Philippe Charpentier
email: philippe.charpentier@math.u-bordeaux1.fr

Yves Dupain
email: yves.dupain@math.u-bordeaux1.fr

Modi Mounkaila
email: modi.mounkaila@yahoo.fr

ESTIMATES FOR SOLUTIONS OF THE ∂ -EQUATION AND APPLICATION TO THE CHARACTERIZATION OF THE ZERO VARIETIES OF THE FUNCTIONS OF THE NEVANLINNA CLASS FOR LINEALLY CONVEX DOMAINS OF FINITE TYPE

Keywords: 2000 Mathematics Subject Classification. 32F17, 32T25, 32T40 lineally convex, finite type, ∂ -equation, Nevanlinna class

In the late ten years, the resolution of the equation ∂ u = f with sharp estimates has been intensively studied for convex domains of finite type in C n by many authors. Generally they used kernels constructed with holomorphic support function satisfying "good" global estimates. In this paper, we consider the case of lineally convex domains. Unfortunately, the method used to obtain global estimates for the support function cannot be carried out in that case. Then we use a kernel that does not gives directly a solution of the ∂ -equation but only a representation formula which allows us to end the resolution of the equation using Kohn's L 2 theory.

As an application we give the characterization of the zero sets of the functions of the Nevanlinna class for lineally convex domains of finite type.

INTRODUCTION

The general notion of extremal basis and the class of "geometrically separated" domains has been introduced in [START_REF]Extremal basis, geometrically separated domains and applications[END_REF]. For such domains it is proved that if there exist "good" plurisubharmonic functions, in which case the domains are called "completely geometrically separated", then sharp estimates on the Bergman and Szegö projections and on the classical invariants metrics can be obtained.

Moreover, using the description of the complex geometry of lineally convex domains of finite type initiated in [START_REF] Conrad | Anisotrope optimale Pseudometriken für lineal konvex Gebeite von endlichem Typ (mit Anwendungen)[END_REF] and the construction of a local support function described in [START_REF] Diederich | Lineally convex domains of finite type: holomorphic support functions[END_REF], it is shown, already in [START_REF]Extremal basis, geometrically separated domains and applications[END_REF], that every lineally convex domain of finite type is completely geometrically separated.

The present paper is a continuation of the study of complex analysis in such domains. We are now interested in the problem of the characterization of the zero sets of functions in the Nevanlinna class. The main result obtained concerns the class of lineally convex domains of finite type and generalizes the results obtained in the case of convex domains ([BCD98, Cum01b, DM01]):

Theorem 1.1. Let Ω be a bounded lineally convex domain of finite type in C n with smooth boundary. Then a divisor in Ω can be defined by a function of the Nevanlinna class of Ω if and only if it satisfies the Blaschke condition.

The general scheme of the proof is identical to the one used in the convex case and consists in three steps. First, for the general case of geometrically separated domains, we prove some "Malliavin conditions" on closed positive (1, 1)-currents Θ and then we solve the equation dw = Θ with good estimates. The third step, which solves the ∂ -equation for (0, 1)-form with L 1 estimates on the boundary, is only done in the case of lineally convex domains of finite type: Theorem 1.2. Let Ω be a bounded lineally convex domain of finite type in C n with smooth boundary. Let f be a (0, 1)-form in Ω whose coefficients are C 1 (Ω) functions and which is ∂ -closed. Then there exists a solution of the equation ∂ u = f , smooth on Ω and continuous on Ω such that u L 1 (∂ Ω) ≤ C||| f ||| k (see Section 2.1 formula (2.3)), the constant C depending only on Ω. In other words, there exists a solution of the equation ∂ b u = f , in the sense of [Sko76], in L 1 (∂ Ω).

SOLUTIONS FOR THE ∂ -EQUATION FOR LINEALLY CONVEX DOMAINS OF FINITE TYPE

First of all, we recall the definition of lineally convex domain: Definition 2.1. A domain Ω in C n , with smooth boundary is said to be lineally convex at a point p ∈ ∂ Ω if there exists a neighborhood W of p such that, for all point z ∈ ∂ Ω ∩W , z + T 10 z ∩ (D ∩W ) = / 0, where T 10 z is the holomorphic tangent space to ∂ Ω at the point z. Furthermore, we always suppose that ∂ Ω is of finite type at every point of ∂ Ω ∩ W . Shrinking W if necessary, we may assume that there exists a C ∞ defining function ρ for Ω and a number η 0 > 0 such that ∇ρ(z) = 0 at every point of W and the level sets {z ∈ W such that ρ(z) = η}, -η 0 ≤ η ≤ η 0 , are lineally convex of finite type.

As we want to obtain global results, we need these properties at every boundary point. Thus, in all our work, by "lineally convex domain" we mean a bounded smooth domain having a (global) defining function satisfying the previous hypothesis at every point of ∂ Ω.

In Section 2.1 we define a punctual anisotropic norm for forms, . k , related to the geometry of the domain (formula (2.2)). With this notation, the main goal of this Section is to prove the following reformulation of Theorem 1.2 for (0, q)-forms:

Theorem 2.1. Let Ω be a smooth bounded lineally convex domain of finite type in C n . Then there exists a constant C > 0 such that, for any smooth ∂ -closed (0, q)-form f , 1 ≤ q ≤ n -1, on Ω there exists a solution u of the equation ∂ u = f , continuous on Ω, such that

∂ Ω |u(z)| dσ (z) ≤ C Ω f (z) k dV (z).
Except for the case of finite type domains in C 2 where such an estimate was proved by D. C. Chang, A. Nagel and E. M. Stein ( [START_REF] Chang | Estimates for the ∂ -Neumann problem for pseudoconvex domains in C 2 of finite type[END_REF]) for the ∂ -Neumann problem, this kind of result was always proved using explicit kernels solving the ∂ -equation. The first result was obtained, independently, by G. M. Henkin and H. Skoda for strictly pseudo-convex domains ( [START_REF] Henkin | Solutions with bounds for the equations of H. Lewy and Poincaré-Lelong. Construction of functions of Nevanlinna class with given zeros in a strongly pseudoconvex domain[END_REF][START_REF] Skoda | Valeurs au bord pour les solutions de l'opérateur ∂ , et caractérisation des zéros des fonctions de la classe de Nevanlinna[END_REF]). Afterward, some generalizations to special pseudo-convex domains of finite type (in dimension n ≥ 3) were obtained by several authors. For example, the case of complex ellipsoids was obtained by A. Bonami and Ph. Charpentier ([BC82]), and, probably the most notable result, the case of convex domains of finite type by J. Bruna Here, we consider the more general case of lineally convex domains of finite type. Our starting point is similar to the one used in [START_REF]Zero sets of functions in the Nevanlinna or the Nevanlinna-Djrbachan classes[END_REF] and [START_REF] Diederich | Zero varieties for the Nevanlinna class on all convex domains of finite type[END_REF]. We try to construct a kernel solving the ∂ -equation following the method described in the classical paper of B. Berndtsson and M. Andersson [START_REF] Berndtsson | Henkin-Ramirez formulas with weight factors[END_REF]. Such kernel is constructed using two forms, s(z, ζ ) and Q(z, ζ ) satisfying some conditions. In particular Q(z, ζ ) is supposed to be holomorphic in z. In many works using these constructions, the forms s and Q (or only Q) are constructed using a holomorphic support function for the domain. In the case of lineally convex domains of finite type such support functions have been constructed by K. Diederich and J. E. Fornaess in [START_REF] Diederich | Lineally convex domains of finite type: holomorphic support functions[END_REF]. Let us denote by

S 0 = ∑ Q 0 i (z, ζ ) (z i -ζ i ) this support function.
If we want to use S 0 to define s and/or Q, a problem appears immediately: some precise global estimates of S 0 are necessary since this function appears in the denominators of the kernels, and Diederich and Fornaess result gives only local estimates (i.e. when the two points z and ζ are close and close to the boundary of the domain). This problem has been noticed previously in the case of convex domains by W. Alexandre in [START_REF] Mounkaila | Construction d'une fonction de support à la Diederich-Fornaess[END_REF] where a modification of the support function is done. Unfortunately, this modification cannot apply for lineally convex domains, the convexity being strongly used to solve a division problem with estimates. Another way to construct a kernel without support function, introduced by A. Cumenge for convex domains in [START_REF]Zero sets of functions in the Nevanlinna or the Nevanlinna-Djrbachan classes[END_REF], is to use the Bergman kernel with the estimates obtained in [START_REF] Mcneal | Estimates on Bergman Kernels of Convex Domains[END_REF]. These needed estimates on the Bergman kernel have been obtained for lineally convex domains in [START_REF]Extremal basis, geometrically separated domains and applications[END_REF] but, once again, the method cannot be carried out for lineally convex domains for the same reason.

Thus we start with the method of Berndtsson and Andersson with Q constructed with S 0 , but the form Q being holomorphic in z only when the two points z and ζ are close (and close to the boundary). Then the construction does not give a kernel solving the ∂ -equation but a representation formula of the following form: if f is a (0, q)-form smooth in Ω, there exist kernels

K(z, ζ ), K 1 (z, ζ ) and P(z, ζ ) such that f (z) = ∂ Ω f (ζ ) ∧ K(z, ζ ) + Ω ∂ f (ζ ) ∧ K 1 (z, ζ ) + Ω f (ζ ) ∧ P(z, ζ ).
In this formula one important point is that, by construction, the kernel

P is C ∞ Ω × Ω . If f is ∂ -closed then the form g = Ω f (ζ ) ∧ P(z, ζ
) is also ∂ -closed, and, by the regularity of P (Lemma 2.2), for all integer r, the Sobolev norm g W r of order r is controlled by C r f L 1 (Ω) . Then, using Kohn's theory ( [START_REF] Kohn | Global regularity for ∂ on weakly pseudo-convex manifolds[END_REF]), it is possible, using a linear operator, to solve the equation ∂ v = g with an estimate of the form v W r ≤ C r f L 1 (Ω) (Lemma 2.3). By Sobolev Lemma, choosing r sufficiently large (depending only on the dimension), there exists a constant C such that, if f ∈ L 1 (Ω), this solution v is continuous on

Ω and v L 1 (∂ Ω) ≤ C f L 1 (Ω) .
Finally, to obtain a solution of the equation ∂ u = f , given by a linear operator, satisfying the desired estimate it suffices to estimate the integral Ω f (ζ ) ∧ K(z, ζ ) which can be done, as we will see, using only the local estimates of the support function S 0 given in [START_REF] Diederich | Lineally convex domains of finite type: holomorphic support functions[END_REF].

Geometry and local support function.

2.1.1. Geometry of lineally convex domains of finite type. Adapting the construction made by J. McNeal for convex domains ( [START_REF] Mcneal | Estimates on Bergman Kernels of Convex Domains[END_REF]) to the case of lineally convex domains of finite type, M. Conrad defined, in [START_REF] Conrad | Anisotrope optimale Pseudometriken für lineal konvex Gebeite von endlichem Typ (mit Anwendungen)[END_REF], the geometry of these domains and, in particular, the notion of extremal basis in this context (note that in his construction the basis are not maximal but minimal, see [START_REF] Hefer | Extremal bases and Hölder Estimates for ∂ on Convex Domains of Finite Type[END_REF][START_REF] Nikolov | On different extremal bases for C-convex domains[END_REF] for more details). Here we will only recall the results which are useful for our purpose. A more detailed exposition is given in [START_REF] Diederich | Hölder estimates on lineally convex domains of finite type[END_REF].

For ζ close to ∂ Ω and ε ≤ ε 0 , ε 0 small, define, for all unitary vector v,

τ (ζ , v, ε) = sup {c such that |ρ (ζ + λ v) -ρ(ζ )| < ε, ∀λ ∈ C, |λ | < c} .
Note that, if v is tangent to the level set of ρ passing through ζ , τ (ζ , v, ε) ε 1/2 (with uniform constant in ζ , v and ε) and that, Ω being of finite type ≤ 2m, τ (ζ , v, ε) ε 1/2m . Let ζ and ε be fixed. Then, an orthonormal basis (v 1 , v 2 , . . . , v n ) is called (ζ , ε)-extremal (or ε-extremal, or simply extremal) if v 1 is the complex normal (to ρ) at ζ , and, for i > 1, v i belongs to the orthogonal space of the vector space generated by (v 1 , . . . , v i-1 ) and minimizes τ (ζ , v, ε) in that space. In association to this extremal basis, we denote

τ(ζ , v i , ε) = τ i (ζ , ε).
Note that there may exist many (ζ , ε)-extremal bases but they all give the same geometry we recall now.

With these notations, one defines polydiscs AP ε (ζ ) by

AP ε (ζ ) = z = ζ + n ∑ k=1 λ k v k such that |λ k | ≤ c 0 Aτ k (ζ , ε) ,
c 0 depending on Ω, P ε (ζ ) being the corresponding polydisc with A = 1 and we also define

d(ζ , z) = inf {ε such that z ∈ P ε (ζ )} .
The fundamental result here is that d is a pseudo-distance which means that, ∀α > 0, there exist constants c(α) and C(α) such that (2.1)

c(α)P ε (ζ ) ⊂ P αε (ζ ) ⊂ C(α)P ε (ζ ) and P c(α)ε (ζ ) ⊂ αP ε (ζ ) ⊂ P C(α)ε (ζ ).
We insist on the fact that this pseudodistance is well defined and is independent of the choice of the extremal bases.

We will make use of the following properties:

(1) Let w = (w 1 , . . . , w n ) be an orthonormal system of coordinates centered at ζ . Then

∂ |α+β | ρ(ζ ) ∂ w α ∂ wβ ε ∏ i τ (ζ , w i , ε) α i +β i , |α + β | ≥ 1.
(2) Let ν be a unit vector. Let

a ν αβ (ζ ) = ∂ α+β ρ ∂ λ α ∂ λ β (ζ + λ ν) |λ =0 . Then ∑ 1≤|α+β |≤2m a ν αβ (ζ ) τ(ζ , ν, ε) α+β ≃ ε. (3) If (v 1 , . . . , v n ) is a (ζ , ε)-extremal basis and γ = ∑ n 1 a j v j = 0, then 1 τ(ζ , γ, ε) ≃ n ∑ j=1 a j τ j (ζ , ε) . (4) If v is a unit vector then: (a) z = ζ + λ v ∈ P δ (ζ ) implies |λ | τ(ζ , v, δ ), (b) z = ζ + λ v with |λ | ≤ τ(ζ , v, δ ) implies z ∈ CP δ (ζ ). (5) τ 1 (ζ , ε) = ε, and, for j > 1 and λ ≥ 1, λ 1/m τ j (ζ , ε) τ j (ζ , λ ε) λ 1/2 τ j (ζ , ε).
Remark. Every lineally convex domain of finite type is completely geometrically separated and the pseudo-distance defined here is equivalent to the one defined in [CD08] using tangent complex vector fields (see Section 7.1 of [START_REF]Extremal basis, geometrically separated domains and applications[END_REF] for some details).

With these notations, we define a punctual anisotropic norm . k for (0, q)-forms with functions coefficients f by

(2.2) f (z) k = sup v i =1 f ; v 1 , . . . , v q (z) ∑ q i=1 k (z, v i )
,

where k (z, v) = δ ∂ Ω (z) τ(z,v,δ ∂ Ω (z))
, δ ∂ Ω (z) being the distance of z to the boundary of Ω. Note that this definition generalizes the definition given in [START_REF]Extremal basis, geometrically separated domains and applications[END_REF] for (0, 1)-forms. Moreover, in the coordinate system associated to an (z, δ (z))-extremal basis, we have

d zI k ≃ min i∈I τ i (z,δ ∂ Ω (z)) δ ∂ Ω (z) , and, if f = ∑ I a I d zI , f k ≃ sup I |a I | min i∈I τ i (z, δ ∂ Ω (z)) δ ∂ Ω (z) .
If f is a (0, q)-form with continuous coefficients, f (z) k is also continuous and we define it's |||.||| k norm by

(2.3) ||| f ||| k = Ω f (z) k dV (z).
2.1.2. The holomorphic support function. In [START_REF] Diederich | Lineally convex domains of finite type: holomorphic support functions[END_REF] the following result is proved:

Theorem 2.2 (K. Diederich & J. E. Fornaess).
Let Ω be a bounded lineally convex domain in C n of finite type 2m with C ∞ boundary.

Then there exist a neighborhood W 0 of the boundary of Ω and, for any

ε > 0 small enough a function S 0 (z, ζ ) ∈ C ∞ (C n ,W 0 )
which is a holomorphic polynomial of degree 2m in z for any ζ ∈ W 0 fixed, such that S 0 (ζ , ζ ) = 0, satisfying the following precise properties: Let M, K > 0 be chosen sufficiently large and ε > 0 sufficiently small. Choose l ζ a family of affine unitary transformations on W 0 translating ζ to 0 and rotating the complex normal n ζ to ρ at ζ to the vector (1, 0, . . . , 0). Then there exists, on W 0 , a family of

holomorphic polynomials A ζ , A ζ (0) = 0, such that, if Φ ζ is defined by Φ -1 ζ (z) 1 = z 1 1 -A ζ (z) , Φ -1 ζ (z) k = z k , k = 2, . . . , n, then (2.4) S 0 l ζ • Φ ζ (ξ ), ζ = ξ 1 + Kξ 2 1 -ε 2m ∑ j=2 M 2 j σ j ∑ |α|= j α=(0,α 1 ,...,α n ) 1 α! ∂ ρ ζ (0) ∂ ξ α ξ α where ρ ζ (ξ ) = ρ l ζ • Φ ζ (ξ ) -ρ(ζ ) and σ j =    1 for j = 0 mod 4 -1 for j = 2 mod 4 0 otherwise .
Moreover, there exist d = d(ε) > 0 and c > 0 such that, if n ζ is the unit real exterior normal to ρ at ζ , for (w 1 , w 2 ) ∈ C 2 and t a unit vector in the holomorphic tangent space to ρ at ζ , for |w| < d, the following estimate holds

(2.5) ℜeS 0 ζ + w 1 n ζ + w 2 t, ζ ≤ ρ ζ + w 1 n ζ + w 2 t -ρ(ζ ) h ζ + w 1 n ζ + w 2 t -εc n ∑ j=2 P j ζ ,t |w 2 | j ,
where h is a positive function bounded away from 0,

P j ζ ,t (w) = P j ζ ζ + w 1 n ζ + w 2 t , with P j ζ (z) = ∑ |α|+|β |= j 1 α!β ! ∂ j ρ(ζ ) ∂ z α ∂ zβ (z -ζ ) α z -ζ β
and, for any polynomial P = ∑ a αβ z α zβ , P = ∑ a αβ .

Remark. In the above Theorem the function S 0 is globally defined for ζ ∈ W 0 , and (2.4) is independent of the choice of l ζ . In particular, as it is stated in [START_REF] Diederich | Lineally convex domains of finite type: holomorphic support functions[END_REF], if we restrict ζ to a small open set in W 0 , the functions l ζ , h and A ζ can be chosen C ∞ in that set (with respect to the two variables ζ and z).

Koppelman formulas.

With the notations used for the holomorphic support function S 0 , we choose

R < d such that A ζ (z) < 10 -1 if |z -ζ | < R and, reducing η 0 if necessary, we may suppose that δ ∂ Ω (ζ ) < η 0 implies ζ ∈ W 0 . Let us define two C ∞ functions χ 1 (z, ζ ) = χ (|z -ζ |) and χ 2 (z) = χ (δ ∂ Ω (z)
) (where δ ∂ Ω denotes the distance to the boundary of Ω) where χ and χ are

C ∞ functions, 0 ≤ χ, χ ≤ 1, such that χ ≡ 1 on [0, R /2] and χ ≡ 0 on [R, +∞[ and χ ≡ 1 on [0, η 0/2] and χ ≡ 0 on [η 0 , +∞[. Then we define χ(z, ζ ) = χ 1 (z, ζ )χ 2 (ζ ) and S(z, ζ ) = χ(z, ζ )S 0 (z, ζ ) -(1 -χ(z, ζ ))|z -ζ | 2 = n ∑ i=1 Q i (z, ζ ) (z i -ζ i ) .
Now we define the two forms s and Q used in [START_REF] Berndtsson | Henkin-Ramirez formulas with weight factors[END_REF] in the construction of the Koppelman formula by

s(z, ζ ) = n ∑ i=1 ζ i -z i d (ζ i -z i ) and Q(z, ζ ) = 1 K 0 ρ(ζ ) n ∑ i=1 Q i (z, ζ )d (ζ i -z i ) ,
where K 0 is a large constant chosen so that

(2.6) ℜe ρ(ζ ) + 1 K 0 S(z, ζ ) < ρ(ζ ) 2 .
Notice that ℜeS ≤ χℜeS 0 ≤ -Cρ(ζ ), by (2.5), so it suffices to take K 0 ≥ 2C . Remark also that, if ζ ∈ ∂ Ω, (2.5) implies ℜeS(z, ζ ) < 0 for z ∈ Ω.

We point out also that Q is not holomorphic in z and that s satisfies

|z -ζ | 2 = | s, z -ζ | ≤ C |z -ζ |, z, ζ ∈ Ω.
Following the construction done in [START_REF] Berndtsson | Henkin-Ramirez formulas with weight factors[END_REF], with G(ξ ) = 1 ξ , we obtain two kernels

(2.7) K(z, ζ ) = C n n-1 ∑ k=0 (n -1)! k! G (k) 1 K 0 ρ(ζ ) S(z, ζ ) + 1 s(z, ζ ) ∧ (dQ) k ∧ (ds) n-k-1 |ζ -z| 2(n-k) and (2.8) P(z, ζ ) = C ′ n G (n) 1 K 0 ρ(ζ ) S(z, ζ ) + 1 (dQ) n
giving the following Koppelman formula:

For all (0, q)-form f , with C 1 ( Ω) coefficients, we have, for z ∈ Ω,

(2.9)

f (z) = ∂ Ω f (ζ ) ∧ K 0 (z, ζ ) + (-1) q+1 ∂ z Ω f (ζ ) ∧ K 1 (z, ζ ) + (-1) q Ω ∂ f (ζ ) ∧ K 2 (z, ζ ) - Ω f (ζ ) ∧ P(z, ζ )
where K 0 (resp. K 1 , resp. K 2 , resp. P) is the component of K of be-degree (0, q) in z and (n, nq -1) in ζ (resp. (0, q -1) in z and (n, nq) in ζ , resp. (0, q) in z and (n, nq -1) in ζ , resp. (0, q) in z and (n, nq) in ζ ).

Moreover, by definition of S, G

(k) (S(z, ζ ) + 1) = c k ρ(ζ ) k+1 1 K 0 S(z,ζ )+ρ(ζ )
k+1 , and, for ζ ∈ ∂ Ω, K 0 (z, ζ ) = 0 so that the first integral in the Koppelman formula disappears, and if f is ∂ -closed (2.9) becomes (2.10)

f (z) = (-1) q+1 ∂ z Ω f (ζ ) ∧ K 1 (z, ζ ) -g, with g = Ω f (ζ ) ∧ P(z, ζ ) and g is ∂ -closed.
To be able to estimate the kernels K 1 and P, we need a fundamental estimate for ρ(ζ

) + 1 K 0 S(z, ζ ) .
Lemma 2.1. There exists K 0 such that, for

ζ ∈ P i ε (z) = P 2 -i ε (z) \ P 2 -i-1 ε (z), we have: (1) ρ(ζ ) + 1 K 0 S(z, ζ ) 2 -i ε, (z, ζ ) ∈ Ω × Ω; (2) 1 K 0 S(ζ , z) 2 -i ε, (z, ζ ) ∈ ∂ Ω × Ω. Proof. Clearly (2) is a special case of (1) (because d(z, ζ ) ≃ d(ζ , z)). Let us prove (1). First note that, by (2.5), if z = ζ + w 1 n ζ + w 2 t, ℜe ρ(ζ ) + 1 K 0 S(z, ζ ) ≤ χ(z, ζ ) c 1 ρ(z) + 1 2 ρ(ζ ) -(1 -χ(z, ζ ))|z, ζ | 2 -c 2 n ∑ j=2 P j ζ ,t |w 2 | j , and then, if |z -ζ | > ε 0 , ℜe ρ(ζ ) + 1 K 0 S(z, ζ ) -c 0 (ε 0 )
, and it is enough to prove the Lemma for ε < ε 0 small enough. Let us denote ε ′ = 2 -i-1 ε and let (w 1 , . . . , w n ) be an ε ′ -extremal basis at ζ . Let us write

z = ζ + ∑ n i=1 λ i w i = ζ + λ 1 w 1 + v 1 = ζ + λ 1 w 1 + v 1 v. Remark that, by (2.6), the result is trivial if |ρ(ζ )| ε ′ . Let us suppose |ρ(ζ )| ≪ ε ′ .
Let 0 < k 0 ≪ 1 be a real number which will be fixed later. Suppose first that, ∀i ≥ 2,

|λ i | < k 0 τ i (ζ , ε ′ ).
Then |λ 1 | ε ′ (property (4) of Section 2.1.1), and, as

m ∑ j=2 M 2 j σ j ∑ |α|= j α 1 =0 ∂ ρ ζ (0) ∂ ξ α ξ α = m ∑ j=2 M 2 j σ j ∑ |α|= j α 1 =0 ∂ ρ(z) ∂ w α |w=0 (λ w) α , by (1) of Section 2.1.1, we have m ∑ j=2 M 2 j σ j ∑ |α|= j α 1 =0 ∂ ρ(z) ∂ w α |w=0 (λ w) α k 2 0 ε ′ .
Using formula (2.4), this gives

ρ(ζ ) + 1 K 0 S 0 (z, ζ ) ≥ ρ(ζ ) + λ 1 1 -A ζ (z) K 0 + Kλ 2 1 1 -A ζ (z) 2 K 0 -C 1 k 2 0 ε ′ ≥ ρ(ζ ) + λ 1 1 -A ζ (z) K 0 -C 2 ε ′ k 2 0 + ε ′ ,
the last inequality following the fact that |λ 1 | ε ′ (property (4) of Section 2.1.1). As A ζ (z) < 10 -1 , it follows that

ρ(ζ ) + 1 K 0 S 0 (z, ζ ) ε ′ ,
if k 0 and ε 0 are chosen sufficiently small. We now fix ε 0 . As ℜe ρ(ζ

) + 1 K 0 S 0 (z, ζ ) < 0, we have ρ(ζ ) + 1 K 0 S(z, ζ ) χ(z, ζ ) ρ(ζ ) + 1 K 0 S 0 (z, ζ ) + (1 -χ(z, ζ )) |z -ζ | 2 ,
and the inequality

|z -ζ | 2 v 1 2 τ(ζ , v, ε) 2 ε ′ (properties (
3) and (5) of Section 2.1.1) proves the Lemma in that case. Suppose then there exists i ≥ 2 such that |λ i | ≥ k 0 τ i (ζ , ε ′ ). By (4) of Section 2.1.1, we have v 1 ≃ τ(ζ , v, ε ′ ), and by (2) of Section 2.1.1, we get (v being tangent to ρ

= ρ(ζ ) at ζ ) ∑ 2≤|α+β |≤m a v αβ (ζ ) τ(ζ , v, ε) α+β = ∑ 1≤|α+β |≤m a v αβ (ζ ) τ(ζ , v, ε) α+β ≃ ε ′ ,
and (2.5) implies

ℜeS 0 (z, ζ ) Cρ(ζ ) -c 1 ε ′ ,
and we conclude by the same argument on |z -ζ | 2 .

Lemma 2.2. If q ≥ 1, all the derivatives, in z, of P(z, ζ ) are uniformly bounded in Ω × Ω.

Proof. Recall P(z, ζ ) = c n ρ(ζ ) n+1 1 K 0 S(z,ζ )+ρ(ζ ) n+1 (dQ) n . If δ ∂ Ω (ζ ) > η 0/2, this is clear by (2.6). Suppose δ ∂ Ω (ζ ) ≤ η 0/2. If |z -ζ | < R /2 then Q is holomorphic
in z and the component of (dQ) n of be-degree (0, q) in z (recall we suppose q ≥ 1) is identically 0 which implies

P = 0. If |z -ζ | ≥ R /2, the preceding Lemmas, give 1 K 0 S(z, ζ ) + ρ(ζ ) (R) 2m (because, for unitary v, τ(z, v, ε) ε 1/2m
), and the Lemma follows easily.

Lemma 2.3. For each positive integer s there exists a linear operator L s and a constant C(s) such that, if g = Ω f (ζ ) ∧ P(z, ζ ) is the ∂ -closed form of (2.10) there exists a solution v s = L s (g) to the equation ∂ v s = g satisfying the estimate

v s s ≤ C(s) f L 1 (Ω) ,
where . s denotes the Sobolev norm of index s in Ω.

Proof. This is an immediate consequence of the preceding Lemma and the results on the weighted ∂ -Neumann problem obtained by J. J. Kohn in [START_REF] Kohn | Global regularity for ∂ on weakly pseudo-convex manifolds[END_REF].

As explained at the end of the introduction of Section 2 this last Lemma shows that to obtain sharp estimates for solution of the ∂ -equation in lineally convex domains of finite type, like the one stated in Theorem 2.1, it is enough to prove that the form

z → Ω f (ζ ) ∧ K 1 (z, ζ ) (c.f.(2.10)) is continuous on Ω and that Ω f (ζ ) ∧ K 1 (z, ζ ) L 1 (∂ Ω) ≤ C||| f ||| k .
This will be done in the next section.

2.3. Proof of Theorem 2.1. By formulas (2.7) and (2.9), we have

K 1 (z, ζ ) = n-1 ∑ k=n-q C ′ k ρ(ζ ) k+1 s ∧ ∂ ζ Q n-q ∧ (∂ zQ) k+q-n ∧ (∂ zs) n-k-1 |z -ζ | 2(n-k) 1 K 0 S(z, ζ ) + ρ(ζ ) k+1 .
A priory, formula (2.9) is only valid for z ∈ Ω. But, as noted by H. Skoda in [START_REF] Skoda | Valeurs au bord pour les solutions de l'opérateur ∂ , et caractérisation des zéros des fonctions de la classe de Nevanlinna[END_REF], the form

Ω f (ζ ) ∧ K 1 (z, ζ )
is continuous in Ω if the kernel K 1 satisfies a condition of uniform integrability:

(2.11)

Ω∩P ε (z) K 1 (z, ζ ) dV (ζ ) = O ε 1/m ,
uniformly for z satisfying δ ∂ Ω (z) ≤ ε and ε small enough. Under these hypothesis (ζ

∈ P ε (z)), Q(z, ζ ) = 1 K 0 ρ(ζ ) ∑ n i=1 Q i (z, ζ )d (ζ i -z i ) is holomorphic in z and K 1 is reduced to K 1 (z, ζ ) = c ρ(ζ ) n-q+1 s ∧ ∂ ζ Q n-q ∧ (∂ zs) q-1 |z -ζ | 2q 1 K 0 S(z, ζ ) + ρ(ζ ) n-q+1 .
To prove (2.11), we use the coordinate system (ζ 1 , . . . , ζ n ) associated to the (z, δ ∂ Ω (z))-extremal basis and the following estimates:

Lemma 2.4. For z close to ∂ Ω, ε small and ζ ∈ P ε (z), we have:

(1) ∂ ρ ∂ ζ i (ζ ) ε τ i (z,ε) ((1) of Section 2.1.1); (2) |Q i (z, ζ )| + |Q i (ζ , z)| ε τ i (z,ε) (see [DF06]); (3) ∂ Q i (z,ζ ) ∂ ζ j ε τ i (z,ε)τ j (z,ε) (see [DF06]). A straightforward calculus shows that ∂ ζ Q n-q = 1 ρ(ζ ) n-q ∑ ′ I=(i 1 ,...,i n-q) J=( j 1 ,..., j n-q) n-q ∏ k=1 ∂ Q i k (z, ζ ) ∂ ζ j k i∈I dζ i j∈J dζ j ± ± 1 ρ(ζ ) n-q+1 ∑ ′ I=(i 1 ,...,i n-q) J=( j 1 ,..., j n-q) ∑ k 0 ∈{1,...,n-q} ∂ ρ(ζ ) ∂ ζ j k 0 Q i k 0 (z, ζ ) ∏ 1≤k≤n-q k =k 0 ∂ Q i k (z, ζ ) ∂ ζ j k i∈I dζ i j∈J dζ j ,
where ∑

′ means that the i r (resp. j r ) are all different.

Suppose

ζ ∈ P 0 ε = P ε (z) \ P ε/2 (z). Then, by Lemma 2.1, |z -ζ | 2q 1 K 0 S(z, ζ ) + ρ(ζ ) n-q+1 ε n-q+1 |z -ζ | 2q
, and, by Lemma 2.4,

K 1 (z, ζ ) ∑ ′ I=(i 1 ,...,i n-q) J=( j 1 ,..., j n-q) (ρ(ζ ) + ε) ε n-q ∏ n-q k=1 τ i k τ j k ε n-q+1 |z -ζ | 2q-1 ∑ ′ I=(i 1 ,...,i n-q) J=( j 1 ,..., j n-q) 1 ∏ n-q k=1 τ i k τ j k |z -ζ | 2q-1 ,
because δ ∂ Ω (z) < ε and using (1) of Section 2.1.1.

The τ i being supposed ordered increasingly, it is easy to see that

P 0 ε 1 |z -ζ | 2q-1 P 0 ε 1 |z -ζ | 2q-1 + ε 2q-1 τ n-q+1 (z, ε) n-q ∏ i=1 τ 2 i (z, ε),
and we obtain

P 0 ε K 1 (z, ζ ) dV (ζ ) τ n-q+1 (z, ε) = O ε 1/m ,
and, if we denote

P i ε = P i ε (z) = P z, 2 -i ε \ P z, 2 -(i+1) ε , P ε K 1 (z, ζ ) dV (ζ ) = ∞ ∑ i=0 P i ε K 1 (z, ζ ) dV (ζ ) ∞ ∑ i=0 O ε2 -i 1/m = O ε 1/m ,
which ends the proof of (2.11).

To finish the proof of Theorem 2.1, by Fubini's Theorem, we have to prove that

∂ Ω K 1 (z, ζ ) ∧ f (ζ ) dσ (z) f (ζ ) k ,
and, by Lemma 2.1, it is enough to see that, for ζ near the boundary and η small enough (to have the reduced form of K 1 ),

∂ Ω∩P η (ζ ) K 1 (z, ζ ) ∧ f (ζ ) dσ (z) f (ζ ) k .
To see it, let us denote

ε = δ ∂ Ω (ζ ), Q 0 ε (ζ ) = P ε (ζ ) and Q i ε (ζ ) = P 2 i ε (ζ ) \ P 2 i-1 ε (ζ ), i ≥ 1,

and let us estimate

∂ Ω∩Q i ε (ζ ) K 1 (z, ζ ) ∧ f (ζ ) dσ (z)
using the coordinate system associated to the ζ , 2 i ε -extremal basis. By the properties of the norm . k it is enough to prove the estimate when the form f is (in the extremal coordinate system) f = dζ I , with |I| = q. Then, denoting I = (i 1 , . . . , i q ) and J = ( j 1 , . . . , j n-q ), I ∪ J = {1, . . . , n}, K

1 (z, ζ ) ∧ f (ζ ) is a sum of expressions of the form W i D , i = 1, 2, with D(ζ , z) = |z -ζ | 2q 1 K 0 S(z, ζ ) + ρ(ζ ) n-q+1
and

W 1 = (ζ m -z m ) ρ(α) n-q ∏ k=1 ∂ Q i k (z, ζ ) ∂ ζ j k n i=1 dζ i ∧ dζ i l∈L |L|=q-1 dz l and, if 1 / ∈ I, W 2 = (ζ m -z m ) ∂ ρ(ζ ) ∂ ζ 1 Q i k 0 (ζ , z) ∏ 1≤k≤n-q k =k 0 ∂ Q i k (z, ζ ) ∂ ζ j k n i=1 dζ i ∧ dζ i l∈L |L|=q-1 dz l .

Now, remarking that

Q i ε (ζ )∩∂ Ω dσ (z) |ζ -z| 2q-1 2 i ετ n-q+1 ζ , 2 i ε n-q ∏ i=2 τ 2
i ζ , 2 i ε , we get, using property (5) of Section 2.1.1,

Q i ε (ζ )∩∂ Ω W 1 (ζ , z) D(ζ , z) ∧ f (ζ ) dσ (z) ρ(ζ ) 2 i ε min j∈I τ j ζ , 2 i ε 2 i ε ρ(ζ ) 2 i ε min j∈I τ j (ζ , ε) ε min j∈I τ j (ζ , δ ∂ Ω (ζ )) 2 i δ ∂ Ω (ζ )
and, because 1 / ∈ I,

Q i ε (ζ )∩∂ Ω W 2 (ζ , z) D(ζ , z) ∧ f (ζ ) dσ (z) min j∈I τ j ζ , 2 i ε 2 i ε min j∈I τ j (ζ , δ ∂ Ω (ζ )) 2 i/2 δ ∂ Ω (ζ ) ,
and the proof is complete.

NONISOTROPIC ESTIMATES OF CLOSED POSITIVE CURRENTS ON GEOMETRICALLY SEPARATED DOMAINS

In this Section, Ω is a pseudo-convex domain in C n with C ∞ boundary which is geometrically separated at a point p 0 ∈ ∂ Ω. To state the main result of this Section we fix some notations.

We denote by ρ a defining function of Ω and

N = 1 |∇ρ| 2 ∑ i ∂ ρ
∂ z i dz i the unit normal defined in a neighborhood U of ∂ Ω. Recall (see [START_REF]Extremal basis, geometrically separated domains and applications[END_REF]) that the hypothesis made on Ω means that there exists a constant K > 0, a neighborhood V = V (p 0 ) ⊂ U of p 0 and a (n -1)-dimensional complex vector space E 0 of (1, 0)-vector fields C ∞ in V , tangent to ρ (i.e. L(ρ) ≡ 0 in V for L ∈ E 0 ) such that, at every point p of V ∩ Ω and for every 0 < δ < δ 0 there exists a (K, p, δ ))-extremal (or (p, δ )-extremal) basis whose elements belong to E 0 . We will denote by E 1 the complex vector space generated by E 0 and N.

For L ∈ E 1 , 0 < ε < δ 0 and z ∈ V ∩ Ω, in [START_REF]Extremal basis, geometrically separated domains and applications[END_REF] we defined the weight F(L, z, ε) as follows:

If L = L τ + a n N, the weight is defined by

F(L, z, ε) = ∑ L ∈L M (L) L (∂ ρ)(z) ε 2/|L | + |a n | 2 ε 2 ,
where M is an integer larger than the type, L M (L) denotes the set of lists L 1 , . . . L k , of length k ≤ M, L j ∈ L τ , L τ , and

L (∂ ρ)(z) = L 1 . . . L k-2 ∂ ρ, L k-1 , L k (z).
Remark. To keep the same notations as in [START_REF]Extremal basis, geometrically separated domains and applications[END_REF], the normal is denoted by L n in opposite of the preceding Section where it was denoted by L 1 . Now we denote τ (L, z, ε) = F (L, z, ε) -1/2 . If v is any non zero vector in C n , and z ∈ V ∩ Ω, there exists a unique vector field

L = L z in E 1 such that L(z) = v. Then we denote τ (z, v, ε) = τ (L, z, ε) and k (z, v, ε) = δ ∂ Ω (z) τ (z, v, ε) ,
where δ ∂ Ω (z) is the distance from z to the boundary of Ω.

When ε = δ ∂ Ω (z) we denote τ (z, v) = τ (z, v, δ ∂ Ω (z)) and k (z, v) = k (z, v, δ ∂ Ω (z)).
To simplify the exposition, we will also use the following terminology. For each point z ∈ V ∩ Ω and each 0

< δ ≤ δ 0 , if (v i ) 1≤i≤n is a basis of C n such that v n = N(z) and (L i ) 1≤i≤n-1 (L i ∈ E 0 ) is a (z, δ )-extremal basis such that L i (z) = v i we will say that (v i ) 1≤i≤n is a (z, δ )-extremal basis.
Suppose now that Θ is a C ∞ (1, 1)-form defined in V ∩ Ω. Then, following [START_REF] Bruna | Zeros varieties for the Nevanlinna class in convex domains of finite type in C n[END_REF], we define, for z ∈ V ∩ Ω,

Θ(z) k = Θ(z) p 0 k = sup u,v∈C * |Θ(z) (u, v)| k(z, u)k(z, v)
and

Θ(z) E = sup u,v∈C * |Θ(z) (u, v)| u v ,
where . denotes the euclidean norm.

Similarly, with the above notations, we extend the notations defined in formulas (2.2) and (2.3) for (0, q)-forms to forms defined in a geometrically separated domain.

The aim of the Section is to prove the following result:

Theorem 3.1. Let Ω be a pseudo-convex domain in C n which is geometrically separated at a boundary point p 0 . Then there exist two neighborhoods V and W ⊂ V of p 0 and a constant C

> 0 such that, if Θ is a smooth positive closed (1, 1)-form defined in V then W δ ∂ Ω (z) Θ(z) k dV (z) ≤ C V δ ∂ Ω (z) Θ(z) E dV (z).
If Ω is a bounded pseudo-convex domain which is geometrically separated at every point of it's boundary, we choose a finite number of points p i , 1

≤ i ≤ N, in ∂ Ω such that the set of the neighborhoods V i = V (p i ) is a covering V of ∂ Ω, V 0 = Ω \ i≥1 V i , and we define Θ(z) V k = max z∈V i Θ(z) i k ,
where, Θ(z

) i k = Θ(z) p i k if i ≥ 1 and Θ(z) 0 k = Θ(z) E .
With this notation, the local result immediately implies a global one: Theorem 3.2. If Ω is a pseudo-convex domain in C n which is geometrically separated at every point of it's boundary, a covering V of ∂ Ω being chosen, there exists a constant C > 0 such that

Ω δ ∂ Ω (z) Θ(z) V k dV (z) ≤ C Ω δ ∂ Ω (z) Θ(z) E dV (z)
for all smooth closed positive (1, 1)-form Θ in Ω.

In fact the two statements are equivalent. In [START_REF]Extremal basis, geometrically separated domains and applications[END_REF] it is proved that if Ω is geometrically separated at p 0 ∈ ∂ Ω then there exists a bounded pseudo-convex domain D with C ∞ boundary contained in Ω which is geometrically separated at every point of it's boundary and whose boundary contains a neighborhood of p 0 in the boundary of Ω. Thus, Theorem 3.1 for Ω follows immediately Theorem 3.2.

We now prove the Theorems. As it is very similar to the proof of the same result for convex domains given in Section 2 of [START_REF] Bruna | Zeros varieties for the Nevanlinna class in convex domains of finite type in C n[END_REF], we will refer to that paper for many details. Precisely, we will only give the mains articulations and the proof of Lemmas where the differences due to the fact that the weights F are defined with vector fields instead of coordinates systems are relevant.

We still use the previous notations for the defining function ρ of Ω, the point p 0 ∈ ∂ Ω, the neighborhood V of p 0 and W a neighborhood of p 0 relatively compact in V .

In Section 3.3 of [START_REF]Extremal basis, geometrically separated domains and applications[END_REF] it is shown that there exists a (z, ε)-adapted coordinate system (ξ i ) i used, in particular, to define "polydiscs" centered at the point z by

P ε (z) = P(z, ε) = {q = (ξ i ) i such that |ξ i | ≤ cτ(z, v i , ε)} , where v i = L i (z), 1 ≤ i ≤ n, (L i ) 1≤i≤n
being the (z, ε)-extremal basis, and c a sufficiently small constant (depending on Ω). Moreover, the set of these polydiscs are associated to a pseudo-distance. With these notations we have

Lemma 3.1. For z ∈ V ∩ Ω and v ∈ C n , v = 0, if w ∈ P ε (z) we have τ(w, v, ε) ≃ τ(z, v, ε).
Proof. Let L z (resp. L w ) the vector field belonging to E 1 such that L z (z) = v (resp. L w (w) = v). Let (L i ) 1≤i≤n be the (z, ε)-extremal basis and let us write

L z = ∑ n i=1 b i L i and L w = ∑ n i=1 b ′ i L i , b i , b ′ i ∈ C. If (ξ i ) 1≤i≤n is the (z, ε)-adapted coordinate system let us write L j (.) = ∑ i a i j (.) ∂ ∂ ξ i
, with a i j (z) = δ j i so that

L z (z) = n ∑ i=1 b i ∂ ∂ ξ i and L w (w) = n ∑ i=1 n ∑ j=1 b ′ j a i j (w) ∂ ∂ ξ i .
In [START_REF]Extremal basis, geometrically separated domains and applications[END_REF] we proved (Proposition 3.6) that,if w ∈ P ε (z),

a i j (w) F 1/2 j (z, ε)F -1/2 i (z, ε).
The basis (L i ) being (z, ε)-extremal we have immediately that F(L z , z, ε) ≃ F(L w , z, ε), and, using another time Proposition 3.6 of [START_REF]Extremal basis, geometrically separated domains and applications[END_REF], that F(L w , z, ε) ≃ F(L w , w, ε) finishing the proof of the Lemma.

The proof of the Theorems is done in three steps.

First step: definition of a family of polydiscs. As the polydiscs P ε (z) are associated to a pseudodistance, there exists a constant M such that, for δ and ε sufficiently small, there exists points z i , 1 ≤ i ≤ n(δ , ε) belonging to the set {ρ = -δ } such that, if

S δ ,ε (z) = S ε (z) = P ε (z) ∩ {ρ = -δ } and S * δ ,ε (z) = S * ε (z) = 2P ε (z) ∩ {ρ = -δ } then (1) {S ε (z i )} i is a covering of W ∩ {ρ = -δ }; (2) for all i, S * ε (z i ) is contained in V ∩ {ρ = -δ };
(3) for all z ∈ {ρ = -δ }, there exist at most M index i such that r ∈ S * ε (z i ). Let α ∈ ]0, 1[ be a parameter that will be fixed later. For ρ = α k , k ≥ k 0 , let z k i 1≤i≤n k be a family of points satisfying these properties for δ = α k and ε = α k /2. The family k≥k 0 z k i , 1 ≤ i ≤ n k will be denoted by (z i ) i . For z close enough to ∂ Ω and δ > 0 sufficiently small, let π δ (z) denote the point where the integral curve of ∇ρ passing through z meets the set {ρ = -δ }. V being small enough, π δ is C ∞ in V and any finite number of derivatives of π δ is bounded independently of δ .

For all z j belonging to

(z i ) i , if -ρ(z j ) = α k we set S j = S α k /2 (z j ), S * j = S * α k /2 (z j ), Q j = w ∈ Ω such that α k ≥ -ρ(w) ≥ α k+1 , π δ (w) ∈ S j , and
Q * j = w ∈ Ω such that α k-1 ≥ -ρ(w) ≥ α k+2 , π δ (w) ∈ S * j .
We suppose k 0 chosen sufficiently large so that j Q * j ⊂ V . Moreover, note that there exist at most 4M index j such that z ∈ Q * j . Second step: some estimates related to the radius τ i .

Lemma 3.2. Let z ∈ V ∩ Ω, v ∈ C n , v = 0, and m an integer larger than the type of p 0 .

(

1) If ε 1 ≥ ε 2 , ε 1 ε 2 1/2 τ (z, v, ε 2 ) τ (z, v, ε 1 ) ε 1 ε 2 1/m τ (z, v, ε 2 ) . (2) If v = ∑ n i=1 w i L i (z), where (L i , 1 ≤ i ≤ n) is the (z, ε)-extremal basis, then n ∑ i=1 |w i | k (z, v i , ε) ≃ k (z, v, ε) . Lemma 3.3. If z = π δ (w), then (1) if δ ∂ Ω (w) ≤ δ ∂ Ω (z), τ(z, v) δ ∂ Ω (w) δ ∂ Ω (z) 1/2 τ(w, v) τ(z, v) δ ∂ Ω (w) δ ∂ Ω (z) 1/m , (2) if δ ∂ Ω (w) ≥ δ ∂ Ω (z), τ(z, v) δ ∂ Ω (w) δ ∂ Ω (z) 1/m τ(w, v) τ(z, v) δ ∂ Ω (w) δ ∂ Ω (z) 1/2 .
Proof. We only need to prove 1., and, by Lemma 3.2 (1), it is enough to prove that

τ (z, v, δ ∂ Ω (z) /2) ≃ τ (w, v, δ ∂ Ω (z) /2) .
This follows the fact that, if L z = ∑ i a i L 0 i and L w = ∑ i b i L 0 i are so that L z (z) = L w (w) = v, then L z = L w + O (δ ∂ Ω (z)), and, thus

F (L z , z, δ ∂ Ω (z) /2) = F (L w , z, δ ∂ Ω (z) /2) + O(1) and F (L w , w, δ ∂ Ω (z) /2) = F (L w , z, δ ∂ Ω (z) /2) + O(1). Lemma 3.4. If (v i ) 1≤i≤n is a (z j , δ ∂ Ω (z j ) /2)-extremal basis then, for w ∈ S * j , sup u,v |Θ(w)(u,v)| k(z j ,u)k(z j ,v) ≃ ∑ n l=1 Θ(w)(v l ,v l ) k(z j ,v l ) 2 .
Proof. The proof is exactly the same as in [START_REF] Bruna | Zeros varieties for the Nevanlinna class in convex domains of finite type in C n[END_REF]. It follows Lemma 3.2 (2) and Cauchy-Schwarz inequality.

Lemma 3.5. Let w be a point in Q * j and ( wi ) i be the coordinates of π -ρ(z j ) (w) in the (z j , δ ∂ Ω (z j ) /2)-adapted coordinate system z i j i .

Then

∂ wr

∂ z l j (w) τ(z j , v r ) τ(z j v l ) , r ≤ n -1, l ≤ n,
where (v i ) 1≤i≤n is (z j , δ ∂ Ω (z j ) /2)-extremal, and

∂ ℑm( wn ) ∂ z l j δ ∂ Ω (z j ) τ(z j , v l ) ,
the constants depending on Ω and α.

Proof. As in [START_REF] Bruna | Zeros varieties for the Nevanlinna class in convex domains of finite type in C n[END_REF], the proof is reduced to the case where ρ(w) = ρ(z j ), then one consider a (w, δ ∂ Ω (z j ) /2)-extremal basis and uses Lemma 3.1.

Third step: proof of Theorem 4.1 and Theorem 3.2. Let ϕ 0 be a C ∞ function supported in [-2, 2] and identically equal to 1 in

[-1, 1], 0 ≤ ϕ 0 ≤ 1. Let Ψ = Ψ α be a C 1 function supported in α 2 , α -1 equal to 1 on [α, 1], 0 ≤ Ψ ≤ 1.
With the notations of Lemma 3.5 we define, for w ∈ Q

* j , ϕ(w) = Ψ α δ ∂ Ω (w) δ ∂ Ω (z j ) ϕ 0 |ℑm( wn )| τ(z j , v n ) n ∏ i=2 ϕ 0 | wi | τ(z j , v i )
.

Applying Stokes's formula to the form (-ρ) 2/m ϕΘ ∧ η l , where η l = i n-1 dz n j n-1 r=1 r =l dz r j ∧ dz r j , on Ω, using the previous Lemmas and a convenient function Ψ α (see [START_REF] Bruna | Zeros varieties for the Nevanlinna class in convex domains of finite type in C n[END_REF][START_REF] Charpentier | Pseudodistances et courants positifs dans les domaines de C 3[END_REF]), for all α > 0, the calculus made p. 404-408 in [START_REF] Bruna | Zeros varieties for the Nevanlinna class in convex domains of finite type in C n[END_REF] leads to the following estimate

Q j δ ∂ Ω (w) Θ(w) k dV (w) ≤ [β (α) + O(diam(Q j ) /α + εC(α)] Q * j δ ∂ Ω (w) Θ(w) k dV (w) +C(ε)C(α) Q * j δ ∂ Ω (w) Θ(w) E dV (w), with lim α→0 β (α) = 0.
To finish the proof of the Theorems choose points p i , 0 ≤ i ≤ N such that the neighborhood V 0 (p i ) is a covering of ∂ Ω. Then choosing α small enough so that β (α)MN is small, then ε small enough so that εC(α)MN is small and finally k 0 large enough so that O(diam(Q j ) /α is small, we get a δ 0 > 0 such that

W ∩{δ ∂ Ω (w)≤δ 0 } δ ∂ Ω (w) Θ(w) k dV (w) V δ ∂ Ω (w) Θ(w) E dV (w) and Ω∩{δ ∂ Ω (w)≤δ 0 } δ ∂ Ω (w) Θ(w) k dV (w) Ω δ ∂ Ω (w) Θ(w) E dV (w)
which proves the Theorems.

NON ISOTROPIC ESTIMATES FOR THE d-OPERATOR FOR GEOMETRICALLY SEPARATED DOMAINS

Theorem 4.1. Let Ω be a pseudo-convex domain in C n which is geometrically separated at a boundary point p 0 . Then there exist two neighborhoods of p 0 , W ⋐ V such that there exists a constant C > 0 such that, for any smooth closed (1, 1)-form Θ in V , there exists a smooth solution w of the equation dw = Θ satisfying the following estimate

W w k dV ≤ C W δ ∂ Ω (z) Θ(z) k dV (z).
The proof of this result is quite standard and we will just adapt the proof made in Section 3 of [START_REF] Bruna | Zeros varieties for the Nevanlinna class in convex domains of finite type in C n[END_REF].

Proof. We choose for V a neighborhood contained in the neighborhood V (p 0 ) defined in Section 3 and sufficiently small so that there exists a neighborhood W ⋐ V of p 0 and a point P ∈ V such that there exists an open set U ⊂ Ω containing (W ∩ Ω) ∪ {P} which is starshaped with respect to P. Solving the equation dw = Θ in U, we can assume, without loss of generality, that the support of Θ does not contain P. By translation, we can also assume that P is the origin in C n . Now, for z ∈ W ∩ Ω, M t : z → tz being the homotopy between 0 and z, Poincaré's formula

w = 1 0 M * t (i z t (Θ)) dt,
where i z t (Θ) denotes the inner contraction of Θ with the field Z t = z t and M * t the pull-back operator, gives a solution w of the equation dw = Θ in W ∩ Ω.

To finish the proof, we have to estimate this solution, and, for this, following the calculus made in [START_REF] Bruna | Zeros varieties for the Nevanlinna class in convex domains of finite type in C n[END_REF] p. 409, we only have to verify the following Lemma:

Lemma. There exists a constant α > 0 such that, for 0

≤ t ≤ 1 and v ∈ C * , τ (tz,tv, δ ∂ Ω (tz)/2) ≥ α δ ∂ Ω (tz) δ ∂ Ω (z) 1/m τ (z, v, δ ∂ Ω (z)/2) .
Proof of the Lemma. By definition of the polydiscs, there exists a constant K (depending only on c) such that z ∈ P Kδ ∂ Ω (tz) (tz). Then, by Lemma 3.1, τ (tz,tv, Kδ ∂ Ω (tz)) ≃ τ (z,tv, Kδ ∂ Ω (tz)), and thus τ (tz,tv, δ ∂ Ω (tz)/2) ≃ K τ (z,tv, δ ∂ Ω (tz)) ≥ τ (z, v, δ ∂ Ω (tz)), because t ≤ 1, and the Lemma follows Lemma 3.2.

From Theorem 4.1 the following global result is easily deduced:

Theorem 4.2. Let Ω be a bounded pseudo-convex domain in C n which is geometrically separated at every point of it's boundary. There exists a constant C > 0 such that, for any smooth closed (1, 1)-form Θ in Ω whose cohomology class in H 2 (Ω, C) is 0, there exists a smooth solution of the equation dw = Θ in Ω satisfying

|||w||| k ≤ C|||Θ||| k .
Here, the |||.||| k norm of a smooth (1, 1)-form Θ is defined using the covering V i defined before the statement of Theorem 3.2:

|||Θ||| k = N ∑ i=0 V i sup u,v∈C * | Θ; u, v | k i (z, u)k i (z, v)
δ ∂ Ω (z)dV (z), with k 0 (z, u) = k 0 (z, v) = 1.

CHARACTERIZATION OF THE ZERO-SETS OF THE FUNCTIONS OF THE NEVANLINNA CLASS FOR LINEALLY CONVEX DOMAINS OF FINITE TYPE

It is a well-known fact that if X is the zero-set of a function f of the Nevanlinna class N(Ω), then it satisfies the Blaschke condition: if X k are the irreducible components of X and n k the multiplicity of f on X k , then

∑ k n k X k δ ∂ Ω dµ k < +∞,
where µ k is the Euclidean measure on the regular part of X k . Classically the data {X k , n k } is called a divisor.

If Θ is the (1, 1)-positive current classically associated to the divisor {X k , n k }, this condition is

Ω δ ∂ Ω d ∑ j,k Θ j,k < +∞.
For more details we refer to [START_REF] Bruna | Zeros varieties for the Nevanlinna class in convex domains of finite type in C n[END_REF].

To prove Theorem 1.1, Ω satisfying automatically some topological condition, by a standard regularization process, it is sufficient to prove that there exists a constant C > 0 such that if Θ is a (1, 1)-closed positive current, C ∞ in Ω, there exists a function u in Ω solution of the equation i∂ ∂ u = Θ satisfying the estimate

∂ Ω |u| dσ ≤ C Ω δ ∂ Ω Θ E .
For details see [START_REF] Bruna | Zeros varieties for the Nevanlinna class in convex domains of finite type in C n[END_REF] and [START_REF] Skoda | Valeurs au bord pour les solutions de l'opérateur ∂ , et caractérisation des zéros des fonctions de la classe de Nevanlinna[END_REF].

As usual, the main part of the proof is done in two steps: resolution of the d-equation and, then, resolution of the ∂ -equation. The second step is done in Section 2. For the first one we use the results of Sections 3 and 4. As this two Sections are written for general geometrically separated domains and in a local context, we give some precisions (see also [START_REF] Conrad | Anisotrope optimale Pseudometriken für lineal konvex Gebeite von endlichem Typ (mit Anwendungen)[END_REF]).

5.1. Non isotropic estimates on closed positive currents in lineally convex domains. Ω being lineally convex of finite type it is completely geometrically separated ( [START_REF]Extremal basis, geometrically separated domains and applications[END_REF]). On the other hand, the proof of the Corollary at the beginning of Section 7.1 of [START_REF]Extremal basis, geometrically separated domains and applications[END_REF] shows that the radius τ(z, v, ε) defined for general geometrically separated domains are equivalents to the ones defined by M. Conrad in [START_REF] Conrad | Anisotrope optimale Pseudometriken für lineal konvex Gebeite von endlichem Typ (mit Anwendungen)[END_REF] (and used in Section 2)

Then, the norm Θ(z) ν k used in Theorem 3.2 is equivalent to the norm Θ(z) k defined with the radius τ i of [START_REF] Conrad | Anisotrope optimale Pseudometriken für lineal konvex Gebeite von endlichem Typ (mit Anwendungen)[END_REF] (and is independent of ν). Thus, Theorem 3.2 means that there exists a constant C 1 such that Ω δ ∂ Ω (z) Θ(z) k dV ≤ C 1 Ω δ ∂ Ω (z) Θ(z) E dV.

Resolution of the d equation in lineally convex domains.

As Ω is lineally convex we have H 2 (Ω, C) = 0 (see for example [START_REF] Conrad | Anisotrope optimale Pseudometriken für lineal konvex Gebeite von endlichem Typ (mit Anwendungen)[END_REF]). Thus, by Theorem 4.2 there exists a smooth form w such that dw = Θ in Ω and Ω w(z) k ≤ C 2 Ω δ ∂ Ω (z) Θ(z) k dV .

REMARKS

The method of resolution of the equation ∂ u = f presented in Section 2 can be used to obtain various other estimates. For example, the estimates obtained for convex domains of finite type in [Cum01a, DFF99, Fis04, Hef04, Ale05, Ale06] can be proved for lineally convex domains using our method (see also [START_REF] Diederich | Hölder estimates on lineally convex domains of finite type[END_REF]).

  , Ph. Charpentier & Y. Dupain, A. Cumenge and K. Diederich & E. Mazzilli ([BCD98, Cum01b, DM01]).


Bijan Mohammadi
email: bijan.mohammadi@univ-montp2.fr

HADAMARD INCOMPLETE SENSITIVITY AND SHAPE OPTIMIZATION

Keywords: Hadamard boundary condition, shape optimization, incomplete sensitivities, turbomachinery

The paper discusses incomplete sensitivity evaluations for shape optimization problems. It also shows how reduced order models can be introduced to extend the validity domain of the approach.

Introduction

For gradient-based shape optimization methods, it is necessary to have an estimation of the derivatives of the discrete cost function with respect to control parameters. When the number of control parameters is large, an adjoint equation is necessary [START_REF] Pironneau | Optimal Shape Design for Elliptic Systems[END_REF][START_REF] Jameson | Optimum aerodynamic design via boundary control[END_REF][START_REF] Giles | Adjoint equations in CFD: duality, boundary conditions and solution behaviour[END_REF]. It is tempting to use a discretization of the adjoint equation of the continuous problem; this however would not account for the discretization errors of the numerical schemes (like numerical dissipation for instance). Automatic differentiation produces the exact derivatives of the discrete cost function. Moreover, in reverse mode, the cost of this evaluation is independent of the number of control parameters as for a standard adjoint method. But, practical issues remain for large codes.

In multi-criteria optimization sensitivity analysis is important to discriminate between Pareto points and this even if a gradient free approach is used. Indeed, the knowledge of sensitivity permits to qualify various points of a Pareto front from the point of view of robustness: two points on a Pareto front can be compared if one considers the sensitivity of the functional with respect to the independent variables which are not control parameter. The robust optimum is the one with lowest sensitivity.

Also, sensitivity evaluation is important because often in simulations information on the uncertainties on the results is more important than the results themselves. For instance, it is essential to be able to identify dominant independent variables in a system. As these will need more accurate monitoring and for which precise measurements should be provided.

This paper is devoted to the concept of incomplete sensitivity in shape optimization. The aim is to avoid the linearization of the state equation. We will see that sometimes this needs a reformulation of the initial problem.

Shape optimization

For the design of a shape S, consider a general situation with a geometrical design or control variable x fixing S(x), auxiliary geometrical parameters q(x) (mesh related informations), a state variable u(q(x)) solution of some state equation F (u(q(x))) = 0 and finally a cost function for optimization J(x, q(x), u(q(x))):

J : x → q(x) → u(q(x)) → J(x, q(x), u(q(x))) (1)
The derivative of J with respect to x is:

∇ x J = J x + J q q x + J u u q q x (2)
The major part of the computing time of this evaluation is due to u q q x in the last term. The classical approach is by adjoint variable where the last term becomes:

J u u q q x = J u (F u ) -1 F q q x = vF q q x
where v si solution of vF u = J u . But, F q q x = F x worths zero except along the shape where it describes the dependency of the boundary conditions on the shape with respect to shape variations. This remark is central for the development of low complexity computational strategies for the gradient. In particular, it means that if the parameterization is chosen such that F x ∼ 0 computing the adjoint variable v is useless. For instance, a typical situation with fluid flows is with zero normal pressure gradient along the shape and admissible shape variation normal to the shape.

A model problem

Let us start with a simple model problem. Consider as cost function J = ǫ n u y (ǫ) and as state equation the following Poisson equation (taking |ǫ| << 1)

-u yy = 1, on ]ǫ, 1[ u(ǫ) = 0, u(1) = 0
which has as solution u(y) = -y 2 /2 + (ǫ + 1)y/2 -ǫ/2. This is a case of a function which has a strong geometrical element and a weak dependance on the geometry via the state u. The gradient of J with respect to ǫ is given by

J ǫ (ǫ) = ǫ n-1 (nu y (ǫ) + ǫu yǫ (ǫ)) = ǫ n-1 2 (-n(ǫ -1) -ǫ)
The second term between parenthesis, -ǫ, is the state linearization contribution which is neglected in incomplete sensitivities. We can see that the sign of the gradient is always correct and the approximation is better for large n.

Hadamard equivalent boundary condition

Above, we mentioned the normal pressure boundary condition. Another important condition is slip or non penetration condition. The Hadamard slip boundary condition accounts for the effects of small shape deformations on the state and is prescribed on the undeformed shape x 1 instead of the slip condition on the deformed shape x 2 .

Denote by n 1 and n 2 the unit normal on the undeformed and deformed shapes. On the later the slipping boundary condition reads: u 2 .n 2 = 0 and on the former If we suppose that the variations of the geometrical quantities dominate the physical ones:

u 2 .n 2 ∼ u 1 .n 1 + u 1 .(n 2 -n 1 ) = 0
This defines an implicit relation for u 1 .n 1 which can be implemented in a iterative resolution procedure:

u p+1 1 .n 1 = -u p 1 .(n 2 -n 1 )
In the same way, an equivalent boundary condition can be derived for the tangential component. These relations give satisfactory results when the shape curvature and the amount of the deformation are not high [START_REF] Mohammadi | Applied Shape Optimization for Fluids[END_REF][START_REF] Bardos | Petites perturbations et équations d'Euler pour l'aéroélasticité[END_REF] and indicate how incomplete sensitivities can be defined neglecting state variations in shape deformation.

Another interesting situation is when one can express the state in the domain as (denote the shape by y s ):

u = w(y -y s )v(w(y -y s )) (3)
Suppose u must satisfy a homogeneous Dirichlet boundary condition on the shape, then w tends to zero with the distance to the shape y -y s and v is selected to satisfy the state equations. Now, sensitivity analysis for a functional such as J = J(y s , u) gives:

∇ ys J = J ys + J u (wv ys + vw ys )
But, w(0) = 0 and one knows the dependency between w and y w . Therefore, in cases where the near-wall dependency of the solution with respect to the distance to the shape is known the sensitivity with respect to shape variations normal to the wall can be obtained without linearizing the state equation. Wall functions for fluids give typical such dependencies.

Incomplete sensitivities

Continuing the analysis above, we have observed that when the shape is regular the last term in ( 2) is small if J is of the form J(x) = shape f (x, q(x))g(u). In other words, it should involve a product of state by geometry quantities. In the analysis for the injection condition we have f = n and g = u.

A middle path between full linearization of the state equation and incomplete sensitivity is to use a reduced complexity or reduced order model ũ(x, u) ∼ u(x) to provide an inexpensive approximation of the missing term in (2):

∇ x J = J x + J q q x + J u ũx u ũ (4)
Reduced order models can come, for instance, from a reduction in dimension of the state equations as in wall functions [START_REF] Mohammadi | Analysis of the k-epsilon turbulence model[END_REF]. They can also be built through learning, assimilation and identification in parametric or non parametric models [START_REF] Krige | A statistical approach to some mine valuations and allied problems at the Witwatersrand[END_REF][START_REF] Chiles | Geostatistics, Modeling Spatial uncertainty[END_REF][START_REF] Hoel | Introduction to Mathematical Statistics[END_REF][START_REF] Jeong | Data Mining for Aerodynamic Design Space[END_REF][START_REF] Kohonen | Self-Organizing Maps[END_REF][START_REF] Kumano | Multidisciplinary design optimization of wing shape with nacelle and pylon[END_REF][START_REF] Lindman | Analysis of Variance in Complex Experimental Designs[END_REF][START_REF] Mandic | Recurrent Neural Networks for Prediction: Architectures, Learning algorithms and Stability[END_REF][START_REF] Spooner | Stable Adaptive Control and Estimation for Nonlinear Systems: Neural and Fuzzy Approximator Techniques[END_REF][START_REF] Veroy | Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: Rigorous reduced-basis a posteriori error bounds[END_REF] using up-to-date techniques to minimize the curse of dimensionality in the sampling needed in building these models [START_REF] Smolyak | Quadrature and interpolation formulas for Tensor Products of Certain Classes of Functions[END_REF][START_REF] Bungartz | Sparse Grids[END_REF][START_REF] Finkel | Quad Trees: A Data Structure for Retrieval on Composite Keys[END_REF][START_REF] Gorban | Principal Manifolds for Data Visualisation and Dimension Reduction[END_REF][START_REF] Jolliffe | Principal Component Analysis[END_REF][START_REF] Kumano | Multidisciplinary Design Optimization of Wing Shape for a Small Jet Aircraft Using Kriging Model[END_REF].

Level set method

Let us make a link between the two previous points and the level set method which is an established technique to represent moving interfaces with a tremendous dedicated literature [START_REF] Osher | Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations[END_REF][START_REF] Peskin | The fluid dynamics of heart valves: experimental, theoretical and computational methods[END_REF][START_REF] Allaire | A level-set method for shape optimization[END_REF]. A parametrization of a boundary Γ by the level set method is based on the zero-level curve of a function ψ (say the signed Euclidean distance to Γ):

Γ = {x ∈ Ω : ψ(x) = 0}, ψ(x) = ± inf y∈Γ |x -y|
with the convention of a plus sign if x ∈ Ω and minus sign otherwise. Hence When the boundary moves with velocity V by a pseudo time step δζ, the shape becomes:

ψ| Γ = 0, ψ| IR d \Ω < 0, ψ Ω > 0 (5)
Γ = {x : ψ(ζ + δζ, x + V δζ) = 0}
This motion can be described by:

∂ψ ∂ζ + V ∇ψ = 0
In the context of optimization we consider V = ∇ ψ J . n where the local normal to the isocontours of ψ is defined by n = ∇ψ/|∇ψ| (where ∇ is with respect to space coordinate). The variation of ψ are then given by

ψ ζ = -∇ ψ J|∇ψ| (6)
Decomposition ( 3) is particularly interesting with a level set parameterization:

u = w(ψ)v
Consider a functional of the form:

J = J(ψ, u(ψ))
with a gradient with respect to ψ given by:

∇ ψ J = J ψ + J u u ψ with u ψ = w ′ v + w(ψ)v ψ
as w(ψ = 0) = 0. Incomplete sensitivity u ψ = w ′ v is exact in this case and can be extended inside the domain if a low order model is known for v. This gradient is used in (6) but it needs regularity control [START_REF] Mohammadi | Global optimization, level set dynamics, incomplete sensitivity and regularity control[END_REF] as shown in figure 1 for a constrained shape optimization problem with level set parameterization. On the other hand, the loss of regularity can be useful in topology optimization [START_REF] Garreau | The Topological Asymptotic for PDE Systems: The Elasticity[END_REF] as it permits for holes to appear.

Multi-criteria problems

This is a situation where incomplete sensitivity has an edge over full gradient calculation. Suppose p functionals j i , i = 1, .., p are involved in a design problem (as the one presented below): min

x j 1 (u(x)), such that j j (u(x)) = 0, j = 2, .., p

The low-complexity of incomplete sensitivity permits to avoid use of penalty in J = i α i j i . Indeed, to get J ′ we evaluate individual incomplete sensitivities j ′ i and use a projection over the subspace orthogonal to constraints: {(.., j ′⊥ j , ..), j = i}. For instance, one can use:

j′ i = j ′ i - j =i (j ′ i , j ′ j )j ′ j then J ′ = j′ imax where i max = Argmin i j′ i .
With a full gradient this would have implied calculating an adjoint variable for each of the constraint v i F u = (j i ) u .

Application

An important class of functionals concerns the aerodynamic force on the shape along an arbitrary direction s:

J = shape [T • n] • s dσ (7)
with T = p I -ν(∇u + ∇u T ) the Newtonian stress tensor. This enters the validity domain of incomplete sensitivities. We present the application of incomplete sensitivity concept to the design of axial blades under geometric and state constraints. Two quantities define the blade functioning: the flow rate Q and the pressure rise △p between in and outlet boundaries (see figure 3-left). The optimization is necessary to improve the fan performances while maintaining strong constraints such as low axial packaging and low sound level emitted. The fan efficiency is defined as:

η = Q△p ΩT r (8)
where T r = shape r [T • n] • e θ dσ is the torque and Ω the rotation rate. The problem of interest is therefore to minimize the torque T r at given Q, Ω and volume. We would like also to increase ∆p. Ω is an independent variable and is given. The volume constraint can be transformed into a boundary integral by denoting X = (x, y, z) t :

V = Ω 1dv = Ω 1 3 ∇.( X)dv = ∂Ω X. ndσ
But, Q and ∆p are not defined on the shape and are therefore outside the application domain of incomplete sensitivity. Q is easy to enforce through the inlet velocity conditions which can be frozen during the design. ∆p on the other hand needs more workout to maintain. This is a typical situation where the original problem is not in the application domain of incomplete sensitivity but can be brought in using the state equations, here the Navier-Stokes system for incompressible flows. Consider the domain in figure 2, using Stokes formula for the steady momentum equation we have:

Γ (u(u.n) + T.n)dσ = 0
A first classical approximation is to neglect viscous effects at inlet and outlet boundaries. Then, using periodicity conditions for lateral boundaries we have:

Γ i u(u.n)dσ + Γo u(u.n)dσ + Γ i pndσ + Γo pndσ + Γw T n w dσ = 0 (9)
Let us denote mean value quantities at inlet and outlet boundaries by u i , u o , p i , p o . The inlet and outlet boundaries have the same length L. From the continuity equation (∇.u = 0) and due to periodicity on lateral boundaries and slip condition on the shape, we have

u i .n i = -u o .n i .
The first component (along x axis) of ( 9) is therefore reduced to:

△p = p o -p i = -C x 1 2 ρ ∞ |u ∞ | 2 c L ( 10
)
where C x = C d cos(β) -C l sin(β) is the horizontal aerodynamic force. Hence, the pressure difference between inlet and outlet boundaries can be expressed through the horizontal aerodynamic force on the blade which is a boundary integral making possible the application of incomplete sensitivities. Let us consider the application of this approximation for the design of a blade for a geometry provided by Valeo Motors and Actuators. It represents a fan with 11 blades at the operating point Q = 1800 m 3 /h and Ω = 1700 rpm. Because this is a real configuration of already optimized blade we do not expect huge improvement. Still it would be interesting see if the design brings some extra performances and also this will be an a posteriori validation of the configuration. The flow solver used is TASCflow with a structured multi-block O-type elliptic mesh having about 140000 nodes and 4000 CAD-Free control points on the blade.

Figure (3-right) shows the history of normalized η, Q and ∆p during the design. One can see that the efficiency increases with a decreasing torque and increasing pressure rise. This proves that the analysis above linking pressure rise to aerodynamic coefficients is valid making possible the use of incomplete sensitivity in this design. This design has also been experimentally validated [START_REF] Stanciu | Low Complexity Models to Improve Incomplete Sensitivities for Shape Optimization[END_REF].

Time dependent problems

Another situation where incomplete sensitivities bring a real relief is for time-dependent applications. In these situations, incomplete sensitivities enable for real time sensitivity definition in the sense that the state and the sensitivities are available simultaneously without the need for solving a backward in time adjoint problem. This avoids the difficulty of intermediate states storage. Indeed, unlike in steady applications where intermediate states can be replaced by the converged state, reducing the storage to one state, in time dependent problems one cannot make this simplification.

We distinguish two situations: when the control is stationary and when the control is time-dependent. A shape optimization problem for unsteady flows is in the first class, while an active flow control problem belongs to the second domain (Below we give an example of buffeting control using Hadamard equivalent boundary conditions and incomplete sensitivities).

The problem of shape optimization for unsteady flows can be formulated as: min S∈O ad J(S, {u(t, q(S)), t ∈ [0, T ]}) [START_REF] Jolliffe | Principal Component Analysis[END_REF] where the state u(t, q(S)) varies in time but not S. The cost function involves the state over a given time interval [0, T ] through for instance:

J(S) = 1 T T 0 j(S, q(S), u(t, q(S)))dt (12)
where j involves instantaneous pressure based lift or drag coefficients:

C d (t) = 1 ρ ∞ | u ∞ | 2 S ( u ∞ . n)p(t, q(S))ds, C l (t) = 1 ρ ∞ | u ∞ | 2 S ( u ⊥ ∞ . n)p(t, q(S))ds
where ρ ∞ and u ∞ denote reference density and velocity vector taken for external flows as far field quantities. The gradient of J is the averaged instantaneous gradients:

J ′ (S) = 1 T T 0 j ′ (S, q(S), u(t, q(S)))dt = 1 T T 0 (j S + j q q S ) + 1 T T 0 j u u S
We need to accumulate the gradient over period [0, T ]. The first term is the incomplete sensitivity. In case the full gradient is required then an adjoint problem is required to compute the remaining terms. Shape optimization for unsteady flows has numerous applications. For instance, noise reduction as the radiated noise is linked to lift and drag time fluctuations [START_REF] Marsden | Optimal aeroacoustic shape design using approximation modelling[END_REF].

Model problem

Let us present the incomplete sensitivity analysis on another model problem for unsteady situations. Consider the following time dependent state equation for u(y, t), -S ≤ y ≤ S, t ≥ 0 in a infinite channel of width 2S.

u t -u yy = F (S, y, t), u(S, t) = u(-S, t) = 0, (13)
with F (S, y, t) = -εω sin(ωt)(S 2 -y 2 ) + 2(1 + ε cos(ωt))

inducing small perturbation in time around a parabolic solution if ε << 1. Indeed, the exact solution for this equation is:

u(y, t) = (S 2 -y 2 )f (t), f (t) = (1 + ε cos(ωt))

Buffeting control with incomplete sensitivities

We consider the flow at transonic regime over an airfoil. At this regime, the wake unsteadinesses forces the shock to move up and down on the upper surface of the airfoil. This is called buffeting. We would like to provide active control by an injection/suction devices placed at mid-chord on the upper and lower surfaces. The amount of injection/suction is defined using instantaneous incomplete sensitivity for J = (C l ) t which aims at removing lift fluctuations. These are set to zero outside the support of the control as shown in figure 4: δS(t) = -ρ∇J(t) χ C

where shape variations are linked to injection/suction by the Hadamard approximation:

u.n(t + δt) = -u.δn(t) + δS(t) δt .n(t)

where n(t) is the normal to the shape at time t and δn(t) is the variation of the normal.

Figure 5 shows the pressure coefficient distributions with and without the control turned on. One can see the control is effective and buffeting has been removed.

Concluding remarks

Hadamard incomplete sensitivity concept has been described. As this is reduced order modelling it has limitations and only applies to special functionals. These must involve product of state by geometry quantities and should be defined over the support of the control parameters (i.e. shape for a shape optimization problem). Aerodynamic coefficients are in this class. It has been shown how to reformulate the initial problem when not entering this validity domain. Again, this is not always possible, but when it is so, it permits to access sensitivity at zero cost. Beyond this, incomplete sensitivities are really helpful for time dependent problems (not reported here) and for multi-criteria situations avoiding the calculation of one adjoint by functional.

Figure 1 :

 1 Figure 1: Effect of regularity control on the level set function during optimization.

Figure 2 :

 2 Figure 2: Sketch of a blade geometry.

Figure 3 :

 3 Figure 3: Axial blade optimization: computational domain (left) and evolution of blade performances during optimization (η/η 0 , △p/△p 0 , T r /T r0 ).

Figure 4 :

 4 Figure 4: Shape variations in time.

Figure 5 :

 5 Figure 5: Instantaneous pressure coefficient over the airfoil. Control off (left) and on (right).

Acknowledgments

The author would like to thank Professors R. Hoppe, J. Sokolowski and A. Zochowski for the organization of the OPTPDE conference in Warsaw on December 2008.

And consider a functional of the form: j(S, t) = S m u y (y = S, t), m ∈ IN * [START_REF] Kumano | Multidisciplinary Design Optimization of Wing Shape for a Small Jet Aircraft Using Kriging Model[END_REF] involving instantaneous state quantities. The sensitivity with respect to S is:

The first term is the 'instantaneous incomplete sensitivity'.

As we have: u y (S, t) = -2Sf (t), and u y S (S, t) = -2f (t)

One can express the different contribution in [START_REF] Kumano | Multidisciplinary design optimization of wing shape with nacelle and pylon[END_REF]:

Comparing with -2(m + 1)S m f (t), one sees that the approximation of the gradient based on this incomplete sensitivity is accurate and its precision increases with m. Most important, the incomplete sensitivity has always the right sign. It is obvious that the analysis still holds if the functional involves a time integral:

Now, if the functional involves an integral over the domain:

J(S, T ) =

(0,T )×(-S,S)

j(y, t) dt dy

One can still make the analysis above and see the importance of different contributions:

(mS m-1 u y (S, t) + S m u y S (y, t)) dt dy

Again, an incomplete evaluation of the sensitivity is accurate because u y S = 0. One also notices that if m is odd the last integral vanishes, even though this integral is cheap to get as it does not involve any state sensitivity with respect to S. Anyway, incomplete sensitivity is efficient but, as we said, it only holds for special functionals.


Christophe Lino

Marc Christie

Efficient Composition for Virtual Camera Control

Keywords: (according to ACM CCS): Mathematics of Computing [G, 1, 6]: Numerical Analysis-Optimization Computer Graphics [I, 3, 6]: Methodology and Techniques-

Automatically positioning a virtual camera in a 3D environment given the specification of visual properties to be satisfied (on-screen layout of subjects, vantage angles, visibility) is a complex and challenging problem. Most approaches tackle the problem by expressing visual properties as constraints or functions to optimize, and rely on computationally expensive search techniques to explore the solution space. We show here how to express and solve the exact on-screen positioning of two or three subjects using a simple and very efficient technique. We express the solution space for each couple of subjects as a 2D manifold surface. We demonstrate how to use this manifold surface to solve Blinn's spacecraft problem with a straightforward algebraic approach. We extend the solution to three subjects and we show how to cast the complex 6D optimization problem tackled by most contributions in the field in a simple 2D optimization on the manifold surface by pruning large portions of the search space. The result is a robust and very efficient technique which finds a wide range of applications in virtual camera control and more generally in computer graphics.

Introduction

A large range of computer graphics applications rely on the computation of viewpoints over 3D scenes that need to display a number of characteristic composition properties (e.g. on-screen positioning, visibility, or vantage angle of one, two or more subjects). Simplified versions of such problems can be tackled with straightforward vector algebra approaches [START_REF] Blinn | Where am I? what am I looking at[END_REF]. More expressive versions are generally casted as non-linear optimization problems in a 7 degree-offreedom search space, i.e. computing the camera position, orientation and field of view, given a number of composition properties expressed and aggregated in a single viewpoint quality function. Given the size of the search space and the computational cost in the evaluation of composition properties (typically the visibility of subjects), this optimization process is a time-consuming task and hampers the use of evolved composition techniques in most applications.

An analysis of the problem and current solutions reveals a central issue. Aggregating quality functions for all properties together reduces the capacities to guide the solving process through the search space, therefore leading to techniques which explore large areas of the search space with-out solutions. One transforms a very specific problem into a general search process for which it is difficult to propose efficient and general heuristics.

In this paper, we address an important visual property in virtual camera composition: the specification of exact onscreen coordinates at which targets should project. We show that the range of solutions for the exact composition of two subjects on the screen is a 2D manifold shaped as a spindle torus and that the manifold can be explored using two meaningful parameters ϕ and θ (see Figure 3). We use this formulation to easily solve Blinn's spacecraft problem [START_REF] Blinn | Where am I? what am I looking at[END_REF], and show how to generalize the approach to three and more subjects. The key contributions of our paper are:

• a novel way to express, as a 2D manifold, the solution space of the exact on-screen composition of two subjects; • an algebraic formulation of on-screen composition for two subjects which solves Blinn's problem [START_REF] Blinn | Where am I? what am I looking at[END_REF] without resorting to an iterative technique; • the expression of virtual camera composition as a search in a 2D space rather than in a 6D space (as performed in most of the literature) for two, three or more targets.

The paper is organized as follows. After presenting the submitted to Eurographics/ ACM SIGGRAPH Symposium on Computer Animation (2012) state of the art in virtual camera composition, we show how the on-screen positioning of two subjects can be expressed in a 2D manifold. We then show how Blinn's spacecraft problem can be solved by using this manifold, and how to extend to three (or more) subjects, before concluding.

Related work

In 1988, Blinn [START_REF] Blinn | Where am I? what am I looking at[END_REF] tried to solve the problem of positioning and orienting a virtual camera to satisfy a given onscreen projection of two subjects with a camera at a distance d of one of the targets, and with some axis (passing through one of the targets) appearing vertical on the screen. By assuming the up vector of the camera was pointing "up" (as much as possible) and using the inverse projection matrix to extract the camera position and orientation from the onscreen projections of subjects, he proposed an iterative algorithm that computes an approximate solution (if one exists) of the problem. To the best of our knowledge, there is no general technique to compute the exact solution (or range of solutions) to this on-screen composition problem.

As a replacement to limitations in Blinn's techniques, researchers have focused on ways to express and solve more complex on-screen compositions. Given the complexity of the problem and the non-linearities in the projection equations, most approaches rely on constraint-based or constrained-optimization techniques in which on-screen properties are expressed as constraints/quality-functions on the camera parameters (see [START_REF] Christie | Camera Control in Computer Graphics[END_REF] for an overview of techniques). Approaches vary in their expressiveness (range of on-screen properties they handle) and efficiency (solving technique employed to search for solutions).

A good illustration is the CamPlan tool [START_REF] Halper | CAMPLAN: A Camera Planning Agent[END_REF] which uses a genetic algorithm to optimize the camera w.r.t. the specified set of visual properties. By encoding the camera parameters as a gene, the authors provide a robust, yet computationally expensive, global search technique. In a similar way, Bares et al. [START_REF] Bares | Virtual 3D Camera Composition from Frame Constraints[END_REF] defined sophisticated camera compositions by creating storyboard frames, indicating how a desired shot should appear and used a heuristic-based optimization technique to compute solutions through a recursive sampling of the search space. Similar approaches in the idea have been followed by [CN05, ACOYL08, BY10, LCL * 10]. Authors design one quality function per on-screen property (see [START_REF] Ranon R | Accurately Measuring the Satisfaction of Visual Properties in Virtual Camera Control[END_REF] which details and compares these properties). The overall quality of a camera configuration is then expressed as a function aggregating the quality of each property and solving techniques explore large regions of the search space leading to significant computationnal costs.

By focusing on the problem of composing two subjects on the screen, our objective is to show that large regions of space can be discarded from the search process. For this purpose, we illustrate the quality of the on-screen positioning of two subjects as a 2D heat map. Figure 1 presents a 2D

( -→ OB, -→ OA) = 2α.
topview of a scene comprizing two targets. Regions around the subjects are colored from blue to green and red, respectively representing low, average and high viewpoint quality. Note how the region with highest quality is restricted to a small continuous portion of the whole search space.

In the following section we show that the entire solution set for the exact on-screen positioning problem can be determined with simple mathematical concepts, and that this model serves as a cornerstone for the resolution of more complex camera composition problems.

Definition 1 The oriented angle between two vectors u and

v is denoted as Ö ( -→ u , -→ v ).
submitted to Eurographics/ ACM SIGGRAPH Symposium on Computer Animation (2012)

Solution in 2D

Let's first consider a 2-subject composition problem in 2D (we study the problem on a plane). In this case the screen dimension is 1D. The goal is to find a 2D camera position P such that two subjects of world positions A and B (A = B) project respectively at normalized screen positions p A , p B ∈ [-1; +1], knowing the camera field of view φ. We here assume that A is the left-most subject (i.e. p A < p B ).

The inscribed angle theorem states that the set of points 2). α is easily expressed from the on-screen distance between A and B and the fixed field of view φ. The arc õ AB goes from A to B in the anticlockwise direction, and the radius r of C is equal to AB 2 sin α . We now introduce two points in the camera coordinate system, projecting respectively at p A and p B on the screen: corresponds to

P satisfying × ( -→ PB, -→ PA) = α is equal to the arc õ AB (excluding A and B) of a circle C of center O passing through A and B, such that × ( -→ OB, -→ OA) = 2α (see figure
Ø ( -→ BP, -→ BB ) = 1 2 Ø ( -→ OP, --→ OB ). The solution for Ø ( -→ OP, -→ OB) is then θ = π ± 2β
, with the restriction that P must lie on the arc õ AB (in red).

The solution of our 2D problem is the arc õ AB such that

α = Ú ( --→ p cam B , --→ p cam A )
To compute the direction vector -→ f of our 2D camera, we know to have

Ö ( -→ PA, -→ f ) = φ 2 • p A and ( -→ PB; -→ f ) = φ 2 • p B
. By using the same theorem we know that the line of direction -→ f and passing through P intersects the circle C at a point

F such that Ø ( -→ OB, -→ OF) = φ • p B
, and is tangent to C when points P and F coincide. The camera orientation is thus algebraically determined from the camera position (see figure 2).

As a conclusion, in 2D, the range of solutions to the exact on-screen composition of two subjects can be described with a single angle θ ∈ ]0 ; 2(πα)[, where θ represents the angle Ø ( -→ OP, -→ OB), P being the position of the camera.

Solution in 3D

Let's now consider this composition problem in 3D. The screen dimension is now 2D, and the goal is to find a 3D camera position P such that the two key subjects A and B (A = B) project respectively at normalized screen positions p A (x A ; y A ) , p B (x B ; y B ) ∈ [-1; +1] 2 , knowing the camera field of view φ and its aspect ratio a. Note that φ define the field of view on the x axis (representing the left to right direction on screen) and that the field of view on the y axis (representing the bottom to top direction on screen) is given by φ a . We also assume that A is the left-most key subject (i.e. either x A < x B , or x A = x B and y A = y B ).

As previously, let us introduce two points described in the submitted to Eurographics/ ACM SIGGRAPH Symposium on Computer Animation (2012) 4), corresponding to a given value θ = θn, and located at a given vertical angle (ϕ 0 = 0) of the manifold. The red circle represents the appearance of v depending on the vertical angle ϕ applied to the viewpoint P. The half-plane (in blue) represents the set of vectors starting from B that appear vertical and pointing "up" on screen ( u is the up vector of the camera when at P, and w represents -→ PB). The solution of Blinn's problem is given by the intersection (E ) of the half-plane and the circle, and provide a mean to compute the rotation ϕ to apply. The camera must satisfy desired exact on-screen positions of A and B, and be d meters away from subject B. An additional constraint is that the vector (in black) starting from B must appear vertical and pointing "up" on screen.

camera coordinate system, projecting respectively at p A and p B on screen:

p cam A x A Sx ; y A Sy ; 1 p cam B x B Sx ; y B Sy ; 1
with Sx = tan φ and Sy = tan φ a . Remember now the previous theorem. By considering a plane P defined by the points A, B, and a third point C not lying on the line (AB), the solution --→ p cam A ) We now have the solution of our problem for an arbitrary plane. Let's consider that point C is the center of the inscribed circle in P that we called O. And let point I be the middle of segment AB. Note that from the definition of the solution in 2D, we know that OI = AB 2 tan α . We can now consider a plane Q, defined by the point I and the normal vector

-→ n = -→ AB.
The set of all possible positions for O is then the circle of center I and radius AB 2 tan α defined in the plane Q. This, together with the solution in an arbitrary plane containing A and B, defines a unique solution set for the camera position (see figure 3).

We now show how to compute the camera orientation that satisfies the composition, assuming the camera position is P (belonging to a plane P). We first initialize the camera orientation as a coordinate system made of three unit vectors: a forward (look-at) vector -→ f i , a right vector -→ r i , and an up vector -→ u i . They are given by submitted to Eurographics/ ACM SIGGRAPH Symposium on Computer Animation (2012)

-→ u i = -→ PB | -→ PB| × - → PA | - → PA| ; -→ f i = -→ PB | -→ PB| + - → PA | - → PA|
scaled to unit length ; and -→ r i = -→ f × -→ u . Note that -→ u i is normal to the supporting plane P, and that -→ f i and -→ r i belong to P.

We then build a quaternion q i from these vectors, which represents a first "default" composition of A and B (i.e. y A = y B = 0 and x A = -x B ). We now compute the rotation qc such that, when applied to q i , points A and B are projected in the appropriate locations on the screen (i.e. p cam A and p cam B ). The solution camera orientation is then given by q = q i • (qc) -1

(1) Note that qc and q i are algebraically determined from respectively the desired on-screen composition, and the camera position.

As a conclusion, in 3D, the range of solutions to the exact on-screen composition of two subjects can be described with two angles ϕ ∈ ]π ; +π] and θ ∈ ]0 ; 2(πα)[, where ϕ represents the angle defining the supporting plane P and θ represents the angle

Ø ( -→ OP, -→ OB)
. Note that it is also possible to describe θ as a ratio defined in ]0 ; 1[ where the ratio 0 maps θ = 0 and the ratio 1 maps θ = 2(πα). This makes θ independent from the value of α.

The position of a viewpoint P ϕ,θ parameterized by (ϕ, θ) can then be computed as follows. We need to compute the reference position O 0 that we consider as the center of the inscribed circle for ϕ = 0 (e.g. such that -→ IO 0 has a z component equal to 0). We then build a quaternion qϕ as the rotation of ϕ radians around the axis -→ AB. The center of the inscribed circle in an arbitrary plane, that we call Oϕ, is then given by Oϕ = (qϕ • -→ IO 0 ) + I A normal of the plane supporting this inscribed circle is

-→ nϕ = --→ OϕA × --→ OϕB.
We then define a quaternion q θ as the rotation of θ radians around the axis -→ nϕ, and finally obtain the position of P ϕ,θ by P ϕ,θ = (q θ • --→ OϕB) + -→ Oϕ

Solution of Blinn's Spacecraft Problem

The problem presented by Blinn [START_REF] Blinn | Where am I? what am I looking at[END_REF] is similar to the following. Where to put the camera and how to set its orientation knowing the result on-screen positions of two key subjects A and B, the distance d between the camera and the subject B, and a given vector (a non-null vector v starting from B) that should appear vertical and pointing "up" on the screen? An illustration of this problem is given in figure 6.

First, we can already state that the solution of this problem belongs to the 2D manifold defined in section 4. Moreover, by using the analytic solution in 2D, the distance constraint is quite simple to resolve. Indeed, let's consider a plane P, and define a point B as the symmetric point to B w.r.t. the center point O on the inscribed circle. We will use the properties between angles and sides in the triangle BPB which is rectangle in P. Let's define the angle β corresponding to

( -→ BP, -→ BB ) = 1 2 ( -→ OP, --→ OB ).
Then we can state that there are 0, 1 or 2 solutions to this distance problem:

• no solution if d = 0 or d > 2r • a single solution if d ∈ ]0 ; AB]: θ = π -2β • a single solution if d = 2r: θ = π • two solutions if d ∈ ]AB ; 2r[: θ = π ± 2β
The resolution method is illustrated in figure 4. This solution in 2D works in 3D since it is valid for every value of the parameter ϕ (vertical angle). The set of solutions to the distance constraint is therefore a circle on the 2D surface such that every point of the circle is at a distance d to B, i.e. this is the set of points such that θ = θn on the manifold. Now let's take a look at the "up" vector constraint. In the following, we will assume that the position of the camera satisfying all the composition constraints is P. As stated above, P belongs to a circle Cn defined by the parameter θ = θn (solution to the distance constraint).

Let's consider -→ u the up vector of the camera (provided from the quaternion q described in equation 1) at position P, and -→ w the vector representing -→ PB. The vector -→ v must appear vertical on screen, thus -→ u , -→ w and -→ v must be coplanar. Moreover -→ v should point "up" on screen. We must thus have -→ u • -→ v > 0. Blinn's problem now corresponds to finding a position P on the circle Cn such that these two constraints are satisfied.

From the last section, we know that the camera coordinates (i.e. its position and orientation) correspond to a rotation of the coordinates of the camera when at a reference position P 0,n (parameterized by (ϕ = 0, θ = θn)) on the circle Cn by an angle of ϕ radians around the axis -→ AB. Our problem is then similar to searching the rotation of ϕ radians around --→ AB to apply to the vector v such that it appears vertical and pointing "up" on screen when the camera is set to P 0,n . Instead of searching the camera position P on the circle Cn, we then build a circle C representing the range of possible positions for the extremity of the vector v, and search a position on C such that the previous property is satisfied.

Before solving this problem, let us define the vectors -→ u 0,n as the up vector of the camera when at P 0,n , --→ w n,0 as the vector ---→ P 0,n B, and -→ v 0 as the vector v rotated by an angle of ϕ radians around --→ AB (i.e. the solution vector). We then solve the problem as follows.

We first define a plane P by a normal vector -→ n = -→ u 0,n × --→ w 0,n (scaled to unit length) and the point P 0,n . By definition, B belongs to P. We then define a half-plane P H ⊂ P delimited by the line L of direction --→ w 0,n passing through P n,0 . The points belonging to P H are located in the direction of -→ u 0,n . This half-plane defines the set of vectors that will appear vertical and pointing "up" on the screen.

We then define an other plane Q by a normal vector submitted to Eurographics/ ACM SIGGRAPH Symposium on Computer Animation (2012) -→ t = -→ AB (scaled to unit length) and a point F given by -

→ F = -→ B + -→ t • | v| • cos(π -γ), where γ = Ö ( -→ t , -→ v ) (N.B.
F belongs to the line (AB)). The circle C is then defined as the circle of center F and radius r = | v| • sin(πγ) in the plane Q. Note that the vertical angle ϕ that is solution to the problem will be given by the angle

Ø ( --→ FE , -→ FE).
The solution of Blinn's problem is then given by the intersection(s) of the half-plane P H and the circle C. In other words, by computing the intersection I of P H and Q (a halfline or the half-plane P H itself), then the intersection(s) of I and the circle C in the plane Q. The figure 5 provides an illustration of this geometric resolution, and the figure 6 provides an example of solution computed with this method.

Solution for 3 or more subjects

The approach is extensible to 3 or more subjects: lets consider 3 subjects with respective positions A, B, C (different from each other). The solution set for each pair of subjects is a 2D manifold. Consequently, any camera position P satisfying the 3-subject composition should belong to the intersection of three manifolds M, M and M respectively defined by couples (A, B), (B,C) and (A,C), and such that the camera orientation at P matches for these three manifolds.

Let first define three angles α, α and α corresponding to M, M and M . Similarly let q, q and q be the orientations (defined as unit quaternions) at a given 3D point P, as defined respectively on M, M and M . Then, if P belongs to M (for instance) it verifies

× ( -→ PB, -→ PA) -α = 0
We propose here an algorithm which searches on the surface of one manifold M the configuration which maximizes the viewpoint quality (defined as a distance to the other manifolds). Typically the on-screen position of two subjects is fixed, and we optimize the on-screen position of the third subject. The cost function to minimize is then min

θ,ϕ × ( -→ PC, -→ PB) -α + × ( -→ PC, -→ PA) -α + 1 - ¬ q , q ¶ 2
The heatmap corresponding to this cost function and an illustration of the solutions of this composition problem are given respectively in figures 7 and 8.

The main advantage of this technique is that it can easily extend to more than 3 subjects. Though the composition may not be strictly satisfied, it enables fixing the on-screen position of two (main) subjects, then optimizing the position of other subjects. All other visual properties can be accounted for in this optimisation on the manifold surface (target size, vantage angle, rectangular framing of targets).

Conclusion

We have introduced a simple parametric model to solve a range of problems that occur in the task of positioning a vir-tual camera given exact on-screen specifications. Our model solves Blinn's spacecraft problem [START_REF] Blinn | Where am I? what am I looking at[END_REF] by using an algebraic formulation rather than an iterative process. It casts camera optimization problems mostly conducted in 6D into searches inside a 2D space on a manifold surface. Interestingly, our model can be easily extended to integrate most of the classical visual properties employed in the litterature [START_REF] Ranon R | Accurately Measuring the Satisfaction of Visual Properties in Virtual Camera Control[END_REF]. For example, size of key subjects (or distance to camera) can be expressed as the set of viewpoints on the manifold which are at a given distance from the camera (resolves as 0, 1 or 2 lines on the manifold surface). In a similar way, vantage angle properties (eg. see the front of a subject) represent sub-regions of the manifold.

By reducing the search space to a manifold where the on-screen location of subjects are exact, we obviously restrict the generality of the technique. However, the benefits in terms of computational cost greatly favors our approach. Though the solution for two subjects appears easy to formulate with vector algebra, it has not been reported before and the model serves as an expressive way on which to build more evolved techniques. The techniques presented in the paper have the potential to replace most of the previous formulations related to camera control with a simpler and more efficient approach, and opens great possibilities to include more evolved on-screen composition techniques in a large range of applications in computer graphics.

Figure 1 :

 1 Figure 1: Heatmap representing the quality of on-screen composition for two subjects (white points) for a region of camera configurations (topview of a the 3D scene). Each colored point represents a camera configuration. Red areas are regions of good quality viewpoints (i.e. good compositions) while blue areas are regions of bad quality viewpoints. Green regions represent average quality viewpoints. Note that the best viewpoints are very local to a given curve around the subjects.

Figure 2 :

 2 Figure 2: The range of solutions for the exact composition of two subjects A and B on the screen is displayed in red. The set of points satisfying a given angle α is exactly the arc õ AB of the inscribed circle centered on O. Moreover×

Figure 3 :Definition 2

 32 Figure 3: Example of 3D solution set for 2 subjects. Subjects A and B are displayed resp. in red and green. Top: view of the solution set (a two parametric space (φ, θ)), for the on-screen projections of A and B, resp. p A (-0.33; -0.33) and p B (+0.33; +0.33). Bottom: the result obtained from the viewpoint parameterized with φ = π 6 and θ = π-α 2 .

Figure 4 :

 4 Figure 4: Illustration of the method used to solve a distanceto-B constraint. The distance BP (P being the camera position) is d meters. The segment BB is a diameter of the inscribed circle of radius r. The angle β = cos -1 d 2r

Figure 5 :

 5 Figure 5: Resolution of Blinn's problem. The vector (in black) starting from B should appear vertical and pointing "up" on screen. Subjects A and B are drawn resp. in red and green. The camera position P is a solution to the distance constraint (see figure4), corresponding to a given value θ = θn, and located at a given vertical angle (ϕ 0 = 0) of the manifold. The red circle represents the appearance of v depending on the vertical angle ϕ applied to the viewpoint P. The half-plane (in blue) represents the set of vectors starting from B that appear vertical and pointing "up" on screen ( u is the up vector of the camera when at P, and w represents -→ PB). The solution of Blinn's problem is given by the intersection (E ) of the half-plane and the circle, and provide a mean to compute the rotation ϕ to apply.

Figure 6 :

 6 Figure 6: Example solution of Blinn's spacecraft problem. Subjects A and B are drawn respectively in red and green.The camera must satisfy desired exact on-screen positions of A and B, and be d meters away from subject B. An additional constraint is that the vector (in black) starting from B must appear vertical and pointing "up" on screen.

Figure 7 :

 7 Figure 7: Heatmap drawn on one of the manifolds. The cost function represents the sum of the values of the other two manifold equations. Red: points close to both other manifolds (drawn in blue and green), thus verifying the on-screen positioning of the 3 subjects. Blue: points farest to both other manifolds. Green: intermediate-distance points.

Figure 8 :

 8 Figure 8: Here are two solutions of the exact on-screen positioning of three subjects A (red), B (green) and C (blue). Their respective on-screen positions are p A (-0.50; +0.25), p B (-0.25; +0.50) and p C (+0.25; +0.50). These solutions correspond to the intersection of three solution set (described in figure 7), each solving the exact on-screen positioning of to of the subjects.