Yves Aurégan
email: yves.auregan@univ-lemans.fr

Ultra-thin low frequency perfect sound absorber with high ratio of active area

Ultra-thin low frequency perfect sound absorber with high ratio of active area

Yves Aurégan

Sound absorption is of great interest to engineers who want to reduce sound reflection in room acoustics or reduce sound emissions. There is a need for innovative acoustic absorbent materials, effective in low frequencies while being able to deal with spatial constraints present in real applications. Innovative ultra-thin materials are also useful tools for the scientific community to manipulate sound waves and obtain negative refraction [1,2], sub-wavelength imaging [3,4], cloaking [5], etc.

Traditional sound absorption structures use perforated and micro-perforated panels covering air or porous materials. [START_REF] Allard | Propagation of sound in porous media: modelling sound absorbing materials[END_REF][START_REF] Maa | [END_REF]. These materials have a low reflection of normal incident waves at frequencies such that the wavelength (λ = c 0 /f where c 0 and f are the sound velocity in air and the frequency) is about four times the thickness of the material H leading to a sub-wavelength ratio r H = λ/H 4. There has been a very significant reduction in the thickness of the absorbent materials [8] by using space-coiling structures [9][10][11][12] of by using slowsound inside the material [13][14][15].

Nevertheless, all these structures have a front plate with small holes leading to a very low open area ratio (σ = area of the orifices on the total area). In this case, since the velocity in the orifices is equal to the acoustic velocity in the incident wave divided by σ, some non-linear effects can easily occur in the orifices when the amplitude of the incident wave becomes large enough [16]. Moreover, in many engineering applications, a grazing flow is present and its effect on the impedance of the material is inversely proportional to σ [17]. For instance, it was experimentally shown [18] that the efficiency of a thin slow-sound metamaterial with σ = 0.023 was divided by 100 in the presence of a grazing flow with a Mach number of 0.2. Therefore, in the case of high sound levels or in the presence of flow, the additional constraint of having a high open area ratio must be added in the design of structures that are thin and absorbent at low frequency.

From this perspective, the use of elastic membranes and decorated elastic membranes as sound absorbers is an interesting option. [19][20][21]. In most of the studied structures, the membrane represents a large part of the active surface and the added mass (platelets) is of smaller size. The maximum absorption appears for an hybrid resonance [20] due to the interaction between two modes of coupled system consisting of the membrane, the platelet and the air cavity. By a proper arrangement of the various parameters very impressive results can be obtained both in reflection [20] and in transmission [22]. At the resonance frequency, the three main parameters of membrane absorbers (equivalent mass, stiffness and damping) are very sensitive to the characteristics of the membrane. By changing one of the properties of the membrane (its tension, its mass density,...) or the size of the added mass, the three equivalent parameters are simultaneously changed. This letter presents an ultra-thin low frequency (UTLF) resonator for which the three parameters (mass, rigidity and damping) can be modified independently of each other. A model has been measured and perfect absorption occurs for a sub-wavelength ratio r H = λ/H = 200 which is, up to now, the highest sub-wavelength ratio experimentally demonstrated for a perfect absorber.

Cavity

This UTLF resonator is ideally composed of three elements displayed on Fig. 1: A volume of air sealed in a cavity where the compression of the air acts as a spring, a mass that moves like a piston in the normal direction (without letting the air trapped in the cavity pass through with the help of a membrane) and a thin resistive layer that dissipates energy. The main difference compared to decorated elastic membranes absorbers [20] is that the membrane plays almost no role in the frequency of perfect absorption. The role of the very flexible membrane is just to guide the motion of the mass and seal the cavity. It will be demonstrated that the membrane also produces some damping.

As the low frequency regime is considered (any axial dimension is much smaller than λ = c 0 /f ), the continuity of the acoustic flow rate is assumed across the resistance and the mass:

v i = v m = v c
where v i , v m and v c are respectively the incident wave velocity, the velocity of the mass and the mean velocity entering in the cavity. For simplicity, a, the difference between the radius of the tube and the radius of the mass, is neglected compared to the tube radius R.

The UTLF resonator can be described by its impedance Z UTLF (normalized by the air characteristic impedance Z 0 = ρ 0 c 0 where ρ 0 is the air density) which is the sum of three terms since the elements are mounted in series. The first term in the impedance is linked to the air compressibility in the cavity of thickness B and can be written Z c = p c /Z 0 v c = c 0 /iωB = 1/iωC where p c is the uniform acoustic pressure in the cavity, ω = 2πf , ω = ω/(2πf R ) where f R is the resonance frequency of the UTLF resonator. The compressibility term

C = 2πf R B c 0 = 2π r B
directly characterizes the sub-wavelength ratio r B = λ R /B at resonance. This value of the cavity impedance is only valid at very low frequencies where ωB/c 0 1. For higher frequencies or higher cavity thickness, a more exact value is Z c = 1/i tan(ωC).

The second term in the impedance is linked to the mass motion subjected to the pressure difference ∆p m between its two faces. The associated impedance is Z m = ∆p m /Z 0 v i = iωm/ρ 0 c 0 S m = iωL where m = ρ m S m e is the mass of the moving part, S m = π(R -a) 2 is the mass area, ρ m is the density of the mass material and e is the mass thickness. It can be noted that the moving mass is connected to the fixed tube by a membrane that acts as a spring but this effect is neglected due to the low value of the membrane stiffness. The last part of the impedance is linked to the resistive layer and is Z R = ∆p R /Z 0 v i = R e where ∆p R is the pressure drop through the resistive layer and R e is the normalized resistance of the resistive layer. As a result, the UTLF impedance can be written

Z UTLF = R e + iωL + 1/iωC.
To be a perfect absorber when ω = 1, the UTLF impedance must perfectly match the characteristic impedance of the air. This is achieved when the two conditions R = 1 and LC = 1 are met. In this case, the normalized UTLF impedance can be written:

Z UTLF = 1 + 1 C iω + 1 iω (1)
showing that the impedance depends only on the subwavelength ratio r B . The absorption coefficient is defined by α = 1 -|(Z -1)/(Z + 1)| 2 and relation (1) shows that it is equal to 1 for ω = 1.

The change rate of α around ω = 1 is linked to the slope of the imaginary part of the impedance which is given by |dZ UTLF /dω| = 2/C for ω = 1. Using a Taylor expansion near the resonance when C is small, the frequency bandwidth ∆f for which α > 0.75 is given by ∆f /f R = 2C/ √ 3. Then, α has an increasingly sharp peak as the sub-wavelength ratio r B increases as it is shown in Fig. 2. In principle, there is no theoretical upper limit for the sub-wavelength ratio but the price to pay to have highly sub-wavelength cavities is to have a very small bandwidth. From a practical point of view (as will be seen later), the increase in the sub-wavelength ratio r B is limited by the losses that always become large when r B increases. When a cavity is made with a tube of thickness B=12 mm, the theoretical value of the resonance frequency is given by

f R = c 0 2π ρ 0 ρ m eB = 111 Hz.
This value can be compared to the measured frequency given in 3(b). A nearly perfect absorption (α = 0.9991) is obtained for f R = 107.25 Hz. The 4% error made on the frequency value is small considering all the simplifications that have been made in the calculations. The value of the compressibility term is C = 2π/269. The bandwidth for which α > 0.75 is ∆f = 2.5 Hz = Cf R . The total height of the material H = 16 mm is the sum of the cavity height B, mass thickness e and thickness of the resistive layer glued on its support (1 mm). Thus, the sub-wavelength ratio of this material is r H = λ/H = 201.6. Up to now, this ratio between wavelength and material thickness is the highest ever measured in an absorbent material. The resonance frequency is very easy to adjust since it depends only on the mass of the moving part and on the height of the cavity. The simplicity of the adjustment comes from the fact that membrane effects can be neglected to compute f R because of the low value of its equivalent stiffness. In the device under study, the resonance frequency of the mass element measured with the membrane alone (without the cavity) is about 35 Hz while it increases to 107.25 Hz in the presence of the cavity. This indicates that the equivalent stiffness of the membrane is 9 times lower than the one of the cavity with a thickness of B=12 mm.

The membrane is involved in the dissipative part of the impedance which is shown in Fig. 4(a). The blue line in this figure gives the dissipative effect when the resistive layer is not present. In this case, the dissipative effects come solely from the membrane. First of all, it can be noted that this dissipation is relatively high (the minimum value of (Z) is 0.71 representing 71% of the target value). It has already been noted in [20] that even a small dissipation in the membrane material could result in significant absorption of the overall system. Second, it can be noted that the dissipation curve appears to result from two different effects. For f >140 Hz, the dissipation increases with frequency. This effect is the only one that is experimentally observed when there is no cavity. This dissipation can therefore be attributed to the mechanical damping resulting from the elongation of the membrane when the mass moves. For f <90 Hz a second phenomenon appears for which the resistance decreases with frequency. It has been experimentally observed that this second resistance increases as the cavity thickness decreases. One possible explanation is that, when the air trapped in the cavity becomes difficult to compress or expand (i.e. for small volume or low frequency), it inflates and aspirates the membrane as schematized in Fig. 4(b), which increases the dissipative phenomena. The combination of the two previous effects induces a minimum of resistance for a frequency close to resonance. When this minimum is smaller than 1, it is possible to add a resistive layer that shifts the resistance to higher values, see the red curve in Fig. 4(b) in order to adapt the resistance exactly to 1 at the resonant frequency. This is what has been done to obtain the perfect absorber whose absorption curve is presented on the Fig. 3(b). In the present case, the addition of a resistive layer only changes the maximum attenuation by a few percent. But for less extreme conditions (λ/B 200), the attenuation due to the membrane can be much lower and the addition of a resistive layer very useful to obtain a perfect absorption.

To conclude, we have seen that it is possible to build a UTLF absorber with very simple elements: mass supported by a very flexible membrane, cavity closed by the membrane and resistive layer. This device can have a large open area to minimize non-linear effects or reduce dependence on grazing flows. The frequency of the UTLF absorber can be easily adjusted by simply changing the mass or thickness of the cavity. With this type of absorber, it is possible to obtain a very high ratio between the wavelength and the thickness of the material but, as a necessary counterpart, the bandwidth is very narrow.

The energy dissipation comes partially from the membrane and partially from the resistive layer. When trying to further reduce the size of the cavity, the resistance of the absorber becomes greater than the characteristic impedance of the air due to the dissipation in the membrane. The dissipation is therefore the limiting factor in the decrease in the size of this type of absorbers. This remark is not specific to this absorber but can be generalized to many resonant sub-wavelength absorbers: It is the damping that limits the sub-wavelength ratio in resonant absorbing devices that are build with three in series elements (mass, spring and damping) like the classical Helmholtz resonator. One possibility to overcome this limit is to add some gain in the system by supplying external energy using, for example, electro-dynamical devices [23,24], thermo-acoustic devices [25] or using the flow [26] which is present in many engineering applications.

FIG. 1 .

 1 FIG. 1. Sketch of an ultra-thin low frequency (UTLF) resonator.

FIG. 2 .

 2 FIG.2. Absorption coefficient α of a perfect resonant absorber as a function of the dimensionless frequency ω for two values of the sub-wavelength ratio: rB = 10 (dashed blue) and rB = 200 (continuous red).

Figure 3 (

 3 Figure 3(a) shows the different elements that composes a manufactured sample of UTLF. The first element is a resistive layer, 1 on Fig.3(a), consisting of a metal wire mesh glued to a short tube (inner radius R= 15 mm, outer radius 19 mm, 1 mm long). Many wire meshes are available with different resistances and the normalized resistance of the one used here was measured at R e = 0.29. The second part is the mass element, 2 on Fig.3(a). The mass is a steel disk (ρ m = 7800 kg.m -3 )with a radius of R -a = 13 mm and a thickness of e = 3 mm. A

FIG. 3 .

 3 FIG. 3. Experimental results:(a) Picture of the elements of the UTLF: 1 Resistive layer, 2 Mass element, 3 Spacer tube to make the cavity, 4 plexiglass cover. (b) Measured absorption coefficient α of the UTLF as a function of frequency. (c) Zoom around the perfect absorption.

FIG. 4 .

 4 FIG. 4. (a) Real part of the impedance (resistance) of the UTLF absorber without the resistive layer (blue) and with the resistive layer (black). (b) Schematic deformation of the membrane.

ACKNOWLEDGMENTS

This work was supported by the "Agence Nationale de la Recherche" international project FlowMatAc No. ANR-15-CE22-0016-01.


Natha Ël

Gawtum Namah

(BOUNDED) TRAVELING COMBUSTION FRONTS WITH DEGENERATE KINETICS

Keywords: October 29, 2019. 2010 Mathematics Subject Classification. Primary 35R35, 80A25, 35C07, 35B10; Secondary 35B27, 80M35 Combustion, free boundary problems, front propagation, bounded traveling wave solutions, degenerate Arrhenius kinetics

We consider the propagation of a flame front in a solid periodic medium. The model is governed by a free boundary system in which the front's velocity depends on the temperature via a kinetic rate which may degenerate. We show the existence of travelling wave solutions which are bounded and global. Previous results by the same authors (cf. [1]) were obtained for essentially positively lower bounded kinetics or eventually which have some very weak degeneracy. Here we consider general degenerate kinetics, including in particular those of Arrhenius type which are commonly used in physics.

Introduction

In this paper, we consider a flame front in a solid periodic medium R x ×R y where the fresh region is a hypograph {x < ξ(y, t)} with a temperature T = T (x, y, t). The evolution of (ξ, T ) will be governed by the free boundary system [START_REF] Alibaud | On the propagation of a periodic flame front by an Arrhenius kinetic[END_REF]    T t -△T = 0,

x < ξ(y, t), t > 0,

ξ t + R(y)K(T ) 1 + ξ 2 y = ξyy 1+ξ 2
y , x = ξ(y, t), t > 0, subject to the boundary conditions

∂T ∂ν = V n , x = ξ(y, t), T (x, y, t) → 0, as x → -∞,

where ν := (1, -ξ y )/ 1 + ξ 2 y is the outward unit normal and V n is the normal velocity of the front. The second equation of [START_REF] Alibaud | On the propagation of a periodic flame front by an Arrhenius kinetic[END_REF] states that the front propagates with a normal velocity V n given by

V n = -R(y)K(T ) -κ,
where κ is the mean curvature and R K is the forcing term. The latter depends on the temperature through a kinetic which is generally of Arrhenius type i.e. of the form

(3) K(T ) = Ae -B T
for some physical positive constants A and B. The heterogeneity is given through the function R which represents the combustion rate. It is typically a periodic step function for a striated medium obtained from a superposition of different materials, which is the most common situation. The heterogeneity may as well come from A, B or other intrinsic parameters such as the diffusivity, etc. Here we will assume that all these parameters are independent of the striations and normalized to one except for R = R(y). We opted for such a simple problem to shed light on the main mathematical difficulties but extensions are possible. For more details and references about the physical model, see [START_REF] Brauner | Propagation of a combustion front in a striated solid medium: a homogenization analysis[END_REF], [START_REF] Chen | Wave propagation under curvature effects in a heterogeneous medium[END_REF] and [START_REF] Alibaud | On the propagation of a periodic flame front by an Arrhenius kinetic[END_REF].

In this work we will focus on the existence of traveling wave solutions (TWS) to (1)- [START_REF] Brauner | Propagation of a combustion front in a striated solid medium: a homogenization analysis[END_REF] i.e. fronts having a constant profile ψ = ψ(y) and moving with a constant speed c > 0. This comes to looking for solutions of the form ξ(y, t) = -c t + ψ(y) and T (x, y, t) = u(x + c t, y).

It is convenient to fix the front through the change of variable x + c t → x which leads to the problem of finding a triplet (c, ψ, u) satisfying ( 4)

       c u x -△u = 0,
x < ψ(y),

∂u ∂ν = c √ 1+ψ 2 y , x = ψ(y), u(x, y) → 0, as x → -∞, and (5)
-c + R(y)K(u) 1 + ψ 2 y = ψ yy 1 + ψ 2 y , x = ψ(y).
This problem has been dealt in [START_REF] Alibaud | On the propagation of a periodic flame front by an Arrhenius kinetic[END_REF] by the present authors for a positively lower bounded combustion rate R and for a positive kinetic K which can eventually very weakly degenerate at zero. This means that K may vanish but not too fast. The worst situation that we were able to consider is when

K(u) ∽ 1 ln 1 u as u → 0.
The existence of a TWS was then proved in [START_REF] Alibaud | On the propagation of a periodic flame front by an Arrhenius kinetic[END_REF] where the speed c > 0 and the profile ψ = ψ(y) is globally defined and bounded. Our aim is to generalize this result to much more degenerate kinetics including those of Arrhenius type [START_REF] Cesaroni | Long-time behavior of the mean curvature flow with periodic forcing[END_REF], which is what seems to be the most realistic. More precisely, we just assume that there are constants R m , R M , and K M such that

(6) R : T → R is measurable with 0 < R m ≤ R(y) ≤ R M , and
K : R + → R + is continuous, nondecreasing with 0 < K(u) < K M ,
where T is the torus R/Z. Let us now define a solution of (4) in the variationnal sense.

Definition 1.

Let Ω := {(x, y); x < ψ(y), y ∈ T}. Then given c ∈ R and ψ ∈ W 1,∞ (T), a function u will be called a variationnal solution of (4) if u ∈ H 1 (Ω) and satisfies

(7) c ˆΩ u x w + ˆΩ ∇u∇w = x=ψ(y) c w 1 + ψ 2 y for all w ∈ H 1 (Ω).
We now state our main result.

Theorem 2. Under (6), there exists a TWS (c, ψ, u) of (1)-( 2) with c > 0, ψ ∈ W 2,∞ (T) and 0 ≤ u ∈ H 1 (Ω). More precisely ψ satisfies (5) a.e. and u is a variationnal solution of (4).

Technically speaking, it is not obvious that the front remains bounded since the forcing term can a priori vanish as u goes to zero. It is indeed well known that for pure geometric propagations of the form

-c + H(y) 1 + ψ 2 y = ψ yy 1 + ψ 2 y , x = ψ(y)
with H not positively lower bounded, one can as well end up with unbounded traveling fronts which in addition may not be globally defined. For more details see [START_REF] Cesaroni | Long-time behavior of the mean curvature flow with periodic forcing[END_REF], where the existence of so-called generalized TWS are discussed by the use of variational techniques. Here, we not only provide a new existence result of TWS for (1)-( 2) with general degenerate kinetics but we also rigorously show that the front is globally defined and bounded.

The rest of the paper is organized as follows. To prove Theorem 2 we will need three key lemmas stated and proved in Sections 2, 3 and 4 respectively. In these preliminary results, we will establish an a priori positive lower bound for the speed c (Lemma 3) and a lower bound for the front's temperature (Lemma 10) based on some adequate monotonicity property of the temperature (Lemma 6). The proof of Theorem 2 will then be done in Section 5.

Positive lower bound for the speed

Here is our first key lemma.

Lemma 3 (Lower bound for c). Assume (6) and let c > 0, ψ ∈ W 2,∞ (T) satisfy (5) a.e. and u ∈ H 1 (Ω) a variationnal solution of (4). Then

c ≥ R m 1 0 K(s) ds.
To prove Lemma 3 we need some technical results. Lemma 4. Let (c, ψ, u) be given as in Lemma 3. Then T u(ψ(y), y) dy = 1.

Proof. Consider any arbitrary L > 0 and take

w(x, y) :=      0 if x ≤ -L, x + L if -L ≤ x ≤ -L + 1, 1 if -L + 1 ≤ x ≤ ψ(y)
in [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF], which is in H 1 (Ω) since ψ is bounded. This gives (8)

T ˆ-L+1 -L c(x + L)u x dx + ˆψ(y) -L+1 cu x dx + ˆ-L+1 -L u x dx dy = c.
After an integration by parts, the sum of the first two terms gives

c T u(ψ(y), y) dy -c T ˆ-L+1 -L u dxdy
where the second integral vanishes when L → +∞ since

0 ≤ c T ˆ-L+1 -L u dxdy ≤ c u L 2 ((-L,-L+1)×T) .
Likewise the third term of (8) goes to zero when L → +∞ and we conclude the proof by passing to the limit in [START_REF] Grisvard | Boundary Value Problems in Non Smooth Domains[END_REF].

We proceed by a second technical lemma.

Lemma 5. Let (c, ψ, u) as in Lemma 3. Then T K(u(ψ(y), y) dy ≥ ´1 0 K(s) ds.

Proof. For any ε > 0, define K ε (u) := 1 ε ´u (u-ε) + K(s) ds. Then K ε is Lipschitz and satisfies [START_REF] Evans | Partial Differential Equations[END_REF]. In particular we can take w(x, y) := K ε (u(x, y)) in [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]. We thus have

c ˆΩ u x K ε (u) + ˆΩ ∇u∇K ε (u) = x=ψ(y) c K ε (u) 1 + ψ 2 y ,
which can be rewritten as

c ˆΩ J ε (u) x dxdy + ˆΩ K ′ ε (u)|∇u| 2 dxdy = T c K ε (u(ψ(y), y)) dy
where J ε (u) := ´u 0 K ε (s) ds. As the second term is nonnegative by [START_REF] Evans | Partial Differential Equations[END_REF] and since

c ˆΩ J ε (u) x dxdy = c T ˆψ(y) -∞ J ε (u) x dx dy = c T J ε (u(ψ(y), y)) dy,
we deduce that

T K ε (u(ψ(y), y)) dy ≥ T J ε (u(ψ(y), y)) dy ≥ J ε T u(ψ(y), y) dy = ˆ1 0 K ε (s) ds
thanks to Jensen's inequality and Lemma 4. We then complete the proof by using the fact that K ε ↑ K as ε ↓ 0.

We are now ready to prove Lemma 3.

Proof of Lemma 3. An integration of ( 5) over T gives

c = T R(y)K(u(ψ(y), y)) 1 + ψ 2 y dy.
We thus have c ≥ R m T K(u(ψ(y), y)) dy and we conclude by Lemma 5.

A monotonicity result

Here is our second key lemma.

Lemma 6 (Monotonicity of u). Let c > 0, ψ ∈ W 2,∞ (T) and u ∈ H 1 (Ω) satisfy (4). Then the function x ∈ (-∞, max ψ] -→ min y:(x,y)∈Ω u(x, y) is nondecreasing. Remark 7.
Recall that u is defined on Ω = {x ≤ ψ(y)}, which explains the above interval of definition.

Remark 8. The min above can be understood in the classical sense since a variationnal solution u ∈ H 1 (Ω) of ( 4) is at least in C(Ω) thanks to classical elliptic regularity results (see e.g. [START_REF] Nittka | Regularity of solutions of linear second order elliptic and parabolic boundary value problems on Lipschitz domains[END_REF] and the references therein or [1, Appendix A.4]).

Proof. Let us take x 0 ∈ (-∞, max ψ) and consider the problem

(9)        c w x -△w = 0, in Ω 0 , ∂w ∂ν = c √ 1+ψ 2 y , on ∂Ω 0 ∩ {x > x 0 }, w = w 0 , on ∂Ω 0 ∩ {x = x 0 },
where Ω 0 := Ω ∩ {x > x 0 } and w 0 := min y u(x 0 , y).

Then the constant function w 0 is a subsolution of ( 9) whereas u is a supersolution. By comparison, (10) u(x, y) ≥ w 0 for all (x, y) ∈ Ω 0 and it follows that min y:(x,y)∈Ω0

u(x, y) ≥ w 0 = min y:(x,y)∈Ω0

u(x 0 , y) for all x ∈ [x 0 , max ψ].

Remark 9. Note that ∂Ω 0 may not be Lipschitz at truncated points. Nevertheless the comparison result used in the above proof holds by standard arguments [START_REF] Brézis | Analyse fonctionnelle : Théorie et Applications[END_REF][START_REF] Grisvard | Boundary Value Problems in Non Smooth Domains[END_REF].

For completeness, a short verification is given in Appendix B.

Positive lower bound for the temperature

In this section we prove a general result concerning the temperature. Consider for this sake the following problem (11)

       c u x -△u = 0, in Ω, ∂u ∂ν = c √ 1+ψ 2 y , x = ψ(y), u(x, y) → 0, as x → -∞,
where c ∈ R and ψ ∈ W 2,∞ (T) satisfy, for some given c 0 , c M and a continuous function G,

   0 < c 0 ≤ c ≤ c M , (12)
|ψ yy | ≤ G(ψ y ) a.e. and min y∈T ψ(y) = 0.

We then claim our last key result.

Lemma 10 (Lower bound on u). Let G ∈ C(R) and 0 < c 0 ≤ c M be given. Then there exists α > 0 such that for all c ∈ R and ψ ∈ W 2,∞ (T) verifying (12), the unique variationnal solution u ∈ H 1 (Ω) of (11) satisfies min y∈T u(0, y) ≥ α.

Proof. Let us define

α := inf c,ψ min y u(0, y); u ∈ H 1 (Ω) solution of (11) ,
the infimum being taken on all c ∈ R and ψ ∈ W 2,∞ (T) satisfying ( 12). The min is to be understood in the classical sense by Remark 8. We have to show that α is positive. Consider for this sake a minimizing triplet (c n , ψ n , u n ) such that (13)

       c n and ψ n ∈ W 2,∞ (T) satisfy (12), u n ∈ H 1 (Ω n := {x < ψ n (y)})

Consider now the limiting domain

Ω ∞ := ∪ n Int(∩ k≥n Ω k ),
where Int is the interior. Note that Ω ∞ is open. Note also that (0, y) ∈ Ω ∞ (the closure of Ω ∞ ) since each ψ n was chosen such that ψ n ≥ 0, so that for all ε > 0, (-ε, y) ∈ Ω ∞ . The analysis which follows will depend on whether (0, y) is on ∂Ω ∞ (the boundary of Ω ∞ ) or in Ω ∞ . In the sequel, B(y, r) := {z ∈ T; d(y, z) < r} will denote a ball of T with d(y, z) := dist(P -1 (y), P -1 (z)) and where P : R → R/Z is the usual projection. We denote similarly by B((x, y), r) the balls of R × T.

First case : (0, y) ∈ ∂Ω ∞ .

Claim 1. In this case we claim that there exist an increasing sequence (n p ) p in N and (w p ) p in T such that (15) w p → y and ψ np (w p ) → 0 as p → ∞.

Indeed B((0, y), 1 p ) intersects Ω c ∞ = ∩ n (∪ k≥n Ω c k )
for any integer p ≥ 1, so that there exists a sequence (x p , y p ) such that (x p , y p ) ∈ B((0, y),

1 p ) ∩ (∪ k≥p Ω c k ). Since B((0, y), 1
p ) is open and (x p , y p ) is the limit of some sequence in ∪ k≥p Ω c k , there exist n p ≥ p and (x p , ỹp ) such that (x p , ỹp ) ∈ B((0, y), 1 p ) ∩ Ω c np . Taking w p = ỹp gives (15). Indeed w p ∈ B(y, 1 p ) and 0 ≤ ψ np (w p ) ≤ xp ≤ 1 p by construction. Moreover the sequence n p can be chosen increasing up to taking subsequences if necessary since n p ≥ p goes to infinity as p → ∞. This proves Claim 1.

We proceed by another claim: Claim 2. For all ε > 0 and p 0 ∈ N, there exist y ∈ B(y, ε) and p ≥ p 0 such that

(16) |ψ ′ np (y)| < ε.
Indeed if the above does not hold, then there exist ε > 0 and p 0 ∈ N such that for any p ≥ p 0 , |ψ ′ np (y)| ≥ ε for all y ∈ B(y, ε). Up to taking a subsequence, we can suppose that for all p, ψ ′ np (y) ≥ ε in B(y, ε). For ε small enough, identifying B(y, ε) with the interval

I := P -1 (B(y, ε)) in R, we have in I ψ np (y -ε) = ψ np (w p ) + ˆy-ε wp ψ ′ np (y) dy ≤ ψ np (w p ) -ε(w p -(y -ε)).
This is not possible because as ψ np ≥ 0 by definition, the LHS is ≥ 0 whereas the RHS goes to -ε 2 < 0 in the limit p → ∞. This proves Claim 2. Now (16) implies that there exist a subsequence (n p k ) k and another sequence ( wk ) k such that wk → y and

ψ ′ np k ( wk ) → 0 as k → ∞.
Recalling (15) and renaming the sequences from the beginning of the proof for simplicity, we have proved that we have (w n ) n and ( wn ) n such that (17) w n → y, wn → y with ψ n (w n ) → 0 and ψ ′ n ( wn ) → 0 as n → ∞. This leads to our next claim below concerning the fronts. Claim 3. There exists ε 0 such that (ψ n ) n is bounded in W 2,∞ (B(y, ε 0 )). In particular ψ n → ψ uniformly in B(y, ε 0 ), up to some subsequence, for some ψ ∈ W 2,∞ (B(y, ε 0 )).

Indeed the existence of ε 0 as well as the uniform bound for ψ ′ n in B(y, ε 0 ) are provided by Lemma 12 in Appendix B and (17). The bounds for ψ n follow from (17) again and the ones for ψ ′′ n from (12). With these bounds in hand, Ascoli-Arzela theorem completes the proof of Claim 3.

To conclude, we will pass to the limit in the boundary problem satisfied by the temperature, cf. (11). In the case where (0, y) ∈ ∂Ω ∞ , we only need to handle the boundary condition on the moving interface {x = ψ n (y), y ∈ B(y, ε 0 )}. Before hand, it is convenient to extend u n onto R × T by ( 18) ũn (x, y)

:= u n (x, y) if (x, y) ∈ Ω n , u n (2ψ n (y) -x, y) otherwise.
Consider now the cylinder Q := {(x, y); y ∈ B(y, ε 0 )}. We then have Claim 4. There exists 0 ≤ ũ ∈ C 1 ( Q) such that ũn → ũ locally uniformly in Q as n → ∞, up to a subsequence, with

∂ ũ ∂ν = c 1 + ψ 2 y on x = ψ(y), y ∈ B(y, ε 0 ) .
To show this claim, apply Lemma 11 of Appendix A with B 0 := B(y, ε 0 ) and B 1 := ∅. Note that (23) holds by ( 14) and Claim 3.

To conclude we have u n (0, y n ) → ũ(0, y) = α as n → ∞, cf. ( 13), ( 14), (17), as well as Claims 3 and 4, and it remains now to show that ũ(0, y) > 0. But as ũ ≥ 0 around (0, y) and ∂ ũ ∂ν (0, y) > 0, we end up with ũ(0, y) > 0. This completes the proof of Lemma 10 when (0, y) ∈ ∂Ω ∞ .

Second case : (0, y) ∈ Ω ∞ . Recalling that Ω ∞ = ∪ n Int(∩ k≥n Ω k ), there exist now n 0 and an open O ⊂ R 2 such that (19) (0, y) ∈ O ⊆ Ω n for all n ≥ n 0 .

Roughly speaking, we will show that u n converges to some nontrivial ũ ≥ 0 which satisfies the first equation of (11) in O. This will imply that α = ũ(0, y) > 0 by an argument of propagation of maximum. The overall idea to get a nontrivial limit is to work eventually in a larger open O with O ⊆ O ⊆ Ω ∞ , so that the nontrivial boundary condition of (11) holds on some part of ∂O.

To construct O, take (z n ) n in T such that z n is a minimizer of ψ n . Considering a subsequence if necessary, we can assume that (20)

ψ n (z n ) = ψ ′ n (z n ) = 0 and z n → z as n → ∞
, for some z ∈ T. The above property looks like (17) and we have the following analogous of Claim 3 whose proof is similar. Claim 5. There exist ε 0 and ψ ∈ W 2,∞ (B(z, ε 0 )) such that (ψ n ) n is bounded in W 2,∞ (B(z, ε 0 )) and ψ n → ψ uniformly in B(z, ε 0 ), up to some subsequence.

Using now (19), there exist η > 0 and ε 1 > 0 such that (-η, η) × B(y, ε 1 ) ⊂ Ω n for all n ≥ n 0 . Recall that Ω n = {x < ψ n (y)} so that we have ψ n (y) ≥ η, for all n ≥ n 0 and y ∈ B(y, ε 1 ).

Using in addition (20) and Claim 5, we note in particular that y = z. This enables us to choose ε 0 and ε 1 (smaller if needed) so that B(z, ε 0 ) ∩ B(y, ε 1 ) = ∅. We can now define O (resp. Õ) as follows:

O (resp. Õ) := {(x, y); x < χ(y) (resp. x < χ(y))} , where χ(y) (resp. χ(y)) :=      ψ(y) (resp. +∞) if y ∈ B(z, ε 0 ), η if y ∈ B(y, ε 1 ), 0 otherwise.
As previously announced, we need to identify both the PDE and the boundary condition of the limiting temperature. We propose Claim 6. There exists

0 ≤ ũ ∈ C 1 ( Õ) ∩ C 2 (O) such that ũn → ũ locally uniformly in Õ as n → ∞,
up to a subsequence, where

(21)    c ũx -△ũ = 0, in O, ∂ ũ ∂ν = c 1+ψ 2 y , x = ψ(y), y ∈ B(z, ε 0 ).
To show this claim, apply again Lemma 11 with now B 0 := B(z, ε 0 ), η as above and B 1 := B(y, ε 1 ), so ( 23) is ensured by Claim 5 and (24) holds by what precedes.

Once again u n (0, y n ) → ũ(0, y) = α and it remains to show that ũ(0, y) > 0. For this sake, note that ũ is not identically zero because c > 0 in the boundary condition of (21). By the strong maximum principle, ũ(0, y) > 0 since it cannot achieve its minimum in the connected open O otherwise it will vanish everywhere in O, cf. [START_REF] Evans | Partial Differential Equations[END_REF]Sec. 6.4.2]. This completes the proof of Lemma 10.

Proof of Theorem 2

Now we are ready to prove our main result. For this sake, let us consider Eqns. ( 4)-( 5) with K(u) replaced by the truncated function

K n (s) := max K(s), 1 n .
By [1, Thm. 3.2], the latter system admits a solution (c n , ψ n , u n ) for any integer n ≥ 1, with c n > 0, ψ n ∈ W 2,∞ (T) and 0 ≤ u n ∈ H 1 ({x < ψ n (y)}). Since ψ n is defined up to an additive constant, we can choose it such that min y ψ n (y) = 0. Now from Lemma 3, we know that

c n ≥ c 0 := R m ˆ1 0 K(s) ds > 0 for all n.
Likewise we have the following upper bound by [5, Thm. 2.1] and Assumption ( 6):

c n ≤ c M := R M K M for all n.
From the front's equation ( 5), we then have

|(ψ n ) yy | ≤ 2c M (1 + ψ 2 ny ) 3 2 .
applying Lemma 10 with G(h) := 2c M (1 + h 2 ) 3 2 leads to the existence of an α > 0, α depending only on c 0 , c M and G, such that min y u n (0, y) ≥ α for all n. Now as x -→ min y u(x, y) is nondecreasing by Lemma 6, we have u n (ψ n (y), y) ≥ α, and consequently by [START_REF] Evans | Partial Differential Equations[END_REF] and the definition of K n , we end up with K n (u n (ψ n (y), y)) ≥ K n (α) ≥ K(α) for all n and y ∈ T.

We proceed by setting

H n (y) := R(y)K n (u n (ψ n (y), y)).
Since K(α) > 0 by [START_REF] Evans | Partial Differential Equations[END_REF], the above uniform positive lower boundedness of (H n ) n enables us to use the results of [START_REF] Alibaud | On the propagation of a periodic flame front by an Arrhenius kinetic[END_REF] to pass to the limit as n → ∞. Indeed an application of [1, Lemmas 2.5 & 2.7] gives the existence of (H, c, ψ, u) limit of (H n , c n , ψ n , u n ), up to a subsequence, where

       c u x -△u = 0, in {x < ψ(y)}, ∂u ∂ν = c √ 1+ψ 2 y , x = ψ(y), u(x, y) → 0, as x → -∞, and
-c + H(y) 1 + ψ 2 y = ψ yy 1 + ψ 2 y , x = ψ(y),
in the sense and with the regularity claimed in Theorem 2. It remains to identify H(y). As H n ⇀ H in L ∞ weak - * (see [START_REF] Alibaud | On the propagation of a periodic flame front by an Arrhenius kinetic[END_REF]), we have

ˆT H n (y)ϕ(y) dy → ˆT H(y)ϕ(y) dy for all ϕ ∈ C ∞ c (T).
But by [1, Lemma 2.8], we have u n (ψ n (y), y) → u(ψ(y), y) for a.e. y ∈ T, so that, by the definition of H n , the assumptions in [START_REF] Evans | Partial Differential Equations[END_REF], and Lebesgue's theorem, we also have ˆT H n (y)ϕ(y) dy → ˆT R(y)K(u(ψ(y), y))ϕ(y) dy for all ϕ ∈ C ∞ c (T).

Finally we end up with H(y) = R(y)K(u(ψ(y), y)) and the proof of Theorem 2 is complete.

Appendix A. A convergence result for the temperature

Let us now state and prove Lemma 11, see below, that we have admitted during the proof of Lemma 10. As before, consider

(c n ) n in R, (ψ n ) n in W 2,∞ (T) and (u n ) n in H 1 (Ω n = {x < ψ n (y)}) such that (22)        c n (u n ) x -△u n = 0, in Ω n ∂un ∂ν = cn √ 1+ψ 2 ny , x = ψ n (y), u n (x, y) → 0, as x → -∞,
as well as

(23)      0 < c n ≤ c M and c n → c > 0 as n → ∞, (ψ n ) n is bounded in W 2,∞ (B 0 ), and
ψ n → ψ uniformly in B 0 as n → ∞,
for some reals c M and c, nonempty open B 0 ⊂ T and ψ ∈ W 2,∞ (B 0 ). Let moreover η > 0 and B 1 ⊂ T be another open, eventually empty, such that (24)

B 0 ∩ B 1 = ∅ and (-η, η) × B 1 ⊂ ∩ n Ω n .
Define then U n := {(x, y); x < χ n (y)}, where

χ n (y) :=      ψ n (y) if y ∈ B 0 , η if y ∈ B 1 , 0
otherwise. Likewise we define the limiting domain U := {(x, y); x < χ(y)}, with χ the same as χ n with ψ n replaced by ψ. We further consider the extended domain Ũ := {(x, y); x < χ(y)} , where χ(y) :=

     +∞ if y ∈ B 0 , η if y ∈ B 1 , 0 otherwise.
Recall that ũn is defined in ( 18) and belongs at least to

C ∩ H 1 (R × T) because u n ∈ C ∩ H 1 (Ω n ) and ψ n ∈ W 1,∞ (T).
Lemma 11. Under the above assumptions, there is 0 Proof. We proceed in several steps.

≤ ũ ∈ C 1 ( Ũ ) ∩ C 2 (U ) such that ũn → ũ
Step 1: A priori estimates. We claim that (26) 0 ≤ u n (x, y) ≤ e cnx ∀(x, y) ∈ Ω n and ´Ωn |∇u n | 2 ≤ c n .

The first estimate follows since the functions 0 and e cnx are respectively sub and supersolution of (22), see e.g. [START_REF] Alibaud | On the propagation of a periodic flame front by an Arrhenius kinetic[END_REF]Appendix A.2]. For the second estimate, take w := u n in [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF], which is a weak formulation of (22) as well, to see that

c n ˆΩ u 2 n /2 x + ˆΩ |∇u n | 2 = x=ψ ( y) u n c n 1 + (ψ ′ n ) 2 dy = c
Ũ ∩ {x < ψ(y), y ∈ B 0 } = U ∩ {y ∈ B 0 },
the LHS of (28) thus converges to ´U (c ũx ϕ + ∇ũ∇ϕ) . Using again the uniform convergence of ψ n to ψ B 0 to handle the RHS, we obtain that

ˆU (c ũx ϕ + ∇ũ∇ϕ) = c B0 ϕ(ψ(y), y) dy ∀ϕ ∈ C ∞ c ( Ũ ).
This proves that ũ ∈ H 1 loc ( Ũ ) is a variational solution of (25). Note that it is nonnegative by (26).

Step 3: Local uniform convergence. When calling for Lemma 11, it was important that ũn → ũ locally uniformly in Ũ , especially around the moving interfaces {x = ψ n (y), y ∈ B 0 }. Let us adapt an idea of [START_REF] Nittka | Regularity of solutions of linear second order elliptic and parabolic boundary value problems on Lipschitz domains[END_REF] to establish this convergence. It consists in considering the problem satisfied by ũn in order to apply standard interior elliptic estimates, cf. also [START_REF] Alibaud | On the propagation of a periodic flame front by an Arrhenius kinetic[END_REF]Appendix A.4].

By [1, Lemma A.5], b n (ũ n ) x -div(A n ∇ũ n ) = -(b n ) x in D ′ (R × T), with b n (x, y) = c n 1 {x<ψn(y)} -c n 1 {x>ψn(y)} , A n (x, y) = 1 0 0 1 1 {x<ψn(y)} - 1 + 4 ψ 2 ny 2 ψ ny 2 ψ ny 1 1 {x>ψn(y)} .
We claim that there are Λ ≥ λ > 0, ν ≥ 0, such that for each n and a.e. x, y ∈ Ũ ,

Λ ≥ A n (x, y) ≥ λ and λ -1 |b n (x, y)| ≤ ν.
For y / ∈ B 0 , this follows once again from the fact Ũ ∩ {y / ∈ B 0 } ⊂ Ω n , so b n = c n and A n = Id everywhere in that region. For y ∈ B 0 , recall that (ψ n ) n is bounded in W 1,∞ (B 0 ) and the proof is the same as in [START_REF] Alibaud | On the propagation of a periodic flame front by an Arrhenius kinetic[END_REF]Lemma A.7]. Now by [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Thm. 8.24] combined with (27), (ũ n ) n is locally equi-Hölder continuous in Ũ and the convergence of ũn to ũ is local uniform, up to taking another subsequence if necessary.

Step 4: C 1 regularity. This step is now standard. For example, let us call for [8, Thm. 2.4.

Appendix B. Technical features

Here is another result used in the proofs. The above set is not empty because |h(y 0 )| ≤ R and h is continuous, so r 0 is well-defined in (0, +∞]. We claim that r 0 ≥ R/C 2R and this will show that any r ≤ R/C 2R fits the lemma. If indeed r 0 < R/C 2R then for all y ∈ B(y 0 , r 0 ), Proof of (10). The domain Ω 0 is not Lipschitz at (x 0 , y 0 ) ∈ ∂Ω 0 whenever ψ(y 0 ) = x 0 and ψ y (y 0 ) = 0. But the variational solution u ∈ H 1 (Ω) of ( 4) is in C 1 (Ω) ∩ C 2 (Ω) by standard elliptic regularity; cf. the fourth step of the proof of Lemma 11. The strong maximum principle [START_REF] Evans | Partial Differential Equations[END_REF]Sec. 6.4.2] then implies that min Ω0 (uw 0 ) is not achieved in Ω 0 or on ∂Ω 0 ∩ {x > x 0 } where ∂(u-w0) ∂ν > 0 pointwise. It is then achieved on ∂Ω 0 ∩ {x = x 0 } where u ≥ w 0 pointwise, including at (x 0 , y 0 ) such as above.

  solves (11) and lim n→∞ min y u n (0, y) = α. Since u n ∈ C(Ω n ) by Remark 8 and T and [c 0 , c M ] are compact, there exist (y n ) n and y in T as well as c ∈ [c 0 , c M ] such that (14) min y u n (0, y) = u n (0, y n ), y n → y and c n → c when n → ∞.

  locally uniformly in Ũ as n → ∞, up to a subsequence, where ψ(y), y ∈ B 0 .

  2.7] to get the regularity around the boundary {x = ψ(y), y ∈ B 0 }. For any arbitrary ρ ∈ C ∞ c ( Ũ ), consider a nonempty C 1,1 open V such that U ∩ supp ρ ⊂ V ⊂ U and rewrite (25) for the function v := ũρ as follows: v -△v = f in V, ∂v ∂νV = g on ∂V, with f = ũρcρũ x -2∇ũ∇ρ -ũ△ρ and on ∂V ∩ {x = ψ(y), y ∈ B 0 }, 0 elsewhere on ∂V , where ν V is the outer unit normal of ∂V . Such a regular open V exists because supp ρ ⊂ Ũ and ψ ∈ W 2,∞ (B 0 ). Using then that ũ ∈ L ∞ ∩ H 1 ( Ũ ), we have f ∈ L 2 (V ) and g ∈ H 1/2

Lemma 12 .

 12 Let G ∈ C(R) and R > 0. There exists r > 0 such that for any h ∈ W 1,∞ (T) and y 0 ∈ T such that |h y | ≤ G(h) a.e. in T and |h(y 0 )| ≤ R, we have |h(y)| ≤ 2R ∀y ∈ B(y 0 , r). Proof. Set C 2R := max{max [-2R,2R] |G|, 1}, let h = h(y) and y 0 be as in the lemma, and define r 0 := sup{r > 0; sup B(y0,r) |h| ≤ 2R}.

  |h(y)| ≤ |h(y0 )| + r 0 |h y | L ∞ (B(y0,r0)) ≤ R + r 0 C 2R < 2R,but one can then choose r 1 > r 0 such that sup B(y0,r1) |h(y)| ≤ 2R and this contradicts the definition of r 0 .It only remains to check that (10) holds, cf. Remark 9.

  → 1 x<ψ(y) a.e. in Ũ ∩ {y ∈ B 0 } by the uniform convergence of ψ n to ψ in B 0 , cf. (23). Noting that

		ˆΩn	ˆΩn			B0
	(28)	c n	(u n ) x ϕ +	∇u n ∇ϕ = c n	ϕ(ψ n (y), y) dy.
	The LHS can be rewritten as
	ˆU∩{y / ∈B0}	(c n (u n ) x ϕ + ∇u n ∇ϕ) +	ˆŨ∩{y∈B0}	(c n (u n ) x ϕ + ∇u n ∇ϕ) 1 x<ψn(y) ,
	since supp ϕ ∩ {y / ∈ B 0 } ⊂ U ∩ {y / ∈ B 0 }. This converges to ˆŨ∩{y∈B0} ˆU∩{y / ∈B0} (c ũx ϕ + ∇ũ∇ϕ) + (c ũx ϕ + ∇ũ∇ϕ) 1 x<ψ(y) ,
	because 1 x<ψn(y)
						n
	by Lemma 4. The proof of (26) is complete since the first term is equal to
	c n ´T u 2 n (ψ n (y), y)/2 dy ≥ 0. Consequently, there is a constant C independent of n such that
					ˆŨ
	(27)		max Ũ |ũ n | ≤ C and	|∇ũ n | 2 ≤ C,
	thanks to (18) and (23). Note that ϕ n is not assumed uniformly bounded outside B 0 , but in that region we use that Ũ ∩ {y / ∈ B 0 } ⊂ Ω n to deduce (27) from (26).
	Step 2: Limiting problem.
	By (27), u n converges weakly to some ũ in H 1 loc ( Ũ ) up to a subsequence. Taking any test ϕ ∈ C ∞ c ( Ũ ) in (22), we have

  (∂V ). It follows that v ∈ H 2 (V ) by [8, Thm. 2.4.2.7]. Since ρ is arbitrary, ũ ∈ H 2 loc (U ) thus a fortiori in H 2 loc ( Ũ ) because ψ ∈ W 2,∞ (B 0 ). By Sobolev's embeddings [4, Cor. IX.14], ∇ũ ∈ L p loc ( Ũ ) for any p ∈ [2, ∞), so f ∈ L p (V ) and g ∈ W 1-1/p,p (∂V ) thanks to trace results [8, Thm. 1.5.1.3]. Here we used that ũ ∈ L ∞ ∩ W 1,p loc ( Ũ ). Applying again [8, Thm. 2.4.2.7], v ∈ W 2,p (V ), ũ ∈ W 2,p loc ( Ũ ), and by [4, Cor. IX.14], ũ ∈ C 1 ( Ũ ). To get the C 2 regularity in U , use e.g. [4, Rem. 26].


Clarisse Bardiot

Clarisse Bardiot Measuring

Merce Cunningham

Measuring Merce Cunningham : a theatre analytics research

Measuring Merce Cunningham : a theatre analytics research Clarisse Bardiot

Abstract :

To paraphrase the culture analytics definition, theatre analytics is the data-driven analysis of theatre, whether it concerns theatre history, drama or mise-en-scène. To understand what quantitative methodologies can bring to the knowledge of theatre, I propose a case study of Merce Cunningham, one of the most influential choreographers of the 20th century. The dataset contains 183 works from 1938 to 2009 (including 86 Dance Capsules) and 347 people. What can we infer from beyond the data about the MCDC history, Cunningham's aesthetics and documentation strategies ?

Paper :
Theatre studies is a largely under-discussed topic in digital humanities research projects. It's lagging behind the first wave of digital humanities scholarship, « focus[ing] on large-scale digitization projects and the establishment of technological infrastructure » [START_REF] Presner | Digital Humanities 2.0: a report on knowledge[END_REF]. Theatre studies remains on the fringe of a growing phenomenon : culture analytics. In the context of big and complex datasets, culture analytics « is the data-driven analysis of culture » (IPAM, 2016). I suggest the expression « theatre analytics » [START_REF] Bardiot | Arts de la scène et culture analytics[END_REF]. To paraphrase the culture analytics definition, theatre analytics is the data-driven analysis of theatre, whether it concerns theatre history [START_REF] Caplan | Reassessing Obscurity: The Case for Big Data in Theatre History[END_REF], drama or mise-en-scène. To understand what quantitative methodologies can bring to the knowledge of theatre, I propose a case study of Merce Cunningham. What can we learn about Merce Cunningham, one of the most influential choreographers of the 20th century, thanks to theatre analytics ? A leader of the American avant-garde throughout his seventy year career from 1938 to 2009, he establishes in 2000, in the twilight of his career, the Merce Cunningham Trust, in order to preserve the integrity of his work. At the same time, he decides to dissolve the Merce Cunningham Dance Company (MCDC) two years after his death and a legacy tour. This is an unprecedented initiative. On one hand, it demonstrates exceptional effort and dedication to document the works. On the other hand, it challenges the ephemeral nature of performing arts : 86 out of 183 choreographies are documented with "digital Dance Capsules" "so that it may be performed in perpetuity" (Dance Capsules, n.d.). By the way, two groups of works are defined : the canon (key works with extensive documentation in order to perform them again and again) ; the auxiliary (minor works with no documentation available to the public and de facto impossible to replay).

The data was collected from the Merce Cunningham Trust website. It concerns theatre production and cast, Dances Capsules documentation and the history of the MCDC. The dataset contains 183 works from 1938 to 2009 (including 86 Dance Capsules) and 347 people. We can identify three main data categories : people, works and documentation. What can we infer from beyond the data about the MCDC history, Cunningham's aesthetics and documentation strategies ?

Measuring means measuring instruments. I used various and complementary tools in order to vary the approaches and analysis of the same dataset : Gephi for network analysis ; Palladio for geographic and temporal representation ; spreadsheet (Excel, Open Office, Datamatic) for statistics analysis. This paper will present the first results of this research, part of it conducted with students during a graduate « introduction to digital humanities » course. Statistical diagrams show three different periods of Cunningham's work ; a stylistic signature with a preference for pieces that are 30 minutes long, and for soli, sextets and works with 13 to 15 dancers ; a general trend towards more dancers and more length ; the special place of soli in order to articulate the canon and the auxiliary ; the organization of documents in the Dance Capsules. Network analysis let me define two different ways of collaboration, the « star » and the « spiral », and raises awareness on pivotal dancers. Geographic representation highlights relations between Europe and the United States.

In a wider historical perspective, it would be interesting to compare these preliminary results with other datasets. One example : two patterns have been identified in the Cunningham collaborations network : the star (figure 1), with discontinuous, centralized collaborations and groups separated from each other; the spiral (figure 2), with continuous, collective collaborations and one group growing organically. The change from the star to the spiral takes place when the company is created. Do these patterns characterize other choreographers and directors careers ? Is the creation of the company the main factor causing the evolution from the first pattern towards the second one ? While a well-worn issue -we do know that the creation of a company plays a crucial role in a career -the fact remains that "theatre analytics" let us visualize the patterns this break constitutes (or maybe not) and define different ways of collaborations.

Figure 1 :

 1 Figure 1 : Merce Cunningham's collaborations network before 1954. The star pattern. Pink, dancers; orange, composers ; green, stage designers ; blue, choreographer.

Figure 2 :

 2 Figure 2 : Merce Cunningham's collaborations network after 1954. The spiral pattern.