Véronique Traverso
email: veronique.traverso@ens-lyon.fr

Fragments of life before: A form of storytelling in psychosocial support sessions for refugee women

Keywords: interactional resources, participation, refugee women, framework, transformation

The space of reflection and discovery that is opened up by these two definitions frames much of Goodwin's work. At least three perspectives are discernible: an essentially classic form of narrative analysis that identifies the components of the story (background information, parenthesis, climax); an interactional description of storytelling as a multi-unit turn, initiated by a preface which allows the introduction to be negotiated and followed by a resumption of turn-taking; and also a praxeological and holistic conception of storytelling as an activity that incorporates the entire interactional scene (rather than just the narrator), the bodies of the participants (rather than just their words), their knowledge that is shared, transmitted, taken up, and transformed, as well as, more broadly, the positions and the construction of identities that the story gives rise to and that structure the social scene. In the first part of this article, I set out what I view as Goodwin's major contributions to interactional approaches to storytelling (namely multimodality, participation, and social organization). Using this framework, I then examine a form of story that recurs during psychosocial support sessions for Syrian refugee women in Beirut, which I call "stories of before." In these accounts, the narrators describe aspects of their life in Syria, most often in comparison with their current life in Lebanon. I endeavor to characterize these narratives and to examine them in light of the way in which, at different levels, they transform the situation in which they are produced. The question that really interests me is: what results from these transformations, in the interaction as well as in the wider context in which they take place? Through a multimodal analysis inspired by Goodwin, I thus seek to understand what women are doing when they produce these stories of before.

Storytelling as a "field of action"

As the above quotes indicate, Goodwin develops a resolutely interactional vision of narratives1 (Sacks 1974(Sacks , 1978(Sacks , 1992;;Jefferson 1978Jefferson , 1988)). This view presents storytelling as a collective activity, which the participants set in motion and then pause in the course of their exchange. The collective dimension of a story is particularly evident in its introduction as well as in the various contributions of the co-participants. 2 To introduce a narrative into the interaction requires that the storyteller obtain from the interlocutors the possibility of producing a long turn (a turn made up of several turns units), which temporarily suspends turn-taking, hence the use of a "preface." 3 The production of the narrative within the interaction is supported by the recipients through their receipt tokens but also by their evaluations which, as well as showing their alignment with the activity, allow them to express their affiliation (Stivers 2008;Mandelbaum 2013). 4 Another form of contribution from the recipients is the production of a "second story," which they also use to show their understanding of what the speaker has narrated. Sacks describes the tasks of the recipient of a story in these terms:

Recipient of a story has as one business to display his understanding of it [. . .], and/or to affiliate to it by showing its particular relevance to him. "Oh God love 'im" is one form such affiliation takes. Another form involves recipient telling a second story, in which recipient figures as teller had figured. (Sacks 1978, 261) These characteristics, briefly summarized here, form the basis of the approach to narrative from which Goodwin proposes changes of perspective, new contributions, and additions. Goodwin's motivation does not seem to have been the search for characteristics that are specific to narratives, distinct from other interactional activities. Within his work, the story appears rather to be an activity whose in-depth study leads to deeper reflection on the nature and forms of recipientship, and on participation and its various modes. Rather than constituting a complete reconfiguration of longstanding perceptions of the narrative, the striking feature of Goodwin's studies on narratives is the way in which he enriches our understanding of the interlocutive scene, bringing into clear focus phenomena that had previously remained invisible or at least had not been described in any detail. Moreover, the stories he studies, of which a few examples appear later in this article (see 1.1 and 1.2), are, in many respects, canonical narratives, in the sense that they present the classic characteristics: displacement toward another time; succession of events starting from an initial situation and manifested in a series of narrative propositions; sequence of events in time underlying an emplotment; and the fall. 5 They are therefore quite unlike the less unified and less well-formed narratives that are very common in conversations and that are now grouped under the label of "small stories":

an umbrella term that captures a gamut of underrepresented narrative activities, such as tellings of ongoing events, future or hypothetical events, and shared (known) events, but it also captures allusions to (previous) tellings, deferrals of tellings, and refusals to tell. (Bamberg and Georgakopoulou 2008, 381) 6 This label was introduced to distinguish small stories from canonical narratives, which De Fina and Tseng describe with reference to the Labovian model:

According to William Labov, [. . .] stories are recapitulations of past events, with a structure involving complications and resolutions, told by a narrator in order to make a point. These canonical narratives recount chronologically ordered events in the past, have well-defined beginnings/middles/ends, and usually revolve around significant incidents. (De Fina and Tseng 2017, 381) Incomplete and imperfectly formed, small stories actually represent a large part of ordinary, everyday narratives, and their study has progressed in parallel with the development of interactional research (e.g., Jefferson 1988;Traverso 1994). While Goodwin's proposals for the analysis of narratives do not engage with the discussion on what is or is not a narrative, they nonetheless constitute a major contribution to the study of this interactional activity, by means of advances in the areas of semiotic resources, the stratification of participants, and the construction of socio-interactional organization.

Semiotic resources

Goodwin's notions of "situated activity systems" (C. Goodwin 1997) and "interactive fields" (C. Goodwin 2015) show that elements at play in an interaction include not only linguistic, discursive, and prosodic dimensions, as well as bodies and their organization in relation to each other, and objects and spatial organization, but also interactional phenomena and practices, and particularly sequential organization as well as various forms of specific and recognizable actions (such as reparations or expressions of disagreement). One example of such systems, where various semiotic resources are interlinked, is in fact the story of the wallpaper, to which Goodwin returns many times over the years (1984,2007,2015,2018).

Extract 1

Taken from C. Goodwin (1984), pages 225-226

In this extract, Goodwin shows how Don, the narrator's husband, who is present as she recounts the faux pas he made, actively participates in the narrative, albeit silently, in the way he visibly organizes his body, the direction of his gaze, his facial expressions, and his laughter. We see this in particular in line 14, the climax of the story, where the narrator inserts syllables of laughter inside the syllables of her words. Simultaneously, but in a different modality, Don produces mimicked laughter, which follows Ann's vocal laughter syllable by syllable. Goodwin describes precisely how Don's facial expression is amplified during "wa(h)llpa(h)p(h)er" (extract 2), resulting in laughter produced simultaneously by two separate bodies (Goodwin 2018, 132).

Extract 2

C. Goodwin (2018), page 132

These very detailed analyses of the resources deployed by participants considerably broaden our understanding of forms of contribution to the narrative, by taking into account not only the speech of the speakers but also their bodies. They also invite us to revisit the question of participation, and in particular to reconsider the enunciative instances of speaker(s) and recipient(s) engaged in narrative activity.

Multiple participants and the stratification of voices

Another of Goodwin's significant contributions to the study of narratives is his discussion, within his own research framework, of proposals emerging from other traditions, and the fact that he merges various dimensions, notably the sequential dimension and the enunciative dimension. His reflection on this area is based on an elaboration of Goffman's (1981) proposals concerning the deconstruction of participants, as well as on a "corporealized" rereading of the polyphony of Valentin Nikolaïevich Voloshinov (Marxism and the Philosophy of Language). In several articles (for example Goodwin 2007Goodwin , 2015)), he discusses the limitations of these two views and puts forward his own vision, which is a temporalized (rather than static) vision, with a high degree of specificity (rather than being global or generic), multimodal (rather than logocentric), and which integrates "into the same world"7 all participants in an interactional situation.

In his 2007 article on the wallpaper story (C. Goodwin 2007), Goodwin shows that, in a Goffmanian interpretation, Ann's speech encompasses a wide series of roles (author, animator, etc.) while the other participants are frozen in a fixed and unchanging status (recipient, official participant, whether addressed or unaddressed, etc.). In contrast, Goodwin's own analysis shows, on the one hand, that Don, who is both the character animated by, and recipient of, Ann's story, is indeed also a coproducer of the story, and on the other hand, that the modes of participation deployed by the "recipients" in the production of the story, as well as the resources they use, change as the story unfolds over time.

The description of these resources and of the complexity of the contributions of the various participants is extended even further in Goodwin's analysis of the speech of his father Chil, who was aphasic as a result of a stroke and in linguistic terms had available only the three words "yes," "no," and "and," as well as a number of lexically empty syllables of various prosodies (C. Goodwin, M.H. Goodwin, and Olsher 2002). His analysis of Chil's account of an earthquake (C. Goodwin 2004(C. Goodwin , 2013) ) during a family gathering highlights not only how Chil manages to have a "second story" told (Sacks 1992), but also how, through his facial expressions and his own body, he co-narrates the story that Helen verbally produces. At a certain point in the story (concerning a painting that almost falls on a baby because of a tremor), Chil expresses his disagreement with Helen:

Extract 3

Goodwin (2004), page 164

When Helen claims the painting had fallen (l. 50), Chil disagrees ("no:" l. 52), raising his hand above his head, before looking at Helen and then Linda (who has just formulated an evaluation, l. 51). The action produced by Chil at this position in the sequence, where an explanation of the disagreement would be expected, functions as a reference to an alternative version of the event, leading the other participants to try to identify it ("started to fall?" l. 57), and ultimately resulting in a collective production of the version of the story desired by Chil. With his analyses of Chil's stories, Goodwin moves further away from his starting points (Goffman and Voloshinov), highlighting a deep interconnection between speaking roles and forms of participation. He describes a "distributed speaker" for each one of us who, like Chil, is able to "co-operatively incorporate in our own actions resources provided by others" (C. Goodwin 2018, 103).

The construction of socio-interactional organization through narrative activity

The last aspect that should be noted here about Goodwin's studies of narratives is the way in which they reflect and contribute to the construction of social organization. He develops his thoughts on the subject with Marjorie Goodwin, based on the latter's work on disagreements among young African Americans. Together, they examined, for example, how a specific type of storytelling builds a specific social organization among the group of girls, by putting participants into positions that push them into acting in a certain way. For example, if Bea tells Julia publicly that Terry made negative comments about Julia, this reported story will put Julia in the position of the "offended party," which will lead her to want to accuse Terry and hold her to account. These "instigating stories" (M.H. Goodwin 1982) set up protagonists and events in a way that is strongly related to the positions that they push the girls in the group to take and the social actions that they push them to carry out. Accordingly, they construct and transform social organizations and the identities of participants.8 This idea of transformation is of major importance in Goodwin's later works. He sees this process at work on different levels, for example when, in a successive turn, participants take up and transform the elements and parts of a previous turn (the "substrates," C. Goodwin 2018); in the fact that a story transforms in situ (i.e., at the time of its production) the socio-interactional organization in place; as well as in the return transformation that the in situ interaction will have on the participants' actions and on themselves:

The process of participating in the interactive field created by a narrative-performing transformative operations on it that display understanding and build appropriate, consequential subsequent action-simultaneously transforms those who perform such operations.

(C. Goodwin 2015, 216)

Once again, in these examples the emphasis is not on the narrative as a series of propositions, nor on the reality, truth, or accuracy of the facts that have been related, but on its effect on the situation in which it is produced, for, with, and between those present.

Contrasting, escaping, and redefining the context by storytelling in a collective

As part of the REMILAS-Liban project9 and as an extension of a series of studies on the construction of identity positions in gatherings (Traverso 2003(Traverso , 2014(Traverso , 2017)), the aim of the following study is to describe what appears to be a particular type of story in the very specific context of psychosocial support sessions for vulnerable refugee women. As well as analyzing these stories, I will seek to show how their production transforms the actual context. My objective is to shed light on what these transformations lead to, not only in the here and now of the interaction (through the sequences they give rise to), but above all in the broader context of the session in which they take place and the social organization that psychosocial support sessions reflect.

Life skills sessions

"Life skills and awareness sessions" fall under the broad category of "mental health and psychosocial support", which is described by humanitarian organizations as "any type of local or outside support that aims to protect or promote psychosocial well-being and/or prevent or treat mental disorder" (InterAgency Standing Committee IASC, 2007). 10 At Nina, the NGO where the sessions studied were recorded, women come to be trained in textile crafts. Their registration in this course also commits them to taking a Life Skills session, where their presence is also monitored. A female psychologist or social worker facilitates these sessions.

The stories that I analyze are therefore produced within a framework that activates identity attributions:11 the very fact that a session is devoted to women's rights, to stress, or to hygiene categorizes the participants as 1) being likely to suffer from issues related to these themes, and 2) not having knowledge about, or at least requiring further information on, these topics. In this article, I focus on a session devoted to hygiene that was attended by two groups of women being trained in two embroidery techniques (machine embroidery or aghabâni, and hand embroidery). The two embroidery trainers arrived during the session. The group is seated around a large table (one of the two trainers is out of the picture). The facilitator (a psychologist) is seated at the end of the table; the women participating in the session are seated by order of arrival and within their embroidery group. Accordingly, the aghabâni group are seated on one side of the table, to the right of the psychologist, and to her left is the hand embroidery group.

Image 1

Translation of captions from left to right: observer, the aghabâni group, embroidery group, psychologist, the aghabâni trainer

The two groups are differentiated not only by their chosen craft, but also by the fact that most of the participants in the aghabâni group are vulnerable Lebanese women, with the exception of one displaced Iraqi woman, while all the women in the hand embroidery group are Syrian.

Stories of before

The women's "stories of before" evoke the country left behind. These are stories from before the war (in Iraq or Syria), from before their exile. They do not relate specific facts or events that took place at a clearly defined time in the past; they concern the habits and routines of life in the country left behind. In recounting these stories, the women highlight the differences in the way of life and living conditions between Syria and Lebanon and point out what was better before, back there. These stories stand out from the types of stories that are most frequently analyzed in the field of migration studies (see among others Pestre 2010; Jacquemet 2009; d'Halluin-Mabillot 2012; Maitilasso 2014; Chambon 2018; Saglio-Yatzimirski 2018; Traverso and Chambon, forthcoming). Indeed, these stories do not conform to the variations and repetitions of the "refugee story" as it must be produced for the organizations that process asylum requests or as it is prepared by the groups that help migrants with their application. Nor do these resemble the narratives solicited in interviews or focus groups, 12 another major category of narratives seen in the field of migration studies. Indeed, these stories of before are not solicited, they emerge; they bring something to the interaction that is unique to the narrators, rather than being their attempt to conform to expectations (of the researcher) or demands (of the protection officer). They take place in a setting where they are not specifically expected or requested, but where conditions nevertheless allow for their production. Formally, they are far from including all the characteristics of the canonical narrative discussed above. One of their defining linguistic markers consists of introductory statements comprising temporal and spatial markers that form a break with the here and now of the interaction ("when I was/we were in. . .," "back then. . ."). They are most often very brief, and do not correspond to the category of the "emplotment," which has a logical model (in the relationship between cause and effect) underpinned by temporal organization (i.e., ordered utterances). Rather, they are based on a model of contrast or opposition. One last important characteristic is that they take place in multiparticipant situations, and that they are themselves collective, even though most of the time only one speaker verbally produces the story form. In this sense, they are truly a field of action (C. Goodwin 2015).

Setting up a collective story: "We had a big house"

The Life Skills session devoted to hygiene begins with a round table during which each woman in turn gives her point of view on what this concept covers and its importance (Traverso and Kerbage 2020). Maha, chosen to speak first, describes the difficulties she has in keeping her children clean. At the end of her turn, when she is about to hand on to her neighbor on her right (image 1 below), another participant (seated opposite her), Maïssa, takes the turn. The extract reproduced below begins at the end of Maha's turn at talk, when Maïssa and then the psychologist produce a few turns, confirming the end of this first turn at talk and formulating their agreement (Maïssa, l. 4) as well as proposing a generalizing statement (the psychologist, l. 5, 7, 9):

Extract 4 13 All interactions take place in Arabic. The original language, in the form of a phonetic transcription, is reproduced only when this is necessary to indicate more precisely the sequentiality of the turn or multimodality. 12 On this point, Maitilasso relates the following anecdote: "'Do you want me to tell you my story? I can do that, I have already done it many times', Thierno explains to me with a smile [. . .] when I tell him about my research and my desire to talk with him about his migratory journey. Thierno's life story is presented as a product, the expression of a literary performance that can be reproduced whenever desired; a product for which there is a strong demand from journalists, ethnographers, and human rights activists" (Maitilasso 2014, 1). My own experience at the NGO Nina also reflects this state of affairs: while the participants (both trainers and beneficiaries) were disconcerted that I did not ask them to tell me about their life but to let me record the training and Life Skills sessions, when I finally conducted interviews (on their experience of training in textile crafts), they found themselves on familiar ground, and I understood how familiar and even pleasing this format was to them. 13 I thank Sophie Harfouche and Maha Barakat for the transcription of the corpora. My warm gratitude also goes to all those involved in NGO Nina, who welcomed me among them for a year and allowed me to make recordings. Sequentially, Maïssa's intervention in line 12 modifies the expected order of turn-taking around the table, since she is opposite Maha rather than next to her, and she was not designated by Maha as the next speaker (image 1). At the same time, she begins her turn with "especially here" (xaṣtan hūn), thus presenting it as an expansion of a previous speech. It is difficult to tell which previous turn Maïssa's turn relates to syntactically, but semantically we can clearly link it to Maha's turn (l. 12) since Maïssa is confirming the difficulty in keeping their children clean, and views this difficulty as particularly linked to "here." Lines 12 to 20 form an introduction to Maïssa's story, which presents only some of the characteristics of a preface (Sacks 1992). In particular, this introduction does not aim to suspend the alternation of turns-at-talk (since in this case the organization of speech is already different from that of a conversation). On the other hand, it sets up a number of favorable conditions for the development of the story. The first is the reorganization of the participation framework. The moment when Maïssa starts to speak is a moment of transition within the round table. The position she chooses within the sequence and the way in which she begins her turn indicate that she is speaking in continuity with Maha's turn. We also observe that Maha aligns herself with Maïssa's turn and immediately affiliates herself with her words (l. 14). Maïssa therefore fundamentally modifies the organization of participation at the level of the interlocutive space (speaker/recipient). She moves from being a recipient to position herself as additional speaker or co-speaker in support of another's speech. In the rest of her turn, she also changes the organization of the group. The collective organization in place until that point was based on (at least) two dimensions of identity. An initial form of organization was visible in the interlocution, with a speaker occupying the floor and an audience, in which the psychologist occupied a special position, being the one who had initiated the round table, the one toward whom the speaker's gaze was directed more than half of the time, and the one who decided whether the turn in progress was sufficient (complete) or not. Another element of the group's organization was linked to the type of training chosen by the women (aghabâni or hand embroidery), with the two groups being seated on either side of the table. Maïssa adds another dimension to the organization of the collective by resorting to the opposition "here, elsewhere" (l. 17-18). She begins by mentioning living spaces (l. 12), then reformulates the idea by speaking of the organization of housing (l. 16-17). The two formulations use a similar syntactic construction: "space {I mean} there isn't" (unfinished statement) -> "the organization the organization of the apartments here (.) it isn't . . ." This expresses a lack, an absence, and projects toward the second part of a binary construction (concerning both the elsewhere and the positive side of the assertion). Then she goes on to contrast (l. 17) "the organization of the apartments here" (nið̣ ām eʃʃiqāq hūn) to "the organization of the big houses as we were in Iraq" (nið̣ ām elbjūt lekbīr mitl ma kənna bi lʕirāq, l. 17-20). This opposition is also marked by a gesture evoking (l. 17 below) the enclosed character of small dwellings in Lebanon-while the term "enclosed" or "small" is not pronounced, it is embodied by the compartmentalized and defined space conveyed by the speaker's two outstretched hands (image 3a, 3b), contrasting with the wide and circular gesture describing the large houses in Iraq (l. 8 below, images 4a, b, c, d, e):

Extract 5 (detailed transcript of extract 4) This part of Maïssa's turn establishes a possible common point (a shared attribute) between the speaker (an Iraqi) and the Syrian participants ("and perhaps in Syria too"). This proposal elicits the agreement of Maha (who nods regularly) as well as that of several of the (Syrian) women, seated opposite Maïssa, who also nod. Following this introduction, Maïssa begins a story of life before (extract 6) by speaking, no longer as a voice supporting Maha's turn, but in the name of an "us" which shares a common experience. She indicates this by the use of the pronoun "us" ("us, our houses, they were uh:," "naḥna buyūtna kānet ø:::"), in a construction with the disjunctive pronoun "naḥna" (us/we) at the beginning of her turn, which reinforces the opposition between the houses of Iraq (and possibly of Syria) and those of Lebanon. The speaker continues, with her gaze directed toward the psychologist, describing what the houses were like "over there": After a description of the houses in Iraq, lines 22-28, Maïssa explains, in line 29, the link with the children of whom Maha spoke. Finally, in line 33, she describes a contrast with what is happening "here" ("not like here," "miʃ mitəl hūn"), where the child "is imprisoned in the house" ("maḥbūs bi lbēt"). The description of places and space is accompanied by numerous gestures which reinforce the contrasts between large spaces (with their various components, cf. baton gestures l. 27-28) and spaces which enclose (retraction of the arms and shoulders, l. 33, images 10, 11). The construction of interlocution by means of the gaze is also interesting. On the first occasion, at line 30, two members of the group (Maha and Samar) use their bodies to express their affiliation with Maïssa's words (they nod their heads), and they reinforce the structuring of the current story: at the statement "so the child was more comfortable playing," which appears to be a kind of conclusion to what precedes, and a possible end of Maïssa's turn (or is at least a marked articulation), they turn to the psychologist, demonstrating that they are waiting for her reaction. The same configuration occurs again at line 36, at the end of Maïssa's turn, which is left open by the speaker as she herself looks toward the women opposite them, while Maha and Samar turn toward the psychologist, again expressing their expectation of a reaction. At line 37, the psychologist produces a simple receipt token, indicating that she will not resume the turn, which allows Maïssa to go on. The essentially silent contributions (except for the very quiet "yes" spoken by Maha, l. 30) and gestural or postural contributions of Maha and Samar also embody the socio-interactional organization set up by Maïssa: these two participants act as active parts of the "we." The affiliation between the participants, which creates a form of collective speaker that represents the voices of those who have left their country, is expressed again at the end of Maïssa's turn: I will not present a detailed analysis of this extract, but will highlight two aspects.

On the one hand, we see that Maïssa again constructs this last part of her contribution on the basis of a contrast ("when we were in Iraq . . .," l. 38, "but here," l. 43), highlighting the difficulties in Beirut and the tranquility of Iraq. At line 49, overlapping with Maïssa's final evaluation, Maha in turn utters an evaluation, thus confirming the closure of the story. This addition is particularly interesting to observe in the light of the processes of resumption, transformation, and recycling of resources that were present in the interaction. It first reproduces Maïssa's syntactic construction of her previous turn (Maïssa: "kamān (0.5) ṣaʕb" "it's difficult," l. 48; Maha: "kamān ø:: ḥabes ḥurijje" "it is also uh:: an imprisonment of freedom," l. 49). It also resumes the same type of construction as in the initiation of Maïssa's turn (l. 12, extract 5), "xaṣtan" "especially" and "kamān" "also" (adverbial expansions) and the same modality i.e., completion of a turn. Finally, the speaker recycles and transforms the lexical root of imprisonment (ḥbs) (used as a noun in "ḥabes ḥurijje" "imprisonment of freedom," l. 49; then as a past participle in "maḥbūs bi lbēt" "imprisoned in the house," employed by Maïssa l. 35, extract 6). Maha's turn at talk leads to a brief affiliative exchange between her and Maïssa, achieving closure of the story: at line 49, Maïssa, who was looking at the psychologist, turns to Maha, then, at line 51, she makes a confirmation (with a "yes" and a nod of her head), before once again uttering her own evaluation, thus integrating Maïssa's contribution into her own speech, while Maha also makes her own ratification (line 52). Both then mark the end of the exchange by looking away from each other, Maha turning to the psychologist, while Maïssa looks up. In summary, we can interpret the speech from line 22 as emanating-through Maïssa's voice-from a collective speaker, who tells a story of life before, accompanied, supported, and completed by the contributions (verbal, vocal, or gestural) of the other participants. This story plays out many processes of resumption, transformation, and echo on the verbal level, as well as the "corporalization" of socio-interactional positions. The story moves the discussion toward common considerations that are shared with the other refugee women in the group, concerning their past life, a life that was more affluent (cf. the list of possessions, lines 27-28) than life in the here and now. This indicates the presence of identity traits that go beyond those of "women requiring Life Skills sessions or women in need of support from groups providing aid to refugees."

Telling a story to differentiate yourself: Aleppo laurel oil soap

The second story, which I will examine very briefly, sets up quite different socio-interlocutive constructions within the group. The aghabâni trainer, Monira, joined the session during the round table . Once each of the participants has expressed herself, the psychologist invites her to speak, and she discusses the curative properties of hot water, which leads her to introduce a story from before:

Extrait 8 Monira develops her story by gazing at the psychologist, while the latter produces continuers (l. 16, 19). At line 26, in an intervention which momentarily interrupts the face-to-face conversation between Monira and the psychologist, Gloria compares the hammam to the sauna, which can also be interpreted as a marker of the contrast between a modern Western reality (the sauna "back home") and a different oriental and traditional reality, of which Monira speaks. 15In the rest of the story, Monira mentions the handcrafted soap (ṣabun Ɣār) that is emblematic of her city, Aleppo. In lines 32-33, she draws attention to it during her turn at talk through a series of processes: gesturing, with a raised finger; glancing, directed at the whole group; self-repair, incorporating a relatively long pause before pronouncing the name. Then, at lines 33-35, with the phrase "I don't know if anyone knows it," she suggests that the reality she is describing is possibly unknown to other participants, including the Syrian women in the group. She thus constructs the group in a very different way to Maïssa's action in the previous story. Here, not only does she imply that the other participants might not know what she is talking about, but when on the contrary the latter claim to know Aleppo soap (l. 36, most of the women in the group nod vigorously, wherever they are sitting around the table and whether or not they are refugees), she goes on to draw attention to the fact that Aleppo soap is very different (and much more hygienic) than local Lebanese soap. In her speech, Monira thus operates lines of distinction that are more complex than what Maïssa did in her story, and which cannot be summed up by the contrast of "here"/"there." On the other hand, rather like Maïssa, she mentions the differences between Aleppo laurel soap and Lebanese handcrafted soap (l. 35-38), emphasizing that the latter does not have the sanitizing qualities of ghâr soap and putting forward a contrast (marked in particular by the opposition of "I" versus "they").

The reception of the stories

Monira's story, like that of Maïssa, is a story that places value on customs from before and that evokes what was better in the country left behind compared to the reality of here. In these two stories, the negative side of the comparison is simply denoted by "here" (hūn), and, rather than "we" or "I," only "they" is spoken. Lebanon, Beirut, and the Lebanese are not explicitly mentioned. In these stories, it is neither the protagonists (see for example the vague use of the pronoun "they" in Monira's account), nor the precise facts that matter, but the fleeting evocation of another time and the emphasis on differences in ways of behaving and living, accompanied by an underlying value judgment which allows us to glimpse the difficulties of living as a refugee. I have focused on the introduction and reception of the stories among the people who are connected to them (having produced the turn that led to the story, or having engaged in it by means of continuers, marks of verbal or gestural agreement, or evaluations, or by collaborating in their closure). But beyond these moments, the space that these stories open up in the general dynamic of the sessions is very quickly closed down in what follows on. Each arouses a reaction from one of the Lebanese women in the group. After Maïssa's story, one woman refutes the point that the story illustrated, arguing that viruses are also caught at home and that therefore "it doesn't make any difference if you deprive them of their freedom." She refutes the accuracy of what has just been said and redirects the discussion toward the theme of hygiene in general. After Monira's story, the same participant takes the floor to make fun of the old customs of the hammam-where "the grandmothers would boil you alive"-throwing doubt on them by pointing out that doctors today recommend, on the contrary, not to treat one's skin aggressively and to wash with mild products. These interventions pick up on some aspect of the narrative as being inaccurate (e.g., "in Lebanon, children must be deprived of their freedom to keep them clean"). They reframe things, and in the course of the sessions these stories of before thus appear as moments, recurrent yet fleeting.

Conclusion

These stories of before are very far removed from being canonical stories or real chronicles of the life of before. They fall back on a past time, but are not organized in the form of a series of events: they evoke a time which does not progress; they use fragments to recall lost ways of living and doing. Through these stories, the women animate voices, both individual and collective, within the collective. Sometimes they argue and openly question the validity of what is said (Traverso 2019; Traverso and Kerbage 2020), but this is quite rare. The question, ultimately, is that of the impact of these stories and the local transformations that they operate on the sessions, on their expectations and their principles, on the themes discussed or on the group. Initially, the analysis of the stories shows that, in the sequential unfurling of the interaction, they are followed by a return to the framework (in terms of subject or format) that was set up at the start. The stories are fleeting, and the space they open up is fragile. More generally, the year-long fieldwork carried out in this NGO did not reveal any significant change in life skills activities, apart from the alterations arising from new directives from funding bodies. However, these stories of before are recurrent and emerge session after session, on different occasions, to express through the motif of contrast the shortcomings and difficulties of life "here." Their recurrence, and the responses they most often elicit from the Lebanese facilitators or participants, seem to point to two aspects, which I will now relate to the very activity of "psychosocial support for refugees and vulnerable people" and all that this implies in terms of constraint, attribution of image, and proposition/imposition of principles, which are often inapplicable in the conditions in which these people live. In interviews, the facilitators, young psychologists, sometimes still in training, or social workers, often express their unease and the difficulty of their task (Traverso 2019). They are well aware that participants attend sessions for very diverse reasons that have little to do with the session content, ranging from the opportunity to leave their homes, meet other people, have their children taken care of while they themselves are attending training, and so on. In practice, many of these women are just waiting for the session to end and sometimes chat with each other, some talk about the topics raised, some dispute what is being said, some tell stories. It is within this context that we must consider these stories of before. Drawing on the work of Charles Goodwin and taking into account the diverse resources, the bodies of the participants, and the establishment of embodied collective roles, my analysis has shown that these stories activate another collective within the group, which, through the story, develops a practice of resistance to the activity and what it is putting in place. This is played out over and over again, in a repetitive way, and provokes, in an equally recurrent way, repartee, reframing, even rebuttals (like "it doesn't make any difference if you deprive them of their freedom"). This repetition resembles a game: a kind of bickering that goes back and forth.

At the same time, the stories also represent narrative escapes, which briefly lead outside the current situation (the Life Skills session) or the more general situation (of being a displaced person). Through their production, the women collectively evoke, remember, and value their customs, as a way of laughing in the face of constraints and difficulties, but also as a way of briefly appropriating to themselves the time that is meant to be dedicated to a session promoting psychosocial well-being.

Translated and edited by Cadenza Academic Translations

Translator: Clare Horackova; Editor: Suzy Bott; Senior editor: Mark Mellor

Transcript conventions

The The Arabic transcript adheres to ARAPI transcript conventions (https://ifpo.hypotheses.org/9335), of which it only uses the interactional tier and the translation tier. In the translation, {} indicates adaptations to facilitate understanding (words added or pragmatic equivalent, for example for discursive markers).

Reference List

  4e -Maïssa, hands relaxed, makes a wide circular gesture 19# +mitlə ma kə#5nna naḥna as we were +gestures toward L with L hand Image 5 -Left hand gesturing to the left "as we were" Right hand gesturing toward Maha "in Syria too"

  Maha and Samar look at MaïssaImage 12 -Maha and Samar have turned their heads toward the psychologist, while Maïssa looks in their direction

  place and these streets and:: (0.9)46KTĪR ṢAʕB jaʕni it's very difficult {in fact} 47 *[(0.5) u kamān tetərki lwa-lwaled *[(0.5) and also that you leave thethe child mah *inclines her head to one side MAI be lbēt kill ennhār kamān *(0.5) [*ṣaʕbin the house all day also *(0.5) *it'

  /icar.cnrs.fr/projets/corinte/documents/2013_Conv_ICOR_250313.pdf. The transcription from Arabic follows ARAPI convention: https://ifpo.hypotheses.org/933. In translation, the sign {} surrounds adaptations aimed at facilitating understanding (additions or translation by pragmatic equivalents, for example for discursive markers). Conventions used for the notation of gestures available on the CORVIS website: http://icar.cnrs.fr/projets/corvis/.

	The *especially #1 here +the #2 space {I mean} there isn't conventions used nods (2.5) *gestures toward her neighbor on her right +G->MAH, raises her hands facing one follow another to describe the space=> MAI isn't isn't [:: 12 # MAI 11 mah mai 13 14 MAH [yes that's true 15 (0.3) mai +G>PSY +G>MAH 17 the organization of the apartments +here +R>PSY it's not like the organization of big houses 19 + as we were +gestures with L hand>left 20 in I+raq and maybe *in Syria al[so mai +G>MAH, gestures with R hand>MAH mah *nods-> 21 MAH [yes al[s*o+ mah -> * mai +G>PSY 21 SAM yes xx[x http:/mah mon nods	ICOR:

14

Transcription formatting uses normal font for transcription, lighter font for translations, and bold grey for multimodal notations. Image 1 -Maha gestures toward her neighbor on the right to indicate it is her turn to speak Image 2 -Maïssa describes the living spaces 16 MAI +{I mean} the ap-+the organization

  transcript conventions used in this paper are a simplified version of the ICOR conventions, whose complete version is available on the website: http://icar.cnrs.fr/projets/corinte/documents/2013_Conv_ICOR_250313.pdf. Conventions for gestures are loosely based on those of Mondada, available on the website CORVIS http://icar.univ-lyon2.fr/projets/corvis/. abc abbreviated names in lowercase letters indicate that the line is devoted to the description of a gesture or movement tourne la tête descriptions of gestures are in grey * * indicates the beginning and end of a gesture or gaze that is described in the next line >> ou --> indicates that the gesture continues during the following lines until the next marker

	(.)	short pauses (untimed) (<0.2s)	(il va)	uncertain transcription
	(2.2)	pauses in seconds	°well°	quiet voice or very quiet (°°well°°)
	&	continuation of the turn by the same	SO	increased volume or other forms of insistence
		participant
	=	latching	[...]	transcriber's cut
	((rire))	double parenthesis: non transcribed	->	before an abbreviated name, points to the turns
		phenomena		commented on in the analysis
	G		gaze
	H		head
	R hand, L	right hand, left hand
	hand
	*, ∆, Ω
	[ ]	beginning and end of overlaps	xxx	inaudible segment
	par-	truncation	/ \	raising/falling intonation
	:	lengthening	.h	inhalation

ABC

le pseudo the participant's pseudo is made of three capital letters.

is used to distinguish the participants in descriptions of gestures

•

  Antaki, Charles, and Sue Widdicombe. 1998. Identities in Talk. London: Sage Publications. • Bamberg, Michael, and Alexandra Georgakopoulou. 2008. "Small Stories as a New Perspective in Narrative and Identity Analysis." Text and Talk 28, no. 3: 377-96. • Bres, Jacques. 2001. "De la textualité narrative en récit oral: l'enchaînement des propositions." Revue québécoise de linguistique 29, no. 1: 23-49. • Chambon, Nicholas. 2018. "Raconter son histoire comme personne: les migrants et leurs récits." Le Sujet Dans la Cité 9, no. 2: 63-74. • De Fina, Anna, and Amelia Tseng. 2017. "Narrative in the Study of Migrants." In The Routledge Handbook of Migration and Language, edited by Suresh Canagarajah, 381-96. London: Routledge. • D'Halluin-Mabillot, Estelle. 2012. Les épreuves de l'asile: Associations et réfugiés face aux

The formulation of this perspective fundamentally revisited the approach of narratological studies to narratives, based on a search for its characteristic features (sequence of narrative propositions, timetable, spatiotemporal markers, emplotment) and its questions such as, for example, that of the "minimal narrative." For a discussion of these structural features of narratives, as well as its interactional features (preface, tellability, evaluation), seeFilliettaz and Revaz (2006), and Traverso (forthcoming). Mandelbaum (2013) provides a synthesis of interactional descriptions of storytelling.

On this point precisely, conversational descriptions of the story depart from the propositions ofLabov and Waletski (1967), which were formed on the basis of stories produced by interviewees whom the interviewer asks to recount an experience where their life was in danger. In that format, unlike in conversations, for example, the narrators do not have to negotiate the introduction of a story, since from the outset they are, as Goodwin says, in the position "of having the rights to an extended floor" (2015: 198). On the Labovian approach, see in particularBres (2001).

As defined by Sacks: "It is an utterance that asks for the right to produce extended talk, and says that the talk will be interesting, as well as doing other things" (1992: 226); see alsoJefferson (1978).

See C. Goodwin (1986), C. Goodwin and M.H. Goodwin (1987) on "receipt tokens" and "assessments" and on the closure value of evaluations.

These characteristics have been widely discussed and research on narrative in social interactions agrees that many of the narratives only imperfectly meet these criteria.

Georgakopoulou explains the blatant opposition seen in this label to that of the "big story" (referring to life stories, biographical stories); its introduction was linked to the need to reveal research bias and to counteract the dominant models of narrative(Georgakopoulou 2015, 256). See alsoGeorgakopoulou (2006).

Taken from Charles Goodwin's review of Goffman's Footing: "In Footing, instead of collaborating together to build talk, speakers and hearers inhabit separate worlds" (2007, 28).

Identities and positions are thus approached in a fundamentally interactional way, i.e., they are viewed as being set up in situ, in both sequential and relational terms (see, among others, Antaki and Widdicombe 1998; Greco, Mondada, and Renaud 2014).

Project funded by Ifpo-Beirut and Saint Joseph University (Beirut).

See http://www.who.int/mental_health/emergencies/guidelines_iasc_mental_health_psychosocial_june_2007.pdf

Greco, Mondada and Renaud (2014); Traverso (2014, 2017); Marranconi and Kerbage (2017); Kerbage and Traverso (2020).

On contrasting representations of the hammam in the Middle East as expressed during sessions, see Traverso(2014, 2017).

politiques du soupçon. Paris: Éditions de l'EHESS. • Filliettaz, Laurent, and Françoise Revaz. 2006. "Actualités du récit dans le champ de la linguistique des discours oraux. Le cas des narrations en situation d'entretien." Protée 34, no. 2-3: 53-66. • Georgakopoulou, Alexandra. 2006. "The Other Side of the Story: Towards a Narrative Analysis of Narratives-in-Interaction." Discourse Studies 8, no. 2: 235-57. • Georgakopoulou, Alexandra, and Anna De Fina, eds. 2015. Handbook of Narrative Analysis. Hoboken: John Wiley and Sons, Inc. • Goffman, Erving. 1981. Forms of Talk. Oxford: Blackwell. • Goodwin, Charles. 1984. "Notes on Story Structure and the Organization of Participation." In Structures of Social Action: Studies in Conversation Analysis, edited by J. Maxwell Atkinson and John Heritage, 225-46. Cambridge: Cambridge University Press. • Goodwin, Charles. 1986. "Between and Within: Alternative Sequential Treatments of Continuers and Assessments." Human Studies 9, no. 2-3: 205-17. • Goodwin, Charles. 1997. "The Blackness of Black." In Discourse, Tools and Reasoning: Essays on Situated Cognition, edited by Lauren B. Resnick, Roger Säljö, Clotilde Pontecorvo, and Barbara Burge, 111-40. Berlin: Springer. • Goodwin, Charles. 2004. "A Competent Speaker Who Can't Speak: The Social Life of Aphasia." Journal of Linguistic Anthropology 14, no. 2: 151-70. • Goodwin, Charles. 2007. "Interactive Footing." In Reporting Talk: Reported Speech in Interaction, edited by Elizabeth Holt and Rebecca Clift, 16-46. Cambridge: Cambridge University Press. • Goodwin, Charles. 2013. "The Co-operative, Transformative Organization of Human Action and Knowledge." Journal of Pragmatics 46, no. 1: 8-23. • Goodwin, Charles. 2015. "Narrative as Talk-in-Interaction." In The Handbook of Narrative Analysis, edited by Anna De Fina and Alexandra Georgakopoulou, 195-218. London: John Wiley and Sons. • Goodwin, Charles. 2017. Co-operative Action. Cambridge: Cambridge University Press. • Goodwin, Charles, Marjorie Harness Goodwin, and David Olsher. 2002. "Producing Sense with Nonsense Syllables: Turn and Sequence in Conversations with a Man with Severe Aphasia." In The Language of Turn and Sequence, edited by Cecilia Ford, Barbara Fox, and Sandra Thompson, 56-80. Oxford: Oxford University Press. • Goodwin, Marjorie Harness. 1982. "'Instigating': Storytelling as Social Process." American Ethnologist 9, no. 4: 799-819. • Goodwin, Charles, and Marjorie Goodwin. 1987. "Concurrent Operations on Talk: Notes on the Interactive Organization of Assessments." Pragmatics 1, no. 1: 1-54. • Greco, Luca, Lorenza Mondada, and Patrick Renaud, eds. 2014. Identités en interaction. Limoges: Lambert Lucas. • Jacquemet, Marco. 2009. "Transcribing Refugees: The Entextualization of Asylum Seekers' Hearings in a Transidiomatic Environment." Text and Talk 29, no. 5: 525-46. • Jefferson, Gail. 1978. "Sequential Aspects of Storytelling in Conversation." In Studies in the Organization of Conversational Interaction, edited by Jim Schenkein, 219-48. New York: Academic Press. • Jefferson, Gail. 1988. "On the Sequential Organization of Troubles-Talk in Ordinary Conversation." Social Problems 35, no. 4: 418-41. • Kerbage, Hala, and Véronique Traverso. 2020. "Life skills and awareness sessions (1): le thème de l'hygiène." Carnet de l'IFPO. https://ifpo.hypotheses.org/10643. • Kerbage, Hala, and Véronique Traverso. 2020. "Life skills et awareness sessions (2): Répondre à son tour." Carnet de l'IFPO. https://ifpo.hypotheses.org/10687. • Kerbage, Hala, and Véronique Traverso. 2020. "Life skills et awareness sessions (3): Faire celle qui sait." Carnet de l'IFPO. https://ifpo.hypotheses.org/10719. • Kerbage, Hala, and Véronique Traverso. 2020. "Life skills et awareness sessions (4): L'hygiène, principes et décalages." Carnet de l'IFPO. https://ifpo.hypotheses.org/10733. • Labov, William, and Joshua Waletsky. 1967. "Narrative Analysis: Oral Versions of Personal

Experience." In Essays on the Verbal and Visual Arts,edited by June Helm,


Dorin Bucur
email: dorin.bucur@univ-savoie.fr

Giuseppe Buttazzo
email: buttazzo@dm.unipi.it

Antoine Henrot
email: henrot@iecn.u-nancy.fr

Minimization of λ 2 (Ω) with a perimeter constraint

Keywords: Dirichlet Laplacian, eigenvalues, perimeter constraint, isoperimetric problem AMS classification: 49Q10, 49J45, 49R50, 35P15, 47A75

We study the problem of minimizing the second Dirichlet eigenvalue for the Laplacian operator among sets of given perimeter. In two dimensions, we prove that the optimum exists, is convex, regular, and its boundary contains exactly two points where the curvature vanishes.

In N dimensions, we prove a more general existence theorem for a class of functionals which is decreasing with respect to set inclusion and γ lower semicontinuous.

Introduction

Let Ω be a bounded open set in R N and let us denote by 0 < λ 1 (Ω) ≤ λ 2 (Ω) ≤ λ 3 (Ω) . . . its eigenvalues for the Laplacian operator with homogeneous Dirichlet boundary condition. Problems linking the shape of a domain to the sequence of its eigenvalues, or to some function of them, are among the most fascinating of mathematical analysis or differential geometry. In particular, problems of minimization of eigenvalues, or combination of eigenvalues, brought about many deep works since the early part of the twentieth century. Actually, this question appears first in the famous book of Lord Rayleigh "The theory of sound". Thanks to some explicit computations and "physical evidence", Lord Rayleigh conjectured that the disk should minimize the first Dirichlet eigenvalue λ 1 of the Laplacian among plane open sets of given area. This result has been proved later by Faber and Krahn using a rearrangement technique. Then, many other similar "isoperimetric problems" have been considered. For a survey on these questions, we refer to the papers [START_REF] Ashbaugh | Isoperimetric inequalities for eigenvalues of the Laplacian, Spectral Theory and Mathematical Physics: a Festschrift in honor of Barry Simon's 60th birthday[END_REF], [START_REF] Belhachmi | Shape optimization problems for eigenvalues of elliptic operators[END_REF], [START_REF] Payne | Isoperimetric inequalities and their applications[END_REF], [START_REF] Yau | Open problems in geometry. Differential geometry: partial differential equations on manifolds[END_REF] and to the recent books [START_REF] Henrot | Extremum Problems for Eigenvalues of Elliptic Operators[END_REF], [START_REF] Kesavan | Symmetrization & applications[END_REF].

Usually, in these minimization problems, one works in the class of sets with a given measure. In this paper, on the contrary we choose to look at similar problems but with a constraint on the perimeter of the competing sets. Apart the mathematical own interest of this question, the reason which led us to consider this problem is the following. Studying the famous gap problem (originally considered in [START_REF] Van Den | On condensation in the free-boson gas and the spectrum of the Laplacian[END_REF], see Section 7 in [START_REF] Ashbaugh | Isoperimetric inequalities for eigenvalues of the Laplacian, Spectral Theory and Mathematical Physics: a Festschrift in honor of Barry Simon's 60th birthday[END_REF] for a comprehensive bibliography on this problem), we were interested in minimizing λ 2 (Ω)λ 1 (Ω), and more generally λ 2 (Ω) -kλ 1 (Ω), with 0 ≤ k ≤ 1, among (convex) open sets of given diameter. Looking at the limiting case k = 0, we realized that the optimal set (which does exist) is a body of constant width. Since all bodies of constant width have the same perimeter in dimension two, we were naturally led to consider the problem of minimizing λ 2 (Ω) among sets of given perimeter. In particular, if the solution was a ball (or more generally a body of constant width), it would give the answer to the previous problem. Unfortunately, as it is shown in Theorem 2.5, it is not the case! The minimizer that we are able to identify and characterize here (at least in two dimension) is a particular regular convex body, with two points on its boundary where the curvature vanishes. It is worth observing that the four following minimization problems for the second eigenvalue have different solutions: with a volume constraint: two identical disks (see [START_REF] Krahn | Über Minimaleigenschaften der Kugel in drei un mehr Dimensionen[END_REF] or [START_REF] Henrot | Extremum Problems for Eigenvalues of Elliptic Operators[END_REF]), with a volume and a convexity constraint: a stadium-like set (see [START_REF] Henrot | Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions[END_REF]), with a perimeter constraint: the convex set described in this paper, with a diameter constraint: we conjecture that the solution is a disk.

Let us remark that the same problems for the first eigenvalue all have the disk as the solution thanks to Faber-Krahn inequality and the classical isoperimetric inequality.

This paper is organized as follows: section 2 is devoted to the complete study of the two-dimensional problem. We first prove the existence of a minimizer and its C ∞ regularity. Then, we give some other qualitative and geometric properties of the minimizer. For that purpose, we use boundary variation (the classical Hadamard's formulae) which leads to an overdetermined boundary value problem, with |gradu 2 | 2 proportional to the curvature of the boundary. We use this boundary condition to prove that the boundary of the optimal domain does not contain any arc of circle and segment and that the curvature of the boundary vanishes at exactly two points. In section 3, we consider the problem in higher dimension; this case is much more complicated since we cannot use the trick of convexification, and actually we conjecture that optimal domains are not convex (see section 4 on open problems). We first give some preliminaries on capacity and γ-convergence (we refer to the book [START_REF] Bucur | Variational Methods in Shape Optimization Problems[END_REF] for all details), then we consider a quite general minimization problem for a class of functionals decreasing with respect to set inclusion and which are γ lower semicontinuous. We work here with measurable sets with bounded perimeter which are included in some fixed bounded domain D. As Theorem 3.6 shows, this relaxed problem is equivalent to the initial problem. For the second eigenvalue of the Laplacian we moreover prove that we can get rid of the assumption that the sets lie in some bounded subset of R N .

2 The two-dimensional case

Existence, regularity

We want to solve the minimization problem

min{λ 2 (Ω), Ω ⊂ R 2 , P (Ω) ≤ c} (1)
where λ 2 (Ω) is the second eigenvalue of the Laplacian with Dirichlet boundary condition on the bounded open set Ω and P (Ω) denotes the perimeter (in the sense of De Giorgi) of Ω. The monotonicity of the eigenvalues of the Dirichlet-Laplacian with respect to the inclusion has two easy consequences:

1. If Ω * denotes the convex hull of Ω, since in two dimensions and for a connected set, P (Ω * ) ≤ P (Ω), it is clear that we can restrict ourselves to look for minimizers in the class of convex sets with perimeter less or equal than c.

2. Obviously, it is equivalent to consider the constraint P (Ω) ≤ c or

P (Ω) = c.
Of course, point 1 above easily implies existence (see Theorem 2.2 below), but is no longer true in higher dimension which makes the existence proof much harder, see Theorem 3.8. For the regularity of optimal domains the following lemma will be used.

Lemma 2.1. If Ω is a minimizer of problem (1), then λ 2 (Ω) is simple.

Proof. The idea of the proof is to show that a double eigenvalue would split under boundary perturbation of the domain, with one of the eigenvalues going down. A very similar result is proved in [START_REF] Henrot | Extremum Problems for Eigenvalues of Elliptic Operators[END_REF]Theorem 2.5.10]. The new difficulties here are the perimeter constraint (instead of the volume) and the fact that the domain Ω is convex, but not necessarily regular. Nevertheless, we know that any eigenfunction of a convex domain is in the Sobolev space H 2 (Ω), see [START_REF] Grisvard | Elliptic Problems in Nonsmooth Domains[END_REF]. Let us assume, for a contradiction, that λ 2 (Ω) is not simple, then it is double because Ω is a convex domain in the plane, see [START_REF] Lin | On the second eigenfunctions of the Laplacian in R 2[END_REF]. Let us recall the result of derivability of eigenvalues in the multiple case (see [START_REF] Cox | Extremal eigenvalue problems for the Laplacian, Recent advances in partial differential equations[END_REF] or [START_REF] Rousselet | Shape design sensitivity of a membrane[END_REF]). Assume that the domain Ω is modified by a regular vector field x → x + tV (x). We will denote by Ω t the image of Ω by this transformation. Of course, Ω t may be not convex but we have actually no convexity constraint (since convexity come for free) and this has no consequence on the differentiability of t → λ 2 (Ω t ). Let us denote by u 2 , u 3 two orthonormal eigenfunctions associated to λ 2 , λ 3 . Then, the first variation of λ 2 (Ω t ), λ 3 (Ω t ) are the repeated eigenvalues of the 2 × 2 matrix

M =    -∂Ω ∂u 2 ∂n 2 V.n dσ -∂Ω ∂u 2 ∂n ∂u 3 ∂n V.n dσ -∂Ω ∂u 2 ∂n ∂u 3 ∂n V.n dσ -∂Ω ∂u 3 ∂n 2 V.n dσ    . (2)
Now, let us introduce the Lagrangian L(Ω) = λ 2 (Ω)+ µP (Ω). As we will see in the proof of Theorem 2.2, the perimeter is differentiable and the derivative is a linear form in V.n supported on ∂Ω (see e.g. [START_REF] Henrot | Variation et Optimisation de Formes[END_REF]Corollary 5.4.16]). We will denote by dP ∂Ω , V.n this derivative. So the Lagrangian L(Ω t ) has a derivative which is the smallest eigenvalue of the matrix M + µ dP ∂Ω , V.n I where I is the identity matrix. Therefore, to reach a contradiction (with the optimality of Ω), it suffices to prove that one can always find a deformation field V such that the smallest eigenvalue of this matrix is negative. Let us consider two points A and B on ∂Ω and two small neighborhoods γ A and γ B of these two points of same length, say 2δ. Let us choose any regular function ϕ(s) defined on (-δ, +δ) (vanishing at the extremities of the interval) and a deformation field V such that

V.n = +ϕ on γ A , V.n = -ϕ on γ B , V.n = 0 elsewhere . Then, the matrix M + µ dP ∂Ω , V.n I splits into two matrices M A -M B
where M A is defined by (and a similar formula for M B ):

M A =    dP γ A , ϕ -γ A ∂u 2 ∂n 2 ϕ dσ -γ A ∂u 2 ∂n ∂u 3 ∂n ϕ dσ -γ A ∂u 2 ∂n ∂u 3 ∂n ϕ dσ dP γ A , ϕ -γ A ∂u 3 ∂n 2 ϕ dσ    . (3)
In particular, it is clear that the exchange of A and B replaces the matrix M A -M B by its opposite. Therefore, the only case where one would be unable to choose two points A, B and a deformation ϕ such that the matrix has a negative eigenvalue is if M A -M B is identically zero for any ϕ. But this implies, in particular

γ A ∂u 2 ∂n ∂u 3 ∂n ϕ dσ = γ B ∂u 2 ∂n ∂u 3 ∂n ϕ dσ (4)
and

γ A ∂u 2 ∂n 2 - ∂u 3 ∂n 2 ϕ dσ = γ B ∂u 2 ∂n 2 - ∂u 3 ∂n 2 ϕ dσ (5)
for any regular ϕ and any points A and B on ∂Ω. This implies that the has to be constant. Since the nodal line of the second eigenfunction touches the boundary in two points (see [START_REF] Melas | On the nodal line of the second eigenfunction of the Laplacian in R 2[END_REF] or [START_REF] Alessandrini | Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains[END_REF]), ∂u 2 ∂n has to change sign. So we get a function belonging to H 1/2 (∂Ω) taking values c and -c on sets of positive measure, which is absurd, unless c = 0. This last issue is impossible by the Holmgren uniqueness theorem.

We are now in a position to prove the existence and regularity of optimal domains for problem (1). Theorem 2.2. There exists a minimizer Ω for problem [START_REF] Adams | Function Spaces and Potential Theory[END_REF] and

Ω is of class C ∞ .
Proof. To show the existence of a solution we use the direct method of calculus of variations. Let Ω n be a minimizing sequence that, according to point 1 above, we can assume made by convex sets. Moreover, Ω n is a bounded sequence because of the perimeter constraint. Therefore, there exists a convex domain Ω and a subsequence still denoted by Ω n such that:

Ω n converges to Ω for the Hausdorff metric and for the L 1 convergence of characteristic functions (see e.g. [START_REF] Henrot | Variation et Optimisation de Formes[END_REF]Theorem 2.4.10]); since Ω n and Ω are convex this implies that Ω n → Ω in the γ-convergence; P (Ω) ≤ c (because of the lower semicontinuity of the perimeter for the L 1 convergence of characteristic functions, see [START_REF] Henrot | Variation et Optimisation de Formes[END_REF]Proposition 2.3.6]); λ 2 (Ω n ) → λ 2 (Ω) (continuity of the eigenvalues for the γ-convergence, see [START_REF] Bucur | Variational Methods in Shape Optimization Problems[END_REF]Proposition 2.4.6] or [START_REF] Henrot | Extremum Problems for Eigenvalues of Elliptic Operators[END_REF]Theorem 2.3.17], see also Section 3.1 below).

Therefore, Ω is a solution of problem (1).

We notice that the limit Ω is a "true" domain (i.e. it is not the empty set); indeed any degenerating sequence, a sequence shrinking to a segment for instance, converges to the empty set, thus the second eigenvalue blows to infinity.

We go on with the proof of regularity, which is classical, see e.g. [START_REF] Chambolle | C ∞ regularity of the free boundary for a two-dimensional optimal compliance problem[END_REF]. We refer also to [START_REF] Brianc ¸on | Regularity of optimal shapes for the Dirichlet's energy with volume constraint[END_REF], [START_REF] Brianc ¸on | Lipschitz continuity of state functions in some optimal shaping[END_REF] and [START_REF] Brianc ¸on | Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints[END_REF] for similar results in a more complicated context. Let us consider (locally) the boundary of ∂Ω as the graph of a (concave) function h(x), with x ∈ (-a, a). We make a perturbation of ∂Ω using a regular function ψ compactly supported in (-a, a), i.e. we look at Ω t whose boundary is h(x)+ tψ(x). The function t → P (Ω t ) is differentiable at t = 0 (see [START_REF] Giusti | Minimal Surfaces and Functions of Bounded Variation[END_REF] or [START_REF] Henrot | Variation et Optimisation de Formes[END_REF]) and its derivative dP (Ω, ψ) at t = 0 is given by:

dP (Ω, ψ) := +a -a h ′ (x)ψ ′ (x) dx 1 + h ′ (x) 2 . ( 6
)
In the same way, thanks to Lemma 2.

dλ 2 (Ω, ψ) := - +a -a |∇u 2 (x, h(x))| 2 ψ(x) dx. (7)
The optimality of Ω implies that there exists a Lagrange multiplier µ such that, for any ψ ∈ C ∞ 0 (-a, a) µdλ 2 (Ω, ψ) + dP (Ω, ψ) = 0 which implies, thanks to ( 6) and [START_REF] Belhachmi | Shape optimization problems for eigenvalues of elliptic operators[END_REF], that h is a solution (in the sense of distributions) of the o.d.e.:

- h ′ (x) 1 + h ′ (x) 2 ′ = µ|∇u 2 (x, h(x))| 2 . ( 8
)
Since u 2 ∈ H 2 (Ω), its first derivatives ∂u 2 ∂x and ∂u 2 ∂y have a trace on ∂Ω which belong to H 1/2 (∂Ω). Now, the Sobolev embedding in one dimension H 1/2 (∂Ω) ֒→ L p (∂Ω) for any p > 1 shows that x → |∇u 2 (x, h(x))| 2 is in L p (-a, a) for any p > 1. Therefore, according to (8), the function h ′ / 1 + h ′ 2 is in W 1,p (-a, a) for any p > 1 (recall that h ′ is bounded because Ω is convex), so it belongs to some Hölder space C 0,α ([-a, a]) (for any α < 1, according to Morrey-Sobolev embedding). Since h ′ is bounded, it follows immediately that h belongs to C 1,α ([-a, a]). Now, we come back to the partial differential equation and use an intermediate Schauder regularity result (see [START_REF] Gilbarg | Intermediate Schauder estimates[END_REF] or the remark after Lemma 6.18 in [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]) to claim that if ∂Ω is of class C 1,α , then the eigenfunction u 2 is C 1,α (Ω) and |∇u 2 | 2 is C 0,α . Then, looking again to the o.d.e. ( 8) and using the same kind of Schauder's regularity result yields that h ∈ C 2,α . We iterate the process, thanks to a classical bootstrap argument, to conclude that h is C ∞ . Remark 2.3. Working harder, it seems possible to prove analyticity of the boundary. It would also give another proof of points 1 and 2 of Theorem 2.5 below.

Qualitative properties

Since we know that the minimizers are of class C ∞ , we can now write rigorously the optimality condition. Under variations of the boundary (replace Ω by Ω t = (I + tV )(Ω)), the shape derivative of the perimeter is given by (see Section 2.1 and [22, Corollary 5.4.16])

dP (Ω; V ) = ∂Ω C V.n dσ
where C is the curvature of the boundary and n the exterior normal vector. Using the expression of the derivative of the eigenvalue given in [START_REF] Belhachmi | Shape optimization problems for eigenvalues of elliptic operators[END_REF] (see also [START_REF] Henrot | Variation et Optimisation de Formes[END_REF]Theorem 5.7.1]), the proportionality of these two derivatives through some Lagrange multiplier yields the existence of a constant µ such that

|∇u 2 | 2 = µC on ∂Ω . (9)
Setting X = (x 1 , x 2 ), multiplying the equality in (9) by X.n and integrating on ∂Ω yields, thanks to Gauss formulae ∂Ω C X.n dσ = P (Ω), and a classical application of the Rellich formulae ∂Ω |∇u 2 | 2 X.n dσ = 2λ 2 (Ω), the value of the Lagrange multiplier. So, we have proved:

Proposition 2.4. Any minimizer Ω satisfies

|∇u 2 (x)| 2 = 2λ 2 (Ω) P (Ω) C(x) , x ∈ ∂Ω ( 10
)
where C(x) is the curvature at point x.

As a consequence, we can state some qualitative properties of the optimal domains. Theorem 2.5. An optimal domain satisfies:

1. Its boundary does not contain any segment.

Its boundary does not contain any arc of circle.

3. Its boundary contains exactly two points where the curvature vanishes.

Proof. An easy consequence of Hopf's lemma (applied to each nodal domain) is that the normal derivative of u 2 only vanishes on ∂Ω at points where the nodal line hits the boundary. Now, we know (see [START_REF] Melas | On the nodal line of the second eigenfunction of the Laplacian in R 2[END_REF] or [START_REF] Alessandrini | Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains[END_REF]) that there are exactly two such points. Then, the first and third items follow immediately from the "over-determined" condition [START_REF] Brianc ¸on | Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints[END_REF]. The second item has already been proved in a similar situation in [START_REF] Henrot | Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions[END_REF]. We repeat the proof here for the sake of completeness. Let us assume that ∂Ω contains a piece of circle γ. According to [START_REF] Brianc ¸on | Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints[END_REF], Ω satisfies the optimality condition

∂u 2 ∂n = c (constant) on γ . ( 11
)
We put the origin at the center of the corresponding disk and we introduce the function w(x, y) = x ∂u 2 ∂y -y ∂u 2 ∂x .

Then, we easily verify that    -∆w = λ 2 w in Ω w = 0 on γ ∂w ∂n = 0 on γ. Now we conclude, using Holmgren uniqueness theorem, that w must vanish in a neighborhood of γ, so in the whole domain by analyticity. Now, it is classical that w = 0 imply that u 2 is radially symmetric in Ω. Indeed, in polar coordinates, w = 0 implies ∂u ∂θ = 0. Therefore Ω would be a disk which is impossible since it would contradict point 3.

3 The N -dimensional case

Preliminaries on capacity and related modes of convergence

We will use the notion of capacity of a subset E of R N , defined by

cap (E) = inf R N (|∇u| 2 + u 2 ) dx : u ∈ U E ,
where U E is the set of all functions u of the Sobolev space H 1 (R N ) such that u ≥ 1 almost everywhere in a neighbourhood of E. Below we summarize the main properties of the capacity and the related convergences. For further details we refer to [START_REF] Bucur | Variational Methods in Shape Optimization Problems[END_REF] or to [START_REF] Henrot | Variation et Optimisation de Formes[END_REF].

If a property P (x) holds for all x ∈ E except for the elements of a set Z ⊂ E with cap (Z) = 0, then we say that P (x) holds quasi-everywhere (shortly q.e.) on E. The expression almost everywhere (shortly a.e.) refers, as usual, to the Lebesgue measure.

A subset Ω of R N is said to be quasi-open if for every ε > 0 there exists an open subset Ω ε of R N , such that cap (Ω ε ∆Ω) < ε, where ∆ denotes the symmetric difference of sets. Equivalently, a quasi-open set Ω can be seen as the set {u > 0} for some function u belonging to the Sobolev space H 1 (R N ). Note that a Sobolev function is only defined quasi-everywhere, so that a quasi-open set Ω does not change if modified by a set of capacity zero.

In this section we fix a bounded open subset D of R N with a Lipschitz boundary, and we consider the class A(D) of all quasi-open subsets of D. For every Ω ∈ A(D) we denote by H 1 0 (Ω) the space of all functions u ∈ H 1 0 (D) such that u = 0 q.e. on D \ Ω, endowed with the Hilbert space structure inherited from H 1 0 (D). This way

H 1 0 (Ω) is a closed subspace of H 1 0 (D).
If Ω is open, then the definition above of H 1 0 (Ω) is equivalent to the usual one (see [START_REF] Adams | Function Spaces and Potential Theory[END_REF]). For Ω ∈ A(D) the linear operator -∆ on H 1 0 (Ω) has discrete spectrum, again denoted by

λ 1 (Ω) ≤ λ 2 (Ω) ≤ λ 3 (Ω) ≤ • • • .
For Ω ∈ A(D) we consider the unique weak solution w Ω ∈ H 1 0 (Ω) of the elliptic problem formally written as

-∆w = 1 in Ω w = 0 on ∂Ω. ( 12
)
Its precise meaning is given via the weak formulation

D ∇w∇φ dx = D φ dx ∀φ ∈ H 1 0 (Ω).
Here, and in the sequel, for every quasi-open set of finite measure we denote by R ω : L 2 (R N ) -→ L 2 (R N ) the operator defined by R ω (f ) = u, where u solves equation ( 12) with the right-hand side f , so w = R ω (1). Now, we introduce two useful convergences for sequences of quasi-open sets.

Definition 3.1. A sequence (Ω n ) of quasi-open sets is said to γ-converge to a quasi-open set Ω if w Ωn → w Ω in L 2 (R N ).
The following facts about γ-convergence are known (see [START_REF] Bucur | Variational Methods in Shape Optimization Problems[END_REF]).

(i) The class A(D), endowed with the γ-convergence, is a metrizable and separable space, but it is not compact.

(ii) The γ-compactification of A(D) can be fully characterized as the class of all capacitary measures on D, that are Borel nonnegative measures, possibly +∞ valued, that vanish on all sets of capacity zero.

(iii) For every integer k the map Ω → λ k (Ω) is a map which is continuous for the γ-convergence.

To overcome the lack of compactness of the γ-convergence, it is convenient to introduce a weaker convergence.

Definition 3.2. A sequence (Ω n ) of quasi-open sets is said to wγ-converge to a quasi-open set Ω if w Ωn → w in L 2 (R N ), and Ω = {w > 0}.
The main facts about wγ-convergence are the following (see [START_REF] Bucur | Variational Methods in Shape Optimization Problems[END_REF]).

(i) The wγ-convergence is compact on the class A(D).

(ii) The wγ-convergence is weaker that the γ-convergence.

(iii) Every functional F (Ω) which is lower semicontinuous for γ-convergence, and decreasing for set inclusion, is lower semicontinuous for wγconvergence too. In particular, for every integer k, the mapping Ω → λ k (Ω) is wγ-lower semicontinuous.

(iv) The Lebesgue measure Ω → |Ω| is a wγ-lower semicontinuous map. This last property can be generalized by the following.

Proposition 3.3. Let f ∈ L 1 (D) be a nonnegative function. Then the mapping Ω → Ω f dx is wγ-lower semicontinuous on A(D).

Proof. Let (Ω n ) be a sequence in A(D) that wγ-converges to some Ω ∈ A(D); this means that w Ωn → w in L 2 (R N ) and that Ω = {w > 0}. Passing to a subsequence we may assume that w Ωn → w a.e. on D. Suppose x ∈ Ω is a point where w Ωn (x) → w(x). Then w(x) > 0, and for n large enough we have that w Ωn (x) > 0. Hence x ∈ Ω n . So we have shown that

1 Ω (x) ≤ lim inf n→+∞ 1 Ωn (x)
for a.e. x ∈ D.

Fatou's lemma now completes the proof.

The link between wγ-convergence and L 1 -convergence is given by the following.

Proposition 3.4. Let (A n ) be a sequence of quasi-open sets which wγconverges to a quasi-open set A, and assume that there exist measurable sets Ω n such that A n ⊂ Ω n , and that (Ω n ) converges in L 1 to a measurable set Ω. Then we have |A \ Ω| = 0.

Proof. By applying Proposition 3.3 with f = 1 D\Ω we obtain

|A \ Ω| ≤ lim inf n→∞ |A n \ Ω| = lim inf n→∞ |A n \ Ω n | = 0,
which concludes the proof.

A general existence result

In this section we consider general shape optimization problems of the form inf

F (Ω) : Ω ⊂ D, P (Ω) ≤ L , (13)
where D is a given bounded open subset of R N with a Lipschitz boundary, and L is a given positive real number. Finally, the cost function F is a map defined on the class A(D) of admissible domains. We assume that:

F is γ-lower semicontinuous on A(D); F is decreasing with respect to set inclusion.

The functional F is then wγ-lower semicontinuous by Section 3.1. Some interesting examples of functionals F satisfying ( 14) are listed below. (i) F (Ω) = Φ λ(Ω) , where λ(Ω) denotes the spectrum of the Dirichlet Laplacian in Ω, that is the sequence λ k (Ω) of the Dirichlet eigenvalues, and the function Φ : R N → R is lower semicontinuous and nondecreasing, in the sense that

λ n k → λ k ∀k ∈ N ⇒ Φ(λ) ≤ lim inf n→∞ Φ(λ n ) λ k ≤ µ k ∀k ∈ N ⇒ Φ(λ) ≤ Φ(µ).
(ii) F (Ω) = cap (D \ Ω), where cap denotes the capacity defined in Section 3.1.

(iii) F (Ω) = D g x, u Ω (x) dx, where g(x, •) is lower semicontinuous and decreasing on R for a.e. x ∈ D, and u Ω denotes the solution of

-∆u = f in Ω u = 0 on ∂Ω with f ∈ H -1 (D) and f ≥ 0.
In order to treat the variational problem [START_REF] Chambolle | C ∞ regularity of the free boundary for a two-dimensional optimal compliance problem[END_REF] it is convenient to extend the definition of the functional F also for measurable sets, where the notion of perimeter has a natural extension. If M ⊂ R N is a measurable set of finite measure, we define

H 1 0 (M ) := {u ∈ H 1 (R N ) : u = 0 a.e. on R N \ M }. (15)
For an arbitrary open set Ω, this definition does not coincide with the usual definition of H 1 0 (Ω). Nevertheless, we point out that it is not restrictive to consider this definition since, for every measurable set M ⊂ R N , there exists a uniquely defined quasi open set ω (see for instance [START_REF] Ashbaugh | On the isoperimetric inequality for the buckling of a clamped plate[END_REF]) such that

H 1 0 (ω) = H 1 0 (M ),
while for a smooth set Ω, (e.g. Lipschitz, see [START_REF] Henrot | Variation et Optimisation de Formes[END_REF]Lemma 3.2.15]) the definition of the spaces H 1 0 (Ω) and H 1 0 (Ω) coincide. With this notion of generalized Sobolev space, one can define

F (M ) := F (ω).
For a nonsmooth open set Ω (say with a crack), we have that H 1 0 (Ω) strictly contains H 1 0 (Ω), which may lead to the idea that l m < l o , where

l m = inf F (M ) : M ⊂ D, P (M ) ≤ L l o = inf F (Ω) : Ω ⊂ D, P (Ω) ≤ L . (16)
In practice, when solving (13), the minimizing sequence will not develop cracks, precisely because by erasing a crack the generalized perimeter is unchanged and the functional decreases as a consequence of its monotonicity (it may remain constant if the crack coincide with some part of the nodal line of lambda 2 ).

Let us notice that for every measurable set M

F (M ) = inf F (A) : A ⊂ M a.e., A ∈ A(D) . (17)
Theorem 3.5. There exists a finite perimeter set M * which solves the variational problem

inf F (M ) : M ⊂ D, P (M ) ≤ L . (18)
Proof. Let (M n ) be a minimizing sequence for problem [START_REF] Giusti | Minimal Surfaces and Functions of Bounded Variation[END_REF]. Since P (M n ) ≤ L we may extract a subsequence (still denoted by (M n )) that converges in L 1 to a set M * with P (M * ) ≤ L.

There are quasi-open sets ω n ⊂ M n a.e. such that

F (ω n ) = F (M n ).
By the compactness of wγ-convergence we may assume that (ω n ) is wγconverging to some quasi-open set ω, and by Proposition 3.4 we have |ω \ M * | = 0. Therefore, we have that

F (M * ) ≤ F (ω) ≤ lim inf n→∞ F (ω n ) = lim inf n→∞ F (M n ).
Hence M * solves the variational problem [START_REF] Giusti | Minimal Surfaces and Functions of Bounded Variation[END_REF].

The relaxed formulation we have chosen (i.e. to work with measurable sets instead of open sets or quasi-open sets) is not a restriction, provided F verifies the following mild γ-continuity property:

ω n ∈ A(D), ω n ⊂ ω, ω n γ → ω =⇒ F (ω n ) → F (ω). ( 19
)
For instance, all spectral functionals F (Ω) = Φ λ(Ω) seen in (i) above fulfill property [START_REF] Grisvard | Elliptic Problems in Nonsmooth Domains[END_REF] provided Φ is continuous; similarly, integral functionals F (Ω) = D g x, u Ω (x) dx seen in (iii) above fulfill property (19) provided g(x, •) is continuous and with quadratic growth.

Theorem 3.6. Assume that F satisfies [START_REF] Grisvard | Elliptic Problems in Nonsmooth Domains[END_REF]. Then, problems [START_REF] Chambolle | C ∞ regularity of the free boundary for a two-dimensional optimal compliance problem[END_REF] and [START_REF] Giusti | Minimal Surfaces and Functions of Bounded Variation[END_REF] are equivalent in the sense that l o = l m , where l 0 and l m are defined in [START_REF] Gilbarg | Intermediate Schauder estimates[END_REF].

Proof. Clearly, l m ≤ l o , since for every quasi open set Ω we have H 1 0 (Ω) ⊂ H 1 0 (ω) where H 1 0 (ω) = H 1 0 (Ω). In order to prove the converse inequality, let M be measurable such that P (M ) ≤ L. There exists a quasi-open set ω ⊂ M a.e. with

H 1 0 (ω) = H 1 0 (M ).
We point out that the measure of M \ ω may be strictly positive, and that P (ω) may be strictly greater than L.

Following the density result of smooth sets into the family of measurable sets [START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs[END_REF]Theorem 3.42], there exists a sequence of smooth sets Ω n , such that

1 Ωn → 1 M in L 1 (R N ) H N -1 (∂Ω n ) -→ P (M ).
Unfortunately, it is immediate to observe that this implies only

F (M ) ≤ lim inf n→∞ F (Ω n ),
while we are seeking precisely the opposite inequality.

We consider the function w = R ω (1), solution of the elliptic problem [START_REF] Bucur | Variational Methods in Shape Optimization Problems[END_REF] in ω, and a sequence (ρ k ) k of convolution kernels. As in the proof of [START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs[END_REF]Theorem 3.42], by the definition of perimeter and the coarea formula we have

P (M ) = R N |∇1 M | = lim k→∞ R N |∇(1 M * ρ k )| dx = lim k→∞ 1 0 P ({1 M * ρ k > t}) dt ≥ 1 0 lim inf k→∞ P ({1 M * ρ k > t}) dt.
For every t ∈ (0, 1) we have

|{1 M * ρ k > t} \ M | ≤ 1 t R N |1 M * ρ k -1 M | dx |M \ {1 M * ρ k > t}| ≤ 1 1 -t R N |1 M * ρ k -1 M | dx, so that 1 {1 M * ρ k >t} converges in L 1 (R N ) to 1 M and lim inf k→∞ P ({1 M * ρ k > t}) ≥ P (M ).
The above inequalities imply that for almost every t ∈ (0, 1)

lim inf k→∞ P ({1 M * ρ k > t}) = P (M ).
For a subsequence (still denoted using the index k) we have

1 {1 M * ρ k >t} → 1 M in L 1 (R N ) P ({1 M * ρ k > t}) → P (M ).
We notice that up to a set of zero measure {w > 0} ⊂ M . We may assume that w ≤ 1 (otherwise we consider in the sequel w w ∞ ). Then we get w

* ρ k ≤ 1 M * ρ k so {w * ρ k > t} ⊂ {1 M * ρ k > t} and F ({1 M * ρ k > t}) ≤ F ({w * ρ k > t}).
Let us prove that lim sup k→∞

F ({w * ρ k > t}) ≤ F ({w > t}). (20)
Thanks to [START_REF] Grisvard | Elliptic Problems in Nonsmooth Domains[END_REF] and to the monotonicity property, it is enough to prove that {w * ρ k > t} ∩ {w > t} γ-converges to {w > t}, so to show (see for instance [START_REF] Bucur | Variational Methods in Shape Optimization Problems[END_REF]Chapters 4,[START_REF] Ashbaugh | On the isoperimetric inequality for the buckling of a clamped plate[END_REF]) that for every ϕ ∈ H 1 0 ({w > t}) there exists a sequence

ϕ k ∈ H 1 0 ({w * ρ k > t}) such that ϕ k → ϕ strongly in H 1 0 .
Using the density result of [START_REF] Maso | Asymptotic behaviour and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators[END_REF], it is enough to choose ϕ ≥ 0 such that ϕ ≤ (w -t) + and take

ϕ k = min{ϕ, (w * ρ k ) -t + }.
Being (20) true for every t, from the convergence F ({w > t}) → F ({w > 0}) as t → 0, by a diagonal procedure we can choose t k → 0 such that lim sup k→∞

F ({w * ρ k > t k }) ≤ F ({w > 0}) = F (ω) = F (M ), and
1 {1 M * ρ k >t k } → 1 M in L 1 (R N ) P ({1 M * ρ k > t k }) → P (M ).
This proves that l o ≤ l m .

The case of the second eigenvalue

As a corollary of Theorem 3.5, since any eigenvalue of the Laplace operator satisfies (14), we have: In this section, we will show that for the second eigenvalue, one can improve the previous Theorem. Actually, we will prove that minimizers exist if D is replaced by all of R N . The proof relies on a concentrationcompactness argument.

Theorem 3.8. Problem inf λ 2 (M ) : M measurable set in R N , P (M ) ≤ L := l m , (22)
has a solution.

Proof. Let (M n ) be a minimizing sequence for problem [START_REF] Henrot | Variation et Optimisation de Formes[END_REF]. We shall use a concentration compactness argument for the resolvent operators (see [START_REF] Bucur | Uniform concentration-compactness for Sobolev spaces on variable domains[END_REF]Theorem 2.2]). Let ω n be the quasi-open sets such that ω n ⊂ M n a.e. and λ 2 (ω n ) = λ 2 (M n ). From the classical isoperimetric inequality, the measures of ω n are uniformly bounded. Consequently, for a subsequence (still denoted using the same index) two situations may occur: compactness and dichotomy.

For a positive Borel measure µ, vanishing on sets with zero capacity, we define (see [START_REF] Bucur | Variational Methods in Shape Optimization Problems[END_REF]) by R µ (f ) the solution of the elliptic problem

-∆u + µu = f in R N , u ∈ H 1 (R N ) ∩ L 2 (µ).
If the compactness issue holds, there exists a measure µ and a sequence of vectors y n ∈ R N such that the resolvent operators R ωn+yn converge strongly in L(L 2 (R N )) to R µ . Since the perimeters of M n are uniformly bounded, we can define (up to subsequences) the sets M k as the limits of M n ∩ B(0, k), and M = ∪ k M k .

Since w ωn+yn converges strongly in L 2 (R N ) to w µ , we get that {w µ > 0} ⊂ M a.e. so that λ 2 (M ) ≤ λ 2 ({w µ > 0}) ≤ λ 2 (µ) = lim n→∞ λ 2 (ω n ).

On the other hand P (M ) = lim k→∞ P (M ∩ B(0, k)) ≤ lim inf k→∞ P (M k ), so that M is a solution to problem [START_REF] Henrot | Variation et Optimisation de Formes[END_REF].

Let us assume that we are in the dichotomy issue. There exists two sequences of quasi open sets such that according to the cylindrical symmetry, it is along a circle. Therefore, one of the curvatures has to be negative at that point.

       ω 1 n ∪ ω 2 n ⊂ ω n ω 1 n = ω n ∩ B(0, R 1 n ), ω 2 n = ω n ∩ (R N \ B(0, R 2 n )), R 2 n -R 1 n → +∞ lim inf n→∞ |ω i n | > 0, i = 1, 2 R ωn -R ω 1 n -R ω 2 n → 0 in L(L 2 (R N )).

Higher eigenvalues

The existence of an optimal domain for higher order eigenvalues under a perimeter constraint is only available when a geometric constraint Ω ⊂ D is imposed; we conjecture the existence of an optimal domain also when D is replaced by R N but, at present, a proof of this fact is still missing.

Connectedness

We believe that optimal domains for problem

inf λ k (Ω) : Ω open set in R N , H N -1 (∂Ω) ≤ L (23)
are connected for every k and every dimension N . Actually, for k = 2, this result is proved in the forthcoming paper [START_REF] Iversen | On the minimization of Dirichlet eigenvalues of the Laplace operator[END_REF]. The idea of the proof consists, first, to show that in the disconnected case, the domain should be the union of two identical balls. Then, a perturbation argument is used: it is shown that the union of two slightly intersecting open balls gives a lower second eigenvalue (keeping the perimeter fixed by dilatation) than two disconnected balls.

product ∂u 2 ∂n ∂u 3 ∂n 2 and the difference ∂u 2 ∂n 2 -∂u 3 ∂n 2

 22232 should be constant a.e. on ∂Ω. As a consequence ∂u 2 ∂n 2

Theorem 3 . 7 .

 37 Let m = 1, 2, 3, • • • , and let D be a bounded open set in R N with a Lipschitz boundary. Then the variational problem inf λ m (M ) : M measurable set in D, P (M ) ≤ L (21) has a minimizer.

Figure 1 :

 1 Figure 1: A possible minimizer obtained numerically (by courtesy of Edouard Oudet)

Acknowledgements. We wish to thank our colleague Michiel van den Berg for the several stimulating discussions we had on the topic.

Let us denote

Since the measures of M n are uniformly bounded, one can suitably increase R 1 n and decrease R 2 n such that lim sup

and all other properties of the dichotomy issue remain valid.

We have that

The first situation is to be excluded since this implies that M 1 n is a minimizing sequence with perimeter less than or equal to some constant α < L, which is absurd. The second situation leads to an optimum consisting on two disjoint balls (this is a consequence of the Faber-Krahn isoperimetric inequality for the first eigenvalue) which is impossible as mentioned in Section 4.4 below.

Further remarks and open questions 4.1 Regularity

We have proved, in any dimension, the existence of a relaxed solution, that is a measurable set with finite perimeter. A further step would consist in proving that this minimizer is regular (for example C ∞ as it happens in two dimensions). It seems to be a difficult issue, in particular because the eigenfunction is not positive, see for example [START_REF] Brianc ¸on | Regularity of optimal shapes for the Dirichlet's energy with volume constraint[END_REF], [START_REF] Brianc ¸on | Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints[END_REF]. Actually, apart the two-dimensional case, at present we do not even know if optimal domains are open sets.

For a similar problem with perimeter penalization the regularity of optimal domains has been proved in [START_REF] Athanasopoulos | An area-Dirichlet integral minimization problem[END_REF].

Symmetry

Numerical simulations, see Figure 1 show that minimizers in two dimensions should have two axes of symmetry (one of these containing the nodal line), but we were unable to prove it. If one can prove that there is a first axis of symmetry which contains the nodal line, the second axis of symmetry comes easily by Steiner symmetrization.

In higher dimensions, we suspect the minimizer to have a cylindrical symmetry and to be not convex. Indeed, assuming C 2 regularity, one can find the same kind of optimality condition as in [START_REF] Brianc ¸on | Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints[END_REF] with the mean curvature instead of the curvature. Since, the gradient of u 2 still vanishes where the nodal surface hits the boundary, the mean curvature has to vanish and,