Olivier Hueber
email: olivier.hueber@unice.fr

The top-down innovative coordination flows in Sophia Antipolis

Keywords: Clusters, entrepreneurship, innovative pole, network externality, agglomeration effect, Top-down, Bottom-up

The theoretical model that has inspired the creation of Sophia Antipolis is governed by a top-down approach. The agglomerations externalities had not sprung up naturally from the dynamics of entreprises located in the cluster. The economic model of Sophia Antipolis is completely different of the traditional innovative district studied by Alfred Marshall (bottom-up approach). The aim of this paper is to understand the coordination mechanisms between enterprises and the main factors of success who made Sophia-Antipolis the largest technology park in the Europe. Such a study presents the Top-down strategy of developpement choosen by the government from the origins of Sophia-Antipolis to promote agglomeration externalities and the increasing returns to adoption gained by firms entering in the park.

22

Introduction

Sophia Antipolis was created by the public authorities to attract high value added activities on the French Riviera, in the aim of strengthening a local economy driven historically by tourism. The creation of Sophia Antipolis forty years ago was, from the beginning, strongly supported by the French government by substantial public investments in telecommunication and transportation infrastructure.

The theoretical model that has inspired the creation of Sophia Antipolis is governed by a top-down approach. By a top-Down approach, we mean an approach built on the sheer power of a centralized processor capable of replicate whatever the human intelligence can do. Such a definition of the top-down approach is inherited from Heyman and Leijonhufvud works (1996) 1 . Concerning the setting-up of Sophia Antipolis, the role of such a centralized power was played by the French ministry of Planning.

The innovative pole of Sophia Antipolis had sprung up from a forrest located among the administrative territories of the cities of Valbonne, Antibes, Biot and Mougins2 . Except many boars and squirrels, this territory had nothing in the seventies that could be qualified as innovative. Sophia Antipolis emerged only due to the strong desire of the French Ministry of planning (top-down approach). The agglomerations externalities, had not sprung up naturally from the dynamics of entreprises located in the cluster. The economic model of Sophia Antipolis is completely different of the traditional innovative district studied by Alfred Marshall (bottom-up approach). Nowadays, the cluster of Sophia-Antipolis is rich of external linkages, but poor of internal relations between the firms. In this local system of Innovation, a large numbers of actors in different sectors are present but any of them is sufficiently dominant to drive the cluster orientations. In this sense, this Local System of Innovation (LSI) is not reliable in the long run. Very few, almost no technological collaborations can be observed. The sustainability of the Sophia-Antipolis cluster does not really depend on the territory. the weakness of the cooperation between companies of the cluster can be partially explained by the local multinational firms which have their branch facilities located in the local system of innovation but at the same time their head office external to the cluster with main decision taken from outside, limiting the potential for local synergies and local collaboration.

Admittedly, a region needs good infrastructure and a pool of educated talent to develop innovative clusters. Nevertheless, public authorities can't manufacture innovation by putting a set of buildings, financial incentives or transportation facilities next to a university and next to research centres.

In Sophia-Antipolis, much more than in other type of innovative poles, the role of incubators is tremendous. The need of effective incubators was felt from the early years of the park. Their role in a top-down approach of innovation lies in promoting the horizontal coordination between firms located in the cluster. In this intention, they have to compensate the lack of industrial and innovative atmosphere.

In this paper we propose to shed light on the necessity to have effective incubators in a technology park like Sophia Antipolis which is characterized by a tremendous lack of coordination between firms.

After presenting the story and key factors of success of Sophia Antipolis (1.), the main coordination channels, between enterprises, are studied (2.). It then become possible to enlight the central role of the incubators in a top-down strategy (3.).

The story of Sophia Antipolis from the early stages

Sophia Antipolis was created ex nihilo in the seventies. The French government of that time did not want any more than the economy of the French Riviera is based solely on tourism.

Admittedly, the emergence of a technological sector on the French Riviera is previous to the creation of Sophia Antipolis. During the period between the two world wars, an aeronautic industry was set up in Cannes and such an industry is still very dynamic nowadays. Currently the Thales Alenia Space Company provides communication satellites, and military rockets. During the same period, another technologic center devoted to electromechanics appeared in Nice. The emergence of the electromechanics industry in Nice with prestigious company like Legrand, was the breeding ground for the implantation of IBM and Texas Instruments on the hinterland of Nice. During the May 1968 protests, the touristic period on the French Riviera was very bad. The public administration in charge of the economic territorial planification, realised that the reliability of the French Riviera cannot be based on the sole tourism sector. Pierre Laffitte was the first to propose in 1969, the creation of a new industrial cluster on the French Riviera devoted to the promotion of the scientific creativity and the technological transfers among innovative firms.

The underlying strategy which has governed the setting-up of Sophia Antipolis includes three main actors namely, private innovative enterprises, the public University, the regional and local public administrations.

In the early seventies, a semi-public company called Symival was created in the aim of structuring the emerging technopole and 5800 acres were acquired on a virgin plateau located on the administrative territories of the cities of Antibes, Valbonne, Mougins, Biot and Roquefort-les-Pins,Villeneuve-Loubet and Vallauris3 .

The implantation of the University in Sophia Antipolis was initiated in 1985 with the transfert and the creation of courses for postgraduates students (Master and Doctorate). Nowadays, 2 500 students from undergraduate to postgraduate levels and belonging to one of the 6 faculties in Sophia Antipolis are studying daily in this technological park.

Sophia Antipolis has been firstly composed by entreprises which had a natural propensity to be easily relocated. Therefore, the earliest establishments of companies were in the sectors of computing and electronics (Air-France, Certiam, Organic, Télésystèmes, Questel, Thomson Sintra and Télémécanique). A second wave of company occured in the eighties with the implantation of Nortel, Lucent, Digital Equipment, Compaq... Nowadays, Sophia Antipolis is decidedly oriented to the knowledge economy. Newly relocated or created enterprises in the technological park product much more immaterial products (software, consulting services…) than real equipments ones. It is in this spirit that the SAP Company was established in early 2000s. 1960 1970 1980 1988 1990 2000 2001 2002 2003 2004 2005 2006 2007 2008 2010 2011 From the early stages of this cluster, newly established firms did not succeed to root their productive activity in the local environment. The geographical proximity did not promote positive externality or any agglomeration effect. The dynamics of this emerging territory was driven by the French ministry of planning through its institution called DATAR.

The coordination of activities in the cluster

The underlying theory of Sophia Antipolis is based on the idea that public authorities or economic development institutions by offering financial incentives, can create hubs of activity in specific industrial sectors by bringing businesses, researchers and suppliers together in buildings or industrial parks. Any government has the desire to promote the setting-up of technological park in which an industrial atmosphere is favorable for the outbreak and growth of firms. Sometimes, the industrial atmosphere emerge naturally from bottom-up coordination mechanisms. Such spontaneous emergence -like in the Italian Industrial districts -of the industrial atmosphere were clearly described by Alfred [START_REF] Marshall | Principles of Economics[END_REF]. The bottom-up approach relies on interacting networks and attempts to make networks evolve [START_REF] Heymann | High Inflation[END_REF]. With a Bottom-Up approach the innovative pole is seen as a network of interacting entreprises, each one with less capability to process information than would be required of a central planning institution set to solve the overall allocation problem for the entire system. The Marshall's conceptualization of the industrial districts teaches us that the benefits of agglomeration come from economic externalities. The externalities are the result of specifics characteristics built up over time and which cannot be transferred or replicated4 . According to Alfred Marshall, enterprises located in the same district may benefit from lower transaction cost or sharing a common labor pool.

Concerning Sophia Antipolis, the French public authorities adopted an economic policy, implemented by the DATAR, which can be qualified as a top-down policy5 . As recalled by Rani Jeanne [START_REF] Dang | Territorial innovation dynamics and integration of SMEs into the collaborative innovation projects of French competitiveness poles: the underlying mechanisms[END_REF], research and innovation in France have always been characterised by the intervention of public authorities. The term of "Colbertist State" is generally used to qualify the French system of innovation and research. The Colbertist state is a model based on the intervention of the state and which gives the priority to major civil programs, that put forward the supremacy of firstly the separation between research and firms and secondly the monopolisation of public supported by some large industrial groups. The Intervention of public authorities promotes the economic development and competitiveness of regions and cities by creating new business opportunities, by fostering entrepreneurship and incubating new innovative companies, by generating knowledge-based jobs, by building attractive spaces for the emerging knowledge workers and by enhancing the synergy between universities and companies.

The Top-down strategy is not a French specificity. Others innovative clusters governed by a top-down approach exist in many countries. For instance, we can mention the "Parc Agrari del Baix Llobregat" in Spain, The "Technologiepark Ostfalen Magdeburg-Barleben" in Germany, The Japan's science city in Tsukuba and the "Silicon Pyramid" in Egypt 3 . Russia is currently setting-up a government-planned "science city" close to Moscow, that it hopes will one day rival Silicon Valley's. Over $200 million are spent by the public authorities to build a technology hub with innovative and competitive companies.

Table 1: Workforce repartition in sophia Antipolis in 2011 123454367 89ABC9A2D7E4F77 789ABCDEFA82E8AA272 562 FB2DEFA8B DB2 !2 DE2 F8 "F82CFD 2 !2 #FF82 $12 %ACCA82 B&F 2 5'2 (BFCDB2D82 A8DB2DEFA82 3$2 %A8 EF82 352 %D8F82 3)2 *F+C8E29AB2ACA99F2 1'2 ,F D8A 2 12 DE2+BA9 FA8 2 1$2 -D2 EDE2 162 (.F2EBD8 +ABEDEFA82 132 /EB 201129AB2D2 EAB2 112 937 7 2222222222222222222222222222222222222222 22222222222222222222 2 $2
The government doles out money to companies that it selects and appoints government administrators for the management of the cluster. Michael [START_REF] Porter | The Competitive Advantage of Nations[END_REF] assets that a governmental planning policy can create something from nothing, turning, for example, a forest (like in Sophia-Antipolis in France) into a technological park including highly competitive, innovative firms. By focusing on the population of enterprises in Sophia Antipolis, we can observe that only four different categories of economic activities represent 70% of the total workforce of the technological park namely, information technology, higher education/research, health science/fine chemicals and building (see table 1).

23,9% of the enterprises in Sophia Antipolis belong to the Information technology (IT) sector but such a percentage matches 45,8% of the total number of the workforce in this technological park (see table 2). More than 300 Enterprises in the sector of the information technologies (like AMADEUS) represent nearly an half of the total workforce of the park. We can cast doubt of the technological aspect of the building activity. If in term of workforce, 70% of the workers are distributed into four different economic activities, the range of the different economic activities in Sophia Antipolis is much wider. Among the five biggest companies, three of them were forced by the public administration to locate in Sophia-Antipolis from the early years of the technological park. These two enterprises are Air-France, Thales (formerly Thomson) and Schneider Automation (formerly Télémécanique) (Table 3). It is somewhat disturbing to note that none of these five companies has its headquarters in Sophia Antipolis. This is a very negative factor for the coordination of productive activities within the technology park.

Admittedly, 65% of the companies in Sophia Antipolis have located their headquarters in the technologic park but the 35% remaining percentage correspond to the biggest companies in term of industrial activity and workforce. In the aim of promoting the IT sector in Sophia Antipolis, the French Government decided to create a pole of competitiveness devoted to secure networks (Solutions communicantes Sécurisées) 6 . Such a policy lies on increasing the French industrial potential and on creating the conditions to foster the emergence of new activities with a high international profile. This strategy is based on active partnership between Small and Medium Sized Enterprises (SMEs), research centres and training institute (either from the public or the private sector). It is expected that such cooperation promotes the emergence of synergies and cooperative efforts. A competitive cluster is a partnership association built on innovative projects. The French poles of competitiveness are inherently considered as devices of skills and knowledge networking that aims to link public researches institutes, Universities, SME's and large firms ("grand comptes") in order build innovative products that can be developed, marketed and can compete at the international level. This type of structure allows SMEs to develop, and researches to be directly applied on the industrial sector.

'2

By referring to Hobday works [START_REF] Hobday | The Project-Based Organisation: An ideal form for managing complex products and systems[END_REF], the French economic policy of poles of competitiveness can be qualified as projects based organisations (PBOs). Each competitive cluster draws up a five-year plan, based on a vision shared by the various stakeholders. With the plan, the competitiveness cluster can develop partnerships between the various stakeholders, constructs shared strategic R&D projects that can benefit from public funding -particularly the Interministeriel Fund (FUI) -and promotes an overall environment favourable to innovation and to the competitiveness cluster's stakeholders.

Governments all over the world have invested huge amount of public money to attract industries they consider strategic 4 . According to [START_REF] Porter | On Competition[END_REF], clusters are defined as: "geographic concentrations of interconnected companies and institutions in a particular field.". Furthermore, [START_REF] Porter | The Competitive Advantage of Nations[END_REF] affirms that the nation's competitiveness is directly linked to the attractiveness of their territorial productive conditions. The main idea of cluster policies promoted by the DATAR in France is very simple: geographic proximity facilitates the collective innovation. In a Local System of Innovation (LSI), positive externalities can emerge more easily. Sophia-Antipolis on the French Riviera is a concrete example of the benefits of the geographical proximity of innovative firms. The "technopolis" was created to attract high value added activities in the region, in the aim of strengthening a local economy driven historically by tourism. The creation of Sophia Antipolis forty years ago was, from the beginning, strongly supported by the French government by substantial public investments in telecommunication and transports infrastructure. There is a phenomenon of agglomeration in any local system of innovation.

Even in competitions, firms tend to gather themselves in a small territory. Agglomeration externalities have positive productivity effects. According to Paul [START_REF] Krugman | Geography and Trade[END_REF], the agglomeration externalities arise from an interaction between economies of scale that generate increasing returns, transportation costs, and regional market potential. The concept of "increasing returns to adoption" (IRA) referring to a chain of development where a system or artefact becomes increasingly more attractive to potential users and its generalization increasingly beneficial to them. A major part of the high technology market is subject to increasing returns to adoption. IRA are also presents in any technological park. Like the model presented by [START_REF] Arthur | Competing Technologies, Increasing Returns, and Lock-In By Historical Events[END_REF], more entreprises in a Local System of Innovation is, the more attractive it becomes. The increasing return to adoption effect is at work.

The best example of a successful positive externality is the birth of the firm Amadeus in Sophia Antipolis. Some 2 035 People are working in Sophia Antipolis for the Amadeus Company. Amadeus emerged thanks to the innovative atmosphere of the Sophia Antipolis technopole. Agglomeration externalities foster labor productivity, rents, and wages in regions with high economic density. Consequently agglomeration externalities can be considered as public goods. Like for any public goods, public authorities must intervene to promote positives externalities and increasing returns to scale.

Despite to appearances and some success stories like Amadeus, the role of the French public authorities for developing positives externalities in Sophia Antipolis was not so successful. Unfortunately, the cluster of Sophia-Antipolis is rich of external linkages, but poor of internal relations between the firms.

1)2

In this local system of Innovation, a large numbers of actors in different sectors are present but any of them is sufficiently dominant to drive the cluster orientations. In this sense, this Local System of Innovation is not reliable in the long run. Very few, almost no technological collaborations can be observed. The sustainability of the Sophia-Antipolis cluster does not really depend on the territory. There is no interdependence among local actors or with the immediate environment. According to Christian [START_REF] Longhi | Networks, collective learning and technology development in innovative high technology region: The case of Sophia1Antipolis[END_REF], the weakness of the cooperation between companies of the cluster can be partially explained by the local multinational firms which have their branch facilities located in the local system of innovation but at the same time their head office external to the cluster with main decision taken from outside, limiting the potential for local synergies and local collaboration. Admittedly, a region needs good infrastructure and a pool of educated talent to develop innovative clusters. Nevertheless, public authorities can't manufacture innovation by putting a set of buildings, financial incentives or transportation facilities next to a university and next to research centres.

The endogenous factors able to promote growth in the technological park are on the one hand, the existence of communities of practice and other hand, strong and reliable interactions between the different communities of practice. Somehow, there is a community of practice within the engineering sector of computing, Information and Communication. There is some mobility of engineers in the various businesses of the park. In addition, many research projects with universities and public research laboratories, promote the emergence of a common culture. This community of practice is reinforced by a rich cultural context (many cultural, religious and sports associations). The English language plays an important role in the emergence of a community of practice in the IT sector. Even if Sophia Antipolis is in France, the working language is English for the main companies in the IT sector. A large number of expat workers strengthens the anglo-saxon character of Sophia-Antipolis. On the French Riviera, in the English Langage, a local radio is broadcasted with specific programs locally set-up, a weekly newspaper and a monthly magazine are published. Moreover, people living and working in Sophia Antipolis can shop at a British bookseller, at a British supermarket and even use the services of many British craftsmen in many different activities.

The Top Down approach and the role of the Incubators

Different types of incubators exist in Sophia-Antipolis. First of all, we have to notice that the term incubator in France is equivalent to a pre-incubator in the anglosaxon sense. Then a start-up which meets success in a French pre-incubator can follow its activities in a business incubator called "pépinière". Since in France the research is mainly held by public structure, the government decided in 1999 to create pre-incubators dedicated to valuation of public research. This type of pre-incubators, called "Allegre incubators" referring to the name of the French education ministry of this period (Prof.Dr. Claude Allègre), have the mission to favour the transfer of technologies developed in the public research laboratories towards the private through new business start-up 8 . These pre-incubators can propose a coaching, the financing of outside services such as a market study or patent registration, an integration to their close networks and a hosting. Their generally principle is a refundable advance in case of success for projects holder. Several Engineer or Business Schools, which are quite wealthy in France and also some public research institutes have created their own pre-incubators. Their aim is to support the projects of new business start-up of their former students or their researchers. Some pre-incubators are dedicated to a specific public (women for example) and their existence is related to the local economic context and they are financed by public actors such as economic development agencies or competitiveness poles. In the end, some private pre-incubators exist. These structures differ from the first ones by their will of profitability. They work as a general rule by taking part with a capital of the company to realize a capital gain. Their economic model having survived with great difficulty the explosion of the internet bubble, the great majority of these incubators disappeared or reconverted as venture capital or in consulting companies. Several other structures (public, private or associative) are involved in the supporting the creation of the new business start-up. They are public or semi-public structures like the Business Innovation Centre (BIC) 9 . The BIC was created by the European Commission to foster innovation in the companies in order to strengthen an economic dynamics leaning on the local potential. BIC are distributed in the whole of the countries of the European Union. 50 technology parks, 33 BIC and 30 pre-incubators for public research are now linked by an association called RETIS. The vision is that these three structures have a complementary role (see chart 3).

In Sophia-Antipolis are located :

•

One 'Allegre' pre-incubator' called "Incubateur PACA Est"

•

The Business Innovation Centre (BIC) called "Antipolis Innovation Campus"

•

The SKEMA Business School pre-incubator

•

The Telecom Paris Tech pre-incubator

•

Several private business centres that host enterprises providing them services What we can observe is that the incubators of Sophia Antipolis are focused on research and innovation. Other structures are dedicated to 'traditional' entrepreneurship (handcraft, commerce and trade). For example a 'Couveuse' is a structure dedicated to support small business projects. An entrepreneur who as such a project can start his activity being supported by the 'couveuse' (council, formation, logistic) and will have very few administrative obligations so that he will be able to test quickly his business model during a short period of few months : he will contract with the 'Couveuse' so that he will be allowed to use the company registration number of the structure as if it was its own. To benefit from this type of device, the project must be backed by a public research laboratory or be winner of the national support for the creation of innovative technology companies, organized by the Ministry of Higher Education and research.Allègre incubators offer assistance for a period of 24 months. http://www.ebn.be/

Externalities from education Very strong

The University of Nice -Sophia Antipolis is the main actor in the promotion of entrepreneurship in the cluster

Networks externalities Weak

The increasing returns to adoption obtained by incubated firms are not really observed. After several decades, Sophia Antipolis cannot be qualified as an innovative network enable to generate positive externalities.

Non pecuniary externalities (information spillover) Medium

An innovative atmosphere exists in the cluster but mainly rely on informal relationships betweens workers. For instance in sports, in cultural or religious events.

Factors of growth

Impact of the incubators Main explanation

Conclusion

The fundamental factor for a successful innovation transfer system in Sophia Antipolis lies in the existence of a public-private partnership. Thanks to a strong commitment of the Government or its local representatives, innovation transfer systems can emerge. The Government is the only legitimate body to guaranty space organization. Consequently, one of the main responsibilities of the public authorities is to facilitate the flow of informations between different actors, to be a network mediator, and to assist firms through the sharing of information. Nevertheless, the successful of such a top-down strategy relies on the ability to set up a reliable coordination between firms located in the cluster.

The French authorities pride themselves on having created in Sophia Antipolis a "Silicon Valley" in comparison with the California's famous technopolis in the United States. Admittedly, in terms of workforce, Sophia Antipolis is the biggest Technological Park in The Europe but the comparison reaches its end to that point. Concerning the famous technological park Silicon Valley in California, even if the public authorities helped, by financial incentives or transportation facilities, to the economic growth of this territory (top-down approach), the innovative atmosphere which had emerged from this park is rooted in the private dynamics of the innovative SMEs (Bottom-up approach).

After four decades of existence, the reliability and the sustainability of Sophia-Antipolis is still to be proven. Admittedly, the incubators, the industrial atmosphere, the University environment and agglomeration effects are very important for the success of the Top-down strategy that governs the development of Sophia Antipolis but all these factors can not alone explain the continuing growth of this park.

The world is full of spaces offering tax incentives, efficient means of transport and a pleasant living environment. The factors of attractiveness of the Sophia Antipolis technological park are actually to be found outside the park. Expatriate workers who arrive in France with their families can benefit from education and health systems virtually free. For a salary equal to their country of origin, their disposable income is much more important. This is one of the main reasons why France is the first home in Europe for foreign direct investment.

Chart 1 :

 1 Employment trends in the technology park of Sophia Antipolis

2

  The Interministerial Regional Planning and Development Committee (CIADT) decided in December 2002 a new industrial strategy based on the development of competitive clusters 7 .

  They were born of the Law on Innovation and Research of the July 12, 1999. 28 incubators certified by the Department are spread throughout France.

Table 2 :

 2 enterprises repartition in Sophia Antipolis in 2011 Through different criteria of segmentation (size, workforce...), we can observe the dominant share of the Information technology sector. Such IT sector includes 325 firms and hires 13.425 people. Even if the average number of employees in the IT sector is 42, about thirty enterprises hire more or less 100 people. The five biggest employers in Sophia Antipolis belong to the IT sector (see table3). Of a total of 325 companies, the 10 biggest enterprises in term of workforce in the IT's sector represent 42.5% of the total workforce of this sector.

	Activity	Entreprises (in %)
	Information technology (IT)	23,9
	Higher Education/research	7,9
	Health sciences/Fine chemicals	7,8
	Building	5,1
	Common services	4,9
	Primary and secondary education	4,3
	Consulting	3,8
	Cleaning	3,6
	Equipment for home/office	3,5
	Miscellanous	3,4
	Health professions	3,2
	Real estate	3,1
	Public transportation	3,0
	Catering	1,9
	Earth Sciences	1,9
	Publishing, advertising, communication	1,7
	Travel agency	1,3
	Cars	1,3e
	Security	1,2
	Hotels	1,1
	Associations, clubs	1,0
	Others (<1% for each sector)	10,8
	Total	100,0

The technological park of Sophia Antipolis is characterized by the predominance of the small and medium sized enterprises (SME's). The number of employees per firm is on average 13 with very large disparities (see Chart 2).

!2

Table 3 :

 3 The 5 biggest employers in Sophia Antipolis11

	D7	CCD234C73937 123454367
			(BAE29F8FEFA82&A+C8E262
			CD8DC8E2A92E2AC+EBF 2
			B B&DEFA82 EC2F8E829AB2EBD&2
			D8E 2DFBF8 2AE 2DB2B8ED 2
	2,23452	3)462	BDF7D 8882
	2952		AF82F8&A&2F82 A8DB2. F8 2
	4:3-;2-2		D8EF .CDBF827DB9DB2 A8DB 2 F8D2D82
	5<5,525252		DED2+BA F82
	2:32D 2		%A8 ED8E 2F82DED2+BA F82F8EBDEFA82
	5B&F25252	3$2	A92 EC 2
			5A9E7DB2A 29AB2E2AC2>
			B&F 2DB5A9E>A F82D82 +D2
	25=252 E2	66)2	EAB 2
			(BAE29F8FEFA82&A+C8E2D82
	27-2"-2:%2>2		A+BDEFA82A927AB7F2DED2+BA F82
	2#9755,:2		EC 29AB2.AE2+D 8B2D829BFE2
	7:"/-,27?42 542	EBD99F 2
			3 F82D82&A+C8E2A92+BABDCCD.2
	5%:73-2		DEACDE2 EC 2D82D AFDE2
	24/,27/:2	53)2	A9E7DB2

Table 4 :

 4 The actors concerned by an entrepreneurial project according to the target activity As pointed out by Romana Rauter, the coordination of economic activities in the cluster lies partly on the capacity of SME's to develop their absorptive capacity. Admittedly, the incubators have a disseminating capacity of knowledge but the SME must be able to absorb such a disseminated knowledge (seeCohen and Levinthal, 1990).

	Activity Targeted	Incubation structure	Actors dedicated
		dedicated
	Handcraft	Couveuses, pépinière des	Chambre des Métiers et de
	(hairdressing,	communautés	l'Artisanat
	plumber, etc…)	d'agglomérations
	Commerce and trade	Couveuses, pépinière des	Economic Chamber of
		communautés	Commerce
		d'agglomérations
	Innovative business	IBEC	Economic Chamber of
			Commerce, OSEO
	Research valorization State Incubators	OSEO

Table 5 :

 5 Impact of the incubators in Sophia-Antipolis on the main factors of growth traditionally at work in any innovative pole

	Factors of	Impact of the
	growth	incubators	Main explanation
			Few incubated firms collaborate together
			and gather information. Very few
	Agglomeration		agglomerations externalities excepting
	externalities	Medium	the Amadeus Company can be observed.

[START_REF] Heymann | High Inflation[END_REF]. P. 156.

2The city of Antibes was named Antipolis at the time of the Ancient Greek. The word Sophia in the Ancient Greek Language means Wisdom.

Syndicat Mixte pour l'aménagement du plateau de Valbonne (Symival)

Externalities are present when the actions of one agent affect the interests of another agent other than by affecting prices.

The "Délégation interministérielle à l'Aménagement du Territoire et à l'Attractivité Régionale" (DATAR) that is the Interministerial Spatial Planning and Regional Attractiveness prepares, promotes and coordinates the policies territorial planning held by the state.

scs.org/ -http://www.pole

2222222222222222222222222222222222222222 22222222222222222222 27 The Interdepartmental Committee for land planning and regional attractiveness (Comité interministériel d'aménagement, de développement du territoire et d'attractivité régionale, CIADT), chaired by the Prime Minister decides the orientation of national policy planning.

The public incubators "Allegre" depend on the Ministry of Higher Education and Research and aims to promote research with the creation of private enterprise.


Emmanuel Creusé
email: emmanuel.creuse@math.univ-lille1.fr

Serge Nicaise
email: serge.nicaise@univ-valenciennes.fr

Emmanuel Verhille
email: emmanuel.verhille@math.univ-lille1.fr

Robust residual a posteriori error estimators for the Reissner-Mindlin eigenvalues system

Keywords: Reissner-Mindlin plate, finite elements, a posteriori error estimators, eigenvalues AMS (MOS) subject classification 74K20, 65M60, 65M15, 65M50

We consider a conforming finite element approximation of the Reissner-Mindlin eigenvalue system, for which a robust a posteriori error estimator for the eigenvector and the eigenvalue errors is proposed. For that purpose, we first perform a robust a priori error analysis without strong regularity assumption. Upper and lower bounds are then obtained up to higher order terms that are superconvergent, provided that the eigenvalue is simple. The convergence rate of the proposed estimator is confirmed by a numerical test.

Introduction

Nowadays, a posteriori error estimators have become an indispensable tool in the context of finite element methods. They are now widely used in order to control the numerical error, as well as to drive the adaptive mesh refinement processes. Many works have been devoted to this topic (see e.g. [START_REF] Ainsworth | A posteriori error estimation in finite element analysis[END_REF][START_REF] Babuška | The finite element method and its reliability[END_REF][START_REF] Neittaanmäki | Reliable methods for computer simulation[END_REF][START_REF] Verfürth | A review of a posteriori error estimation and adaptive mesh-refinement techniques[END_REF] for general monographies). Considering the Reissner-Mindlin system, several kind of suitable finite elements exist, and a well known task to overcome is to avoid the so-called "shear locking effect", by using properly defined operators at the discrete level. In the literature, if a lot of papers have already been devoted to the a priori error analysis of this system, far less references can be found on its a posteriori error analysis (see e.g. [START_REF] Beirão Da Veiga | A-priori and a-posteriori error analysis for a family of Reissner-Mindlin plate elements[END_REF][START_REF] Carstensen | Residual-based a posteriori error estimate for a nonconforming Reissner-Mindlin plate finite element[END_REF][START_REF] Carstensen | A posteriori error analysis for conforming MITC elements for Reissner-Mindlin plates[END_REF][START_REF] Carstensen | Residual-based a posteriori error estimate for a mixed Reißner-Mindlin plate finite element method[END_REF][START_REF] Frolov | Guaranteed functional error estimates for the Reissner-Mindlin plate problem[END_REF][START_REF] Hu | A posteriori error analysis of finite element methods for Reissner-Mindlin plates[END_REF][START_REF] Liberman | A posteriori error estimator for a mixed finite element method for Reissner-Mindlin plate[END_REF][START_REF] Lovadina | A posteriori error analysis of the linked interpolation technique for plate bending problems[END_REF]).

In this work, we are specifically interested in the Reissner-Mindlin eigenvalues system, corresponding to the modeling of a vibration plate problem. Our goal is to derive an a posteriori estimator which is robust with respect to the plate thickness parameter t, efficient and also explicitly computable. To our best knowledge, only the a priori analysis of this eigenvalue problem in a regular context is up to now available (see [START_REF] Durán | Error estimates for low-order isoparametric quadrilateral finite elements for plates[END_REF][START_REF] Durán | Approximation of the vibration modes of a plate by Reissner-Mindlin equations[END_REF][START_REF] Durán | Finite element analysis of the vibration problem of a plate coupled with a fluid[END_REF][START_REF] Gómez | On the vibrations of a plate with a concentrated mass and very small thickness[END_REF][START_REF] Hernández | Approximation of the vibration modes of a plate coupled with a fluid by low-order isoparametric finite elements[END_REF][START_REF] Lovadina | Approximation of the buckling problem for Reissner-Mindlin plates[END_REF] for an overview on this topic). We have here in mind to extend it to the non regular context, and, with these results in hand, to provide a relevant a posteriori error estimator. For similar results for the Laplace equation, we refer to [START_REF] Larson | A posteriori and a priori error analysis for finite element approximations of self-adjoint elliptic eigenvalue problems[END_REF][START_REF] Durán | A posteriori error estimates for the finite element approximation of eigenvalue problems[END_REF].

The outline of the paper is as follows : In Section 1, we recall the Reissner-Mindlin eigenvalues system and its discretization. Section 2 gives an a priori error analysis whithout strong regularity assumptions, that constitutes its originality. Section 3 is devoted to some preliminary results in order to prove the upper bound of the a posteriori estimator. This one directly follows and is detailed in section 4. We then give an a posteriori estimate for the eigenvalues error in section 5. The lower bound is developped in section 6 and leads to the efficiency of our estimator. Finally, some numerical tests are presented in section 7, that confirm its requested behavior.

The boundary value problem and its discretization

Let Ω be a bounded open domain of R 2 with a Lipschitz boundary that we suppose to be polygonal. Assuming that the plate is clamped, its free vibration modes are solutions of the following problem (called Reissner-Mindlin eigenvalue problem) : Given t a fixed positive real number that represents the thickness of the plate, find a non-trivial solution (ω, φ) ∈ H 1 0 ( Ω) × H 1 0 ( Ω) 2 and νt > 0 such that for all (ṽ, ψ) ∈ H 1 0 ( Ω) × H 1 0 ( Ω) 2 we have :

t 3 ã( φ, ψ) + ζ t Ω( ∇ω -φ) • ( ∇ṽ -ψ) dx = ν 2 t t Ω ρ ω ṽ dx + t 3 12 Ω ρ φ • ψ dx , ( 1
)
where νt is the angular vibration frequency, ρ is the density of the plate and ã( φ, ψ) = Ω C ε( φ) : ε( ψ)dx.

Here, the operator : denotes the term-by term tensor product and

ε( φ) = 1 2 ( ∇ φ + ( ∇ φ) T ),
where ∇ denotes the usual gradient operator over Ω. C is the elasticity tensor given by

C ε( φ) = E 12(1 + ν) ε( φ) + E ν 12(1 -ν 2 )
tr(ε( φ)) I,

where E and ν are respectively the Young modulus and the the Poisson coefficient of the material. We also define

ζ = E k 2(1 + ν) ,
where k is the shear correction factor usually equal to 5/6 [START_REF] Durán | Approximation of the vibration modes of a plate by Reissner-Mindlin equations[END_REF]. Now, in order to perform an a posteriori error analysis that do not depend on the chosen unit of length problem [START_REF] Ainsworth | A posteriori error estimation in finite element analysis[END_REF] has to be given in its dimensionless formulation. To do it, we introduce a density as well as a length scale of reference, respectively denoted by ρ and L (the latest being in the order of the diameter of the domain Ω). We consequently define the dimensionless variables and unknowns x, ρ, φ and ω by :

x = L x, ρ = ρ ρ, φ = φ and ω = L ω.

Considering the case of the constant density (ρ ≡ ρ so that ρ ≡ 1), problem [START_REF] Ainsworth | A posteriori error estimation in finite element analysis[END_REF] in which the eigenvector is normalized is now equivalent to find a non-trivial (ω, φ) ∈ H 1 0 (Ω) 2 × H 1 0 (Ω) and α t > 0 such that for all (v, ψ) ∈ H 1 0 (Ω) × H 1 0 (Ω) 2 we have :

           a(φ, ψ) + Ω γ • (∇v -ψ) dx = α t Ω ω v dx + t 2 12 Ω φ • ψ dx , Ω ω 2 dx + t 2 12 Ω φ • φ dx = 1, (2)
where we note : a(φ, ψ) = Ω Cε(φ) : ε(ψ)dx, Cε(φ) = 2µ ε(φ) + λ tr(ε(φ)) I,

with µ = 1 24(1 + ν) , λ = ν 12(1 -ν 2 )
, and ε(φ) = 1 2 (∇φ + (∇φ) T ).

Defining ζ = ζ/E, the dimensionless variables and parameters arising in [START_REF] Arnold | A uniformly accurate finite element method for the Reissner-Mindlin plate[END_REF] are given by :

t = t/L, γ = ζ t 2 (∇ω -φ) and α t = ρ ν 2 t L 2 E t 2 .
From now on, the parameter t is supposed to be in the interval (0, t max ] with t max > 0 fixed. In the following, (• , •) D stands for the usual inner product in (any power of) L 2 (D). For shortness the L 2 (D)-norm is denoted by • D . For s ≥ 0, the usual norm and seminorm of H s (D) are respectively denoted by • s,D and | • | s,D and the usual norm on

H -s (D) = (H s 0 (Ω)) ′ is denoted • -s,D .
For all these notations, in the case D = Ω, the index Ω is dropped. The usual Poincaré-Friedrichs constant in Ω is the smallest positive constant c F such that ||φ|| ≤ c F |φ| 1 ∀φ ∈ H 1 0 (Ω) 2 . By Korn's inequality [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF], a is an inner product on H 1 0 (Ω) 2 equivalent to the usual one. Indeed, defining the energy norm || • || C by

ψ 2 C = a(ψ, ψ) ∀ ψ ∈ H 1 0 (Ω) 2 ,
it can be shown (see [START_REF] Creusé | Robust equilibrated a posteriori error estimators for the Reissner-Mindlin system[END_REF]) that

|ψ| 2 1 ≤ 1 µ ψ 2 C ∀ ψ ∈ H 1 0 (Ω) 2 . ( 3
)
Let us now consider a discretization of (2) based on a conforming triangulation T h of Ω composed of triangles. We assume that this triangulation is regular, i.e., for any element T ∈ T h , the ratio h T /ρ T is bounded by a constant σ > 0 independent of T and of the mesh size h = max

T ∈T h h T
, where h T is the diameter of T and ρ T the diameter of its largest inscribed ball. We consider on this triangulation classical conforming finite element spaces

W h × Θ h such that W h ⊂ W ℓ,h := v h ∈ C 0 ( Ω); v h = 0 on ∂Ω and v h|T ∈ P ℓ (T ) ∀ T ∈ T h ⊂ H 1 0 (Ω), Θ h ⊂ W ℓ,h × W ℓ,h ⊂ H 1 0 (Ω) × H 1 0 (Ω)
, for some positive integer ℓ, where P ℓ (T ) is the space of polynomials of degree at most l defined on T . The discrete formulation of the Reissner-Mindlin eigenvalue problem is now to find (ω h , φ h ) ∈ W h × Θ h and α t,h > 0 such that

     a(φ h , ψ h ) + (γ h , ∇v h -R h ψ h ) = α t,h (ω h , v h ) + t 2 12 (φ h , ψ h ) , ∀ (v h , ψ h ) ∈ W h × Θ h , ω h 2 + t 2 12 φ h 2 = 1, (4) with γ h = ζt -2 (∇ω h -R h φ h ). (5)
Here, R h denotes the reduction integration operator in the context of shear-locking with values in the so-called discrete shear force space Γ h which depends on the involved finite element [START_REF] Bathe | A four-node plate bending element based on Mindlin-Reissner plate theory and a mixed interpolation[END_REF][START_REF] Brezzi | Mixed-interpolated elements for Reissner-Mindlin plates[END_REF][START_REF] Durán | Error estimates for low-order isoparametric quadrilateral finite elements for plates[END_REF][START_REF] Durán | On mixed finite element methods for the Reissner-Mindlin plate model[END_REF][START_REF] Stenberg | An hp error analysis of MITC plate elements[END_REF]. We assume moreover that

R h ψ h ∈ H 0 (rot, Ω) ∀ψ h ∈ Θ h , where H 0 (rot, Ω) = {v ∈ L 2 (Ω) 2 ; rot v ∈ L 2 (Ω) and v • τ = 0 on ∂Ω}, equipped with the norm v 2 H(rot,Ω) = v 2 + rot v 2 .
Here, for any

v = (v 1 , v 2 ) T ∈ L 2 (Ω) 2 , rot v = ∂v 2 /∂x -∂v 1 /
∂y and τ is the unit tangent vector along ∂Ω. In this work, R h is defined as the interpolation operator from Θ h on the H 0 (rot, Ω) conforming lower-order Nedelec finite element space [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF].

In this paper, we consider the lowest order MITC element (also called the Duran Liberman element) for which W h and Θ h are defined by

W h = v h ∈ C 0 ( Ω); v h = 0 on ∂Ω and v h|T ∈ P 1 (T ) ∀ T ∈ T h , Θ h = W 2 h ⊕ B h
, where B h is the edge bubble space (see [START_REF] Carstensen | A posteriori error analysis for conforming MITC elements for Reissner-Mindlin plates[END_REF][START_REF] Durán | On mixed finite element methods for the Reissner-Mindlin plate model[END_REF] for more details). In that case, Γ h is chosen as the lowest order Nédélec finite element space, namely

Γ h = σ ∈ H 0 (rot, Ω); σ |T ∈ P 0 (T ) 2 ⊕ P 0 (T )(x 2 , -x 1 ) ⊤ ∀ T ∈ T h ,
and the reduction operator R h is the associated interpolation operator that is characterized as follows: for any

ψ ∈ H 0 (rot, Ω), R h ψ is the unique element in Γ h satisfying E (R h ψ -ψ) • τ E ds = 0,
for all edges E of T and any T ∈ T h . The advantage of this element is that it is locking free (see [START_REF] Durán | On mixed finite element methods for the Reissner-Mindlin plate model[END_REF] for a robust a priori estimate). Other examples are also possible, we refer to Table 1 of [START_REF] Carstensen | A posteriori error analysis for conforming MITC elements for Reissner-Mindlin plates[END_REF] for a comprehensive list. In that case, our a posteriori error analysis is valid, but the robust a priori error analysis remains open for some of these elements (for instance, the MITC3 element).

By the usual Helmholtz decomposition of any H 0 (rot, Ω) vector field [8, p. 299], for any

φ h ∈ Θ h there exist z ∈ H 1 0 (Ω) and β ∈ H 1 0 (Ω) 2 such that, (R h -I)φ h = ∇z -β, (6)
as well as a constant C > 0 such that

z 2 1 + β 2 1 ≤ C (R h -I)φ h 2 H(rot,Ω) . (7)
More precisely, if we introduce the constant c R such that

|β| 1 ≤ c R rot(R h -I)φ h , then we have C = (1 + c 2 F )(1 + c 2 R + c 2 R c 2 F ).
If (ω, φ) is the solution of ( 2) and (ω h , φ h ) the one of (4), the usual error e ev h is defined as

(e ev h ) 2 = |ω -ω h | 2 1 + |φ -φ h | 2 1 + ζ -1 t 2 γ -γ h 2 + ζ -2 t 4 rot(γ -γ h ) 2 + γ -γ h 2 -1 . (8)
The residuals are defined as follows :

Res 1 (v) = (α t,h ω h , v) -(γ h , ∇v) for any v ∈ H 1 0 (Ω), (9)
Res 2 (ψ) = -a(φ h , ψ) + (γ h , ψ) + t 2 12 (α t,h φ h , ψ) for any ψ ∈ H 1 0 (Ω) 2 . ( 10
)
We finally need to introduce the following mesh-dependent norm. For all (ψ, v) ∈

H 1 0 (Ω) × H 1 0 (Ω) 2 , we define | (ψ, v)| 2 1,h = ∇ψ 2 + T ∈T h 1 t 2 + h 2 T ∇v -ψ 2 T . (11)
For any functional F defined on H 1 0 (Ω) × H 1 0 (Ω) 2 , the dual norm associated with ( 11) is classically defined by

| F | -1,h = sup (ψ,v)∈H 1 0 (Ω)×H 1 0 (Ω) 2 \{0} F (ψ, v) | (ψ, v)| 1,h . (12)
In the following, the notation a b and a ∼ b mean the existence of positive constants c 1 and c 2 , which are independent of the mesh size, of the plate thickness parameter t, of the quantities a and b under consideration and of the coefficients of the operators such that a c 2 b and c 1 b a c 2 b, respectively. The constants may in particular depend on the aspect ratio σ of the mesh. We denote by ω T the union of elements T ′ ∈ T h that share at least a node with T and by ω E the union of elements having in common the edge E.

Finally, E h denote the set of interiors edges in T h and, for any edge E ∈ E h , we define by h E its length and by n E a fixed unit normal vector to E.

Robust a priori estimations

This section is devoted to an a priori error analysis of the Reissner-Mindlin eigenvalue problem. This subject is the origin of a lot of works (see e.g. [START_REF] Durán | Error estimates for low-order isoparametric quadrilateral finite elements for plates[END_REF], [START_REF] Durán | Approximation of the vibration modes of a plate by Reissner-Mindlin equations[END_REF], [START_REF] Durán | Finite element analysis of the vibration problem of a plate coupled with a fluid[END_REF], [START_REF] Gómez | On the vibrations of a plate with a concentrated mass and very small thickness[END_REF], [START_REF] Hernández | Approximation of the vibration modes of a plate coupled with a fluid by low-order isoparametric finite elements[END_REF], [START_REF] Lovadina | Approximation of the buckling problem for Reissner-Mindlin plates[END_REF]) in the smooth case, in the sense that the domain is supposed to have a smooth boundary or to be a convex polygon. Here we want to perform a similar analysis without the convexity assumption. This requires to revisit the whole results with less regular solutions. We first start with robust a priori estimates for the Reissner-Mindlin system with data in L 2 (Ω) and then give their consequence to the eigenvalue problem.

Robust a priori estimates for the Reissner-Mindlin system

As suggested before, we need to determine the regularity properties and to give uniform estimates of the solution of the Reissner-Mindlin system with L 2 right-hand side. For this purpose, let us consider the following problem : Given g ∈ L 2 (Ω) and ϕ ∈ L 2 (Ω) 2 , find

(β t , w t ) ∈ H 1 0 (Ω) 2 × H 1 0 (Ω) such that for all (η, v) ∈ H 1 0 (Ω) 2 × H 1 0 (Ω)    a(β t , η) + (τ t , ∇v -η) = (g, v) + t 2 12 (ϕ, η), τ t = ζt -2 (∇w t -β t ). ( 13
)
This problem has a unique solution in H 1 0 (Ω) 2 × H 1 0 (Ω) since the bilinear form

((β, w), (η, v)) → a(β, η) + ζt -2 (∇w -β, ∇v -η), is coercive in H 1 0 (Ω) 2 × H 1 0 (Ω).
For such a problem we have the following regularity result with robust a priori estimates (in the regular case, see Theorem 7.1 of [START_REF] Arnold | A uniformly accurate finite element method for the Reissner-Mindlin plate[END_REF]).

Theorem 2.1 There exists ε 0 ∈ (0, 1 2 ] such that for all ε ∈ (0, ε 0 ], (β t , w t ) ∈ H 3/2+ε (Ω) 2 × H 3/2+ε (Ω) with

β t 3/2+ε + w t 3/2+ε + τ t -1/2+ε + t τ t 1/2+ε g + t 2 ϕ . (14)
Proof: As in [START_REF] Arnold | A uniformly accurate finite element method for the Reissner-Mindlin plate[END_REF], we see that

(β t , w t ) ∈ H 1 0 (Ω) 2 × H 1 0 (Ω) is the unique solution of (13) if and only if (r, β t , p t , w t ) ∈ H 1 0 (Ω) × H 1 0 (Ω) 2 × Ĥ1 (Ω) × H 1 0 (Ω) is solution of the triangular system        ζ(∇r, ∇µ) = (g, µ), ∀µ ∈ H 1 0 (Ω), a(β t , ψ) -ζ(curl p t , ψ) = ζ(∇r, ψ) + t 2
12 (ϕ, ψ), ∀ψ ∈ H 1 0 (Ω) 2 , -(β t , curl q) -t 2 (curl p t , curl q) = 0, ∀q ∈ Ĥ1 (Ω), (∇w t , ∇s) = (β t + t 2 ∇r t , ∇s), ∀s ∈ H 1 0 (Ω), [START_REF] Durán | On mixed finite element methods for the Reissner-Mindlin plate model[END_REF] with the relation t -2 (∇w t -β t ) = ∇r + curl p t , and the notation

Ĥ1 (Ω) = H 1 (Ω) ∩ L2 (Ω), L2 (Ω) = {q ∈ L 2 (Ω) : Ω q(x) dx = 0}.
Now we divide the proof is different steps:

1) The first problem in ( 15) is a Dirichlet problem in Ω with a L 2 (Ω) datum, therefore by [START_REF] Grisvard | Singularities in boundary value problems[END_REF]Corollary 2.4.4], there exists ε ∆ ∈ (0, 1 2 ] such that r ∈ H 3/2+ε (Ω), for all ε ∈ (0, ε ∆ ] and r 3/2+ε g .

2) We now look at the system in (β t , p t ) that by taking the difference between the second and the third line of (15) (multiplied by ζ) takes the form

a(β t , ψ)-ζ(curl p t , ψ)+ζ(β t , curl q)+ζt 2 (curl p t , curl q) = F, ψ , ∀(ψ, q) ∈ H 1 0 (Ω) 2 × Ĥ1 (Ω), (17)
where here F := ζ∇r+ t 2 12 ϕ. Again this problem has a unique solution for any F ∈ H -1 (Ω) 2 since the left-hand side is coercive in H 1 0 (Ω) 2 × Ĥ1 (Ω). By taking (ψ, q) = (β t , p t ) and using Korn's inequality, we get

β t 2 1 + t 2 curl p t 2 F -1 β t 1 ,
and therefore

β t 1 + t curl p t F -1 . (18)
But by taking ψ = 0 in [START_REF] Durán | Approximation of the vibration modes of a plate by Reissner-Mindlin equations[END_REF], we get

(β t , curl q) + t 2 (curl p t , curl q) = 0, ∀q ∈ H 1 (Ω), (19)
since the curl of a constant function is zero. By integration by parts, we get equivalently (curl β t , q) = -t 2 (curl p t , curl q), ∀q ∈ H 1 (Ω).

By Cauchy-Schwarz's inequality in the right-hand side, we obtain

curl β t -1 = sup q∈H 1 0 (Ω),q =0 |(curl β t , q)| q 1 t 2 curl p t ,
and by ( 18) we arrive at curl

β t -1 t F -1 . (20)
We now look at an estimate of the L 2 -norm of p t . For that purpose, we notice that by Corollary I.2.4 of [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF] there exists φ 0 ∈ H 1 0 (Ω) 2 such that div φ 0 = p t in Ω, and φ 0 1 p t .

Therefore the function ψ 0 = (-φ 0 2 , φ 0 1 ) belongs to H 1 0 (Ω) 2 and satisfies

curl ψ 0 = div φ 0 = p t in Ω, (21)
as well as

ψ 0 1 p t . ( 22
)
Using as test function in [START_REF] Durán | Approximation of the vibration modes of a plate by Reissner-Mindlin equations[END_REF] the pair (ψ 0 , 0), we get

ζ(curl p t , ψ 0 ) = a(β t , ψ 0 ) -F, ψ 0 .
Using ( 21) and Green's formula, we obtain

ζ p t 2 = ζ(p t , curl ψ 0 ) = ζ(curl p t , ψ 0 ) = a(β t , ψ 0 ) -F, ψ 0 .
Using Cauchy-Schwarz's inequality, ( 18) and ( 22), we arrive at

p t F -1 . (23)
Let us now introduce the mapping A 0 as follows

A 0 : H -1 (Ω) 2 → H 1 0 (Ω) 2 × L2 (Ω) : F → (β 0 , p 0 ), where (β 0 , p 0 ) ∈ H 1 0 (Ω) 2 × L2 (Ω)
is the unique solution of the Stokes like system (that formally corresponds to [START_REF] Durán | Approximation of the vibration modes of a plate by Reissner-Mindlin equations[END_REF]

with t = 0) a(β 0 , ψ) -ζ(p 0 , curl ψ) = F, ψ , ∀ψ ∈ H 1 0 (Ω) 2 , (curl β 0 , q) = 0, ∀q ∈ L2 (Ω). ( 24
)
Clearly (see [2, p. 1288]) A 0 is an isomorphism and consequently for all t ∈ (0, t max ] we can consider the mapping

B t : H 1 0 (Ω) 2 × L2 (Ω) → L2 (Ω) × H -1 (Ω) : (β 0 , p 0 ) → (p t , curl β t )
where (β t , p t ) is the unique solution of ( 17) with right-hand side F = A -1 0 (β 0 , p 0 ). First we notice that the estimates ( 20) and [START_REF] Grisvard | Singularité en elasticité[END_REF] imply that B t is uniformly (in t) bounded in the sense that

p t + t -1 curl β t -1 β 0 1 + p 0 .
On the other hand the proof of Theorem 7.1 of [START_REF] Arnold | A uniformly accurate finite element method for the Reissner-Mindlin plate[END_REF] shows that B t is also uniformly bounded from

H 2 (Ω) 2 ∩ H 1 0 (Ω) 2 × Ĥ1 (Ω) to Ĥ1 (Ω) × L 2 (Ω) in the sense that p 1 + t -1 curl β t 0 β 0 2 + p 0 1 ,
reminding that curl β 0 = 0. Therefore by interpolation, the mapping B t is uniformly bounded from

H 1+s (Ω) 2 ∩ H 1 0 (Ω) 2 × Ĥs (Ω) to Ĥs (Ω) × H s-1 (Ω), for all s ∈ [0, 1], s = 1/2 (for s = 1/2,
the statement is also valid but the target space should be changed into Ĥ1/2 (Ω) × ( H1/2 (Ω)) ′ ) with the estimate

p s + t -1 curl β t s-1 β 0 1+s + p 0 s . (25)
Let us show that this implies that there exists ε 0 ∈ (0, 1 2 ] such that for all ε ∈ (0, ε 0 ] (β t , p t ) belongs to H 3/2+ε (Ω) 2 × H 3/2+ε (Ω) with the estimate

β t 3/2+ε + p t 1/2+ε + t p 3/2+ε F -1/2+ε . (26)
Indeed by [START_REF] Grisvard | Singularities in boundary value problems[END_REF]Theorem 6.2.3] and [34, section 6.2], there exists ε S ∈ (0, 1 2 ] such that for all ε ∈ (0,

ε S ], A 0 is an isomorphism from H -1/2+ε (Ω) into H 3/2+ε (Ω) 2 ∩ H 1 0 (Ω) 2 × H 1/2+ε (Ω) ∩ L2 (Ω).
Hence by the property (25) of B t with s = 1/2 + ε, we get

p t 1/2+ε + t -1 curl β t -1/2+ε F -1/2+ε . (27)
At this stage, we can look at β t ∈ H 1 0 (Ω) 2 solution of the elasticity system

a(β t , ψ) = F + ζ curl p t , ψ ∀ψ ∈ H 1 0 (Ω) 2 ,
and using [23, Thm 6.1] and [34, section 6.1], there exists ε L ∈ (0, 1 2 ] such that for all ε ∈ (0, ε L ], 2 with the estimate

β t ∈ H 3/2+ε (Ω) 2 if F + ζ curl p t ∈ H -1/2+ε (Ω)
β t 3/2+ε F + ζ curl p t -1/2+ε . (28)
In a second step, as [START_REF] Durán | A posteriori error estimates for the finite element approximation of eigenvalue problems[END_REF] means that

p t ∈ Ĥ1 (Ω) is the unique solution of the Neumann problem ∆p t = t -2 curl β t in Ω, ∂ n p t = 0 on ∂Ω.
Hence if ε ∈ (0, ε ∆ ] by [13, ?], we find that p t belongs to H 3/2+ε (Ω) with the estimate

p t 3/2+ε t -2 curl β t -1/2+ε .
Consequently for ε 0 ≤ min{ε ∆ , ε S }, by ( 27), we get

t p t 3/2+ε F -1/2+ε . (29)
The estimate [START_REF] Hu | A posteriori error analysis of finite element methods for Reissner-Mindlin plates[END_REF] then follows from ( 27), ( 28) and ( 29) by choosing ε 0 = min{ε S , ε L , ε ∆ }.

Coming back to problem [START_REF] Durán | On mixed finite element methods for the Reissner-Mindlin plate model[END_REF], the right-hand side of ( 17) is given by F := ζ∇r + t 2 12 ϕ. Hence by ( 16) and ( 26), for all ε ∈ (0, ε f ], (β t , p t ) belongs to H 3/2+ε (Ω) 2 × H 3/2+ε (Ω) with the estimate

β t 3/2+ε + p t 1/2+ε + t p 3/2+ε g + t 2 ϕ . (30)
3) The last identity in [START_REF] Durán | On mixed finite element methods for the Reissner-Mindlin plate model[END_REF] means that w t ∈ H 1 0 (Ω) can be seen as the unique solution of (∇w t , ∇s) = (β t + t 2 ∇r t , ∇s), ∀s ∈ H 1 0 (Ω),

Hence for all ε ∈ (0, ε f ], w t belongs to H 3/2+ε (Ω) with the estimate

w t 3/2+ε β t + t 2 ∇r t -1/2+ε .
Combined with ( 16) and [START_REF] Liberman | A posteriori error estimator for a mixed finite element method for Reissner-Mindlin plate[END_REF] we have obtained

w t 3/2+ε g + t 2 ϕ . (31)
Finally recalling that τ t = ζt -2 (∇w t -β t ) = ζ(∇r + curl p t ), the estimate ( 14) is a simple consequence of ( 16), ( 30) and (31).

Robust a priori error estimates for the eigenvalue problem

In order to perform the error analysis between the exact eigenvalues of (2) and their approximation (eigenvalues of ( 4)), it is convenient to introduce the operator

T t : L 2 (Ω) 2 × L 2 (Ω) → L 2 (Ω) 2 × L 2 (Ω) : (ϕ, g) → T t (ϕ, g) = (β t , w t ), where (β t , w t ) ∈ H 1 0 (Ω) 2 × H 1 0 (Ω)
is the unique solution of (13) with datum (ϕ, g). As the bilinear form a introduced before is symmetric, T t is a selfadjoint and compact operator from L 2 (Ω) 2 × L 2 (Ω) into itself equipped with the natural inner product and norm

|(ϕ, g)| 2 t = t 2 12 ϕ 2 + g 2 .
Furthermore α t is an eigenvalue of (2) if and only if 1 αt is an eigenvalue of T t . As t → 0 (cfr. [START_REF] Brezzi | Mixed and hybrid finite element methods[END_REF]), the solution (β t , w t ) of ( 13) converges to (β 0 , w 0 ) ∈ H 1 0 (Ω) 2 ×H 1 0 (Ω), where (β 0 , p 0 ) is the unique solution of [START_REF] Grisvard | Singularities in boundary value problems[END_REF],

w 0 ∈ H 2 0 (Ω) is the unique solution of 1 12(1 + ν) ∆ 2 w 0 = f in Ω. Setting τ 0 = ζ(∇r + curl p 0 ) (that belongs to H 0 (curl, Ω) ′ ), it holds a(β 0 , ψ) + τ 0 , ∇v -ψ = (g, v), ∀(η, v) ∈ H 1 0 (Ω) 2 × H 1 0 (Ω), β 0 = ∇w 0 . (32)
Let us notice that the regularity results from Theorem 2.1 only yield τ 0 ∈ H -1/2+ε (Ω) for some ε ∈ (0, 1/2].

As before we define the operator T 0 by

T 0 : L 2 (Ω) 2 × L 2 (Ω) → L 2 (Ω) 2 × L 2 (Ω) : (ϕ, g) → T 0 (ϕ, g) = (β 0 , w 0 ).
The first aim is to prove that T t tends to T 0 as t goes to zero even in the non-convex case (see Lemma 3.1 of [START_REF] Durán | Approximation of the vibration modes of a plate by Reissner-Mindlin equations[END_REF] in the convex case):

Lemma 2.2 For all (ϕ, g) ∈ L 2 (Ω) 2 × L 2 (Ω), it holds (T t -T 0 )(ϕ, g) H 1 0 (Ω) 2 ×H 1 0 (Ω) √ t|(ϕ, g)| t .
Proof: Subtracting (32) to [START_REF] Dauge | Elliptic boundary value problems on corner domains -smoothness and asymptotics of solutions[END_REF] we have

a(β t -β 0 , ψ) + τ t -τ 0 , ∇v -ψ = t 2 12 (ϕ, η), ∀(η, v) ∈ H 1 0 (Ω) 2 × H 1 0 (Ω).
Hence taking η = β t -β 0 and v = w t -w 0 , we find

a(β t -β 0 , β t -β 0 ) = t 2 12 (ϕ, β t -β 0 ) - t 2 ζ τ t -τ 0 , τ t .
Using the (uniform) coerciveness of a, Cauchy-Schwarz's inequality and the a priori estimate ( 14), we get

β t -β 0 2 1 ≤ t 2 12 ϕ β t -β 0 + t 2 ζ τ t -τ 0 -1/2+ε τ t 1/2-ε t|(ϕ, g)| t β t -β 0 1 + t|(ϕ, g)| 2
t . Hence Young's inequality leads to

β t -β 0 1 √ t|(ϕ, g)| t . (33)
Observing that

∇(w t -w 0 ) = β t -β 0 + t 2 ζ τ t ,
we get

∇(w t -w 0 ) ≤ β t -β 0 + t 2 ζ τ t β t -β 0 + t 2 τ t 1/2+ε .
The conclusion then follows from the previous estimate ( 33) and ( 14).

Once such a convergence result is obtained by standard perturbation arguments (see for instance [START_REF] Kato | Perturbation theory for linear operators[END_REF] and [START_REF] Durán | Approximation of the vibration modes of a plate by Reissner-Mindlin equations[END_REF] for its application to the Reissner-Mindlin system), we obtain the next result. Lemma 2.3 Let µ 0 > 0 be a fixed eigenvalue of T 0 of algebraic multiplicity m and let D be a open disc of the complex plane centred at µ 0 that contains no other element of the spectrum of T 0 . Then there exists t 0 > 0 (depending on µ 0 ) such that for all t ∈ (0, t 0 ], T t contains exactly m eigenvalues in D (repeated according to their algebraic multiplicities). In particular µ 0 is the limit of eigenvalues of T t . Furthermore if µ 0 is a simple eigenvalue of T 0 , then T t has a simple eigenvalue µ t in D for all t ≤ t 0 and the distance of µ t to the remainder of the spectrum of T t remains uniformly bounded from below.

We are now ready to prove some convergence results between exact eigenvalues and eigenvectors and discrete ones: Theorem 2.4 Let µ 0 > 0 be a simple eigenvalue of T 0 and fix t 0 small enough such that T t fulfils the properties of Lemma 2.3, in particular denote by µ t its eigenvalue that converges to µ 0 . Let α t = 1 µt that is a simple eigenvalue of problem [START_REF] Arnold | A uniformly accurate finite element method for the Reissner-Mindlin plate[END_REF] and let (ω, φ) ∈ H 1 0 (Ω) 2 ×H 1 0 (Ω) be its corresponding normalized eigenvector, i.e., |(ω, φ)| t = 1. Then there exist h 0 > 0, and ε ∈ (0, 1 2 ] such that for all h < h 0 , the discrete problem (4) has a unique eigenvalue α t,h that converges to α t as h goes to zero. Furthermore if

(ω h , φ h ) ∈ W h × Θ h is the corresponding normalized eigenvector, i.e., |(ω h , φ h )| t = 1, then one has φ -φ h 1 + ω -ω h 1 h 1/2+ε , (34) φ -φ h + ω -ω h h 1+2ε , ( 35
) |α t -α t,h | h 1+2ε . ( 36
) Proof: Given (ϕ, g) ∈ L 2 (Ω) 2 × L 2 (Ω), we consider (β t,h , w t,h ) ∈ W h × Θ h solution of    a(β t,h , η h ) + (τ t,h , ∇v h -R h η h ) = (g, v h ) + t 2 12 (ϕ, η h ) , ∀ (v h , η h ) ∈ W h × Θ h τ t,h = ζt -2 (∇w t,h -R h β t,h ).
and the mapping

T t,h : L 2 (Ω) 2 × L 2 (Ω) → L 2 (Ω) 2 × L 2 (Ω) : (ϕ, g) → T t,h (ϕ, g) = (β t,h , w t,h ).
As in Lemma 3.2 of [START_REF] Durán | Approximation of the vibration modes of a plate by Reissner-Mindlin equations[END_REF], we prove that for all (ϕ, g)

∈ L 2 (Ω) 2 × L 2 (Ω), it holds (T t -T t,h )(ϕ, g) H 1 0 (Ω) 2 ×H 1 0 (Ω) h 1/2+ε |(ϕ, g)| t . (37)
Indeed the only difference is to use the estimate

β t -β t,h 1 + t τ t -τ t,h h 1/2+ε ( β t 3/2+ε + t τ t 1/2+ε + τ t -1/2+ε ). ( 38
)
If this estimate holds then by ( 14) we will get

β t -β t,h 1 + t τ t -τ t,h h 1/2+ε ( g + t 2 ϕ ),
and consequently

w t -w t,h 1 h 1/2+ε ( g + t 2 ϕ ), (39)
since

∇(w t -w t,h ) = β t -R h β t,h + t 2 ζ (τ t -τ t,h ) = β t -R h β t + R h (β t -β t,h ) + t 2 ζ (τ t -τ t,h ).
Hence using standard propertries of R h we get

∇(w t -w t,h ) ≤ β t -R h β t + R h (β t -β t,h ) + t 2 ζ τ t -τ t,h h β t 1 + β t -β t,h 1 + t τ t -τ t,h ,
which yields (39) thanks to (38) and ( 14).

To prove (38) we adapt Lemma 3.1 of [START_REF] Durán | On mixed finite element methods for the Reissner-Mindlin plate model[END_REF] to our setting by proving that for any

( β, ŵ) ∈ W h × Θ h , setting τ = ζt -2 (∇ ŵ -R h β), we have β -β t,h 1 + t τ -τ t,h 1 β -β t 1 + t τ -τ t + h 1/2+ε γ t -1/2+ε (40)
Indeed as in Lemma 3.1 of [START_REF] Durán | On mixed finite element methods for the Reissner-Mindlin plate model[END_REF], we may write

a( β -β t,h , β -β t,h ) + t 2 ζ (τ -τ t,h , τ -τ t,h ) = a( β -β t , β -β t,h ) + t 2 ζ (τ -τ t , τ -τ t,h ) + (γ t , β -β t,h -R h ( β -β t,h ).
Hence by the uniform coerciveness of a, Young's inequality and Cauchy-Schwarz's inequality we get

β -β t,h 2
1 + t 2 τ -τ t,h 2 β -β t 2 1 + t 2 τ -τ t 2 + γ t -1/2+ε β -β t,h -R h ( β -β t,h ) 1/2-ε .
Hence using the estimate

η -R h η 1/2-ε h 1/2+ε η 1 , (41)
again by Young's inequality we arrive at (40). This estimate (40), ( 14) and the arguments of Corollary 3.2 of [START_REF] Durán | On mixed finite element methods for the Reissner-Mindlin plate model[END_REF] lead to (38).

The estimate (37) and Theorem 7.1 of [START_REF] Babuška | Eigenvalue problems[END_REF] lead to [START_REF] Nicaise | Regularity of the solutions of elliptic systems in polyhedral domains[END_REF] due to Lemma 2.3. Since β t belongs to H 1 0 (Ω) 2 , we can use the same duality argument than the one from Lemma 3.4 of [START_REF] Durán | Approximation of the vibration modes of a plate by Reissner-Mindlin equations[END_REF] thanks to Lemma 3.3 of [START_REF] Durán | Approximation of the vibration modes of a plate by Reissner-Mindlin equations[END_REF] and get

β t -β t,h + w t -w t,h h 1+2ε ( g + t 2 ϕ ).
In other words, for all (ϕ, g)

∈ L 2 (Ω) 2 × L 2 (Ω), it holds (T t -T t,h )(ϕ, g) L 2 (Ω) 2 ×L 2 (Ω) h 1+2ε |(ϕ, g)| t . (42)
and ( 35) follows as before.

In order to prove the relation [START_REF] Verfürth | A review of a posteriori error estimation and adaptive mesh-refinement techniques[END_REF], we use the same argument than in Theorem 2.2 of [START_REF] Durán | Approximation of the vibration modes of a plate by Reissner-Mindlin equations[END_REF], namely applying Remark 7.5 of [START_REF] Babuška | Eigenvalue problems[END_REF], we have

|µ t -µ t,h | ≤ C(|(T t -T t,h )(β t , w t )| t + |(T t -T t,h )(β t , w t )| 2 t ),
where C is a positive constant depending on the inverse of the distance from µ t to the remainder of the spectrum of T t . Hence by Lemma 2.3 and (42) we obtain

|µ t -µ t,h | h 1+2ε . As α t = 1 µ t and α t,h = 1 µ t,h
, we arrive at (36).

Remark 2.5

If Ω is convex, then we can take ε = 1/2, and we recover standard results presented in most existing works (e.g. [START_REF] Durán | Error estimates for low-order isoparametric quadrilateral finite elements for plates[END_REF], [START_REF] Durán | Approximation of the vibration modes of a plate by Reissner-Mindlin equations[END_REF], [START_REF] Durán | Finite element analysis of the vibration problem of a plate coupled with a fluid[END_REF], [START_REF] Gómez | On the vibrations of a plate with a concentrated mass and very small thickness[END_REF], [START_REF] Hernández | Approximation of the vibration modes of a plate coupled with a fluid by low-order isoparametric finite elements[END_REF], [START_REF] Lovadina | Approximation of the buckling problem for Reissner-Mindlin plates[END_REF]).

Preliminary results

The aim of this section is to prove three lemmas which will be used in the following of the paper. The proofs of Lemmas 3.2 and 3.3 are close (but non identical) to the ones of [START_REF] Carstensen | A posteriori error analysis for conforming MITC elements for Reissner-Mindlin plates[END_REF] and [START_REF] Creusé | Robust equilibrated a posteriori error estimators for the Reissner-Mindlin system[END_REF]. Nevertheless, we give them for the sake of completeness.

Lemma 3.1 We have γ -γ h 2 -1 ≤ 6 (µ + λ) φ -φ h 2 C + 3 Res 2 2 -1 + 3 c 2 F t 2 12 α t φ -α t,h φ h 2 . ( 43
)
Proof: First, it can be shown that for any ψ ∈ (H 1 0 (Ω)) 2 (cf [START_REF] Creusé | Robust equilibrated a posteriori error estimators for the Reissner-Mindlin system[END_REF]),

ψ 2 C ≤ 2 (µ + λ)|ψ| 2 1 ,
hence by ( 2), ( 4) and the definition of Res 2 , we get

(γ -γ h , ψ) = a(φ, ψ) -α t t 2 12 (φ, ψ) -(γ h , ψ) = a(φ -φ h , ψ) -Res 2 (ψ) - t 2 12 (α t φ -α t,h φ h , ψ) ≤ φ -φ h C ψ C + Res 2 -1 |ψ| 1 + t 2 12 α t φ -α t,h φ h ψ ≤ (2 (µ + λ)) 1/2 φ -φ h C + Res 2 -1 + c F t 2 12 α t φ -α t,h φ h |ψ| 1 .
By the definition of the norm in H -1 (Ω), we conclude that

γ -γ h 2 -1 ≤ (2 (µ + λ)) 1/2 φ -φ h C + Res 2 -1 + c F t 2 12 α t φ -α t,h φ h 2 ≤ 6 (µ + λ) φ -φ h 2 C + 3 Res 2 2 -1 + 3 c 2 F t 2 12 α t φ -α t,h φ h 2 . Lemma 3.2 φ -φ h 2 C + ζ -1 t 2 γ -γ h 2 = Res 1 (ω -ω h + z) + Res 2 (φ -φ h + β) -a(φ -φ h , β) +(α t ω -α t,h ω h , ω -ω h + z) + t 2 12 (α t φ -α t,h φ h , φ -φ h + β),
where z and β are the functions appearing in the Helmholtz decomposition [START_REF] Beirão Da Veiga | A-priori and a-posteriori error analysis for a family of Reissner-Mindlin plate elements[END_REF].

Proof: First, ( 2) and ( 6) lead to

(γ -γ h , (R h -I)φ h ) = (γ -γ h , ∇z -β) = (γ, ∇z -β) -(γ h , ∇z -β) = α t (ω, z) + α t t 2 12 (φ, β) -a(φ, β) -(γ h , ∇z -β) = α t (ω, z) + α t t 2 12 (φ, β) -a(φ -φ h , β) -a(φ h , β) -(γ h , ∇z -β).
As γ = ζt -2 (∇ω -φ) and γ h = ζt -2 (∇ω h -R h φ h ), we may write

φ-φ h 2 C +ζ -1 t 2 γ-γ h 2 = a(φ-φ h , φ-φ h )+(γ-γ h , (∇ω-∇ω h )-(φ-φ h ))+(γ-γ h , (R h -I)φ h ).
By the previous identity and the definition of Res 1 and Res 2 , we obtain

φ -φ h 2 C + ζ -1 t 2 γ -γ h 2 = α t (ω, ω -ω h ) + α t t 2 12 (φ, φ -φ h ) -a(φ h , φ -φ h ) -(γ h , ∇(ω -ω h )) +(γ h , φ -φ h ) + α t (ω, z) + α t t 2 12 (φ, β) -a(φ -φ h , β) -a(φ h , β) -(γ h , ∇z -β) = Res 2 (φ -φ h + β) - t 2 12 (α t,h φ h , φ -φ h + β) -(γ h , ∇(ω -ω h + z)) -a(φ -φ h , β) +α t (ω, ω -ω h + z) + t 2 12 (α t φ, φ -φ h + β) = Res 2 (φ -φ h + β) + Res 1 (ω -ω h + z) -(α t,h ω h , ω -ω h + z) -a(φ -φ h , β) +α t (ω, ω -ω h + z) + t 2 12 (α t φ -α t,h φ h , φ -φ h + β) + α t (ω, ω -ω h + z) = Res 2 (φ -φ h + β) + Res 1 (ω -ω h + z) -a(φ -φ h , β) +(α t ω -α t,h ω h , ω -ω h + z) + t 2 12 (α t φ -α t,h φ h , φ -φ h + β).
This proves the requested identity.

Lemma 3.3 1 2 φ -φ h + β 2 C + 1 2 φ -φ h 2 C + 1 2 ζ -1 t 2 γ -γ h 2 + 1 2 T ∈T h ζ t 2 + h 2 T ∇(ω -ω h + z) -(φ -φ h + β) 2 T ≤ (α t ω -α t,h ω h , ω -ω h + z) + t 2 12 (α t φ -φ t,h φ h , φ -φ h + β) + Res 1 (ω -ω h + z) + Res 2 (φ -φ h + β) + 1 2 β 2 C .
Proof: Because of (6), we first remark that

γ -γ h = ζt -2 (∇ω -∇ω h -φ + φ h + ∇z -β),
so that we have for all T ∈ T h

∇(ω -ω h + z) -(φ -φ h + β) 2 T ≤ ζ -2 t 4 γ -γ h 2 T .
This estimate implies that

1 2 φ -φ h + β 2 C + 1 2 φ -φ h 2 C + 1 2 ζ -1 t 2 γ -γ h 2 + 1 2 T ∈T h ζ t 2 + h 2 T ∇(ω -ω h + z) -(φ -φ h + β) 2 T ≤ 1 2 φ -φ h + β 2 C + 1 2 φ -φ h 2 C + 1 2 ζ -1 t 2 γ -γ h 2 + 1 2 ζ -1 t 2 T ∈T h γ -γ h 2 T ≤ ζ -1 t 2 γ -γ h 2 + 1 2 a(φ -φ h + β, φ -φ h + β) + 1 2 φ -φ h 2 C = ζ -1 t 2 γ -γ h 2 + 1 2 φ -φ h 2 C + 2a(φ -φ h , β) + β 2 C + 1 2 φ -φ h 2 C = φ -φ h 2 C + ζ -1 t 2 γ -γ h 2 + 1 2 β 2 C + a(φ -φ h , β).
The conclusion follows from Lemma 3.2.

4 Reliability of the estimator Theorem 4.1 Let us consider 0 < ε < 1/2, as well as two parameters ν 1 > 0 and ν 2 > 0.

Moreover, let us define

B(ε) = max 3 µ + c 2 F 1 ε + ε -1 µ(1 -2ε) + 6(µ + λ); 1 + t 2 ζ(1 -2ε) .
Then,

(e ev h ) 2 ≤ A 1 | Res 1 | 2 -1,h + A 2 Res 2 2 -1 + A 3 φ -φ h + β 2 C + A 4 φ h -R h φ h 2 H(rot,Ω) - T ∈T h A T 5 ∇(ω -ω h + z) -(φ -φ h + β) 2 T + A 6 t 2 12 α t φ -α t,h φ h 2 +A 7 (α t ω -α t,h ω h , ω -ω h + z) + t 2 12 (α t φ -α t,h φ h , φ -φ h + β) , (44)
with

A 1 = ν 1 B(ε) 2 ; A 2 = ν 2 B(ε) 2 + 3; A 3 = 1 µ 1 ν 1 + 1 ν 2 -B(ε); A 4 = max 2 ε -1 1 -2ε ; 2 + 2B(ε)(µ + λ)c 2 R ; A T 5 = ζB(ε) t 2 + h 2 T - 1 ν 1 (t 2 + h 2 T ) , ∀ T ∈ T h ; A 6 = 3 c 2 F ; A 7 = 2 B(ε).
Proof: The proof is similar to the one of [12, Theorem 1] so we only give a sketch of it. Using [12, Lemma 2], we have

(e ev h ) 2 ≤ t 2 ζ(1 -2ε) ζ -1 t 2 γ -γ h 2 - 1 -2/ε 1 -2ε φ h -R h φ h 2 - 1 -1/ε -ε 1 -2ε φ -φ h 2 +|φ -φ h | 2 1 + ζ -1 t 2 γ -γ h 2 + ζ -2 t 4 rot(γ -γ h ) 2 + γ -γ h 2 -1 ≤ 1 -c 2 F 1 -1/ε -ε 1 -2ε |φ -φ h | 2 1 + 1 + t 2 ζ(1 -2ε) ζ -1 t 2 γ -γ h 2 - 1 -2/ε 1 -2ε φ h -R h φ h 2 + ζ -2 t 4 rot(γ -γ h ) 2 + γ -γ h 2 -1 ≤ 1 µ + c 2 F 1 ε + ε -1 µ(1 -2ε) φ -φ h 2 C + 1 + t 2 ζ(1 -2ε) ζ -1 t 2 γ -γ h 2 + 2 ε -1 1 -2ε φ h -R h φ h 2 + ζ -2 t 4 rot(γ -γ h ) 2 + γ -γ h 2 -1 .
Then, because of Lemma 3.1 as well as

ζ -2 t 4 rot(γ -γ h ) 2 ≤ 2 µ φ -φ h 2 C + 2 rot(φ h -R h φ h ) 2 ,
we obtain

(e ev h ) 2 ≤ 3 µ + c 2 F 1 ε + ε -1 µ(1 -2ε) + 6(µ + λ) φ -φ h 2 C + 2 rot(φ h -R h φ h ) 2 + 1 + t 2 ζ(1 -2ε) ζ -1 t 2 γ -γ h 2 + 2 ε -1 1 -2ε φ h -R h φ h 2 +3 Res 2 2 -1 + 3 c 2 F t 2 12 α t φ -α t,h φ h 2 .
By the definition of B(ε) as well as Lemma 3.3, we get

(e ev h ) 2 ≤ B(ε) 2(α t ω -α t,h ω h , ω -ω h + z) + 2 t 2 12 (α t φ -φ t,h φ h , φ -φ h + β) +2 Res 1 (ω -ω h + z) + 2 Res 2 (φ -φ h + β) + β 2 C -φ -φ h + β 2 C - T ∈T h ζ t 2 + h 2 T ∇(ω -ω h + z) -(φ -φ h + β) 2 T + 3 Res 2 2 -1 + 2 ε -1 1 -2ε φ h -R h φ h 2 + 2 rot(φ h -R h φ h ) 2 +3 c 2 F t 2 12 α t φ -α t,h φ h 2 .
We notice that

Res 1 (ω -ω h + z) ≤ | Res 1 | -1,h | (ψ, ω -ω h + z)| 1,h ∀ ψ ∈ H 1 0 (Ω) 2 , Res 2 (φ -φ h + β) ≤ Res 2 -1 |φ -φ h + β| 1 .
where, here and below, with a small abuse of notation, we use the extension (by zero) of the linear operator Res 1 to the whole of H 1 0 (Ω) × H 1 0 (Ω) 2 :

Res 1 : H 1 0 (Ω) × H 1 0 (Ω) 2 -→ R : (v, ψ) -→ Res 1 (v).
Introducing now the parameters ν 1 > 0 and ν 2 > 0 and using two times Young's inequality lead to

(e ev h ) 2 ≤ ν 1 B(ε) 2 | Res 1 | 2 -1,h + 1 ν 1 | (ψ, ω -ω h + z)| 2 1,h + ν 2 B(ε) 2 Res 2 2 -1 + 1 ν 2 |φ -φ h + β| 2 1 -B(ε) φ -φ h + β 2 C + B(ε) β 2 C + 2 ε -1 1 -2ε φ h -R h φ h 2 + 3 Res 2 2 -1 + 2 rot(φ h -R h φ h ) 2 - T ∈T h ζB(ε) t 2 + h 2 T ∇(ω -ω h + z) -(φ -φ h + β) 2 T +2B(ε) (α t ω -α t,h ω h , ω -ω h + z) + t 2 12 (α t φ -α t,h φ h , φ -φ h + β) +3c 2 F t 2 12 α t φ -α t,h φ h 2 .
Finally, choosing ψ = φ -φ h + β, we get

| (ψ, ω -ω h + z)| 2 1,h = ∇(φ -φ h + β) 2 + T ∈T h 1 t 2 + h 2 T ∇(ω -ω h + z) -(φ -φ h + β) 2 T ,
and the previous inequality yields (44).

Corollary 4.2 It holds

(e ev h ) 2 Res 1 | 2 -1,h + Res 2 2 -1 + φ h -R h φ h 2 H(rot,Ω) (α t ω -α t,h ω h , ω -ω h + z) + t 2 12 (α t φ -α t,h φ h , φ -φ h + β) + t 4 α t φ -α t,h φ h 2 .
Proof: Assuming 1 -2ε > 0, the parameters ν 1 and ν 2 arising in the values of A 3 and A T 5 in (44) are first chosen such that A 3 ≤ 0 and A T 5 ≥ 0 for all T ∈ T h . Namely we take

ν 1 = ν 2 = 2 κ/B(ε) with κ = max 1 µ , 1 2 ζ ,
and we obtain

(e ev h ) 2 ≤ Ã1 | Res 1 | 2 -1,h + Ã2 Res 2 2 -1 + Ã4 φ h -R h φ h 2 H(rot,Ω) + Ã6 (α t ω -α t,h ω h , ω -ω h + z) + t 2 12 (α t φ -α t,h φ h , φ -φ h + β) + Ã7 t 2 12 α t φ -α t,h φ h 2 , (45)
with Ã1 = 2κB(ε);

Ã2 = 2κB(ε) + 3; Ã4 = max 2 ε -1 1 -2ε ; 2 + 2B(ε)(µ + λ)c 2 R ; Ã6 = 2B(ε); Ã7 = 3 c 2
F . We conclude by taking any ε ∈ (0, 1 2 ).

Lemma 4.3

(α t ω -α t,h ω h , ω -ω h + z) + t 2 12 (α t φ -α t,h φ h , φ -φ h + β) ≤ c 2 F C 1 + c 2 F 1/2 α t ω -α t,h ω h + t 2 12 α t φ -α t,h φ h φ h -R h φ h H(rot,Ω) + α t + α t,h 2 ω -ω h 2 + t 2 12 φ -φ h 2 .

20

Proof: Clearly we have

(α t ω -α t,h ω h , ω -ω h + z) + t 2 12 (α t φ -α t,h φ h , φ -φ h + β) = (α t ω -α t,h ω h , ω -ω h ) + t 2 12 (α t φ -α t,h φ h , φ -φ h ) +(α t ω -α t,h ω h , z) + t 2 12 (α t φ -α t,h φ h , β). ( 46
)
We can notice that for all v ∈ H 1 0 (Ω) or v ∈ H 1 0 (Ω) 2 , we have

v ≤ c 2 F 1 + c 2 F 1 2 v 1 .
Using Cauchy-Schwarz's inequality and ( 7), we have :

(α t ω -α t,h ω h , z) + t 2 12 (α t φ -α t,h φ h , β) ≤ α t ω -α t,h ω h z + t 2 12 α t φ -α t,h φ h β ≤ α t ω -α t,h ω h c 2 F 1 + c 2 F 1/2 z 1 + t 2 12 α t φ -α t,h φ h c 2 F 1 + c 2 F 1/2 β 1 ≤ c 2 F C 1 + c 2 F 1/2 α t ω -α t,h ω h 2 + t 2 12 α t φ -α t,h φ h 2 1/2 φ h -R h φ h H(rot,Ω) .
(47) For the other term in the right-hand side of (46), we have by the normalization of the eigenvectors :

(α t ω -α t,h ω h , ω -ω h ) + t 2 12 (α t φ -α t,h φ h , φ -φ h ) = α t ω 2 -(α t + α t,h )(ω, ω h ) + α t,h ω h 2 + t 2 12 α t φ 2 -(α t + α t,h )(φ, φ h ) + α t,h φ h 2 = (α t + α t,h ) 1 -(ω, ω h ) - t 2 12 (φ, φ h ) .
But we also have :

ω -ω h 2 + t 2 12 φ -φ h 2 = ω 2 -2(ω, ω h ) + ω h 2 + t 2 12 φ 2 -2(φ, φ h ) + φ h 2 = 2 -2(ω, ω h ) -2 t 2 12 (φ, φ h ).
(48)

Hence :

(α t ω -α t,h ω h , ω -ω h ) + t 2 12 (α t φ -α t,h φ h , φ -φ h ) = α t + α t,h 2 ω -ω h 2 + t 2 12 φ -φ h 2 .
(49) Using ( 47) and ( 49) into (46), Lemma 4.3 holds. Now, it remains to bound each of the two residuals.

Lemma 4.4 With the notations ( 9) and ( 12), we have

| Res 1 | 2 -1,h T ∈T h h 2 T (t 2 + h 2 T ) α t,h ω h + divγ h 2 T + E∈E h h E (t 2 + h 2 E ) [γ h ] E • n E 2 E . ( 50
)
where [γ h ] E is the jump of γ h across E defined by :

[γ h ] E = γ h | T + -γ h | T -with E = T + ∩ T -. Proof: Let v ∈ H 1 0 (Ω).
Using standard Green formula into each element of the triangulation, we get : [START_REF] Carstensen | A posteriori error analysis for conforming MITC elements for Reissner-Mindlin plates[END_REF]) be such that, for all T ∈ T h :

Res 1 (v) = (α t,h ω h , v) -(γ h , ∇v) = T ∈T h T (α t,h ω h + divγ h )v - E∈∂T E γ h • n E v = T ∈T h T (α t,h ω h + divγ h )v - E∈E h E [γ h ] E • n E v, Let v I ∈ S 1 0 (T h ) = {v ∈ H 1 0 (Ω) : ∀ T ∈ T h , v| T ∈ P 1 (T )} (cf. estimate (3.6) of
v -v I T h T { ∇v -ψ ω T + h T ∇ψ ω T } ∀ ψ ∈ H 1 0 (Ω) 2 ; v -v I E h 1/2 E { ∇v -ψ ω E + h E ∇ψ ω E } ∀ ψ ∈ H 1 0 (Ω) 2 .
We can notice that Res 1 (v I ) = 0, and consequently :

Res 1 (v) = Res 1 (v -v I ) = T ∈T h T (α t,h ω h + divγ h )(v -v I ) - E∈E h E [γ h ] E • n E (v -v I ) ≤ T ∈T h α t,h ω h + divγ h T v -v I T + E∈E h [γ h ] E • n E E v -v I E T ∈T h h T t 2 + h 2 T α t,h ω h + divγ h T 1 t 2 + h 2 T { ∇v -ψ ω T + h T ∇ψ ω T } 22 + E∈E h h 1/2 E t 2 + h 2 E [γ h ] E • n E E 1 t 2 + h 2 E { ∇v -ψ ω E + h T ∇ψ ω E } T ∈T h h 2 T (t 2 + h 2 T ) α t,h ω h + divγ h 2 T 1/2 | (ψ, v)| 1,h + E∈E h h E (t 2 + h 2 E ) [γ h ] E • n E 2 E 1/2 | (ψ, v)| 1,h ,
so that (50) holds.

Lemma 4.5 With the notation [START_REF] Carstensen | A posteriori error analysis for conforming MITC elements for Reissner-Mindlin plates[END_REF], we have

Res 2 2 -1 T ∈T h h 2 T divCε(φ h ) + γ h + t 2 12 α t,h φ h 2 T + E∈E h h E [Cε(φ h )] E n E 2 E + µ 2 h (γ h ), (51)
where µ h (γ h ) = sup

η h ∈S 1 0 (T h ) 2 \{0} |(γ h , (I -R h )η h )| |η h | 1 .
Proof: Let ψ ∈ Θ. Using standard Green formula into each element of the triangulation, we get :

Res 2 (ψ) = -a(φ h , ψ) + (γ h , ψ) + t 2 12 (α t,h φ h , ψ) = T ∈T h T (divCε(φ h ) + γ h + t 2 12 α t,h φ h ) • ψ - E∈∂T E Cε(φ h )n E • ψ = T ∈T h T (divCε(φ h ) + γ h + t 2 12 α t,h φ h ) • ψ - E∈E h E [Cε(φ h )] E n E • ψ, Let ψ I ∈ S 1 0 (T h ) 2 ⊂ Θ h (cf. [1, Theorem 1.7]) be such that |ψ I | 1 |ψ| 1 and for all T ∈ T h : ψ -ψ I T h T ∇ψ ω T ; ψ -ψ I E h 1/2 E ∇ψ ω E .
We can notice that Res 2 (ψ I ) = (γ h , (I -R h )ψ I ), which implies :

Res 2 (ψ) = Res 2 (ψ -ψ I ) + (γ h , (I -R h )ψ I ) = T ∈T h T (divCε(φ h ) + γ h + t 2 12 α t,h φ h ) • (ψ -ψ I ) - E∈E h E [Cε(φ h )] E n E • (ψ -ψ I ) + (γ h , (I -R h )ψ I ) ≤ T ∈T h divCε(φ h ) + γ h + t 2 12 α t,h φ h T ψ -ψ I T + E∈E h [Cε(φ h )] E n E E ψ -ψ I E + |(γ h , (I -R h )ψ I )| |ψ I | 1 |ψ I | 1 T ∈T h h T divCε(φ h ) + γ h + t 2 12 α t,h φ h T ∇ψ ω T + E∈E h h 1/2 E [Cε(φ h )] E n E E ∇ψ ω E + µ h (γ h )|ψ I | 1 . (52)
Then, using the estimate |ψ I | 1 |ψ| 1 and the discrete Cauchy-Schwarz inequality, we get

Res 2 (ψ) T ∈T h h 2 T divCε(φ h ) + γ h + t 2 12 α t,h φ h 2 T 1/2 |ψ| 1 + E∈E h h E [Cε(φ h )] E n E 2 E 1/2 |ψ| 1 + µ h (γ h )|ψ| 1 .
The definition of the norm of H -1 (Ω) leads to (51).

Theorem 4. [START_REF] Beirão Da Veiga | A-priori and a-posteriori error analysis for a family of Reissner-Mindlin plate elements[END_REF] We have

(e ev h ) 2 T ∈T h h 2 T (t 2 + h 2 T ) α t,h ω h + divγ h 2 T + E∈E h h E (t 2 + h 2 E ) [γ h ] E • n E 2 E + T ∈T h h 2 T divCε(φ h ) + γ h + t 2 12 α t,h φ h 2 T + E∈E h h E [Cε(φ h )] E n E 2 E + φ h -R h φ h 2 H(rot,Ω) + µ h (γ h ) 2 + t 2 12 α t φ -α t,h φ h 2 + α t ω -α t,h ω h 2 + α t + α t,h 2 ω -ω h 2 + t 2 12 φ -φ h 2 . (53)
Proof: The theorem is a direct consequence of Corollary 4.2, Lemma 4.3, Lemma 4.4 and Lemma 4.5.

Remark 4.7 From a practical point of view, the estimate (53) is not useful since the L 2 (Ω) norm of the error is still present in the right-hand-side (see theorem 3.1 of [START_REF] Durán | A posteriori error estimates for the finite element approximation of eigenvalue problems[END_REF] for a similar phenomenom for the Laplace equation). However, the terms containing the exact solution in the right hand side of (53) are neglectible if the eigenvalue is simple (cf. Theorem 2.4). This is the subject of the following corollary.

Corollary 4.8 (Reliability of the estimator) Assume that α t is a simple eigenvalue, then we have :

(e ev h ) 2 T ∈T h h 2 T (t 2 + h 2 T ) α t,h ω h + divγ h 2 T + E∈E(Ω) h E (t 2 + h 2 E ) [γ h ] E • n E 2 E + T ∈T h h 2 T divCε(φ h ) + γ h + t 2 12 α t,h φ h 2 T + E∈E(Ω) h E [Cε(φ h )] E n E 2 E + φ h -R h φ h 2 H(rot,Ω) + µ h (γ h ) 2 + h.o.t.
where h.o.t. corresponds to higher order terms.

Proof: Using Cauchy-Schwarz's inequality, we get :

t 2 12 α t φ -α t,h φ h 2 + α t ω -α t,h ω h 2 = (α t ω -α t,h ω h , α t (ω -ω h ) + (α t -α t,h ) ω h ) + t 2 12 2 (α t φ -α t,h φ h , α t (φ -φ h ) + (α t -α t,h )φ h ) ≤ α t (α t ω -α t,h ω h , ω -ω h ) + α t t 2 12 2 (α t φ -α t,h φ h , φ -φ h ) +|α t -α t,h | α t ω -α t,h ω h ω h + t 2 12 2 α t φ -α t,h φ h φ h .
But, by (49), we have :

(α t ω -α t,h ω h , ω -ω h ) + t 2 12 2 (α t φ -α t,h φ h , φ -φ h ) ≤ max 1; t 2 12 (α t ω -α t,h ω h , ω -ω h ) + t 2 12 (α t φ -α t,h φ h , φ -φ h ) ≤ max 1; t 2 12
α t + α t,h 2 ω -ω h 2 + t 2 12 φ -φ h 2 .
By a discrete Cauchy-Schwarz inequality and the normalization of the discrete solutions given by (4), we get :

α t ω -α t,h ω h ω h + t 2 12 2 α t φ -α t,h φ h φ h ≤ α t ω -α t,h ω h 2 + t 2 12 2 α t φ -α t,h φ h 2 1/2 ω h 2 + t 2 12 2 φ h 2 1/2 ≤ α t ω -α t,h ω h 2 + t 2 12 2 α t φ -α t,h φ h 2 1/2 max 1; t 2 12 1/2 . 25
Therefore, using Young's inequality with a parameter δ > 0 :

t 2 12 α t φ -α t,h φ h 2 + α t ω -α t,h ω h 2 ≤ α t max 1; t 2 12
α t + α t,h 2 ω -ω h 2 + t 2 12 φ -φ h 2 +|α t -α t,h | max 1; t 2 12 1/2 α t ω -α t,h ω h 2 + t 2 12 2 α t φ -α t,h φ h 2 1/2 ≤ α t max 1; t 2 12
α t + α t,h 2 ω -ω h 2 + t 2 12 φ -φ h 2 + δ 2 |α t -α t,h | 2 max 1; t 2 12 + 1 2δ α t ω -α t,h ω h 2 + t 2 12 2 α t φ -α t,h φ h 2 .
Choosing δ = 1, we get :

t 2 12 α t φ -α t,h φ h 2 + α t ω -α t,h ω h 2 ≤ α t (α t + α t,h ) max 1; t 2 12 ω -ω h 2 + t 2 12 φ -φ h 2 + |α t -α t,h | 2 max 1; t 2 12 .
As ω -ω h

A posteriori estimate for the eigenvalue error

Let η ev be the estimator presented in Corollary 4.8 :

η 2 ev = T ∈T h h 2 T (t 2 + h 2 T ) α t,h ω h + divγ h 2 T + E∈E h h E (t 2 + h 2 E ) [γ h ] E • n E 2 E + T ∈T h h 2 T divCε(φ h ) + γ h + t 2 12 α t,h φ h 2 T + E∈E h h E [Cε(φ h )] E n E 2 E + φ h -R h φ h 2 H(rot,Ω) + µ h (γ h ) 2 .
Theorem 5.1 We have :

|α t -α t,h | η 2 ev + ζt -2 φ h -R h φ h 2 + (γ h , φ h -R h φ h ) + T 2 ex , ( 54
)
where

T 2 ex = t 2 12 α t φ -α t,h φ h 2 + α t ω-α t,h ω h 2 + α t + α t,h 2 ω -ω h 2 + t 2 12 φ -φ h 2
is the term containing the exact solution.

Proof: We recall that (ω, φ, α t ) (resp. (ω h , φ h , α t,h )) is the solution of problem (2) (resp. ( 4)). Then we have :

φ -φ h 2 C + ζ -1 t 2 γ -γ h 2 = φ 2 C + φ h 2 C -2a(φ, φ h ) + ζ -1 t 2 γ 2 + γ h 2 -2(γ, γ h ) = φ 2 C + ζ -1 t 2 γ 2 + φ h 2 C + ζ -1 t 2 γ h 2 -2 a(φ, φ h ) + ζ -1 t 2 (γ, γ h ) = α t + α t,h -2 a(φ, φ h ) + ζ -1 t 2 (γ, γ h ) , (55)
as well as

a(φ, φ h ) + ζ -1 t 2 (γ, γ h ) = a(φ, φ h ) + (γ, ∇ω h -R h φ h ) = a(φ, φ h ) + (γ, ∇ω h -φ h ) + (γ, φ h -R h φ h ) = α t (ω, ω h ) + t 2 12 (φ, φ h ) + (γ, φ h -R h φ h ). (56)
Then, from the relations (55) and (56), we have :

φ -φ h 2 C + ζ -1 t 2 γ -γ h 2 α t + α t,h -2 α t (ω, ω h ) + t 2 12 (φ, φ h ) + (γ, φ h -R h φ h ) ,
so that from (48) :

φ -φ h 2 C + ζ -1 t 2 γ -γ h 2 = α t + α t,h + α t ω -ω h 2 + t 2 12 φ -φ h 2 -2 -2(γ, φ h -R h φ h ) = α t,h -α t + α t ω -ω h 2 + t 2 12 φ -φ h 2 -2(γ, φ h -R h φ h ),
In other words, noticing that α t > 0, we have :

α t,h -α t = φ -φ h 2 C + ζ -1 t 2 γ -γ h 2 -α t ω -ω h 2 + t 2 12 φ -φ h 2 +2(γ, φ h -R h φ h ) = φ -φ h 2 C + ζ -1 t 2 γ -γ h 2 -α t ω -ω h 2 + t 2 12 φ -φ h 2 +2(γ -γ h , φ h -R h φ h ) + 2(γ h , φ h -R h φ h ) ≤ φ -φ h 2 C + ζ -1 t 2 γ -γ h 2 -α t ω -ω h 2 + t 2 12 φ -φ h 2 +2 γ -γ h φ h -R h φ h + 2(γ h , φ h -R h φ h ) ≤ φ -φ h 2 C + ζ -1 t 2 γ -γ h 2 -α t ω -ω h 2 + t 2 12 φ -φ h 2 +ζ -1 t 2 γ -γ h 2 + ζt -2 φ h -R h φ h 2 + 2(γ h , φ h -R h φ h ) ≤ 2[ φ -φ h 2 C + ζ -1 t 2 γ -γ h 2 ] -α t ω -ω h 2 + t 2 12 φ -φ h 2 +ζt -2 φ h -R h φ h 2 + 2(γ h , φ h -R h φ h ) ≤ 2(e ev h ) 2 + ζt -2 φ h -R h φ h 2 + 2(γ h , φ h -R h φ h ).
Using Theorem 4.6, we obtain :

α t,h -α t ≤ C 1 (η 2 ev + T 2 ex ) + ζt -2 φ h -R h φ h 2 + 2(γ h , φ h -R h φ h ), (57)
for some C 1 > 0 (independent of t and h).

In order to obtain an evaluation of the error eigenvalues, we must now evaluate α t -α t,h to finally control the quantity |α t -α t,h |. All we have to do is to repeat the previous arguments replacing (56) by the identity

a(φ, φ h ) + ζ -1 t 2 (γ, γ h ) = -Res 1 (ω) -Res 2 (φ) + α t,h (ω h , ω) + t 2 12 (φ h , φ) ,
that directly follows from the definition of Res 1 and Res 2 . Furthermore by ( 2) and ( 4), we see that

α t = a(φ, φ) + ζ -1 t 2 γ 2 , α t,h = a(φ h , φ h ) + ζ -1 t 2 γ h 2 .
These two identities and the normalization in ( 2) and ( 4) lead to

α t -α t,h = φ -φ h 2 C + ζ -1 t 2 γ -γ h 2 -α t,h ω -ω h 2 + t 2 12 φ -φ h 2 -2 Res 1 (ω) -2 Res 2 (φ).
Then, using the fact that Res 1 (ω h )+ Res 2 (φ h ) = (γ h , φ h -R h φ h ) and inserting the functions β and z from the Helmholtz decomposition (6) we get

α t -α t,h = φ -φ h 2 C + ζ -1 t 2 γ -γ h 2 -α t,h ω -ω h 2 + t 2 12 φ -φ h 2 28 -2 Res 1 (ω -ω h + z) + (γ h , ∇z) -(α t,h ω h , z) -2 Res 2 (φ -φ h + β) + (γ h , φ h -R h φ h ) + a(φ h , β) -(γ h , β) - t 2 12 (α t,h φ h , β) = φ -φ h 2 C + ζ -1 t 2 γ -γ h 2 -α t,h ω -ω h 2 + t 2 12 φ -φ h 2 -2 Res 1 (ω -ω h + z) + Res 2 (φ -φ h + β) + (γ h , φ h -R h φ h ) +a(φ h -φ, β) + (γ -γ h , φ h -R h φ h ) + (α t ω -α t,h ω h , z) + t 2 12 (α t φ -α t,h φ h , β) ,
the last identity following from [START_REF] Arnold | A uniformly accurate finite element method for the Reissner-Mindlin plate[END_REF]. By Cauchy-Schwarz's inequality, we get

α t -α t,h ≤ φ -φ h 2 C + ζ -1 t 2 γ -γ h 2 -α t,h ω -ω h 2 + t 2 12 φ -φ h 2 +2 Res 1 (ω -ω h + z) + Res 2 (φ -φ h + β) + |(γ h , φ h -R h φ h )| + φ h -φ C β C + ζ -1/2 t γ -γ h ζ 1/2 t -1 φ h -R h φ h + α t ω -α t,h ω h 2 + t 2 12 2 α t φ -α t,h φ h 2 1/2 z 2 + β 2 1/2 .
Noticing that α t,h > 0 and using the reliability of the estimator presented in section 4 we obtain :

α t -α t,h ≤ C 2 (η 2 ev + T 2 ex ) + ζt -2 φ h -R h φ h 2 + 2(γ h , φ h -R h φ h ), (58)
for some C 2 > 0 (independent of t and h). Hence (54) is a direct consequence of the estimates (57) and (58).

Similarly to Corollary 4.8, we have :

Corollary 5.2 Assume that α t is a simple eigenvalue, then we have :

|α t -α t,h | η 2 ev + ζt -2 φ h -R h φ h 2 + (γ h , φ h -R h φ h ) + h.o.t. Remark

Efficiency of the estimator

In order to prove the efficiency of the estimator, each part of it (except the terms involving the exact solutions) has now to be bounded by the error e ev h up to a multiplicative constant.

Lemma 6.1

(R h -I)φ h 2 H(rot,Ω) ζ -2 t 4 γ -γ h 2 Ω + |ω -ω h | 2 1 +|φ -φ h | 2 1 + ζ -2 t 4 rot(γ -γ h ) 2 . Proof: Since (R h -I)φ h = ζ -1 t 2 (γ -γ h ) -∇(ω -ω h ) + (φ -φ h ), we have R h -I)φ h ≤ ζ -1 t 2 γ -γ h + |ω -ω h | 1 + φ -φ h ,
and with the Poincaré-Friedrichs inequality, we get

(R h -I)φ h 2 ζ -2 t 4 γ -γ h 2 + |ω -ω h | 2 1 + |φ -φ h | 2 1 .
Moreover, we have

rot(φ h -R h φ h ) 2 ζ -2 t 4 rot(γ -γ h ) 2 + |φ -φ h | 2 1 ,
so that lemma 6.1 holds.

Lemma 6.2 We have

T ∈T h h 2 T (t 2 + h 2 T ) α t,h ω h + divγ h 2 T T ∈T h h 2 T (t 2 + h 2 T ) α t,h ω h -α t ω 2 T +t 2 γ -γ h 2 + γ -γ h 2 -1 . (59)
and

E∈E(Ω) h E (h 2 E + t 2 ) [γ h ] E • n E 2 E T ∈T h h 2 T (t 2 + h 2 T ) α t ω -α t,h ω h 2 T +t 2 γ -γ h 2 + γ -γ h 2 -1 . ( 60
) Proof: Let v T = b 2 T (α t,h ω h + divγ h ) for all T ∈ T h , b
T being the classical element bubble function. So, we get by the elementwise inverse estimates :

α t,h ω h + divγ h 2 T (α t,h ω h + divγ h , v T ) T = (α t,h ω h , v T ) T -(γ h , ∇v T ) T = (α t,h ω h -α t ω, v T ) T + (γ -γ h , ∇v T ) T . (61) h 4 T α t,h ω h -α t ω 2 T 1/2 .
(64) Using Young's inequality in this last estimate, we get (59). For all interior edge E = T + ∩T -, we define the classical edge bubble function b

E ∈ H 1 0 (ω E ) such as supp b E = ω E , 0 ≤ b E ≤ 1 = max x∈E b E , ω E b E dx ≈ h 2 E , E b E ds ≈ h E , ∇b E L 2 (T ± ) h -1 E b E L 2 (T ± ) , |∇b E | H 1 (ω E ) h -2 E b E L 2 (ω E ) .
For all E ∈ E h , we define w

E = b 2 E ([γ h ] E • n E ) ∈ H 2 0 (ω E ) 2 with w E 2 ω E h E [γ h ] E • n E 2 E . So, we obtain : [γ h ] E • n E 2 E ([γ h ] E • n E , w E ) E = (divγ h , w E ) ω E + (γ h , ∇w E ) ω E = (α t,h ω h + divγ h , w E ) ω E + (γ h , ∇w E ) ω E -(α t,h ω h , w E ) ω E = (α t,h ω h + divγ h , w E ) ω E -(γ -γ h , ∇w E ) ω E + (α t ω -α t,h ω h , w E ) ω E .
Using a discrete Cauchy-Schwarz inequality and the regularity of the mesh, we have by summation

E∈E h h E (h 2 E + t 2 ) [γ h ] E • n E 2 E E∈E h (α t,h ω h + divγ h , h E (h 2 E + t 2 )w E ) ω E + (α t ω -α t,h ω h , h E (h 2 E + t 2 )w E ) ω E -γ -γ h , E∈E h h E (h 2 E + t 2 )∇w E ω E T ∈T h h 2 T (t 2 + h 2 T ) α t,h ω h + divγ h 2 T 1/2 E∈E h (h 2 E + t 2 ) w E 2 ω E 1/2 + T ∈T h h 2 T (t 2 + h 2 T ) α t ω -α t,h ω h 2 T 1/2 E∈E h (h 2 E + t 2 ) w E 2 ω E 1/2 + γ -γ h -1 ∇ E∈E h h 3 E ∇w E + t γ -γ h E∈E(Ω) th E ∇w E . (65)
By the following inverse estimate :

∇w E ω E + h E |∇w E | 1,ω E h -1/2 E [γ h ] E • n E E ,
we get :

∇ E∈E h h 3 E ∇w E 2 E∈E h h 6 E |∇w E | 2 1 E∈E h h 3 E [γ h ] E • n E 2 E (66) ≤ E∈E h h E (t 2 + h 2 E ) [γ h ] E • n E 2 E .
The same kind of argument give :

E∈E h th E ∇w E 2 E∈E h t 2 h 2 E ∇w E 2 ω E E∈E h t 2 h E [γ h ] E • n E 2 E (67) ≤ E∈E h h E (t 2 + h 2 E ) [γ h ] E • n E 2 E .
By (65), (66) and (67), we obtain :

E∈E h h E (h 2 E + t 2 ) [γ h ] E • n E Theorem 6.5 We have T ∈T h h 2 T (t 2 + h 2 T ) α t,h ω h + divγ h 2 T + E∈E h h E (t 2 + h 2 E ) [γ h ] E • n E 2 E + φ h -R h φ h 2 H(rot,Ω) + T ∈T h h 2 T divCε(φ h ) + γ h + t 2 12 α t,h φ h 2 T + E∈E h h E [Cε(φ h )] E n E 2 E + µ h (γ h ) 2 |ω -ω h | 2 1 + |φ -φ h | 2 1 + (1 + ζ -1 t 2 )ζ -1 t 2 γ -γ h 2 + γ -γ h -1 +ζ -2 t

Numerical validation

Here we illustrate and validate our theoretical results by a simple computational example.

Let Ω be the unit square ]0, 1[ 2 . We take ν = 0.3, k = 5/6 and t = 0.1. The meshes we use are uniform ones composed of n 2 squares, each of them being cut into 8 triangles as displayed the Figure 1 for n = 4. The refinement strategy is an uniform one so that the value of the mesh size h between two consecutive meshes is twice smaller.

Before evaluating the a posteriori error estimator, we compute ω 1 t,h by :

ω 1 t,h = 2(1 + ν) t α 1 t,h ,
where α 1 t,h is the first computed approximated eigenvalue. In fact, this rescaling process is done in order to allow some comparisons with some bibliography data [START_REF] Dawe | Rayleigh-Ritz vibration analysis of Mindlin plates[END_REF][START_REF] Durán | Approximation of the vibration modes of a plate by Reissner-Mindlin equations[END_REF][START_REF] Huang | A nine node Lagrangian Mindlin plate element with enhanced shear interpolation[END_REF]. Table 1 displays the obtained values for different values of n, and shows that ω 1 t,h converges when h goes towards zero, as theoretically expected. 2 with the ones obtained by the Huang and Hinton method in [START_REF] Huang | A nine node Lagrangian Mindlin plate element with enhanced shear interpolation[END_REF] (column HH), the Dawe and Roufaeil method in [START_REF] Dawe | Rayleigh-Ritz vibration analysis of Mindlin plates[END_REF] (column DR) and the Durán, Hervella-Nieto, Liberman, Rodríguez and Solomin method in [START_REF] Durán | Approximation of the vibration modes of a plate by Reissner-Mindlin equations[END_REF] (column DHLRS). Our value is clearly in good agreement with these references, even if from Table 2 it can noticed that it is the smallest one. This can be explained by the fact that our mesh resolution is finer. Indeed, in We plot in Figure 2 the evolution of the computed estimator η ev as well as its different contributions η h,i , i = 1..6 versus h. First, it can be seen that the contributions η h,4 and η h,6 converge at order 3 and that the contributions η h,3 and η h,5 converge at order 2. Moreover, it is clear that the main part of η ev is η h,2 . Nevertheless, we can also remark that the convergence rate of η h,2 (resp. η h,1 ) starts for coarse meshes near to 2 (resp. 3). This behaviour can easily be explained by the definition of η h,2 (resp. η h,1 ) when h is larger than t. As soon as h becomes smaller than t, the convergence rate equal to 1 for η h,2 (resp. 2 for η h,1 ) is recovered as it can be observed in Figure 2 for the finest meshes.

Figure 1 :

 1 Figure 1: Mesh level corresponding to n = 4 and h = √ 2/8.

  n

Figure 2 :

 2 Figure 2: Estimators convergence rate, t = 0.1.

  2 , φ -φ h 2 and |α t -α t,h | 2 are superconvergent (cf. Theorem 2.4), then

	t 2 12 perconvergent. α t φ -α t,h φ h	2	+ α t ω -α t,h ω h	2 +	α t + α t,h 2	ω -ω h	2 +	t 2 12	φ -φ h	2	are su-

  5.3 The term (γ h , φ h -R h φ h ) can be evaluated numerically. However, it can be bounded by γ h φ h -R h φ h . We further can numerically remark that the term φ h -R h φ h converges faster than the estimator : hence if α t is a simple eigenvalue, we can claim that the term |α

t -α t,h | is superconvergent (since it is bounded by the square of η ev up to higher order terms) and the relation (iii) given in Theorem 2.4 is recovered.

  4 rot(γ-γ h ) 2 + T ∈T h h 2 T (t 2 + h 2 T ) α t ω -α t,h ω h The proof is a direct consequence of lemma 6.1, 6.2, 6.3 and 6.4. Like the Corollaries 4.8 and 5.2, we have Corollary 6.6 (Efficiency of the estimator) Assume that the eigenvalue α t is simple, then

									2 T
	+	t 2 12 T ∈T h	h 2 T α t φ -α t,h φ h	2 T +	t 2 12	α t φ -α t,h φ h	2	.
	Proof: T ∈T h	h 2 T divCε(φ h ) + γ h +	t 2 12	α t,h φ h	2 T	+	E∈E h	h

T ∈T h h 2 T (t 2 + h 2 T ) α t,h ω h + divγ h 2 T + E∈E h h E (t 2 + h 2 E ) [γ h ] E • n E 2 E + φ h -R h φ h 2 H(rot,Ω) + E [Cε(φ h )] E n E 2 E + µ h (γ h ) 2 |ω -ω h | 2 1 + |φ -φ h | 2 1 + (1 + ζ -1 t 2 )ζ -1 t 2 γ -γ h 2 + γ -γ h -1

+ζ -2 t 4 rot(γ -γ h ) 2 + h.o.t.

Table 1 :

 1 2.3391 1.6992 1.6129 1.5934 1.5886 1.5874 1.5871 Values of the first approximated eigenvalue ω 1 t,h . Now, our result on the finest grid (n = 128) is compared in Table

	2	4	8	16	32	64	128
	ω 1 t,h

Table 3 ,

 3 it can be observed similar results for similar mesh resolutions.

	HH	DR DHLRS Our result
	1.591 1.594 1.5913	1.587

Table 2 :

 2 The first value of ω 1 t,h . (Value obtained with the finest mesh avalaible in each paper).To verify the estimator reliability presented in section 5, the error estimator η ev is defined by :η 2 ev = η 2 h,1 + η 2 h,2 + η 2 h,3 + η 2 h,4 + η 2 h,5 + η 2 h,6, where the different contributions are given by :

	η 2 h,1 =		h 2 T (t 2 + h 2 T ) α t,h ω h + divγ h	2 T ,
		T ∈T h
	η 2 h,2 =	E∈E h	h E (t 2 + h 2 E ) [γ h ] E • n E	2 E ,
	η 2 h,3 =			t 2 12	α t,h φ h	2 T	,	(72)

T ∈T h h 2 T divCε(φ h ) + γ h +

Table 3 :

 3 The first value of ω 1 t,h . Comparison with DHLRS for different mesh resolutions.

	η 2 h,4 =	h E [Cε(φ h )] E n E	2 E ,
	E∈E h
	η 2 h,5 = φ h -R h φ h η 2 h,6 = µ 2	2 H(rot,Ω) ,

h (γ h ).

Acknowledgements

The authors are grateful to the referee for her/his helpful comments and suggestions which allowed to improve the obtained results.

Proof: We just need to use the same arguments as the proof of lemma 6.2. Lemma 6.4

Proof: We recall the definition of µ h (γ h ) :

The Korn and Friedrichs inequalities lead to (71).