviors of PMOS-BBICS are omitted in this section, although their results have been also taken into account in analysis of section 4 and 5. The NMOS-BBICS's basic structure in Fig. 2 is composed of a latch (transistors 5, 6, 7, and 8) that is responsible for amplifying the anomalous transient currents coming from the bulk "NMOS_Bulk" of the monitored block (e.g. "virtual_ gnd_1" of system's block 1 in Fig. 1 (c)). Higher gain of amplification is obtained by increasing factors X and Y, hence higher BBICS's sensitivity in detecting transient faults is also determined in terms of these design factors.

BBICS's latch has, moreover, the function of memorizing a flag in case of a transient fault within a defined current rangei.e. "Flag_N" in Fig. 2 keeping V DD level. On the other hand, as soon as the flag of fault is processed by higher instances of the system, BBICS's latch must be reset (through the input "Reset" in Fig. 2) to detect other transient faults.

Low-power features

Previous subsection highlights that growing factor X of Fig. 2's transistors 6 and 8 allows improving the BBICS sensitivity in detecting transient faults. However, it also contributes considerably to the increase of static power consumption, which is today responsible for up to 50% of the power dissipation of systems based on ultra-deep submicron technologies [START_REF] Kim | Leakage Current: Moore's Law Meets Static Power[END_REF].

In fact, if higher BBICS sensitivity is desired, the sensor amplification has to be increased by higher X, i.e. greater diffusion widths and decreased v th of transistors 6 and 8. As consequence, lower v th makes possible "Flag_N" switches (from GND to V DD level) with lower amplitudes of anomalous transient currents on "NMOS_bulk". The sensor is thus able to detect smaller profiles of transient faults (I fault from section 2) that propagate from the "NMOS_bulk" (through transistor 2) to the node "Flag_N". On the other hand, following from equation ( 2), lower v th results in considerably higher I sub , and thus greater static power consumption.

One could argue that V gs of transistors 6 and 8 are zero in fault-free condition, and then from equation ( 2) I sub would be negligible. However, we have to consider that there is a very small voltage drop on nodes "Flag_N" and "node1" (complement) due to transistor 5 and 7, which act as high ohmic transistors (large channel lengths), creating an offset on V gs of transistors 6 and 8. In addition, a small offset is also induced on V sb of monitored transistors (e.g. T11, T12, and T13 in Fig. 1 (c)), and thus, from ( 3) and ( 2), their v th are slightly lower, and their I sub are higher.

Our solution for reducing this static power consumption is introducing transistor 9 such as illustrated in Fig. 2 (a). It allows the utilization of a sleep-mode when the system is left on standby. Transistor 9 is, in this case, set "on", making a less resistive path between the node "NMOS_bulk" and GND. Consequently, V gs of transistors 6 and 8 and V sb of monitored transistors approach zero, I sub becomes much lower, and thus the static power consumption is drastically reduced.

Furthermore, based on simulation experiments by using a 32-nm CMOS technology [START_REF]Predictive Technology Model (PTM) Web site[END_REF], we identified that the costly transistors 9, 10, and 11 from original BBICS in Fig. 2 (b) are not necessary to efficiently and quickly detect short and long duration transient faults.

Method for calibrating bulk built-in current sensors

The previous section defines the two factors X and Y in Fig. 2 that allow calibrating the amplification of the anomalous transient current, and thus, adjusting the BBICS sensitivity in detecting transient faults. This section defines a calibration method that searches, at design time and in function of X, for the smallest and the largest Y able to detect a minimum transient-fault profile (I fault in section 2).

In order to find these factors Y max and Y min , several sensor versions are designed in function of different values for Y and X. Transistor-level simulations are performed with the injection of a single transient fault. Firstly, for a determined X, if the simulation result for a chosen value Y shows that a flag is not latched (i.e. the transient fault could not be detected), it can be concluded that the chosen Y is lesser than Y min . Then, greater amplification is required and another higher Y must be tested up to a simulation result illustrating a sensor version that detects (a) "sibbics_esref2012" (b) "bbics_tns2008" the fault. In contrast, if the simulation result shows that a flag is always latched even before the fault occurrence, the sensor is not capable of identifying a fault, and the chosen value Y is greater than Y max . Then, other lower Y must be tested to search a value that makes a good detection sensor. This calibration method, thus, allows finding, for any technology, the optimal BBICS design factors within a range between Y max and Y min .

The definition of the smallest transient-fault profile I fault that the sensor must be capable of detecting is given in function of a chosen reasonable minimum amplitude and duration. A minimum profile must be able to propagate without masking through a combinational logic and result in soft error (i.e. a bit-flip of a storage element). A larger profile than the smallest chosen one will be then also detectable since BBICS are capable of amplifying and latching its higher and longer induced voltages. Evidently, the largest profile of transient fault is limited by the highest current that does not cross the technology specification for causing permanent on the circuit.

The shortest durations of transient faults induced by radiation effects on integrated circuits are of the order of tens of picoseconds [START_REF] Ferlet-Cavrois | Statistical analysis of the charge collected in SOI and bulk devices under heavy ion and proton irradiation-implications for digital SETs[END_REF]. Hence, this paper had defined the minimum profile of transient fault that the sensors must detect as a double-exponential transient current I fault with duration of 50ps (measured at the half amplitude of the current). Moreover, the resultant transient voltage must have minimum detectable amplitude of 50% of V DD .

Simulation results

Transistor-level simulations were performed for comparing the proposed "sibbics_esref2012" with the previous BBICS labelled as "bbics_tns2008". The circuit versions were designed with V DD of 0.9V and in nominal conditions of a 32-nm CMOS technology [START_REF]Predictive Technology Model (PTM) Web site[END_REF]. Table 1 illustrates the resultant values for Y max and Y min of the schemes in Fig. 2 by using the method of section 4.

Fig. 3 shows the normalized power consumption and area of a reference circuit (10 chains of 10 inverters) in which all its transistors are monitored by only one BBICS cell composed of two parts (PMOS and NMOS). The "bbics_tns2008" and "sibbics_esref2012" versions were designed with Y max , Y min , and different X. In addition, Fig. 3 illustrates the results when the sleep mode is active. The 200 transistors of the chains were designed with minimum size to analyze the technology's smallest capacitances, which represent the most sensitive nodes. Thus, this casestudy circuit allows inducing the smallest profiles of transient fault as well as evaluating the minimum detection sensitivity of the BBICS. We can conclude from Fig. 3 that "bbics_tns2008" versions present the largest power consumption and area overheads. On the contrary, "sibbics_esref2012" versions have smaller area and very lower power consumption, which is further reduced during sleep-mode. Note that the results allow defining the minimum and maximum increases since our calibration method searches for the smallest and the largest possible Y. Moreover, the method identifies the optimal trade-off between power and area.

Finally, Fig. 4 illustrates that the response times of "sibbics_esref2012" versions are at least equivalent to "bbics_tns2008" versions, and they can be of the order of hundreds of picoseconds.

Conclusions

The paper presents a promising solution for detecting transient faults in integrated circuits. With no performance penalty and much lower overheads in terms of area and power consumption than state-of-the-art's CED techniques, the SIBBICS are perfectly suitable for system design flows based on CMOS standard-cell layout approach. In fact, the standard cells of commercial libraries are never modified but a set of SIBBICS cells is designed in function of them. Ongoing works are the fabrication of a prototype and the validation of the approach by using a laser beam to inject transient faults.

Table 1

Results of the calibration method for a profile of fault: the minimum and maximum factors Y for a range of X. It means, in function of X, the largest and the smallest sensor versions that can detect a transient fault with amplitude of 50% of V DD and 50ps of duration. Y max : 23.6 17.7 14.7 12.7 11.3 10.2 9.3 8.6 8.0 7.5 7.0 6.6 6.3 6.0 5.7 5.5 5.2 5.0 Y min : 10.1 8.1 6.3 5.3 4.7 4.3 4.0 3.8 3.6 3.5 3.3 3.3 3.2 3.1 3.0 2.9 2.9 2.8 PMOS Y max : 74.2 47.2 34.7 27.2 22.3 19.2 16.4 14.4 12.9 11.7 10.7 9.8 9.1 8.5 7.9 7.4 7.0 6.7

Y min : 45.9 26.2 18.1 13.7 12.4 11.1 8.1 7.3 6.5 5.9 5.4 5.0 4.7 4. Y max : 70.6 46.1 34.1 27.0 22.2 18.8 16.3 14.4 12.9 11.7 10.6 9.8 9.1 8.4 7.9 7.4 7.0 6.6

Y min : 29.7 10.1 7.3 5.8 5.3 4.8 4.4 4.1 3.9 3.8 3.8 3.6 3.5 3.4 3.3 3.2 3.2 3.1 PMOS Y max : 74.2 47.2 34.7 27.3 22.4 18.9 16.4 14.5 12.9 11.7 10.7 9.8 9.1 8.5 7.9 7.4 7.0 6.7

Y min : 44.1 25.1 17.5 13.2 10.7 9.2 7.9 7.1 6.5 5.8 5.4 5.0 4.7 4.4 4.2 4.0 3.8 3.7

Fig. 1 .

 1 Fig.1. The two cases of transient faults in a CMOS inverter perturbed by an anomalous current "I fault " in (a) and (b), and "K" blocks of a system protected by "K" BBICS in (c) Fig.1(c) summarizes the strategy for protecting system's blocks against transient faults in pull-down networks by using BBICS. An equivalent strategy must be taken for detecting transient faults in pull-up networks as well. Note that in such a strategy the connection between the monitored circuit (e.g. system's block 1) and the BBICS circuitry (e.g. NMOS-BBICS 1) is done via metal from the body-ties of each monitored transistor (e.g. T11, T12, and T13) up to the input of the BBICS circuitry. Thereby, the peak of the anomalous transient current (i.e. the transient fault) is almost not attenuated, ensuring thus an efficient detection[START_REF] Wirth | Bulk built in current sensors for single event transient detection in deep-submicron technologies[END_REF]. In fact, this very small attenuation is a function of the local distance between the struck zone of the monitored transistor and its body-tie.The work in[START_REF] Lisboa | Using Built-in Sensors to Cope with Long Duration Transient Faults in Future Technologies[END_REF] shows that area overheads imposed by BBICS's mechanisms for protecting adder circuits can be up to 13.4 % without impact on the system's operating frequency. The costs therefore are considerably smaller than the ones due to classic CED schemes[START_REF] Bastos | How to Sample Results of Concurrent Error Detection Schemes in Transient Fault Scenarios?[END_REF]. Moreover, BBICS approach is much more efficient for dealing with transient faults of long duration and multiple faults. Nevertheless, the negative issue of existing BBICS is the elevated power consumption to provide high detection sensitivity in ultra-deep submicron technologies[START_REF] Neto | Using Bulk Built-in Current Sensors to Detect Soft Errors[END_REF][6][START_REF] Zhang | A new bulk bult-in current sensing circuit for single-event transient detection[END_REF].

Fig. 2 .

 2 Fig. 2. The proposed NMOS sleep-mode improved BBICS circuit in (a) and the original NMOS-BBICS version [6] in (b)

Fig. 3 .

 3 Fig. 3. Normalized power consumption and area of a reference circuit composed of 100 NMOS and 100 PMOS monitored by one NMOS-BBICS and one PMOS-BBICS. The BBICS versions are defined in function of design factors X, Y max , Y min . For instance, "sibbics_esref2012 (Ymin, Sleep-Mode ON)" version with X=5 represents the optimal design. When the sleep mode in active state, this design version imposes around 25% of power consumption and area overhead over the reference circuit due to the SBBICS implementation with a factor X=5 and its corresponding Y min = 5.3 (NMOS) and Y min = 10.7 (PMOS).

Fig. 4 .

 4 Fig. 4. Response times of PMOS-BICS versions for raising a flag due to a transient fault with amplitude of 50% of V DD and of duration, the response times for NMOS-BICS versions are omitted here but they are always lower