Timothée Jamin

Charlotte Py

Eric Falcon

Instability of the origami of a ferrofluid drop in a magnetic field

Keywords: numbers: 47.65.Cb, 46.32.+x, 68.08.-p

Capillary origami is the wrapping of an usual fluid drop by a planar elastic membrane due to the interplay between capillary and elastic forces. Here, we use a drop of magnetic fluid whose shape is known to strongly depend on an applied magnetic field. We study the quasi-static and dynamical behaviors of such a magnetic capillary origami. We report the observation of an overturning instability that the origami undergoes at a critical magnetic field. This instability is triggered by an interplay between magnetic and gravitational energies in agreement with the theory presented here. Additional effects of elasticity and capillarity on this instability are also discussed.

Generally, a solid in contact with a static liquid interface is undeformed by surface tension forces at large scales. However, for sub-millimetric scales or when gravity is negligible, capillary forces may deform an elastic structure in various domains (see [1] for a review): adhesion (coalesence of wet hairs [START_REF] Cohen | Proc. Natl. Acad. Sci. USA[END_REF]), biological systems (floating flowers [START_REF] Reis | [END_REF]), or industrial applications (microscale fabrication such as 3D photovoltaic cells or MEMS [1]). The most well-known phenomenon involving the interplay between elasticity and capillarity is capillary origami. It concerns the spontaneous wrapping of a liquid droplet by a planar elastic sheet when the capillary forces dominate the restoring elastic ones [4,5]. Various folding shapes (spherical, cubic or triangular encapsulation) can then be tuned from the geometry of the initial flat membrane. A new challenge is to accurately control the folding and unfolding of the origami without the use of mechanical part. Such a capillary origami control has been recently performed by means of an electric field and could lead to potential applications to digital displays [6,7]. However, the capillary origami is generally considered as quasi-static phenomenon and dynamical studies are rare. To our knowledge, only one experiment concerns the dynamical selection of the final shape of the capillary origami when a drop impacts an elastic membrane [START_REF] Rivetti | proceedings of 13ème Rencontre du Non Linéaire[END_REF].

In this Letter, a drop of magnetic fluid is deposited on a flat elastic membrane and is submitted to a magnetic field. The shape of the ferrofluid drop is known to strongly depend on the applied magnetic field [START_REF] Arkhipenko | [END_REF][10][START_REF] Blums | Magnetic Fluids[END_REF]. The wrapping of the drop by the thin elastic membrane (origami) is thus expected to be strongly modified by the magnetic field. Both quasi-static and dynamical behaviors of such a magnetic capillary origami are reported here. The most striking one is the observation of an overturning instability that the origami undergoes at a critical magnetic field. This instability is shown to arise from the interplay between magnetic and gravitational energies.

The ferrofluid used is an ionic aqueous suspension synthesized with 12.4% by volume of maghemite particles (Fe 2 0 3 ; 7 ±0.3 nm in diameter) [START_REF]The ferrofluid synthesis has been performed by D. Talbot (LI2C, Univ Paris 6) according to Massart's method: R. Massart[END_REF]. The properties of this magnetic fluid are: density ρ = 1550 kg/m 3 , surface tension γ = 43 ± 3 mN/m, initial magnetic susceptibility χ i = 0.75, magnetic saturation M sat = 36 × 10 3 A/m, and dynamic viscosity 1.4 ×10 -3 N s/m 2 . The elastic membranes are made of polydimethylsiloxane (PDMS -Dow Corning Sylgard 184), a 10:1 polymer/curing agent mix. The PDMS is spin coated at rotation rates in the range 1800 -2800 rpm over Emery polishing paper (grit 0) with average roughness of 192 µm. The thickness h of the membrane depends on the rotation rate and ranges from 50 to 100 µm. Its roughness significantly reduces the adhesion of the membrane on the substrate [6]. The ferrofluid drop alone or wrapped by the elastic membrane (origami) is then placed between two horizontal coils, 25 cm (resp. 53 cm) in inner (resp. in outer) diameter, 7 cm far apart. A dc current, I, is supplied to the coils in series by a power supply (50 V/50 A). The vertical magnetic induction, B, generated is up to 780 G and is measured by a Hall probe located near the drop. The magnitude of B is proportional to I and is controlled either manually or by means of a ramp generator of typical speed rate 0.4 A/s. Deformations of the drop are visualized from a side view by means of a 45 degree mirror reflecting a diffuse lighting towards a high-resolution camera (Pixelink 2208 × 3000 pixels) located above the drop, and are recorded up to a 150 Hz sampling.

We first study the deformation of a ferrofluid drop alone deposited on a superhydrophobic surface and submitted to a vertical magnetic field. As shown in the inset of Fig. 1, the drop is quasi-spherical at B = 0 (a). When B is increased, the drop lengthens in the direction of B and its shape changes continuously from a semi-ellipsoid (b), then a pointed ellipsoid (c) up to a sharp tip (d). By analogy with a semi-ellipsoid, we define a (resp. b) as the semi-major (resp. minor) axis of the deformed drop. a is measured from the top of the drop to the horizontal plane where its width is largest, whereas b is its half-maximal width (see inset of Fig. 1). Both are measured on the image of the drop with a 5% accuracy. The aspect ratio a/b of the drop is plotted in Fig. 1 as Typeset by REVT E X a function of the magnetic Bond number, Bo m . This dimensionless number characterizes the order of magnitude of the ratio between magnetic and capillary energies [START_REF] Blums | Magnetic Fluids[END_REF]. Bo m ≡ χ(B)B 2 R 0 /(µ 0 γ) with R 0 the initial drop radius at B = 0, µ 0 = 4π × 10 -7 H/m the magnetic permeability of the vacuum, and χ(B) the magnetic susceptibility of the ferrofluid which is a known decreasing function of B with χ i ≡ χ(B = 0) [START_REF] Rosensweig | Ferrohydrodynamics[END_REF]. The elongation of the drop is found to strongly increase with Bo m since the ferrofluid tends to align towards the direction of B. Moreover, as shown in Fig. 1, both curves performed for two different R 0 superimpose underlying that a small drop is less deformed than a larger one for the same applied B. Note that such a deformation has been also observed when the ferrofluid drop is surrounded by a non miscible fluid of almost same density leading to a full ellipsoidal shape: a/b is then found to increase continuously with B for small χ i ∼ 1 [START_REF] Arkhipenko | [END_REF] whereas for large one (χ i 20) a discontinuous deformation occurs [10]. The elongation of a full ellipsoidal droplet can be described theoretically in the limit of a linear [10,[START_REF] Blums | Magnetic Fluids[END_REF] or nonlinear [START_REF] Ivanov | [END_REF] drop magnetization. To our knowledge, no analogous analytical computation exists for a ferrofluid drop with a semi-ellipsoid shape (as the one in Fig. 1).

Let us now focus on the behavior of a ferrofluid drop wrapped by an elastic membrane and submitted to a magnetic field. A geometric shape of the membrane is manually cut out from the PDMS layer and placed on the superhydrophobic surface. The membrane shape is chosen to be an equilateral triangle of side L (5 ≤ L ≤ 15 mm) such that the closed state of the origami will have a pyramid-like shape [4]. A typical experiment is as follows. First, at B = 0, a ferrofluid drop is deposited on the planar membrane. Second, the volume of the drop is adjusted such that, due to the competition between capillary and elastic forces, the three corners of the membrane touch themselves and wrap the spherical drop as shown in Fig. 2a. When B is turned on and is increased, the drop lengthens vertically (Fig. 2b), then exhibits a pyramidal shape at higher B due to the presence of the membrane (see Fig. 2c). For a critical value B c , corresponding to a critical Bond, Bo c m , the origami suddenly overturns, then oscillates around its vertical position before stopping its rocking motion (see movie in [15]). This leads to a new static configuration of the origami (see Fig. 2d): the drop now forms a cone at its top, and rests on one corner of the membrane, the two other corners wrapping it horizontally. If B is now decreased, the origami falls forward, and never returns to its initial configuration thus showing a hysteretic behavior.

To quantitatively study the above instability, the maximum height z max of the magnetic capillary origami is measured during a linear temporal increase of B at a tunable rate ranging from 3 G/s to 40 G/s. This rate is slow enough to consider a quasi-static evolution of B (see below). Figure 3 shows the dimensionless height z max /R 0 as a function of Bo m for the origami and for a ferrofluid drop alone (R 0 being the initial radius). For the drop alone, z max /R 0 is a continuous function of Bo m . For the origami, a jump of z max /R 0 is observed for a critical Bond number, Bo c m , corresponding to the threshold of the overturning instability for which the origami shape changes from a pyramid (Fig. 2c) to a cone-like shape (Fig. 2d). Due to inertia, this jump in z max /R 0 is accompanied by an overshoot (z max /R 0 reaching a maximum higher than the equilibrium) followed by underdamped oscillations (see bottom inset of Fig. 3 and movie in [15]). Moreover, just after the instability occurring at Bo c m , the origami in its new configuration has a height close to the one of the drop alone meaning that the stress applied by the membrane has relaxed. When iterating this experiment for different sizes L of the triangular membrane of fixed thickness h, the onset of this instability Bo c m is found to increase with L, as shown in the top inset of Fig. 3.

We first explain these above observations by dimensional scaling laws. At the onset of the instability, an increase of the origami height is observed (see Fig. 3) underlying that gravitational energy E g should be taken into account. One has E g ∼ ρV gz mc with z mc the mass center of the origami and V the volume of the ferrofluid. One can assume z mc ∼ L, and V ∼ L 3 since the initial configuration of the origami is quasispherical, and thus E g ∼ L 4 . The magnetic energy writes E m ∼ µ 0 M HV ∼ L 3 B 2 χ(B)/µ 0 since the magnetization M ∼ χ(H)H and the magnetic field H = B/µ 0 [START_REF] Rosensweig | Ferrohydrodynamics[END_REF]. The balance between the magnetic and gravitational energies thus leads to the critical magnetic field B c at the instability threshold such that χ(B c )B 2 c ∼ L. Thus, Bo c m ∼ χ(B c )B 2 c is expected to increase with L as observed in the inset of Fig. 3. Bo c m is also expected to be independent of the membrane thickness h as found experimentally. A similar scaling law analysis balancing the magnetic energy with the capillary energy, E c , or the elastic one, E e , leads for both cases to inverse predictions, i.e. a decrease of Bo c m with L since E c ∼ γL 2 , and E e ∼ Eh 3 κ 2 L 2 (E being the Young modulus of the membrane of curvature κ ∼ 1/L). Thus, the origami overturning occurring at B c is triggered by a competition between magnetic and gravitational energies. Note that the role of capillarity and elasticity is non zero. When B = 0, it drives the origami formation when E c E e (i.e. for L Eh 3 /γ [4]), and they both deform the ferrofluid shape when B = 0. us now characterize more quantitatively this instability. To do that, one has to compute the magnetic energy E m ≡ -µ 0 V H0 0 M (H)dH 0 with M the ferrofluid magnetization assumed uniform within the drop [START_REF] Ivanov | [END_REF]. H is the magnetic field within the ferrofluid droplet that depends on the applied magnetic field H 0 through the implicit equation H = H 0 -DM (H), D being the demagnetization coefficient that depends on the ferrofluid shape [START_REF] Rosensweig | Ferrohydrodynamics[END_REF]. For a full ellipsoid, it is known that D = 1-e 2 2e 3 [ln 1+e 1-e -2e] with e = 1 -(b/a) 2 the eccentricity, and a (resp. b) the semi-major (resp. minor) axis [10,[START_REF] Blums | Magnetic Fluids[END_REF]. For instance, in the case of an ellipsoidal drop alone, D = 0 for a drop infinitely stretched in the direction of H 0 , D = 1/3 for a spherical drop, and D = 1 for an infinitely flat drop normal to H 0 . The magnetic energy E m thus depends on the external magnetic field H 0 and on the droplet shape through its magnetization M (H) since H is a function of both D and H 0 . Consequently, the computation of E m demands to integrate M (H) over the external magnetic field H 0 . We estimate M (H) by using the usual Langevin's expression:

M (H) = M sat L(3χ i H/M sat ) where L(x) ≡ coth (x) -1/x [13].
The initial magnetic susceptibility, χ i ≡ dM dH | H→0 , and the magnetic saturation, M sat = M (H)| H→∞ , are given by the ferrofluid properties (see above).

The variation of the magnetic energy during the instability, occurring at the critical field

H 0 = H c , reads ∆E m = -µ 0 V Hc 0 ∆M (H)dH c , where ∆M ≡ M (H 2 ) - M (H 1 )
is the variation of M before and after the instability (denoted by subscripts 1 and 2 respectively).

H 1 = H c -D 1 M (H 1 ) and H 2 = H c -D 2 M (H 2 )
are the magnetic field within the drop before and after the instability. Since the origami undergoes a vertical elongation,

For different sizes, L, of the membrane, ∆z mc is deduced from the measurements of z mc before and after the instability occurring at B c . The origami aspect ratio a/b is also measured in the same way as for the ferrofluid drop alone on a plane (see above). For such a semiellipsoid, no theoretical expression for the demagnetization coefficient D is known. However, substituting the measured value a/b of the semi-ellipsoid-like origami into the above expression of D(a/b) for a full ellipsoid leads to a measured value of the demagnetization coefficient D m . The quantity Hc 0 ∆M m (H)dH c is experimentally found as follows (m denoting a measured quantity). For each H c ≡ B c /µ 0 corresponding to each value of L, one computes ∆M m (H) by using the experimental values of D m before and after the instability, by numerically solving the above implicit equations given H a and H b , and finally by iterating H 0 from 0 to H c . Fig. 4 then shows ∆z mc as a function of Hc 0 ∆M m (H)dH c for different values of L. Both quantities are found proportional, and with a constant µ 0 /(2ρg), i.e. half the coefficient predicted in Eq. (1). Since magnetic energy mostly focuses near drop regions of small curvature (tip-like effect), the value of the magnetic energy for a semi-ellipsoid can be assumed to be half of the one of the full ellipsoid, i.e. ∆M = ∆M m /2. The experimental curve is therefore in good agreement with the predictions of Eq. ( 1) with no adjustable parameter. Finally, ∆z mc is measured to be independent of the thickness h at fixed L, emphasizing that elasticity does not play a significant role during this overturning instability.

Capillarity and elasticity are at the origin of the capillary origami formation but are not involved in the threshold of the gravity-magnetic instability reported here. However, capillarity and elasticity effects can appear in the origami configuration before the instability: dewetting of the drop in the vicinity of the strongest curvature of the membrane (see left arrow in the top inset of Fig. 5), as well as curvature of a ridge of the pyramid can occur (see right arrow). Both phenomena result from elasticity and capillarity since they depend on the membrane thickness h and length L according to the phase diagram shown in Fig. 5. For thin enough membranes and/or long enough ones, the elasticity energy ∼ h 3 L 0 is much smaller than the capillary ∼ h 0 L 2 and magnetic ∼ h 0 L 3 ones. It thus leads to very small curvature radii of the membrane that rigidify the origami, and consequently favours the ferrofluid dewetting (see zone 3 in Fig. 5). For thinner h and/or longer L, as elasticity becomes negligible with respect to magnetic effects, the membrane bends inward and follows the elongated ferrofluid shape, leading to the curvature of the pyramid ridges (see zone 4). These additional effects of elasticity and capillarity on the overturning instability deserves further studies. Note also that if an adhesion force exists between the substrate and the membrane, the instability leads to an anomalous final state: one corner of the membrane points up while the two others wrap the drop at its base (see bottom inset of Fig. 5).

Finally, the instability reported here could have some interest for potential applications (see Ref. [1]). Indeed, the overturning instability is controlled in a non intrusive way and its scaling law is suitable for miniaturization as lower critical magnetic fields are required for smaller size membranes.

FIG. 1 .

 1 FIG. 1. (Color online) Aspect ratio a/b of a ferrofluid drop on a superhydrophobic substrate versus the magnetic Bond number Bom, for two values of the initial drop radius: R0 = 1.2 (△) and 0.93 mm ( ). Inset: Photo of the ferrofluid drop deformation for different vertical magnetic fields: B = 0 (a), 260 (b), 520 (c), and 780 G (d) corresponding to Bom = 0, 11, 31, and 53 respectively. R0 = 1.2 mm. Bar scale is 2 mm.

FIG. 2 .

 2 FIG. 2. Deformation and instability of the magnetic capillary origami as the vertical magnetic field is increased: B = 0 (a), 150 (b), 400 (c), and 600 G (d) corresponding to Bom = 0, 10, 48, and 84 respectively. An overturning instability occurs between pictures (c) and (d) (see text and movie in Ref. [15]). R0 = 2.7 mm. L = 10 mm. h = 65 µm. Bar scale is 2 mm.

FIG. 3 .

 3 FIG. 3. (Color online) Height of the origami [red (light gray) dots] rescaled by its initial radius (R0 = 2.3 mm), as a function of the magnetic Bond number Bom, and the corresponding evolution for a ferrofluid drop alone (black dots, R0 = 2 mm). L = 8.4 mm. h = 56 µm. A jump in the height of the origami is observed for a critical Bond Bo c m = 25.3, corresponding to the instability shown between Fig. 2c and 2d. Bottom inset: zoom near the instability region. Top inset: Bo c m versus the size L of the elastic membrane.

FIG. 4 .

 4 FIG. 4. (Color online) Variation of the height of the origami mass center, ∆zmc, during the instability as a function of Hc 0 ∆Mm(H)dHc for different sizes L of the membrane: L = 7.8 (•), 8.5 ( ), 9.4 ( ), and 10 mm ( ). Membrane thickness h = 65 µm. Solid line has a slope µ0/(2ρg).

FIG. 5 .

 5 FIG. 5. (Color online) Main: Phase diagram of the origami configuration before instability as a function of membrane length, L, and thickness, h: no origami ( ), usual origami ( ), origami with dewetting only (⋄), with ridge curvature only (•), with ridge curvature and dewetting (×). Top inset: ferrofluid dewetting (see left arrow) and ridge curvature (see right arrow) before instability. Bottom inset: "Anomalous" final state after the instability when adhesion is involved.

We thank J.-C. Bacri for fruitful discussions, J. Servais, A. Lantheaume, and M. Piñeirua for their technical help. This work has been supported by ANR Turbonde BLAN07-3-197846.


Alessio Porretta

Laurent Véron

Separable solutions of quasilinear Lane-Emden equations

Keywords: 2010 Mathematics Subject Classification. 35J92, 35J60, 47H11, 58C30 quasilinear elliptic equations, p-Laplacian, cones, Leray-Schauder degree

For 0 < p -1 < q and either ǫ = 1 or ǫ = -1, we prove the existence of solutions of -∆ p u = ǫu q in a cone C S , with vertex 0 and opening S, vanishing on ∂C S , under the form u(x) = |x| -β ω( x |x| ). The problem reduces to a quasilinear elliptic equation on S and existence is based upon degree theory and homotopy methods. We also obtain a non-existence result in some critical case by an integral type identity.

Introduction

It is well established that the description of the boundary behavior of positive singular solutions of Lane-Emden equations -∆u = ǫu q (1.1) with q > 1 in a domain Ω ⊂ R N is greatly helped by using specific separable solutions of the same equation. This was performed in 1991 by Gmira-Véron [START_REF] Gmira | Boundary singularities of solutions of some nonlinear elliptic equations[END_REF] in the case ǫ = -1 and more recently by Bidaut-Véron-Ponce-Véron [START_REF] Bidaut-Véron | Isolated boundary singularities of semilinear elliptic equations[END_REF] in the case ǫ = 1. If the domain is assumed to be a cone C S = {x ∈ R N \ {0} : x/|x| ∈ S} with vertex 0 and opening S S N -1 (the unit sphere in R N ), separable solutions of (1.1) vanishing on ∂C S \ {0} were of the form

u(x) = |x| -2 q-1 ω(x/|x|), (1.2)
with ω satisfying -∆ ′ ω -ℓ q,N ω -ǫω q = 0 in S,

1 vanishing on ∂S and where ℓ q,N = 2 q-1 2q q-1 -N and ∆ ′ is the Laplace-Beltrami operator on S N -1 . To this equation is associated the functional

J(φ) := S 1 2 |∇ ′ φ| 2 - ℓ q,N 2 φ 2 - ǫ q + 1 |φ| q+1 dv g , (1.4)
where ∇ ′ is the covariant derivative on S N -1 . In the case ǫ = 1, non-existence of a non-trivial positive solution of (1.3) when ℓ q,N ≥ λ S (the first eigenvalue of -∆ ′ in W 1,2 0 (S)) follows by multiplying the equation by the first eigenfunction and integrating over S; existence holds when ℓ q,N < λ S and q < N +1 N -3 by classical variational methods, and again non-existence holds when q ≥ N +1 N -3 and S ⊂ S N -1 + is starshaped by using an integral identity [3, Th 2.1,Cor 2.1]. When ǫ = -1, non-existence of a non-trivial solution of (1.3) when ℓ q,N ≤ λ S is obtained by multiplying the equation by ω and integrating over S, while existence when ℓ q,N > λ S follows by minimizing J over W 1,2 0 (S) ∩ L q+1 (S). In this paper we investigate similar questions for the quasilinear Lane-Emden equations -div |∇u| p-2 ∇u = ǫu q in C S ,

where S is a smooth subset of S N -1 , q > p -1 > 0 and ǫ = ±1 and we look for positive solutions u, vanishing on ∂C S \ {0}, under the separable form

u(x) = |x| -β ω(x/|x|). (1.6)
It is straightforward to check that u is a solution of (1.5) provided

β = β q := p q + 1 -p (1.7)
and ω is a positive solution of -div β 2 q ω 2 + |∇ ′ ω| 2 (p-2)/2 ∇ ′ ω -β q λ(β q ) β 2 q ω 2 + |∇ ′ ω| 2 (p-2)/2 ω = ǫω q (1.8) in S vanishing on ∂S, where div(•) is the divergence operator defined according to the intrinsic metric g and where we have set λ(β) = β(p -1) + p -N.

(1.9)

If ǫ = 0, it is now well-known that positive p-harmonic functions in C S vanishing on ∂C S exist under the form (1.6), and either they are regular at 0 and β = -βS < 0, or they are singular and β = β S > 0, where the values of βS , β S are unique. In this case ω = ωS or ω S is a solution of -div β 2 ω 2 + |∇ ′ ω| 2 (p-2)/2 ∇ ′ ω -βλ(β) β 2 ω 2 + |∇ ′ ω| 2 (p-2)/2 ω = 0 (1.10) in S, where β = βS or β S . The existence of ( βS , ωS ) is due to Tolksdorf in a pioneering work [START_REF] Tolksdorff | On the Dirichlet problem for quasilinear equations in domains with conical boundary points[END_REF]. Tolksdorf's method has been adapted by Véron [20] in order to prove the existence of (β S , ω S ). Later on Porretta and Véron [START_REF] Porretta | Separable p-harmonic functions in a cone and related quasilinear equations on manifolds[END_REF] obtained a more general proof of the existence of such couples. Notice that β S (as well as βS ) is uniquely determined while ω is unique up to homothety. In both cases the proofs rely on strong maximum principle. When p = 2, existence of a nontrivial solution in the case ǫ = 1 is obtained in [START_REF] Bidaut-Véron | Separable solutions of some quasilinear equations with source reaction[END_REF] when N = 2 and β q < β S by a dynamical system approach; while if ǫ = -1 and β q > β S , such an existence is proved in [START_REF] Véron | Some existence and uniqueness results for solution of some quasilinear elliptic equations on compact Riemannian manifolds[END_REF] by a suitable adaptation of Tolksdorf's construction. Notice that no functional can be associated to (1.8), excepted in the case q = q * = N p N -p -1. In such a case (1.8) is the Euler-Lagrange equation for the functional

J q (φ) := S 1 p β 2 q * φ 2 + |∇ ′ φ| 2 p 2 - ǫ q * + 1 |φ| q * +1 dv g , (1.11)
and existence of a non-trivial solution of (1.8) with ǫ = 1 is derived from the mountain pass theorem. In all the other cases variational techniques cannot be used and have to be replaced by topological methods based upon Leray-Schauder degree. Define q c by

q c = q c,p = (N -1)p N -1-p -1 if p < N -1 ∞ if p ≥ N -1,
then we prove the following results:

I Let ǫ = 1.
Assume p > 1, q < q c and β q < β S , then (1.8) admits a positive solution in S vanishing on ∂S.

II Let ǫ = -1. Assume p > 1 and β q > β S , then (1.8) admits a unique positive solution in S vanishing on ∂S.

The result I is based upon sharp Liouville theorems for solutions of (1.5) in R N or R N + respectively due to Serrin-Zou [START_REF] Serrin | Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities[END_REF] and Zou [START_REF] Zou | A priori estimates and existence for quasi-linear elliptic equations[END_REF]. In the case of II, the existence part is already known, but we give here a simpler form than the one in [START_REF] Véron | Some existence and uniqueness results for solution of some quasilinear elliptic equations on compact Riemannian manifolds[END_REF], using a topological deformation acting on the exponent p. In the case ǫ = 1, the result is optimal in the case q = q c ; indeed, using an integral identity, we also prove

III Let ǫ = 1, S S N -1
+ be a starshaped domain and 1 < p < N -1. If q = q c , then (1.8) admits no positive solution in S vanishing on ∂S.

Notice that when p = 2 an integral identity was used in [START_REF] Bidaut-Véron | Isolated boundary singularities of semilinear elliptic equations[END_REF] to prove non existence for all q ≥ q c,2 . The form which is derived in the case p = 2 is much more complicated and we prove non-existence only in the case q = q c,p .

Finally, the constraint β q < β S in I (respectivey, β q > β S in II) is sharp. When ǫ = 1, the non-existence of positive solutions of (1.8) when β q ≥ β S has been proved in [START_REF] Bidaut-Véron | Separable solutions of some quasilinear equations with source reaction[END_REF]. The method is based upon strong maximum principle. When ǫ = -1 a somewhat similar method is used in [START_REF] Véron | Singular p-harmonic functions and related quasilinear equations on manifolds[END_REF] and yields to non-existence results when β q ≤ β S . Notice that the obtention of such results when p = 2 is straightforward.

Nonexistence for the reaction problem

Let S be a bounded C 2 sub-domain of S N -1 . We consider the positive solutions in S of

-div β 2 ω 2 + |∇ ′ ω| 2 (p-2)/2 ∇ ′ ω -βλ(β) β 2 ω 2 + |∇ ′ ω| 2 (p-2)/2 ω = ω q (2.1)
vanishing on ∂S. Recall that λ(β) is given by (1.9) and that, in connection with problem (1.5), we have interest in the special case where β = β q is given by (1.7). The following Pohozaev-type identity, which is valid for any β, is the key for nonexistence. We denote by S N -1 + the half sphere. Proposition 2.1 Let S S N -1 be a C 2 domain and φ the first eigenfunction of

-∆ ′ in W 1,2 0 (S N -1 + ). If ω ∈ W 1,p 0 (S) ∩ C(S)
is a positive solution in S of (2.1), and if we set Ω = (β 2 ω 2 + |∇ ′ ω| 2 ) 1/2 , then the following identity holds

1 - 1 p ∂S |ω ν | p φ ν dS = A S ω q+1 φ dσ + B S Ω p-2 |∇ ′ ω| 2 φ dσ + C S Ω p-2 ω 2 φ dσ, (2.2
) with A = A(β) := - N -1 q + 1 -β(pβ + p -N ) (2.3)
B = B(β) := N -1 -p p + β(pβ + p -N ), (2.4)
C = C(β) := β 2 N -1 p -(pβ + p -N )λ(β) . (2.5)
In order to prove Proposition 2.1, we start with the following lemma.

Lemma 2.1 Let S ⊂ S N -1 be a C 2 domain and φ ∈ C 2 (S). If ω ∈ W 1,p 0 (S) ∩ C(S) is a positive solution of (2.1) in S, we have:

1 - 1 p ∂S |ω ν | p φ ν dS = S ∆ ′ φ q + 1 -β(pβ + p -N )φ ω q+1 dσ - 1 p S Ω p ∆ ′ φ dσ + S Ω p-2 D 2 φ(∇ ′ ω, ∇ ′ ω)dσ + β(pβ + p -N ) S Ω p-2 |∇ ′ ω| 2 φ dσ -β 2 (pβ + p -N )λ(β) S Ω p-2 ω 2 φ dσ.
(2.6)

Proof. By the regularity theory of p-Laplace type equations (see e.g. [START_REF] Dibenedetto | C 1+α local regularity of weak solutions of degenerate elliptic equations[END_REF], [START_REF] Tolksdorff | Regularity for a More General Class of Quasilinear Elliptic Equations[END_REF] and the Appendix in [START_REF] Porretta | Separable p-harmonic functions in a cone and related quasilinear equations on manifolds[END_REF]) it turns out that ω ∈ C 1,γ (S) for some γ ∈ (0, 1), and since

β 2 ω 2 + |∇ ′ ω| 2 > 0 in
:= (β 2 ω 2 + |∇ ′ ω| 2 ) 1/2 ) S Ω p-2 1 2 ∇ ′ |∇ ′ ω| 2 , ∇ ′ φ + D 2 φ(∇ ′ ω, ∇ ′ ω) ζ dσ + S Ω p-2 ∇ ′ ω, ∇ζ ∇ ′ ω, ∇ ′ φ dσ = βλ(β) S Ω p-2 ω ∇ ′ ω, ∇ ′ φ ζ dσ + 1 q + 1 S ∇ ′ ω q+1 , ∇ ′ φ ζ dσ. Since Ω p-2 1 2 ∇ ′ |∇ ′ ω| 2 , ∇ ′ φ = 1 p ∇ ′ Ω p , ∇φ -β 2 Ω p-2 ω ∇ ′ ω, ∇ ′ φ
we obtain, due to (1.9),

1 p S ∇ ′ Ω p , ∇ ′ φ ζ dσ + S Ω p-2 D 2 φ(∇ ′ ω, ∇ ′ ω) ζ dσ + S Ω p-2 ∇ ′ ω, ∇ ′ ζ ∇ ′ ω, ∇ ′ φ dσ = β(pβ + p -N ) S Ω p-2 ω ∇ ′ ω, ∇ ′ φ ζ dσ + 1 q + 1 S ∇ ′ ω q+1 , ∇ ′ φ ζ dσ.
Integrating by parts the first and last term we get

- 1 p S Ω p ∇ ′ φ, ∇ ′ ζ dσ + 1 q + 1 S ω q+1 ∇ ′ φ, ∇ ′ ζ dσ + S ω q+1 q + 1 - Ω p p ∆ ′ φ ζ dσ + S Ω p-2 D 2 φ(∇ ′ ω, ∇ ′ ω) ζ dσ + S Ω p-2 ∇ ′ ω, ∇ ′ ζ ∇ ′ ω, ∇ ′ φ dσ = β(pβ + p -N ) S Ω p-2 ω ∇ ′ ω, ∇ ′ φ ζ dσ.
(2.7) Now we choose ζ = ζ δ , where ζ δ is a sequence of C 1 compactly supported functions such that ζ δ (σ) → 1 for every σ ∈ S and |∇ ′ ζ δ | is bounded in L 1 (S). It is easy to see by integration by parts that we have for every continuous vector field

F ∈ C(S) S F, ∇ ′ ζ δ dσ → - ∂S F, ν(σ) dσ
where ν is the outward unit normal on ∂S. We take ζ = ζ δ in (2.7) and we let δ → 0. Using that ω ∈ C 1 (S) and that, by Hopf lemma, ω ν := ∇ ′ ω, ν(σ) < 0 we can actually pass to the limit in the integrals containing ∇ ′ ζ δ . Recalling that ω = 0 and

∇ ′ ω = -|ω ν |ν on ∂S we obtain 1 - 1 p ∂S |ω ν | p φ ν dS = S ω q+1 q + 1 - Ω p p ∆ ′ φ dσ + S Ω p-2 D 2 φ(∇ ′ ω, ∇ ′ ω)dσ -β(pβ + p -N ) S Ω p-2 ω ∇ ′ ω, ∇ ′ φ dσ.
(2.8) Multiplying (2.1) by ωφ we derive

S Ω p-2 ω ∇ ′ ω, ∇ ′ φ dσ = - S Ω p-2 |∇ ′ ω| 2 φ dσ + βλ(β) S Ω p-2 ω 2 φ dσ + S ω q+1 φ dσ, so that (2.8) becomes, replacing its last term, 1 - 1 p ∂S |ω ν | p φ ν dS = S ω q+1 q + 1 - Ω p p ∆ ′ φ dσ + S Ω p-2 D 2 φ(∇ ′ ω, ∇ ′ ω)dσ -β(pβ + p -N ) S ω q+1 φ dσ + β(pβ + p -N ) S Ω p-2 |∇ ′ ω| 2 φ -β 2 (pβ + p -N )λ(β) S Ω p-2 ω 2 φ dσ.
which is (2.6).

Proof of Proposition 2.1. We use Lemma 2.1 choosing in (2.6) φ to be the first

eigenfunction of -∆ ′ in W 1,2 0 (S N -1 + ). Since ∆ ′ φ = (1 -N )φ, D 2 φ = -φg 0 , we get 1 - 1 p ∂S |ω ν | p φ ν dS = - S N -1 q + 1 + β(pβ + p -N ) ω q+1 φ dσ + N -1 p S Ω p φ dσ - S Ω p-2 |∇ ′ ω| 2 φ dσ + β(pβ + p -N ) S Ω p-2 |∇ ′ ω| 2 φ dσ -β 2 (pβ + p -N )λ(β) S Ω p-2 ω 2 φ.
(2.9)

Then, using also the definition of Ω, (2.2) follows, with A, B and C given by (2.3)-(2.5).

We shall say that a C 2 domain S ⊂ S N -1 + is starshaped if there exists a spherical harmonic φ of degree 1 such that φ > 0 on S and for any a ∈ ∂S,

∇φ, ν a ≤ 0 (2.10)
where ν a is the unit outward normal vector to ∂S at a in the tangent plane T a to S N -1 . It also means that there exists some x 0 ∈ S such that the geodesic connecting x 0 and a remains inside S.

Theorem 2.1 Assume that 1 < p < N -1, q = q c and S ⊂ S N -1 + is starshaped. Then (2.1) admits no positive solution in S vanishing on ∂S.

Proof. Recall that in (1.8) we have β q = p q-(p-1) , hence different values of q are in one-to-one correspondence with different values of β. We first notice that, if q = q c the corresponding critical β is given by

β c := p q c -(p -1) = N -1 -p p . ( 2

.11)

We use now Proposition 2.1 with β = β q and we analyze the values of the coefficients A, B, C given by (2.3)-(2.5) as functions of β. First of all, since q + 1 = p(1+β) β , we have

A = - (N -1)β p(1 + β) -β(pβ + p -N ) = - β (β + 1) N -1 p + p(β + 1) 2 -N (β + 1)
and since from (2.11) we have

β c + 1 = N -1 p we deduce A = - β (β + 1) p β + 1 - 1 p (β -β c ) .
Still using (2.11), we also get

B = β c + β(p(β -β c ) -1) = (β -β c )(βp -1) .
Finally, using (1.9) and (2.11) we have

C = β 2 N -1 p -(pβ + p -N )((p -1)β + p -N ) = β 2 (β c + 1 -(p(β -β c ) -1)(p(β -β c ) -(β + 1))) = β 2 (β -β c )(1 -p) pβ -1 -p(N -p) p-1
.

(2.12)

Therefore A ≥ 0, B ≥ 0 and C ≥ 0 can be obtained only if q = q c , i.e. β = β c , in which case A = B = C = 0. Since φ ν ≤ 0 because S is star-shaped, we deduce from (2.2) that |ω ν | p φ ν = 0 on ∂S. Unless ω is identically zero, we have ω ν < 0 by Hopf boundary lemma. Then φ ν ≡ 0, and using the equation satisfied by φ and Gauss formula, we derive

λ S S φdσ = 0 =⇒ φ ≡ 0 in S,
which is impossible since φ > 0 in S N -1

+

. This proves the first assertion.

Remark. If p = 2, it is proved in [START_REF] Bidaut-Véron | Isolated boundary singularities of semilinear elliptic equations[END_REF] that the nonexistence result of Theorem 2.1 holds for every q ≥ q c , which suggests that our result above is not optimal. The proof in [START_REF] Bidaut-Véron | Isolated boundary singularities of semilinear elliptic equations[END_REF] cannot be applied here since the term S Ω p-2 ω ∇ ′ ω, ∇ ′ φ dσ is completely integrable only if p = 2. However, we conjecture that, even when p = 2, the conclusion of Theorem 2.1 holds under the more general condition q ≥ q c .

Remark. If we assume that p = 2, the proof of Theorem 2.1 relies on the existence of a positive function φ in S, satisfying (2.10) on ∂S and

∆ ′ φ (q + 1)φ -β(pβ + p -N ) ≥ 0, (2.13) pD 2 φ(ξ, ξ) -∆ ′ φ pφ + β(pβ + p -N ) ≥ 0 ∀ξ ∈ S N -1 , (2.14)
and

- ∆ ′ φ pφ -(pβ + p -N )((p -1)β + p -N ) ≥ 0. (2.15)
Remark 2.1 For the sake of completeness, we recall the non-existence result obtained in [2, Th 1]:

Let ǫ = 1 and 0 < p -1 < q. If β q ≥ β S , there exists no positive solution of (1.8) in S which vanishes on ∂S.

Existence for the reaction problem

Concerning the problem with reaction we consider a more general statement than Theorem I, replacing the sphere by a complete d-dimensional Riemannian manifold (M, g) and suppose that S is a relatively compact smooth open domain of M . We denote by ∇ := ∇ g the gradient of a function identified with its covariant derivatives and by div := div g the intrinsic divergence operator acting on vector fields. The following result is proved in [START_REF] Porretta | Separable p-harmonic functions in a cone and related quasilinear equations on manifolds[END_REF].

Theorem 3.1 For any β > 0 there exists a unique Λ β > 0 and a unique (up to an homothety) positive function

ω β ∈ C 2 (S) ∩ C 1 (S) solution of      -div β 2 ω 2 β + |∇ω β | 2 p-2 2 ∇ω β = βΛ β β 2 ω 2 β + |∇ω β | 2 p-2 2 ω β in S ω β = 0 on ∂S. ( 3
0 < β < β S ⇐⇒ Λ β -β(p -1) > Λ β S -β S (p -1) = p -d -1 Therefore, if we set λ(β) = β(p -1) + p -d -1, we deduce that 0 < β < β S ⇐⇒ Λ β > λ(β). (3.2)
Let us now prove the existence of solutions for the reaction problem.

Theorem 3.2 Assume 1 < p < d and p -1 < q < q c := pd/(d -p) -1. Then for any 0 < β < β S , there exists a positive function ω ∈ C(S) ∩ C 2 (S) satisfying -div (β 2 ω 2 + |∇ω| 2 ) p 2 -1 ∇ω = βλ(β)(β 2 ω 2 + |∇ω| 2 ) p 2 -1 ω + ω q in S ω = 0 on ∂S, (3.3) where λ(β) = β(p -1) + p -d -1.
In order to prove Theorem 3.2, we use topological arguments as it is often needed in a non-variational setting. In particular, following a strategy similar as in [START_REF] Quaas | Existence results for nonproper elliptic equation involving the Pucci's Operator[END_REF], our proof is based upon the following fixed point theorem which is only one possible consequence of Leray-Schauder degree theory to compute the fixed point index of compact mappings. Such results were developed mostly by Krasnoselskii ([9]), we refer to Proposition 2.1 and Remark 2.1 in [START_REF] De Figuereido | A priori estimates and existence of positive solutions of semilinear elliptic equations[END_REF] for the statement below. Theorem 3.3 Let X be a Banach space and K ⊂ X a closed cone with non empty interior. Let F : K × R + → K be a compact mapping, and let Φ(u) = F (u, 0) (compact mapping from K into K). Assume the following holds: there exist R 1 < R 2 and T > 0 such that (i) u = sΦ(u) for every s ∈ [0, 1] and every u: u = R 1 .

(ii) F (u, t) = u for every (u, t): u ≤ R 2 and t ≥ T . (iii) F (u, t) = u for every u: u = R 2 and every t ≥ 0. Then, the mapping Φ has a fixed point u such that R 1 < u < R 2 .

We also recall the following non-existence results respectively due to Serrin and Zou [START_REF] Serrin | Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities[END_REF], and Zou [START_REF] Zou | A priori estimates and existence for quasi-linear elliptic equations[END_REF].

Theorem 3.4 Assume 1 < p < d and p -1 < q < q c . Then there exists no C 1 positive solution of

-∆ p u = u q (3.4) in R d .
Theorem 3.5 Assume 1 < p < d and p -1 < q < q c . Then there exists no C 1 positive solution of

-∆ p u = u q (3.5) in R d + := {x = (x 1 , ..., x d ) : x d > 0} vanishing on ∂R d + := {x = (x 1 , ..., x d ) : x d = 0}.
Proof of Theorem 3.2. Define the operator A in W 1,p 0 (S) as

A(ω) := -div g (β 2 ω 2 + |∇ω| 2 ) p 2 -1 ∇ω + β 2 ω(β 2 ω 2 + |∇ω| 2 ) p 2 -1 .
Note that A is the derivative of the functional

J(w) = 1 p S (β 2 ω 2 + |∇ω| 2 ) p 2 dv g
Since J is strictly convex, then A is a strictly monotone operator from W 1,p 0 (S) into W -1,p ′ (S), henceforth its inverse is well defined and continuous [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF]. In order to apply Theorem 3.3, we denote by X = C 1 0 (S), the closure of C 1 0 (S) in C 1 (S). Clearly X ⊂ W 1,p 0 (S), with continuous imbedding, if it is endowed with its natural norm ||.||

X := ||.|| C 1 (S) . Furthermore, since ∂S is C 2 , C 1 (S) ∩ W 1,p 0 (S) = C 1 0 (S). If
K is the cone of nonnegative functions in S, it has a nonempty interior. For t > 0, we set

F (ω, t) := A -1 β (λ(β) + β + t) ω(β 2 ω 2 + |∇ω| 2 ) p 2 -1 + (ω + t) q .
Note that

Φ(ω) := F (ω, 0) = A -1 β (λ(β) + β) ω(β 2 ω 2 + |∇ω| 2 ) p 2 -1 + ω q ;
henceforth any nontrivial fixed point for Φ would solve problem (3.3).

We have to verify the assumptions of Theorem 3.3. First of all, the compactness of F (ω, t). If we set F (ω, t) = φ, then it means that φ ∈ W 1,p 0 (S) satisfies

-div g (β 2 φ 2 + |∇φ| 2 ) p 2 -1 ∇φ + β 2 φ(β 2 φ 2 + |∇φ| 2 ) p 2 -1 = β (λ(β) + β + t) ω(β 2 ω 2 + |∇ω| 2 ) p 2 -1 + (ω + t) q (3.6)
Thus, if we assume that ω belongs to a bounded set in K ∩ X, the right-hand side of (3.6) is bounded in C(S). Thus, by standard regularity estimates up to the boundary for p-Laplace type operators (see [START_REF] Porretta | Separable p-harmonic functions in a cone and related quasilinear equations on manifolds[END_REF]Appendix] and [START_REF] Dibenedetto | C 1+α local regularity of weak solutions of degenerate elliptic equations[END_REF], [START_REF] Tolksdorff | Regularity for a More General Class of Quasilinear Elliptic Equations[END_REF]), φ remains bounded in C 1,α (S) and therefore relatively compact in C 1 (S). It remains to show that conditions (i)-(iii) of Theorem 3.3 hold.

Step 1: Condition (i) holds. We proceed by contradiction in supposing that there exists a sequence {s n } ⊂ [0, 1] such that for any n ∈ N the following problem

       -div g (β 2 ω 2 + |∇ω| 2 ) p 2 -1 ∇ω + β 2 (β 2 ω 2 + |∇ω| 2 ) p 2 -1 ω = s p-1 β(λ(β) + β)(β 2 ω 2 + |∇ω| 2 ) p 2 -1 ω + s p-1 n ω q in S ω = 0 on ∂S, (3.7
) admits a positive solution ω n , and that there holds

ω n X → 0 as n → ∞. Set w n = ω n / ω n , then w n solves        -div g (β 2 w 2 n + |∇w n | 2 ) p 2 -1 ∇w n + β 2 w n (β 2 w 2 n + |∇w n | 2 ) p 2 -1 = s p-1 n β(λ(β) + β)(β 2 w 2 n + |∇w n | 2 ) p 2 -1 w n + s p-1 n w q n w n q-(p-1) X in S
w n = 0 on ∂S Up to subsequences, we assume that s n → s for some s ∈ [0, 1]. Using compactness arguments we deduce that w n will converge strongly in C 1 (S) to some positive function w such that w X = 1 and which solves

       -div g (β 2 w 2 + |∇w| 2 ) p 2 -1 ∇w = β s p-1 λ(β) + (s p-1 -1)β (β 2 w 2 + |∇w| 2 ) p 2 -1 w in S w = 0 on ∂S (3.8)
Using Theorem 3.1, we derive Λ β = s p-1 λ(β)+(s p-1 -1)β. Since β < β S , λ(β) < Λ β by (3.2). Therefore, as s ≤ 1, we get

s p-1 λ(β) + (s p-1 -1)β ≤ s p-1 λ(β) < Λ β ,
which is a contradiction. Consequently, there exists R 1 > 0 such that for any s ∈ [0, 1], there holds ω = sΦ(ω) for any ω such that ω X = R 1 .

Step 2: Condition (ii) holds. Consider the first eigenvalue λ 1,β associated with the operator A, i.e.

λ 1,β = min S (β 2 ω 2 + |∇ω| 2 ) p 2 dv g : ω ∈ W 1,p 0 (S) , S |ω| p dv g = 1 (3.9)
Note that for t large enough, we have λ(β) + β + t ≥ 0, hence, using that q > p -1, we can find T > 0 such that

β (λ(β) + β + t) ω(β 2 ω 2 + |∇ω| 2 ) p 2 -1 + (ω + t) q ≥ (λ 1 + δ)ω p-1 ∀t ≥ T , ∀ω ≥ 0 .
Therefore, if t ≥ T and F (ω, t) = ω we deduce that ω = 0 and satisfies

-div g (β 2 ω 2 + |∇ω| 2 ) p 2 -1 ∇ω + β 2 ω(β 2 ω 2 + |∇ω| 2 ) p 2 -1 ≥ (λ 1,β + δ)ω p-1 in S ω = 0 on ∂S
The existence of a positive super-solution with λ 1,β + δ would make it possible to construct a positive solution as well. But since λ 1,β is an isolated eigenvalue (see Appendix) this yields a contradiction. Therefore, for t ≥ T the equation F (ω, t) = ω has no solution at all. Note that T only depends on λ 1 , β.

Step 3: Condition (iii) holds. Since we proved that (ii) holds independently on the choice of R 2 , it is enough to show that (iii) holds for every t ≤ T . This is done if we have the existence of universal a priori estimates, i.e. if we can prove the existence of a constant R 2 such that for any t ≤ T every positive solution of

         -div g (β 2 ω 2 + |∇ω| 2 ) p 2 -1 ∇ω + β 2 ω(β 2 ω 2 + |∇ω| 2 ) p 2 -1 = β(λ(β) + β + t)(β 2 ω 2 + |∇ω| 2 ) p 2 -1 ω + (ω + t) q in S ω = 0 on ∂S satisfies ω < R 2 .
The crucial step is to prove that there exist universal a priori estimates for the L ∞ -norm (a bound for the W 1,p 0 -norm would follow immediately, and then a bound in X from the regularity theory). A standard procedure is to reach this result reasoning by contradiction and using a blow-up argument. Indeed, if a universal bound does not exist, there exist a sequence of solutions ω n and t n ≤ T such that

ω n ∞ → ∞ .
Let σ n be the (local coordinates of) maximum points of ω n ; up to subsequences, we

have σ n → σ 0 ∈ S. Setting M n = ω n -q-(p-1) p ∞ , define v n (y) = ω n (σ n + M n y) ω n ∞ = M p q-(p-1) n ω n (σ n + M n y)
Then v n is a sequence of uniformly bounded solutions, which will be locally compact in the C 1 -topology. Rescaling the equation and passing to the limit in n we find out that the limit function v is positive and satisfies the equation

-∆ p v = c 0 v q
for some constant c 0 (coming out from the local expression of Laplace-Beltrami operator). Depending whether σ 0 ∈ S or σ 0 ∈ ∂S, the equation would take place in either R d or in the half space R d + , where d = N -1, in which case v vanishes on ∂R d

+ . Since p -1 < q < q c , this contradicts either Theorem 3.4, or Theorem 3.5 because, by construction, we have v(0) = 1.

Remark. In the case p = 2, existence is proved in [START_REF] Bidaut-Véron | Isolated boundary singularities of semilinear elliptic equations[END_REF] using a standard variational method. It is also proved that, if (M, g) = (S d , g 0 ) (the standard sphere), and if S is a spherical cap with center a, any positive solution of

∆ ′ ω + β(β + 1 -d))ω + ω q = 0 in S ω = 0 on ∂S, (3.10)
depends only on the angle θ from a. Furthermore, uniqueness is proved by a delicate analysis of the non-autonomous second order O.D.E. satisfied by ω. In the case p = 2 and assuming always that S is a spherical cap of (S d , g 0 ), it is still possible to construct a radial (i.e. depending only on θ) positive solution of (3.3): it suffices to restrict the functional analysis framework to radial functions. However, there are two interesting open questions the answer to which would be important: (i) Are all positive solutions of (3.3) radial ? (ii) Is there uniqueness of positive radial solutions of (3.3)?

Existence for the absorption problem

Let us now consider the absorption problem, namely (1.8) with ǫ = -1. We give an existence result which extends the previous ones obtained in [START_REF] Véron | Some existence and uniqueness results for solution of some quasilinear elliptic equations on compact Riemannian manifolds[END_REF], with a simpler proof.

Theorem 4.1 Assume 0 < p -1 < q. Then for any β > β S , there exists a unique positive function ω ∈ C(S) ∩ C 2 (S) satisfying

-div g (β 2 ω 2 + |∇ω| 2 ) p 2 -1 ∇ω = βλ(β)(β 2 ω 2 + |∇ω| 2 ) p 2 -1 ω -ω q in S ω = 0 on ∂S, (4.1) where λ(β) = β(p -1) + p -d -1.
Before proving Theorem 4.1, we will need the following lemma. Proof. By Theorem 3.1, Λ β is uniquely defined for any fixed p > 1. To emphasize the dependence of Λ β on p, let us denote it now by Λ β,p . The continuity of Λ β,p with respect to p can be proved in the same way as we proved (see Proposition 2.4 in [START_REF] Porretta | Separable p-harmonic functions in a cone and related quasilinear equations on manifolds[END_REF]) the continuity of Λ β,p with respect to β. Thus, we only sketch the argument, which relies on the construction itself of Λ β,p . Indeed, we proved in [START_REF] Porretta | Separable p-harmonic functions in a cone and related quasilinear equations on manifolds[END_REF] that Λ β,p is the unique constant such that there exists a function v ∈ C 2 (S) satisfying

     -∆ g v -(p -2) D 2 v∇v.∇v 1 + |∇v| 2 + β(p -1)|∇v| 2 = -Λ β,p in S lim σ→∂S v(σ) = ∞. (4.2)
If we normalize v by setting, for example, v(σ 0 ) = 0 for some σ 0 ∈ S, then v is unique. Moreover v ∈ C 2 (S) and v satisfies estimates in W 1,∞ loc (S) which are uniform as β ∈ (0, ∞) and p ∈ (1, ∞) vary in compact sets. It is also easy to check (see [START_REF] Porretta | Separable p-harmonic functions in a cone and related quasilinear equations on manifolds[END_REF]) that Λ β,p remains bounded whenever β varies in a compact set of (0, ∞) and p vary in a compact set of (1, ∞). The estimates obtained on v and ∇v imply that, whenever β n or p n are convergent sequences, the sequence of corresponding solutions v n of (4.2) (such that v n (σ 0 ) = 0) is relatively compact (locally uniformly in C 1 ). The equation (4.2) turns out then to be stable (including the boundary estimates); finally, the uniqueness property of Λ β,p , and of the associated (normalized) solution v, implies the continuity of Λ β,p with respect to both β and p.

Let now β S,p be the spectral exponent defined by the equation

Λ β,p = β(p -1) + p -d -1 (4.3)
First of all note that when p lies in a compact set in (1, ∞), then necessarily β S,p is bounded. Indeed, since Λ β,p ≤ Λ 1,p whenever β ≥ 1, we have that

β S (p -1) + p -d -1 ≤ Λ 1,p if β S ≥ 1, so that β S ≤ 1 + 1 p -1 (Λ 1,p -(p -d -1)) .
Therefore, if p belongs to a compact set in (1, ∞), then β S remains also in a bounded set. Now, if p n → p 0 , setting β n = β S,pn , we have that β n is bounded and, up to subsequences, it is convergent to some β 0 . From (4.3), we deduce that Λ βn,pn is bounded, which implies that β n cannot converge to zero, hence β 0 > 0. Then, using the continuity of Λ β,p , we can pass to the limit in (4.3) and we deduce that β 0 is the spectral exponent with p = p 0 , i.e. β 0 = β S,p 0 . This proves that β S,p is continuous with respect to p.

We are now ready to prove Theorem 4.1.

Proof of Theorem 4.1.

Step 1: construction of a solution. We use similar ideas as in the proof of Theorem 3.2, i.e. a topological degree argument. On the Banach space X = C 1 0 (S) (endowed with its natural norm) with positive cone K, we set

B(ω) = -div g (β 2 ω 2 + |∇ω| 2 ) p 2 -1 ∇ω + β 2 (β 2 ω 2 + |∇ω| 2 ) p 2 -1 ω + |ω| q-1 ω (4.4) Ψ(ω) := B -1 β (λ(β) + β) (β 2 ω 2 + |∇ω| 2 ) p 2 -1 ω + .
Clearly, Ψ(w) = w implies that w ≥ 0 and solves (4.1). Then, it is enough to prove the existence of a non trivial fixed point for Ψ. Observe that, as in Theorem 3.2, Ψ is a continuous compact operator in X thanks to the C 1,α estimates for p-Laplace operators, and Ψ(K) ⊂ K.

We now wish to compute the degree of I -Ψ. First of all we consider, if R is sufficiently large, deg(I -Ψ, B + R , 0) where

B + R = B R ∩ K. To this purpose, define, for t ∈ [0, 1], Ψ * (ω, t) = tΨ(ω). Then Ψ * is a compact map on X × [0, 1] and if Ψ * (ω, t) = ω, we have -div g (β 2 ω 2 + |∇ω| 2 ) p 2 -1 ∇ω + β 2 (β 2 ω 2 + |∇ω| 2 ) p 2 -1 ω + 1 t q-(p-1) ω q = t p-1 β (λ(β) + β) (β 2 ω 2 + |∇ω| 2 ) p 2 -1 ω. (4.5)
We get, by the maximum principle,

ω t q-(p-1) ∞ ≤ t p-1 β p-1 (λ(β) + β) ≤ β p-1 (λ(β) + β) .
Since t ≤ 1, we deduce in particular that ω ∞ is bounded independently on t. Then, we have

1 t q-(p-1) ω q ≤ ω t q-(p-1) ∞ ω p-1 ∞ ≤ C ω p-1 ∞ ≤ C .
Multiplying by ω we obtain a similar bound for ω W 1,p 0 (S) , and the regularity theory for p-Laplace type equations yields a further estimate on ∇ω ∞ . Therefore, we conclude that there exists a constant M , independent on t ∈ [0, 1], such that tΨ(ω) = ω implies ω X ≤ M . As a consequence, if R is sufficiently large we have tΨ(ω) = ω on ∂B R . We deduce that deg(

I -tΨ, B + R , 0) is constant. Therefore deg(I -Ψ, B + R , 0) = deg(I -tΨ, B + R , 0) = deg(I, B + R , 0) = 1 . (4.6)
Next, we compute deg(I -Ψ, B + r , 0) for small r. We set

B t (ω) = -div g (β 2 ω 2 + |∇ω| 2 ) p 2 -1 ∇ω + β 2 (β 2 ω 2 + |∇ω| 2 ) p 2 -1 ω + t|ω| q-1 ω (4.7) and F (ω, t) := B t -1 β (λ(β) + β) ω + (β 2 ω 2 + |∇ω| 2 ) p 2 -1 .
Again, we have Ψ(•) = F (•, 1). We claim that there exists a small r > 0 such that F (ω, t) = ω for every t ∈ [0, 1] and ω ∈ ∂B r . Indeed, reasoning by contradiction, if this were not true there would exist a nonnegative sequence ω n such that 0 = ω n → 0, and t n ∈ [0, 1] such that F (ω n , t n ) = ω n , which means that

-div g (β 2 ω 2 n + |∇ω n | 2 ) p 2 -1 ∇ω n + β 2 (β 2 ω 2 n + |∇ω n | 2 ) p 2 -1 ω n + t n ω q n = β (λ(β) + β) ω n (β 2 ω 2 n + |∇ω n | 2 ) p 2 -1
Dividing by ω n p-1 and letting n → ∞, we find that ωn ωn would converge to some function ŵ such that ŵ ≥ 0, ŵ = 1 and

-div g (β 2 ω2 + |∇ω| 2 ) p 2 -1 ∇ω + β 2 (β 2 ω2 + |∇ω| 2 ) p 2 -1 ω = β (λ(β) + β) ω(β 2 ω2 + |∇ω| 2 ) p 2 -1
By Theorem 3.1 this means that λ(β) = Λ β , which is not possible since λ(β) > Λ β because β > β S (see Remark 3.1). Therefore, we conclude that F (ω, t) = ω for every t ∈ [0, 1] and ω ∈ ∂B r provided r is sufficiently small. We deduce that deg(I -F (•, t), B r , 0) is constant and in particular deg(I -Ψ, B + r , 0) = deg(I -F (•, 0), B + r , 0) .

In order to compute this degree, we perform an homotopy acting on p and β by setting p t = 2t + (1 -t)p and by taking β t so that t → β t is continuous on [0, 1], β 0 = β, β t > β S,pt for every t ∈ [0, 1] (where β S,pt is the spectral exponent for S with p = p t ) and β 1 > 0 is large enough. It follows from Lemma 4.2 that β S,pt is a continuous function of t and remains bounded as t ∈ [0, 1]. Therefore, a similar choice of function β t is possible. In the space C 1 0 (S) we define the mapping C t by

C t (ω) := -div g (β 2 t ω 2 + |∇ω| 2 ) p t 2 -1 ∇ω + β 2 t (β 2 t ω 2 + |∇ω| 2 ) p t 2 -1 ω. (4.8) We set F (ω, t) = C -1 t β t (λ(β t ) + β t )(β 2 t ω 2 + |∇ω| 2 ) p t 2 -1 ω . (4.9)
Combining the Tolksdorf's construction [START_REF] Tolksdorff | Regularity for a More General Class of Quasilinear Elliptic Equations[END_REF] which shows the uniformity with respect to p t of the C 1,α estimates (with α = α t ∈ (0, 1)), with the perturbation method of [13, Th A1], we obtain that (ω, t) → F (ω, t) is compact in C 1 0 (S) × [0, 1]. Since β t > β S,pt , clearly I -F (., t) does not vanish on ω X = r for any r > 0 which implies that

deg(I -Ψ, B + r , 0) = deg(I -F (•, 0), B + r , 0) = deg(I -F (•, 1), B + r , 0). But I -F (•, 1) = I -β 1 (λ(β 1 ) + β 1 )(-∆ g + β 2 1 ) -1 . (4.10)
Since -∆ g has only one eigenvalue in S with positive eigenfunction and multiplicity one, choosing β 1 large in a way that λ(β 1 )β 1 > λ 1 (S) it follows that

deg(I -F (•, 1), B + r , 0) = -1 = deg(I -Ψ, B + r , 0) . To conclude, since we have deg(I -Ψ, B + R \ B + r , 0) = deg(I -Ψ, B + R , 0) -deg(I -Ψ, B + r , 0) = 0
we deduce the existence of some ω such that r < ω < R which is a solution of (4.1).

Step 2: uniqueness. If ω is any positive solution, then β 2 ω 2 + |∇ω 2 | is positive in S. This is obvious in S and it is a consequence of Hopf boundary lemma on ∂S.

Let ω and ω be two positive solutions. Either the two functions are ordered or their graphs intersect. Since all the solutions are positive in S and satisfy Hopf boundary lemma, we can define θ := inf{s ≥ 1 : sω ≥ ω}, and denote ω * := θω. Either the graphs of ω and ω * := θω are tangent at some interior point α ∈ S, or ω * > ω in S and there exists α ∈ ∂S such that ω ν (α) = ω * ν (α) < 0. We put w = ω -ω * and use local coordinates (σ 1 , ..., σ d ) on M near α. We denote by g = (g ij ) the metric tensor on M and g jk its contravariant components. Then, for any ϕ ∈ C 1 (S),

|∇ϕ| 2 = j,k g jk ∂ϕ ∂σ j ∂ϕ ∂σ k = ∇ϕ, ∇ϕ g . If X = (X 1 , ...X d ) ∈ C 1 (T M
) is a vector field, if we lower indices by setting

X ℓ = i g ℓi X i , then div g X = 1 |g| ℓ ∂ ∂σ ℓ |g|X ℓ = 1 |g| ℓ,i ∂ ∂σ ℓ |g|g ℓi X i .
By the mean value theorem applied to

t → Φ(t) = β 2 (ω * + tw) 2 + |∇(ω * + tw)| 2 ( p 2 -1) (ω * + tw) t = 0, 1,
we have, for some t ∈ (0, 1),

(β 2 ω 2 + |∇ω| 2 ) ( p 2 -1) ω -(β 2 ω * 2 + |∇ω * | 2 ) ( p 2 -1) ω * = j a j ∂w ∂σ j + bw, where b = β 2 (ω * + tw) 2 + |∇(ω * + tw)| 2 ( p 2 -2) (p -1)β 2 (ω * + tw) 2 + |∇(ω * + tw)| 2 and a j = (p -2) β 2 (ω * + tw) 2 + |∇(ω * + tw)| 2 ( p 2 -2) (ω * + tw) k g jk ∂(ω * + tw) ∂σ k Considering now t → Φ i (t) = β 2 (ω * + tw) 2 + |∇(ω * + tw)| 2 ( p 2 -1) ∂(ω * + tw) ∂σ i t = 0, 1,
we see that there exists some t i ∈ (0, 1) such that

(β 2 ω 2 + |∇ω| 2 ) ( p 2 -1) ∂ω ∂σ i -(β 2 ω * 2 + |∇ω * | 2 ) ( p 2 -1) ∂ω * ∂σ i = j a ij ∂w ∂σ j + b i w,
where

b i = (p -2) β 2 (ω * + t i w) 2 + |∇(ω * + t i w)| 2 ( p 2 -2) β 2 (ω * + t i w) ∂(ω * + t i w) ∂σ i and a ij = (p -2) β 2 (ω * + t i w) 2 + |∇(ω * + t i w)| 2 ( p 2 -2) ∂(ω * + t i w) ∂σ i k g jk ∂(ω * + t i w) ∂σ k +δ j i β 2 (ω * + t i w) 2 + |∇(ω * + t i w)| 2 ( p 2 -1)
.

Set P = ω * (α) = ω(α) and Q = ∇ω * (α) = ∇ω(α). Then P 2 + |Q| 2 > 0 and b i (α) = (p -2) β 2 P 2 + |Q| 2 ( p 2 -2) β 2 P Q i , and
a ij (α) = β 2 P 2 + |Q| 2 p 2 -2 δ j i (β 2 P 2 + |Q| 2 ) + (p -2)Q i k g jk Q k .
Because ω * is a supersolution for (4.1), the function w satisfies

- 1 |g| ℓ,j ∂ ∂σ ℓ A jℓ ∂w ∂σ j + i C i ∂w ∂σ i + Dw ≤ 0 (4.11)
where the C i and D are continuous functions and

A jℓ = |g| i g ℓi a ij .
The matrix (a ij )(a) is symmetric definite and positive since it is the Hessian of Therefore the matrix (A jℓ ) keeps the same property in a neighborhood of a. Since w is nonpositive and vanishes at some a ∈ S or w < 0 and w ν = 0 at some boundary point, it follows from the strong maximum principle or Hopf boundary lemma (see [START_REF] Protter | Maximum principles in differential equations[END_REF]) that w ≡ 0, i.e. θω = ω. This implies that actually θ = 1 and ω = ω.

Appendix

We prove here the following result The function |ω| is also a minimizer for λ 1,β , thus it is a positive solution of (5.2). By Harnack inequality [START_REF] Serrin | Local behaviour of solutions of quasilinear equations[END_REF], for any compact subset K of S, there exists which implies that ω is an eigenfunction associated with λ 1,β . We observe that ω n cannot have constant sign. Indeed, if we had that ω n is positive in Ω, we could proceed as in the proof of Theorem 4.1-Step 2; up to rescaling ω n , we could assume that w = ω -ω n is nonpositive, is not zero, and the graphs of ω and ω n are tangent. In that case, using (5.2) and (5.

C K such that |ω|(σ 1 ) |ω|(σ 2 ) ≤ C K ∀σ i ∈ K, i = 1, 2.

. 1 )

 1 The mapping β → Λ β is continuous and decreasing, and the spectral exponent β S is the unique β > 0 such that Λ β S = β S (p -1) + p -d -1. Remark 3.1 Let us notice that the monotone character of β → Λ β implies that

Lemma 4 . 2

 42 For β > 0 and p > 1, let Λ β and β S be defined by Theorem 3.1. Then both Λ β and β S are continuous functions of p, varying in (1, ∞).

x = (x 1 ,

 1

Theorem 5 . 1 2 - 1 2 - 1

 512121 Let S be a subdomain of a complete d-dimensional Riemannian manifold (M, g). If β > 0 and p > 1, the first eigenvalue λ 1,β of the operator ω → -div((β 2 ω 2 + |∇ω| 2 ) p ∇ω) + β 2 ω(β 2 ω 2 + |∇ω| 2 ) p in W 1,p0 (S) is isolated. Furthermore any corresponding eigenfunction has constant sign.Proof. The proof is an adaptation of the original one due to Anane and Lindqvist when β = 0. We recall thatλ 1,β = inf S (β 2 ω 2 + |∇ω| 2 ) p 2 dv g : ω ∈ W 1,p 0 (S), |ω| p dv g = 1 ,(5.1)and that there exists ω ∈ W 1,p 0 (S) ∩ C 1,α (S) such that -div((β 2 ω 2 + |∇ω| 2 ) p 2 -1 ∇ω) + β 2 ω(β 2 ω 2 + |∇ω| 2 ) p 2 -1 = λ 1,β |ω| p-2 ω in S. (5.2)

2 - 1 2 - 1 =

 2121 Thus any minimizer ω must keep a constant sign in S. If λ 1,β is not isolated, there exists a decreasing sequence {µ n } of real numbers converging to λ 1,β and a sequence of functions ω n ∈ W 1,p 0 (S), solutions of-div(β 2 ω 2 n + |∇ω n | 2 ) p ∇ω n ) + β 2 ω n (β 2 ω 2 n + |∇ω n | 2 ) p µ n |ω n | p-2 ω n in S (5.3) such that ω n L p (S) = 1. By standard compactness and regularity results, we can assume that ω n → ω weakly in W 1,p 0 (S) and strongly in L p (S). Thus S (β 2 ω 2 + |∇ω| 2 ) p 2 dv g ≤ lim inf n→∞ S (β 2 ω 2 n + |∇ω n | 2 ) p 2 dv g = λ 1,β

  [START_REF] Bidaut-Véron | Isolated boundary singularities of semilinear elliptic equations[END_REF], we see that w satisfies a nondegenerate elliptic equation (as in (4.11)), and we obtain a contradiction either by the strict maximum principle or by Hopf lemma. Thus, any eigenfunction ω n must change sign in Ω. Set S + n = {σ ∈ S : ω n (σ) > 0} and S - n = {σ ∈ S : ω n (σ) < 0}. Clearly, for 0 < θ < 1,S ± n (β 2 ω 2 n + |∇ω n | 2 ) p 2 dv g ≥ (1 -θ)β p S ± n |ω n | p dv g + θ S ± n |∇ω n | p dv g .

It follows from (5.3), multiplying by ω + n , that

Since for some suitable q > p (for example q = p * if p < d, or any p < q < ∞ if p ≥ d)

Similarly we get, multiplying (5.3) by ω - n , that

It follows that the two sets S ± = lim sup n→∞ S ± n have positive measure. Since ω ≥ 0 on S + and ω ≤ 0 on S -, we derive a contradiction with the fact that any eigenfunction corresponding to λ 1,β has constant sign.


Sylvain Kubicki
email: sylvain.kubicki@tudor.lu

Conrad Boton
email: conrad.boton@tudor.lu

4D-BASED TEACHING OF HIGH-RISE STRUCTURAL PRINCIPLES

Keywords: High-rise, Structural principle, 4D simulation, 4D multi-view, Pedagogy

4D modeling and visualization is more and more used in construction education curricula. The main interest of this technology is its ability to simulate real sequencing of construction tasks in order to confront the student with real-life construction management issues. This article presents a course for architecture and construction engineers about the teaching of structural principles of high-rise buildings and the particularity of their design and construction. The students use 4D simulation as a support to the analysis of structural characteristics of the studied projects. Pedagogical interest is highlighted in the article. The emphasis is particularly put on the visualization of 3D/4D models. A metamodel of multi-view is proposed and enables the modeling of particular 4D multiple views in order to reuse it. The aim of this work is to build a method assisting students when choosing visualization parameters for realizing 4D simulations.

INTRODUCTION

The University of Liège in Belgium proposes curriculums for the initial training of engineer-architects and construction engineers. Specific master's courses are developed for the teaching of organizational issues of construction projects and the management of complex projects' teams. Some issues could be noticed regarding some particular types of construction projects or management techniques:

-The design principles of high-rise construction projects are not really addressed in the current curriculum. Moreover the necessary multi-expertise of project management methods during design and construction phases is not part of usual architects/engineers trainings. -When coming to the teaching of planning methods, traditional planning courses sometime appears as disconnected from reality for students. Indeed they are not really aware of the "insitu" life of construction projects. One can recognize that it is usually difficult to address 1) the issue of high-rise design, 2) the technical aspects of tall buildings structures and 3) the characteristics of high-rise construction processes, within a single and short-duration course. From these statements an original pedagogical scenario inspired by the recent advances in the Construction IT research community is proposed. Research works about 4D models and simulations and their application in pedagogy are considered. Indeed 4D modeling and simulation appears as promising technology for helping students to better analyze high-rise buildings design and construction, and especially their structural principles. The paper firstly addresses 4D technology and its place in education curricula. Then a specific course is described as well as some student's results regarding 4D use and pedagogical feedback. Further research about describing 4D visualization is proposed in order to improve visualization of 4D models by the students. Possible application is finally illustrated through an example.

LITERATURE REVIEW: 4D TECHNOLOGIES IN EDUCATION

The emerging 4D applications

4D modeling and simulation consists in linking a 3D model with construction project planning in order to visualize the construction process over time. It can be used in many phases of a construction project including design & analyzes, scheduling and management [START_REF] Tommelein | Knowledge-based assembly of simulation networks using construction designs, plans and methods[END_REF]. Previous research works have shown the interest of 4D simulation use [START_REF] Hartmann | Areas of Application for 3D and 4D Models on Construction Projects[END_REF]. They stated that it has a significant impact on the project realization by increasing productivity, reducing interferences on the ground, enabling easier pre-production and time efficiency [START_REF] Staub-French | 3D and 4D modeling for design and construction coordination: issues and lessons learned[END_REF]. Moreover, 4D users can communicate and coordinate better and reach better performance [START_REF] Dawood | Measuring the effectiveness of 4D planning as a valuable communication tool[END_REF]). In the pre-construction phase, the use of 4D simulations is useful to compare the constructability of working methods, to visually identify conflicts and clashes, and also as visual tool for different actors to discuss and to plan the progress of works [START_REF] Mahalingam | An evaluation of the applicability of 4D CAD on construction projects[END_REF]. At date, the use of 4D simulation has considerably been diversified and many innovative applications were proposed since its introduction two decades ago. In addition to the simulation of the construction steps and the identification of clashes and conflicts, 4D CAD enables nowadays optimization of equipment space requirements including the site logistics deployment [START_REF] Akinci | Assessment of the Capabilities of a Commercial 4D CAD System to Visualize Equipment Space Requirements on Construction Sites[END_REF][START_REF] Heesom | Trends of 4D CAD applications for construction planning[END_REF]. It can also be used to simulate excavation and earthworks [START_REF] Yabuki | A Management System for Cut and Fill Earthworks Based on 4D CAD and EVMS[END_REF][START_REF] Shah | Automatic generation of progress profiles for earthwork operations using 4D visualisation model[END_REF], to manage oil rig management [START_REF] Doucet | Advanced mid-water tools for 4D marine data fusion and analysis[END_REF]) and for urban communication as well [START_REF] Stellingwerff | 3D/4D Communication Tools for Facilitators in Public Participation[END_REF]. To be more generic, one can consider that it has been shown that 4D simulations are very useful when used to visually, identify and compare "items". Thus, it is more and more used as visual support to education.

4D in pedagogical curriculums

Construction projects management courses are very diversified in architecture, engineering or construction curricula [START_REF] Dietz | Education for Construction[END_REF]. In architecture curriculums it is usually stated that architecture students do not gain much practical knowledge of construction management and methods [START_REF] Clayton | Virtual construction of architecture using 3D CAD and simulation[END_REF]. With the emergence of computer supporting 3D building modeling, innovative methods are more and more explored in order to improve the construction education experience. [START_REF] Clayton | Virtual construction of architecture using 3D CAD and simulation[END_REF]) described a virtual construction exercise experience with students using 3D CAD and simulation. He concluded that virtual construction is very interesting to teach construction management to students more easily through addressing many learning situations or projects examples. [START_REF] Perdomo | Interactive 3D Visualization As A Tool For Construction Education[END_REF]) presented a study with Virginia Tech architecture and building construction department that was investigating the advantage for education between 2D drawings and 3D representation in understanding construction assemblies and details. The results were very positive. But in curricula addressing complex "construction environments" like bridges, towers or high-rise buildings construction, it is important to decrease the lack of technical subject's integration. Indeed, such environments require treating scale, quickness, working at multiple levels of detail, dealing with design variability, and realistic representation of the work [START_REF] Russell | Visualizing high-rise building construction strategies using linear scheduling and 4D CAD[END_REF]. And students have to understand difficulties related to the conduct of such project as well as decisions that have to be taken. So, it is important to include the temporal dimension related to the scheduling, in order to explore and analyze the constructability of working methods. 4D simulations appeared to be an innovative solution and some teachers previously implemented it in their courses. [START_REF] Kang | Development of Web-based Interactive 4D Block-Tower Model for Construction Planning and Scheduling Education[END_REF]) presented a webbased interactive 4D block tower model for construction planning and scheduling education and showed 4D visualization interests for education. [START_REF] Sampaio | Virtual Reality technology applied in Civil Engineering education[END_REF]) demonstrated through many examples that virtual reality, including 4D CAD, can be useful in teaching material elaboration. [START_REF] Wang | Assessment of 4D modeling for schedule visualization in construction engineering education[END_REF]) described a study to assess the value of using 4D modeling in construction engineering education and compared results from two different 4D processes that are traditional 4D and virtual construction simulator. Both processes were found valuable for student learning experience improvement.

TEACHING OF HIGH-RISE STRUCTURE PRINCIPLES THROUGH 4D SIMULATION

The following parts address the course called "Project Management", proposed at University of Liège (Belgium) to construction engineer and architect-engineer in their Master curricula. Firstly the context of this training and the methods used are explained. Then some representative results related to 4D simulations performed by the students are described.

Pedagogical context

The original aim of "project management" course in Master curricula is to confront the students with the complexity inherent to large-scale construction projects. When addressing "large scale" project, the focus is both on the complexity of the design project itself (through its architectural, structural, equipment-related… dimensions) and on the organization and project management methods that are required for involving viewpoints of multiple experts. Since 2009, high-rise building are chosen as large scale project for this course, and the teaching team aims at addressing issues related to their design (especially their structural principles) and to the management of the construction.

The pedagogical scenario retained for this course consists of both theoretical courses and practical works. Practical works are themselves divided into two stages: single-student work and teamwork.

-Theoretical courses aim at providing students with basic knowledge in the fields of high-rise buildings and project management. Structural constraints and common solutions are the main topics of courses. A typology, notably inspired from [START_REF] Sev | Recent trends in tall building design[END_REF], is presented as a support to learning. Vertical transportation systems, as well as façade techniques, are also addressed. The second part of the course introduces project management especially in terms of organizations of actors and coordination mechanisms. Finally IT-supported coordination is explained through existing IT systems and emerging ones. -Practical works consist of analyzing high-rise buildings projects. In a first stage (4-5 working weeks), each student is expected to analyze various aspects of a project based on the theoretical inputs. They also have to model a regular storey of the building using Google SketchUp™, as an introductory exercise (most of construction engineers are not aware of modeling softwares). Then in a second stage, students are grouped within 3 or 4 teams. Each team chooses an interesting building and has to realize a complete analysis about structural principle and construction process. Then they have to propose a 4D simulation "scenario" which aims to provide a "didactic understanding" of building structure and construction. 4D modeling and simulation themselves are the final part of the teamwork. 3D modeling is realized with Google SketchUp ™, and 4D modeling and simulation is enabled thanks to the 4D Virtual Builder© (Nys 2010) plug-in for Google SketchUp ™. This course has been taught two times, during the fall semesters of 2010 and 2011 academic years. 14 students were involved in 2010, and 12 students participated in 2011. In 2010, 3 teams worked on New-York Times Building (New-York), Sears Tower (Chicago) and Debis Tower (Berlin). In 2011, 4 teams worked on Caja Madrid, Opernturn (Frankfurt), Shard London Tower and World Financial Center (Shanghai).

Research methodological approach

Beyond the course itself, the paper aims at 1) validating the interest of 4D technologies as support to the teaching of high-rise principles as well as 2) finding ways of improvement for guiding students when they are confronted to modeling and simulating their analyzed projects. The methodological approach followed in the paper consists in presenting some results in order to describe students' works and the way they perform 4D simulations. From these works we present the pedagogical findings and qualitatively assess on the usefulness of 4D in the framework of the course. Then we suggest going deeper in the analysis of proposed simulations, especially in terms of (multi) visualization. The analysis enables modeling visualization properties. It allows us to open prospects on the guidance that we could provide to students having to perform 4D simulation, especially for choosing modeling and visualization properties for their simulations.

4D simulations: results from 2009 and 2010 sessions

In this section some results from the 2009 and 2010 sessions are presented. High-rise projectsselected by the students-are briefly exposed and the 4D simulation strategy is described, as well as the obtained 4D visualization. For the purpose of the following analysis and proposition, 4D simulations presented below are considered as independent of the software really used to model or visualize it, but as a composition of 3D + Time views.

Caja Madrid Tower (2010 session)

The Caja Madrid project (Figure 1 This simulation aims at showing the construction of the structural elements of the tower that are mainly: core, metallic structure and glazing. It also shows the element's bearing level using a shade of color ranging from yellow (for the least bearing elements) to bright red (the most bearing elements). The blue color is used for glazing representation. The floors are considered identical. Time is represented by a list of selectable dates in which the current date is highlighted. The current date is also displayed at the bottom of this list. The simulation uses zoom, perspective and extended zoom to improve understanding.

Debis Tower (2009 session)

The Debis Tower (Figure 1, right) (http://library-towers.wikidot.com/debis-tower, last accessed 2011/05/06) is a 106 m-high building located in Potsdamer platz, Berlin. Construction completed in 1997. The simulation of Debis tower made by students is based on a simplified planning and shows the offset construction of bearing elements (core, columns, slabs) and facade elements. Elements are visible from their execution dates and appear according to a specific color system: green for columns, blue for slabs, red for central core. Importance is placed on transparency applied to facade elements and slabs to allow visualizing of the structural principle inside the building. In this simulation, the time is not the subject of a special visualization mode. Current date is simply displayed. This simulation uses zoom, perspective and sectional views to facilitate understanding of specific informations.

New-York Times Building (2009 session)

Ney York Times building (Figure 2, left) (http://newyorktimesbuilding.com/, last accessed 2011/05/06) is 228 m-high tower, designed by Renzo Piano and completed in 2007. The structural principle consists of a concrete core associated with steel structure for the floors. Simulation of the New York Times Building construction focuses on building components (flooring, core, columns and steel beams, bracing, façade and roof construction). In this simulation scenario, the building is considered as a set of floors that are not identical. Each floor is composed of a certain number of given components. Thus the first floor contains four types of objects, while levels from 2 to 4 are composed of 6 types of objects, such as level 5. Floors from 8 to 52 contain 8 types of objects such as higher levels. Emphasis is given on structural elements (core, columns and steel beams) that are represented with red color. Other items are represented in lighter or darker gray. The green color represents the elements during construction. Partial transparency is used on the facade elements to allow viewing inside the building.Zoom, extended zoom and perspective are used to facilitate understanding of certain construction details. Towards the end, the orbit is used to allows viewing a realistic appearance of the complete building. In this simulation, the time is not subject of a special visualization mode. The current date is simply displayed.

Shanghai World Financial Center Tower (2010 session)

The Shanghai World Financial Tower (Figure 2, right) is a 492 m-high tower located in the district of Pudong (http://en.wikipedia.org/wiki/Shanghai_World_Financial_Center, last accessed 2011/05/06). The project started in 1997, but the construction completed in 2078. The principle used by students to simulate the construction of the Shanghai World Financial Center Tower aims to bring up the simultaneous work of four different teams. Then, it uses a system of colors to show offset in the groups building works. The central core construction is represented by white color. Yellow is associated with other bearing structural elements and green color represents concrete pouring while glazing is shown with blue color. Items under construction are in red color. The floors are not standard. From time to time, the zoom is used to show the construction of out-riggers or other complex elements. The camera position also changes to improve visibility. Time is represented by a list of selectable dates in which the current date is highlighted. The current date is also displayed at the bottom of the list.

MAIN FINDINGS

Pedagogical feedback

The course presented above is experimental in the engineer-architect and construction-engineer curriculums of University of Liège. It was designed in the continuity of the previous course of project management, which was dedicated to the understanding of particular constraints related to the planning and design of large-scale projects. The main hypothesis is to benefit of 4D modeling and simulation technologies. The feedbacks from teaching team are the following:

-Firstly, the use of 3D modeling tools like Google SketchUp™ is possible and valuable, also when students are not familiar with 3D modeling (it is the case of the construction-engineers students). SketchUp™ is rapidly understood and usable by all of the students. -4D modeling of high-rise building (although the aim is not to provide a very fine-grained planning) requires a deep understanding of structural principle, because it impacts the skeleton of the construction planning (i.e. Work Breakdown Structure). The pedagogical team appreciates the transparency and completeness of students' analyses. This is due to the need of clearness when modeling buildings' main structural 3D elements as well as planning's WBS. -Finally, as high-rise construction documentation is usually difficult to obtain (planning asrealized, detailed plans etc.) students have to infer both structural principles and construction planning. It requires them both making hypotheses on the design as well as finding evidences of construction procedures (photos, webcam, or TV documentary). The exercise then becomes original compared to classical "planning" or "structure" courses and students get easily engaged. The previous 4D simulations examples show that many different scenarios can be proposed by the students in order to realize a construction 4D simulation. These scenarios depend on the aim of the simulation, what they want to visualize as well as the need of interaction.

Synthesis of 4D simulation results

Some common properties can be identified to describe the simulations, especially in terms of visualization. In Table 1, some of these properties are identified on the basis of the 4 examples.

Caja Madrid Tower simulation

Debis Tower simulation

New-York Times Building simulation

World Financial Tower simulation

Building modeling principles

Floor type

Standard Not standard Not standard Not standard

Shown elements

Central core, Metal It is then possible to describe and to compare 4D simulations according to these identified visualization-related properties. In oder to be able to do it precisely and accurately, it is important to enable modeling visualizations in a structured way. Model Driven Engineering approach recommends the use of meta-models, thus each model has to be conformed to its meta-model [START_REF] Favre | Towards a Basic Theory to Model Driven Engineering[END_REF]. A « 4D simulation view" can be considered as a coordinated set of multiple views (CMV): the association of a 3D view and a planning view, coordinated through appropriate interaction principles.

TOWARDS MODELING 4D MULTI-VIEWS FOR PEDAGOGY

5.1 A metamodel for describing 4D multi-views "A multiple view system uses two or more distinct views to support the investigation of a single conceptual entity" [START_REF] Wang-Baldonado | Guidelines for Using Multiple Views in Information Visualization[END_REF] and "coordinated multiple views" concept can be used when operations on the views are coordinated [START_REF] Roberts | State of the Art: Coordinated & Multiple Views in Exploratory Visualization[END_REF]. So 4D views can be considered as coordinated multiple views [START_REF] Boton | Method to design coordinated multiple views adapted to user's business requirements in 4D collaborative tools in AEC[END_REF], composed of 3D view and planning view where activities are linked to building elements. 3D and planning views are both business views, visualization modes that practitioners use in their daily work. To be able to model 4D we first need an appropriate metamodel. Indeed, Model Driven Engineering (MDE) approach recommends the use of metamodels to define domain languages, thus models have to be conformed to their metamodel.

In order to propose a coordinated multiple views metamodel, we rely on the eight guidelines proposed by [START_REF] Wang-Baldonado | Guidelines for Using Multiple Views in Information Visualization[END_REF] for the design of multiple view systems and the 2x3 taxonomy of multiple window coordination from [START_REF] North | A taxonomy of multiple window coordinations[END_REF] The aim while developing such a metamodel is to enable the description of several 4D multi-views, their storage and the possible comparison of multi-views on the basis of selection criteria. The next part suggests an example of metamodel application aiming at describing a new 4D multi-view inspired from students' work.

Example of metamodel application

The following example suggests an innovative 4D multi-view based on the Shanghai World Financial Center Tower project analyses performed by the students in the 2010 session. One important aim of the students was to represent the simultaneous work of four different teams of workers. To make this possible and more expressive, considering 4D simulations as CMV visualizations and using the CMV metamodel ( §5.1), the proposed approach enables choosing adapted visualization modes for 3D view as well as for planning view, and to model them. In this example, we rely on the 3D model developed by the students. But, for the planning representation, instead of the list of date used by students, it seems better to use the "line of balance" representation mode. Indeed, this planning visualization technique enables to consider both time and location in the same view. It is then easier to identify clearly the construction offset of the four different teams. In such a planning view, colors system could even be the same as for the 3D representation (grey for central core, yellow for metal structure, light blue for concrete pouring and dark blue for glazing). A mockup of the appearance of such a 4D CMV is shown in Fig 4. Relying on the proposed metamodel, such a 4D multi-view could be described by the following model.

Figure 5: Model of suggested 4D multi-view.

CONCLUSION

This article describes a pedagogical course for architect and construction engineers of the University of Liège about the structural principles of high-rise buildings. Within this course, students experiment the use of 3D and 4D tools as support to the analysis of some projects' structures and construction processes. Authors conclude that 4D modeling and simulation of analyzed projects enables a better understanding by the students. Students' attention is particularly guided to the making of choices during 3D modeling as well as the scenography proposed for 4D visualization: colors used, camera and 3D views proposed and relationship to the timeline of construction. After two sessions of this course the authors note that such choices are wide, and that students do not enough know what 4D simulation is to correctly setup their own scenography. Then, they propose consolidating 4D multiviews and their components through models dedicated to the purpose of "structural principles didactic". A metamodeling approach aiming at describing coordinated multiple views, and previously experimented to 4D multiple views supporting collaboration during pre-construction phase is reused. An example finally shows how an "invented" 4D multiple view can be described relying on the metamodel. The limit of this approach is that it is extensively theoretical. Then an automatic implementation of the model-based generation of 4D multi-views is not possible today. However the authors underline their prospects towards a repository of 4D multi-views. Indeed the metamodeling approach enables the description of structured models of 4D multi-views, by the way of conceptual modeling tools for editing and storing models. Then it could be envisaged to store such models of 4D multi-views in order to reuse it from one session to another and possibly to improve it, or provide more case studies.

Sharing is also a prospect and universities could share their models to the construction education community.

Figure 1 :

 1 Figure 1: Caja Madrid's (left) and Debis Tower's (right) 4D simulations.

Figure 2 :

 2 Figure 2: NY Times building's (left) and Shanghai World Financial Center's (right) 4D simulations.

  , left) (http://en.wikipedia.org/wiki/Caja_Madrid, last accessed 2011/05/06) is a 250 m-high office tower located in Madrid (Spain) and designed by Foster & Partners. Construction started in 2004 until 2008. Three groups of stairs' blocks are constructed around two main concrete cores.

Figure 4 :

 4 Figure 4: Suggested 4D multi-view.

Table 1 :

 1 4D modeling and visualization properties.

				Central core, Steel	Core, Steel struct.,
				columns and beams,	Concreting,
			, Slabs,	Bracing, Facade,	Bracing, Outrigger,
			Facades	Circulations, Roof.	Glazing
	Color semantic
	Execution	Green color for	Not applicable	Green color for	Red color for
	state	ongoing works		ongoing works	ongoing works
	Building	-Color scale	-Green color	-Red color represents	-Yellow color
	element	corresponds to	represents columns	structure elements	represents structure
		elements bearing	-Blue color		elements
		capacity	represents slabs		-Green represents
		-Blue color	-Red color		concreting
		represents glazing	represents central		-Blue represents
			core		glazing
	Transparency -Full transparency	-Full transparency	-Full transparency	-Full transparency
		for not started	for not started	for not started	for not started
		elements	elements	elements	elements
		-Partial	-Partial		-Partial
		transparency on	transparency on		transparency on
		glazing	glazing and roofs		glazing
	Time (planning)
	Time view	-Dates list with	-Current date	-Current date	-Dates list with
		current date	displayed	displayed	current date
		highlighted			highlighted
		-Current date			-Current date
		displayed			displayed
	Camera
	Camera	Zoom, Extended	Zoom, Perspective,	Zoom, extended	Zoom, Orbit
	principles	zoom, Perspective	Section	zoom, Perspective,
				Orbit

  . Proposed metamodel is depicted on Fig 3. This metamodel reuse the main classes of the business view metamodel and create associations between them. Coordinated multiple views are composed of 2 to n business views with coordination mechanisms.

	Figure 3: Coordinated Multiple View metamodel.
	According to these mechanisms selecting item(s) in one view can lead to selecting/highlighting item(s)
	in other view(s), navigating one view can lead to navigating other views, or selecting items in one
	view can lead to navigating other views.

ACKNOWLEDGMENTS

The authors thank the students of the 2009-2010 and 2010-2011 sessions as well as the pedagogical team of the University of Liège, Applied Sciences Faculty, ArGEnCo department. We also thank D-Studio firm for providing the SketchUp plug-in "xD Virtual Builder" and participating to the experiment. The paper has been co-written by a PhD student beneficiary of a PhD AFR grant from Fonds National de la Recherche (FNR), Luxembourg.