Muriel Gilardone

Isabelle Guerin

Jane Palier

The weight of institutions on women's capabilities How far can microfinance help?

The second part of the paper explores one technical issue: how can microfinance make the Capability Approach operational? Microfinance appears more and more as a tool of emancipation and promotion of the feminine condition. The aimed objectives are multiple: increase women's incomes and facilitate their financial independence, stabilize their entrepreneurship activity notably by training, but also, and maybe especially, improve their status within the family and strengthen the respect of themselves, and finally, favour their capacities of auto-organization and thus, their capacities of expression and claiming. In theory, microfinance appears as a means to increase women capabilities. How is it in practice? Available results of impact studies incite to be careful: microfinance can free women of certain links of dependence but also forge new kinds of dependence and subordination, so strengthening disparities among men and women but also among women themselves. Consequently, it is crucial to link microfinance programmes with institutions transformations.

Introduction

The starting point of the reflection is the fact that cultural features of a society are generally unfavourable to women. Even when women formally have the same rights and liberty as men (which is still not effective everywhere), mentalities are strongly rooted in traditions and existing social norms keep oppressing women. As legal opportunities are essential but not sufficient, it is indispensable to act on informal institutions to reinforce women's capabilities -to use Martha Nussbaum and Amartya Sen's concept. By "informal institutions", we mean the set of rules of conduct, norms, routines, traditions which are deeply settled down in every society and implicitly command its organization. One could also call it "social institutions"; the aim here is just to point out that laws are only "the visible part of the iceberg".

Concerning the assessment of women's quality of life, there are many theoretical approaches. For a long time, welfare economics or development economics has been underlain by utilitarianism only concerned with the social maximization of utility, construed as the satisfaction of preference or desire. The limits of this approach have largely been demonstrated, particularly its indifference to welfare distribution and the fact that preferences and tastes are endogenously shaped by laws, traditions and institutions, tending to reproduce inequalities2 . For this reason, the paper will focus on Martha Nussbaum's work, even if a priori her universal list of capabilities seems to ignore the important role played by institutions. One can indeed wonder if her search for ethical objectivity isn't as inadequate as the universal calculus generality of utilitarianists, which she criticises for his lack of consideration for particular contexts and histories. Isn't it meaningless to discuss about individual lives outside the institutional context?

In fact it is clear that Martha Nussbaum knows perfectly well how women's capabilities are reinforced -or weakened -by existing norms. And that is the reason why she refuses desire-based approaches as well as relativists views, which use the local criteria of goodness -often unjust to women -to assess their well-being. As preferences are "adaptative", it seems necessary to base the studies about women's capabilities on "transcultural norms, justifiable by reference to reasons of universal human validity, by reference to which we may appropriately criticise different local conceptions of the good"3 . And her universal list of capabilities -sort of objective basis -allows putting women's lack of liberty to the fore. Why women would have fewer requirements as men would in term of liberty?

The first step of our reflection about the improvement of women's quality of life is therefore to point out what central capabilities are hardly accessible to them. Then it is crucial to understand what institutions -formal and more especially informal -are incompatible with women flourishing in order to act well on it. Their freedom of agency is inescapably qualified and constraint by the social, political and economic opportunities that are available to them. One has to recognize the importance of both individual agency and social arrangements. We will here particularly focus on opportunities of economic participation for women which help greatly to foster their own initiatives in overcoming their deprivations.

I. Capabilities, economic freedom and institutions

In what sense the Capability Approach is relevant?

As Martha Nussbaum notice, "any living culture […] contains relatively powerful voices, relatively silent voices, and voices that cannot speak at all in the public space" 1 . Women are often in the second category, indeed even in the third, simply because their being women.

Women have to face gender-related problems depending on the political and economic circumstances, which are very different from a society to another. But all over the world they are more or less concerned with questions such as rape, sexual harassment, hunger, sex-selective infanticide and abortion, denial of the right to work, sex discrimination in family laws especially in the distribution of food, education and health-care 2 .

If some women try to improve their relatively bad situation, most of them are brought up to lack autonomy and self-respect, which prevent them from being aware of their subordination. For this reason, desire-based approaches are unable to analyse sex inequalities, "since it is difficult to desire what one cannot imagine as a possibility" 3 . For example, John Kenneth Galbraith, in his analysis of poverty, notices that one of the most important causes of the persistence of poverty is "accommodation", that is to say the absence of effort to escape poverty 4 . And he criticises economists who generally think that the search for material progress is natural. On the contrary, he demonstrates that the trend to prefer resignation to frustrated hope is not only understandable, but it is rational [START_REF] Galbraith John | The Nature of Mass Poverty[END_REF]. Although the standard utilitarian approach assumes it, preferences are not always reliable indicators of life quality. Thus, rather than looking only at women's stated satisfactions and preferences, which may be deformed by intimidation, lack of information, and habit, Martha Nussbaum urges us to look at their actual "functioning". More exactly, she is interested in the space of functionings, the various things a person may value doing (or being), e.g. the "capability" of a person.

The capability approach to quality of life assessment has been pioneered within economics by Amartya Sen. It is a will to give a good guidance to governments, offering basic constitutional principles to promote liberty for each individual. Judging individual well-being, neither in terms of commodities consumed nor in terms of the central metric of utilities, it focuses either on the realized functionings or the set of alternatives a person has. It is a "perspective of freedom in a positive sense: who can do what" 5 . By establishing a list of "central human functional capabilities", Martha Nussbaum goes further in the will to describe "a life in which the dignity of the human being is not violated by hunger or fear or 1 NUSSBAUM [1999a], p. 8. 2 See for example INTERNATIONAL CENTER FOR RESEARCH ON WOMEN (ICRW), [1999]. 3 NUSSBAUM, SEN [1993], p. 5. 4 According to Galbraith, recognizing the reality of accommodation is very different of accepting poverty as an inescapable phenomenon. It is above all the mean to understand why many efforts apparently promising to relieve poverty have failed. 5 SEN [1984], p. 376.

the absence of opportunities" 1 . This list is an attempt to answer the question "what are the functions without which we would regard a life as not, or not fully, human?" 2 and proposes a set of interrelated and indivisible capabilities, conceived as human rights.

According to her, moving all citizens above a basic threshold of capability should be taken as a central social goal. We notice that, even if functioning is the effective way of living, capability, and not functioning, is the political goal. In conformity with a form of liberal philosophy (inspired by Kant, Mill and Aristotle), Martha Nussbaum consider that, in virtue of the basic human capacities for choice and reasoning, citizens must be left free to determine their course after that. This respect for choice is crucial for her, but depends of many factors like the access to a minimal subsistence, the level of education, health and also employment, seen as a source of opportunity and empowerment. Indeed, "People who have to fight for the most basic things are precluded by that struggle from exercising their agency in other more fulfilling and socially fruitful ways." 3

Economic freedom: a good means for women's empowerment

Our opportunities and prospect depend crucially on what institutions exist and how they function. At the core of the capability approach, there are the ideas of equal human worth and dignity, which are strongly connected to the idea of liberty. It is clearly opposed to many existing value systems which are highly paternalistic, particularly toward women. Many value systems treat women indeed as unequal under the law, as lacking full civil capacity, as not having the property rights, associative liberties, and employment rights of males. This distinct sex bias against the females all over the world is much more sharper and more widespread in the poorer Third World economics 4 .

More over, as "the family is often the decision-making unit for work and consumption" 5 , the existence of substantial intra-family disparities has serious implication on women's capabilities. According to Amartya Sen, in order to understand the distribution of "entitlements" within the family, there has to be a clearer analysis of the existence of both cooperative and conflicting elements in family relations. He relates this problem to what Nash calls "the bargaining problem":

The essence of the problem is that there are many cooperative outcomes -beneficial to all the parties compared with non-cooperation -but the different parties have strictly conflicting interests in the choice among the set of efficient cooperative arrangements. […] Second, the bargaining problem of finding a particular cooperative solution, yielding a particular distribution of benefits, will be sensitive to various parameters, including the respective powers of the different members of the family, given, for example, by the nature of the "fall-back" positions if there should be a breakdown. 6 How is established the bargaining power within the family 7 ? We can get to this question better if we approach it via three somewhat more concrete questions. Who does what? Who is allowed to consume what? Who decides of what? And it is likely that men have typically better bargaining power, related to better outside job opportunities, that leads to a corresponding more favourable 1 See NUSSBAUM [1999a], pp. 41-42.

2 NUSSBAUM [1999a], p. 39.

3 NUSSBAUM [1999a], p. 19. 4 For example, the Government of India amended the Hindu Personal Law Code in the Fifties in order to ban bigamy, facilitate divorce, counter child marriage, etc. Under the law, girls and boys can now inherit equally but only if there is no will so that in practice cultural traditions of male inheritance are maintained. [DAS GUPTA Monica et alii, 2000]. 5 SEN [1984], p. 360. 6 See [START_REF] Sen Amartya | Resources, Values and Development[END_REF], p. 375. 7 See for example [START_REF] Kabeer Naila | Conflicts over credit: re-evaluating the empowerment potential of loans to women in rural Bangladesh[END_REF]. Her main conclusion about conflicting findings in impact assessment of microcredit programme is that differences lie in understandings of intra-household power relations. Results of empowerment evaluation depend on to which priority is given: positive views focus more on outcomes whereas negative verdict is more based on change processes. Besides implicit pre-notions infer results, i.e. negative studies give more priority to conflict inside households than cooperation and insist more on individualized behaviour. cooperative outcome for the men. Economic independence seems definitely to be more powerful than a contribution which is vital for the family, but not remunerated. Indeed, a woman's perceived contribution to the well-being of the family unit is often determined by her ability to work outside, e.g. her wage, and this determines, in turn, her bargaining position within the family unit. As Martha Nussbaum analyses it, "the wage labor outside the home is highly correlated with a woman's ability to command food and other goods within it"1 . Sometimes the connection is direct, for example in the case of Indian widows starving because of norms that prevent them form earning money. Sometimes the connection is more indirect, through a perception of a woman's importance to the future of the family. In any case, cash wages are usually perceived as making a great contribution to the family's well-being, whereas housework is not.

Thus, since they work outside, women's participation becomes more visible. And because their independence increases, their voices become more audible. Finally, it helps them to stop being only supporters of ends of others, and to subordinate their interests to the larger goal of the family unit. Finally the evolution of their status modifies the generally accepted ideas about their role of reproducers and caregivers. This idea doesn't mean that individual welfare is independent of the welfare of the rest of the family. There is a fundamental difference between the concepts of separateness and independence concerning individual welfare [Sen, 1984, p. 362]. In fact individual welfares may be interdependent, but distinct. The problem concerning women comes from their common propensity to subordinate themselves to others and to sacrifice their well-being for that of their family. This trend is obviously the consequence of long habits of domination and subordination. And Martha Nussbaum links it to Marx concept of exploitation:

"Marx, like his bourgeois forebears, holds that it is profoundly wrong to subordinate the ends of some individuals to those of others. That is at the core of what exploitation is, to treat a person as a mere object for the use of others." 2As the capability approach is promoting a society in which individuals are treated as each worthy of regard, it has to take account of what is valuable in given society. And nowadays, employment is a central value in people's existence. It not only gives an income to the employed, but it also gives him or her "the recognition of being engaged in something worth his while" [Sen, 1984, p. 242]. This fact has not always been; it is historically and culturally dated. For example, "in ancient Greece, there was a common aristocratic prejudice against earning wages. The ancient Greek gentleman was characterized by "leisure" -meaning that he did not have to work for a living" [Nussbaum, 1999a, p. 278]. For Aristotle, the idea of freedom, and dignity, is highly connected to the capacity of not being preoccupied with the baser things in life. And at this period, one could be a citizen only if he was not constrained by economic problems, leaving work to slaves and home management to women. Politics was the central value and dignity stake; economics was seen as a contemptible domain3 . And again one can notice that women were confined to a devaluated sphere.

In short, "it is likely that women's subordination will not be adequately addressed as long as women are confined to a sphere traditionally devaluated, linked with a low "perceived well-being contribution" [Nussbaum, 1999a, p.52]. That is why their ability, or even their inclination, to accept outside employment is crucial for their empowerment.

How institutions lead women's work to perverse effect

A reciprocity relation exists between prejudices and social status. The traditional gendered division of labour -male breadwinner, female homecare -leads to a belief that, on the one hand, males are naturally "autonomous", capable of practical reasoning, independent and self-sufficient, allegedly good at political deliberation. On the other hand, females are considered as naturally focused on pleasing others and good at caring for others, but dependent on males because of their incapacity for practical reasoning.

Consequently of the inequalities of education, or sexist discriminations in the job market, the traditional arrangement that often emerges is women doing housework and being able to take up outside work only if it is additional. Even when women are employed, it is not only often added to long hours of unpaid household labour, but it is generally in a restricted range of jobs offering law pay and law respect. And as Martha Nussbaum underlines it, "the very perpetuation of separate spheres of responsibility might reinforce subordination" [Nussbaum, 1999a, p. 51].

However Amartya Sen explains that the traditional gendered division labour corresponds to what he calls "the social technology"1 [START_REF] Sen Amartya | On Ethics and Economics[END_REF]. Generally, is considered as "work" a production in compensation of which a wage is given. Does it mean that unpaid work is unproductive? If it is often perceived as such, unpaid work is nevertheless necessary to the social organization and to allow what is seen as "real production". In the one hand, it is obvious that workers have to take sustenance, to survive and to reproduce when they are at home. But in the other hand, activities that allow the satisfaction of these vital needs are rarely considered as contributing to the production. Thus, since women work outside their home, there is a sort of imbalance in the "social technology", because they have less time for their traditional activities. As mentalities change slowly, the "division of labour" is strongly rooted in traditions and often becomes an "accumulation of labour" for women. The evolution is long until another equilibrium could emerge and improve women's quality of life. And this evolution is enormously influenced by social arrangements for education, health care, microcredit, and other public policies.

For example, the way a society organises childcare and work in the home is a key issue in terms of the position of women, and men, on the labour market. The model differs radically from country to country. The integration of small child's mothers into the labour market can be facilitated by the welfare state, particularly the well-developed child care systems, nursery schools, or crèches. This is especially the case in Nordic countries or in France. Whereas German society does still support the "hausfrau" model by joint family income taxation and women's pension calculated on the basis of their husband income, many young German women are choosing paid employment despite the fact that child care costs swallow almost their entire salary 2 . This case shows clearly that wage is not the only thing at stake in employment.

More over it is clear that, still because of "informal institutions", women have a very limited access to credit, which prevent them to set out their own business. For this reason, microfinance has an important role to play in order to enhance their capability by offering an economic opportunity, but also illustrates the complex link between economic freedom and real freedom. As we will see in the following part, microfinance appears as a means to improve economic freedom of women, but if others institutional changes do not take place at the same time, this economic freedom does not favour necessarily their autonomy and their real freedom.

II. Microfinance and Capabilities

In the history of the credit to the "poor", targeting women is rather new. While they were deliberately excluded from the first experiments which took place throughout the XX-th century [START_REF] Fournier | Les coopératives d'épargne et de crédit en Afrique[END_REF], today they represent an important part of the clientele. According to the statistics of the IFPRI (International Food Policy Research Institute) concerning Africa, Asia and Latin America, the women represent 45 % of the clientele [START_REF] Lapenu | Distribution, Growth and Performance of Microfinance Institutions in Africa, Asia and Latin America[END_REF]. Concerning western Africa, the last database built by the International Labour Office gives similar results (44%), and on 174 inventoried microfinance intermediaries, 46 (that is 26 %) are exclusively intended for women [BIT / BCEAO, 1999].

To recognize this shape of disparity is completely justifiable and justified. However, this focalisation on women lifts several questions. First of all, what kind of women does it involve ? Not only the category "woman" is very diverse -consequently, the criteria "female targeting" is not sufficient -but still more, there is no reason that microfinance, taken remotely, is enough to promote autonomy and women freedom.

Microfinance impacts : contrasted results

Let us begin with a first observation : except our own fieldwork -an impact study led in Senegal, on a women program set up by the Crédit Mutuel -different researches have shown the positive effects of microfinance, even if they are contrasted. This contrast holds as much in the variety of the socioeconomic context, in the profile of the borrowers, that in the offer of credit, and probably as well, in the methods of data collection1 .

Improvement in quality of life

Because of high rate of profitability (on average, between 50 and 100 %), microcredit generates profits which can be important. A high part of these profits -about 50 % on average -and often a part of the credit, is allocated to expenditure which improve appreciably day to day life, even if they can be considered as "unproductive". This observation illustrates the priorities of the borrowers. For example in Senegal, among the borrowers, the poorest are more worried by expenditure of food, clothing, health, schooling and ceremonies, whereas the richest invest to develop their commercial activity, build their house, send their children abroad.

Several longitudinal studies show that the effects are sometimes of long term. This kind of effect has been observed in several fields : food security in Ghana, education in Cameroon [START_REF] Schrieder | Impact of Poverty Reduction[END_REF], health, clothing and construction in Bangladesh [Doligez and Bissonais, 1996] and in Guinea [START_REF] Doligez | Dix ans d'études de l'impact de la microfinance : synthèse de quelques observations de terrain[END_REF]. In Niger, women invest first and foremost in the dowry of their daughters. This, normally, assures them a more respectable marriage and thus a better future [START_REF] Schrieder | Impact of Poverty Reduction[END_REF]. Sometimes, one also observes that the effects exceed the domestic scale and concerns local and regional markets. In Guinea for example, 30% of incomes led by the credit are assigned to the housing upgradation, what revitalizes strongly the local market of the house building [START_REF] Doligez | Dix ans d'études de l'impact de la microfinance : synthèse de quelques observations de terrain[END_REF]. In Senegal, the credit allows women to master better some distribution channels by bypassing one or several intermediaries (for example in fish or textile markets etc.).

Dependence decrease

Paul Mosley and David Hulme estimated the impact on thirteen microfinance intermediaries in seven Asian countries [1998]. Their works show a net correlation between the levels of initial incomes of the borrower and the increase of income led by the credit: the more they are situated below the poverty line and the more the led incomes are weak -when they are not negative -borrower having been forced to get into debt to pay back. Consequently, the researchers are rather sceptic towards a massive generalization of microfinance which would not take into account the financial instability of the most deprived. They suggest an adaptation of microfinance services, by combining credit, saving and insurance : this combination may allow borrowers to be less exposed to the risk. Our observations in Senegal go in the same way. The poorest women are overrepresented in the cases of loss i.e. when the profits of the activity are not enough to pay back. Conversely, it is for the richest that the margins are the most important. The result is not nevertheless dramatic since the women adhere, in a formal way or not, to a system of insurance. Here in this particular case in Senegal, the programme, probably wrongly, foreseen nothing of such, but the women self-insure mutually, using informal financial systems like ROSCAs.

Besides, if one argues in terms of well being and independence feeling and not strictly in terms of incomes, it is for the most vulnerable that the impact is really significant and becomes completely convincing. In India and in the Sri Lanka, the SEWA Bank (the bank of the Self-Employed Women's Association) allows to the most deprived women, belonging to the lowest castes, to settle former debts, so terminating a relation of quasi-exploitation [START_REF] Schrieder | Impact of Poverty Reduction[END_REF]. For the customers of the Grameen Bank in Bangladesh, those of the village banking in the dogon country and of Kafo Jiginew in Mali, or still those of the plain of Rushes in Vietnam, microcredit allows above all to avoid capital reduction or expensive credit like usurer ones [START_REF] Doligez | Etude bibliographique, Programme régional d'appui aux opérations de crédit décentralisé[END_REF]. We did the same report in Senegal. The women feel " more comfortable ", " less bound ". They also say that they "manage" better, and that they less have "to beg". Now a woman who "does not beg" is more respected. Before microcredit, either the commercial activity had no existence, or it was financed from a very expensive credit or still from a sale of farm produces normally intended for the domestic consumption and very often, profits are partially dedicated to the repayment of certain debts.

Commercial activities stabilization

Still in Senegal, the most important effect concerns the stabilization of the commercial activity of those who, up to there, were regularly obliged to stop their commercial activity. This result goes against a lot of impact studies which concluded that microfinance has no effect in terms of accumulation. Not only consumer needs absorb a good part of the profits, but borrowers often prefer the diversification rather than to risk economies of scale by betting on a single activity. One understands easily that in a context of strong uncertainty, it is more reassuring to opt for a distribution of the risks. For others, if there is no accumulation, it is simply that they absolutely have no ambition to widen their business, nor even to stabilize it. For example in Senegal, for certain women, commercial activity remains a punctual and temporary answer to a determined well need (a ceremony, the school re-opening, etc.). All the same it is possible to attend certain forms of accumulation by capitalization of the commercial working capitals, or at least a stabilization of the activity. This process is possible in particular when borrowers have an access repeated to the credit of short term [START_REF] Doligez | Dix ans d'études de l'impact de la microfinance : synthèse de quelques observations de terrain[END_REF], and provided that it is not transformed into relation of dependence towards the microfinance intermediary [START_REF] Diarra | Femmes et micro-économie au Niger : le petit crédit[END_REF]]. When the credit is intended for groups of borrowers, it happens as well that the group use microcredit to develop a regular activity of credit towards its members. It is what we noticed in Senegal.

Strengthening the weight of obligations

Microfinance is thus a means to improve the every day life, to limit the dependence and to stabilize commercial activities. But it can also lead to strengthen the weight of obligations. Fist of all, the question of 'who control the credit' is crucial. Concerning the women, their husbands or other male members of the family can be incited to appropriate the credit. For example, according to a research carried out in Bangladesh (four microfinance programmes are studied), 10% of the borrowers of the Grameen Bank have little, even no control over the use of their credit, this proportion goes from 40 to 60 % for the three other programmes [START_REF] Goetz A.-M | Who Takes the Credit ? Gender, Power and Control Over Loans Use in Rural Credit Programs in Bangladesh[END_REF]. Concerning the customers of the Grameen Bank and the way they decided to apply for a microcredit, other field researches reveal the determining influence of the husband. This influence is probably precious when it incites the women to challenge the tradition; but sometimes it is similar to a real manipulation: the woman is finally only a "way" to obtain a loan without being involved in the reimbursement [START_REF] Rahman | Microcredit Initiatives for Equitable and Sustainable Development : Who Pays ?[END_REF]. The difficulties met by the small project of rural credit in the Burkina Faso illustrate another shape of drift bound to women targeting. In a Sahelian context where the opportunities of commercial activities are limited, for women as well as for men, the emergence of a female concurrence towards male entrepreneurship collided with male resistance and one can wonder if targeting women was really relevant. In other contexts, where male patriarchy is very strong, and where women are not allowed to enter public spaces and have to stay at home, for example certain regions of Niger, several microfinance experiences excluding men led to bad results. The absence of dialogue with the male have indeed provoked a strong destabilization of local social relationships [START_REF] Schrieder | Impact of Poverty Reduction[END_REF]. Finally, a last risk have to be mentioned: microcredit may lead to weigh down even more their responsibilities, the men putting forward this "feminine privilege" to disengage from their own obligations.

Ambivalence of the collective approach

Women are more and more numerous to have a right to credit access, provided that they join in groups and provided that they mutually stand surety. IFPRI statistics show that collective loan with joint-liability is mainly adopted for women borrowers, who represent 80 % of the clientele, while they represent less than a third of individual loan's clientele [START_REF] Lapenu | Distribution, Growth and Performance of Microfinance Institutions in Africa, Asia and Latin America[END_REF]. It is a question of efficiency: through costs reductions, collective loan is supposed to assure efficiency and maybe viability. It is also a question of mobilization of the female civil society: through the collective management of credit, one hopes to strengthen their capacities of auto-organization. Certainly, the capacity of auto-organization of local groups, notably concerning women, is remarkable. However, a romantic and fonctionnalist approach of collective action is not sufficient. A collective loan can favour individual autonomy as well as denying it by consolidating existing links of dependence, even by forging new kinds of dependence. It may strengthen social networks but also destabilize them. It may support collective competence of organization and management but also arouse and encourage the creation of fictitious or monopolistic groups.

The heterogeneousness of female groups forbids any generalization. First of all, this diversity depends of the context, urban or rural, and of the size of the groups. For example in Senegal, some are very small, with only a dozen of members, whereas the others have several hundreds, in particular in rural areas. This variety depends also of what activities are led: local business but also import / export, education, organization of religious or political events by way of "talks", moments of exchanges and discussions about extremely varied subjects. Of this variety ensue various degrees of intellectual and financial autonomy, social cohesion and economic dynamism. If certain groups are based on social traditional networks of mutual aid, it is certain that others built up themselves only to get financing or sit the political authority of someone. At this first risk, one can add that of a strong hierarchy. Nobody will go to pressure a group president who uses the credit to finance the marriage of her daughter. For that reason, lots of women prefer individual loan. Conversely, women can be urged to adhere and to benefit from a credit while they do not need it. For the choice of the activity financed by the credit, they imitate those who are successful, but without benefiting the same advantages (free transport, a well-established clientele, etc.).

A need to couple economic action and political action

Microfinance neither replaces state intervention nor market transactions

These contrasted results must not lead to give up the idea of a specific female supply of microfinance services, they just urge to be vigilant as for different possible dangers and risks. First, bottom-up development has limits: when a group is well-equilibrated between its own activities and its external relations, access to credit can become a useful tool to improve women's autonomy. But the delegation of responsibilities to groups must not lead to consider groups only as a whole without taking intra-groups relations into account. Once again, socio-cultural norms and power relations must not be ignored. When collective interest is more powerful than individual interest of members, it can become coercive for them and lead to negative outcomes: applying for a loan without need, launching unviable business enterprise, getting indebted from pawnbrokers to pay back at all costs, etc. Second, the recognition of the role played by grass-root organizations as women groups must not lead to underestimate the responsibility of the national state and the international organisations, such as the International Monetary Fund or the World Bank. Supporting and strengthening people-based initiatives is legitimate and justified. But NGOs' goal (or other grass-roots institutions) is neither to compensate for state disengagement nor to compensate for the costs of structural adjustment policies. Nor can they act without a minimal institutional framework neither can they replace state interventions or market transactions.

The risk of microfinance, already denounced by some authors, would be that public authorities take advantage of it to disengage more, giving the priority to the integration of the poor in market mechanisms -and this in expenses of their access of economic and social basic rights -while betting on the capacity of NGOs and the auto-organization of local populations to face the problems of poverty. Feminist movements were the first ones to denounce the dangers of this approach. They do not question the legitimacy of microfinance but plead in favour of its political anchoring. According to these movements, microfinance projects are justified only if strategic links are established with other forces of change, among which networks and women's movements, as well as organizations of defence and lobbies for women [START_REF] Mayoux | Microfinance and the Empowerment of Women. A Review of the Key Issue[END_REF]Hofmann and Marius-Gnanou, 2001]. The International Labour Office adopted the same position by recognizing that microfinance has no vocation to eliminate poverty problems, especially female poverty, and that it is justifiable only if it is integrated into a more global reflection on the valuation of the fundamental rights, in particular for women [BIT, 1999b].

Political anchoring of microfinance

To advocate political anchoring is all the more relevant that microfinance can facilitate women mobilization through collective discussion, debate and action. Thus women can accede to public space and public voice, which can be favourable to real freedom of women. Laws can't impose all the gender-related issues. Whatever the context is, even when state interventions can influence the level of gender equity achieved by legal ways or public schemes, only "social dialogue" or public debate can deeply transform mentalities and traditions or cultural norms. As we noticed above, women are not only often deprived of the same rights as men but most of the time they are also not aware of their subordination and deprivation. Once again, only social dialogue can allow making personal choices and needs being expressed or emerging. That's why whatever the policy sets up to fight inequalities, it must associate economic freedom and political freedom.

How arousing this "social dialogue"? To rely on mass-mobilisation of people in traditional forms of commitment as trade union or political party is no longer the only solution. People, and more specifically women, need to be involved in shorter commitments in order to resolve concrete problems. It's why local spaces of discussion where everybody can give his opinion and debate with others about daily life preoccupations can be a means for this "social dialogue" 1 . But it is not sufficient. Women have still to want to promote institutional change and not only to resolve practical problems. At a local level, women's groups are able to influence public policies and can contribute to local development. But going beyond this local level is one of the most delicate aspects of this issue.

It can be illustrated by a constitutional change which occurs in India in 1992 2 . This initiative set up a three-tier system of decentralized governance for spreading local democracy in the country through elective bodies but it also represent a powerful strategy to increase women's participation in the political process because the legislation reserve one third of all seats in the local bodies for women 3 . This new system, which ensures political representation of women, is no longer a contested issue, for example, in Karnataka state, elected women exceed the 33.3 per cent quota 4 . As with all innovations, this must be treated with care. First, powerful and deep-rooted patriarchal norms in Indian society prevent women from a real equal sharing of powers: reservation of seats is not yet a qualitative change but still just a quantitative one. Second, the vast majority of elected women comes from "elite" which goes together with high-education level. They belong to families with privileged economic and sociocultural status or where men have already political positions. One can notice that a quota of seats is also reserved to depressed castes, but candidacy selection is often conformed to the "elite". This 1 See Bernard EME et Jean-Louis LAVILLE [1994] about the concept of " espaces publics de proximité ". 2 The 73rd Constitutional Amendment Act for Panchayati Raj (rural) and the 74th Constitutional Amendment Act for Municipal (urban) governance in 1992. Additional information are available in the website of the Urban Development Ministry http://urbanindia.nic.in and the Plan Commission http://www.nic.in/ninthplan/vol1/v1c6-2.htm 3 According to a World Bank report, there are, spread among 25 states and 7 union territories, 3586 urban local bodies and 234 078 rural local bodies. [BANQUE MONDIALE, (2000), Le développement au seuil du XXIe siècle : rapport sur le développement dans le monde 1999-2000, Paris, Eska, 329p.] 4 See SEKHER Madhushree, [2001]. pessimistic point of view must be counterbalance by the fact that women have now a "voice" in the local political arena and a role in decision-making: they can acquire useful political and administrative skills which is one step towards empowerment or improved capabilities. The challenge is to transform this women presence in the local bodies into effective participation in the governance process. Besides, efforts to widen seat reservation to the federal parliament and state legislatives have not yet fructified due to the reluctance within the male-dominated political system to share power with women 1 .

Women's organisations and microfinance in India 2

One of the most significant examples of political anchoring seems to be that of Indian Microfinance Institutions (MFIs). By and large, their main goal is multidimensional development of their members: provision of financial services is only one tool among others to alleviate poverty and to empower women so they also implement programmes to improve education, health, housing, etc. For example, the two essential elements of Baroda Citizen Council (BCC)3 strategy are participation by the community in efforts to improve its standard of living with as much reliance as possible on its own initiatives and provision of technical, managerial and other services in order to encourage self-help initiatives. BCC share " the belief that social improvement does not come until the people involved believe that improvement is possible. "4 Similarly, " the main philosophy guiding the promoters of SPMS [Sri Padmavathy Mahila Abyudaya Sangam, a federation of women's Self Help Groups in Tirupati, Andhra Pradesh] is that poor women should be organized and helped in capacity-building, so that they can solve their own problems. "5 Governance structure of these Indian MFIs (as Working Women's Forum, an NGO in Chennai, Tamil Nadu) reflects their philosophy guiding their action. They are classified as "people's organisation" where full ownership rests with its members. WWF and SPMS explicitly claim to be women organisations created by women and for women. Recurring key terms used by Indian MFIs to describe their philosophy are women's empowerment and self-reliance by capacity-building and self-help.

Since the 1970s, the Self employment women association (SEWA6 ) fights for "serving" the condition of the women workers from the informal sector (small storekeepers, peddlers, homeworkers, etc.7 ) as a labour union registered in 1972. These home or street workers, not salaried, are "invisible" because their labour and income are ignored by statistics (economical disempowerment). Besides, they are without own social status but "women of" or "daughter of"(social disempowerment). SEWA's two main goals are to organize women for full employment (employment providing work security, income security, food security and social security) at the household level and for selfreliance. To achieve these goals, SEWA have chosen an integrated and holistic approach which attempts to strengthen them, covers the risks they face and reduces insecurity. The SEWA considerably widened its activities, while pursuing the same objective -to improve the autonomy of the women -and with the same method, widely inspired of the gandhienne philosophy. It acts simultaneously on the political, economic, social and cultural plans by articulating local action and institutional change [Hofmann and Marius-Gnanou, 2001;Landing, 2001]. SEWA's Strategy involves struggles (against the many injustices women face, for visibility and recognition, for political and 1 Ibid.

2 Since the independence in 1947, Indian associations who champion women's cause have been trying to make existing laws enforced: laws banning widow-immolation or sati, (1829), enabling Hindu widow to remarry (1856), banning female infanticide (1870) and against child marriage (1891,1929) and to fight against murdered daughter-in-law (because of inadequate dowry), rapes, child labour, girls' prostitution, etc. They recognize favourable Indian legislation but regret its weak application. Besides, as noticed above, because of deep-rooted cultural norms, women "prefer" staying married even if they are beaten rather than divorcing implying being rejected by her family and community (joint family and status of rural Indian women are discussed in MARIUS-GNANOU Kamala, (1998)). economic leadership) and constructive actions by creating employment (cooperatives of production), developing financial services (the SEWA Bank was recognized by the Indian banking authorities in 1974), capacity-building training, health and child care, housing services, insurance, etc. One of these activities concerns educational actions, with the Academy of the SEWA, the explicit objective of which is to strengthen women's capacities in collective management and in political action. Political strategy is oriented towards collaboration with public or private institutions either in specific projects and events (slum upgradation by building drainage and sanitary connections, drought in 1995, etc.) or in building partnership to change existing structures in the long term. At international level, this association works with the International Labour Organization: one of her main achievement was the adoption of an ILO convention on home-workers [START_REF] Krauss | SEWA -Inde. L'accès au pouvoir et l'expérience de la SEWA[END_REF].

Conclusion

Women who "can" have economic independence, in the sense that no law prevents them, may be prevented simply by lacking assets, or access to credit. As Martha Nussbaum writes it "liberty is not just a matter of having rights on paper, it requires being in a material position to exercise those rights" [Nussbaum, 2001, p.1]. For this reason, microcredit organizations have an important role to play to improve women's capabilities by offering them real opportunities. And if it concerns economical opportunities, it is not disconnected from social and political spheres. We agree with Amartya Sen when he says that there is strong evidence that economic and political freedoms help to reinforce one another.

Besides, the strong resistances against women's participation in the market economy that still exist are not only unjust, but also unefficient. Eliminating that discrimination is indeed not only a question of justice, promoting equality between sexes, but it is also a means to increase national income. There is now overwhelming evidence that women's empowerment, through schooling and employment opportunities among other things, carries a financial and social benefit for all -men, women and children. It has an immediate effect on children's nutritional status, health and development, on the mother's health and on the size of the family. Thus, it is more influential than economic growth in moderating fertility rates and reducing child mortality. For this reason, it cannot be assumed that women's work outside the home has perverse effects on children well-being. On the contrary, as their bargaining power increases within the family, women are more able to plan the family's future and to invest in child health and education. As Martha Nussbaum writes it, "the disability imposed by childbearing on a member of the labor force is to a large extent socially constructed, above all by the absence of support for childcare, from the public sphere, from employers, and from male partners."1 Legal guarantees established by the state, as well as opportunities created by non governmental organizations, help women to have real access to some functunnings and do not erode agency. They create "a framework within which people can develop and exercise agency" [Nussbaum, 1999a, p. 19]. But the state that thinks to guaranty people rights effectively has to recognize norms beyond the small menu of basic rights. As Amartya Sen underlines it, beyond the rights concerning their wellbeing, it is crucial to confer women a role of agent in change [Sen, 1999]. In order to effectively improve their condition, they shouldn't be passive addressees of reforms concerning their status, but dynamic initiators of social transformations affecting their existence as well as the one of men.

Local traditions and practices are often very complex and interconnected. Changing one rule of conduct inexorably affects other institutions' way of working. That is why imposing cultural changes only by the way of legislation can be very perverse. As we tried to show it in this paper, microcredit organizations that target women shouldn't act without a public debate. New laws, new opportunities for women have to be well understood and admitted by them, and maybe above all by men, to be applied in a way that is not too far away from the initial intention. Microfinance is not blessed with the ability to right the power imbalances which results from inequalities in the way society treats men and women. It may be able to make such a contribution, but to do so require a clear commitment and strategic approach to ensuring it does.

The authors are from Centre Auguste & Léon Walras, Lyon, France.

For example, see SEN A. K.[1999], chap.

3 NUSSBAUM Martha C., "Non-relative Virtues: An Aristotelian Approach", in NUSSBAUM,[START_REF] Sen Amartya | Éthique et économie [On Ethics and Economics[END_REF], p. 243.

NUSSBAUM [1999a], p. 88.

[START_REF] Nussbaum Martha | Women and Work -The capability approach[END_REF], p.

3 This idea has largely been developped by Hannah Arendt in her book The Human Condition[1958]. And one of her fear was to observe how the modern society became brutally a "society of workers" while technical progress allows to work less. For her, that relatively new ideology that glorifies work is due to economists such as Adam Smith or even Karl Marx. She notices that, in their writtings, work is seen either as the cause of wealth or as a central value.

See SEN [1991], p.

in the French translation of 1993, Presses Universitaires de France. 2 About this question, see "Child care, domestic services, employment and gender inequality", National Institute for Working Life[2000], http://www.niwl.se/wl2000/workshop23/article_en.asp.

As mentioned above, see for example[START_REF] Kabeer Naila | Conflicts over credit: re-evaluating the empowerment potential of loans to women in rural Bangladesh[END_REF] about the reasons why recent evaluations of the empowerment potential of microcredit programs in Bangladesh have arrived at very conflicting conclusions.

An NGO in Vadadora (Gujarat) formerly known as Baroda.

WORLD SANITATION PROGRAM, (1999b).

WORLD SANITATION PROGRAM, (1999d).

There are a lot of published reports and other type of documentation about SEWA, some of them are reported in our references.

See PRÜGL Elisabeth and TINKER Irene (1997) for a presentation of SEWA case as example of successful organization which supports home-based workers by combining microentreprise development and union organizing.

NUSSBAUM [1999a], p. 51.


Simona Rota Nodari
email: rotanodari@ceremade.dauphine.fr

PERTURBATION METHOD FOR PARTICLELIKE SOLUTIONS OF EINSTEIN-DIRAC EQUATIONS

The aim of this work is to prove by a perturbation method the existence of solutions of the coupled Einstein-Dirac equations for a static, spherically symmetric system of two fermions in a singlet spinor state. We relate the solutions of our equations to those of the nonlinear Choquard equation and we show that the nondegenerate solution of Choquard's equation generates solutions for Einstein-Dirac equations.

Introduction

In this paper, we study the coupled Einstein-Dirac equations for a static, spherically symmetric system of two fermions in a singlet spinor state. Using numerical methods, F. Finster, J. Smoller and ST. Yau found, in [START_REF] Finster | Particlelike solutions of the Einstein-Dirac equations[END_REF], particlelike solutions; our goal is to give a rigorous proof of their existence by a perturbation method 1 .

The Einstein-Dirac equations take the form (Dm)ψ = 0 (1.1)

R i j - 1 2 Rδ i j = -8πT i j (1.2)
where D denotes the Dirac operator, ψ is the wave function of a fermion of mass m, R i j is the Ricci curvature tensor, R indicates the scalar curvature and, finally, T i j is the energy-momentum tensor of the Dirac particle. In [START_REF] Finster | Particlelike solutions of the Einstein-Dirac equations[END_REF], Finster, Smoller and Yau work with the Dirac operator into a static, spherically symmetric space-time where the metric, in polar coordinates (t, r, ϑ, ϕ), is given by (1.3)

g ij = diag 1 T 2 , - 1 A , -r 2 , -r 2 sin 2 ϑ (1.4) g ij = diag T 2 , -A, - 1 r 2 , - 1 r 2 sin 2 ϑ
with A = A(r), T = T (r) positive functions; so, the Dirac operator can be written as

(1.5) D = iγ t ∂ t + γ r i∂ r + i r 1 -A -1/2 - i 2
T ′ T + iγ ϑ ∂ ϑ + iγ ϕ ∂ ϕ
Date: October 2, 2009. 1 After completing this work, we learned from professor Joel Smoller that Erik J. Bird had proved the existence of small solutions of the Einstein-Dirac equations in his doctoral thesis in 2005 [START_REF] Bird | A proof of existence of particle-like solutions of Einstein Dirac Equations[END_REF]. His method is quite different from ours: he uses Schauder's fixed point theorem. with γ t = T γ0 (1.6) γ r = √ A γ1 cos ϑ + γ2 sin ϑ cos ϕ + γ3 sin ϑ sin ϕ (1.7)

γ ϑ = 1 r
-γ 1 sin ϑ + γ2 cos ϑ cos ϕ + γ3 cos ϑ sin ϕ (1.8)

γ ϕ = 1 r sin ϑ -γ 2 sin ϕ + γ3 cos ϕ (1.9)
where γi are the Dirac matrices in Minkowski space (see [START_REF] Finster | Particlelike solutions of the Einstein-Dirac equations[END_REF]).

Moreover, Finster, Smoller and Yau are looking for solutions taking the form

(1.10) ψ = e -iωt r -1 T 1/2     Φ 1 1 0 iΦ 2 σ r 1 0     ,
where σ r = σ1 cos ϑ + σ2 sin ϑ cos ϕ + σ3 sin ϑ sin ϕ is a linear combination of the Pauli matrices σi and Φ 1 (r), Φ 2 (r) are radial real functions. We remind also that the energy-momentum tensor is obtained as the variation of the classical Dirac action

S = ψ(D -m)ψ |g| d 4 x
and takes the form

T i j = 1 r 2 diag 2ωT 2 |Φ| 2 , -2ωT 2 |Φ| 2 + 4T 1 r Φ 1 Φ 2 + 2mT Φ 2 1 -Φ 2 2 , -2T 1 r Φ 1 Φ 2 , -2T 1 r Φ 1 Φ 2
(see [START_REF] Finster | Particlelike solutions of the Einstein-Dirac equations[END_REF] for more details). In this case, the coupled Einstein-Dirac equations can be written as

√ AΦ ′ 1 = 1 r Φ 1 -(ωT + m)Φ 2 (1.11) √ AΦ ′ 2 = (ωT -m)Φ 1 - 1 r Φ 2 (1.12) rA ′ = 1 -A -16πωT 2 Φ 2 1 + Φ 2 2 (1.13) 2rA T ′ T = A -1 -16πωT 2 Φ 2 1 + Φ 2 2 + 32π 1 r T Φ 1 Φ 2 + +16πmT Φ 2 1 -Φ 2 2 (1.14)
with the normalization condition

(1.15) ∞ 0 |Φ| 2 T √ A dr = 1 4π .
In order that the metric be asymptotically Minkowskian, Finster, Smoller and Yau assume that lim r→∞

T (r) = 1.
Finally, they also require that the solutions have finite (ADM) mass; namely

lim r→∞ r 2 (1 -A(r)) < ∞.
In this paper, we will prove the existence of solutions of (1.1-1.2) in the form (1.10) by a perturbation method. In particular, we follow the idea described by Ounaies in [START_REF] Ounaies | Perturbation method for a class of non linear Dirac equations[END_REF] (see also [START_REF] Guan | Solitary Wave Solutions for the Nonlinear Dirac Equations[END_REF] for a rigorous existence proof of nonlinear Dirac solitons based on Ounaies' approach). Ounaies, by a perturbation parameter, relates the solutions of a nonlinear Dirac equation to those of nonlinear Schrödinger equation. Imitating the idea of Ounaies, we relate the solutions of ours equations to those of nonlinear Choquard's equation (see [START_REF] Lieb | Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation[END_REF], [START_REF] Lions | The Choquard equation and related questions[END_REF] for more details on Choquard's equation) and we obtain the following result.

Theorem 1.1. Given 0 < ω < m such that mω is sufficiently small, there exists a non trivial solution of (1.11-1.14).

In Section 2, we solve the Einstein-Dirac equations by means of the perturbation method suggested by Ounaies; in particular in the first subsection we describe a useful rescaling and some properties of the operators involved, whereas in the second subsection we prove the existence of solutions generated by the solution of the Choquard equation.

Perturbation method for Einstein-Dirac equations

First of all, we observe that writing T (r) = 1 + t(r) and using equation (1.13), the coupled Einstein-Dirac equations become

√ AΦ ′ 1 = 1 r Φ 1 -(ω + m)Φ 2 -ωtΦ 2 (2.1) √ AΦ ′ 2 = (ω -m)Φ 1 + ωtΦ 1 - 1 r Φ 2 (2.2) 2rAt ′ = (A -1)(1 + t) -16πω(1 + t) 3 Φ 2 1 + Φ 2 2 + +32π 1 r (1 + t) 2 Φ 1 Φ 2 + 16πm(1 + t) 2 Φ 2 1 -Φ 2 2 (2.3)
where

(2.4) A(r) = 1 - 16πω r r 0 (1 + t(s)) 2 Φ 1 (s) 2 + Φ 2 (s) 2 ds := 1 - 16πω r Q(r).
Furthermore, because we want A(r) > 0, we have that the following condition must be satisfied

(2.5) 0 ≤ Q(r) r < 1 16πω
for all r ∈ (0, ∞). Now, to find a solution of the equations (2.1-2.3), we exploit the idea used by Ounaies in [START_REF] Ounaies | Perturbation method for a class of non linear Dirac equations[END_REF]. In particular, we proceed as follow: in a first step we use a rescaling argument to transform (2.1-2.3) in a perturbed system of the form (2.6)

       A (ε, ϕ, χ, τ ) d dr ϕ -1 r ϕ + 2mχ + K 1 (ε, ϕ, χ, τ ) = 0 A (ε, ϕ, χ, τ ) d dr χ + 1 r χ + ϕ -mϕτ + K 2 (ε, ϕ, χ, τ ) = 0 A (ε, ϕ, χ, τ ) d dr τ + 8πm r 2 r 0 ϕ 2 ds + K 3 (ε, ϕ, χ, τ ) = 0
where ϕ, χ, τ : (0, ∞) → R. Second, we relate the solutions of (2.6) to those of the nonlinear system (2.7)

         -d 2 dr 2 ϕ + 2mϕ -16πm 3 ∞ 0 ϕ 2 max(r,s) ds ϕ = 0 χ(r) = 1 2m 1 r ϕ -d dr ϕ τ (r) = 8πm ∞ 0 ϕ 2 max(r,s) ds.
We remark that ϕ is a solution of (2.7) if and only if u(x) = ϕ(|x|) |x| solves the nonlinear Choquard equation

(2.8) -△u + 2mu -4m 3 R 3 |u(y)| 2 |x-y| dy u = 0 in H 1 R 3 .
To prove this fact it's enough to remind that for a radial function ρ,

△ρ = 1 r 2 d dr r 2 d dr ρ and (| • | ⋆ ρ) (x) = 4π ∞ 0 s 2 ρ(s) max(r, s) ds with r = |x|.
We observe also that if we write

u(x) v(x) = r -1     ϕ(r) 1 0 iχ(r)σ r 1 0     with r = |x|, (ϕ, χ) is a solution of (2.7) if and only if (u(x), v(x)) solve (2.9) -△u + 2mu -4m 3 R 3 |u(y)| 2 |x-y| dy u = 0 v = -iσ∇u 2m in R 3 where σ∇ = 3 i=1
σi ∂ i . It's well known that Choquard's equation (2.8) has a unique radial, positive solution u 0 with |u 0 | 2 = N for some N > 0 given. Furthermore, u 0 is infinitely differentiable and goes to zero at infinity; more precisely there exist some positive constants

C δ,η such that |D η (u 0 )| ≤ C δ,η exp(-δ|x|) for x ∈ R 3 . At last, u 0 ∈ H 1 (R 3
) is a radial nondegenerate solution; by this we mean that the linearization of (2.8) around u 0 has a trivial nullspace in L 2 r (R 3 ). In particular, the linear operator L given by [START_REF] Lieb | Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation[END_REF], [START_REF] Lions | The Choquard equation and related questions[END_REF], [START_REF] Lenzmann | Uniqueness of ground states for pseudo-relativistic Hartree equations[END_REF] for more details). The main idea is that the solutions of (2.6) are the zeros of a

Lξ = -△ξ + 2mξ -4m 3 R 3 |u 0 (y)| 2 |x -y| dy ξ -8m 3 R 3 ξ(y)u 0 (y) |x -y| dy u 0 satisfies ker L = {0} when L is restricted to L 2 r (R 3 ) (see
C 1 operator D : R × X ϕ × X χ × X τ → Y ϕ × Y χ × Y τ . If we denote by D ϕ,χ,τ (ε, ϕ, χ, τ ) the derivative of D(ε, •, •, •)
, by (ϕ 0 , χ 0 , τ 0 ) the ground state solution of (2.7) and we observe that D ϕ,χ,τ (ε, ϕ 0 , χ 0 , τ 0 ) is an isomorphism, the application of the implicit function theorem (see [START_REF] Renardy | An Introduction to Partial Differential Equations[END_REF]) yields the following result, which is equivalent to theorem 1.1.

Theorem 2.1. Let (ϕ 0 , χ 0 , τ 0 ) be the ground state solution of (2.7), then there exists δ > 0 and a function η ∈ C((0, δ), X ϕ ×X χ ×X τ ) such that η(0) = (ϕ 0 , χ 0 , τ 0 ) and (ε, η(ε)) is a solution of (2.6), for 0 ≤ ε < δ.

2.1.

Rescaling. In this subsection we are going to introduce the new variable (ϕ, χ, τ ) such that Φ 1 (r) = αϕ(λr), Φ 2 (r) = βχ(λr) and t(r) = γτ (λr), where Φ 1 , Φ 2 , t satisfy (2.1-2.3) and α, β, γ, λ > 0 are constants to be chosen later.

Using the explicit expressions of A, given in (2.4), we have

A(Φ 1 , Φ 2 , t) = 1 - 16πωα 2 r r 0 (1 + γτ ) 2 ϕ 2 + β α χ 2 ds := A α,β,γ (ϕ, χ, τ ) (2.10) It's now clear that if Φ 1 , Φ 2 , t satisfy (2.1-2.3), then ϕ, χ, τ satisfy the system (2.11)        A α,β,γ αλ β d dr ϕ -αλ β 1 r ϕ + (m + ω)χ + ωγτ χ = 0 A α,β,γ d dr χ + 1 r χ + α βλ (m -ω)ϕ -αγ βλ ωτ ϕ = 0 2A α,β,γ r d dr τ + K α,β,γ = 0 with K α,β,γ (ϕ, χ, τ ) = 16πω r α 2 γ r 0 (1 + γτ ) 2 ϕ 2 + β α χ 2 ds (1 + γτ ) + +16πω α 2 γ (1 + γτ ) 3 ϕ 2 + β α χ 2 -32π λαβ γ 1 r (1 + γτ ) 2 ϕχ + -16πm α 2 γ (1 + γτ ) 2 ϕ 2 - β α χ 2 .

By adding the conditions

α βλ (m-ω) = 1, αλ β = 1, αγ βλ = 1, α 2 γ = 1 and m-ω ≥ 0, we obtain α = (m -ω) 1/2 , λ = (m -ω) 1/2 , β = m -ω and γ = m -ω. Denoting ε = m -ω, (2.11) is equivalent to (2.12)        A (ε, ϕ, χ, τ ) d dr ϕ -1 r ϕ + 2mχ + K 1 (ε, ϕ, χ, τ ) = 0 A (ε, ϕ, χ, τ ) d dr χ + 1 r χ + ϕ -mϕτ + K 2 (ε, ϕ, χ, τ ) = 0 A (ε, ϕ, χ, τ ) d dr τ + 8πm r 2 r 0 ϕ 2 ds + K 3 (ε, ϕ, χ, τ ) = 0 where A (ε, ϕ, χ, τ ), K 1 (ε, ϕ, χ, τ ), K 2 (ε, ϕ, χ, τ ) and K 3 (ε, ϕ, χ, τ ) are defined by A (ε, ϕ, χ, τ ) = 1 - 16π(m -ε)ε r r 0 (1 + ετ ) 2 ϕ 2 + εχ 2 ds; (2.13) K 1 (ε, ϕ, χ, τ ) = -εχ + ε(m -ε)τ χ; (2.14) K 2 (ε, ϕ, χ, τ ) = ετ ϕ; (2.15) and K 3 (ε, ϕ, χ, τ ) = 8πmε r 2 r 0 χ 2 ds + 16πmε r 2 r 0 τ ϕ 2 + εχ 2 ds + + 8πmε 2 r 2 r 0 τ 2 ϕ 2 + εχ 2 ds + - 8πε r 2 r 0 (1 + ετ ) 2 ϕ 2 + εχ 2 ds τ + + 8π(m -ε)ε r 2 r 0 (1 + ετ ) 2 ϕ 2 + εχ 2 ds τ + +16πmε χ 2 r + 8πmε(3τ + 3ετ 2 + ε 2 τ 3 ) ϕ 2 + εχ 2 r + -8πε(1 + ετ ) 3 ϕ 2 + εχ 2 r -16πε(1 + ετ ) 2 ϕχ r 2 + -8πmε(2τ + ετ 2 ) ϕ 2 -εχ 2 r . (2.16)
For ε = 0, (2.12) becomes (2.17)

       d dr ϕ -1 r ϕ + 2mχ = 0 d dr χ + 1 r χ + ϕ -mϕτ = 0 d dr τ + 8πm r 2 r 0 ϕ 2 ds = 0 that is equivalent to (2.18)          -d 2 dr 2 ϕ + 2mϕ -16πm 3 ∞ 0 ϕ 2 max(r,s) ds ϕ = 0 χ(r) = 1 2m 1 r ϕ -d dr ϕ τ (r) = 8πm ∞ 0 ϕ 2 max(r,s) ds.
Then, we denote by (ϕ 0 , χ 0 , τ 0 ) a solution of (2.18); in particular

χ 0 (r) = - r 2m d dr ϕ 0 r τ 0 (r) = 8πm ∞ 0 ϕ 2 0 max(r, s) ds.
Now, to obtain a solution of (2.12) from (ϕ 0 , χ 0 , τ 0 ), we define the operators L 1 :

R×X ϕ ×X χ ×X τ → Y ϕ , L 2 : R×X ϕ ×X χ ×X τ → Y χ , L 3 : R×X ϕ ×X χ ×X τ → Y τ and D : R × X ϕ × X χ × X τ → Y ϕ × Y χ × Y τ by L 1 (ε, ϕ, χ, τ ) = A (ε, ϕ, χ, τ ) 1 r d dr ϕ - ϕ r 2 + 2m χ r + 1 r K 1 (ε, ϕ, χ, τ ) L 2 (ε, ϕ, χ, τ ) = A (ε, ϕ, χ, τ ) 1 r d dr χ + χ r 2 + ϕ r -m ϕ r τ + 1 r K 2 (ε, ϕ, χ, τ ) L 3 (ε, ϕ, χ, τ ) = A (ε, ϕ, χ, τ ) d dr τ + 8πm r 2 r 0 ϕ 2 ds + K 3 (ε, ϕ, χ, τ ) and D(ε, ϕ, χ, τ ) = (L 1 (ε, ϕ, χ, τ ), L 2 (ε, ϕ, χ, τ ), L 3 (ε, ϕ, χ, τ ))
where

X ϕ = ϕ : (0, ∞) → R ϕ(|x|) |x| 1 0 ∈ H 1 R 3 , R 2 X χ = χ : (0, ∞) → R χ(|x|) |x| σ r 1 0 ∈ H 1 R 3 , C 2 X τ = τ : (0, ∞) → R lim r→∞ τ (r) → 0, d dr τ ∈ L 1 ((0, ∞), dr) Y ϕ = Y χ = L 2 R 3 Y τ = L 1 ((0, ∞), dr).
Furthermore we define the following norms:

ϕ Xϕ = ϕ(|x|) |x| H 1 (R 3 ) , χ Xχ = χ(|x|) |x| σ r 1 0 H 1 (R 3 ) , τ Xτ = d dr τ L 1 ((0,∞),dr)
.

It's well known that

H 1 R 3 ֒→ L q R 3 2 ≤ q ≤ 6 X τ ֒→ L ∞ ((0, ∞), dr) .
Moreover, using Hardy's inequality

R 3 |f | 2 |x| 2 dx ≤ 4 R 3 |∇f | 2 dx,
we get the following properties:

(2.19) ρ ∈ H 1 ((0, ∞), dr) ֒→ L ∞ ((0, ∞), dr) ρ r ∈ L 2 ((0, ∞), dr) . ∀ρ ∈ X ϕ , ∀ρ ∈ X χ .
Since the operator A(ε, ϕ, χ, τ ) must be strictly positive, we consider B ϕ , B χ , B τ , defined as the balls of the spaces X ϕ , X χ , X τ , and ε 1 , ε 2 , depending on m and on the radius of B ϕ , B χ , B τ , such that

1 - 16π(m -ε)ε r r 0 (1 + ετ ) 2 ϕ 2 + εχ 2 ds ≥ δ > 0 for all (ε, ϕ, χ, τ ) ∈ (-ε 1 , ε 2 ) × B ϕ × B χ × B τ . The existence of ε 1 , ε 2 is assured by the fact that ϕ, χ, τ are bounded; in particular, if ε ≥ 0, 1 - 16π(m -ε)ε r r 0 (1 + ετ ) 2 ϕ 2 + εχ 2 ds ≥ ≥ 1 -20mε ϕ 2 Xϕ -8mε 2 5 τ Xτ ϕ 2 Xϕ + χ 2 Xχ + -4mε 3 τ Xτ 5 τ Xτ ϕ 2 Xϕ + 4 χ 2 Xχ -8mε 4 τ 2 Xτ χ 2 Xχ ,
then there exists ε 2 > 0 such that A(ε, ϕ, χ, τ ) > 0 for all ε ∈ [0, ε 2 ). In the same way, if ε < 0,

1 - 16π(m -ε)ε r r 0 (1 + ετ ) 2 ϕ 2 + εχ 2 ds ≥ ≥ 1 -8mε 2 χ 2 Xχ -8m|ε| 3 χ 2 Xχ 1 + 2 τ Xτ + -8mε 4 χ 2 Xχ τ Xτ 2 + 1 τ Xτ -8m|ε| 5 τ 2 Xτ χ 2 Xχ , then there exists ε 1 > 0 such that A(ε, ϕ, χ, τ ) > 0 for all ε ∈ (-ε 1 , 0). Lemma 2.2. The operators L 1 , L 2 ∈ C 1 ((-ε 1 , ε 2 ) × B ϕ × B χ × B τ , Y ϕ ) and L 3 ∈ C 1 ((-ε 1 , ε 2 ) × B ϕ × B χ × B τ , Y τ ).
Before starting the proof of the lemma we observe that for a radial function ρ such that ρ r ∈ H 1 r R 3 we have

(2.20) |ρ(r)| ≤ r 1/2 d dr ρ(r) r L 2 rad .
We remind that

H 1 r R 3 = u ∈ H 1 R 3 | u is radial .
Proof. We begin with L 3 ; first, we have to prove that it is well defined in Y τ = L 1 ((0, ∞), dr). We remark that

|L 3 (ε, ϕ, χ, τ )| ≤ C 1 d dr τ + C 2 r 2 r 0 ϕ 2 + εχ 2 ds + C 3 r ϕ 2 + εχ 2 + + C 4 r 2 |ϕχ| + C 5 r ϕ 2 -
ϕ 2 + εχ 2 , 1 r ϕ 2 -εχ 2 ∈ Y τ . Finally, ∞ 0 |ϕ| r |χ| r dr ≤ C ϕ r L 2 ((0,∞)) χ r L 2 ((0,∞)) < +∞ thanks to (2.19), then 1 r 2 ϕχ ∈ Y τ . Now, we have to prove that L 3 (ε, ϕ, χ, τ ) is C 1 ; by classical arguments, it's enough to show that for (h 1 , h 2 , h 3 ) ∈ B ϕ × B χ × B τ ∂ ∂ϕ (L 3 (ε, ϕ, χ, τ )) h 1 ∈ Y τ , ∂ ∂χ (L 3 (ε, ϕ, χ, τ )) h 2 ∈ Y τ , ∂ ∂τ (L 3 (ε, ϕ, χ, τ )) h 3 ∈ Y τ . We begin with ∂ ∂ϕ (L 3 (ε, ϕ, χ, τ )), ∂ ∂ϕ (L 3 (ε, ϕ, χ, τ )) h 1 = ∂ ∂ϕ (A (ε, ϕ, χ, τ )) h 1 d dr τ + + 16π(m -ε) r 2 r 0 (1 + ετ ) 2 ϕh 1 ds (1 + ετ ) + +16π(m -ε)(1 + ετ ) 3 ϕh 1 r -16πε(1 + ετ ) 2 h 1 χ r 2 + -16πm(1 + ετ ) 2 ϕh 1 r ; for ∂ ∂χ (L 3 (ε, ϕ, χ, τ )), ∂ ∂χ (L 3 (ε, ϕ, χ, τ )) h 2 = ∂ ∂χ (A (ε, ϕ, χ, τ )) h 2 d dr τ + + 16π(m -ε)ε r 2 r 0 (1 + ετ ) 2 χh 2 ds (1 + ετ ) + +16π(m -ε)ε(1 + ετ ) 3 χh 2 r -16πε(1 + ετ ) 2 ϕh 2 r 2 + -16πmε(1 + ετ ) 2 χh 2 r and, finally, ∂ ∂τ (L 3 (ε, ϕ, χ, τ )) h 3 = ∂ ∂τ (A (ε, ϕ, χ, τ )) h 3 d dr τ + +A (ε, ϕ, χ, τ ) d dr h 3 + + 16π(m -ε)ε r 2 r 0 (1 + ετ )h 3 ϕ 2 + εχ 2 ds (1 + ετ ) + + 8π(m -ε)ε r 2 r 0 (1 + ετ ) 2 ϕ 2 + εχ 2 ds h 3 + +24π(m -ε)ε(1 + ετ ) 2 ϕ 2 + εχ 2 r h 3 -32πε 2 (1 + ετ ) ϕχ r 2 h 3 + -16πmε(1 + ετ ) ϕ 2 -εχ 2 r h 3 .
First of all, we remark that if ϕ, h

1 ∈ B ϕ , χ, h 2 ∈ B χ and τ, h 3 ∈ B τ , then ∂ ∂ϕ (A (ε, ϕ, χ, τ )) h 1 , ∂ ∂χ (A (ε, ϕ, χ, τ )) h 2 and ∂ ∂τ (A (ε, ϕ, χ, τ )) h 3 are bounded. So, we have that ∂L 3 ∂ϕ h 1 ≤ C 1 d dr τ + C 2 r 2 r 0 |ϕh 1 | ds + C 3 |ϕh 1 | r + C 4 |h 1 χ| r 2 ∂L 3 ∂χ h 2 ≤ C 5 d dr τ + C 6 r 2 r 0 |χh 2 | ds + C 7 |χh 2 | r + C 8 |ϕh 2 | r 2 ∂L 3 ∂τ h 3 ≤ C 9 d dr τ + C 10 d dr h 3 + C 11 r 2 r 0 ϕ 2 + εχ 2 ds + C 12 ϕ 2 + εχ 2 r + +C 13 |ϕχ| r 2 + C 14 ϕ 2 -εχ 2 r
with C i positive constants. With exactly the same arguments used above, we conclude that

∞ 0 ∂ ∂ϕ (L 3 (ε, ϕ, χ, τ )) h 1 dr < +∞ ∞ 0 ∂ ∂χ (L 3 (ε, ϕ, χ, τ )) h 2 dr < +∞ ∞ 0 ∂ ∂τ (L 3 (ε, ϕ, χ, τ )) h 3 dr < +∞ if (ε, ϕ, χ, τ ) ∈ (-ε 1 , ε 2 ) × B ϕ × B χ × B τ , and (h 1 , h 2 , h 3 ) ∈ B ϕ × B χ × B τ . Furthermore ∂L3 ∂ϕ , ∂L3
∂χ and ∂L3 ∂τ are continuous; thus the proof for L 3 . We consider now L 1 ; first, we have to prove that it is well defined in Y ϕ . We observe that

|L 1 (ε, ϕ, χ, τ )| ≤ C 1 1 r d dr ϕ + ϕ r 2 + C 2 χ r with C 1 , C 2 positive constants then, L 1 (ε, ϕ, χ, τ ) ∈ L 2 R 3 , thanks to conditions (2.19). Now, we have to prove that L 1 (ε, ϕ, χ, τ ) is C 1 ; by classical arguments, it's enough to show that for (h 1 , h 2 , h 3 ) ∈ B ϕ × B χ × B τ ∂ ∂ϕ (L 1 (ε, ϕ, χ, τ )) h 1 ∈ Y ϕ , ∂ ∂χ (L 1 (ε, ϕ, χ, τ )) h 2 ∈ Y ϕ , ∂ ∂τ (L 1 (ε, ϕ, χ, τ )) h 3 ∈ Y ϕ .
By a straightforward computation, we find out

∂L 1 ∂ϕ h 1 = 1 2 A -1/2 ∂A ∂ϕ h 1 1 r d dr ϕ + A 1/2 1 r d dr h 1 - h 1 r 2 , ∂L 1 ∂χ h 2 = 1 2 A -1/2 ∂A ∂χ h 2 1 r d dr ϕ + (2m -ε) h 2 r + ε(m -ε)τ h 2 r , ∂L 1 ∂τ h 3 = 1 2 A -1/2 ∂A ∂τ h 3 1 r d dr ϕ + ε(m -ε)h 3 χ r ;
and, using the positivity of A,

∂L 1 ∂ϕ h 1 ≤ C 1 1 r d dr ϕ + C 2 1 r d dr h 1 + h 1 r 2 ∂L 1 ∂χ h 2 ≤ C 3 1 r d dr ϕ + C 4 h 2 r ∂L 3 ∂τ h 3 ≤ C 5 1 r d dr ϕ + C 6 χ r .
with C i positive constants. Then, we can conclude that

R 3 ∂ ∂ϕ (L 3 (ε, ϕ, χ, τ )) h 1 2 dx < +∞ R 3 ∂ ∂χ (L 3 (ε, ϕ, χ, τ )) h 2 2 dx < +∞ R 3 ∂ ∂τ (L 3 (ε, ϕ, χ, τ )) h 3 2 dx < +∞ if (ε, ϕ, χ, τ ) ∈ (-ε 1 , ε 2 ) × B ϕ × B χ × B τ , and (h 1 , h 2 , h 3 ) ∈ B ϕ × B χ × B τ .
Furthermore ∂L1 ∂ϕ , ∂L1 ∂χ and ∂L1 ∂τ are continuous; thus the proof for L 1 and with the same arguments for L 2 .

2.2. Branches generated by solutions of Choquard equation. In this subsection, we show that a solution φ 0 = (ϕ 0 , χ 0 , τ 0 ) of (2.7) can generate a local branch of solutions of (2.6).

First, we linearize the operator D on (ϕ, χ, τ ) around (0, φ 0 )

D ϕ,χ,τ (0, φ 0 )(h, k, l) =     1 r d dr h -h r 2 + 2m k r 1 r d dr k + k r 2 + h r -m h r τ 0 -m ϕ0 r l d dr l + 16πm r 2 r 0 ϕ 0 h ds     .
Now, if we prove that D ϕ,χ,τ (0, φ 0 ) is an isomorphism, the implicit function theorem can be applied and we can find solutions of (2.6) near the ground state φ 0 .

Lemma 2.3. We define the operator V :

X ϕ × X χ → Y ϕ × Y χ , by V (ϕ, χ) = 1 r d dr ϕ -1 r 2 ϕ + 2m 1 r χ 1 r d dr χ + 1 r 2 χ + 1 r ϕ , then V is an isomorphism of X ϕ × X χ onto Y ϕ × Y χ .
This lemma is obvious if we remind that L 2 (R 3 , C 4 ) can be written as the direct sum of partial wave subspaces and that the Dirac operator leaves invariant all these subspaces (see [START_REF] Thaller | The Dirac Equation[END_REF]). So, thanks to lemma 2.3 of [START_REF] Ounaies | Perturbation method for a class of non linear Dirac equations[END_REF], we know that V :

H 1 R 3 , C 2 × H 1 R 3 , C 2 → L 2 R 3 , C 2 × L 2 R 3 , C 2 defined by V (u, v) = iσ∇u + 2mv -iσ∇v + u is an isomorphism of H 1 R 3 , C 2 × H 1 R 3 , C 2 onto L 2 R 3 , C 2 × L 2 R 3 , C 2
and then V is an isomorphism of each partial wave subspace. In particular, V coincide with V on the partial wave subspace X ϕ × X χ .

Lemma 2.4. We define the operator W :

X ϕ × X χ × X τ → Y ϕ × Y χ × Y τ , by W (h, k, l) =     1 r d dr h -h r 2 + 2m k r 1 r d dr k + k r 2 + h r -m ϕ0 r l d dr l     , then W is an isomorphism of X ϕ × X χ × X τ onto Y ϕ × Y χ × Y τ .
Proof. First we prove that W is one to one. We observe that W (h, k, l) = 0 if and only if (h, k, l) satisfies

       1 r d dr h -h r 2 + 2m k r = 0 1 r d dr k + k r 2 + h r -m ϕ0 r l = 0 d dr l = 0 in Y ϕ × Y χ × Y τ .
In particular, we must have l ≡ 0 and (h, k) solution of

1 r d dr h -h r 2 + 2m k r = 0 1 r d dr k + k r 2 + h r = 0 that is equivalent to V (h, k) = 0. So, thank to lemma 2.3, h ≡ k ≡ 0 and W is one to one in Y ϕ × Y χ × Y τ .
Secondly, we have to prove that for f

= (f 1 , f 2 , f 3 ) ∈ Y ϕ × Y χ × Y τ , there exists (h, k, l) ∈ X ϕ × X χ × X τ such that W (h, k, l) = f . This means that the system        1 r d dr h -h r 2 + 2m k r = f 1 1 r d dr k + k r 2 + h r -m ϕ0 r l = f 2 d dr l = f 3 has a solution in X ϕ × X χ × X τ for all (f 1 , f 2 , f 3 ) ∈ Y ϕ × Y χ × Y τ . We observe that ∀f 3 ∈ L 1 ((0, ∞), dr) there exist l * (r) = - ∞ r f 3 ds such that d dr l * = f 3 ; furthermore l * ∈ X τ . So, we have to show that (2.21) 1 r d dr h -h r 2 + 2m k r = f 1 1 r d dr k + k r 2 + h r = f 2 + m ϕ0 r l * has a solution in X ϕ × X χ for all (f 1 , f 2 ) ∈ Y ϕ × Y χ .
Now, we remark that ϕ0 r l * ∈ L 2 R 3 and then, thanks to lemma 2.3, (2.21) has a solution in

X ϕ × X χ for all (f 1 , f 2 ) ∈ Y ϕ × Y χ . In conclusion W is an isomorphism of X ϕ × X χ × X τ onto Y ϕ × Y χ × Y τ .
Finally, we observe that D ϕ,χ,τ (0, φ 0 )(h, k, l) can be written as

(2.22) D ϕ,χ,τ (0, φ 0 )(h, k, l) = W (h, k, l) + S(h) with (2.23) S(h) =     0 -m h r τ 0 16πm r 2 r 0 ϕ 0 h ds     .
Theorem 2.5. Let φ 0 be the ground state solution of (2.7), then there exists δ > 0 and a function η ∈ C((0, δ), X ϕ × X χ × X τ ) such that η(0) = φ 0 and D(ε, η(ε)) = 0 for 0 ≤ ε < δ.

Proof. Since D(0, φ 0 ) = 0 and D is continuously differentiable in a neighborhood of (0, φ 0 ), to apply the implicit function theorem we have to prove that D ϕ,χ,τ (0,

φ 0 ) is an isomorphism of X ϕ × X χ × X τ onto Y ϕ × Y χ × Y τ .
We observe that D ϕ,χ,τ (0, φ 0 )(h, k, l) = 0 if and only if (h, k, l) satisfies (2.24) It's well known that the unique solution of the first equation of (2.26) in H 1 r (R 3 ) is ξ ≡ 0 (see [START_REF] Lenzmann | Uniqueness of ground states for pseudo-relativistic Hartree equations[END_REF] for more details) and that implies ζ ≡ l ≡ 0. So the unique solution of (2.24) is h ≡ k ≡ l ≡ 0 and D ϕ,χ,τ (0, φ 0 ) is one to one in X ϕ × X χ × X τ .

       d dr h -h r + 2mk = 0 d dr k + k r + h -mhτ 0 -mϕ 0 l = 0 d dr l + 16πm r 2 r 0 ϕ 0 h ds = 0 that means (2.25)        -d 2 dr 2 h + 2mh -16πm 3 ∞ 0 ϕ 2 0 max(r,s) ds h -32πm 3
Next, if we show that S(h) is a compact operator, we have that D ϕ,χ,τ (0, φ 0 ) is a one to one operator that can be written as a sum of an isomorphism and a compact operator and then it's an isomorphism. First, we can easily see that T (h) = 1 r 2 r 0 ϕ 0 h ds is a compact operator from X ϕ on Y τ ; in particular, we use the fact that H 1 r R 3 is compactly embedded in L q R 3 , for 2 < q < 6, to prove that for any bounded sequence {h n } ⊂ X ϕ , the sequence {T (h n )} ⊂ Y τ contains a Cauchy subsequence. Second, we have to show that the operator h r τ 0 from X ϕ to L 2 R 3 is compact. If hn r is a bounded sequence in H 1 (R 3 ) then hn r τ 0 is precompact on L 2 loc (R 3 ), thanks to compact Sobolev embedding and, since τ 0 (r) → 0 when r → +∞, we can conclude that hn r τ 0 is precompact on L 2 (R 3 ). So S(h) is a compact operator from X ϕ on Y ϕ × Y χ × Y τ and D ϕ,χ,τ (0,

φ 0 ) is an isomorphism of X ϕ × X χ × X τ onto Y ϕ × Y χ × Y τ .
In conclusion, we can apply the implicit function theorem to find that there exists δ > 0 and a function η ∈ C((0, δ), X ϕ × X χ × X τ ) such that η(0) = φ 0 and D(ε, η(ε)) = 0 for 0 ≤ ε < δ.

εχ 2 where C 1 ,ϕ 2 + 0 ϕ 2 2 r 0 ϕ 2 +

 2120222 C 2 , C 3 , C 4 , C 5 are positive constants and, by definition, we have that d dr τ ∈ L 1 ((0, ∞), dr). εχ 2 ds dr = ∞ + εχ 2 s ds < +∞, using Hölder's inequality, then 1 r εχ 2 ds ∈ Y τ . In the same way, we can conclude that 1 r

-4m 3 R 3 |u0(y)| 2 |x-y| dy ξ -8m 3 R 3 ξ

 33233 ,s) ds Now, if we write ξ(x) = h(|x|) |x| and we remind that ϕ 0 (|x|) = |x|u 0 (x) with u 0 solution of (2.8), we have that (h, k, l) is a solution of (2.25) if ξ(x) ζ(x) = r -1

Acknowledgment. The author would like to thank professor Eric Séré for helpful discussions and useful comments.


STAMP: SMTP server Topological Analysis by Message headers Parsing

Emmanuel Lochin1,2 1 CNRS ; LAAS ; 7 avenue du colonel Roche, F-31077 Toulouse, France 2 Université de Toulouse ; UPS, INSA, INP, ISAE ; LAAS ; F-31077 Toulouse, France emmanuel.lochin@isae.fr

Abstract-This paper presents STAMP: a tool to analyse SMTP servers overlay topology. STAMP builds a weighted and oriented graph from an email database, an email log or just a simple email header allowing a post-analysis of the SMTP overlay structure and the identification of the paths used by an email. The objective of this tool is twofold. STAMP allows to automatically perform an analysis of the SMTP topology for debugging (e.g, message delay, emails loop, ...) and for metrology purposes. While several traceroute-like measurements projects try to map the Internet, to the best of our knowledge, no tool allows to drive an analysis of the SMTP overlay network. Thus, the goal of the resulting graph is to develop methods (from graph theory, statistical analysis, ...) to identify relaying problems. We aim to explore the impact of IP network problems over emails delivery (and respectively: emails' traffic over IP networks) in conjunction with IP measurements driven synchronously. In the present paper, we introduce the design and the measurement methodology of the STAMP software and as second contribution, bring out to the networking community the tool and some measurements databases.

I. INTRODUCTION

Email messages delivered using SMTP usually contain headers which describe the travel path of a message between senders and recipients [START_REF] David | Standard for the format of ARPA Internet text messages[END_REF]. However, most email users are not concerned by the transmission of data. Indeed, most of email client programs tend to hide these records. The information, gathered during the transit of emails between the source and destination, include the time they were received and transmitted by any intermediate mail transfer agents (MTA). Despite of the relative lack of interest of these information for the end-user, their analysis can reveal problems (such as servers path that cause message delay) that can be used to bench the performances of an email provider. Furthermore, it brings to ITs, where complex mail relay topology is enabled, to verify the correct forwarding rules of their email traffic. This analysis might also be useful for the identification of network problems and to detect anomalies that may indicate email server problems. For instance: a lack of email received, not even junk emails, during a certain period; a wrong configuration of a greylisting server; a loop or a down overlay link.

All these facts have motivated the use of STAMP. STAMP drives a passive analysis of email headers and draws a weighted and oriented graph from an email database, an email log or just a simple email header. STAMP allows to collect emails, within a database, at a frequency and a duration set by the user and then treats the data collected. STAMP is developed in Python 1 and can be easily modified and enhanced. As a primary extension, STAMP interfaces the Google location API [2] that enables a geolocalisation of the identified STMP servers over Google Map.

Despite the undeniable interest of this tool for network administrators, its development has been preliminary motivated to investigate STMP overlay network dynamic. As a result in the following, we focus on the utilisation of this tool for metrology purpose and present the way an email database is constructed and analysed. Section II presents the measurements method used to collect the information analysed and motivates the reason of this choice. We then explain how the graph is drawn and introduce a second functionality allowing to prune non pertinent data from this representation (section III). Finally, we present the expected future work (section IV). Several tools attempt to draw an Internet map for the sake of a better understanding of the Internet topology dynamic. However, research work also show that measurements obtained might contain several bias [START_REF] Lakhina | Sampling biases in IP topology measurements[END_REF] mainly due to the way data are collected (in particular when distributed measurements are enabled). In a recent work, the authors in [START_REF] Latapy | A Radar for the Internet[END_REF] propose to avoid these issues by using a different approach: they focus on what a given host sees the topology around itself (similar to a traceroute measurement). This kind of measure, called ego-centered view, allows to perform periodic rounds of measurements in order to obtain information on the changing of the topology. STAMP follows this measurement scheme. Indeed, this method, done with a predefined root servers, allows to collect information over a long period and to facilitate the statistical analysis and the comparison of the results obtained.

II. MEASUREMENTS METHOD AND DATA COLLECTION

The measurements setting are done through a simple interface (see Figure 1) that allows to configure both sender account and recipient (Figure 1.b); to start and stop email sending and to collect the data (Figure 1.a). First, the user must define the root of the graph tree (in general, the name of the user STMP server). Then, after the end of data collection, STAMP draws a graph starting from the first STMP server set as the root of the tree. The following section presents the graphs obtained.

III. RESULTING GRAPH

We built a dataset of emails' headers by sending emails from a french Internet Service Provider (ISP) messages to several recipients hosted in Yahoo France, Brazil, Mexico, Canada, Chile, Argentina, Colombia and in Google Mail, Hotmail, ... Each database represents a collection of hundred (or more) emails sent. All data are anonymized and once a time a database is built, the results remains accessible through the STAMP interface until deletion by the user. This allows to draw simultaneous topologies for comparison purpose.

Figure 2 presents the resulting graph obtained with Yahoo Brazil database. Each row of Figure 2 represents a transfer level. We balanced the arches (depending on the number of emails) with the following colors code:

• In red: for emails which rarely travel through this server;

• In blue: emails that always transit through this server;

• In purple: emails that often transit through it;

• In green : emails that loops between two servers. This color identifies either a wrong configuration or a server that acts like a spam filter (this is typically the case in Figure 2). We can note that in this experiment, the top of the graph is quite complex and difficult to interpret. To ease the analysis, STAMP can group servers following predefined criterion by selecting servers to group and prune from the servers lists. The rank of a server is defined by its position in the email header. Then, we establish the position in a row of a given server as a function of its predecessors and followers. As some servers can transfer messages to lower rows servers, grouped servers might not be necessarily deleted. Therefore, a recursive procedure checks the list after this first processing. First, we suppress all the servers that do not receive any message, and then, we lower servers if possible. That means we put the servers in the row just after the earliest predecessor. Finally, we suppress empty rows.

This filtering process is done over the servers in the same group and obey to the following criterion:

• They must have a common domain name;

• They must be in the same order of magnitude of frequency usage;

• They must have the same number of followers. Figure 3 illustrates the results obtained. The STAMP interface also enables statistics on demand of the approached mean and variance of the time needed to reach a server assuming that clocks are synchronized (we noticed during our experiments that most of commercial emails provider were NTP synchronized). Eventually, we have to notice that STAMP is developed in a object oriented way which allows to add others analysis methods.

IV. CONCLUSION

This paper has presented STAMP. A powerful tool able to build a graph of an SMTP overlay topology for debugging or metrology purpose. Following this development, we are currently analysing several dataset that would lead to a better understanding of this particular overlay network. We hope to generate an interest from the networking community to use this tool in order to investigate Internet measurements issues.

Fig. 1 .

 1 Fig. 1. The STAMP interface

Fig. 2 .

 2 Fig. 2. Topology obtained with yahoo.com.br

Fig. 3 .

 3 Fig. 3. Grouped topology obtained with yahoo.com.br

See http://manu.lochin.net/stamp/

V. ACKNOWLEDGEMENTS

The author would like to thank Florian Cazes, Yves de la Broise and Jean-Charles Tropato for the development of the STAMP software during their internship at ISAE.