Pascal Bégout
email: pascal.begout@math.cnrs.fr

Jesús Ildefonso Díaz

A sharper energy method for the localization of the support to some stationary Schr ödinger equations with a singular nonlinearity

Keywords: energy method, Schrödinger equation, solutions with compact support 1

We prove the compactness of the support of the solution of some stationary Schrödinger equations with a singular nonlinear order term. We present here a sharper version of some energy methods previously used in the literature and, in particular, by the authors. Contents 1 Introduction 2 2 From suitable local inequalities to the vanishing of the involved complex functions on some small ball 3 3 A general framework of applications related to the Schrödinger operator 5 4 Proofs of the main results 6 5 Application to the localization property to the case of Neumann boundary conditions 10 References 14

Introduction

Since the beginnings of the eighties of the last century, it is already well-known that the absence of the maximum principle for the case of systems and higher order nonlinear partial differential equations was one of the main motivations of the introduction of suitable energy methods allowing to conclude the compactness of the support of their solutions (see, e.g., the presentation made in the monograph Antontsev, Díaz and Shmarev [START_REF] Antontsev | Energy methods for free boundary problems: Applications to nonlinear PDEs and fluid mechanics[END_REF]).

The application of such type of methods to the case of nonlinear Schrödinger equations with a singular zero order term required some important improvements of the method. That was the main object of the previous author's papers of Bégout and Díaz [START_REF] Bégout | On a nonlinear Schrödinger equation with a localizing effect[END_REF][START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF].

The main goal of this new paper is to present a sharper version of the mentioned method potentially able to be applied to many other problems related to this type of Schrödinger equations such as the study of self-similar solutions, case of Neumann boundary conditions, presence of nonlocal terms (such as, for instance, in Hartree-Fock theory: Cazenave [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]), etc., which can not be treated with the mere technique presented in Bégout and Díaz [START_REF] Bégout | On a nonlinear Schrödinger equation with a localizing effect[END_REF][START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF]. As a matter of fact, the concrete application of this sharper energy method to the concrete case of self-similar solutions of the evolution Schrödinger problem requires many additional arguments justifying the special structure of those solutions, reason why we decided to present it in a separated work (Bégout and Díaz [4]). We send the reader to Bégout and Díaz [START_REF] Bégout | Self-similar solutions with compactly supported profile of some nonlinear Schrödinger equations[END_REF] for a long description of the important role of the compactness of the solution in this context and for many other references related to this qualitative property of the solution. This paper is organized as follows. Below, we give some notations which will be used throughout this paper. In Section 2, we give the precise "localization" estimates which imply a solution of a partial differential equation to be compactly supported see Theorems 2.1 and 2.2, and especially estimates (2.1) and (2.3) . In Section 3, we give a tool which permits, from a solution of some partial differential equation, to establish the "localization" estimate (Theorem 3.1). The results of these two sections are proved in Section 4. In Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF], localization property is studied for the complex-valued equation

-∆u + a|u| -(1-m) u + bu = F, in Ω. (1.1)
We also study this property here, but with a change of notation (see Remark 5.1 below for the motivation of this change). Section 5 is devoted to the study of the localization property of the solutions of equation (1.1), in the same spirit as Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF], but with the homogeneous Theorem 5.6 below with Theorem 3.5 in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF]). Finally, at the end of the paper, we treat equation (1.1) with the homogeneous Dirichlet boundary condition (Remark 5.8). We state the same results as in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF], but with now the weaker assumption F ∈ L 2 (Ω).

Before ending this section, we shall indicate here some of the notations used throughout. We write i 2 = -1. We denote by z the conjugate of the complex number z. For 1 p ∞, p ′ is the conjugate of p defined by

1 p + 1 p ′ = 1. For j, k ∈ Z with j < k, j, k = [j, k] ∩ Z.
We denote by Γ the boundary of a nonempty subset Ω ⊆ R N and Ω c = R N \ Ω its complement. Unless if specified, any function lying in a functional space L p (Ω), W m,p (Ω), etc is supposed to be a complex-valued function L p (Ω; C), W m,p (Ω; C), etc . For a Banach space E, we denote by E ⋆ its topological dual and by . , . E ⋆ ,E ∈ R the E ⋆ -E duality product. In particular, for any

T ∈ L p ′ (Ω) and ϕ ∈ L p (Ω) with 1 p < ∞, T, ϕ L p ′ (Ω),L p (Ω) = Re Ω T (x)ϕ(x)dx.
As usual, we denote by C auxiliary positive constants, and sometimes, for positive parameters a 1 , . . . , a n , write C(a 1 , . . . , a n ) to indicate that the constant C continuously depends only on a 1 , . . . , a n (this convention also holds for constants which are not denoted by "C").

2 From suitable local inequalities to the vanishing of the involved complex functions on some small ball

In this section, we establish some results improving the presentation of some energy methods of Antontsev, Díaz and Shmarev [START_REF] Antontsev | Energy methods for free boundary problems: Applications to nonlinear PDEs and fluid mechanics[END_REF] which allow to prove localization properties of solutions of a general class of nonlinear partial differential equations (Section 5, Remark 5.8 below and Bégout and Díaz [START_REF] Bégout | Self-similar solutions with compactly supported profile of some nonlinear Schrödinger equations[END_REF]).

In contrast to the presentation in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF] (see e.g. Theorem 1.1), the following statement does not need any information on the second order equation but it will merely use a suitable balance between the total local energy (diffusion + absorption local energies) and the local boundary flux.

This will be crucial for the applicability of the method to cases for which the techniques of Bégout and Díaz [START_REF] Bégout | On a nonlinear Schrödinger equation with a localizing effect[END_REF][START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF] can not be applied.

Theorem 2.1. Assume 0 < m < 1 and let N ∈ N. Then there exists C = C(N, m) satisfying the following property: let x 0 ∈ R N , ρ 0 > 0 and u ∈ H 1 loc B(x 0 , ρ 0 ) . If there exist L > 0 and M > 0 such that for almost every ρ ∈ (0, ρ 0 ),

∇u 2 L 2 (B(x0,ρ)) + L u m+1 L m+1 (B(x0,ρ)) M S(x0,ρ) u∇u. x -x 0 |x -x 0 | dσ , (2.1)
then u |B(x0,ρmax) ≡ 0, where

ρ ν max = ρ ν 0 -CM 2 max 1, 1 L 2 max ρ ν-1 0 , 1 × min τ ∈( m+1 2 ,1] E(ρ 0 ) γ(τ ) max{b(ρ 0 ) µ(τ ) , b(ρ 0 ) η(τ ) } 2τ -(1 + m) + , (2.2)
and where,

E(ρ 0 ) = ∇u 2 L 2 (B(x0,ρ0)) , b(ρ 0 ) = u m+1 L m+1 (B(x0,ρ0)) , k = 2(1 + m) + N (1 -m), ν = k m+1 > 2, γ(τ ) = 2τ -(1 + m) k ∈ (0, 1), µ(τ ) = 2(1 -τ ) k , η(τ ) = 1 -m 1 + m -γ(τ ) > 0.
for any τ ∈ m+1 2 , 1 .

Here and in what follows, r + = max{0, r} denotes the positive part of the real number r. For x 0 ∈ R N and r > 0, B(x 0 , r) is the open ball of R N of center x 0 and radius r, S(x 0 , r) is its boundary and B(x 0 , r) is its closure. Finally, σ is the surface measure on a sphere. A sharper estimate, in the same line of extension of the applicability of the techniques of Bégout and Díaz [START_REF] Bégout | On a nonlinear Schrödinger equation with a localizing effect[END_REF][START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF] indicated before, can be obtained under some additional assumption on F.

Theorem 2.2. Let 0 < m < 1, x 0 ∈ R N , ρ 1 > ρ 0 > 0, F ∈ L 2 B(x 0 , ρ 1
) and u ∈ H 1 loc B(x 0 , ρ 1 ) . If there exist L > 0 and M > 0 such that for almost every ρ ∈ (0, ρ 1 ),

∇u 2 L 2 (B(x0,ρ)) + L u m+1 L m+1 (B(x0,ρ)) + L u 2 L 2 (B(x0,ρ)) M S(x0,ρ) u∇u. x -x 0 |x -x 0 | dσ + B(x0,ρ) |F (x)u(x)|dx , (2.3)
then there exist E ⋆ > 0 and ε ⋆ > 0 satisfying the following property:

if ∇u 2 L 2 (B(x0,ρ1)) < E ⋆ and F 2 L 2 (B(x0,ρ)) ε ⋆ (ρ -ρ 0 ) + p , ∀ρ ∈ (0, ρ 1 ), (2.4
)

where p = 2(1+m)+N (1-m) 1-m , then u |B(x0,ρ0) ≡ 0.
In other words, with the notation of Theorem 2.1,

ρ max = ρ 0 .
Remark 2.3. We may estimate E ⋆ and ε ⋆ as

E ⋆ = E ⋆ u -1 L m+1 (B(x0,ρ1)) , ρ 1 , ρ 0 ρ 1 , L M , N, m , ε ⋆ = ε ⋆ u -1 L m+1 (B(x0,ρ1)) , ρ 0 ρ 1 , L M , N, m .
The dependence on 1 δ means that if δ goes to 0 then E ⋆ and ε ⋆ may be very large. Note that p = 1 γ(1) , where γ is the function defined in Theorem 2.1.

L 1 loc ([0, ρ 0 ); R)
and so is defined almost everywhere in (0, ρ 0 ). Consequently, by Hölder's inequality, the right-hand side in (2.3) is defined almost everywhere in (0, ρ 1 ).

A general framework of applications related to the Schrödinger operator

The following result will be applied later to many concrete equations associated to the Schrödinger operator.

Theorem 3.1. Let Ω ⊂ R N be a nonempty open subset of R N , let x 0 ∈ Ω, let ρ 0 > 0, let 1 p 1 , . . . , p n1 , q 1 , . . . , q n2 < ∞, let F ∈ L 1 loc (Ω) be such that F |Ω∩B(x0,ρ0) ∈ L 2 Ω ∩ B(x 0 , ρ 0 ) and let f ∈ C   n2 k=1 L q k loc (Ω); n1 j=1 L p ′ j loc (Ω)   . Let u ∈ H 1 loc (Ω) ∩ L pj loc (Ω) ∩ L q k loc (Ω), for any (j, k) ∈ 1, n 1 × 1, n 2
, be any solution to the complexvalued equation

-∆u + f (u) = F, in D ′ (Ω).
(3.1)

If ρ 0 > dist(x 0 , Γ) then assume further that f ∈ C   n2 k=1 L q k (Ω); n1 j=1 L p ′ j (Ω)   , u ∈ H 1 0 (Ω), u |Ω∩B(x0,ρ0) ∈ L pj Ω ∩ B(x 0 , ρ 0 ) ∩ L q k Ω ∩ B(x 0 , ρ 0 ) , for any (j, k) ∈ 1, n 1 × 1, n 2 . Set for every ρ ∈ [0, ρ 0 ), I(ρ) = Ω∩S(x0,ρ) u∇u. x -x 0 |x -x 0 | dσ , J(ρ) = Ω∩B(x0,ρ) |F (x)u(x)|dx, (3.2)
w(ρ) = Ω∩S(x0,ρ) u∇u. x -x 0 |x -x 0 | dσ, I Re (ρ) = Re w(ρ) , I Im (ρ) = Im w(ρ) . (3.3)
Then we have,

I, J, I Re , I Im ∈ C([0, ρ 0 ); R), (3.4
)

∇u 2 L 2 (Ω∩B(x0,ρ)) + Re    Ω∩B(x0,ρ) f (u)udx    = Re    F (x)u(x)dx    + I Re (ρ), (3.5) Im    Ω∩B(x0,ρ) f (u)udx    = Im    Ω∩B(x0,ρ) F (x)u(x)dx    + I Im (ρ), (3.6)
for any ρ ∈ [0, ρ 0 ).

Remark 3.2. One easily sees that if ρ 0 < dist(x 0 , Γ) then I, J, I Re , I Im ∈ C([0, ρ 0 ]; R).

Example 3.3. We give some functions f for which Theorem 3.1 applies.

1) Typically, we apply Theorem 3.1 to

f (u) = a|u| -(1-m) u + bu + V u, with (a, b) ∈ C 2 , V ∈ L ∞ loc (Ω) and 0 < m < 1. One easily checks that, f ∈ C L 2 loc (Ω) ∩ L m+1 loc (Ω); L 2 loc (Ω) + L m+1 m loc (Ω) .
If in addition, V ∈ L ∞ (Ω) then one also has,

f ∈ C L 2 (Ω) ∩ L m+1 (Ω); L 2 (Ω) + L m+1 m (Ω) . Let z ∈ C \ {0}. Since |z| -(1-m) z = |z| m , it is understood in the above example that |z| -(1-m) z = 0 when z = 0. 2) Hartree-Fock type equations. Let V ∈ L p (R N ; R) + L ∞ (R N ; R), with min 1, N 2 < p < ∞ and let W ∈ L q (R N ; R) + L ∞ (R N ; R), with min 1, N 4 < q < ∞. Set r = 2p p-1 , s = 4q q-1 , E = L 2 (R N ) ∩ L 4 (R N ) ∩ L r (R N ) ∩ L s (R N ), f (u) = V u + (W ⋆ |u| 2 )u,
for any u ∈ H 1 (R N ). Then H 1 (R N ) ֒→ E with dense embedding and, by density of

D(R N ) in spaces L m (R N ), for any m ∈ [1, ∞),
we have

E ⋆ = L 2 (R N ) + L 4 3 (R N ) + L r ′ (R N ) + L s ′ (R N ), f ∈ C E; E ⋆ , f ∈ C H 1 (R N ); H -1 (R N ) .
See Cazenave [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]

Proofs of the main results

Before proceeding to the proof of Theorems 2.1 and 2.2, we recall the well-known Young's inequality.

For any real x 0, y 0, λ > 1 and ε > 0, one has

xy 1 λ ′ ε λ ′ x λ ′ + 1 λ ε -λ y λ . (4.1)
Proof of Theorems 2.1 and 2.2. We write ρ ⋆ = ρ 0 , for the proof of Theorem 2.1 and ρ ⋆ = ρ 1 , for the proof of Theorem 2.2. Let us introduce some notations. Let ρ ∈ (0, ρ ⋆ ). We set

E(ρ) = ∇u 2 L 2 (B(x0,ρ)) , b(ρ) = u m+1 L m+1 (B(x0,ρ)) , a(ρ) = u 2 L 2 (B(x0,ρ)) , θ = (1+m)+N (1-m) k ∈ (0, 1), ℓ = 1 θ(1+m) , δ = k 2(1+m) .
We may assume that u ∈ H 1 B(x 0 , ρ ⋆ ) . Indeed, the case u ∈ H 1 loc B(x 0 , ρ ⋆ ) can be treated by following the method in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF] (see the end of Step 6, p.49, for Theorem 2.1 and the end of Step 7, p.50, for Theorem 2.2. We now proceed with the proof in 3 steps.

Step

1. E ∈ W 1,1 (0, ρ ⋆ ), for a.e. ρ ∈ (0, ρ ⋆ ), E ′ (ρ) = ∇u 2 L 2 (S(x0,ρ))
and

E(ρ) + b(ρ) 1 2 K 1 (τ )ρ -(ν-1) E ′ (ρ) 1 2 (E(ρ) + b(ρ)) γ(τ )+1 2 + (L 1 M ) 2 F 2 L 2 (B(x0,ρ)) , (4.2)
where

K 1 (τ ) = C(N, m)L 2 1 M 2 max ρ ν-1 ⋆ , 1 max{b(ρ ⋆ ) µ(τ ) , b(ρ ⋆ ) η(τ ) } and L 1 = max 1, 1 L .
By the first lines of Step 2, p.47, in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF], we only have to show (4.2). Let ρ ∈ (0, ρ ⋆ ).

We have to slightly modify the proof of Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF]. Indeed, since F ∈ L 2 , we need of the term u 2 L 2 . We have,

S(x0,ρ) u∇u. x -x 0 |x -x 0 | dσ E ′ (ρ) 1 2 u L 2 (S(x0,ρ)) , (4.3)
u L 2 (S(x0,ρ)) C(N, m) ∇u L 2 (B(x0,ρ)) + ρ -δ u L m+1 (B(x0,ρ)) θ u 1-θ L m+1 (B(x0,ρ)) . (4.4)
See Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF]: estimates (7.11), p.47, and (7.12), p.48. Putting together (2.1) (for Theorem 2.1), (2.3) (for Theorem 2.2), (4.3) and (4.4), we obtain,

E(ρ) + b(ρ) + κa(ρ) CL 1 M E ′ (ρ) 1 2 E(ρ) 1 2 + ρ -δ b(ρ) 1 m+1 θ b(ρ) 1-θ m+1 + L 1 M B(x0,ρ) |F (x)u(x)|dx, (4.5)
where κ = 0, in the case of Theorem 2.1 and where κ = 1, in the case of Theorem 2.2. In the case of

|F (x)u(x)|dx L 1 M 2 F 2 L 2 (B(x0,ρ)) + 1 2L 1 M a(ρ), (4.6)
for any ρ ∈ (0, ρ ⋆ ). Putting together (4.5) and (4.6), we obtain for both theorems, for a.e. ρ ∈ (0, ρ ⋆ ),

E(ρ) + b(ρ) C 0 L 1 M E ′ (ρ) 1 2 E(ρ) 1 2 + ρ -δ b(ρ) 1 m+1 θ b(ρ) 1-θ m+1 + (L 1 M ) 2 F 2 L 2 (B(x0,ρ)) . (4.7)
Let τ ∈ m+1 2 , 1 and let ρ ∈ (0, ρ ⋆ ). A straightforward calculation yields

E(ρ) 1 2 + ρ -δ b(ρ) 1 m+1 b(ρ) 1-θ θ(m+1) = E(ρ) 1 2 b(ρ) 1-θ θ(m+1) + ρ -δ b(ρ) 1 θ(m+1) = E(ρ) 1 2 b(ρ) τ (1-θ)ℓ b(ρ) (1-τ )(1-θ)ℓ + ρ -δ b(ρ) 1 2 +τ (1-θ)ℓ b(ρ) ℓ-τ (1-θ)ℓ-1 2 2ρ -δ max ρ δ ⋆ , 1 K 2 2 (τ ) 1 2θ (E(ρ) + b(ρ)) 1 2 +τ (1-θ)ℓ ,
where

K 2 2 (τ ) = max{b(ρ ⋆ ) µ(τ ) , b(ρ ⋆ ) η(τ ) }, since µ(τ ) 2θ = (1 -τ )(1 -θ)ℓ and η(τ ) 2θ = ℓ -τ (1 -θ)ℓ -1 2 .
Hence (4.2) follows from (4.7) and the above estimate with

K 1 (τ ) = 16C 2 0 L 2 1 M 2 K 2 2 (τ ) max ρ ν-1 ⋆ , 1 , since 2δθ = ν -1 and θ 1 2 + τ (1 -θ)ℓ = γ(τ )+1 2 .
Step 2. For any τ ∈ m+1 2 , 1 and for a.e. ρ ∈ (0, ρ ⋆ ),

0 E(ρ) 1-γ(τ ) K 1 (τ )ρ -(ν-1) E ′ (ρ) + (2L 1 M ) 2(1-γ(τ )) F 2(1-γ(τ )) L 2 (B(x0,ρ)) .
Following Step 4, p.48, in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF] but with Young's inequality (4.1) applied with x =

1 2 K 1 (τ )ρ -(ν-1) E ′ (ρ) 1 2 , y = (E(ρ) + b(ρ)) γ(τ )+1 2 , λ = λ(τ ) = 2 γ(τ )+1 and ε = ε(τ ) = (γ(τ ) + 1) 1 λ(τ ) ,
Step 2 follows from the estimates

E(ρ) + b(ρ) 1 2 K 1 (τ )ρ -(ν-1) E ′ (ρ) 1 2 (E(ρ) + b(ρ)) γ(τ )+1 2 + (L 1 M ) 2 F 2 L 2 (B(x0,ρ)) , C(τ ) 2 λ(τ ) λ(τ )-1 K 1 (τ )ρ -(ν-1) E ′ (ρ) 1 1-γ(τ ) + 1 2 (E(ρ) + b(ρ)) + (L 1 M ) 2 F 2 L 2 (B(x0,ρ)) , 1 2 K 1 (τ )ρ -(ν-1) E ′ (ρ) 1 1-γ(τ ) + 1 2 (E(ρ) + b(ρ)) + (L 1 M ) 2 F 2 L 2 (B(x0,ρ)) , C(τ ) = λ(τ ) -1 λ(τ ) ε(τ ) λ(τ ) λ(τ )-1 < λ m+1 2 -1 λ m+1 2 (γ(τ ) + 1) 1 λ(τ )-1 < 1 2 2 1 λ(τ )-1 < 1 2 2 λ(τ ) λ(τ )-1 .
Step 3. Conclusion.

Now, following from

Step 5 to Step 7, p.48-50, in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF], where estimate (7.16) therein has to be replaced with estimate of the above Step 2 and where the mapping ρ -→ F (ρ) has to be replaced with the new function ρ

-→ (2L 1 M ) 2(1-γ) F 2(1-γ) L 2 (B(x0,ρ))
, we prove Theorems 2.1 and 2.2. This achieves the proof.

Proof of Theorem 3.1. If ρ 0 > dist(x 0 , Γ) then u ∈ H 1 0 (Ω). So we may extend u by 0 on Ω c ∩ B(x 0 , ρ 0 ). Denoting u this extension, we have u ∈ H 1 0 Ω ∪ B(x 0 , ρ 0 ) . We first consider the case where ρ 0 = dist(x 0 , Γ). We deal with ρ 0 = dist(x 0 , Γ) at the end of the proof. It follows that J ∈ C([0, ρ 0 ]; R) and by Cauchy-Schwarz's inequality, I ∈ L 1 (0, ρ 0 ). Thus, I, J, I Re , I Im are defined almost everywhere on (0, ρ 0 ). It follows from (3.1) that,

∇u, ∇ϕ D ′ (Ω),D(Ω) + f (u), ϕ D ′ (Ω),D(Ω) = F, ϕ D ′ (Ω),D(Ω) , (4.8)
for any ϕ ∈ D(Ω). Let ρ ∈ (0, ρ 0 ). For any n ∈ N, n > 1 ρ , we define

ψ n ∈ W 1,∞ (R; R) by ∀t ∈ R, ψ n (t) =          1, if |t| ∈ 0, ρ -1 n , n(ρ -|t|), if |t| ∈ ρ -1 n , ρ , 0, if |t| ∈ [ρ, ∞),
and we set ϕ n (x) = ψ n (|xx 0 |) u(x) and ϕ n = ϕ n|Ω , for almost every x ∈ Ω ∪ B(x 0 , ρ 0 ). We easily check that for any (j, k) ∈ 1,

n 1 × 1, n 2 , ϕ n|Ω∩B(x0,ρ0) ∈ H 1 0 Ω ∩ B(x 0 , ρ 0 ) ∩ L pj Ω ∩ B(x 0 , ρ 0 ) ∩ L q k Ω ∩ B(x 0 , ρ 0 ) , ϕ n ∈ H 1 0 Ω ∪ B(x 0 , ρ 0 ) ∩ L pj Ω ∪ B(x 0 , ρ 0 ) ∩ L q k Ω ∪ B(x 0 , ρ 0 ) , ϕ n ∈ H 1 0 (Ω) ∩ L pj (Ω) ∩ L q k (Ω).
Then there exists (ϕ m n ) m∈N ⊂ D(Ω) such that for any (n, m) ∈ N 2 , supp ϕ m n ⊂ Ω ∩ B(x 0 , ρ 0 ) and

ϕ m n H 1 0 (Ω)∩L p j (Ω)∩L q k (Ω) ---------------→ m-→∞ ϕ n ,
for any (j, k) ∈ 1, n 1 × 1, n 2 . Consequently, ϕ = ϕ n are admissible test functions in (4.8). We have,

∇u, ∇ϕ n L 2 (Ω),L 2 (Ω) = ∇ u, ∇ ϕ n L 2 (Ω∪B(x0,ρ0)),L 2 (Ω∪B(x0,ρ0)) = B(x0,ρ) ψ n |x -x 0 | |∇ u| 2 dx -n Re    ρ ρ-1 n    S(x0,r) u∇ u. x -x 0 |x -x 0 | dσ    dr    ,
where we introduced the spherical coordinates (r, σ) at the last line. We now let n ր ∞. Using the Lebesgue's dominated convergence Theorem and recalling that

I Re ∈ L 1 ((0, ρ 0 ); R), we obtain lim n→∞ ∇u, ∇ϕ n L 2 (Ω),L 2 (Ω) = ∇u 2 L 2 (Ω∩B(x0,ρ)) -I Re (ρ). (4.9)
Proceeding as above but also with ϕ = iϕ n , we get lim

n→∞ ∇u, i∇ϕ n L 2 ,L 2 = -I Im (ρ) and lim n→∞ f (u), ϕ n F ⋆ ,E = Re A(u) , lim n→∞ f (u), iϕ n F ⋆ ,E = Im A(u) , lim n→∞ F, ϕ n L 2 ,L 2 = Re B(u) , lim n→∞ F, iϕ n L 2 ,L 2 = Im B(u) .
where E = 5 Application to the localization property to the case of Neumann boundary conditions

In Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF], the authors study the localization property for equation (5.6) below with the homogeneous Dirichlet boundary condition (see, for instance, Theorem 3.5 in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF]). In Theorem 5.6 below, we show that the same property holds with the homogeneous Neumann boundary condition. Before, we need to prove that solutions exist. This can be found in Bégout and Díaz [START_REF] Bégout | Existence of weak solutions to some stationary Schrödinger equations with singular nonlinearity[END_REF].

Note that from Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF]

with possibly b = 0, and we worked with Assumptions (5.5) are made to prove the existence and the localization property of solutions to

-i∆u + a|u| -(1-m) u + bu = F .
-∆u + a|u| -(1-m) u + bu = F, in L 2 (Ω). (5.6)
For uniqueness, the hypotheses are the following (Theorem 2.10 in Bégout and Díaz [START_REF] Bégout | Existence of weak solutions to some stationary Schrödinger equations with singular nonlinearity[END_REF]).

Assumption 5.2 (Uniqueness). Assume that (a, b) ∈ C 2 satisfies one of the two following conditions.

1) a = 0, Re(a) 0 and Re(ab) 0.

2) b = 0, Re(b) 0 and a = kb, for some k 0.

A geometric interpretation of (5.5) and 1) of Assumption 5.2 is given in Section 6 in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF], modulus a rotation in the complex plane. Now, we give some results about equation (5.6) when (a, b) ∈ C 2 satisfies (5.5). If furthermore (a, b) satisfies Assumption 5.2 then the solution of (5.7) is unique. Let v ∈ H 1 (Ω) be any solution to (5.7). Then v ∈ H 2 loc (Ω). In addition,

v H 1 (Ω) M F L 2 (Ω) , (5.8)
where M = M (|a|, |b|). Finally, if for some α ∈ (0, m], F ∈ C 0,α loc (Ω) then u ∈ C 2,α loc (Ω).

Symmetry Property 5.4. If furthermore, for any R ∈ SO N (R), RΩ = Ω and if F is spherically symmetric then we may construct a solution which is additionally spherically symmetric. For N = 1, this means that if F is an even (respectively, an odd) function then u is also an even (respectively, an odd) function.

Here and in what follows, SO N (R) denotes the special orthogonal group of R N . be a solution to (5.7). If uniqueness holds for the problem (5.7)1 , supp F is a compact set and

F L 2 (Ω) δ 0 then supp u ⊂ K(ε) ⊂ Ω, where
K(ε) = x ∈ R N ; ∃y ∈ supp F such that |x -y| ε ,
which is compact.

The proof relies on the following lemma.

⋆ > 0. If F |Ω∩B(x0,ρ⋆) ∈ L 2 Ω ∩ B(x 0 , ρ ⋆ ) then for any ρ ∈ [0, ρ ⋆ ), ∇u 2 L 2 (Ω∩B(x0,ρ)) + L u m+1 L m+1 (Ω∩B(x0,ρ)) + L u 2 L 2 (Ω∩B(x0,ρ)) M Ω∩S(x0,ρ) u∇u. x -x 0 |x -x 0 | dσ + Ω∩B(x0,ρ) |F (x)u(x)|dx , (5.10)
where it is additionally assumed that u ∈ H 1 0 (Ω) if ρ ⋆ > dist(x 0 , Γ).

Proof. Let x 0 ∈ Ω and let ρ ⋆ > 0. We set for every ρ ∈ [0, ρ ⋆ ),

I(ρ) = Ω∩S(x0,ρ)
u∇u.

x It follows that supp u ⊂ K(ε) and u ∈ H 1 0 (Ω) is a solution to (5.7). By uniqueness assumption, u = u so that supp u ⊂ K(ε) ⊂ Ω, which is the desired result.

Remark 5.8. In Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF], the authors study existence, uniqueness, smoothness and localization property for the equations (5.6) with an external source F belonging to L m+1 m (Ω) with 0 < m < 1 (see, for instance, Theorem 3.5 in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF]). Below, we explain how the same results hold true with the weaker assumption F ∈ L 2 (Ω). Indeed, when |Ω| < ∞ and 0 < m < 1, L m+1 m (Ω) ֒→ L 2 (Ω) and L m+1 m (Ω) = L 2 (Ω). Results of existence can be found in Bégout and Díaz [START_REF] Bégout | Existence of weak solutions to some stationary Schrödinger equations with singular nonlinearity[END_REF]

Theorem 2 . 2 ,

 22 we apply (4.1) with x = |F |, y = |u|, λ = 2 and ε = √ L 1 M , and we get B(x0,ρ)

L

  qj (Ω), F = n1 j=1 L pj (Ω), A(u) = Ω∩B(x0,ρ) f (u) u dx and B(u) = Ω∩B(x0,ρ) F (x)u(x)dx. Estimates (3.5) and (3.6) then follow from (4.9) and these five last estimates. Since all terms in (3.5) and (3.6) are continuous on [0, ρ 0 ], except eventually I Re and I Im , we deduce that I Re and I Im are continuous and (3.5) and (3.6) hold for any ρ ∈ [0, ρ 0 ]. The case ρ 0 = dist(x 0 , Γ) follows from the above proof applied with ρ n 0 = ρ 0 -1 n in place of ρ 0 and letting n ր ∞.

Remark 5 . 1 .

 51 to this paper, there was a slight change of notation. See Remark 5.1 below. In the context of the paper of Bégout and Díaz [3], we can establish an existence result with the homogeneous Neumann boundary condition (instead of the homogeneous Dirichlet condition) and F ∈ L 2 (Ω) instead of F ∈ L m+1 m (Ω) . In Bégout and Díaz [3], we introduced the set, A = C \ z ∈ C; Re(z) = 0 and Im(z) 0 , and assumed that ( a, b) ∈ C 2 satisfies, ( a, b) ∈ A × A and              Re( a)Re( b) 0, or Re( a)Re( b) < 0 and Im( b) > Re( b) Re( a) Im( a),

  But here in order to follow a closer notation with most of the works dealing with Schrödinger equations, we do not work any more with this equation but with,-∆u + a|u| -(1-m) u + bu = F,and b = 0. This means that we choose, a = ia, b = ib and F = iF. Then assumptions on (a, b) are changed by the fact that for z = iz, Re(z) = Re(-i z) = Im( z), (5.2) Im(z) = Im(-i z) = -Re( z). (5.3) It follows that the set A and (5.1) become, A = C \ z ∈ C; Re(z) 0 and Im(z) = 0 , b) ∈ C 2 satisfies (5.1) ⇐⇒ (a, b) ∈ C 2 satisfies (5.5) .

Corollary 5 . 3 (

 53 Neumann boundary conditions). Let Ω be a nonempty bounded open subset of R N having a C 1 boundary, let ν be the outward unit normal vector to Γ, let 0 < m < 1 and let (a, b) ∈ C 2 satisfies (5.5). For any F ∈ L 2 (Ω), there exists at least one solutionu ∈ H 1 (Ω) to    -∆u + a|u| -(1-m) u + bu = F, in L 2 (Ω),

Remark 5 . 5 .Theorem 5 . 6 .

 5556 One easily checks that if (a, b) ∈ A 2 satisfies Re(a) 0 and Re(ab) 0 then (a, b) ∈ C 2 verifies (5.5). In this case, uniqueness assumptions imply existence assumptions. Proof of Corollary 5.3 and Symmetry Property 5.4. The result comes from Bégout and Díaz [5]: Theorem 2.8 (existence and symmetry property), Theorem 2.10 (uniqueness), Theorem 2.9 a priori estimate (5.8) and Theorem 2.12 (local smoothness). Concerning the support of solution of (5.7) we have: Let Ω be a nonempty bounded open subset of R N having a C 1 boundary, let 0 < m < 1 and let (a, b) ∈ C 2 satisfies (5.5). Then there exists ε ⋆ > 0 such that for any 0 < ε ε ⋆ , there exists δ 0 = δ 0 (ε, |a|, |b|, N, m) > 0 satisfying the following property. Let F ∈ L 2 (Ω) and let u ∈ H 1 (Ω)

Lemma 5 . 7 .

 57 Let Ω ⊂ R N be a nonempty open subset of R N , let 0 < m < 1 and let (a, b) ∈ C 2 satisfies (5.5). Let F ∈ L 1 loc (Ω) and let u ∈ H 1 loc (Ω) be any solution to -∆u + a|u| -(1-m) u + bu = F, in D ′ (Ω). (5.9) Then there exist two positive constants L = L(|a|, |b|) and M = M (|a|, |b|) satisfying the following property. Let x 0 ∈ Ω and ρ

-x 0 2 L 2 ( 2 L 2 ( 2 L 2 (

 0222222 |xx 0 | dσ and J(ρ) = Ω∩B(x0,ρ) |F (x)u(x)|dx. It follows from Theorem 3.1 that I, J ∈ C([0, ρ ⋆ ); R) and ∇u Ω∩B(x0,ρ)) + Re(a) u m+1 L m+1 (Ω∩B(x0,ρ)) + Re(b) u Ω∩B(x0,ρ)) I(ρ) + J(ρ), (5.11) Im(a) u m+1 L m+1 (Ω∩B(x0,ρ)) + Im(b) u Ω∩B(x0,ρ))I(ρ) + J(ρ),(5.12)for any ρ ∈ [0, ρ ⋆ ). Estimate (5.10) then follows from (5.11), (5.12) and Lemma 4.5 from Bégout and Díaz[START_REF] Bégout | Existence of weak solutions to some stationary Schrödinger equations with singular nonlinearity[END_REF] with δ = 0.Proof of Theorem 5.6. Let F ∈ L 2 (Ω) with supp F ⊂ Ω and let u ∈ H 1 (Ω) a solution to (5.7) be given by Theorem 5.3. Set K = supp F andO(ε) = x ∈ R N ; ∃y ∈ K such that |x -y| < ε . Then K(ε) = O(ε). Let ε ⋆ > 0 be small enough to have K(5ε ⋆ ) ⊂ Ω and let ε ∈ (0, ε ⋆ ]. Let L andM be given by Lemma 5.7 applied with ρ ⋆ = 2ε. By Theorem 2.1 and estimate (5.8) in Corollary 5.3 above, there exists δ 0 = δ 0 (ε, |a|, |b|, N, m) > 0 such that if F L 2 (Ω) δ 0 then u |B(x0,ε) ≡ 0, for anyx 0 ∈ Ω such that B(x 0 , 2ε) ∩ K = ∅ and B(x 0 , 2ε) ⊂ Ω. One easily sees that B(x 0 , 2ε) ∩ K = ∅, for anyx 0 ∈ K(2ε) c ∩ K(3ε). We deduce that for any x 0 ∈ K(2ε) c ∩ K(3ε), u |B(x0,ε) ≡ 0. By compactness, there exist n ∈ N and x 1 , . . . , x n ∈ K(2ε) c ∩ K(3ε) such that,K(ε) c ∩ O(4ε) ⊂ n j=1 B(x j , ε) ⊂ n j=1 B(x j , 2ε) ⊂ K(5ε) ⊂ Ω.It follows that u |K(ε) c ∩O(4ε) ≡ 0. Let us define u in Ω by, u = u, in O(2ε), 0, in Ω \ O(2ε).

Ω∩B(x0,ρ)

which is the case, for instance, if (a, b) ∈ C

satisfies Assumption 5.2.

† The research of J.I. Díaz was partially supported by the project ref. MTM200806208 of the DGISPI (Spain) and the Research Group MOMAT (Ref. 910480) supported by UCM. He has received also support from the ITN FIRST of the Seventh Framework Program of the European Community's (grant agreement number 238702) 2010 Mathematics

jointly to some others additional results. Hypotheses on (a, b) ∈ C 2 are the same as in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF], except we have to require b = 0. Note that from Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF] to the present paper, there was a change of notation. See Remark 5.1 for precision. Throughout this remark, equation Indeed, by Bégout and Díaz [START_REF] Bégout | Existence of weak solutions to some stationary Schrödinger equations with singular nonlinearity[END_REF] (Theorems 2.8, 2.9, 2.10 and 2.12), these results hold but with u ∈ H 2 loc (Ω) and

(Ω), (4.1) and (4.2) in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF]. Concerning the localization property, Theorems 3.1 and 3.5 in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF] still hold true but with F ∈ L 2 (Ω) and

instead of (3.1) in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF]. The proofs are essentially the same where we use Lemma 5.7 and (5.13) above instead of (4.1) in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF]. It follows that Theorems 1.1 and 1.2 in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF] can be easily adapted with the obvious modifications.


Jules Barbier

Michel Carré

Barbier

1951 Le Conte

José Corti Manuela Mohr

Rethinking the vampire: the fantastic on the puppet stage

Keywords: fantastic, Hoffmann, devil, vampire, interiorisation, psychologisation In

This article examines the new production of The Tales of Hoffmann at the Salzburger Puppentheater under the direction of Philippe Brunner (2019). Afterlife figures such as the devil, the vampire and the ghost are not always identifiable figures located in one single body. They are interiorised, becoming an expression of psychological distress, and spread to many characters. The staging with string puppets in Salzburg dematerialises the afterlife figures and helps rethinking their traditional characteristics.

materialise in Hoffmann's conception of the fantastic, they appear more interiorized in the libretto, as an expression of inner turmoil. How does puppetry deal with these figures, and to what extent do the specificities of string puppets, the dramaturgy and the staging affect the questions and challenges that come with the particular interaction of the vampirical and the diabolical figure in the Austrian production?

The dramatic experience interacts with literary fantastic but works in its very own way when it comes to vampires. Although the devil is explicitely invoked and incarnated by the male antagonists, the vampire is not an identifiable character. His presence is discreet. Research on 19 th century fantastic literature has pointed out the transformations of the diabolical figure becoming a literary motif [START_REF] Milner | Le Diable dans la littérature française[END_REF]) and the assimilation of vampires and ghosts [START_REF] Sangsue | Vampires, fantômes et apparitions : nouveaux essais de pneumatologie littéraire[END_REF]. However, we are interested in the implications of the staging of Hoffmann's works at a string puppet theatre in the 21 st century. Therefore, we will see how details of the staging lead to a rethinking of the vampirical figure and the specific possibilities of string puppets to explore mental, invisible processes. The devil and the vampire are strongly related. The staging with string puppets can ask in what way they lead to a special configuration of the fantastic genre and the vampirical figure. A puppet theatre choosing Offenbach's opera shows that the puppet can capture a fantastic world in a unique way while at the same time having to find solutions to the staging of afterlife figures evoking ontological issues. 3

Evil afterlife figures: from Hoffmann to Offenbach

In Hoffmann's fantastic stories, which came to France in 1829 translated by Adolphe Loève-Veimars, the devil is strongly related to the Doppelgänger. The devil functions as the hero's antagonist and is responsible for the division of the individual who becomes a stranger to itself. A part of the individual's identity is repressed, allowing the evil Doppelgänger to come forward. The devil maintains his traditional characteristics : he tempts innocent souls, promising them what they desire in exchange for their soul. Medardus, the protagonist of The Devil's Elixirs (1815), drinks from the potions he was told to keep ; his uncanny Doppelgänger then feels the need to drink blood.

Hoffmann was familiar with the vampirical figure. He discreetly associates the devil and the vampire, an aspect one also observes in Offenbach's opera and in Philippe Brunner's production at the Marionettentheater. Hoffmann is interested in showing how madness is induced by stories about the devil and the vampire, i.e. how words and narratives cause loss of control or acts of violence associated to these afterlife figures.

In every act of the libretto, a diabolical opponent apparently prevents Hoffmann from being with the woman he loves. However, a closer look reveals that several characters combine multiple diabolical traits as well as characteristics of other afterlife figures. Therefore, they defy the limits between each other and contaminate Hoffmann by their presence. The characters between life and death question stable identities, but Offenbach's opera does not refer to the vampire like the Austrian string puppets. Hence, the vampirical figure is not displayed in the same way. This offers the possibility to highlight particular consequences of the interaction of the devil and the vampire in the libretto, and to track the specificities of Brunner's work.

The story which forms the framework of the libretto is set in master Luther's tavern. The owner's name relegates to the religious sphere. Martin Luther was feared as a dangerous opponent of the catholic doctrine. According to Luther, God does not bargain: the human being has to take responsibility for his actions in the present, as we cannot change the past. The libretto does not emphasize Luther's potential to be an antagonist. As we will see, Brunner confers more depth to Luther and works on the diabolical opponent influencing character dynamics by playing with light and colours.

Lindorf is Hoffmann's opponent. Irritated and egocentric ("Le conseiller Lindorf, parbleu ! Tu ne connais pas le conseiller Lindorf ?"4 (Salzburger Marionettentheater, 2019, 1) ["The councillor Lindorf, by Jove! Do you not know the councillor Lindorf?"]), he is dislikeable from the very beginning. Lindorf subjects the servant Andres to a rude interrogation about the beautiful singer Stella. The competition for Stella's love favours Hoffmann ("Mon rival est aimé, je ne le suis pas"

(2) ["My rival is loved, and I am not"]), but he fakes carelessness and still has an ace up his sleeve: "[J]e suis vieux, mais je suis vif !" (2) ["I am old, but full of life"]. He will never tell his exact age, but his vividness is evident. Lindorf never stops conspiring, observing and plotting. He unites a strange power in a body defying physical limits. He therefore describes himself as an afterlife figure. Given that the vampire has to be active in order to provide for himself, movement is a component of a vampire's definition. 5 Musical Leitmotive emphasize the diabolical traits of Hoffmann's opponents. When Lindorf and others appear and the word "devil" is pronounced, a specific bass motif sounds. This arrangement by Offenbach (Oeser 1977, 25) links the afterlife figure to the music and transforms it into an obsessively recurrent mental issue, the idée fixe (idem).

After the Kleinzach-song, Hoffmann offers to talk about "ces folles amours" (Salzburger Marionettentheater, 2019, 5) ["This mad love"]. Of course, it was the song's lyrics that made him think about the past. However, there is another reason: as an afterlife figure, Lindorf who is also on stage brings back memories of another figure between life and death, that is Olympia the automaton.

Coppélius enters when Hoffmann is alone at Spalanzani's house, as if he had waited on purpose to catch an unstable and enamoured soul. Dapertutto also prowls around, waiting for the right moment to strike. Both resemble the devil and the vampire. Dapertutto uses Giulietta to get to the reflections she substracts from those who trust her. Maybe he makes her work for him because he does not have a reflection or a shadow either. According to some superstitious beliefs, vampires can lack both. He makes Hoffmann wonder whether he is "le jouet de l'enfer" (6) ["Hell's toy"]; he maliciously comments on his "pâleur mortelle" (idem) ["deathly pale"]. Hoffmann's pale skin can indicate vampirisation. Nevertheless, the devil and the vampire are not just deathbringers. They can offer immortality or a longer life. Religious and mythological accounts state that the devil wanted to create life, for which he was punished by God. In the 19 th century, fewer people believe in him or picture him as a hideous creature. As the devil's religious power fades, his power in the artistic field rises. He transforms into a familiar and human-like character (Milner 2007, 48), stimulating new aesthetical interpretations for this role. The vampire is one of the demon's maleficient creatures. A pact with the devil can lead to a vampirical condition of the victim. He is also related to ghosts since the 19 th century, a general term including vampirical figures (Sangsue 2018, 7). The vampire is a cursed being: he suffers from his condition he did not choose deliberately. The ambivalence of the devil and the vampire is highly stressed in Brunner's production by means connected to the possibilities of the puppet.

Spalanzani is the creator of a woman without blood in her veins. Hoffmann says: "[L]a vie manque à ce regard, le sang à ce visage" (Salzburger Marionettentheater, 2019, 8) ["These eyes are missing life, this face is missing blood"], as if Spalanzani has drunk the blood of his daughter, which makes him a vampire. His manipulative power becomes manifest in the way he arranges Olympia's performance in front of the ball guests. He tells them that his daughter will do everything they want although it is really him who pushes and drags Olympia to the right place and makes her speak and sing at the right moment. Nevertheless, Spalanzani and Coppélius both have to concede defeat.

Spalanzani cannot earn any money with Olympia destroyed by Coppélius in a rage (we will see how Brunner modifies this detail and inserts the effects in a broader perspective), and Coppélius cannot claim his 500 ducats from the bankrupt Jew Elias. Neither Hoffmann nor his antagonists win.

Antonia's mother emerges from a life-sized portrait, a fantastic topos. Antonia reaches out to her in a moment of despair, after having already shown signs of a mysterious illness. At first, the stage direction indicates: "[s]a mère lui apparaît" (29) ["Her mother appears before her eyes"]. The following stage directions call the deceased mother "une voix" (idem) ["a voice"] and "le fantôme" (30) ["the ghost"] even after Antonia has explicitely identified her. The lack of specificity of the libretto and the repeated identification audible for the public and other characters create a discrepancy. With the mother rising from the dead and the daughter being killed, Miracle combines the evil powers of the devil and the vampire. But the mother is herself a ghost, a vampire and a devil 6 . Crespel sees in Antonia his deceased wife and says "quelle illusion me poursuit" (23) ["what illusion persecutes me"], but the German translation in Salzburg's bilingual libretto refers to the ghostly figure: "illusion" is translated "Spuk" meaning spook or ghost. The dead mother sucks the life out of her own offspring. Antonia's music stresses that her feelings do not come from her: she expresses herself through songs that could have been part of her mother's repertoire. 7 Living characters, an automaton, figures between life and death, visible and invisible figures appear together, questioning categories and limits between various forms of existence. Intrinsically fantastic (Plassard 2014, 13), the puppet can intervene in such a configuration in its very own way.

It becomes an expression for particular bodies and mental states by rethinking mechanicity and hybrid life forms, thus broadening the range of expressions of the vampirical afterlife figures.

We have considered the evolutions and specificities of the afterlife figures from E.T.A. Hoffmann to Offenbach, and presented the main male characters who, in Brunner's production, seem to be Hoffmann's antagonists and have certain traits in common. In the following, we will see how Brunner uses concepts and implications related to the imaginary world of the machine.

6 This particularity shows that the diabolical figure is not limited to male characters. 7 "Antonias Gefühlsäußerungen sind aus zweiter Hand. Sie singt von Gefühlen, sie äußert sich in Kunstprodukten, […] There she commits entirely to her performance, as if she was on a podium or a stage, with an aria in the style of Donizetti which, at the limit of triteness, has obviously been extracted from her mother's repertoire, who has also been a singer"]. Translation mine.

Machineries and mechanisms

In Brunner's production, Olympia is the only character who is not a string puppet but a remotecontrolled, partly mechanical partly electronical figure. Olympia's evilness is much more problematic because she is staged like a vampire's victim who takes revenge on her torturers who are at the same time perpetrators and victims with a complex inner life. Her two "fathers", Spalanzani and Coppélius, show more evil traits related to afterlife figures through specific staging choices. The puppet transforms the afterlife figures with respect to questions and challenges about identity and relationships. Olympia's artificiality and her state between life and death are dealt with in a way that does not leave room for doubt. For this, the zombie figure is used: Olympia mostly holds her arms up in front of her, and this gesture resembles the typical posture of a zombie [Fig. 1].

The all-invading presence of artificiality establishes a dialogue between Offenbach's opera and the puppet. Artificiality introduces considerations on inner processes and incertainty of identity.

The projections on the back of the stage during the first scenes of the Olympia act are black and white anthropomorphic figures. The second figure to the left seems to be a woman in distress on a table under which scientific instruments can be seen. Science resembles torture: this detail is part of a staging which makes Olympia a strong female figure trying to scotch her cruel fathers' plans.

When Coppélius enters the stage,8 Olympia is not directly illuminated any more. The attention is directed towards Coppélius, meaning that there exists a conflict between these two. It is almost as if Olympia wanted to hide from him. The semi-darkness surrounding her gives her a life-like appearance because there is not enough light for the audience to determine whether she moves or not. Ambiguity is therefore a threat for human beings and their psychological integrity, but an opportunity for puppets (Wolfson 2018, 212). As Olympia seems most powerful with little light, she evokes the vampire who cannot tolerate sunlight.

During the argument between Spalanzani and Coppélius, the automaton's eyes constitute an important part of the latter's reasoning. As Olympia has his eyes, 9 he should be considered her father and get his fair share. A kind of a horrific genetic creation process is evoked. Spalanzani declares that Olympia is his daughter, suggesting that Olympia inherited mostly his genetic make-up.

Coppélius then makes a spyglass appear in the air which flows over Hoffmann's head. When the hero turns around, the audience discoveres that he wears a pair of pink, round-shaped glasses. A comical moment of surprise ensues. The expression "to wear rose-coloured glasses" is taken literally to state that Hoffmann sees Olympia with different eyes, namely Coppélius' realitydistorting ones. The hero's glasses are related to the love he feels for Olympia, but they also take on a darker meaning because a part of the whole world remains hidden from his conscience, as

Hoffmann never puts them down. This lasts until Hoffmann loses his "eyes", triggering Olympia to lose hers. As soon as he does not wear his glasses any more, Olympias eyes pop out of her head and keep dangling under her face. The very special lighting chosen for Olympia at that moment creates pitch-black shadows surrounding her eyes, which accentuate her empty eye sockets. She almost looks like a demon. This has an absolutely shocking effect which persists until she leaves the scene.

Cochenille pushes Olympia from the stage; a green light flashes several times, then something explodes and the stage goes dark. In the libretto, Coppélius destroys Olympia right after Hoffmann breaks his glasses. However, in Brunner's production, Olympia is not destroyed, 10 only broken. The staging shows that the enchantment is over and suggests Olympia's rebellion against her fathers.

During Coppélius' presence on stage, an enormous single eye pulsates on the back of the stage.

Circles and gear hang from the ceiling, one looks like a clock and another like a steering wheel.

They convey a mechanical imagery pointing Olympia's artificial nature, but the clock and the wheel 9 Coppélius may have given her the eyes he had previously stolen from children. From the opening of the opera, eyes are an important motive. The devil and the vampire are at the same time predators and seducers; this is why they are often associated to cats. The performance begins with two lights resembling two cat's eyes shining in the dark. T he eyes have a slightly different colour, announcing Coppélius' assembly of "eyes". The left eye closes before the right one, almost as if the animal winked, like a huge, dangerous creature foreshadowing evil things to come and setting the tone for afterlife figures. 10 What is more, the public discovers a strange figure in Spalanzani's laboratory. It seems to be a female body in a glass case. Could it be Olympia's prototype or even a newer version of her?

are related respectively to time and space. Olympia defies both (an automaton cannot die and she can roll anywhere very quickly without getting tired), just like devils and vampires do. When Olympia moves, a mechanical whir can be heard. Spalanzani makes her execute movements which only make sense for humans, such as the hand movements following her voice going up and down.

What serves as facilitation to real humans is no more than a reproduction of their imperfection.

Olympia never risks singing out of tune, but only being out of batteries. Suddenly, she bends forward and her voice gets deeper and deeper. This is a very intriguing detail. 11 Quickly, Spalanzani has to straighten her back until her voice goes up. Another unexpectedly comical scene occurs when Olympia leaves the scene rolling backwards: Spalanzani panicks and rushes behind the stage where he pushes her in the right direction until Olympia comes out rolling forward again. One has the impression that Olympa sabotages her father's plan. When she forces him to catch her, Spalanzani adopts a posture similar to a puppeteer or a hypnotist, putting his arms in front of him and fixing

Olympia with his eyes. The comical moments of the staging are not an indication of parody. They offer a comic relief and make funny scenes significant with respect to the way specific staging choices point towards essential ontological issues related to afterlife figures.

In Brunner's production, Spalanzani is not quite the mean and carefree villain figure he appears to be in the libretto. He has to overlook, and is in charge of, everything and everyone. It is only natural, then, that he is very agitated, walks around very fast and waves his arms a lot. Secretly tormented, Spalanzani often looks directly to the guests, encouraging them to admire his daughter.

As he tries to induce the "right" opinions, his ambition is to enter and to control their feelings. He is not only craving approval, as the libretto states, but also seems scared because his identity as a proud father of an irreproachable daughter is at stake. During Olympia's performance, Spalanzani leans far backwards several times, remaining in a position that only a puppet can maintain. His position is important because even him, Olympia's creator, seems overwhelmed by his creature.

11 Philippe Brunner confirms that Olympia's voice going deeper is used in order to make the audience understand that her batteries are dying. He wanted to find an acoustic expression for her technical problem.

Hoffmann's friend Niklaus approaches Olympia from behind her back after Spalanzani has stepped aside for a second, but as soon as he sees Niklaus, he comes running and chases him away.

Spalanzani goes as far as to stand between Niklaus and Olympia so that the young man cannot see her at close range. But with the guests to enchant and Niklaus to monitor, Spalanzani has so much to do that he calls Cochenille for help who guides the guests to another room whilst Spalanzani prevents Niklaus from coming too close. Finally, Niklaus gives up and follows the guests; however, he never attempts to inform Hoffmann about the doubts he obviously has. When Hoffmann follows Olympia around, Niklaus makes no convincing effort to prevent his friend from doing so.

The afterlife figures are reconfigurated and function as catalysts of the plot and relationship dynamics. Making characters side with the antagonists or suggesting the hero's betrayal by those he calls his friends underlines the dynamics of vampirism. The staging reinterprets the affinities of afterlife figures who give new impulsions to the puppet stage. As we have seen how the imaginary world of machines and mechanisms influences characters, the stage design and the character dynamics, we will show how a vampirical dimension invests these elements of Brunner's production.

Vampirical machinations

Nathanaël's entrance in the first act makes him appear suspicious. He hides behind the back of the Muse; when she turns, he stays behind her, his back against hers, and they turn around by 180 degrees at the same time. The Muse transforms into Nathanaël in order to stay next to Hoffmann because she feels neglected. Nevertheless, this movement seems vicious because he hides behind another string puppet. Entering the stage whilst hiding behind something is ominous, as this is also the way Dapertutto 12 comes on stage, for example. Nathanaël then flows into the dark back of the stage by making a huge step in the air. When everyone enters the tavern in the first act, Lindorf stays at his separate table at the left of the stage. He is partially hidden in the shadows [Fig. 2], illuminated only by a bluish light which confers him an eerie aura. At first, he is the only one of the characters present on stage who is in a blue luminous spot. Later however, the same light is also directed towards Luther. The bluish cold colour contrasts with a warmer yellow used for the other figures. This is indeed a relevant aspect because Luther defects to Hoffmann's enemy and will not protect him from Lindorf's racketeering taking places at his tavern. At the end of the first act, those who remain visible on stage when the lights slowly fade are Lindorf and Luther. Back in the tavern after the Antonia act, a warm, yellow light illuminates everyone but them.

Also in the first act, when Hoffmann's mind is about to wander to the woman he once loved, Nathanaël stands behind him and thinks he is going mad. Hoffmann's respiration is difficult, as he is in emotional distress, but Nathanaël bent over his neck seems to analyse him (he does not understand what is going on with his friend and interrupts his thinking) as if he could see into his head and take it off, making his puppet-status once again obvious. Traditionally, the vampire bites the victim in the neck; therefore his position suggests that a vampirical force threatens Hoffmann.

Indeed, as Hoffmann sings, two yong men to the right seem to whisper to each other. This gesture points towards a conspiracy or at least a lack of compassion and interest in Hoffmann's feelings.

Finally, in the last act, Nathanaël leaves Hoffmann alone, who falls on his knees in despair.

Hoffmann struggles to be a real part of a community or a group. 13 As the vampire is a solitary figure, the staging makes him become more and more motionless and do things that question his mental health.

At the beginning of the second act, Hoffmann is alone on the dark stage. He seems to walk with caution, putting his entire sole of foot on the ground, feeling his way. He walks slowly and is ultimately swallowed by darkness. The string puppets who put their whole foot on the ground when they walk convey Hoffmann's disoriented and lost state. Physical properties of the puppet's body gain importance as they give an insight in a character's mind, and the foreboding of this scene indicates that Hoffmann faces, and will lose against, an ominous force.

Then a little light illuminates the hero who converses with someone. The scene is still very dark, therefore his interlocutor only becomes apparent when the light gets brighter: it's a statue! However, Hoffmann is clearly turned towards it and makes gestures in its direction. He even tries to speak to another statue right next to the first one before noticing a third statue, ending the dialogue and disapprearing briefly from the stage. The statues are illuminated one by one as Hoffmann tries to speak to the the first and the second one. This is an eloquent staging element. It blurs the lines between animate and inanimate figures even before Olympia appears. The confusion is not limited to her but emerges as a larger concept. Thus, this scene can account for amalgamations of categories and characters. In fact, Hoffmann always needs to interact in order to exist. He never managed to distance himself from the women he loved: interestingly, in some versions of Offenbach's opera (Oeser 1977, 24) and also in Brunner's production, Hoffmann has "not a single monologue, solo aria without other listeners, or commentary on himself to himself" (idem) ["[K]einen einzigen Monolog, keine Solo-Arie ohne Lauscher, keine Aussage über sich selbst zu sich selbst"]. He only emerges as a character through others. In the middle of the second act, Hoffmann aligns with the three statues, standing as immobile as them. He stands between the first (the smallest) and the second statue (a little taller) so that their four heads form a perfect straight line guiding the eye towards the centre of the stage where the great eye pulsates. Hoffmann finds his place between these inanimate figures, leading to an exchange of characteristics: he becomes more like a statue whereas the statues seem more alive.

At the beginning of the Giulietta act, Hoffmann is in a dream-like state. He contemplates without moving. The servant has to shake his arm in order to guide him to Giulietta in the seashell.

Hoffmann then sings for the guests with a glass of wine in his hand. He wants to give a toast but nobody else has a glass. Hoffmann is excluded from the group. At the end of the act, the servant because her father fears for her health. She dies singing, accompanied by Miracle. Of the three family members, the mother is the last one to be seen on stage: at first, Antonia is illuminated, then Again, Crespel bends backwards into an improbable position stressing how ambiguous their relationship is. 15 The hypnosis performed by the doctor receives a particular expression with regard to string puppets when he juggles with his potions: the flacons fly away from his hands and come back like yo-yos. He uses them as if they were weapons, pushing Crespel into a corner. Then

Miracle lays his back on the piano, still juggling, like a possessed.

Some figures are conscious about their puppet identity. Crespel's servant Frantz looks at his bent leg, holds his arm above it and moves the arm and the leg as if he knew he was a string puppet, pulling an invisible string to move the leg up and down. Hoffmann himself makes a strange and unique movement in the first act: he rapidly moves his head to the left and to the right several times.

In E.T.A. Hoffmann's Sandman, Nathanaël is afraid of the mean man visiting his family's house who could unscrew his head. He imagines being treated like a puppet; emotional suffering and puppet identity are strongly linked. The intertextuality underlines how string puppets not only execute movements impossible for real humans, but also use their puppet identity, made explicit, to explore how individuals influence each other's inner life. When the puppet signals that it knows about its puppet identity, a loss of control becomes evident, closely linked to vampirisation and hypnosis.

In Brunner's production, numerous degrees of animation and life and death are displayed. Not everything that comes to life has a physical body (Kleinzach materializes; he becomes an animated character moving like Hoffmann's antagonists), and not every character has exactly one identity.

Julie Postel has identified the concept of an existence between bodies in contemporary puppet plays [START_REF] Postel | Marionnette infigurée et persistance de la matière[END_REF]. The possibility of the fantastic at the theatre has recently attracted the attention of 15 Once he tries to get up from the piano chair, but Miracle just has to turn in order to make him sit again. Then, when an eerie green light shines brighter and brighter and leaves green reflections on Miracle's cheek, Crespel makes an effort to attack the doctor and get him out of the house, but all he achieves is the extinction of the green light. Miracle's power cannot be defeated by Crespel. It is interesting to note that the green colour is linked to death. For example, in the Giulietta act, the green lights are the last thing we see on stage after Giulietta has been poisoned.

certain scholars (Bionda 2016) but Brunner's production challenges already existing concepts and theories because it requires a thorough reflection on the interiorisation of afterlife figures and because it transposes literary procedures on the scene, acknowledging the specific forms of presence and movement of the string puppet. Characters resemble other characters or objects; the exchange of characteristics could support the hypothesis that the puppet allows to soften up the strict distinctions between the figures of afterlife which are not located in one puppet body: they appear between characters or between a figure and an object. The string puppet dematerialises the devil and the vampire. The female figures also question the validity of categories and limits.

Olympia does so by her hybrid nature, Giulietta by stealing a part of an individual's identity, and Antonia by resembling her mother so much that she practically relives her life. Brunner's production can rearrange, reinterpret and rethink afterlife figures: string puppets become a particular mode of communication conferring them new opportunities, asking new questions through an already existing text.

  Antonia's father. Only then the portrait of the mother becomes visible. It is slightly inclined to the left, an early sign of her animation. When the mother moves, Miracle touches the frame and a light behind the upper part of the frame goes on. Rather than in a portrait, the mother seems to be in a cage or a showcase. During the examination performed by Miracle, Antonia is offstage. The doctor waves with his hands and mumbles Antonia's answers to his questions. Miracle's behaviour imitates a séance or a suggestion session. A very intense moment is created as a crystal blue light illuminates a door opening abruptly, then closing noisily. When the door is open, Antonia's voice is louder and strident. She is singing in pain because her favourite activity has been transformed into a harmful one. Finally, as Antonia lies dead on the floor and Hoffmann enters, he does not look at her or touch her. His friend Niklaus 14 guides him to a chair. Only Antonia's father exteriorises his grief by throwing himself over his daughter.The interactions between Crespel and Miracle show how the puppet, the vampirical figure and the reflection on mental states are intertwined. When the doctor arrives, they greet each other with la bise, an occasion for Miracle to get dangerously close to his neck -he could bite him like vampires do. From this point on, Miracle enchants Crespel who adopts an impossible posture defying the laws of physics, leaning backwards like a plank in an angle of about 45 degrees. Antonia's father behaves as if he was possessed. Then the doctor suddenly glides behind the chair Crespel is sitting in, although he has walked normally before. The string puppets can glide and therefore underline the supernatural or treacherous nature of a character. They achieve a unique effect making Miracle a supernatural afterlife figure flying just above the ground. Behind the chair, the doctor resembles a devil with horns because of his hair, but also a puppeteer because of his posture. They both walk with their head turned to the right, their arms slightly open and stretched out. Crespel is under Miracle's hypnotising influence. Whereas the doctor had to stay very close to his victim in the beginning, he can now control him over a greater distance, from one end of the stage to the other.

Fig. 2

 2 Fig. 2

"We are able to realize a whole number of Hoffmann's visionary shapes along with figments of Offenbach's lively imagination that are inevitably unattainable on 'full-size' opera stages. The irreality associated with the Tales of Hoffmann is thus a challenge"[START_REF] Ludwig | Some Reflections on the Production of 'The Tales of Hoffmann[END_REF]).

The German translation of Salzburg's bilingual libretto says "zum Teufel" (1), a significant choice, as Lindorf invokes the devil twice during his short interaction with Andres: "[V]a-t-en au diable !" ["Go to hell!"]. Translation mine.

"[L]e déplacement est consubstantiel au vampire" ("[movement defines the vampire]").Sangsue, 2018, 46.

The libretto indicates that Coppélius carries a bag. On the puppet stage, he does not have one, but carries everything in his coat pockets. Coats and capes are the traditional attire of vampires.

When he first appears, smoke rises from his body, as if he was a diabolical creature smelling like sulphur.

In the last act at the tavern, the young men form a group and sway simultaneously to the music, but Hoffmann stands farther away from everyone. Again, Hoffmann stands out as a solitary figure who cannot integrate any social group. As an outsider, he is vampirised and becomes the very same thing that has destroyed his life.

Niklaus has arrived on scene with a huge jump over the harpsichord. Those who can fly are suspicious figures.

kills Giulietta with a lime-green, poisoned drink. As both Hoffmann and his antagonist carry a glass, they become complementary figures. The limits between individuals are problematic; figures communicate and correspond, their characteristics overlap.

In the following scene, Dapertutto hides behind a statue and glides on stage. He shows himself and small lights at the statues go on. As a diabolical figure, he reminds us that "Lucifer" means "lightbringer". The light in Dapertutto's chest glows in a bright yellow when Giulietta appears. He is a torn figure with a burning desire, as the round-shaped light covers exactly the spot of his heart.

When his chest light is on, the lights of the statues are off. An incompatibility of those lights is suggested; as they cannot shine at the same time, an inner conflict emerges between Dapertutto's vulnerable side (he loves Giulietta) and his evil side (he is Lucifer). The staging makes afterlife figures more complex by giving them contradictory, tormenting feelings.

An interesting difference between the libretto and the staging concerns the duel between Hoffmann and Schlemil. According to the libretto, the former takes Dapertutto's sword and kills the latter, but on stage, Dapertutto makes two swords appear from the ground (where Hell might be, this underlines his diabolical nature). The swords kill Schlemil without Hoffmann ever touching them.

This makes Dapertutto stand out as a figure with magical powers he uses for his own interests, related to the devil as well as to the puppeteer, the vampire and the mesmerist, insofar as he controls other characters and objects.

How the characters' bodies are positioned to one another says a lot about their relationship. In Brunner's production, character dynamics expresses ambiguous feelings. This translates to parallel back and forth body movements: characters are trapped in a hesitation between attraction and repulsion. These contradictory emotions are vividly expressed when several characters approach and retreat again and again in a synchronised way, or when one character repeatedly approaches another one who backs off. This is what happens at Crespel's music room and is related to hypnosis that induces a state of mind similar to the victim of an afterlife figure. Antonia is forbidden to sing


A Rocher

lnfluence of major PWR radiochemical pollutants on dose rates and dosimetry -lmportance of efficiency of retention techniques

Introduction

In addition to usual systems contamination mostly due to materials generalised corrosion, some pollutions can lead to ( over-contamination >.

The major pollutants responsible for over-contarnination in EDF plants are : Co 60 coming from stellites degradation, Ag 110 m coming fromcontrol rod piercing, and Sb 724 coming from some pumps of auxiliary systems.

Given the importance of over-contamination with Íespect to dosimetry, representing 10 to 30 7o of outages dosimetry in the concerned units, a specific program is being set up in France, in the context of the ALARA project. The goal is to optimise diagnosis, surveillance, prevention and remedies to fight against penalising pollutions.

Pliolity is set on prevention.

A plan of action aimed at eradicating hot spots in PWR was set up in 1997. It consists of three complementary phases: prevention, surveillance and curative action.

Prevention

Prevention consists in avoiding the appearance of hot spots. It concerns all units and is based mainly on the following actions: ¡ limiting the use of cobalt based components in contact with effluent from the primary circuit ;

. removing stellite particles produced during certain maintenance operations (e.g. lapping) ;

. filtering effluent liable to carry hot spots outside the Reactor Building. The simplified diagram below illustrates the principles of the preventive filtration methods proposed (Fig. 1).

Programme to eradicate hot spots (Co 60) carried out in French PWR units

In most cases, hot spots [l] are due to particles of cobalt activated by a neutrons flux (Co 60) mainly from stellite surfaced equipment (rich in cobalt) [2] in the primary cooling circuit (valves, pumps, internals, etc.).

Cobalt 60 emits, in particular, two types of high energy gamma radiation t3l (Y radiation of 1.17 MeV and y radiation of 1.33 MeV given off by decay). Its decay period is 5.3 years ; it therefore generates high equivalent dose rates over many years if it is not removed from the cooling circuits.

The contribution of hot spots to shutdown dosimetry may appeal' to be marginal across all French reactors (2To 4 7a) but is significant (15 to25 7o) for the units affected. This excess dosimetry is a significant factor, particularly for the most exposed workers (the population with a dosimetry of 2 20 mSv/year). Approximately ten French units have been affected by this phenomenon over the last 12 years.

Surveillance

Surveillance is designed to inform the site, as early as possible, of the presence of hot spots (mapping) in order to take the appropriate measures to prevent their propagation and/or to eradicate them.

Curative actions

The curative actions proposed are gathered into a set of sheets, approved by operators and which are broken down into two parts. Actions to be taken in the reactor pool (whether filled with water or not) and actions to be taken on the cooling circuits (flushing, biological protcction, etc.).

Since no warning signs have yet been identified, to indicate the appearance of hot spots, it was decided to concentrate on preventive filtering, trapping hot spots as close as possible to their source to eliminate them.

Metastable silver 110 contamination in Tricastin NPP

Context

In some 900 MWe PWR units, the primary and auxiliary systems are subject to metastable silver 110

(Ag 110 m) contamination, due to silver 109 neutronic activation. Silver comes from control rods untightness (Ag, Cd, In parts).

Depending on the environment chemical and physical

properties (pH, redox potential), silver can be :

in ionic form in solution ; in colloïd form ; in metallic form.

Ag 110 m systems contamination can be very penalising since it impacts :

dosimetry integrated during maintenance operations ; shutdown schedule ; -control rods management (examination and rejection).

It can represent 5 to l5 o/o of a shutdown total dosimetry.

Systems most sensitive to silver contamination are :

the Residual Heat Removal System and CVCS coolers ; -the purification system downstream the CVCS demineralisers ;

the reactor cavity.

Prevenîive actions

Preventive actions to control silver root source are :

control rods examination ; and primary Ag I l0 m volumic activity Ag 110 m activity level likely to be observed as the primary coolant is oxygenated during the end of cycle The correlation between Ag l10m activity released during operation and activity integrated during primary coolant oxygenation allows to forecast the activity level likely to be observed during the end of cycle shutdown.

Remedial actions

Remedial actions depend on Ag I l0 m activity released during cycle operation or during shutdown. They depend on :

purification during end of cycle shutdown ; elution of activity confined in dead legs ; other complementary actions (system cleaning).

Primary coolant purification : Ag 110 m purification is performed on a dedicated ion exchanger operating at the maximum possible flowrate. Depending on the estimated Ag 110 m activity level and on the number of end of cycle shutdowns < lived > by the demineraliser, a second demineraliser can be set into service. Such a configuration allows to improve purification and to prevent from a possible unefficiency of the shutdown dedicated demineraliser.

Crossover legs drainage: Forecasting Ag I 10 m activity level allows to get ready, before shutdown, for a specific operation to decrease Ag I l0 m volumic activity confìned in the main primary system crossover legs, prior to reactor cavity filling and refueling.

Usually, crossover legs drainage is perforrned afier core unloading.

In the presence of Ag I l0 rn, scheduling a crossover legs rinsing prior to system drainage allows to elute the activity confined in them, and hence to lower the risks for contamination of the water during cavity filling. Ag I l0 m final volumic activity of the water in the reactor cavity can be reduced by a factor of 3, leading to limited dose integrated during fuel handling operations.

Complementary øctions : Other remedial actions can be undertaken depending on the types of operations to be performed during the end of cycle outage (cleaning of some systems parts, such as Residual Heat Removal System and CVCS coolers).

Coolers contamination occurs mostly during outages. Contamination of CVCS pumps and valves dowstream the purification system is mostly due to the lithiated CVCS demineraliser unefficiency in purifying Ag 110 m during opelation.

A pretreatment of the effluents prior to release can also be performed. It consists in precipitating silver together with aluminum or iron hydroxide. It allows to catch Ag I l0 m on filters prior to purification on demineralisers and prevents resins contamination. Such a treatment leads to Ag I l0 m final volumic activity lower than 0.5 MBq/t.

Antimony pollution

In some French PWRs, l22Sb and l'osb volume activity peaks are observed. These peaks, sometimes greater than I l0 GBq/t, can be higher than ssco peaks. They impact dosirnetry and waste management.

l22sb and 'tosb are two multigamma emitters with radioactive half-life of 2.7 days and 60.2 days lespectively. Thanks to its radioactive half-life, t"Sb has a low impact on health physics, but it allows to determine antimony sources.

Antimony soLtrces

Two sources have been identified in French PWRs :

L Beryllium-Antimony source rod failure, but there are source rods only for the first reactor cycles ; IL bearing wear of pumps in the Boron Recycle System. These bearings contain graphite impregnated with about l0 7o of antimony' Some activated antimony grams can explain such peaks of l22sb and ''asb. So, about 7 grams can lead to a 'tosb peak of 100 GBq/t.

Antimony pollution impact

This important antimony released activity has an impact on dosimetry and waste management.

'2asb deposited activities on the out-of-core surfaces increase a lot as it is shown in table 1.

Table I : RCS recontamination after a 70 GBq/t b peak of Thus, the l2aSb contribution to the dose rate can reach l0 7o in some parts of circuits. And the dosimetry for maintenance during a plant shutdown can increase by about 5 7o because of the l2asb contamination. 481 Remedies To avoid antimony pollution, the incriminated pump bearings have been replaced by pump bearings without antimony. As it is indicated in the table below (table 2), that allows to make the l24sb peaks decrease. During oxygenation, antimony is soluble (< 0.45 pm). If an antimony pollution occurs, to improve purification by the ion exchange resins, reactor coolant lithium content has to be inferior to 1 ppm [4]. In order to obtain an acid medium, purification can be carried out thanks to nonsaturated lithium cation bed demineralizers of the Boron Recycle System. Nevertheless, it should be noted that an acid reactor coolant seems to increase antimony deposition on the primary circuit surfaces [4].

Conclusion

The presence of some pollutions in the systems (Co 60, Ag 110 m, Sb 124) leads to significant increase of shutdown dosimetry (10 to 30 Vo) in affected units.

Early detection of pollutants, in operation as well as during shutdown, is the best way to set up appropriate solutions. The early replacement of possibly polluting materials, the implementation of a specific chemistry or the choice of specific resins or filters are among the available answers to minimise over-contamination risks in PWR units.

At the present time, an optimisation of the chemical and physical properties of the environment during shutdown is being examined, in order to minimise contamination. Studies on filters and resins types and on the impact of purification flowrate are being performed.

Minimising systems contamination by retaining pollutants such as Ag 110 m is of the major importance for EDF.

Fig. 1 :

 1 Fig. 1 : Preventive filtration methods

  shutdown can be forecasted by correlating Ag 110 m mean activity during operation to Ag 110 m maximum activity at end of cycle shutdown oxygenation (fig a). Hence, Ag 1l0m maximum volumic activity as the prilnary coolant is oxygenated matches the total activity released during the outage.

  r:1"""i;", 'n 't r¡r re

Fig

  Fig. a : Ag I l0 m activity in Tricastin 2 unit

  Bearing replacement during the 14 cycle

  References tll A. Rocher et al, Programme to eradicate hot spots (Co-60) carried out in French PWR units, JAIF International Conference on W'ater Chemistry in Nuclear Power Plants, Kashiwasaki 1998 tzl S. Anthoni, Synthèse Points Chauds Origine et Comportement, Note 96041 -CEA/DNR/DEC/SECA/L'IC 1996 t3l K-H. Neeb, The Radio of Nuclear Power Plants with Light Water Reactors p 153-154, Walter de Gruyter, Berlin, New York 1997 Í41 Hauser, "Retour d'expérience sur la pollution en antimoine des tranches du Bugey,

  It was the case inTricastin 2, at the end of cycle 17 shutdown, as Ag 110 m volumic activity reached 16000M8q/t during primary coolant oxygenation. Such an activity called for It consists in comparing the on-going cycle volumic activity with that of the previous cycles.

	monitoring
	during operation.
	Control rods examination .' Eddy current and
	ultrasonic non destructive control rod examination is
	performed every three cycles. Doubtful rods or rods with
	piercing signals are systematically replaced with either
	chrome or nitride coated rods.
	However', such examinations do not allow for detection of fast evolution kinetics flaws.

unscheduled examination of the control rods, revealing an early piercing of l0 standard rods, which were rejected and remplaced with coated rods.

Primary Ag I l0 m volumic activity monitoring : A tight monitoring of Ag I 10 m volumic activity in operation can allow to anticipate an anomaly before the shutdown WATER CHEMISTRY OF NUCLEAR REACTOR SYSTEMS 8 transient.

Water Chemistry of Nuclear Reacfor Sysfems 8,BNES, 2001