Jérôme Duriez
email: jerome.duriez@ucalgary.ca

Richard Wan

Jérôme Duriez ¨richard Wan

Contact angle mechanical influence in wet granular soils

Keywords:

come

Introduction

Granular soils encountered in geotechnical engineering often encompass several immiscible fluids within their pore network such as in situations related to oil and gas production, non-aqueous pollutant transport in groundwater, frozen soils, or simply, classical unsaturated soil mechanics dealing with a mixture of air and water.

The presence of such fluid mixture is an important issue since it greatly influences the soil's mechanical behaviour, as illustrated by the classical example of sandcastles that require wet sand to hold. Restricting ourselves to ternary mixtures with one granular solid phase and two fluid phases, one key parameter that controls the various fluid distributions is the contact (or wetting) angle θ describing the contact between the solid phase and the wetting fluid phase along the so-called contact lines where the three phases intersect. The contact angle value is highly dependent on the particular case of interest. For spherical particles, wet samples of glass beads may show low contact angle values (ď 10 0 ) [30,[START_REF] Scheel | Morphological clues to wet granular pile stability[END_REF], while the contact angle between steel balls and water is around 50 0 [START_REF] Soulié | Influence of liquid bridges on the mechanical behaviour of polydisperse granular materials[END_REF]. As far as real sandy soils are concerned, contact angle values reaching 50 0 and more have been reported in [START_REF] Bachmann | Modified sessile drop method for assessing initial soilwater contact angle of sandy soil[END_REF][START_REF] Liu | Capillary rise method for the measurement of the contact angle of soils[END_REF]. Adding to complexity, the contact angle is also dependent on the hydraulic loading path; its value showing a hysteresis between extreme limits that correspond to receding or advancing contact lines. Such hysteresis has now be shown to affect the fluid phase distribution [START_REF] Mani | Role of contact-angle hysteresis for fluid transport in wet granular matter[END_REF].

Focusing on low degrees of saturation within the socalled pendular regime, the fluid distribution is characterized by isolated capillary bridges, i.e. menisci, consisting of the wetting fluid (considered from now on to be a liquid) and that form between solid particles pairs. Several capillary bridge analyses have been proposed that accommodate for any contact angle value, giving the possibility to investigate the contact angle influence at this microscopic scale [START_REF] Hotta | The capillary binding force of a liquid bridge[END_REF][START_REF] Lian | A theoretical study of the liquid bridge forces between two rigid spherical bodies[END_REF][START_REF] Molenkamp | Interactions between two rough spheres, water bridge and water vapour[END_REF][START_REF] Soulié | Influence of liquid bridges on the mechanical behaviour of polydisperse granular materials[END_REF][START_REF] Lechman | Capillary force and water retention between two uneven-sized particles[END_REF]. From these studies, the contact angle is known to affect the liquid volume-capillary pressure relationship and the capillary force magnitude, for instance. Such microscopic effects necessarily also reflect into the macroscopic behaviour of soil samples. However, a proper macroscopic analysis of the contact angle influence seems to be at best limited in the literature. Straightforward experiments are prevented by the major difficulties measuring contact angle at a soil sample scale, compared with the ideal case of droplets lying on a flat surface [START_REF] Bachmann | Modified sessile drop method for assessing initial soilwater contact angle of sandy soil[END_REF][START_REF] Czachor | Modelling the effect of pore structure and wetting angles on capillary rise in soils having different wettabilities[END_REF]. On the other hand, only few macroscopic modelling approaches include a contact angle analysis such as done in [START_REF] Chenu | Organic matter influence on clay wettability and soil aggregate stability[END_REF][START_REF] Czachor | Modelling the effect of pore structure and wetting angles on capillary rise in soils having different wettabilities[END_REF][START_REF] Mani | Role of contact-angle hysteresis for fluid transport in wet granular matter[END_REF]. Among these studies, the focus is mostly set on the description of two-phase pore flow within a fixed packing [START_REF] Czachor | Modelling the effect of pore structure and wetting angles on capillary rise in soils having different wettabilities[END_REF][START_REF] Mani | Role of contact-angle hysteresis for fluid transport in wet granular matter[END_REF]. Finally, the assumption of a zero contact angle is still routinely adopted in most macroscopic-scale numerical modelling of unsaturated soils in a mechanical framework [START_REF] Gili | Microstructural deformation mechanisms of unsaturated granular soils[END_REF][START_REF] Richefeu | Stress transmission in wet granular materials[END_REF][START_REF] Scholtès | Micromechanics of granular materials with capillary effects[END_REF][START_REF] Mani | Liquid migration in sheared unsaturated granular media[END_REF][START_REF] Wang | Anisotropy of a tensorial Bishop's coefficient for wetted granular materials[END_REF][START_REF] Than | Basic mechanical properties of wet granular materials: A DEM study[END_REF]. As such, the objective of the present paper is to assess through careful modelling how contact angle influences the mechanical behaviour of wet granular assemblies, in the pendular regime.

Contact angle mechanical influence in wet granular soils [START_REF] Bathurst | Observations on stressforce-fabric relationships in idealized granular materials[END_REF] We herein consider an idealized granular material with solid particles being spherical in shape. Such assumption eases the menisci computation from the Laplace-Young equation. Both the solid and liquid phases show in the pendular regime a discrete microstructure, with distinct solid grains and liquid bridges. In connection with this discrete microstructure, it is convenient to choose the Discrete Element Method (DEM) as a modelling approach, as done in [START_REF] Gili | Microstructural deformation mechanisms of unsaturated granular soils[END_REF][START_REF] Richefeu | Stress transmission in wet granular materials[END_REF][START_REF] Scholtès | Micromechanics of granular materials with capillary effects[END_REF][START_REF] Mani | Liquid migration in sheared unsaturated granular media[END_REF][START_REF] Wang | Anisotropy of a tensorial Bishop's coefficient for wetted granular materials[END_REF][START_REF] Than | Basic mechanical properties of wet granular materials: A DEM study[END_REF]. The DEM describes any internal force using interaction forces between the discrete elements that correspond to spherical solid particles. As such, the wetting and non-wetting (from now on, a gas) fluids are indirectly incorporated in the model through the resultant capillary forces that act on the particle pairs bonded by a capillary bridge.

Because the capillary force depends on the liquid bridge geometry, we begin with Section 2 that gives details of a liquid bridge determination for various contact angles that is inspired from previous works on monosized particles pairs [START_REF] Hotta | The capillary binding force of a liquid bridge[END_REF][START_REF] Lian | A theoretical study of the liquid bridge forces between two rigid spherical bodies[END_REF]. Next, the results of Section 2 are implemented in the open source DEM code Yade [START_REF] Šmilauer | Yade Documentation[END_REF] in line with Scholtès et al. [START_REF] Scholtès | Micromechanics of granular materials with capillary effects[END_REF] who restricted the computations to zero contact angles. The DEM implementation leads to a macroscopic analysis of the contact angle influence in Section 3. Finally, Section 4 analyzes in depth the results of Section 3, focusing on the cap-illary stresses that encompass the mechanical actions existing in a wet assembly due to the fluid mixture.

Liquid bridge determination

Considering a polydispersed idealized granular material, capillary bridges form in between two dissimilar spherical particles of radii R 1 , R 2 with radius ratio r " R 2 {R 1 ě 1. Three-dimensional axisymmetric conditions hold when neglecting gravity g, i.e. for low Bond numbers B " pρ l ´ρg q g R 2 2 {γ with ρ l , ρ g being respectively the liquid and gas densities, g the gravity and γ the liquid-gas surface tension. It is thus convenient to use 3D cylindrical coordinates pρ, φ, zq with z as the axis of rotational symmetry defined by angle φ, such that a function ζpzq defines the liquid bridge surface tρ " ζpzq; φ P r0; 2πs; z P r0; z f su where z f is the distance between the two three-phase contact lines. Fig. 1 illustrates such a liquid bridge between two spherical particles: δ 1 and δ 2 refer to the half-filling angles, and

θ is the contact (wetting) angle. R 2 R 1 θ ρ=ζ(z) θ δ 2 δ 1 z z=0 z=z f
Fig. 1 Liquid bridge geometry

It is convenient to normalize all lengths through division by the largest radius R 2 which serves as a length scale so that normalized quantities are z ˚" z{R 2 , ρ ˚" ρ{R 2 and ζ ˚pz ˚q " ζpR 2 z ˚q{R 2 . Thus, the liquid bridge surface is described in the dimensionless space as tρ ˚" ζ ˚pz ˚q; φ P r0; 2πs; z ˚P r0; z f su.

Generally speaking, liquid-gas interfaces always obey the Laplace-Young equation given as:

u g ´ul " u c " γ div n (1)
with the normal n oriented towards the liquid, u g and u l as the gas and liquid pressure respectively, and u c the capillary pressure. Since adsorbed liquid layers are negligible for granular soils, the capillary pressure u c also corresponds to the matric suction.

The Laplace-Young equation serves as a partial differential equation describing the liquid bridge configuration. In order to obtain the latter, Eq. ( 1 i.e.

u c R 2 γ " u c " ζ ˚2 `1 `ζ˚1 2 ˘3{2 ´1 ζ ˚`1 `ζ˚1 2 ˘1{2 (2)
which introduces a dimensionless capillary pressure u c .

Eq. ( 2) might be approximately solved using a toroidal expression for ζ ˚from two constant principal radii with the meridian liquid profile being simply a circular arc [START_REF] Haines | Studies in the physical properties of soils: II. A note on the cohesion developed by capillary forces in an ideal soil[END_REF][START_REF] Fisher | On the capillary forces in an ideal soil; correction of formulae given by W. B. Haines[END_REF]. However, this approximation leads to a nonconstant surface curvature locally as defined by div n, which, strictly speaking, contradicts the Laplace-Young equation (1) [START_REF] Hotta | The capillary binding force of a liquid bridge[END_REF][START_REF] Lian | A theoretical study of the liquid bridge forces between two rigid spherical bodies[END_REF].

Numerical procedure

Similar to [START_REF] Lian | A theoretical study of the liquid bridge forces between two rigid spherical bodies[END_REF] which dealt with only monosized spherical particles, the current numerical procedure to solve Eq. ( 2) considers a particular half-filling angle δ 1 on the smallest particle and the dimensionless capillary pressure u c . Starting from the smaller particle on the left with boundary condition ζ 0 " sinpδ 1 q{r, the liquid bridge profile ζ ˚pz ˚q is incrementally computed using a second order Taylor series expansion:

ζ ˚`z ˚" pi `1q ∆z ˚˘" ζ i`1 "ζ i `∆z ˚ζ˚1 i `1 2 ∆z ˚2 ζ ˚2 i (3)
The first derivative ζ ˚1 i is computed using a second order finite difference approximation, Eq. ( 4), that is suited to describe the evolving profile slope, contrary to the always positive expression presented in [START_REF] Lian | A theoretical study of the liquid bridge forces between two rigid spherical bodies[END_REF]. The initial value ζ ˚1 0 is a distinct case and is obtained from the wetting of the smallest solid particle along the left contact line:

ζ ˚1 i " $ ' ' & ' ' % ´1 tanpδ 1 `θq for i " 0 ζ i ´ζi ´1 ∆z ˚`1 2 ∆z ˚ζ˚2 i´1 else (4)
Appearing in Eqs. ( 3) and ( 4), the second derivative

ζ ˚2 is expressed in terms of ζ ˚and ζ ˚1 from Laplace-
Young Eq. ( 2), i.e.

ζ ˚2 i " 1 `ζ˚1 i 2 ζ i `uc ´1 `ζ˚1 i 2 ¯3{2 (5)
As such, the liquid bridge profile is obtained by applying successively Eqs. ( 4), ( 5) and (3) until the right contact line, defined by ζ f " sinpδ 2 q, is reached for z ˚" z f . Prior to that, the filling angle δ 2 on the largest particle is deduced by expressing Eq. ( 6) at both contact lines and solving numerically for δ 2 :

ζ 1 `ζ˚1 2 ˘1{2 `uc 2 ζ ˚2 " C " cst (6)
The l.h.s. of Eq. ( 6) is the dimensionless expression of the capillary force resulting from capillary pressure and surface tension actions along any meniscus crosssection, in particular the left or right wetted particles surfaces. Obtained from making a variable substitution in the Laplace-Young equation as given in [START_REF] Lian | A theoretical study of the liquid bridge forces between two rigid spherical bodies[END_REF], Eq. ( 6) basically expresses meniscus force equilibrium.

Also, an increment ∆z ˚is chosen to be small enough (2 ˆ10 ´6 typically) so as not to have any influence on the numerical results. In particular, this ensures both stability and accuracy of the explicit scheme related to Eqs. (3)- [START_REF] Bisschop | A physical model for liquid capillary bridges between adsorptive solid spheres: The nodoid of Plateau[END_REF].

Determining the liquid bridge profile ζ ˚pz ˚q, z ˚P r0; z f s, following the above numerical procedure ulti-mately leads to a comprehensive geometrical description of the liquid bridge and other characteristics. For instance, the dimensionless interparticle distance d ˚"

d{R 2 and liquid volume V ˚" V l {R 2 3 are readily computed excluding the solid volumes:

d ˚" z f ´1 r p1 ´cos δ 1 q ´p1 ´cos δ 2 q (7)
and

V ˚" f ř i"1 π ζ i 2 ∆z ˚´π 3 r 3 pcos 3 δ 1 ´3 cos δ 1 `2q ´π 3 pcos 3 δ 2 ´3 cos δ 2 `2q (8)
Furthermore, the dimensionless capillary force is directly obtained by satisfying Eq. ( 6) along either contact line of the liquid bridge:

F ˚" F cap 2 π γ R 2 " C (9)
Finally, the liquid bridge description is completed by the calculation of the dimensionless meniscus orientation tensor π

˚: π ˚" 1 R 2 2 ż Sm n b n dS (10)
that is obtained from the normal n, the local orientation tensor n b n, and a numerical integration along the liquid bridge external surface S m , see Appendix 1.

Such meniscus orientation tensor enters into the description of the internal surface tension forces within the liquid bridge surface [START_REF] Chateau | Homogenization of a nonsaturated porous medium: Hill's lemma and applications[END_REF][START_REF] Chateau | Micromechanics of unsaturated granular media[END_REF][START_REF] Gray | Analysis of the solid phase stress tensor in multiphase porous media[END_REF], and represents a pertinent tensorial quantity that was not provided by previous analyses [START_REF] Lian | A theoretical study of the liquid bridge forces between two rigid spherical bodies[END_REF][START_REF] Scholtès | Micromechanics of granular materials with capillary effects[END_REF]. In particular, the inclusion of π ˚facilitates the stress analysis in Section 4 without using any assumption on the meniscus orientation. It is worth noting that the expression of π ˚in the local (meniscus) basis pe ρ , e φ , e z q is diagonal and axisymmetric with π ρρ " π φφ ‰ π zz because of the meniscus shape. Further insight is gained by considering the trace of π ˚which turns out to be the dimensionless meniscus surface S m:

trpπ ˚q " 2 π ρρ `πz z " 1 R 2 2 ż Sm ||n|| 2 dS " S m R 2 2 " S m (11)
In the end, the numerical procedures described in the above lead to a schema in which, for given values of normalized suction u c , contact angle θ, and radius ratio r, the complete set of liquid bridge configurational parameters td ˚, V ˚, F ˚, δ 1 , δ 2 , π ρρ , π zz u is obtained by sweeping through various filling angles δ 1 P r0 0 ; 90 0 ´θs.

Values of δ 1 greater than p90 0 ´θq are not considered since they would correspond to concave liquid bridges (negative suctions). Also, numerical computations may give configurations that present negative volume and/or particles distance. Such non-physical solutions correspond to impossible liquid bridges and are thus disregarded.

Liquid bridge stability

When analyzing granular assemblies, a key variable that provides insights into the nature of the liquid bridge solution is the inter-particle distance. As such, liquid bridge configurations obtained in the previous sub-section 2.1 are depicted according to d ˚in Fig. 2, which shows two possible bridge configurations for the same distance and same tr; θ; u c u parameters. These two solutions both obey the force equilibrium condition expressed by Laplace-Young equation and correspond to stable or unstable distinct equilibrium states [START_REF] Padday | The stability of axisymmetric menisci[END_REF][START_REF] Lian | A theoretical study of the liquid bridge forces between two rigid spherical bodies[END_REF][START_REF] Lowry | Capillary surfaces: Stability from families of equilibria with application to the liquid bridge[END_REF]. Several free energy expressions have been proposed to assess the stability of these different configurations [START_REF] Bisschop | A physical model for liquid capillary bridges between adsorptive solid spheres: The nodoid of Plateau[END_REF][START_REF] Lian | A theoretical study of the liquid bridge forces between two rigid spherical bodies[END_REF]. However, these expressions rely on a constant liquid volume assumption, whereas constant pressure conditions prevail herein. As this disparity in state description affects stability properties [START_REF] Lowry | Capillary surfaces: Stability from families of equilibria with application to the liquid bridge[END_REF], we consider here the following more suitable dimensionless free energy E ˚whose derivation is given in Appendix 2:

E ˚" E γ R 2 2 " u c V ˚`2 π « f ř i"1 ζ i b 1 `ζ˚1 i 2 ∆z cos θ ˆ1 ´cos δ 1 r 2 `1 ´cos δ 2 ˙ff (12)
The highlights of the dimensionless free energy expressed in Eq. ( 12) include surface energies for liquid-gas as well as liquid-solid and solid-gas interfaces, considering constant solid surfaces for the particles. Specific volume

Contact angle influence at the liquid bridge scale

The determination of capillary bridge configurations for different contact angles offers first micromechanical insights in the contact angle influence on wet granular packings. At the microscopic scale, and for given dimensionless suction, interparticle distance, and relative radii parameters, an increasing contact angle decreases the wetted surfaces, the meniscus volume and surface, and the capillary force, see Fig. 4. Such decreases in liquid volume and capillary forces with the contact angle have also been described in [START_REF] Molenkamp | Interactions between two rough spheres, water bridge and water vapour[END_REF][START_REF] Lechman | Capillary force and water retention between two uneven-sized particles[END_REF].

Comparison with empirical relations

As an alternative to systematically solve Laplace-Young equation, several empirical relations for the capillary force have been proposed in the literature [START_REF] Chateau | Micromechanics of unsaturated granular media[END_REF][START_REF] Richefeu | Stress transmission in wet granular materials[END_REF][START_REF] Willett | Capillary bridges between two spherical bodies[END_REF].

These are now compared with the results of the presented numerical procedure.

Pitois et al. [START_REF] Chateau | Micromechanics of unsaturated granular media[END_REF] derived the following Eq. ( 13) for the capillary force in the case of a mono-sized particle pair and assuming a cylindrical (flat) liquid bridge profile.

The following relation has been experimentally verified for θ " 10 0 , i.e.

F ˚" cos θ ¨1 ´1 c 1 `2 V π d ˚2 ‹ ‹ ' (13)
For such a mono-sized particle pair, a fairly good agreement is found between our solution and Eq. ( 13) for a wide range of contact angles as illustrated in Figs. 5(a)

and 5(b). Consistent with other experimental comparisons presented in [START_REF] Chateau | Micromechanics of unsaturated granular media[END_REF], the agreement between the solution of Laplace-Young equation and Eq. ( 13) is most satisfactory for small liquid volumes (Fig. 5 Eq. ( 13) in the polydisperse case for which the flat profile assumption never holds.

In order to account for polydispersity, Richefeu et al. [START_REF] Richefeu | Stress transmission in wet granular materials[END_REF] proposed Eq. ( 14) for macroscopic DEM simulations purposes, i.e.

F ˚" cos θ ? r exp ˜´r 0.9 a V ˚p1 `rq{2 d ˚¸(14)
However, for the cases herein considered, the comparison between Eq. ( 14) and our numerical solutions shows a poorer agreement for the polydisperse cases than for the monodisperse ones, see Fig. 6.

A last comparison is made with the relation proposed by Willett et al. [START_REF] Willett | Capillary bridges between two spherical bodies[END_REF], detailed in Appendix 3 due to its complexity, that has been used for DEM simulations in [START_REF] Mani | Liquid migration in sheared unsaturated granular media[END_REF]. Among the cases here tested, there is a very good agreement for only contacting particles, irrespective of the volume and radius ratio. Then, deviations increase according to the interparticle distance, see Fig. where liquid bridges are computed from Section 2. Thus, the corresponding liquid distribution conforms with uniform capillary pressure conditions that are specific to thermodynamic equilibrium and also considered by [START_REF] Scholtès | Micromechanics of granular materials with capillary effects[END_REF][START_REF] Wang | Anisotropy of a tensorial Bishop's coefficient for wetted granular materials[END_REF]. On the other hand, contrary to [START_REF] Gili | Microstructural deformation mechanisms of unsaturated granular soils[END_REF][START_REF] Melnikov | Micromechanical failure analysis of wet granular matter[END_REF], the model does not include any pore flow computations. Thus, some phenomena known to affect the fluid distribution in unsaturated conditions, such as the ink-bottle effect, are not included in the model. Also, the consideration of a constant uniform contact angle value for each simulation disregards the possibility for contact angle hysteresis that is also known to impact the fluid distribution [START_REF] Mani | Role of contact-angle hysteresis for fluid transport in wet granular matter[END_REF]. However, the liquid phase distribution that results from such numerical modelling paradigm has been favourably compared with experiments in terms of Soil-Water Characteristic Curves in a previous paper [START_REF] Wan | A tensorial description of stresses in triphasic granular materials with interfaces[END_REF].

Capillary bridges, with their associated capillary forces, are thus defined for all contacting and separated particles for which a solution of the Laplace-Young equation can be found. As such, contrary to many other DEM studies [START_REF] Richefeu | Stress transmission in wet granular materials[END_REF][START_REF] Mani | Liquid migration in sheared unsaturated granular media[END_REF][START_REF] Than | Basic mechanical properties of wet granular materials: A DEM study[END_REF][START_REF] Melnikov | Micromechanical failure analysis of wet granular matter[END_REF], the capillary force is not obtained from any phenomenological relations such as the ones discussed in section 2.4.

The standard DEM calculations then carry on with classical frictional contact forces being applied between contacting particles in addition to the above-mentioned capillary forces. In other words, along the normal direction, repulsive contact forces evolve with a fictitious overlap according to linear elasticity. It is to be understood that the numerical overlap corresponds in reality to slight changes in shape when the contact force increases between two actual spherical particles. By design, such deformations in DEM calculations are restricted to small values, and these are thus not accounted for during capillary force determination in the contact regime. In addition to the normal component of the contact force, a tangential component is expressed from the tangential relative displacements using a linear elastic-plastic relation. As usual, the contact law only requires three parameters: Y and P that govern the normal and tangential stiffnesses, and a contact friction angle ϕ. However, the particle size distribution of the sample (Fig. 8) becomes another parameter in the model since capillary force computations are sizedependent, and hence a characteristic length scale is introduced. All parameter values are given in Table 1, and we refer to a previous work [START_REF] Wan | A tensorial description of stresses in triphasic granular materials with interfaces[END_REF] for more details concerning Y , P , ϕ. The considered surface tension is the one developing between air and water at ambient temperature. Note finally that there is a total number of 20,000 particles making up the numerical sample, which is high enough to have no influence on the results. ter vapour condensates into liquid primarily over contacting solid surfaces, the consideration of menisci at contacts only may be related to a pseudo primary wetting path. On the other hand, considering menisci for admissible particle distances, i.e. as long as a solution to the Laplace-Young equation can be found, is presented as a pseudo primary drying path. However, it is reminded that there is no simulation of the poral flow of the two fluids, which leads the model to neglect some mechanisms such as the ink-bottle effect that are responsible for hysteresis during physical hydraulic paths.

As such, a limited hysteresis is obtained between the pseudo-wetting and pseudo-drying SWCCs, see Fig. 9. It is noted that the contact angle inevitably influences the SWCC since it is directly connected to the volumetric-suction V ˚pu c q relationship at the liquid bridge scale. As seen in Fig. 9, there is a significant difference in SWCC for contact angles greater than θ " 20 0 . Indeed, an increasing contact angle reduces the suction for a given degree of saturation with the two being interchangeable. This result further suggests that soils are less sensitive to unsaturated conditions for higher contact angles, as the next sections will demonstrate in detail.

Contact angle influence on strength

The association between the contact angle value and the mechanical consequences of unsaturated conditions is investigated into more details considering the apparent shear strength of wet samples. For each contact an-gle value, and different applied suctions, two triaxial i.e. axisymmetric compressions, with 10 and 20 kPa confining pressure, are carried out in order to determine the corresponding apparent shear strength as interpreted using a Mohr-Coulomb criterion. As has been observed experimentally [START_REF] Pierrat | Effect of moisture on the yield locus of granular materials: theory of shift[END_REF]30], the macroscopic friction angle φ is unaffected by unsaturated conditions and corresponds to the friction angle developed in dry conditions: all φ values are equal to 30.1˘0.3 0 . The correspondence between φ, that has been measured for dry conditions in previous works [START_REF] Duriez | Effective stress in unsaturated granular materials: micro-mechanical insights[END_REF][START_REF] Wan | A tensorial description of stresses in triphasic granular materials with interfaces[END_REF], with the local friction angle between two discrete elements, ϕ, is purely coincidental. Disregarding the constant friction angle, the shear strength is then quantified through the apparent cohesion c (Fig. 10). We recall first that the cohesion values depend on the chosen particle size distribution: greater cohesions would be obtained for another distribution involving smaller particles. Also, in line with the differences in the SWCCs previously shown in Fig. 9, the apparent cohesion of the unsaturated samples significantly decreases with the contact angle, independent of the suction or degree of saturation as depicted in Fig. 10. Such a decrease in the apparent cohesion is attributed to the lower attractive capillary forces that were observed for higher contact angles back in Section 2.3 (Fig. 4).

Apparent cohesion for different contact angles

We notice that the strength as expressed in terms of the apparent cohesion globally increases with the degree of saturation, i.e. decreasing suction, for a given contact angle. This trend is typical of granular materials with low degrees of saturation, as has been observed experimentally [START_REF] Pierrat | Effect of moisture on the yield locus of granular materials: theory of shift[END_REF]30]. The explanation resides in the underlying microscopic mechanism where the capillary force associated with a liquid bridge decreases in the range of high capillary pressures, see Fig. 11. This is equivalent to increases in capillary forces with water content at the low end of water contents within the pendular regime. See also [START_REF] Molenkamp | Interactions between two rough spheres, water bridge and water vapour[END_REF] for r " 1

The case of a liquid bridge between two contacting particles is nonetheless different, displaying a monotonic increase of the capillary force with decreased liquid saturation, i.e. an increase in suction. It turns out that this difference in mechanical behaviour confers a significant role to liquid bridges between distant particles that have been previously argued to not contribute to the stress transmission inside an unsaturated granular soil [30], and thereafter neglected in DEM models similar to the one here considered [START_REF] Wan | Micromechanical analysis of force transport in wet granular soils[END_REF].

In the present paper, it is chosen to simulate mechanical loadings from an initial state including menisci at contacts only, and thereafter consider menisci even if contact is lost, but ensuring Laplace-Young equation can be solved. This includes the instance when initially contacting spheres start separating, while new menisci also form at new contacts. For the purpose of emphasiz-ing the role of stretched menisci between separated particles, apparent cohesions are also measured with a simplified version of the model that discards menisci and capillary forces as soon as initially contacting spheres separate. As expected, lower cohesions are measured with the simplified model due to a lower number of liquid bridges and attractive forces inside the sample, see Fig. 12. More importantly, the cohesion trends with water content are also affected with the simplified model showing a monotonic decrease with water content that contradicts experimental evidence [START_REF] Pierrat | Effect of moisture on the yield locus of granular materials: theory of shift[END_REF]30] or other numerical models [START_REF] Melnikov | Micromechanical failure analysis of wet granular matter[END_REF]. Apart from the previously discussed strength changes, the contact angle has negligible influence on the overall constitutive behaviour, see Fig. 13 and 14. It is clear that virtually the same residual states and strain behaviour are observed for a given suction, irrespective of the contact angle. Initial stiffnesses are also found to be unaffected. In order to get further insights in the role played by the contact angle in wet conditions e.g., on the apparent shear strength, attention is now focused on the nature of the stress state of an unsaturated soil.

Stress state of wet soils

As a starting point, the various stress contributions to the total stress σ in an unsaturated soil are briefly presented. Alluding to the various internal forces that exist

V (%, >0 in comp.) θ=0 °θ=20 °θ=40 °θ=60 °Fig. 40 -12 -10 -8 -6 -4 -2 0 ε 1 (%) ε
14 Contact angle influence during triaxial loadings with 300 kPa suction within a wet granular material, these contributions necessarily implicate contact forces between solid particles as well as fluid pressures and surface tension forces.

The contact stress tensor σ cont accounting for the contact forces is given by the celebrated Love-Weber formula [START_REF] Love | A treatise on the mathematical theory of elasticity[END_REF][START_REF] Weber | Recherches concernant les contraintes intergranulaires dans les milieux pulvérulents[END_REF]:

σ cont " 1 V ÿ c. 1,2 f 2 b l 12 (15)
Eq. ( 15) considers all contacting particles pairs 1-2 in the REV V , with f 2 the contact force as sustained by 2, and l 12 the branch vector connecting the center of 1 to that of 2.

Whereas the contact stress tensor equals to the total net stresses in dry conditions, a capillary stress tensor σ cap arises in unsaturated conditions, accounting for the stress contributions from the fluid phases and the existing interactions between the various phases:

σ cap " σ ´ug δ ´σcont (16)
Note that the capillary stress terminology, used also by Scholtès et al. [START_REF] Scholtès | Micromechanics of granular materials with capillary effects[END_REF], corresponds to the suction stress as coined by Lu & Likos [START_REF] Lu | Suction stress characteristic curve for unsaturated soil[END_REF]. From micromechanics, and under quasi-static conditions, the expression for σ cap is obtained from the different mechanical actions related to fluid pressures and surface tension forces [START_REF] Chateau | Homogenization of a nonsaturated porous medium: Hill's lemma and applications[END_REF][START_REF] Chateau | Micromechanics of unsaturated granular media[END_REF]:

σ cap " ´uc χ ´γ ξ (17)
χ " Fig. [START_REF] Gray | Analysis of the solid phase stress tensor in multiphase porous media[END_REF] shows the various characteristics of a liquid bridge between two particles that appear in the capillary stress expression [START_REF] Hotta | The capillary binding force of a liquid bridge[END_REF]. The latter encompasses all internal forces within the fluid volumes in terms of an isotropic capillary pressure term ´uc V l {V δ, as well as the internal forces within the so-called contractile skin [START_REF] Fredlund | Stress state variables for unsaturated soils[END_REF] formed of all liquid-gas interfaces S lg " ř S m in terms of surface tension forces oriented along the surface projection tensor pδ ´n b nq [START_REF] Chateau | Homogenization of a nonsaturated porous medium: Hill's lemma and applications[END_REF][START_REF] Chateau | Micromechanics of unsaturated granular media[END_REF][START_REF] Gray | Analysis of the solid phase stress tensor in multiphase porous media[END_REF]. Eq. ( 17) also includes interactions terms between the different phases: first, the non-isotropic fluids action on the solid through the capillary pressure u c acting along the wetted surfaces S l p of the solid particles p showing radii R p ;

1 V ˜Vl δ `ÿ p R p ż S l
and second, surface tension forces γ e dl as sustained by solid particles along the contact lines Γ p from the contractile skin. The vector n is the outward external normal to the solid particles in the last term of [START_REF] Hotta | The capillary binding force of a liquid bridge[END_REF].

Previous DEM modelling approaches customarily use an alternate expression for σ cap that is based on resultant capillary forces only [30,[START_REF] Scholtès | Micromechanics of granular materials with capillary effects[END_REF][START_REF] Wang | Anisotropy of a tensorial Bishop's coefficient for wetted granular materials[END_REF][START_REF] Than | Basic mechanical properties of wet granular materials: A DEM study[END_REF]. Interestingly, the comprehensive description of the solid and fluid microstructure by the DEM model enables one to compute the capillary stresses from Eq. ( 17) as well. Because the internal forces within an actual wet granular material differ in nature from point forces, the choice is made here to only consider Eq. ( 17) that reflects the distributed nature of e.g. liquid pressure, instead of rely-ing on resultant point forces. The discussion here is not trivial because internal stresses and resultant forces are not interchangeable, as any medium sustaining stresses while in equilibrium -under zero resultant forceillustrates it. However, this issue is outside the scope of this paper and other publications by the authors present further details and explanations [START_REF] Wan | A tensorial description of stresses in triphasic granular materials with interfaces[END_REF].

From the capillary stresses expression [START_REF] Hotta | The capillary binding force of a liquid bridge[END_REF], it is noteworthy that the capillary stresses systematically include in the general case a deviatoric part depending on the distribution of fluids implicating S l p , S lg , Γ p and interface orientations, which extends the classical Bishop's theory [START_REF] Bishop | Some aspects of effective stress in saturated and partly saturated soils[END_REF]. Such deviatoric stress contribution from the fluids has previously been investigated for a zero contact angle, e.g. by the authors using a slightly different expression for σ cap [START_REF] Wan | Micromechanical analysis of force transport in wet granular soils[END_REF][START_REF] Duriez | Effective stress in unsaturated granular materials: micro-mechanical insights[END_REF][START_REF] Wan | A tensorial description of stresses in triphasic granular materials with interfaces[END_REF]. Because the contact angle governs the whole fluid distribution -for instance the contact line orientation and the ş Γp e b n dl term -it is natural to expect that the contact angle affects the capillary stresses.

Capillary stresses during hydraulic loading

Capillary stresses are first measured along a pseudoprimary drying path simulated under constant isotropic (total) stresses p " 10 kPa. At this stage, the sample is isotropic, and the liquid distribution including menisci between contacting and separated particles is also isotropic. As such, σ cap is spherical and is completely characterized by its mean pressure p cap . We have namely:

p cap " trpσ cap q 3 " p χ `pγ p χ " ´uc 3 V ˜3 V l `ř p R p S l p pγ " ´γ 3 V ˜2 S lg `ř p R p Γ p sin θ ¸(18)
Irrespective of the degree of saturation, an increasing contact angle leads to a decrease (in absolute value) of the capillary stress p cap due to the two-fluid mixture, see Fig. 16. The decline in p cap with θ follows a decreasing (in absolute value) p χ that is not counterbalanced by an increasing p γ . As for p χ , higher contact angles induce lower suctions for a given water content (Fig. 9), then lower p χ . As for p γ , its direct dependence on the contact angle with sin θ appearing Eq. ( 18) leads p γ to increase with θ.

For a given contact angle, changing the degree of saturation, i.e. suction, has a minor influence on the capillary stress p cap because of the opposite trends of p χ pS r q and p γ pS r q. Note finally that apart from very low water contents, with negligible interface surfaces and wetted contours, a significant amount of the capillary stresses is due to p γ that accounts for surface tension forces. 2). Contrary to the previous isotropic example, induced anisotropy here takes place in the initially isotropic sample upon the deviatoric mechanical loading. As menisci form along new contacts following induced anisotropy in the granular packing, an anisotropic liquid bridge distribution develops which gives way to deviatoric capillary stresses q cap [START_REF] Wan | Micromechanical analysis of force transport in wet granular soils[END_REF][START_REF] Duriez | Effective stress in unsaturated granular materials: micro-mechanical insights[END_REF][START_REF] Wan | A tensorial description of stresses in triphasic granular materials with interfaces[END_REF].

Turning to the influence of contact angle, it is found that higher values of θ decrease the intensity of capillary stresses with regards to both p cap and q cap for all suction values, see Fig. 17 and18. Also, the deviatoric nature of σ cap -as measured by η cap " |q cap {p cap |slightly increases with the contact angle for low suctions, Fig. 17(c). On the other hand, for high suction, the deviatoric nature of σ cap does not depend anymore on the contact angle, as evidenced in Fig. 18(c). Such high suction prevents menisci to exist between separated particles, and there is in this case equivalence between the liquid bridge and the contact distributions, hence between the deviatoric nature of σ cap and the contact anisotropy. Contrary to the stress state, the straining of the packing can be considered as insensitive to the contact angle, as suggested previously in Fig. 13

Deviatoric nature

Fig. 18 Capillary stresses during triaxial loadings with 300 kPa suction and 14. This explains why η cap does not depend on the contact angle for u c " 300 kPa. Note that the apparent ultimate value for η cap is related to an existing critical state for the contact anisotropy [START_REF] Bathurst | Observations on stressforce-fabric relationships in idealized granular materials[END_REF][START_REF] Yunus | Numerical local analysis of relevant internal variables for constitutive modelling of granular materials[END_REF].

Failure description from capillary stresses

The distinct limit stress states as a function on wettability that we alluded to back in Section 3.3, are directly related to the corresponding capillary stresses observed in section 4.3. However, a unified failure description is still possible, irrespective of the contact angle values.

Such unified failure description is only obtained when the stress limit states (maxima of η " q{p) are expressed in terms of the contact stress tensor σ cont " σ ´σcap (neglecting u g ), instead of the total stresses σ that are necessarily affected by the fluid mixture and the contact angle. Shear strength data from triaxial compressions under 10 and 20 kPa confining pressures and several degrees of saturation in r0.01%; 10%s illustrate clearly that the consideration of σ cont leads to a unique plastic limit criterion, for all contact angles and degrees of saturation, see Fig. 19. This interesting result illustrates the validity of the single effective stress concept to describe the failure of granular materials in both dry and unsaturated conditions as discussed e.g. by Alonso et al. [START_REF] Alonso | A microstructurally based effective stress for unsaturated soils[END_REF]. However, we do not address here the more complex issue of a comprehensive stress-strain behaviour. Also, this study confirms previous numerical results pertaining to the pendular regime with θ " 0 0 [START_REF] Scholtès | Micromechanics of granular materials with capillary effects[END_REF][START_REF] Duriez | Effective stress in unsaturated granular materials: micro-mechanical insights[END_REF][START_REF] Wan | A tensorial description of stresses in triphasic granular materials with interfaces[END_REF], as well as experimental works [START_REF] Lu | Suction stress characteristic curve for unsaturated soil[END_REF][START_REF] Alonso | A microstructurally based effective stress for unsaturated soils[END_REF].

Conclusion

The contact angle θ affects the mixture of fluids in multiphasic granular soils, which, in turn, impacts on the mechanical behaviour of the soil. In order to assess this contact angle mechanical influence, we proposed a DEM model allowing a macroscopic mechanical analysis for any θ value. The DEM model relies on a systematic numerical solution of Laplace-Young equation, so that capillary bridges are precisely described in terms of capillary force, volume, surface and orientation tensor.

In line with the microscopic contact angle influence at the capillary bridge scale, macroscale mechanical simulations performed with the DEM model show that the stress state of macroscopic soil samples is less sensitive to unsaturated conditions for higher contact angles. As a matter of fact, higher θ values reduce the suction i.e. capillary pressure as well as the apparent shear strength increase classically associated with unsaturated conditions. Such contact angle influence arises from a decrease in the capillary stresses according to θ.

As such, the classical zero contact angle assumption is shown to be non-conservative for the mechanical analysis of wet granular soils.

Appendix 1: Liquid bridge orientation tensor

Considering a meniscus-related orientation basis pe ρ , e φ , e z q (Fig. 1), the liquid bridge external normal n and the local orientation tensor n b n are:

n " 1 a 1 `ζ˚1 2 ¨1 0 ´ζ˚1 ‹ ‹ ‹ ‹ ‹ ‹ ' (19)
n b n " 1 1 `ζ˚1 2 ¨1 0 ´ζ˚1 0 0 0 ´ζ˚1 0 ζ ˚12 ‹ ‹ ‹ ‹ ‹ ‹ ' (20)
From the discrete description of the liquid bride profile pz i , ζ i q and a numerical integration along the profilez i P r0; z f s -the dimensionless meniscus orientation tensor π ˚is obtained as: surface free energy E int is present along the interfaces between liquid and gas, liquid and solid, and solid and gas [START_REF] Padday | The stability of axisymmetric menisci[END_REF][START_REF] Bisschop | A physical model for liquid capillary bridges between adsorptive solid spheres: The nodoid of Plateau[END_REF]. We denote A, A ls and A sg the respective areas; and γ, γ ls and γ sg the respective surface tensions. Using Young's equation γ cos θ " γ sg ´γls , and the area decomposition A s " A ls `Asg , it transpires directly that:

E int " γ A `γls A ls `γsg A sg " γ pA ´Als cos θq `γsg A s

The wetted area A ls is:

A ls " 2 π `R1 2 p1 ´cos δ 1 q `R2 2 p1 ´cos δ 2 q ˘(23)

Through classical differential geometry, the profile ζ ˚pz ˚q gives the axisymmetric meniscus area:

A " 2 π R 2 2 z f ż 0 ζ ˚pz ˚q b 1 `ζ˚1 pz ˚q2 dz « 2 π R 2 2 f ÿ i"1 ζ i b 1 `ζ˚1 i 2 ∆z ˚(24)
liquid and gaseous fluid volumes include the free energy E f " ´uf V f (f " l, g). We consider the capillary bridge forming in a constant surrounding volume V " V s `Vl `Vg , such that:

E f " ´ug V g ´ul V l " ´ug pV ´Vs q `uc V l (25)
no energy is associated with the rigid spheres. The same assumption leads to consider A s as constant in Eq. [START_REF] Lowry | Capillary surfaces: Stability from families of equilibria with application to the liquid bridge[END_REF].

Omitting the constant terms γ sg A s and u g pV ´Vs q, the expression of E int `Ef in a dimensionless form -E ˚" pE int `Ef q{pγ R 2 2 q -gives Eq. ( 12).

  ) is classically rewritten in the following dimensionless form accounting for the 3D axisymmetric situation, and denoting ζ ˚1 and ζ ˚2 as the first and second derivatives of ζ ˚pz ˚q,

Fig. 2 Fig. 3

 23 Fig.2Possible liquid bridge configurations according to the particles distance; for r " 2, θ " 20 0 and different suctions

Fig. 4

 4 Fig. 4 Contact angle influence at the liquid bridge scale. For r " 3 and u c " 200

7 . 10 ´4Fig. 5 10 ´3Fig. 6 10 ´3Fig. 7

 7105106107 Fig.5Comparison with Equ. 13, from[START_REF] Chateau | Micromechanics of unsaturated granular media[END_REF]

Fig. 8

 8 Fig. 8 Particle size distribution of the DEM sample

3. 2

 2 Contact angle influence on SWCCThe computation of all liquid bridge volumes in the DEM calculations provides the possibility of obtaining the macroscale Soil-Water Characteristic Curves (SWCCs) for uniform suction conditions. Being dependent on the chosen particle size distribution, the SWCCs are generated for different contact angles under an isotropic stress of p " 10 kPa -this value has little influence on the final result -and imposing different suction values to the numerical sample. For each contact angle, two distinct SWCCs are determined by computing menisci either between contacting particles only, or between both contacting and separated particles. As wa-

9

  Pseudo-wetting and pseudo-drying SWCCs for different contact angles

Fig. 11

 11 Fig. 11 Dimensionless capillary force according to dimensionless suction for different distances and r " 2.5; θ " 20 0 .

  Fig. 12 considers the case of θ " 0 0 but the exact same trends have been measured for other contact angle values. It is for the above reason that we

Fig. 12

 12 Fig.[START_REF] Fredlund | Stress state variables for unsaturated soils[END_REF] Apparent cohesion for θ " 0 0 accounting for menisci between distant particles ("cont. + dist." data set), or not ("cont. only" data set)

p n b n dS ξ " 1 VFig. 15

 115 Fig.[START_REF] Gray | Analysis of the solid phase stress tensor in multiphase porous media[END_REF] Liquid bridge characteristics associated with Eq.[START_REF] Hotta | The capillary binding force of a liquid bridge[END_REF]

Fig. 16

 16 Fig.[START_REF] Haines | Studies in the physical properties of soils: II. A note on the cohesion developed by capillary forces in an ideal soil[END_REF] Capillary stresses during the hydraulic loading of an isotropic sample (Sr " 0 as well as pχ " 0 are both never reached limits)

Fig. 17

 17 Fig. 17Capillary stresses during triaxial loadings with 20

  Fig. 17Capillary stresses during triaxial loadings with 20 kPa suction

  Using contact stresses σ cont " σ ´σcap

Fig.

  Fig. Failure description during triaxial compressions

Appendix 2 :

 2 Free energy of liquid bridge configurationsThe capillary bridge depicted in Fig.1include free energy of different types:

Table 1

 1 Model parameters

	Y	P	ϕ	Dmax D min	D 50	γ	θ
	(MPa)	(-)	( 0 )	(-)	(mm) (N/m)	( 0 )
	50	0.5 30	3	0.058	0.073	P r0; 60s

Table 2

 2 Initial degree of saturation (%) of considered triaxial loadings (with 20 kPa confining pressure)

Acknowledgements This work is supported by the Natural Science and Engineering Research Council of Canada and Foundation Computer Modelling Group within the framework of a Government-Industry Partnership (NSERC-CRD) Grant toward the fundamental understanding of complex multiphasic granular media.

Appendix 3: Comparison with Willett's model

With respect to our notations, the capillary force expression proposed by Willett [START_REF] Willett | Capillary bridges between two spherical bodies[END_REF] applies to a dimensionless force equal to F ˚p1 `rq{2. In [START_REF] Willett | Capillary bridges between two spherical bodies[END_REF], the relation is expressed according to a dimensionless distance S that corresponds, here, to d ˚a2{pV ˚p1 `rqq. Namely, the equation proposed by Willett finally reads:

26)

The coefficients f i depend on the contact angle and a dimensionless volume that corresponds here, with respect to our notations, to V ˚p1 `rq 3 {8. Their exact expressions can be found in the Appendices of [START_REF] Willett | Capillary bridges between two spherical bodies[END_REF] or [START_REF] Gladkyy | Comparison of different capillary bridge models for application in the discrete element method[END_REF].


Samy Skander Bahoura

ABOUT BREZIS-MERLE PROBLEM WITH HOLDERIAN CONDITION: THE CASE OF ONE OR TWO BLOW-UP POINTS

We consider the following problem on open set Ω of R 2 :

We give a quantization analysis of the previous problem under the conditions:

and

On the other hand, if we assume that

or, V i s-holderian with 1/2 < s ≤ 1, and,

then we have a compactness result, namely:

sup

where A is the holderian constant of V i .

INTRODUCTION AND MAIN RESULTS

We set ∆ = ∂ 11 + ∂ 22 on open set Ω of R 2 with a smooth boundary. We consider the following problem on Ω ⊂ R 2 :

(P ) -∆u i = V i e ui in Ω ⊂ R 2 , u i = 0 in ∂Ω.
We assume that, Ω e ui dy ≤ C, and,

0 ≤ V i ≤ b < +∞
The above equation is called, the Prescribed Scalar Curvature equation, in relation with conformal change of metrics. The function V i is the prescribed curvature.

Here, we try to find some a priori estimates for sequences of the previous problem. Equations of this type were studied by many authors, see . One can see in [START_REF] Brezis | Uniform estimates and blow-up bihavior for solutions of -∆u = V e u in two dimensions[END_REF] different results for the solutions of those type of equations with or without boundaries conditions and, with minimal conditions on V , for example we suppose V i ≥ 0 and V i ∈ L p (Ω) or V i e ui ∈ L p (Ω) with p ∈ [1, +∞].

Among other results, we can see in [START_REF] Brezis | Uniform estimates and blow-up bihavior for solutions of -∆u = V e u in two dimensions[END_REF], the following important Theorem, Theorem A(Brezis-Merle [START_REF] Brezis | Uniform estimates and blow-up bihavior for solutions of -∆u = V e u in two dimensions[END_REF]).If (u i ) i and (V i ) i are two sequences of functions relatively to the previous problem (P ) with, 0 < a ≤ V i ≤ b < +∞, then, for all compact set K of Ω, sup K u i ≤ c = c(a, b, K, Ω).

We can find in [START_REF] Brezis | Uniform estimates and blow-up bihavior for solutions of -∆u = V e u in two dimensions[END_REF] an interior estimate if we assume a = 0, but we need an assumption on the integral of e ui . We have in [START_REF] Brezis | Uniform estimates and blow-up bihavior for solutions of -∆u = V e u in two dimensions[END_REF]:

Theorem B (Brezis-Merle [START_REF] Brezis | Uniform estimates and blow-up bihavior for solutions of -∆u = V e u in two dimensions[END_REF]).If (u i ) i and (V i ) i are two sequences of functions relatively to the previous problem (P ) with, 0 ≤ V i ≤ b < +∞, and, Ω e ui dy ≤ C, then, for all compact set K of Ω, sup

K u i ≤ c = c(b, C, K, Ω).
If, we assume V with more regularity, we can have another type of estimates, sup + inf. It was proved, by Shafrir, see [START_REF] Shafrir | A sup+inf inequality for the equation -∆u = V e u[END_REF], that, if (u i ) i , (V i ) i are two sequences of functions solutions of the previous equation without assumption on the boundary and, 0 < a ≤ V i ≤ b < +∞, then we have the following interior estimate:

C a b sup K u i + inf Ω u i ≤ c = c(a, b, K, Ω).
One can see in [START_REF] Chen | A sharp sup+inf inequality for a nonlinear elliptic equation in R 2[END_REF], an explicit value of C a b = a b . In his proof, Shafrir has used a blowup function, the Stokes formula and an isoperimetric inequality, see [START_REF] Bandle | Isoperimetric inequalities and Applications[END_REF]. For Chen-Lin, they have used the blow-up analysis combined with some geometric type inequality for the integral curvature.

Now, if we suppose (V i ) i uniformly Lipschitzian with A the Lipschitz constant, then, C(a/b) = 1 and c = c(a, b, A, K, Ω), see Brezis-Li-Shafrir [START_REF] Brezis | A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities[END_REF]. This result was extended for Hölderian sequences (V i ) i by Chen-Lin, see [START_REF] Chen | A sharp sup+inf inequality for a nonlinear elliptic equation in R 2[END_REF]. Also, we have in [START_REF] Yy | Harnack Type Inequality: the Method of Moving Planes[END_REF] an extension of the Brezis-Li-Shafrir result to compact Riemann surface without boundary. We have in [START_REF] Yy | Blow-up Analysis for Solutions of -∆u = V e u in Dimension Two[END_REF] explicit form, (8πm, m ∈ N * exactly), for the numbers in front of the Dirac masses, when the solutions blowup. Here, the notion of isolated blow-up point is used. Also, we have in [START_REF] Chen | Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces[END_REF] and [START_REF] Zhang | Blowup solutions of some nonlinear elliptic equations involving exponential nonlinearities[END_REF] refined estimates near the isolated blow-up points and the bubbling behavior of the blow-up sequences.

In [START_REF] Brezis | Uniform estimates and blow-up bihavior for solutions of -∆u = V e u in two dimensions[END_REF], Brezis and Merle proposed the following Problem:

Problem (Brezis-Merle [START_REF] Brezis | Uniform estimates and blow-up bihavior for solutions of -∆u = V e u in two dimensions[END_REF]).If (u i ) i and (V i ) i are two sequences of functions relatively to the previous problem (P ) with,

0 ≤ V i → V in C 0 ( Ω). Ω e ui dy ≤ C,

Is it possible to prove that:

sup

Ω u i ≤ c = c(C, V, Ω) ?
Here, we assume more regularity on V i , we suppose that

V i ≥ 0 is C s (s-holderian) 1/2 < s ≤ 1) .
We give the answer where bC < 24π.

On other hand, in our work we give a complete caracterization of the blow-up analysis on the boundary.

In the similar way, we have in dimension n ≥ 3, with different methods, some a priori estimates of the type sup × inf for equation of the type:

-∆u + n -2 4(n -1) R g (x)u = V (x)u (n+2)/(n-2) on M.
where R g is the scalar curvature of a riemannian manifold M , and V is a function. The operator ∆ = ∇ i (∇ i ) is the Laplace-Beltrami operator on M . When V ≡ 1 and M compact, the previous equation is the Yamabe equation. T. Aubin and R. Scheon solved the Yamabe problem, see for example [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF]. Also, we can have an idea on the Yamabe Problem in [START_REF] Lee | The Yamabe problem[END_REF]. If V is not a constant function, the previous equation is called a prescribing curvature equation, we have many existence results see also [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF]. Now, if we look at the problem of a priori bound for the previous equation, we can see in [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF][START_REF] Bahoura | Harnack inequalities for Yamabe type equations[END_REF][START_REF] Bahoura | Lower bounds for sup+inf and sup × inf and an extension of Chen-Lin result in dimension 3[END_REF][START_REF] Bahoura | Estimations uniformes pour l'quation de Yamabe en dimensions 5 et 6[END_REF][START_REF] Chen | Estimates of the conformal scalar curvature equation via the method of moving planes[END_REF][START_REF] Yy | Prescribing scalar curvature on Sn and related Problems[END_REF][START_REF] Yy | Yamabe Type Equations On Three Dimensional Riemannian Manifolds[END_REF] some results concerning the sup × inf type of inequalities when the manifold M is the sphere or more generality a locally conformally flat manifold. For these results, the moving-plane was used, we refer to [START_REF] Caffarelli | Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth[END_REF][START_REF] Gidas | Symmetry and related properties via the maximum principle[END_REF][START_REF] Yy | A Harnack type inequality for the Yamabe equation in low dimensions[END_REF] to have an idea on this method and some applications of this method.

Also, there are similar problems defined on complex manifolds for the Complex Monge-Ampere equation, see [START_REF] Siu | The existence of Kahler-Einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group[END_REF][START_REF] Tian | A Harnack type inequality for certain complex Monge-Ampre equations[END_REF], with various inequalitites of type sup + inf.

Our main results are: Theorem 1.1. Assume Ω = B 1 (0), and,

u i (x i ) = sup B1(0) u i → +∞.
There is a finite number of sequences (x k i ) i , (δ k i ), 0 ≤ k ≤ m, such that:

(x 0 i ) i ≡ (x i ) i , δ 0 i = δ i = d(x i , ∂B 1 (0)) → 0, and each δ k i is of order d(x k i , ∂B 1 (0)). and, u i (x k i ) = sup B1(0)-∪ k-1 j=0 B(x j i ,δ j i ǫ) u i → +∞, u i (x k i ) + 2 log δ k i → +∞, ∀ ǫ > 0, sup B1(0)-∪ m j=0 B(x j i ,δ j i ǫ) u i ≤ C ǫ ∀ ǫ > 0, lim sup i→+∞ B(x k i ,δ k i ǫ)
V i e ui dy ≥ 4π > 0.

If we assume:

V i → V in C 0 ( B1 (0)), then, ∀ ǫ > 0, lim sup i→+∞ B(x k i ,δ k i ǫ) V i e ui dy = 8πm k , m k ∈ N * .
And, thus, we have the following convergence in the sense of distributions:

B1(0) V i e ui dy → B1(0) V e u dy + m k=0 8πm ′ k δ x k 0 , m ′ k ∈ N * , x k 0 ∈ ∂B 1 (0).
Theorem 1.2. Assume that:

B1(0)
V i e ui dy ≤ 4π, Then,

u i (x i ) = sup B1(0) u i ≤ c = c(b, C), Theorem 1.3. Assume that, V i is uniformly s-holderian with 1/2 < s ≤ 1, and, B1 (0)
V i e ui dy ≤ 24π -ǫ, ǫ > 0, then we have:

sup

Ω u i ≤ c = c(b, C, A, s, ǫ, Ω).
where A is the holderian constant of V i .

Question 1: (a Bartolucci type result; one holderian singularity): with the same technique, assume that:

V i (x) = (1 + x s 1 )W i (x)
for example and 0 ∈ ∂Ω with W i uniformly lipschitzian and , 0 < s ≤ 1, can one conclude with the Pohozaev identity that the sequence is compact ? here we extend the case 0 < s ≤ 1/2. Question 2: (the limit case s = 1/2) assume that V i is uniformly 1/2holderian with A i the holderian constant and suppose that A i → 0, can one conclude with the blow-up technique that the sequence of the solutions u i is compact ?

PROOFS OF THE RESULTS

Proofs of the theorems: Without loss of generality, we can assume that Ω = B 1 (0) the unit ball centered on the origin. We assume that:

u i ∈ W 1,1 0 (Ω).
According to the work of Brezis-Merle, e kui ∈ L 1 for all k > 2 and the elliptic estimates and the Sobolev embedding gives:

u i ∈ W 2,k (Ω) ∩ C 1,ǫ ( Ω).
Here, G is the Green function of the Laplacian with Dirichlet condition on B 1 (0). We have (in complex notation):

G(x, y) = 1 2π log |1 -xy| |x -y| ,
we can write:

v i (x) = B1 (0)
G(x, y)V i (y)e ui(y) dy,

Remark that, we can write:

F (x) = B1(0) - 1 2π log |x -y|V i (y)e ui(y) dy,
By a result in Gilbarg-Trudinger (cahpter 4, Newtonnian potential), we have F ∈ C 1 ( Ω) Also, we have:

G(x) = B1(0) - 1 2π log |1 -xy|V i (y)e ui(y) dy,
For x near 0, G is smooth by the usuual differentability theorem. For x = 0, we can write: K(x) = F (1/x) which is C 1 by a result of Gilbarg-Trudinger. Combining the two last results, we have v i is C 1 and by the maximum principle we have v i ≡ u i . We use the fact that G is real to write

∂ x G = ∂x Ḡ = ∂x G to have the dirivative of u i . G(x, y) = 1 2π log |1 -xy| |x -y| ,
we can write:

u i (x) = B1(0)
G(x, y)V i (y)e ui(y) dy,

We can compute (in complex notation) ∂ x G and ∂ x u i :

∂ x G(x, y) = 1 -|y| 2 (x -y)(xȳ -1) , ∂ x u i (x) = B1(0) ∂ x G(x, y)V i (y)e ui(y) dy = B1(0) 1 -|y| 2 (x -y)(xȳ -1)
V i (y)e ui(y) dy we write,

u i (x i ) = Ω G(x i , y)V i (y)e ui(y) dx = Ω-B(xi,δi/2) G(x i , y)V i e ui(y) dy+ B(xi,δi/2)
G(x i , y)V i e ui(y) dy

According to the maximum principle, the harmonic function G(x i , .) on Ω -B(x i , δ i /2) take its maximum on the boundary of B(x i , δ i /2), we can compute this maximum:

G(x i , y i ) = 1 2π log |1 -xi y i | |x i -y i | = 1 2π log |1 -xi (x i + δ i θ i )| |δ i /2| = 1 2π log 2(|(1+|x i |)+x i θ i |) < +∞ with |θ i | = 1/2.
Thus,

u i (x i ) ≤ C + B(xi,δi/2) G(x i , y)V i e ui(y) dy ≤ C + e ui(xi) B(xi,δi/2) G(x i , y)dy
Now, we compute B(xi,δi/2) G(x i , y)dy we set in polar coordinates,

y = x i + δ i tθ
we find:

B(xi,δi/2) G(x i , y)dy = B(xi,δi/2) 1 2π log |1 -xi y| |x i -y| = 1 2π 2π 0 1/2 0 δ 2 i log |1 -xi (x i + δ i θ)| (δ i t)/2 tdtdθ = = 1 2π 2π 0 1/2 0 δ 2 i (log 2(|1 + |x i | + tx i θ|) -log t)tdtdθ ≤ Cδ 2 i .
Thus, xi) , We can conclude that:

u i (x i ) ≤ C + Cδ 2 i e ui(xi) , which we can write, because u i (x i ) → +∞, u i (x i ) ≤ C ′ δ 2 i e ui(
u i (x i ) + 2 log δ i → +∞.
Now, consider the following function :

v i (y) = u i (x i + δ i y) + 2 log δ i , y ∈ B(0, 1/2)
The function satisfies all conditions of the Brezis-Merle hypothesis, we can conclude that, on each compact set:

v i → -∞
we can assume, without loss of generality that for 1/2 > ǫ > 0, we have:

v i → -∞, y ∈ B(0, 2ǫ) -B(0, ǫ), Lemma 2.1. For all 1/4 > ǫ > 0, we have: sup B(xi,(3/2)δiǫ)-B(xi,δiǫ) u i ≤ C ǫ .
Proof of the lemma Let t ′ i and t i the points of B(x i , 2δ i ǫ) -B(x i , (1/2)δ i ǫ) and B(x i , (3/2)δ i ǫ) -B(x i , δ i ǫ) respectively where u i takes its maximum.

According to the Brezis-Merle work, we have:

u i (t ′ i ) + 2 log δ i → -∞ We write, u i (t i ) = Ω G(t i , y)V i (y)e ui(y) dx = Ω-B(xi,2δiǫ) G(t i , y)V i e ui(y) dy+ + B(xi,2δiǫ)-B(xi,(1/2)δiǫ) G(t i , y)V i e ui(y) dy+ + B(xi,(1/2)δiǫ) G(t i , y)V i e ui(y) dy
But, in the first and the third integrale, the point t i is far from the singularity x i and we know that the Green function is bounded. For the second integrale, after a change of variable, we can see that this integale is bounded by (we take the supremum in the annulus and use Brezis-Merle theorem)

δ 2 i e ui(t ′ i ) × I j
where I j is a Jensen integrale (of the form

1 0 2π 0 (log(|1 + |x i | + tθ|) -log |θ i -tθ|)tdtdθ which is bounded ).
we conclude the lemma.

From the lemma, we see that far from the singularity the sequence is bounded, thus if we take the supremum on the set B 1 (0) -B(x i , δ i ǫ) we can see that this supremum is bounded and thus the sequence of functions is uniformly bounded or tends to infinity and we use the same arguments as for x i to conclude that around this point and far from the singularity, the seqence is bounded.

The process will be finished , because, according to Brezis-Merle estimate, around each supremum constructed and tending to infinity, we have:

∀ ǫ > 0, lim sup i→+∞ B(xi,δiǫ) V i e ui dy ≥ 4π > 0.
Finaly, with this construction, we have a finite number of "exterior "blow-up points and outside the singularities the sequence is bounded uniformly, for example, in the case of one "exterior" blow-up point, we have:

u i (x i ) → +∞ ∀ ǫ > 0, sup B1(0)-B(xi,δiǫ) u i ≤ C ǫ ∀ ǫ > 0, lim sup i→+∞ B(xi,δiǫ)
V i e ui dy ≥ 4π > 0.

x i → x 0 ∈ ∂B 1 (0).

We have the following lemma:

Lemma 2.2. Each δ k i is of order d(x k i , ∂B 1 (0))
. Namely: there is a positive constant C > 0 such that for ǫ > 0 small enough:

δ k i ≤ d(x k i , ∂B 1 (0)) ≤ (2 + C ǫ )δ k i .
Proof of the lemma Now, if we suppose that there is another "exterior" blow-up (t i ) i , we have, because (u i ) i is uniformly bounded in a neighborhood of ∂B(x i , δ i ǫ), we have :

d(t i , ∂B(x i , δ i ǫ)) ≥ δ i ǫ If we set, δ ′ i = d(t i , ∂(B 1 (0) -B(x i , δ i ǫ))) = inf{d(t i , ∂B(x i , δ i ǫ)), d(t i , ∂(B 1 (0)))} then, δ ′ i is of order d(t i , ∂B 1 (0)).
To see this, we write:

d(t i , ∂B 1 (0)) ≤ d(t i , ∂B(x i , δ i ǫ)) + d(∂B(x i , δ i ǫ), x i ) + d(x i , ∂B 1 (0)),
Thus,

d(t i , ∂B 1 (0)) d(t i , ∂B(x i , δ i ǫ)) ≤ 2 + 1 ǫ ,
Thus,

δ ′ i ≤ d(t i , ∂B 1 (0)) ≤ δ ′ i (2 + 1 ǫ ).
Now, the general case follow by induction. We use the same argument for three, four,..., n blow-up points.

We have, by induction and, here we use the fact that u i is uniformly bounded outside a small ball centered at x j i , j = 0, . . . , k -1:

δ j i ≤ d(x j i , ∂B 1 (0)) ≤ C 1 δ j i , j = 0, . . . , k -1, . d(x k i , ∂B(x j i , δ j i ǫ/2)) ≥ ǫδ j i , ǫ > 0, j = 0, . . . , k -1, .
and let's consider x k i such that:

u i (x k i ) = sup B1(0)-∪ k-1 j=0 B(x j i ,δ j i ǫ) u i → +∞,
take,

δ k i = inf{d(x k i , ∂B 1 (0)), d(x k i , ∂(B 1 (0) -∪ k-1 j=0 B(x j i , δ j i ǫ/2))}, if, we have, δ k i = d(x k i , ∂B(x j i , δ j i ǫ/2)), j ∈ {0, . . . , k -1}. Then, δ k i ≤ d(x k i , ∂B 1 (0)) ≤ ≤ d(x k i , ∂B(x j i , δ j i ǫ/2)) + d(∂B(x j i , δ j i ǫ/2), x j i ) + d(x j i , ∂B 1 (0)) ≤ (2 + C 1 ǫ )δ k i .
To apply lemma 2.1 for m blow-up points, we use an induction: We do directly the same approch for t i as x i by using directly the Green function of the unit ball.

If we look to the blow-up points, we can see, with this work that, after finite steps, the sequence will be bounded outside a finite number of balls , because of Brezis-Merle estimate (corollary 4 of Brezis-Merle's paper):

∀ ǫ > 0, lim sup i→+∞ B(x k i ,δ k i ǫ)
V i e ui dy ≥ 4π > 0.

Here, we can take the functions:

u k i (y) = u i (x k i + δ k i y) + 2 log δ k i , (By corollary 4 of Brezis-Merle's paper if lim sup i→+∞ B(x k i ,δ k i ǫ) V i e ui dy ≤ 4π -ǫ 0 < 4π, then (u k i )
+ is locally uniformly bounded, which in contradiction with u k i (0) → +∞). Finaly, we can say that, there is a finite number of sequences (x k i ) i , (δ k i ), 0 ≤ k ≤ m, such that:

(x 0 i ) i ≡ (x i ) i , δ 0 i = δ i = d(x i , ∂B 1 (0)), (x 1 i ) i ≡ (t i ) i , δ 1 i = δ ′ i = d(t i , ∂(B 1 (0) -B(x i , δ i ǫ)
), and each δ k i is of order d(x k i , ∂B 1 (0)). and,

u i (x k i ) = sup B1(0)-∪ k-1 j=0 B(x j i ,δ j i ǫ) u i → +∞, u i (x k i ) + 2 log δ k i → +∞, ∀ ǫ > 0, sup B1(0)-∪ m j=0 B(x j i ,δ j i ǫ) u i ≤ C ǫ ∀ ǫ > 0, lim sup i→+∞ B(x k i ,δ k i ǫ)
V i e ui dy ≥ 4π > 0.

The work of YY.Li-I.Shafrir With the previous method, we have a finite number of "exterior" blow-up points (perhaps the same) and the sequences tend to the boundary. With the aid of proposition 1 of the paper of Li-Shafrir, we see that around each exterior blow-up, we have a finite number of "interior" blow-ups. Around, each exterior blow-up, we have after rescaling with δ k i , the same situation as around a fixed ball with positive radius. If we assume:

V i → V in C 0 ( B1 (0)), then, ∀ ǫ > 0, lim sup i→+∞ B(x k i ,δ k i ǫ) V i e ui dy = 8πm k , m k ∈ N * .
And, thus, we have the following convergence in the sense of distributions:

B1(0)
V i e ui dy →

B1(0) V e u dy + m k=0 8πm ′ k δ x k 0 , m ′ k ∈ N * , x k 0 ∈ ∂B 1 (0).
Consequence 1: Proof of theorem 2 Assume that:

B1(0)
V i e ui dy ≤ 4π, Then, if the sequence blow-up, there is one and only one blow-up point and we have:

u i (x i ) = sup B1(0) u i → +∞, u i (x i ) + 2 log δ i → +∞, ∀ ǫ > 0, sup B1(0)-B(xi,δiǫ) u i ≤ C ǫ
We set,

r i = e -ui(xi)/2 ,
The blow-up function is locally bounded thus, r 2 i e ui ≤ C on B(x i , 2r i ). We write:

u i (x i ) = Ω-B(xi,δiǫ) G(x i , y)V i e ui(y) dy+ B(xi,δiǫ) G(x i , y)V i e ui(y) dy ≤ C ǫ + B(xi,δiǫ)
G(x i , y)V i e ui(y) dy we have:

B(xi,δiǫ) G(x i , y)V i e ui(y) dy = B(xi,δiǫ)-B(xi,2ri) G(x i , y)V i e ui(y) dy+ B(xi,2ri) G(x i , y)V i e ui(y) dy
We use the maximum principle on B(x i , δ i ǫ) -B(x i , 2r i ) and the explicit formula of G to prove that:

G(x i , y) ≤ C + 1 2π log δ i r i = C + 1 4π (u i (x i ) + 2 log δ i ).
On B(x i , 2r i ) we use the fact that:

r 2 i e ui ≤ C and the explicit formula for G to have:

B(xi,2ri) G(x i , y)V i e ui(y) dy ≤ C + 1 2π log δ i r i B(xi,2ri)
V i e ui(y) dy.

We conclude that:

u i (x i ) ≤ C + 1 2π log δ i r i B(xi,δiǫ)
V i e ui(y) dy.

which we can write as:

u i (x i ) ≤ C + 1 4π (u i (x i ) + 2 log δ i ) B(xi,δiǫ)
V i e ui(y) dy.

Our hypothesis on the integrale of V i e ui imply that:

log δ i ≥ -C,
in other words, we have uniformly,

d(x i , ∂B 1 (0)) = δ i ≥ e -C > 0.
this contredicts the fact that (x i ) tends to the boundary. The sequence (u i ) is bounded in this case.

We can see that the case:

B1(0)
V i e ui dy ≤ 4π, is optimal, because Brezis-Merle have proved that, there is a counterexample of blow-up sequence with:

B1(0)
V i e ui dy = 4πA > 4π.

Consequence 2: using a Pohozaev-type identity, proof of theorem 3 By a conformal transformation, we can assume that our domain Ω = B + is a half ball centered at the origin, B + = {x, |x| ≤ 1, x 1 ≥ 0}. In this case the normal at the boundary is ν = (-1, 0) and u i (0, x 2 ) ≡ 0. Also, we set x i the blow-up point and x 2 i = (0, x 2 i ) and x 1 i = (x 1 i , 0) respectevely the second and the first part of x i . Let ∂B + the part of the boundary for which u i and its derivatives are uniformly bounded and thus converge to the corresponding function.

The case of one blow-up point:

Theorem 2.3. If V i is s-Holderian with 1/2 < s ≤ 1 and, Ω V i e ui dy ≤ 16π -ǫ, ǫ > 0,
we have :

V i (x i ) Ω e ui dy -V (0) Ω e u dy = o(1)
which means that there is no blow-up points.

Proof of the theorem

The Pohozaev identity gives us the following formula:

Ω < (x -x i 2 )|∇u i > (-∆u i )dy = Ω < (x -x i 2 )|∇u i > V i e ui dy = A i A i = ∂B + < (x -x i 2 )|∇u i >< ν|∇u i > dσ + ∂B + < (x -x i 2 )|ν > |∇u i | 2 dσ
We can write it as:

Ω < (x -x i 2 )|∇u i > (V i -V i (x i ))e ui dy = A i + V i (x i ) Ω < (x -x i 2 )|∇u i > e ui dy = = A i + V i (x i ) Ω < (x -x i 2 )|∇(e ui ) > dy
And, if we integrate by part the second term, we have (because x 1 = 0 on the boundary and ν 2 = 0):

Ω < (x -x i 2 )|∇u i > (V i -V i (x i ))e ui dy = -2V i (x i ) Ω e ui dy + B i
where B i is,

B i = V i (x i ) ∂B + < (x -x i 2 )
|ν > e ui dy applying the same procedure to u, we can write:

-2V i (x i ) Ω e ui dy+2V (0) Ω e u dy = Ω < (x-x i 2 )|∇u i > (V i -V i (x i ))e ui dy- Ω < (x-x i 2 )|∇u > (V -V (0))e u dy+ +(A i -A) + (B i -B),
where A and B are,

A = ∂B + < (x -x i 2 )|∇u >< ν|∇u > dσ + ∂B + < (x -x i 2 )|ν > |∇u| 2 dσ B = V (0) ∂B + < (x -x i 2 )|ν > e u dy
and, because of the uniform convergence of u i and its derivatives on ∂B + , we have:

A i -A = o(1) and B i -B = o(1)
which we can write as:

V i (x i ) Ω e ui dy -V (0) Ω e u dy = Ω < (x -x i 2 )|∇(u i -u) > (V i -V i (x i ))e ui dy+ + Ω < (x -x i 2 )|∇u > (V i -V i (x i ))(e ui -e u )dy+ + Ω < (x -x i 2 )|∇u > (V i -V i (x i ) -(V -V (0)))e u dy + o(1)
We can write the second term as:

Ω < (x-x i 2 )|∇u > (V i -V i (x i ))(e ui -e u )dy = Ω-B(0,ǫ) < (x-x i 2 )|∇u > (V i -V i (x i ))(e ui -e u )dy+ + B(0,ǫ) < (x -x i 2 )|∇u > (V i -V i (x i ))(e ui -e u )dy = o(1),
because of the uniform convergence of u i to u outside a region which contain the blow-up and the uniform convergence of V i . For the third integral we have the same result:

Ω < (x -x i 2 )|∇u > (V i -V i (x i ) -(V -V (0)))e u dy = o(1),
because of the uniform convergence of V i to V . Now, we look to the first integral:

Ω < (x -x i 2 )|∇(u i -u) > (V i -V i (x i ))e ui dy,
we can write it as:

Ω < (x-x i 2 )|∇(u i -u) > (V i -V i (x i ))e ui dy = Ω < (x-x i )|∇(u i -u) > (V i -V i (x i ))e ui dy+ + Ω < x i 1 |∇(u i -u) > (V i -V i (x i
))e ui dy, Thus, we have proved by using the Pohozaev identity the following equality, in the case of one blow-up (the term o(1)):

Ω < (x -x i )|∇(u i -u) > (V i -V i (x i ))e ui dy+ + Ω < x i 1 |∇(u i -u) > (V i -V i (x i ))e ui dy = = 2V i (x i ) Ω e ui dy -2V (0) Ω e u dy + o(1)
We can see, because of the uniform boundedness of u i outside B(x i , δ i ǫ) and the fact that :

||∇(u i -u)|| 1 = o(1),
it is sufficient to look to the integral on B(x i , δ i ǫ).

Assume that we are in the case of one blow-up, it must be (x i ) and isolated, we can write the following inequality as a consequence of YY.Li-I.Shafrir result:

u i (x) + 2 log |x -x i | ≤ C,
We use this fact and the fact that V i is s-holderian to have that, on B(x i , δ i ǫ),

|(x -x i )(V i -V i (x i ))e ui | ≤ C |x -x i | 1-s ∈ L (2-ǫ ′ )/(1-s) , ∀ ǫ ′ > 0,
and, we use the fact that:

||∇(u i -u)|| q = o(1), ∀ 1 ≤ q < 2
to conclude by the Holder inequality that for 0 < s ≤ 1 :

B(xi,δiǫ) < (x -x i )|∇(u i -u) > (V i -V i (x i ))e ui dy = o(1),
For the other integral, namely:

B(xi,δiǫ) < x i 1 |∇(u i -u) > (V i -V i (x i ))e ui dy,
We use the fact that, because our domain is a half ball, and the sup + inf inequality to have:

x i 1 = δ i , u i (x) + 4 log δ i ≤ C
and,

e (s/2)ui(x) ≤ |x -x i | -s , |V i -V i (x i )| ≤ |x -x i | s ,
Finaly, we have:

| B(xi,δiǫ) < x i 1 |∇(u i -u) > (V i -V i (x i ))e ui dy| ≤ C B(xi,δiǫ) |∇(u i -u)|e ((3/4)-(s/2))ui ,
But in the second member, for 1/2 < s ≤ 1, we have q s = 1/(3/4 -s/2) > 2 and thus q ′ s < 2 and,

e ((3/4)-(s/2))ui ∈ L qs ||∇(u i -u)|| q ′ s = o(1), ∀ 1 ≤ q ′ s < 2, one conclude that: B(xi,δiǫ) < x i 1 |∇(u i -u) > (V i -V i (x i ))e ui dy = o(1)
Finaly, with this method, we conclude that, in the case of one blow-up point and V i is s-Holderian with 1/2 < s ≤ 1 :

V i (x i ) Ω e ui dy -V (0) Ω e u dy = o(1)
which means that there is no blow-up, which is a contradiction. Finaly, for one blow-up point and V i is s-Holderian with 1/2 < s ≤ 1, the sequence (u i ) is uniformly bounded on Ω.

The case of two blow-up points:

Theorem 2.4. If V i is s-Holderian with 1/2 < s ≤ 1 and, Ω V i e ui dy ≤ 24π -ǫ, ǫ > 0,
we have :

V i (x i ) Ω e ui dy -V (0) Ω e u dy = o(1)
which means that there is no blow-up points.

Proof of the Theorem

The case of two "interior" blow-up points: As in the previous case, we assume that Ω = B + is the half ball. We have two "interior" blow-up points x i and y i :

|y i -x i | ≤ δ i ǫ,
We use a Pohozaev type identity:

Ω < (x -x i 2 )|∇u i > (-∆u i )dy = Ω < (x -x i 2 )|∇u i > V i e ui dy = A i
with A i the regular part of the identity (on which the uniform convergence holds).

A i = ∂B + < (x -x i 2 )|∇u i >< ν|∇u i > dσ + ∂B + < (x -x i 2 )|ν > |∇u i | 2 dσ
We divide our domain in two domain Ω i 1 and Ω i 2 such that:

Ω i 1 = {x, |x -x i | ≤ |x -y i |}, Ω i 2 = {x, |x -x i | ≥ |x -y i |}. We set, D i = {x, |x -x i | = |x -y i |}.
We write:

A i = Ω i 1 < (x-x i 2 )|∇u i > (V i -V i (x i ))e ui dy + Ω i 2 < (x-x i 2 )|∇u i > (V i -V i (y i ))e ui dy+ +V i (x i ) Ω i 1 < (x -x i 2 )|∇u i > e ui dy + V i (y i ) Ω i 2 < (x -x i 2 )|∇u i > e ui dy.
As for the case of one blow-up point, it is sufficient to consider terms which contain the difference ∇(u i -u).

We can write the last addition as (after using ∇(u i -u)) :

V i (x i ) Ω < (x -x i 2 )|∇u i > e ui dy - Ω < (x -x i 2 )|∇u > e u dy + +(V i (y i ) -V i (x i )) Ω i 2 < (x -x i 2 )|∇(u i -u) > e ui dy.
First of all, we consider the term (which equal, after integration by part to ):

V i (x i ) Ω < (x -x i 2 )|∇u i > e ui dy - Ω < (x -x i 2 )|∇u > e u dy = = -2V i (x i ) Ω e ui dy + 2V (0) Ω e u dy + (B i -B)
with the same notation for B i and B as for the previous case.

Case 1: suppose that, |x -

y i | ≥ |x i -y i |, thus |V i (x i ) -V i (y i )| ≤ |x i -y i | s ≤ |x -y i | s Thus, |(V i (y i )-V i (x i )) Ω i 2 ∩{x,|x-xi|≥|x-yi|}. < (x-x i 2 )|∇(u i -u) > e ui dy| ≤ Ω i 2 |x-y i | 1+s |∇(u i -u)|e ui dy+ +|y i 2 -x i 2 | Ω i 2 |x -y i | s |∇(u i -u)|e ui dy + |y i 1 | Ω i 2 |x -y i | s |∇(u i -u)|e ui dy But, |y i -x i | ≤ δ i ǫ,
x i 1 = δ i we use the same method (with the sup + inf inequality) to prove that for 1 ≥ s > 1/2 the two integrals converges to 0.

Case 2: suppose that, |x -

y i | ≤ |x i -y i |,
We do integration by parts, we have one part on D i and the other one on the circle with center y i . In fact, we have intersection of convex 2-dimensional domains, which is convex (Lipschitz domain or a domain with finite nimber of singularity) and thus one can apply the Stokes or Green-Riemann formulas.

(V i (y i ) -V i (x i )) Ω i 2 ∩{x,|x-yi|≤|xi-yi|}. < (x -x i 2 )|∇(e ui ) > dy = = (V i (y i ) -V i (x i )) Di∩{x,|x-yi|≤|xi-yi|}. < (x -x i 2 )|ν > e ui dy+ +(V i (y i ) -V i (x i )) {x,|x-yi|=|xi-yi|}∩{x,|x-yi|≤|x-xi|} < (x -x i 2 )|ν > e ui dy+ +2(V i (y i ) -V i (x i )) {x,|x-yi|≤|xi-yi|} e ui dy
We set:

I 1 = (V i (y i ) -V i (x i )) Di∩{x,|x-yi|≤|xi-yi|}. < (x -x i 2 )
|ν > e ui dy,

I 2 = (V i (y i ) -V i (x i )) {x,|x-xi|=|xi-yi|}∩{x,|x-yi|≤|x-xi|} < (x -x i 2 )
|ν > e ui dy Lemma 2.5. We have:

I 1 = o(1),
and,

I 2 = o(1).

Proof of the lemma

For I 1 , we have:

|V i (x i ) -V i (y i )| ≤ 2C|x -y i | s , |I 1 | ≤ C Di∩{x,|x-yi|≤|xi-yi|}. | < (x -y i )|ν > ||x -y i | s e ui + +|x i 2 -y i 2 | Di∩{x,|x-yi|≤|xi-yi|}. |x -y i | s e ui dy+ +|y i 1 | Di∩{x,|x-yi|≤|xi-yi|}.
|x -y i | s e ui dy But,

x i 1 = δ i , |y i -x i | ≤ δ i ǫ, u i (x) + 4 log δ i ≤ C
and, e (3/4)ui(x) ≤ |x -y i | -3/2 , Thus,

|I 1 | ≤ Di∩{x,|x-yi|≤|xi-yi|}. |x -y i | s-1 + +C Di∩{x,|x-yi|≤|xi-yi|}.
|x -y i | (-3/2)+s dy,

If we set t 0 = (x i + y i )/2, we have on one part of D i :

|x -t 0 | ≤ |x -y i | = |x -x i | ≤ |x i -y i |,
by a change of variable u = x -t 0 on the line D i , we can compute the two last integrals directly, to have, for 1 ≥ s > 1/2:

|I 1 | ≤ C(|x i -y i | s + |x i -y i | s-(1/2) ) = o(1),
For I 2 we have:

I 2 = (V i (y i ) -V i (x i )) {x,|x-yi|=|xi-yi|}∩{x,|x-xi|≤|xi-yi|} < (x -x i 2 )
|ν > e ui dy and,

|V i (x i ) -V i (y i )| ≤ 2C|x -y i | s , |I 2 | ≤ C {x,|x-yi|=|xi-yi|}∩{x,|x-yi|≤|x-xi|}. | < (x -y i )|ν > ||x -y i | s e ui + +|x i 2 -y i 2 | {x,|x-yi|=|xi-yi|}∩{x,|x-yi|≤|x-xi|}. |x -y i | s e ui dy+ +|y i 1 | {x,|x-yi|=|xi-yi|}∩{x,|x-xi|≤|x-xi|}.
|x -y i | s e ui dy with the same method as for I 1 we have:

|I 2 | ≤ C {x,|x-yi|=|xi-yi|}∩{x,|x-yi|≤|x-xi|}. |x -y i | s-1 + + {x,|x-yi|=|xi-yi|}∩{x,|x-yi|≤|x-xi|}.
|x -y i | -(3/2)+s dy, Finaly, we have:

|I 2 | ≤ C(|x i -y i | s + |x i -y i | s-(1/2) ) = o(1),
The case of two "exterior" blow-up points:

Let (x i ) i and (t i ) i two sequences of "exterior" blow-up points. If d(x i , t i ) = O(δ i ) or d(x i , t i ) = O(δ ′
i ) then we use the same technique as for two interior blow-up with the Pohozaev identity. In this case the sup + inf inequality holds, because d(x i , t i ) is of order δ i or δ ′ i . Assume that:

d(x i , t i ) δ i → +∞ and d(x i , t i ) δ ′ i → +∞
In this case, we assume that, we are on the half ball.(In fact one consider the intersection of disks and half plane which is convex, and we take its image by the conformal map. In this case we have a domain on which we can apply the Stokes formula). By a conformal transformation, f , we can assume that our two sequences are on the unit ball. First of all, we use the Pohozaev identity on the half ball as for the previous cases, but our domain change, we have one part is vertical, the second part is a part of the boundary of the unit ball, in which the sequences (u i ) and (∂u i ) i are uniformly bounded and converge to the corresponding function, and the third part of boundary, is a regular curve D ′ i such that its image by f is the mediatrice D i of the segment (x i , t i ). In the Pohozaev identity, we have a terms of type:

D ′ i < (x -x i 2 )|∇u i >< ν|∇u i > dσ + D ′ i < (x -x i 2 )|ν > |∇u i | 2 dσ
But if we integrate on the rest of the domain and if we use the Pohozeav identity on this second domain and we replace x i 2 by t i 2 , the integral on D ′ i is :

-

D ′ i < (x -t i 2 )|∇u i >< ν|∇u i > dσ - D ′ i < (x -t i 2 )|ν > |∇u i | 2 dσ
If, we add the two integral, we find:

D ′ i < (x i 2 -t i 2 )|∇u i >< ν|∇u i > dσ + D ′ i < (x i 2 -t i 2 )|ν > |∇u i | 2 dσ
We have the same techniques as for the previous cases ("interior" blow-up), except the fact that here, we use the Pohozaev identity on two differents domains which the union is our half ball.( In fact the image by a conformal map of a convex domain, intersection of two disks and a half plane. (Intersection of the unit disk and D(x 0 , ǫ) and the mediatrice of [x i , t i ], here, x 0 is the "blow-up" point, and we apply the conformal map f )). And apply the Green-Riemann theorem for smooth domains with finite number of singular points on the boundary. Or directly the Stokes theorem with the fact that we have a Lipschitz domain because it is the image of a Lipschitz domain by a conformal map (Hofmann-Mitrea-Taylor).

Remark that, here, because we have the two conditions on d(x i , t i ) and δ i , δ ′ i , the mediatrice of [x i , t i ] is close to a fixed segment [0, x 0 ]. (use angles for this fact).

To conclude, we must show that this last integral is close to 0 as i tends to +∞. By a conformal map, it is sufficient to prove that the corresponding integral on the unit ball on D i tends to 0. Without loss of generality, we can assume here that we work on the unit ball (for this integral).

On the unit ball, with the Dirichlet condition, the Green function is (in complex notation) :

G(x, y) = 1 2π log |1 -xy| |x -y| ,
we can write:

u i (x) = B1 (0)
G(x, y)V i (y)e ui(y) dy,

We can compute (in complex notation) ∂ x G and ∂ x u i :

∂ x G(x, y) = 1 -|y| 2 (x -y)(xȳ -1) , ∂ x u i (x) = B1(0) ∂ x G(x, y)V i (y)e ui(y) dy = B1(0) 1 -|y| 2 (x -y)(xȳ -1) V i (y)e ui(y) dy Let t i 0 = (x i + t i )/2. We assume that |x -t i 0 | ≤ 1 -ǫ and |t i 0 | ≥ 1 -(ǫ/2). Proposition 2.6. 1) For ((1/2) + ǫ)|x i -t i | ≤ |x -t i 0 | ≤ 1 -ǫ we have, |∂ x u i (x)| ≤ C ′ + C δ i |x i -t i | 1 |x -t i 0 | = C ′ + o(1) |x -t i 0 | . 2) For |x -t i 0 | ≤ ((1/2) -ǫ)|x i -t i | we have, |∂ x u i (x)| ≤ C ′ + C δ i |x i -t i | 1 |x i -t i 0 | = C ′ + o(1) |x i -t i 0 | . with o(1) → 0 as i → +∞ . 3) For ((1/2) -ǫ)|x i -t i | ≤ |x -t i 0 | ≤ ((1/2) + ǫ)|x i -t i | we have, |x i -t i ||∇u i | L ∞ (Di∩{((1/2)-ǫ)|xi-ti|≤|x-t i 0 |≤((1/2)+ǫ)|xi-ti|} ≤ C.

Proof of the proposition:

To estimate ∂ x u i on D i , we divide the last integral in three parts: V i (y)e ui(y) dy+

∂ x u i (x) =
+ B(ti,δ ′ i ǫ)
1 -|y| 2 (x -y)(xȳ -1)

V i (y)e ui(y) dy Let us set:

I 1 = B1(0)-(B(xi,δiǫ)∪B(ti,δ ′ i ǫ))
1 -|y| 2 (x -y)(xȳ -1)

V i (y)e ui(y) dy

I 2 = B(xi,δiǫ)
1 -|y| 2 (x -y)(xȳ -1)

V i (y)e ui(y) dy,

I 3 = B(ti,δ ′ i ǫ)
1 -|y| 2 (x -y)(xȳ -1)

V i (y)e ui(y) dy

For the first integral, because u i ≤ C on B 1 (0) -(B(x i , δ i ǫ) ∪ B(t i , δ ′ i ǫ)), we have: V i (y)e ui(y) dy Case 1:

|I 1 | ≤ C B1(0) 1 -|y| 2 |x -y||xȳ -1| dy, But, 1 ≥ |x| = |x -t i 0 + t i 0 | ≥ |t i 0 | -|x -t i 0 | ≥ 1 -(ǫ/2) -(1 -ǫ) = ǫ/
((1/2) + ǫ)|x i -t i | ≤ |x -t i 0 | ≤ 1 -ǫ:
In this case we have: and, thus,

1 -|y| 2 = 1 -|x i + δ i z| 2 = δ i (2 + o( 1
D ′ i < (x i 2 -t i 2 )|∇u i >< ν|∇u i > dσ + D ′ i < (x i 2 -t i 2 )|ν > |∇u i | 2 dσ = o(1)
here, we used the previous estimates with i → +∞ and ǫ → 0 (for the previous case 3).

B1( 0

 0 )-(B(xi,δiǫ)∪B(ti,δ ′ i ǫ)) 1 -|y| 2 (x -y)(xȳ -1)V i (y)e ui(y) dy++ B(xi,δiǫ) 1 -|y| 2 (x -y)(xȳ -1)

2 ,

 2 thus, we can write:|I 1 | ≤ C B1(0) 1 -|y| 2 |x -y||x||ȳ -1/x| dy,and, we use the fact that:|ȳ -1/x| ≥ ||ȳ| -1/|x|| ≥ |1/|x| -|y|| ≥ (1 -|y|), To have: |∂ x u i (x)| ≤ |I 2 | + |I 3 | + C B1(0) 1 + |y| |x -y| dy = |I 2 | + |I 3 | + C ′ ,Now, we look to the second and third integrals, it is sufficient to consider the first one :I 2 = B(xi,δiǫ) 1 -|y| 2 (x -y)(xȳ -1)

  )), and, |x -y| = |x -t i 0 + t i 0 -y i -δ i z| ≥ (ǫ/2)|x i -t i |, and,|xȳ -1| = |((x -t i 0 + t i 0 -x i ) + x i )(x i + δ i z) -1| ≥ (ǫ/2)|x -t i 0 |, Thus, |∂ x u i (x)| ≤ C ′ + C δ i |x i -t i | → 0 as i → +∞ . Case 2: |x -t i 0 | ≤ ((1/2) -ǫ)|x i -t i |:In this case, we have:1 -|y| 2 = 1 -|x i + δ i z| 2 = δ i (2 + o(1)),and,|x -y| = |x -t i 0 + t i 0 -y i -δ i z| ≥ (ǫ/2)|x i -t i |, and, |xȳ -1| = |((x -t i 0 + t i 0 -x i ) + x i )(x i + δ i z) -1| ≥ (ǫ/2)|x i -t i 0 |, Thus, |∂ x u i (x)| ≤ C ′ + C δ i |x i -t i | → 0 as i → +∞ . Case 3: ((1/2) -ǫ)|x i -t i | ≤ |x -t i 0 | ≤ ((1/2) + ǫ)|x i -t i |: Let ti 0 the point of D i such that | ti 0 -t i 0 | = 1/2(|x i -t i |).We use the fact that the function:v i (t) = u i ( ti 0 + (|x i -t i 0 |/4)t), is uniformly bounded for |t| ≤ 1 and is a solution of PDE which is uniformly bounded on |t| ≤ 1. By the elliptic estimates we have:|x i -t i ||∇u i | L ∞ (Di∩{((1/2)-ǫ)|xi-ti|≤|x-t i 0 |≤((1/2)+ǫ)|xi-ti|} ≤ C. Thus, we use the previous cases to compute the following integral:

	with, o(1) 1 |x i -t i 0 |	= C ′ +	o(1) |x i -t i 0 |	.
	with, o(1)
	1 |x -t i 0 |	= C ′ +	o(1) |x -t i 0 |	.

Di < (x i 2 -t i 2 )|∇u i >< ν|∇u i > dσ + Di < (x i 2 -t i 2 )|ν > |∇u i | 2 dσ = o(1)