Frédéric & Fonteneau
email: frederic.fonteneau@gmail.com

Timothée R Cook
email: timothee.cook@gmail.com

-New

New data on gastrointestinal helminths in shags (Phalacrocorax verrucosus) at Kerguelen Archipelago

Keywords: Nematoda, Cestoda, Acanthocephala, Southern Ocean, Cormorant, Seabird

To date, the knowledge of the helminth communities of Antarctic birds is scarce or fragmented. Knowledge about diseases and parasites is crucial for understanding and managing ecosystems, particularly in isolated areas where host species are more sensitive to new diseases or parasite infections. It has been showed that variations in rate of parasitism may occur between populations of host species. Two major non exclusive hypotheses have been proposed to explain such variability: exposure to parasitism and, perhaps more important, life history strategies of hosts. We studied the helminth community of the Kerguelen Shag Phalacrocorax verrucosus, an endemic seabird species of the Kerguelen Archipelago. We provide new data on the helminths infecting this species from partial or complete digestive tracts of two birds. Two nematodes (Contracaecum rudolphii s.l. and Ingliseria cirrohamata) were found free or attached to the wall of the proventriculus of birds, while the acanthocephalan Corynosoma sp. and the cestode species Tetrabothrius sp. occurred in the intestine of the shags. The genus Tetrabothrius is reported for the first time in Kerguelen Shags and in this area. The analysis of stomach contents from 41 live Kerguelen Shag individuals revealed infection by Contracaecum nematodes. The proportion of infected birds differed between colonies, possibly in relation to differential exposure to infected fish hosts.

Introduction

Parasites can affect the population dynamics of their hosts. This may be determinant for isolated populations, and endemic or endangered species (e.g. Warner 1968) as a result of less developed immune defences [START_REF] Lindström | Immunological investments reflect parasite abundance in island populations of Darwin's finches[END_REF]). To a large extent, bird and mammal species living in the Antarctic and sub Antarctic regions are endemic. To date, knowledge of the parasite fauna of these definitive hosts is virtually unknown [START_REF] Barbosa | Health of Antarctic birds: a review of their parasites, pathogens and diseases[END_REF], as are the processes of infection in these areas.

The Blue-eyed Shag complex is a group of 13 species exhibiting close morphological and behavioural characters, and which are loosely distributed around the Southern Ocean between roughly 40°S and 70°S (Orta 1992), in more or less complete isolation from one another. The Kerguelen Shag Phalacrocorax verrucosus (Cabanis, 1875) (Phalacrocoracidae: Cormorants) is a member of this species complex and is endemic to the Kerguelen Archipelago. This seabird is a marine piscivorous diver which feeds on fish from the Notothenioid sub-order [START_REF] Cook | Can foraging ecology drive the evolution of body size in a diving endotherm?[END_REF], a group of benthic fish evolved in the Southern Ocean. Its population is estimated around 7000 breeding pairs (Weimerskirch et al. 1988), although this figure is probably approximate. The helminth fauna of this bird host is not well known. Three species are currently the only recorded parasites in Kerguelen Shags: the nematodes Ingliseria cirrohamata [START_REF] Linstow | Report on the entozoan collected by HMS Challenger during the years 1873-76[END_REF] and Contracaecum rudolphii s.l. Hartwich, 1964, both first described in the Kerguelen Shag in the late 19 th century [START_REF] Linstow | Report on the entozoan collected by HMS Challenger during the years 1873-76[END_REF]) and confirmed later [START_REF] Cram | Bird parasites of the suborders Strongylata, Ascaridata and Spirurata[END_REF]Johnston and Mawson 1945) and one acanthocephalan species (Corynosoma clavatum [START_REF] Goss | Platyhelminth and Acanthocephalan parasites of local shags[END_REF], described in the mid-20 th century (Edmonds 1957). The aim of this work is to contribute to the knowledge of the gastrointestinal helminth community of Kerguelen Shags, and more largely of Antarctic and Subantarctic seabirds, and to discuss potential sources of infection.

Materials and methods

The study was carried out at the Kerguelen Archipelago (49°S, 69°E) during the Austral summer from December 2005 to March 2006. Parasites came from two sources: live bird regurgitations and dead bird carcasses. Forty-one spontaneously regurgitated stomach contents were collected while handling 41 nesting birds upon their return from the sea (contents weighing less than 18g were not kept for analysis). Stomach contents came from the Pointe Suzanne (n=18) and Sourcils Noirs (n=23) shag breeding colonies (colonies 30 km distant from one another, Fig. 1). In addition, the entire digestive tract of a fresh adult Kerguelen Shag found dead at Ratmanoff locality and the stomach (intestine too damaged for analyses) of one immature bird collected dead at Mayès were studied for parasitical examination (Fig. 1). After collection, carcasses and regurgitations were rapidly frozen and conserved at -20°C until examination.

Samples were thawed at the laboratory. All worms were collected from regurgitates while these were sorted for diet analysis [START_REF] Cook | Can foraging ecology drive the evolution of body size in a diving endotherm?[END_REF]) using a binocular magnifying glass (× 2.5). Contents of digestive tracts were macroscopically examined. All worms collected in regurgitates and digestive tracts were counted, fixed in 10% formalin, and stored in 70% alcohol before identification to genus or species level whenever possible. Parasite identifications were carried out using a list of specific bibliography on helminths of Subantarctic and Antarctic seabirds (e.g. [START_REF] Cram | Bird parasites of the suborders Strongylata, Ascaridata and Spirurata[END_REF]Johnston 1937a,b;[START_REF] Diaz | Redescription of Ingliseria cirrohamata (Linstow, 1988) (Nematoda: Acuariidae), including new hosts and geographical records[END_REF] or Austral seabirds (e.g. [START_REF] Goss | Platyhelminth and Acanthocephalan parasites of local shags[END_REF].

Occurrence and intensity of infection in nematodes were tested using Chi-square test and

Mann-Whitney test respectively

The study was approved by the French Ethic Committee of the Institut Polaire Paul-Emile Victor (IPEV) and by the Polar Environment Committee of the Terres Australes et Antarctiques Françaises (TAAF).

Results

During our study, only the nematode Contracaecum rudolphii sensus lato (s.l.) was found in the 41 regurgitates of nesting adults. In total, 34.1% of regurgitates contained parasites.

Occurrence of infection was significantly different between colonies (Chi-square test: χ 2 (1) = 4.36,p<0.05) (Table 1). Infection ranged between 0 and 28 nematodes per regurgitate and was also significantly different between colonies (Mann-Whitney test: U(39) = 129.5, Z = -2.02, p <0.05) (Table 1). Four helminth genera (of which two were identifiable to the species level) were found in the digestive tract of the two carcasses of Kerguelen Shags. Two nematode species, Contracaecum rudolphii s.l. and Ingliseria cirrohamata, were found free or attached to the wall of the proventriculus of birds. Two genera, the acanthocephalan

Discussion

In all, only four genera of helminths (in which two species) were found in the Kerguelen Shag and more examinations will be necessary to confirm these results and to identify at specific level some of the species found. Despite a modest sample size, the helminth community of the digestive tract of the Kerguelen Shag appears poor compared to that of more northerly and continental Phalacrocoracidae species in which up to 17 species can occur, as in the Great Cormorant Phalacrocorax carbo in Europe (e.g. Švažas et al. 2011), the Double-crested Cormorant Phalacrocorax auritus in North America and the Neotropic Cormorant Phalacrocorax brasilianus in South America (e.g. [START_REF] Fedynich | Helminth community structure and pattern in sympatric populations of Double-crested and Neotropic cormorants[END_REF]). The helminth richness in Kerguelen Shags is in accordance with the relatively low number of parasites species (from 3 to 6) found in other Sub-Antarctic and Antarctic piscivorous birds [START_REF] Fonteneau | Gastrointestinal helminths of King penguins (Aptenodytes patagonicus) at Crozet Archipelago[END_REF]Vidal et al. 2012;[START_REF] Diaz | Gastrointestinal helminths of Gentoo penguins (Pygoscelis papua) from Stranger Point, 25 de Mayo/King George Island, Antarctica[END_REF]. Such latitudinal differences in helminth diversity could result from the global tropic-to-pole diversity gradient (Turner 2004) and isolation of Sub-Antarctic and Antarctic species, leading to lower parasite richness in these regions.

Tetrabothrius sp. was present in the intestine of the adult Kerguelen Shag. This is the first report of this cestode genus in the Kerguelen Shag and in this geographical area. This genus occurs also in two other Phalacrocoracidae birds (the Double-crested Cormorant Phalacrocorax auritus and in the Antarctic Shag Phalacrocorax bransfieldensis) [START_REF] Hoberg | Tetrabothrius shinni sp. nov. (Eucestoda) from Phalacrocorax atriceps bransfieldensis (Pelecaniformes) in Antarctica with comments on morphological variation, host-parasite biogeography, and evolution[END_REF]) and throughout a wide range of Sub-Antarctic and Antarctic bird host species, like penguins (e.g. [START_REF] Fonteneau | Gastrointestinal helminths of King penguins (Aptenodytes patagonicus) at Crozet Archipelago[END_REF]Vidal et al. 2012) and petrels (e.g. [START_REF] Jones | Notes on parasites in penguins (Spheniscidae) and petrels (Procellariidae) in the Antarctic and subantarctic[END_REF]). More investigations should be conducted to identify the status of the Tetrabothrius species infecting the Kerguelen Shag compared to other Tetrabothrius species found in other seabirds.

Our study reports for the second time acanthocephalans of the genus Corynosoma from the intestine of the adult Kerguelen Shag. Further investigations should be conducted to identify the status of the Corynosoma sp. found here compared to Corynosoma clavatum previously described in the Kerguelen Shag (Edmonds 1957).

The Acuariidae nematode Ingliseria cirrohamata was first described in the Kerguelen Shag at Kerguelen Archipelago [START_REF] Linstow | Report on the entozoan collected by HMS Challenger during the years 1873-76[END_REF][START_REF] Cram | Bird parasites of the suborders Strongylata, Ascaridata and Spirurata[END_REF]Johnston & Mawson 1945).

Recently, this nematode was re-described with new samples from this type host and locality and in three other bird species (the Imperial Shag Phalacrocorax albiventer, the Neotropic Cormorant and the Magellanic Cormorant Phalacrocorax magellanicus) from two other geographical localities (Argentina and Chile) [START_REF] Diaz | Redescription of Ingliseria cirrohamata (Linstow, 1988) (Nematoda: Acuariidae), including new hosts and geographical records[END_REF]. In view of our results, infection intensity would seem more important in the Kerguelen Shag (up to 207 specimens in one bird) than in the three other definitive hosts (no more than four specimens per bird) [START_REF] Diaz | Redescription of Ingliseria cirrohamata (Linstow, 1988) (Nematoda: Acuariidae), including new hosts and geographical records[END_REF]). However, no regurgitations of Kerguelen Shags contained Ingliseria cirrohamata and only one of the two carcasses was infected. This nematode does not seem a common parasite in this host. Indeed, only two specimens were found in the stomach of one bird during the first parasite exploration in this host species [START_REF] Linstow | Report on the entozoan collected by HMS Challenger during the years 1873-76[END_REF][START_REF] Cram | Bird parasites of the suborders Strongylata, Ascaridata and Spirurata[END_REF]) and later only one specimen was obtained in one of seven examined birds (Johnston and Mawson 1945).

Nematodes of the genus Contracaecum (Anisakidae) are known to commonly infect the Phalacrocoracidae family (e.g. [START_REF] Fedynich | Helminth community structure and pattern in sympatric populations of Double-crested and Neotropic cormorants[END_REF][START_REF] Abollo | Anisakid infection in the European shag Phalacrocorax aristotelis aristotelis[END_REF][START_REF] Dezfuli | Intra-and interspecific density-dependant effects on growth in helminth parasites of the cormorant, Phalacrocorax carbo sinensis[END_REF][START_REF] Garbin | Genetic and morphological evidences for the existence of a new species of Contracaecum (Nematoda: Anisakidae) parasite of Phalacrocorax brasilianus (Gmelin) from Chile and its genetic relationships with congeners from fish-eating birds[END_REF]). Contracaecum rudolphii s.l. was described infecting the Kerguelen Shag [START_REF] Linstow | Report on the entozoan collected by HMS Challenger during the years 1873-76[END_REF]. It is now known that Contracaecum rudolphii s.l. is formed by a complex of several sibling species (e.g. Mattiucci et al. 2008;[START_REF] D'amelio | Two new species of Contracaecum Railliet & Henry, 1912 (Nematoda: Anisakidae), C. fagerholmi n. sp. and C. rudolphii F from the brown pelican Pelecanus occidentalis in the northern Gulf of Mexico[END_REF]. The Contracaecum species found in the Kerguelen Shag during the present study was genetically analysed and identified as a new sibling species differing from those Contracaecum rudolphii s.l. known in other birds (Mattiucci et al. 2009). Further studies are needed to determine which morphological characteristics can be used to distinguish this species in the Contracaecum rudolphii complex in order to assign a formal name to it. We found this nematode at all sites, which is consistent with a previous study which reported it at "various localities in Kerguelen" (Johnston and Mawson 1945), suggesting it is distributed throughout the archipelago.

Contracaecum nematodes have been reported attached to the wall of the proventriculus of some piscivorous birds. This fastening of nematodes is thought to be an adaptation to counter the abrasive effect of fish bones resulting from digestion [START_REF] Huizinga | Contracaeciasis in Pelecaniform birds[END_REF][START_REF] Liu | Gastric ulcers associated with Contracaecum spp. (Nematoda: Ascaroidea) in a Steller sea lion and a White pelican[END_REF]. Thus, this attachment seems to be effective only towards the end of the digestive period. During the digestive process, most nematodes are normally free and mixed to the stomach content. As regurgitates in the present study come exclusively from birds just returning from fishing areas, we assumed that nematodes were free in the stomach at the moment of regurgitation. Nevertheless, as spontaneous regurgitations represented a fraction of the stomach content, the number of nematodes collected per bird amounted only to a certain percentage of all the nematodes actually present in the whole stomach. Contracaecum rudolphii s.l. was found in the stomachs of the two carcasses from the present study, and in all seven birds examined previously by Johnston and Mawson (1945). However, occurrence of this nematode in regurgitations ranged only from 17 to 48%, depending on the colony. Also, Contracaecum numbers per regurgitations (from 0 to 28) were low compared to the numbers of nematodes found in the stomachs of the two carcasses (from 101 to 289). Obviously, the regurgitation method does not provide an absolute figure of the parasite load in the bird's stomach, but an estimate of the number per gram of food. The prevalence and intensity at Sourcils Noir colony were higher than at Pointe Suzanne despite the average mass of stomach content being lower at the former, suggesting that infection estimates were not biased by sample size. Even by extrapolating the number of parasites in each sample to the total theoretical mass of food per content, the number of parasites does not approach the number found in the stomach of the carcasses. Several hypotheses can be suggested to explain this discrepancy. First, not all Contracaecum are detached from the proventriculus when the bird's stomach is full. Second, the deaths of the two Kerguelen Shags found dead could be related to their high infections in Contracaecum. The first description of this nematode genus in Kerguelen Shags reports a bird with only 36 nematodes [START_REF] Linstow | Report on the entozoan collected by HMS Challenger during the years 1873-76[END_REF], suggesting that the numbers found in the two carcasses could be indeed higher than normal. Future investigations should focus on testing the efficiency of parasitical study of regurgitations as a nondestructive method to evaluate infection rates of stomach nematodes in piscivorous birds.

The diet of Kerguelen Shags is comprised of nearly 90% of nothotenioid fish species (Table 1), namely Lepidonotothen mizops, Notothenia cyanobrancha and Harpagifer sp. [START_REF] Cook | Can foraging ecology drive the evolution of body size in a diving endotherm?[END_REF]. In Sub-Antarctic and Antarctic areas, some nototheniid fish, amongst which the three genera Lepidonotothen, Harpagifer and Notothenia, were mentioned as vectors for larval stages of Contracaecum nematodes (e.g. [START_REF] Laskowski | The helminth fauna of some notothenioid fishes collected from the shelf of Argentine Islands, West Antarctica[END_REF]Rokicki et al. 2009). The dominance of Lepidonotothen mizops at Sourcils Noirs, the colony with the highest infection rates of Contracaecum in stomach contents (Table 1), points to a possible determining role of that fish species as a vector of Contracaecum rudolphii s.l., even if Notothenia cyanobrancha is also known as a possible vector for Contracaecum larvae in waters off Kerguelen Archipelago (Johnston and Mawson 1945). Fish of the family Nototheniidae, which are identified vectors for larval stages of ancanthocephalans of the genus Corynosoma (see Zdzitowiecki 1986 for review) and for cestodes of the genus Tetrabothrius [START_REF] Hoberg | Tetrabothrius shinni sp. nov. (Eucestoda) from Phalacrocorax atriceps bransfieldensis (Pelecaniformes) in Antarctica with comments on morphological variation, host-parasite biogeography, and evolution[END_REF], could also play a similar role in the helminth infection of the Kerguelen Shags and other birds in Sub-Antarctic areas. Little is known about the life cycle of the genus Ingliseria. As all the definitive hosts in which this nematode species were described are fish-eating birds [START_REF] Diaz | Redescription of Ingliseria cirrohamata (Linstow, 1988) (Nematoda: Acuariidae), including new hosts and geographical records[END_REF], fish are also likely vector for Ingliseria cirrohamata.

Table 1. Characteristics of Kerguelen Shag stomach contents: prey and infection rates. The three most abundant prey species (amounting to almost 90% of all prey items) are presented (data from [START_REF] Cook | Can foraging ecology drive the evolution of body size in a diving endotherm?[END_REF]. Occurrence is defined as the occurrence of the fish or parasite in the content relative to the total number of contents. Proportion is defined as the proportion of individuals from any prey species relative to the total number of prey items in all contents. Values are presented as average (± SD) and range. -13 -

  Corynosoma sp. and the cestode Tetrabothrius sp., occurred in the intestine of the shags. The digestive tract of the adult bird contained 1050 Tetrabothrius sp., 101 Contracaecum rudolphii s.l., 207 Ingliseria cirrohamata and 161 Corynosoma sp. The stomach of the fresh immature bird contained 289 Contracaecum rudolphii s.l.

Fig. 1 .

 1 Fig. 1. Localisation of sampled sites at Kerguelen Islands during the breeding season 2005-2006. Mayès and Ratmanoff correspond to the names of the sites where the two dead shags were found near the shore. Pointe Suzanne and Sourcils Noirs correspond to the two shag breeding colonies where the regurgitated stomach contents were sampled.

  Fonteneau, F. & Cook, T.R. -New data on gastrointestinal helminths in shags (Phalacrocorax verrucosus) at Kerguelen Archipelago.

		Sourcils Noirs	Pointe Suzanne
	Stomach content mass (g)	59 ± 38 (18-192)	98 ± 43 (24-179)
	Harpagifer sp. occurrence (%)	46.1	38.9
	Harpagifer sp. proportion (%)	10.0	7.2
	Notothenia cyanobrancha occurrence (%)	19.2	94.4
	Notothenia cyanobrancha proportion (%)	3.5	74.0
	Lepidonotothen mizops occurrence (%)	92.3	22.2
	Lepidonotothen mizops proportion (%)	72.6	7.2
	Contracaecum rudolphii s.l. occurrence (%)	47.8	16.7
	Contracaecum rudolphii s.l. per content	4.2 ± 7.3 (0-28)	0.2 ± 0.5 (0-2)
	Contracaecum rudolphii s.l. per gram	0.083 ± 0.182 (0-0.840) 0.003 ± 0.007 (0-0.025)

Acknowledgements

The authors are grateful to the French Polar Institute (Institut Paul-Emile Victor -IPEV) and the Terres Australes et Antarctiques Françaises for financial and logistical means in the field. Field work was approved by the institute's ethics committee and was conducted under IPEV research program number 394 (Diving Seabirds, coordinator: Charles-André Bost).

Thanks to John Mike Kinsella for his precious assistance in the identification of parasites and to two anonymous referees to their constructive comments and suggestions.


Saïma Khenissy

Olivier Rey
email: rey@math.polytechnique.fr

A criterion for existence of solutions to the supercritical Bahri-Coron's problem

We consider the supercritical elliptic problem -∆u = u 5+ε , u > 0 in Ω; u = 0 on ∂Ω with Ω a smooth bounded domain in R 3 , and ε > 0 a small number. Denoting by G the Green's function of -∆ on Ω with Dirichlet boundary conditions, and by H its regular part, we show that a nontrivial relative homology between the level sets ϕ b and ϕ a of ϕ, a < b < 0, ϕ(x, y) = H(x, x) 1/2 H(y, y) 1/2 -G(x, y), implies the existence, for ε small enough, of a solution to the problem which blows up, as ε goes to 0, at two points x, y such that a ≤ ϕ(x, y) ≤ b, ∇ϕ(x, y) = 0.

Introduction

Let us consider the nonlinear elliptic problem (P q )

-∆u = u q , u > 0 in Ω u = 0 on ∂Ω where 1 < q < +∞, and Ω is a smooth and bounded domain in R N , N ≥ 3.

When q is subcritical, i.e. q < N +2 N -2 , the mountain pass lemma proves the existence of a solution to (P q ) for any domain Ω. In the case q = N +2 N -2 , Pohozaev's identity [START_REF] Pohozaev | Eigenfunctions of the equation ∆u + λf (u) = 0[END_REF] shows that problem (P q ) has no solution as Ω is starshaped. On the other hand Kazdan and Warner [START_REF] Kazdan | Remarks on some quasilinear elliptic equations[END_REF] proved that a solution exists in the special case Ω is an annulus, and Bahri and Coron [START_REF] Bahri | On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain[END_REF] showed that a nontrivial topology of the domain, in the sense H 2n-1 (Ω; Q) = 0 or H n (Ω; Z/2Z) = 0 for some n ∈ N * , implies that (P q ) has a solution. When q > N +2 N -2 , Passaseo [START_REF] Passaseo | New nonexistence results for elliptic equations with supercritical nonlinearity[END_REF][START_REF] Passaseo | Non trivial solutions of elliptic equations with supercritical exponent in contractible domains[END_REF] gives, for N ≥ 4, an example of a topologically nontrivial domain for which no solution of (P q ) exists, provided that q > N +1

N -3 . The present paper is interested in the slightly supercritical case, i.e. q = N +2 N -2 + ε, with ε > 0 a small number. More precisely, we consider, for ξ ∈ R N and λ > 0, the functions

U ξ,λ (x) = α N λ λ 2 + |x -ξ| 2 N -2 2 , x ∈ R N
Functions U ξ,λ are approximate solutions to (P p ) when ξ ∈ Ω and λ goes to zero. This fact suggests looking for solutions to (P q ), q close to p, in a neighbourhood of the U ξ,λ 's, that is for solutions writing as

u = k i=1 U ξi,λi + v
with k ∈ N * and v small in some norm.

When q = p-ε, Bahri, Li and Rey [START_REF] Bahri | On a variational problem with lack of compactness: the topological effect of the critical points at infinity[END_REF], Rey [START_REF] Rey | The topological impact of critical points in a variational problem with lack of compactness: the dimension 3[END_REF] proved that such solutions exist. In particular, a solution exists for k = 1 and ε small enough, which blows up at a critical point of the Robin function H(x, x) as ε goes to 0.

When q = p + ε, the situation comes out to be different. Ben Ayed, Grossi, El Mehdi and Rey [START_REF] Ben Ayed | A nonexistence result of single Peaked Solutions to a supercritical nonlinear problem[END_REF] showed that there is no solution blowing up at a single point as ε goes to 0. However, Del Pino, Felmer and Musso [START_REF] Del Pino | Two-bubble solutions in the super-critical Bahri-Coron's problem[END_REF] proved, under certain topological conditions, the existence of such solutions when k = 2. To state their result, some notations have to be introduced.

We denote by G the Green's function of the operator (-∆) with Dirichlet boundary conditions on Ω, by H its regular part, i.e.

G(x, y)

= b N 1 |x -y| N -2 -H(x, y) with b N = (N -2)σ N -1
-1 , σ N -1 is the measure of the (N -1)-dimensional unit sphere, and

∆ x H = 0 in Ω × Ω H(x, y) = b N 1 |x-y| N -2 on ∂(Ω × Ω) For (ξ 1 , ξ 2 ) ∈ Ω × Ω we define ϕ(ξ 1 , ξ 2 ) = H 1 2 (ξ 1 , ξ 1 )H 1 2 (ξ 2 , ξ 2 ) -G(ξ 1 , ξ 2 ).
(1.1)

H d (B) denotes the d-th cohomology group with integral coefficients of B ⊂ Ω, and i * : H * (Ω) → H * (B) the homomorphism induced by the inclusion i : B → Ω. We have :

Theorem [START_REF] Del Pino | Two-bubble solutions in the super-critical Bahri-Coron's problem[END_REF]. Assume 3 ≤ N ≤ 6 and let Ω be a bounded domain with smooth boundary in R N . Suppose there exists a compact manifold M ⊂ Ω and an integer d ≥ 1 such that ϕ < 0 on M × M, i * : H d (Ω) → H d (M) is nontrivial, and either d is odd or H 2d (Ω) = 0.

Then there exists ε 0 > 0 such that, for any 0 < ε < ε 0 , problem (P p+ε ) has at least one solution u ε . Moreover, let C be the component of the set where ϕ < 0 which contains M × M. Then, given any sequence (ε n ) going to zero, there is a subsequence, which we denote in the same way, and a critical point (ξ 1 , ξ 2 ) ∈ C of the function ϕ such that u εn (x) → 0 on compact subsets of Ω \ {ξ 1 , ξ 2 } and such that for any δ > 0 sup |x-ξi|<δ u εn (x) → +∞, as ε → 0, i = 1, 2.

Examples are given of domains satisfying the topological assumptions of the theorem. Namely, a fixed domain D in R N , from which a subdomain ω is excised, ω contained in a ball of sufficiently small radius. When N = 3, another example consists of an arbitrary domain D, from which a solid torus is excised, with sufficiently small cross-section.

In the present work, we consider the case N = 3. We prove again the existence of the twobubble solutions to (P 5+ε ), blowing up at two points as ε goes to 0, with some modifications of the method used in [START_REF] Del Pino | Two-bubble solutions in the super-critical Bahri-Coron's problem[END_REF], leading to a simpler and more natural assumption. Indeed, denoting by

ϕ a = {x ∈ Ω × Ω / ϕ(x) ≤ a}
the level sets of ϕ, we prove : Theorem 1.1 N = 3. Assume that a < b < 0, b is not a critical value of ϕ, and the relative homology H * (ϕ b , ϕ a ) is nontrivial. Then, for ε small enough, there exists a sequence of solutions to (P 5+ε ) which blows up at ξ 1 , ξ 2 as ε goes to zero, with

(ξ 1 , ξ 2 ) a critical point of ϕ such that a < ϕ(ξ 1 , ξ 2 ) < b.
Furthermore, denoting by

∆ = {(x 1 , x 2 ) ∈ Ω × Ω / x 1 = x 2 }
the diagonal of Ω × Ω, we have :

Corollary 1.1 N = 3.
Assume that b ≤ 0 is not a critical value of ϕ and H * (ϕ b , ∆) = 0. Then, the problem (P 5+ε ) has a solution as described in Theorem 1.1, with ϕ(ξ 1 , ξ 2 ) < b.

The corollary follows from the fact that ϕ a retracts by deformation on ∆, for a small enough, as proved at the end of the paper.

Remarks.

1) The arguments are still valid for N = 4. However, for sake of simplicity, we restrict ourselves in this paper to the case N = 3. Dimensions N ≥ 5 could be treated using the same tools as in [START_REF] Rey | On elliptic Neumann problems with supercritical exponent[END_REF].

2) The problem has a variational structure. Let J q denote a functional whose critical points are solutions to (P q . For slightly subcritical exponents, i.e. q = N +2 N -2 -ε, the difference of topology induced by the solutions blowing up at two points, as ε goes to zero, between the level sets of J q , is linked to the relative topology between Ω 2 and ϕ -= {x ∈ Ω × Ω / ϕ(x) ≤ 0} [START_REF] Bahri | On a variational problem with lack of compactness: the topological effect of the critical points at infinity[END_REF]. Here, the relevant quantity is the relative topology between ϕ -and ∆. We know that for expanding annuli-domains A η , η < |x| < η -1 , ϕ -is homotopically equivalent to A η × A η , whereas for thin annuli-domains ϕ -is homotopically equivalent to ∆ [START_REF] Ould Ahmadou | Computation of the difference of topology at infinity for Yamabe-type problems on annuli-domains, I & II[END_REF]. In one case, we obtain blowing up solutions for the supercritical exponent going to the critical one, in the other case blowing up solutions for the subcritical exponent going to the critical one.

Our approach to prove Theorem 1.1 is similar to Del Pino, Felmer and Musso's one, with appropriate modifications. In Section 2, we recall some results of [START_REF] Del Pino | Two-bubble solutions in the super-critical Bahri-Coron's problem[END_REF] which lead to a finitedimensional reduction of the problem, together with an asymptotic expansion of the reduced functional. Section 3 is devoted to the additional results which are required in Section 4 to prove Theorem 1.1.

Known results : the finite dimensional reduction

We consider the energy functional, defined in H 1 0 (Ω) ∩ L 6+ε (Ω), whose positive critical points are solutions to the problem (P ε ) := (P 5+ε )

J ε (u) = 1 2 Ω |∇u| 2 - 1 6 + ε Ω u 6+ε + .
We look for solutions to (P ε ) in a neighbourhood of functions

U = U 1 + U 2 , U i = U ξi,λi denoting the H 1 0 (Ω)-projection of U i = U ξi,λi introduced in Section 1. Let δ > 0 be a small number, we set O δ (Ω) = (ξ 1 , ξ 2 ) ∈ Ω × Ω : |ξ 1 -ξ 2 | > δ, dist(ξ i , ∂Ω) > δ, i = 1, 2 and
λ i = cΛ 2 i ε, i = 1, 2, δ < Λ i < δ -1 (2.1) with c = b 3 R 3 U 6 6α 3 R 3 U 5 = 1 128 U = U 0,1 .
We also define

Ψ(ξ 1 , ξ 2 , Λ 1 , Λ 2 ) = 1 2 H(ξ 1 , ξ 1 )Λ 2 1 + H(ξ 2 , ξ 2 )Λ 2 2 -2G(ξ 1 , ξ 2 )Λ 1 Λ 2 + ln Λ 1 Λ 2 .
(2.2)

Expanding J ε (U 1 + U 2 ) with respect to ε, we find ( [START_REF] Bahri | On a variational problem with lack of compactness: the topological effect of the critical points at infinity[END_REF], [START_REF] Del Pino | Two-bubble solutions in the super-critical Bahri-Coron's problem[END_REF]) :

Lemma 2.1 There exist constants C 1 > 0, C 2 > 0, C 3 such that, for any δ > 0

J ε (U 1 + U 2 ) = C 1 + C 2 ε ln ε + C 3 ε + C 2 εΨ(ξ 1 , ξ 2 , Λ 1 , Λ 2 ) + o(ε) uniformly with respect to (ξ 1 , ξ 2 , Λ 1 , Λ 2 ) ∈ O δ (Ω)×]δ, δ -1 [ 2 .
This expansion holds in C 2 -norm with respect to the variables ξ and Λ in the considered domain.

Remark. Writing U = U 0,1 , constants C 1 , C 2 and C 3 are defined as

C 1 = 2 1 3 R 3 U 6 = √ 3 π 2 2 C 2 = 1 6 R 3 U 6 = √ 3 π 2 8 C 3 = 1 6 1 3 + ln c R 3 U 6 - 1 3 R 3 U 6 ln U . (2.3)
In view of this result, the strategy to prove the theorem consists in reducing the original problem of finding a critical point of J ε to a finite dimensional one in the variables ξ 1 , ξ 2 , Λ 1 , Λ 2 , such that the critical points of Ψ will provide us with the solutions that we are looking for. We first perform a rescaling. We set Ω ε = Ω/ε and δ > 0 being fixed, we consider points ξ

′ i ∈ Ω ε , numbers Λ i > 0, i = 1, 2, such that |ξ ′ 1 -ξ ′ 2 | > δ ε , dist(ξ ′ i , ∂Ω ε ) > d ε , δ < Λ i < δ -1 . (2.4)
We define the functions

V i = U ξ ′ i ,Λ * i = α N Λ * i (Λ * i ) 2 + | • -ξ ′ i | 2 1 2 Λ * i = cΛ 2 i .
As previously, we define the projections onto H 1 0 (Ω ε ) of these functions, namely the functions V i given as the unique solutions of

-∆V i = V 5 i in Ω ε V i = 0 on ∂Ω ε .
Let us denote

V = V 1 + V 2 V = V 1 + V 2 and (ξ ′ 1 , ξ ′ 2 , Λ 1 , Λ 2 ) = (ξ ′ , Λ).
Our aim is finding a solution to the problem in a neighbourhood of V , for appropriate ξ ′ and Λ. In order to reduce the problem to a finite dimensional one, we first solve the linearized equation at V . We define, for i = 1, 2, the functions

Z ij = ∂V i ∂ξ ′ ij 1 ≤ j ≤ 3 Z i4 = ∂V i ∂Λ * i
which span the kernel of the linearized problem at V i on R 3 when ε = 0, and their H 1 0 (Ω ε )projections Z ij , i.e. the unique solutions of

∆Z ij = ∆Z ij in Ω ε Z ij = 0 on ∂Ω ε .
Then, following [START_REF] Del Pino | Two-bubble solutions in the super-critical Bahri-Coron's problem[END_REF], we consider the problem : h ∈ L ∞ (Ω ε ) being given, find a function φ which satisfies

   ∆φ + (5 + ε)V 4+ε φ = h + i,j c ij V 4 i Z ij in Ω ε φ = 0 on ∂Ω ε < V 4 i Z ij , φ > = 0 for all i, j (2.5)
for some numbers c i,j , < •, • > denoting the scalar product in L 2 (Ω ε ) (we notice that the orthogonality to

V 4 i Z ij in L 2 (Ω ε ) is equivalent to the orthogonality to Z ij in H 1 0 (Ω ε )).
Existence and uniqueness of φ follows from the implicit functions theorem, in suitable functional spaces. Actually, for ψ a function defined on Ω ε , we consider the following weighted L ∞ -norms

ψ * = sup x∈Ωε (1 + |x -ξ ′ 1 | 2 ) -1 2 + (1 + |x -ξ ′ 2 | 2 ) -1 2 -1 ψ(x)
and

ψ * * = sup x∈Ωε (1 + |x -ξ ′ 1 | 2 ) -1 2 + (1 + |x -ξ ′ 2 | 2 ) -1 2 -4 ψ(x) .
These norms are equivalent to (V ) -1 ψ ∞ and (V ) -4 ψ ∞ respectively, uniformly with respect to the points and numbers satisfying (2.4).

We recall the following result (see Propositions 4.1 and 4.2 in [START_REF] Del Pino | Two-bubble solutions in the super-critical Bahri-Coron's problem[END_REF]) :

Proposition 2.1 Assume that conditions (2.4) hold. There exists ε 0 > 0 and a constant C > 0 independent of ε, ξ ′ , Λ, such that for all 0 < ε < ε 0 and all h ∈ L ∞ (Ω ε ), problem (2.5) has a unique solution φ ≡ L ε (h), which satisfies

L ε (h) * ≤ C h * * |c ij | ≤ C h * * . (2.6)
Moreover, the map (ξ

′ , Λ) → L ε (h) is C 1 and D (ξ ′ ,Λ) L ε (h) * ≤ C h * * . (2.7)
A first order correction to consider, between V and a solution to the original problem, is given by

ψ = -L ε (R ε ) R ε = V 5+ε -V 5 1 -V 5 2 . (2.8)
Estimating R ε * * -see 3.22 below -provides us, through Proposition 2.1, with an estimate of ψ * . Namely, there exists a constant C, independent of ε and ξ ′ , Λ satisfying (2.4) such that ψ * ≤ Cε.

(2.9)

Still following [START_REF] Del Pino | Two-bubble solutions in the super-critical Bahri-Coron's problem[END_REF], the next step is considering the nonlinear problem of finding φ such that, for some numbers c ij , the following holds

   ∆(V + ψ + φ) + (V + ψ + φ) 5+ε + = i,j c ij V 4 i Z ij in Ω ε φ = 0 on ∂Ω ε < V 4 i Z ij , φ > = 0 for all i, j. (2.10)
Setting

N ε (η) = (V + η) 5+ε + -V 5+ε -(5 + ε)V 4+ε η (2.11)
system (2.10) writes as

   ∆φ + (5 + ε)V 4+ε φ = -N ε (ψ + φ) + i,j c ij V 4 i Z ij in Ω ε φ = 0 on ∂Ω ε < V 4 i Z ij , φ > = 0
for all i, j.

(2.12) Lemma 5.1 in [START_REF] Del Pino | Two-bubble solutions in the super-critical Bahri-Coron's problem[END_REF] provides us with the following estimate, when conditions (2.4) hold : there is a positive constant C such that, for any sufficiently small ε and φ * ≤ 1

N ε (ψ + φ) * * ≤ C( φ 2 * + ε 2 ). (2.13)
Then, applying a fixed point theorem to the map

A ε from F = {φ ∈ H 1 0 ∩ L ∞ (Ω ε ) : φ * ≤ ε} to H 1 0 ∩ L ∞ (Ω ε ) defined as A ε (φ) = -L ε (N ε (φ + ψ))
which is contracting in norm • * , we obtain (see Propositions 5.1 and 5.2 in [START_REF] Del Pino | Two-bubble solutions in the super-critical Bahri-Coron's problem[END_REF]) :

Proposition 2.2 Assume that conditions (2.4) hold. For ε small enough, there exists a unique

solution φ = φ(ξ ′ , Λ) to problem (2.12) in F . Moreover, (ξ ′ , Λ) → φ(ξ ′ , Λ) is C 1
, and there exists a constant C > 0 independent of ε, ξ', Λ, such that

D (ξ ′ ,Λ) φ * ≤ Cε.
(2.14)

Remark. Actually, φ = φ(ξ ′ , Λ) and its first derivatives satisfy the estimates

φ * ≤ Cε 2 D (ξ ′ ,Λ) φ * ≤ Cε 2 . (2.15)
The first inequality is a consequence of (2.13) and the definition of φ as a fixed point of A ε . The second one is proved in the next section -see Lemma 3.3.

To come back to the original problem, we consider the rescaled functions defined in Ω

ψ(ξ, Λ)(x) = ε -ζ ψ(ε -1 ξ, Λ)(ε -1 x) ζ = 1 2 + 1 2 ε (2.16) φ(ξ, Λ)(x) = ε -ζ φ(ε -1 ξ, Λ)(ε -1 x) (2.17) and Û (ξ, Λ)(x) = ε -ζ V (ε -1 x) = ε ( 1 2 -ζ) (U 1 (x) + U 2 (x)) U i (x) = U ξi,λi (x) (2.18)
with

λ i = cΛ 2 i ε and ξ i = εξ ′ i ∈ O δ (Ω).
Lastly, we define

I ε (ξ, Λ) ≡ J ε ( Û + ψ + φ)(ξ, Λ) . (2.19)
Previous results provide us with the following basic assertion (see Lemma 6.1 in [START_REF] Del Pino | Two-bubble solutions in the super-critical Bahri-Coron's problem[END_REF]) :

Proposition 2.3 The function u = Û + ψ + φ is a solution to problem (P ε ) if and only if (ξ, Λ) is a critical point of I ε .
Consequently we are led, for proving the theorem, to find a critical point of I ε . We establish before, in the next section, some results about the second derivatives of I ε .

3 Improved results : the second derivatives of I In this section, we prove a C 2 -expansion of I ε -see (3.33) below. For this purpose, we first show that inequalities (2.7) and (2.14) are still valid for the second derivatives of L ε and φ respectively.

Lemma 3.1 Assume that assumptions of Proposition 2.1 are satisfied. L ε (h) is C 2 with respect to Λ, ξ
′ , and there is a constant

C independent of ε, ξ ′ , Λ, such that D 2 (ξ ′ ,Λ) L ε (h) * ≤ C h * * . (3.1) Proof. For given h ∈ L ∞ (Ω ε ), we recall that problem    ∆φ + (5 + ε)V 4+ε φ = h + i,j c ij V 4 i Z ij in Ω ε φ = 0 on ∂Ω ε < V 4 i Z ij , φ > = 0 for all i, j (3.2)
has a unique solution φ = L ε (h) for some numbers c ij , with

L ε (h) * ≤ C h * * |c ij | ≤ C h * * . (3.3)
Let us write

Z = ∂ ξ ′ φ
at least formally. Differentiating the first and third equations in (3.2) with respect to ξ ′ , we find

∆Z + (5 + ε)V 4+ε Z = -(5 + ε)(∂ ξ ′ V 4+ε )φ + i,j d ij V 4 i Z ij + i,j c ij ∂ ξ ′ (V 4 i Z ij ) in Ω ε (3.4) with d ij = ∂ ξ ′ c ij and < ∂ ξ ′ (V 4 i Z ij ), φ > + < V 4 i Z ij , Z >= 0 for all i, j. (3.5)
We decompose Z, writing

Z = Z + lk b lk Z lk with < V 4 i Z ij , Z >= 0 ∀i, j. It follows from (3.5) that lk b lk < V 4 i Z ij , Z lk >= -< ∂ ξ ′ (V 4 i Z ij ), φ > ∀i, j. (3.6)
This defines a linear system in the b lk 's which is almost diagonal, with uniformly bounded coefficients since, as ε goes to 0, we have

< V 4 i Z ij , Z lk >= δ i,l δ j,k R N U 4 0,Λ * i ∂U 0,Λ * i ∂y j 2 + o(1) (3.7)
with y j = x j , 1 ≤ j ≤ 3, and y 4 = Λ * i . Such a system is uniquely solvable, and estimating the right hand side provides us with b lk = O( φ * ). Whence, using (3.3)

b lk = O( h * * ). (3.8)
On the other hand, using (3.4) we find that Z satisfies the equation

∆ Z + (5 + ε)V 4+ε Z = f + i,j d ij V 4 i Z ij in Ω ε (3.9) with f = -(5 + ε)(∂ ξ ′ V 4+ε )φ - i,j b ij [∆Z ij + (5 + ε)V 4+ε Z ij ] + i,j c ij ∂ ξ ′ (V 4 i Z ij ). (3.10)
It is easily checked, taking account of (3.3) and (3.8), that

f * * ≤ C h * * .
Then, we deduce from Proposition 2.1 that Z ≡ L ε (f ) and d ij satisfy

Z * ≤ C h * * |d ij | ≤ C h * * . (3.11)
Thus, using again estimate (3.8) we obtain

Z * ≤ C h * * . (3.12)
Once these estimates are known, one easily check that

Z = L ε (f ) + + lk b lk Z lk
with the b lk 's defined by (3.6) and f defined by (3.10) is definitely the derivative of φ with respect to ξ ′ .

We have now to estimate the second derivatives of φ with respect to ξ ′ . Differentiating (3.4) and (3.5) with respect to ξ ′ , and writing formally

W = ∂ ξ ′ Z = ∂ 2 ξ ′ 2 φ we find ∆W + (5 + ε)V 4+ε W = -2(5 + ε)(∂ ξ ′ V 4+ε )Z -(5 + ε)(∂ 2 ξ ′ V 4+ε )φ + i,j e ij V 4 i Z ij + 2d ij ∂ ξ ′ (V 4 i Z ij ) + c ij ∂ 2 ξ ′ (V 4 i Z ij ) (3.13) with e ij = ∂ ξ ′ d ij and < ∂ 2 ξ ′ 2 (V 4 i Z ij ), φ > +2 < ∂ ξ ′ (V 4 i Z ij ), Z > + < V 4 i Z ij , W >= 0 ∀i, j. (3.14)
We proceed as previously, writing W as

W = W + l,k a lk Z lk with < V 4 i Z ij , W >= 0 ∀i, j. It follows from (3.14) that l,k a lk < V 4 i Z ij , Z lk >= -< ∂ 2 ξ ′ 2 (V 4 i Z ij ), φ > -2 < ∂ ξ ′ (V 4 i Z ij ), Z > ∀i, j. (3.15)
(3.7) implies again that (3.15) is a linear system in the a lk 's which is almost diagonal, and (3.3), (3.12) show that

|a lk | ≤ C( φ * + Z * ) ≤ C ′ h * * . (3.16)
On the other hand, we have

(∂ ξ ′ V 4+ε )Z * * ≤ C Z * ≤ C ′ h * * (∂ 2 ξ ′ 2 V 4+ε )φ * * ≤ C φ * ≤ C ′ h * * (3.17)
and

∂ ξ ′ (V 4 i Z ij ) * * ≤ C ∂ 2 ξ ′ 2 (V 4 i Z ij ) * * ≤ C. (3.18)
Going back to (3.13), we see that W satisfies Finally, estimate (3.16) yields

   ∆ W + (5 + ε)V 4+ε W = g + i,j e ij V 4 i Z ij in Ω ε W = 0 on ∂Ω ε < V 4 i Z ij , W > = 0 for all i, j (3.19) with g = - i,j a ij [∆Z ij + (5 + ε)V 4+ε Z ij ] + 2 i,j d ij ∂ ξ ′ (V 4 i Z ij ) + i,j c ij ∂ 2 ξ ′ 2 (V 4 i Z ij ) -(5 + ε)(∂ 2 ξ ′ 2 V 4+ε )φ -2(5 + ε)(∂ ξ ′ V 4+ε )Z. ( 3
W * ≤ C h * * .
Once again, these estimates show that W = L ε (g) + l,k a lk Z lk , where the a lk 's are defined by (3.15) and g by (3.20), is indeed the second derivative of φ with respect to ξ ′ . The other first and second derivatives of φ, involving the Λ i 's, may be treated in the same way. This concludes the proof of Lemma 3.1.

Lemma 3.2 The map (ξ ′ , Λ) → ψ(ξ ′ , Λ) is C 2
for the norm • * , and there exists C independent of ε and ξ', Λ satisfying (2.4) such that

ψ * ≤ Cε D (ξ ′ ,Λ) ψ * ≤ Cε D 2 (ξ ′ ,Λ) ψ * ≤ Cε. (3.21)
Proof. We recall that ψ is defined as

ψ = -L ε (R ε ) R ε = (V 1 + V 2 ) 5+ε -V 5 1 -V 5 2 .
The smoothness of (ξ ′ , Λ) → ψ(ξ ′ , Λ) follows from the smoothness of R ε with respect to (ξ ′ , Λ) and Lemma 3.1. Moreover, in view of Proposition 2.1 and Lemma 3.1, it is sufficient for establishing (3.21) to prove

R ε * * ≤ Cε D (ξ ′ ,Λ) R ε * * ≤ Cε D 2 (ξ ′ ,Λ) R ε * * ≤ Cε (3.22)
as ε goes to zero. Let us estimate R ε * * . In the regions |x -ξ ′ i | ≤ δ/ε, for small δ > 0, we write (for i = 1)

R ε = V 5+ε 1 + O(V 4+ε 1 V 2 ) -V 5 1 -V 5 2 ≤ CεV 5 1 | ln V 1 | + O(V 4+ε 1 V 2 ) + O(ε 5 ) whence |V -4 R ε | ≤ Cε.
In the exterior of these two regions, we see that |R ε | ≤ Cε 5 , and since

• * * is equivalent to V -4
• ∞ , the first inequality in (3.22) follows. Next, we write

∂ ξ ′ 1 R ε = (5 + ε)V 4+ε (∂ ξ ′ 1 V 1 ) -5V 4 1 (∂ ξ ′ 1 V 1 ). (3.23) Setting V 1 = V 1 -f 1 , we get ∂ ξ ′ 1 R ε = 5(V 4+ε -V 4 1 )(∂ ξ ′ 1 V 1 ) -5V 4+ε (∂ ξ ′ 1 f 1 ) + εV 4+ε (∂ ξ ′ 1 V 1 ).
Arguing as above, we have in the region |x -ξ

′ 1 | ≤ δ/ε, for small δ > 0 V 4+ε -V 4 1 = V 4+ε 1 -V 4 1 + O(V 3 1 V 2 ) = V 4+ε 1 -V 4 1 + O(V 3 1 f 1 + V 3 1 V 2 ) = εV 4 1 ln V 1 + O(ε 2 V 4 1 (ln V 1 ) 2 + V 3 1 f 1 + V 3 1 V 2 )
and, using the fact that

V 2 = O(ε) in the considered region, |∂ ξ ′ 1 V 1 | ≤ CV 2 1
and, through the maximum principle, f 1 = O(ε) in Ω ε , uniformly with respect to the parameters, provided that (2.4) is satisfied, we obtain

V -4 ∂ ξ ′ 1 R ε ≤ Cε.
In the region |x -ξ

′ 2 | ≤ δ/ε, for small δ > 0 V -4 |(V 4+ε -V 4 1 )(∂ ξ ′ 1 V 1 )| ≤ CV 2 1 ≤ Cε 2
whence again, in that region

V -4 |∂ ξ ′ 1 R ε | ≤ Cε.
In the exterior of the previous regions we derive from (3.23) that

|∂ ξ ′ 1 R ε | ≤ CV 6 ≤ Cε 6
from which we deduce

∂ ξ ′ R ε * * ≤ Cε. (3.24)
In order to estimate ∂ 2 ξ ′ Λ R ε * * , we write

∂ 2 ξ ′ 1 Λ1 R ε = (5 + ε)(4 + ε)(∂ ξ ′ 1 V 1 )(∂ Λ1 V 1 )(V 1 + V 2 ) 3+ε + (5 + ε)(V 1 + V 2 ) 4+ε (∂ 2 ξ ′ 1 Λ1 V 1 ) -5V 4 1 (∂ 2 ξ ′ 1 Λ1 V 1 ) -20(∂ ξ ′ 1 V 1 )(∂ Λ1 V 1 )V 3 1 . Away from ξ ′ 1 , noticing that ∂ Λ1 V 1 = O(V 1 ), ∂ 2 ξ ′ 1 Λ1 V 1 = O(V 2 1 ) in Ω ε (and the same estimates hold for V 1 instead of V 1 ) |∂ 2 ξ ′ 1 Λ1 R ε | ≤ C(ε 3 V 3 + ε 2 V 4 + ε 6 ) whence in this region, since V ≥ Cε in Ω ε V -4 |∂ 2 ξ ′ 1 Λ1 R ε | ≤ Cε 2 . Looking at the region |x -ξ ′ 1 | ≤ δ/ε, for small δ > 0, we write ∂ 2 ξ ′ 1 Λ1 R ε in the following way ∂ 2 ξ ′ 1 Λ1 R ε = 20 (∂ ξ ′ 1 V 1 )(∂ Λ1 V 1 )(V 1 + V 2 ) 3+ε -(∂ ξ ′ 1 V 1 )(∂ Λ1 V 1 )V 3 1 + 5 (V 1 + V 2 ) 4+ε (∂ 2 ξ ′ 1 Λ1 V 1 ) -V 4 1 (∂ 2 ξ ′ 1 Λ1 V 1 ) + ε(9 + ε)(∂ ξ ′ 1 V 1 )(∂ Λ1 V 1 )(V 1 + V 2 ) 3+ε + ε(V 1 + V 2 ) 4+ε (∂ 2 ξ ′ 1 Λ1 V 1 ) = I 1 + I 2 + I 3 + I 4 .
We have

V -4 (|I 3 | + |I 4 |) ≤ CεV 2 ≤ Cε. Writing V 1 = V 1 -f 1 ,
we have also

I 1 = (∂ ξ ′ 1 V 1 )(∂ Λ1 V 1 ) V 3+ε -V 3 1 + (∂ ξ ′ 1 f 1 )(∂ Λ1 f 1 ) -(∂ ξ ′ 1 V 1 )(∂ Λ1 f 1 ) -(∂ Λ1 V 1 )(∂ ξ ′ 1 f 1 ) V 3+ε and (∂ ξ ′ 1 V 1 )(∂ Λ1 V 1 ) V 3+ε -V 3 1 = O εV 4 |V 3 1 ln V 1 | + V 5 (f 1 + V 2 ) . Consequently V -4 |I 1 | ≤ Cε and similarly V -4 |I 2 | ≤ Cε.
The computations for estimating ∂ 2 ξ ′ 1 Λ2 R ε * * are simpler and we omit them. Finally, we obtain

∂ 2 ξ ′ Λ R ε * * ≤ Cε. (3.25) ∂ Λ R ε * * , ∂ 2 ξ ′ 2 R ε * * and ∂ 2 Λ 2 R ε *
D (ξ ′ ,Λ) φ * ≤ Cε 2 D 2 (ξ ′ ,Λ) φ * ≤ Cε 2 . (3.26)
Proof. We recall that φ given by Proposition 2.2 is defined through the relation

φ = -L ε (N ε (φ + ψ)) (3.27)
and

N ε ( φ) = N ε (ξ ′ , Λ, φ) = (V + φ) 5+ε + -V 5+ε -(5 + ε)V 4+ε φ. (3.28) Setting O ε δ = ξ ′ = (ξ ′ 1 , ξ ′ 2 ) ∈ Ω ε × Ω ε : |ξ ′ 1 -ξ ′ 2 | > δ/ε, d(ξ ′ i , ∂Ω) > δ/ε, i = 1, 2 and F = { φ ∈ H 1 0 (Ω ε ) : φ * ≤ ε}, we define the map B : O ε δ × ]δ, δ -1 [ 2 × F → L ∞ * (Ω ε ) as B(ξ ′ , Λ, φ) ≡ φ + L ε (N ε ( φ + ψ)).
We have

∂ φB(ξ ′ , Λ, φ)[θ] = θ + L ε (∂ φN ε )(ξ ′ , Λ, φ + ψ) θ ≡ θ + M (θ) = (I + M )(θ)
and, according to Proposition 2.1

M (θ) * ≤ C (∂ φN ε )(ξ ′ , Λ, φ + ψ)θ * * ≤ C V -3 (∂ φN ε )(ξ ′ , Λ, φ + ψ) ∞ θ * .
(3.28) yields

(∂ φN ε )(ξ ′ , Λ, φ) = (5 + ε)[(V + φ) 4+ε + -V 4+ε ] (3.29)
whence, using (3.21) and φ ∈ F

V -3 |(∂ φN ε )(ξ ′ , Λ, φ + ψ)| ≤ Cε (3.30)
and M (θ) * ≤ Cε θ * . As a consequence, for ε small enough, the linear operator

∂ φB(ξ ′ , Λ, φ) is invertible in L ∞ * (Ω ε ), with uniformly bounded inverse. As (ξ ′ , Λ, φ) → B(ξ ′ , Λ, φ) is C 2 , (ξ ′ , Λ) → φ(ξ ′ , Λ) defined by B(ξ ′ , Λ, φ(ξ ′ , Λ)) = 0 is also a C 2 -map.
Let us prove (3.26). Derivating B(ξ ′ , Λ, φ(ξ ′ , Λ)) = 0 with respect to ξ ′ , we find

∂ ξ ′ φ = [∂ φB] -1 [∂ ξ ′ B] = [∂ φB] -1 (∂ ξ ′ L ε ) N ε (φ + ψ) + L ε (∂ ξ ′ N ε )(φ + ψ) +L ε (∂ φN ε )(φ + ψ) ∂ ξ ′ ψ
whence, in view of Proposition 2.1

∂ ξ ′ φ * ≤ C N ε (φ + ψ) * * + (∂ ξ ′ N ε )(φ + ψ) * * + (∂ φN ε )(φ + ψ) ∂ ξ ′ ψ * * .
From (3.28), (3.21) and (2.14) we deduce that

N ε (φ + ψ) * * ≤ Cε 2 . (3.31)
We also compute, recalling that |∂ ξ ′ V | ≤ CV 2 , and assuming that φ * ≤ Cε

V -4 |(∂ ξ ′ N ε )(ξ ′ , Λ, φ)| ≤ CV -2 (V + φ) 4+ε + -V 4+ε -(4 + ε)V 3+ε φ ≤ C φ 2 * whence (∂ ξ ′ N ε )(ξ ′ , Λ, φ + ψ) * * ≤ Cε 2 .
Lastly, using (3.30) and (3.21)

(∂ φN ε )(φ + ψ) ∂ ξ ′ ψ * * ≤ C V -3 (∂ φN ε )(φ + ψ) ∞ ∂ ξ ′ ψ * ≤ Cε 2
and finally we obtain

∂ ξ ′ φ * ≤ Cε 2 .
The estimate for ∂ Λ φ is obtained in the same way. We turn now to the second derivatives. We concentrate our attention on ∂ 2 ξ ′ Λ φ, since the estimates for ∂ 2 Λ 2 φ and ∂ 2 ξ ′ 2 φ follow from identical arguments. Derivating B(ξ ′ , Λ, φ(ξ ′ , Λ)) = 0 with respect to ξ ′ and Λ, we find

∂ 2 ξ ′ Λ φ = [∂ φB] -1 ∂ 2 φ2 B.∂ ξ ′ φ.∂ Λ φ + ∂ 2 φξ ′ B.∂ Λ φ + ∂ 2 φΛ B.∂ ξ ′ φ + ∂ 2 ξ ′ Λ B (3.
∂ 2 φ2 B.∂ ξ ′ φ.∂ Λ φ = L ε ∂ 2 φ2 N ε .∂ ξ ′ φ.∂ Λ φ = L ε (5 + ε)(4 + ε)(V + φ + ψ) 3+ε + (∂ ξ ′ φ)(∂ Λ φ)
and in view of Proposition 2.2, we have to estimate

(V + φ + ψ) 3+ε + (∂ ξ ′ φ)(∂ Λ φ) * * ≤ V -2 (V + φ + ψ) 3+ε + ∞ ∂ ξ ′ φ * ∂ Λ φ * ≤ Cε 2
using (3.21) and Proposition 2.2. Estimating the other terms in the same way provides us with the announced result.

We are now able to prove the main result of this Section. From Proposition 2.3, we know that u = Û + ψ + φ defined through (2.16) (2.17) (2.18) is a solution to the initial problem if and only if (ξ, Λ) is a critical point of I ε = J ε ( Û + ψ + φ).

Proposition 3.1 We have the expansion

I ε (ξ, Λ) = C 1 + C 2 ε ln ε + C 3 ε + C 2 εΨ(ξ, Λ) + εθ ε (ξ, Λ) (3.33)
with θ ε , D ξ,Λ θ ε and D 2 ξ,Λ θ ε going to zero as ε goes to zero, uniformly with respect to ξ, Λ satisfying (2.4).

Proof. We claim that

I ε (ξ, Λ) = J ε ( Û ) + ερ ε (ξ, Λ) (3.34)
with ρ ε , D (ξ,Λ) ρ ε and D 2 (ξ,Λ) ρ ε going to zero as ε goes to zero, uniformly with respect to ξ, Λ satisfying (2.4). Then, expansion (3.33) is obtained estimating J ε ( Û ). Indeed, we have

ε 2ζ-1 J ε ( Û ) = ε 2ζ-1 J ε (ε 1 2 -ζ (U 1 + U 2 )) = J ε (U 1 + U 2 ) + 1 -ε ε 2 6 + ε Ω (U 1 + U 2 ) 6+ε = J ε (U 1 + U 2 ) + 1 6 - ε 2 ln ε + o(ε) 2 R 3 U 6+ε + o(ε ln |ε|) = J ε (U 1 + U 2 ) - √ 3 π 2 8 ε ln ε + o(ε) whence J ε ( Û ) = 1 + 1 4 ε ln ε + o(ε) J ε (U 1 + U 2 ) - √ 3 π 2 8 ε ln ε + o(ε)
and the desired expansion for J ε ( Û ) using Lemma 2.1 and (2.3). The expansions for the derivatives of J ε ( Û ) are obtained in the same way.

We turn now to the proof of claim (3.34). Validity of such an expansion for I ε and its first derivatives has already been proved in [START_REF] Del Pino | Two-bubble solutions in the super-critical Bahri-Coron's problem[END_REF] (Proposition 6.1). Moreover, it can easily be found again from the arguments used below, concerning the second derivatives. Actually, we shall concentrate our attention on the second derivative with respect to ξ, Λ, for the second derivatives with respect to Λ 2 and ξ 2 may be treated exactly in the same way. The proof is composed of two steps : we show that

∂ 2 ξΛ [I(ξ, Λ) -J ε ( Û + ψ)] = o(ε) (3.35)
and

∂ 2 ξΛ [J ε ( Û + ψ) -J ε ( Û )] = o(ε). (3.36)
Let us prove (3.35). From a Taylor expansion and the fact that J ′ ε ( Û + ψ + φ)[φ] = 0, we have

I(ξ,Λ) -J ε ( Û + ψ) = J ε ( Û + ψ + φ) -J ε ( Û + ψ) = 1 0 D 2 J ε ( Û + ψ + t φ)[ φ, φ]tdt = ε 1-2ζ 1 0 Ωε N ε (φ + ψ)φ + Ωε (5 + ε) V 4+ε -(V + ψ + tφ) 4+ε + φ 2 tdt. (3.37) We recall that ζ = (2 + 1 2 ε) -1 < 1/2.
Differentiating twice with respect to ξ, Λ, we find

∂ 2 ξΛ [I(ξ, Λ) -J ε ( Û + ψ)] = ε -2ζ 1 0 Ωε ∂ 2 ξ ′ Λ [N ε (φ + ψ)φ] - Ωε (5 + ε)∂ 2 ξ ′ Λ (V + ψ + tφ) 4+ε + -V 4+ε φ 2 tdt (3.38) with ξ ′ i = ξ i /ε. Let us estimate the last integral. Setting ϕ = ψ + tφ H = H ε (ξ ′ , Λ) = (V + ϕ) 4+ε + -V 4+ε
we have

∂ 2 ξ ′ Λ (Hφ 2 ) =2(∂ ξ ′ φ)(∂ Λ φ)H + 2φ(∂ 2 ξ ′ Λ φ)H + 2φ(∂ Λ φ)(∂ ξ ′ H) + 2φ(∂ Λ H)(∂ ξ ′ φ) + φ 2 (∂ 2 ξ ′ Λ H) with ∂ ξ ′ H = (4 + ε)[(V + ϕ) 3+ε + -V 3+ε ](∂ ξ ′ V ) + (4 + ε)(V + ϕ) 3+ε + (∂ ξ ′ ϕ)
and

∂ 2 ξ ′ Λ H = (4 + ε)(3 + ε)[(V + ϕ) 2+ε + -V 2+ε ](∂ ξ ′ V )(∂ Λ V ) + (4 + ε)[(V + ϕ) 3+ε + -V 3+ε ](∂ 2 ξ ′ Λ V ) + (4 + ε)(3 + ε)(V + ϕ) 2+ε + (∂ Λ ϕ)[(∂ ξ ′ V ) + (∂ ξ ′ ϕ)] + (4 + ε)(3 + ε)(V + ϕ) 2+ε + (∂ ξ ′ ϕ)(∂ Λ V ) + (4 + ε)(V + ϕ) 3+ε + (∂ 2 ξ ′ Λ ϕ).
One the one hand, we deduce from the definition of V

|∂ Λ V | ≤ CV |∂ ξ ′ V | ≤ CV 2 |∂ 2 ξ ′ Λ | ≤ CV 2 (3.39)
uniformly in Ω ε , with C independent of ε and ξ ′ , Λ satisfying (2.4). One the other hand, (3.21) and (3.26) provide us with estimates of ψ and φ in norm • * . Lastly, we notice that for any γ > 3

Ωε V γ = O(1) (3.40)
as ε goes to zero, uniformly with respect to ξ ′ , Λ satisfying (2.4). Using these informations, straightforward computations yield

ε -2ζ Ωε ∂ 2 ξ ′ Λ (Hφ 2 ) = o(ε).
The same arguments applied to the first integral in (3.38) lead to a similar estimate, establishing (3.35). We turn now to the proof of (3.36). A Taylor expansion gives

J ε ( Û + ψ) -J ε ( Û ) =ε 1-2ζ I ε (V + ψ) -I ε (V ) =ε 1-2ζ DI ε (V )[ψ] + 1 0 (1 -t)D 2 I ε (V + tψ)[ψ, ψ] . As DI ε (V )[ψ] = -Ωε (∆V + V 5+ε )ψ and ∆V = ∆(V 1 + V 2 ) = -V 5 1 -V 5 2 = R ε -V 5+ε , we obtain DI ε (V )[ψ] = - Ωε R ε ψ.
On the other hand

D 2 I ε (V + tψ)[ψ, ψ] = Ωε |∇ψ| 2 -(5 + ε) Ωε (V + tψ) 4+ε + ψ 2 .
Then, integration by parts and ψ = -L ε (R ε ) yield

D 2 I ε (V + tψ)[ψ, ψ] = Ωε R ε ψ -(5 + ε) Ωε (V + tψ) 4+ε + -V 4+ε ψ 2 .
Consequently

J ε ( Û + ψ) -J ε ( Û ) = ε 1-2ζ - 1 2 Ωε R ε ψ -(5 + ε) 1 0 (1 -t)( Ωε (V + tψ) 4+ε + -V 4+ε ψ 2 )dt
and

∂ 2 ξΛ J ε ( Û + ψ) -J ε ( Û ) = ε -2ζ - 1 2 ∂ 2 ξ ′ Λ Ωε R ε ψ -(5 + ε) 1 0 (1 -t)( Ωε ∂ 2 ξ ′ Λ (V + tψ) 4+ε + -V 4+ε ψ 2 )dt .
We first consider the last integral. Denoting by

K = K ε (ξ ′ , Λ)(x) = (V + tψ) 4+ε + -V 4+ε
we have

∂ 2 ξ ′ Λ (Kψ 2 ) =2(∂ ξ ′ ψ)(∂ Λ ψ)K + 2ψ(∂ 2 ξ ′ Λ ψ)K + 2ψ(∂ Λ ψ)(∂ ξ ′ K) + 2ψ(∂ Λ K)(∂ ξ ′ ψ) + ψ 2 (∂ 2 ξ ′ Λ K) with ∂ ξ ′ K = (4 + ε)[(V + tψ) 3+ε + -V 3+ε ](∂ ξ ′ V ) + (4 + ε)t(V + tψ) 3+ε + (∂ ξ ′ ψ).
A similar expression holds for ∂ Λ K, and

∂ 2 ξ ′ Λ K =(4 + ε)(3 + ε)[(V + tψ) 2+ε + -V 2+ε ](∂ ξ ′ V )(∂ Λ V ) + (4 + ε)(3 + ε)(V + tψ) 2+ε + t(∂ Λ ψ)(∂ ξ ′ V ) + (4 + ε)[(V + tψ) 3+ε + -V 3+ε ](∂ 2 ξ ′ Λ V ) + (4 + ε)(3 + ε)t(V + tψ) 2+ε + (∂ ξ ′ ψ)(∂ Λ V ) + (4 + ε)(3 + ε)t 2 (V + tψ) 2+ε + (∂ ξ ′ ψ)(∂ Λ ψ) + (4 + ε)t(V + tψ) 3+ε + (∂ 2 ξ ′ Λ ψ).
Once again, the estimate

ε -2ζ Ωε ψ 2 (∂ 2 ξ ′ Λ K) = o(ε)
follows directly from (3.21), (3.26), (3.39) and (3.40).

In order to complete the proof of (3.36), it only remains to estimate the quantity

ε -2ζ ∂ 2 ξ ′ Λ Ωε R ε ψ = ε -2ζ Ωε R ε (∂ 2 ξ ′ Λ ψ) + (∂ Λ ψ)(∂ ξ ′ R ε ) + (∂ ξ ′ ψ)(∂ Λ R ε ) + ψ(∂ 2 ξ ′ Λ R ε ). (3.41)
Among the four integral terms which occur on the right hand side of (3.41), let us consider, for example, the first one

I ε = ε -2ζ Ωε R ε (∂ 2 ξ ′ Λ ψ).
Let R > 0, and

Ω ′ ε = Ω ε \ B(ξ ′ 1 , R) ∪ B(ξ ′ 2 , R) . We have ε -2ζ Ω ′ ε R ε (∂ 2 ξ ′ Λ ψ) = O ε -1 ∂ 2 ξ ′ Λ ψ * R ε * * Ω ′ ε V 5 = O ε Ω ′ ε V 5 using (3.22
) and (3.21), and

Ω ′ ε V 5 ≤ C +∞ R r 2 (1 + r 2 ) 2 dr ≤ C ′ R with C ′ a constant independent of R. Therefore ε -2ζ Ω ′ ε R ε (∂ 2 ξ ′ Λ ψ) = O ε R . (3.42)
On the other hand, according to the definition (2.8) of R ε there exists a constant c i such that

ε -1 R ε (ξ ′ i + x) -→ V 5 0,Λ * i ln V 0,Λ * i + c i V 4 0,Λ * i uniformly on B(ξ ′ i , R) (with Λ * i = (c N Λ 2 i ) 1 N -2 ). We recall that ψ = -L ε (R ε ) is the unique solution of    ∆ψ + (5 + ε)V 4+ε ψ = -R ε + i,j c ij V 4 i Z ij in Ω ε ψ = 0 on ∂Ω ε < V 4 i Z ij , ψ > = 0 for all i, j for some numbers c ij , with |c ij | ≤ C R ε * * ≤ C ′ ε.
Then, proceeding as in [START_REF] Wei | Asymptotic behavior of least energy solutions to a semilinear Dirichlet problem near the critical Sobolev exponent[END_REF] (proof of Proposition 3.3), we obtain that, up to a subsequence, ε -1 ψ converges uniformly in B(ξ

′ i , R), as ε goes to 0, to a solution Φ i of      ∆Φ + 5V 4 0,Λ * i Φ = -V 5 0,Λ * i ln V 0,Λ * i -c i V 4 0,Λ * i + i,j d ij V 4 0,Λ * i Z ij in R 3 < V 4 0,Λ * i Z ij , Φ > = 0 1 ≤ j ≤ 4 |Φ| ≤ CV 0,Λ * i (3.43)
for some number d ij . Multiplying the equation by Z ij , 1 ≤ j ≤ 3, and integrating in R 3 , the orthogonality relations satisfied by Φ and oddness of Z ij with respect to the variable (x -ξ ′ i ) j yield d ij = 0, 1 ≤ j ≤ 3. On the other hand, the kernel of the operator L i = ∆ + 5V 4 0,Λ * i in W 2,r (R 3 ), 3 < r < +∞, is spanned by the Z ij 's, 1 ≤ j ≤ 4 (Lemma 2.3 in [START_REF] Wei | Asymptotic behavior of least energy solutions to a semilinear Dirichlet problem near the critical Sobolev exponent[END_REF]). |Φ i | ≤ CV 0,Λ * i implies that Φ i ∈ W 2,r (R 3 ) for r large enough. Then, if Φ i and Φ ′ i are two solutions of (3.43), Φ i -Φ ′ i ∈ Ker L i , and the orthogonality relations < V

4 0,Λ * i Z ij , Φ i -Φ ′ i >= 0, 1 ≤ j ≤ 4, imply that Φ i = Φ ′ i .
Since the solution Φ i of (3.43) is unique, it is radial (otherwise, a rotation of Φ i would provide us with another solution). Proceeding in the same way, we would prove that, up to a subsequence

ε -1 (∂ 2 ξ ′ Λ ψ) -→ ∂ 2 ξ ′ Λ Φ i uniformly B(ξ ′ i , R). ∂ 2 ξ ′ Λ Φ i being odd in the variables (x -ξ ′ i
) j , we finally see that

ε -1 B(ξ ′ i ,R) R ε (∂ 2 ξ ′ Λ ψ) dx = o(ε) as ε goes to 0.
Since R may be chosen as large as desired, this result, together with (3.42), shows that

I ε = o(ε).
The other terms in (3.41) may be treated in the same way, completing the proof of (3.36).

4 Proof of Theorem 1.1

According to the statement of Theorem 1.1, we assume in this section that there exists a and b, a < b < 0, such that H * (ϕ b , ϕ a ) is nontrivial. In order to prove Theorem 1.1, we have to show that for ε small enough, (P ε ) has a solution, which blows up at two points ξ 1 , ξ 2 as ε goes to zero, with a ≤ ϕ(ξ 1 , ξ 2 ) ≤ b and ∇ϕ(ξ 1 , ξ 2 ) = 0.

In view of Proposition 2.3 and Proposition 3.1, we have to prove the existence of a critical point of

I ε (ξ, Λ) = C 1 + C 3 ε + C 2 εΨ(ξ, Λ) + εθ ε (ξ, Λ) (4.1)
with θ ε = o(1), D (ξ,Λ) θ ε = o(1), D 2 (ξ,Λ) θ ε = o(1) as ε goes to 0, uniformly with respect to (ξ, Λ) ∈ O δ (Ω)×]δ, δ -1 [ 2 , and Ψ is given by (2.2).

First, we remark that for any ξ ∈ Ω 2 , such that ϕ(ξ) < 0, Λ -→ Ψ(ξ, Λ) has a unique critical point Λ(ξ) in (R * + ) 2 , such that

Λ2 i (ξ) = - H(ξ j , ξ j ) 1/2 H(ξ i , ξ i ) 1/2 1 ϕ(ξ) i, j = 1, 2 i = j. (4.2)
Note that ξ ∈ O δ1 and ϕ(ξ) < -δ 2 , with δ 1 , δ 2 strictly positive constants, imply the existence of

δ 3 > 0 such that δ 3 < Λi (ξ) < δ -1 3 , i = 1, 2. Note also that (∂ 2 Λ 2 1 Ψ) (∂ 2 Λ1Λ2 Ψ) (∂ 2 Λ1Λ2 Ψ) (∂ 2 Λ 2 2 Ψ) ξ, Λ(ξ) = 4H(ξ 1 , ξ 1 ) 1/2 H(ξ 2 , ξ 2 ) 1/2 ϕ(ξ) < -δ 4 < 0
for some δ 4 > 0. Therefore, the implicit functions theorem provides us, for ξ ∈ O δ1 , ϕ(ξ) < -δ 2 and ε small enough, with the existence of Λ(ξ) close to Λ(ξ) in C 1 -norm as ε goes to 0, such that

∂ Λ I ε ξ, Λ(ξ) = 0.
Then, in view of (4.1), finding a critical point of (ξ, Λ) -→ I ε (ξ, Λ) reduces to finding a critical point of ξ -→ Ĩε (ξ), with Ĩε (ξ) = Ψ ξ, Λ(ξ) + θε (ξ) Arguing by contradiction, we assume that for any δ 1 > 0, Ĩε has no critical point in the subset ξ ∈ O δ1 (Ω), a < ϕ(ξ) < b. We note that ϕ has isolated critical values. This property follows from the analyticity of ϕ on Ω 2 \ ∆, and the fact that (ξ n ) being a sequence of Ω 2 \ ∆ such that ξ n → ∆, ϕ ′ (ξ n ) = 0, ϕ(ξ n ) cannot be bounded (see the proof of Corollary 1.1 below). Then, assuming that b is not a critical value of ϕ, ϕ has no critical value c in (a, a + η] ∪ (b -η, b] for some η > 0 sufficiently small. Consequently, ϕ b retracts by deformation onto ϕ b-η , ϕ a+η retracts by deformation onto ϕ a , and H * ϕ b-η , ϕ a+η = 0 (on the boundary of Ω 2 , -∇ϕ points inward, see Lemma 4.1 below).

We are going to use the gradient of Ĩε to build a continuous deformation of ϕ b-η onto ϕ a+η , whence a contradiction. As Ĩε is not defined on whole ϕ b , we shall use the gradient of ϕ in the complementary regions.

We notice that a < 0 and δ 0 > 0 being given, ϕ(ξ) > a and d(ξ i , ∂Ω) > δ 0 , i = 1, 2, imply that |ξ 1 -ξ 2 | > δ ′ 0 , with δ ′ 0 a strictly positive constant. For δ 0 > 0 small enough and d(x, ∂Ω) ≤ 2δ 0 , we denote by n x the outward normal to ∂Ω at x ′ , with |x -x ′ | = Min y∈∂Ω |x -y|. We have the following lemma:

Lemma 4.1 Let a < b < 0, ξ ∈ Ω 2 such that a ≤ ϕ(ξ) ≤ b and d(ξ i , ∂Ω) = Min j=1,2 d(ξ j , ∂Ω) ≤ 2δ 0 .
Then, for δ 0 > 0 small enough, we have

∂ ξi ϕ(ξ).n ξi > 0.
Before proving this lemma, let us complete the proof of Theorem 1.1.

Proof of Theorem 1.1 completed. We consider

ζ ∈ C ∞ (Ω × Ω), 0 ≤ ζ ≤ 1, such that ζ(ξ) = 1 if d(ξ i , ∂Ω) ≥ 2δ 0 i = 1, 2 ζ(ξ) = 0 if d(ξ i , ∂Ω) ≤ δ 0 i = 1 or 2.
We set

F = ζ∇ Ĩε + (1 -ζ)∇ϕ
and we consider the differential flow

d dt ξ(t) = -F ξ(t) , ξ(0) = ξ 0 , a ≤ ϕ(ξ 0 ) < -δ 2 . (4.4)
According to the assumption on Ĩε , (4.3) and Lemma 4.1, F (ξ) does not vanish for a ≤ ϕ(ξ 0 ) < -δ 2 . On one hand, if d(ξ i , ∂Ω) ≤ 2δ 0 , i = 1 or 2, (4.3) yields

d dt ϕ(ξ) = 2ζ(ξ) |∇ϕ(ξ)| 2 ϕ(ξ) -1 -ζ(ξ) |∇ϕ(ξ)| 2 + o ε (1) < η 0 < 0
provided that ε is small enough, as Lemma 4.1 shows. Lemma 4.1 also shows that the orbits do not meet the boundary of Ω 2 . On the other hand, if d(ξ i , ∂Ω) ≥ 2δ 0 , i = 1, 2, and a < ϕ(ξ) ≤ b, we have

d dt Ĩε (ξ) = -|∇ Ĩε (ξ)| 2 < 0. ( 4.5)
From (4.3) we deduce that if ξ 0 ∈ ϕ b-η , the orbit ξ(t) with ξ 0 as initial datum satisfies ϕ ξ(t) ∈ ϕ b for any t, provided that ε is small enough. Therefore, (4.5) is valid along the orbit, and (4.3) proves that for ε sufficiently small, there is some t such that ϕ ξ(t) = a + η. Finally, composing the flow with a retraction of ϕ b onto ϕ b-η , we obtain a continuous deformation of ϕ b-η onto ϕ a+η , a contradiction with H * (ϕ b-η , ϕ a+η ) = 0.

Actually, it is to be noticed that the previous arguments provide us, for ε small enough, with a nontrivial solution u ε of -∆u = u 5+ε + in Ω, u = 0 on ∂Ω. Then, the strong maximum principle ensures that u ε > 0 in Ω. The concentration, as ε goes to zero, up to a subsequence, of u ε at two points ξ 1 , ξ 2 such that a < ϕ(ξ 1 , ξ 2 ) < b and ∇ϕ(ξ 1 , ξ 2 ) = 0 is a consequence of the construction of u ε and (4.3).

Proof of Lemma 4.1. We prove the result for any dimension N ≥ 3. From [START_REF] Rey | A multiplicity result for a variational problem with lack of compactness[END_REF], we know the uniform expansion with respect to y ∈ Ω This expansion shows that d x , d y and |x -y| are of the same order as these quantities go to 0. Then [START_REF] Rey | A multiplicity result for a variational problem with lack of compactness[END_REF] provides us with the expansions Proof of Corollary 1.1. We have to prove that for a small enough, ϕ a retracts by deformation on ∆. We are going to show that ∇ϕ(x, y).(y -x, x -y) < 0 when ϕ(x, y) < a, with a small enough. (4.8) According to the definition of ϕ, we have : ϕ(x, y) = H(x, x) 1/2 H(y, y)

1 W

 1 .20) Estimates (3.3), (3.11), (3.16), (3.17) and (3.18) prove that g * * ≤ C h * * whence, in view of Proposition 2.* ≤ C h * * .

  32) and we have to estimate each term of the right hand side in norm • * * . Let us consider the first term. According to the definition of B and (3.28)

  and θε = o(1), ∇ θε = o(1) as ε goes to 0, uniformly with respect toξ ∈ O δ1 (Ω), ϕ(ξ) < -δ 2 . Λ(ξ) being C 1 -close to Λ(ξ), (4.2) yields Ĩε (ξ) = -2 ln -ϕ(ξ) -1 + θε (ξ) (4.3)with θε = o(1), ∇ θε = o(1) as ε goes to 0, uniformly with respect to ξ ∈ O δ1 (Ω), ϕ(ξ) < -δ 2 .

H 6 ) 1 2 1 d

 611 (x, y) = b N |x -y + 2d x n x | N -2 + o 1 d N -2 x as d x = d(x, ∂Ω) goes to zero. In particular H(x, x) = b N 2 N -2 d N -2Assume that a ≤ ϕ(x, y) ≤ b and d x goes to 0. Then |x -y| has also to go to 0, and we have the expansionϕ(x, y) = b N y| N -2 + 1 |x -y + 2d x n x | N -2 + o

d y -d x |x -y| N + d x + d y |x -y| 2 + 4d x d y N 2 + o 1 d N - 1 x.

 2211 y) = -(N -1)b N d y -d x |x -y| N + d x + d y |x -y| 2 + 4d x d yThis last quantity is clearly strictly positive as d x ≤ d y and d x goes to 0.

(4. 11 ) 1 d N - 2 x 2 N 1 d N - 1 x 1 (

 11122111 If x and y remain far from the boundary, say d(x, ∂Ω), d(y, ∂Ω) > δ > 0, H and its derivatives remain bounded, (4.9) implies that bN |x-y| N -2 > -a, and (4.11) yields ∇ϕ(x, y).(y -x, x -y) < 2(N -2) a + C δ < 0 for a small enough.If x and y approach the boundary, (4.9) implies that |x -y| goes to 0, (4.7) that d x and d y are of the same order and |x -y|/d x is bounded. Moreover,[START_REF] Rey | A multiplicity result for a variational problem with lack of compactness[END_REF] provides us with the estimate∂H ∂x (x, y) = -(N -2)b N x -y -2 n x .(x -y) n x -2d x n x |x -y + 2d x n x | N + o as d x → 0and a similar expansion holds for ∂H ∂y (x, y). Therefore, taking account of (4.6) and (4.10), we find∂ϕ ∂x (x, y) = (N -2)b N n x |x -y| N + x -y -2n x .(x -y)n x -2d x n x |x -y + 2d x n x | N + o as d x → 0.The same expression holds for the derivative of ϕ with respect to y, interchanging the roles of x and y. Note that we can writey = x -(d y -d x ) n x + τ + o(d x ) with τ.n x = 0, |τ | = O(|x -y|). Noticing that n y = n x + o(1), we obtain ∇ϕ(x, y).(y -x, x -y) = 2(N -2)b N -1 |x -y| N -2 -(d x -d y ) 2 -d y )2 -|x -y| 2 (d x + d y ) 2 + |τ | 2 d x goes to 0, |x -y|/d x bounded, since dx+dy) N .

  * may be estimated in the same way, concluding the proof of the lemma. , Λ) provided by Proposition 2.2 is C 2 for the norm • * , and there exists C independent of ε and ξ', Λ satisfying (2.4) such that

	Lemma 3.3 The map (ξ

′ , Λ) → φ(ξ ′


Jawhar Elgueder
email: jawhar.el_gueder@utt.fr

Florent Cochennec
email: florent.cochennec@utt.fr

Lionel Roucoules
email: lionel.roucoules@ensam.fr

Emmanuelle Rouhaud
email: emmanuelle.rouhaud@utt.fr

Product-process interface for effective product design and manufacturing in a DFM approach

Keywords: virtual engineering, manufacturing process selection, manufacturing simulation

In order to tackle a continuous improvement of virtual engineering, product modelling has to integrate always more knowledge that refer to every decision taken during the product development process. Those decisions have to be related to the assessment of the whole product lifecycle. This paper particularly addresses the domain of product's industrialisation that aims at selecting the manufacturing processes. This selection must currently be done as soon as possible and has to be strongly linked with product definition and CAD 1 modelling. This paper presents first some new results concerning a product-process interface to integrate manufacturing information in the product model and how it leads the definition of the CAD model. Secondly this interface, that also manages specific information coming from the manufacturing process (tolerances, stresses gradient…), is used to improve the whole manufacturing process plan simulation. This process plan has, indeed, to track every material transformation issued from each manufacturing operation.

Key words: product-process interface / DFM /

On s'intéresse tout particulièrement dans cette communication aux résultats relatifs au DFM (Design For Manufacturing) et aux choix des procédés de fabrication. L'intégration des données issues du choix des procédés (exemple de donnée : tolérances, gradients de contraintes…) et le lien avec le procédé de fabrication sont formalisés grâce à une interface produit-procédés. Cette interface fera que les contraintes relatives aux procédés de fabrication seront intégrées au plus tôt dans le processus de conception tout en gardant une émergence progressive de la solution du produit.

Mots clés : conception pour la fabrication / intégration produit-procédés / choix procédés de fabrication / ingénierie simultanée

Introduction

For almost 30 years CAD systems have been developed and improved to currently reach very powerful features to support product's forms modelling. Nevertheless they are actually presented and used as one of the central systems that make the design process a geometric centric approach. This approach has shown its great interest in industry to tackle the problem of digitizing hand-done drawing or to improve the CAD-CAM 2 links and to enhance the process plan activity. Nowadays, the CAD model also finds an interest to improve the digital mock-up used during a decision making process for instance. However current CAD systems are not able to manage all the information related to the product definition. This information as mentioned in [START_REF] Krause | Product Modelling[END_REF] has to be related to the whole lifecycle (from requirement specifications to dismantling information). The product, and its CAD model, is then defined, as far as possible, taken into account "X" constraints as assumed in a DFX 3 approach. One of the domains that have to be integrated in design is manufacturing (i.e. DFM). That means that manufacturing activities have to be assessed concurrently to the product development and the CAD modelling activity.

Once the CAD done, manufacturing processes can be detailed. As far manufacturing simulation is concerned, CAD model is seen as input and software tools have to simulate the behaviour of the materials flow during each manufacturing operation (ex : forging, casting, machining….).

The main issue of that design approach remains in the fact that:

-The CAD model is almost never defined taking into account manufacturing information.

-The manufacturing simulations do not take into account the history of the whole process planning. The input CAD is very often seen as virgin of any previous manufacturing operation.

This paper gives some results to manage the whole manufacturing process plan information and to integrate those data (i.e. knowledge synthesis approach) in the CAD model that is, then, constructed with respect to a more adequate DFM approach.

The second part introduces the design approach and the main concepts used to breakdown the product and its CAD model. It also gives the product- 2 Computer Aided Manufacturing 3 Design For X: design approach able to take into account activity information (e.g. manufacturing, assembly…) during the product development.

process interface concepts used to tackle the information synthesis.

The third part gives some ideas and results to manage the manufacturing information of the global process in order to use it during the whole manufacturing simulation process.

Finally the conclusion and the perspectives for further work are enounced.

Objectives, context and concepts of the DFM approach

The fundaments of authors' DFM approach are the integration of manufacturing information4 constraints and data at the earliest stage of design. The developed model of integration (i.e. productprocess interface model) is based on the research work done by Roucoules and Skander [START_REF] Roucoules | Manufacturing process selection and integration in product design. Analysis and synthesis approaches[END_REF]. They showed that taking manufacturing information into account as soon as possible in the design process is of great interest for manufacturing process selection. That indeed supports the emergence of product geometry [START_REF] Roucoules | Knowledge intensive approach towards multiple product modelling and geometry emergence to foster cooperative design[END_REF] and goes towards a limited number of iterations between design and manufacturing decisions; the term of "right the first time" is used for such approaches versus the approaches of "do until right".

Considering that the manufacturing domain is extended to other product lifecycle phases (e.g. assembly, recycling, dismantling, etc.), the assumption is that the design process should then be centred on multiple-views product modelling and expert analyses instead of being CAD centric. One of the main issues of that CAD centric approach remains in the unique product breakdown that does not reflect the design intends of every expert designers involved in the design group. Figure 1 shows the features breakdown used to obtain the CAD model. Obviously, this breakdown does not represent what should or could be the real manufacturing process plan. It does not have any sense for the engineers in charge of the manufacturing activities. The general context of authors' research work lies on the multiple views product breakdown concepts proposed in [START_REF] Tichkiewitch | Specifications on integrated design methodology using a multi-view product model[END_REF]. As presented in [START_REF] Roucoules | CoDE: a Cooperative Design Environment. A new generation of CAD systems[END_REF], the first design step consists in the definition of functional surfaces to achieve design requirements. These functional surfaces can emerge from specific "Function-Structure" analysis that describes every product specifications as energetic flows in the product structure. One example based on FBS [START_REF] Gero | Design prototypes: a knowledge representation schema for design[END_REF] and bondgraph concepts [START_REF] Thoma | Introduction to Bondgraphs and their Application[END_REF] is given in [START_REF] Meyer | Case study of a MEMS switch supported by a FBS and DFM framework[END_REF]. The second steps aims at adding (i.e. integrating) lifecycle information to this first product description. This approach is often called "design by least commitment".

Skander et al. [START_REF] Skander | Méthode et modèle DFM pour le choix des procédés et l'intégration des contraintes 8[END_REF] treat the activity of "manufacturing processes selection" (i.e. manufacturing expertise on figure 2) and then proposed to apply the Design For Manufacturing approach as soon as the first functional surface is defined. They thus propose a specific product model based on an adaptation of the skin and skeleton concepts [START_REF] Muh-Cerng Wu | A skeleton approach for modelling assembly products[END_REF][START_REF] Tollenaere | A part description model for the preliminary design, Advanced CAD/CAM Systems -State-of-the-art and future trends in feature technology[END_REF] to allow the "X" constraints integration (see figure 2), and specifically the manufacturing constraints integration [14 and 11]. This specific product model can be seen as an "interface model" used to specify, vulgarize the product information issued from different activities (i.e. expertises) (e.g. "technological components selection" or "manufacturing processes selection"). These interface models (e.g. product-process interface) are translated into a collaborative multiple views definition of the product.

The central "product modelling" concepts, and specifically the "relation" concept, are then used to link and/or propagate data from different expertises.

Product Model Interface model

Objectives of the DFM approach

Once the first functional surfaces are specified, the design actor in charge of the industrialisation should wonder about which manufacturing processes would be eligible for generating these surfaces. Many industrial and research studies have been done to characterise product-process relationships (e.g. [START_REF] Boothroyd | Product design for manufacture and assembly[END_REF]). Skander et al. proposed to translate these product-process relationships in specific skin and skeleton attributes in order to analyse the correlation between product specifications and the process-resulting product characteristics. Then, the translation of the energetic flows definition in specific skin and skeleton attributes will lead to the creation of a technological interface model (see figure 2) and the translation of the product-process relationships in a same way will lead to the creation of manufacturing interface model corresponding to the product alternatives resulting from the analysis of all available manufacturing processes capabilities.

Checking the consistency of the data contained in these two interface models will then imply the acceptance of some product-process alternatives and the reject of some others. The acceptance criteria are based on the fact that the data obtained during the product-process constraints identification must be sufficiently pertinent to define the process capabilities.

The DFM activity is detailed in figure 3. The first task (A1) aims at analysing the requirements specification using energetic flows and specific technological interface model as presented on figure 2. Once this task achieved, designers have to find product-process alternatives in which the manufacturing constraints are integrated (A2). The DFM output is then a list of products with respect to available manufacturing plans. The selection of the final product-process alternatives is not treated in the presented approach. Indeed, such a choice is led by economic criterions and depends on many external factors as the factory production capabilities, the lead-time of the production… The authors are nevertheless convinced that the proposition of product-process alternatives in which manufacturing constraints have been integrated brings solid arguments to the process selection activity.

A1

Product-process interface modelling

As mentioned above the integration of manufacturing information is based on a specific product-process interface. That model comes from the assumption that every manufacturing operation is based on a material flow. Those flows (cf. Figure 4) are then defined with:

-Sections defining the initial and final surfaces through which the material is going (i.e. transversal surfaces).

-A trajectory on which the material is formed.

-An envelope surface which is generated. Those surfaces are in the added and removed processes categories equal to the envelope surface.

Beyond very good results presented in [START_REF] Skander | Design and manufacturing interface modelling for manufacturing processes selection and knowledge synthesis in design[END_REF] that concerns the current results of that approach for nominal aspects, figure 8 gives the novelties of that paper. The new results concern the capabilities of that product-process interface:

-To manage product tolerances coming from manufacturing operations. Each level of tolerancing features (dimensional tolerances, form tolerances and roughness) is concerned. Figure 8 shows how those features are integrated in the product-process interface (i.e. manufacturing skeleton) characteristics.

-To manage material heterogeneity coming from manufacturing operations. It is also obvious that material flows (cf. above assumption) generate some gradients inside the manufactured product. Those gradients (called in the following "heterogeneities") can, for instance, come from (cf. Figure 8):

o Thermal phenomena in the skeleton's sections that come from a cooling phase which is not always homogeneous during casting operations.

o Mechanical stresses gradient on the skeleton's trajectory coming from high deformation in forging operations.

Another example of that heterogeneity (i.e. residual stresses) is given on the following section. It is based on peen forming process. More details can nevertheless be found in [START_REF] Cochennec | Mechanical Analysis to identify knowledge for a DFM approach. Application to Shot Peen-forming process[END_REF].

Application of product-process interface to the peen-forming process

The peen-forming process is a cold-work forming process mainly used in the aeronautical and aerospace industry to form large metallic panels (cf. Figure 5). The concept is to project balls on the part in order to create some local plastic deformation. The global elastic equilibrium then generates geometrical deformation.

Figure 5. Illustration of the peen forming process

It presents many advantages for this kind of application: none spring-back problems are encountered; the parts can be formed at ambient temperature, the process induces little metallurgical modifications and none dilatational dispersions; the residual stresses states are partially mastered; a good reproducibility can be achieved [START_REF] Ramati | Single piece wing skin utilization via advanced peen forming technologies[END_REF]. Being used for more than fifty years, this process is still under industrial and research development. Many analytical and numerical models are proposed in the literature for predicting the geometrical distortions induced [START_REF] Guagliano | Relating Almen intensity to residual stresses induced by shot-peening: a numerical approach[END_REF], [START_REF] Grasty | Shot peen forming sheet metal: finite element prediction of deformed shape[END_REF], [START_REF] Homer | Aircaft wing skin contouring by shot-peening[END_REF] and [START_REF] Han | Combined finite/discrete element and explicit/implicit simulations of peen forming process[END_REF]. These models are based on the numerical introduction of equivalent plastic strains as a boundary condition of a finite element problem, which implies that the plastic strain fields induced by the treatment must be known. Some models have been proposed to predict the residual stress fields induced by known peening parameters [START_REF] Rouhaud | Handbook on Residual Stress[END_REF] but these models are still to be developed in order to complete the state of knowledge of the process. These studies are indeed depending on the treated materials and on the peening parameters retained for the treatment. The actual state of knowledge makes thus difficult to plan the forming phases and trials and tests are still a needed way to achieve a specific geometry. This section treats the use of mechanical analysis to identify the product-process interface (i.e. material flow as presented in 2.3) as presented in [START_REF] Skander | Méthode et modèle DFM pour le choix des procédés et l'intégration des contraintes 8[END_REF] in order to integrate, as soon as possible, peen forming information in the product definition following the general design approach presented in 2.2.

The Peen Forming process specificity lies on the fact that the material flow induces an elastic response of the sheet blank which generates the global distortion. Indeed, contrary to classical forming processes as stamping for example, only gentle curved shapes can be obtained due to the fact that the forming mechanism is based on elastic deformations and not chiefly on plastic ones. Then, the forming origin is the incompatible plastic strain field induced by the shot impacts while the forming mechanism involved lies on the elastic strains resulting from the material compatibility condition.

The authors decided as a first assumption to model the material flow taking only into account the plastic strains induced by the treatment, this data being the starting point of the study of the distortions induced. Three basic curving attributes must be defined to cover the process capabilities: cylindrical, spherical and saddle shaped, the combination of these three attributes for the description of a large sheet metal being of course thinkable. Let us concentrate on the spherical form attribute, which is the simplest one. An illustration of a manufacturing skeleton and its corresponding manufacturing skin is given in figure 6.

Illustration of the product-process interface in the DFM approach

Keeping in mind the CAD model presented on figure 1 and taken into account the previously presented product-process interface, the manufacturing product breakdown would be the following (cf. Figure 7):

-An extrusion operation as primary process. Tolerances are integrated in the section of the extrusion skeleton. (Step 1) -Three machining operations as secondary processes. (Step 2)

Figure 7. Illustration of the proposed DFM approach

The CAD model is then created according to manufacturing information (i.e. manufacturing skeleton) that leads the CAD breakdown and all the information related to product tolerances (as presented on Figure 8). So far we have presented how product-process interface is used in a DFM approach. The second goal is to take into account this new information of material heterogeneity (cf. figure 8) to better simulate each manufacturing operation. Every simulation can then, indeed, integrate an initial state with respect to the history of previous operations of the process plan. It is then compulsory to model every gradient of information (ex: stresses coming from forging, casting…) coming from this history.

Manufacturing Data management

Figure 9 gives an overview of a KBE5 application developed to manage the global process plan with respect to the previously presented product-process interface. A Manufacturing process database is used to guide the user on his choices and to complement the CAD systems by adding the engineering knowledge that drives the product design process.

That application proposes via its Graphic User Interface to manage both process and product information. The main functions offered by this application are:

-To select manufacturing process that could respect the requirements specification coming from the first step of the design approach (cf. 2.).

-To define every manufacturing operation parameters. This is, so far, done manually by the user according to his experience and the final part he wants to create.

-To define, via a database, product features based on manufacturing skeleton. o The tolerances on the product coming from manufacturing capability.

o The product's material behaviour (ex: stresses gradient) coming from material flows.

The final structure breakdown therefore gives every product alternatives according to manufacturing process plan alternatives (cf. breakdown tree on Figure 9) chosen by the user. It is important to note that each manufacturing alternative provides a CAD alternative and different material heterogeneity. The evolution of the CAD after each manufacturing operation with respect to that heterogeneity and to the simulation is then also different for each alternative. That why it is nowadays important to manage all the manufacturing information.

The data model of the KBE application is currently implemented using OCAF6 package encapsulated in MFC 7 objects and Open CASCADE 3D viewer.

Manufacturing data management and simulation

Based on this KBE application it is then possible to know what is the exact initial state of the product before each manufacturing operation simulation. This initial state obviously encapsulates the product behaviour issued from previous manufacturing operations. Indeed each manufacturing interface (i.e. manufacturing skeleton) of the data structure

Figure 1 . 2 . 1

 121 Figure 1. Incoherency between CAD model breakdown and manufacturing breakdown

Figure 2 .

 2 Figure 2. Product modelling for "X" constraints integration

Figure 3 .

 3 Figure3. The DFM activity schematisation[START_REF] Roucoules | Manufacturing process selection and integration in product design. Analysis and synthesis approaches[END_REF]

Figure 4 .

 4 Figure 4. Material flow definition for productprocess interface Based on that flow (called manufacturing skeleton) the material can be added (ex: injection), removed (ex: machining) or deformed (ex: forging) to obtain the final part surfaces (called manufacturing skin).Those surfaces are in the added and removed processes categories equal to the envelope surface.

[Figure 6 .

 6 Figure 6. Illustration of manufacturing skeleton concepts in a peen formed product case

Figure 8 .

 8 Figure 8. Example of product information issued from manufacturing process and managed by the product-process interface

o

  That includes: The emergence of the product CAD model integrating all the manufacturing variability.

Figure 9 .

 9 Figure 9. Overview of the KBE application

Computer Aided Design

Information is used in this work as both "new data" that complete product or process definition or "constraints" that is used to reduce the range of value of an existing data. Some details can be found in[START_REF] Roucoules | Knowledge intensive approach towards multiple product modelling and geometry emergence to foster cooperative design[END_REF].

KBE: Knowledge Based Engineering. Software developed in order to link CAD systems and Knowledge database

Open CASCADE Application Framework

Microsoft Foundation Components

gives that information.

As presented in the figure 10, the difficulty currently remains in transferring each gradient from the KBE data management structure to the initial model of the simulation (most often Finite Element Simulation). Manufacturing skeletons are, indeed, not based on meshing and the gradient of information have then to be linked to topological parameters that have a strong meaning for manufacturing experts. That is not the case of any meshes that are only dedicated to specific simulation models.

Keeping the link between manufacturing parameters and product information is very useful to notify every change concerning product definition that can therefore be quickly propagated to manufacturing information without processing any new FEA.

The proposed solution based on the presented product-process interface is to link information gradient to each manufacturing skeleton which is represented by topological features and linked to manufacturing parameters (cf. Figure 10); each skeleton being adequate for each material flow of the given manufacturing operation. In very complicated cases for which information gradient cannot be explicit, a specific mesh could be associated to skeleton features; each mesh being also adequate to the specific material flow of the manufacturing operation. This model based on material flow modelling with respect to skeleton and skin concepts is first used to integrate manufacturing information as soon as possible in the product design process (i.e. "by least commitments design approach"). This integration strongly leads the CAD modelling and by the way focuses the design process on expert designers' knowledge and not on CAD model any more.

The second objective of that interface model is to manage manufacturing information linked to product characteristics (ex: topology, tolerances, material behaviour…). It is then easy to use that link to simulate manufacturing processes taking into account the evolution of product characteristics with respect to the manufacturing plan. The whole history of each manufacturing operation is then linked to the product definition that is not currently the case in CAD centric design approach.

The main perspectives for future work concern:

-The achievement of the KBE application in order to test more complicated cases. The current developments are related to the implementation of a