Alessandro Annoni
email: lars.bernard|alessandro.annoni@jrc.it

Lars Bernard

John Douglas
email: j.douglas@brgm.fr

Joseph Greenwood
email: joe.greenwood@ordnancesurvey.co.uk

Irene Laiz

Michael Lloyd

Zoheir Sabeur
email: zoheir.sabeur@bmtcordah.com

Anne-Marie Sassen
email: anne-marie.sassen@atosorigin.com

Jean- Jacques Serrano

Thomas Usländer
email: thomas.uslaender@iitb.fraunhofer.de

Alessandro Annoni1

Lars Bernard1

John Douglas2

Joseph Greenwood3

Irene Laiz4

Michael Lloyd5

Zoheir Sabeur4

Anne-Marie Sassen6

Jean-Jacques Serrano2

Thomas Usländer7

Orchestra: Developing a Unified Open Architecture for Risk Management Applications

Keywords:

Due to organizational and technological barriers, actors involved in the management of natural or man-made risks cannot cooperate efficiently. In an attempt to solve some of these problems, the European Commission has made "Improving risk management" one of its strategic objectives of the IST programme. The integrated project Orchestra is one of the projects that recently started in this area. The main goal of Orchestra is to design and implement an open service oriented software architecture that will improve the interoperability among actors involved in multi-risk management. In this paper we will describe the goals of Orchestra and explain some of the key characteristics of the project. These are

• The chosen design process of the Orchestra Architecture.

• How to further improve geospatial information and standards for dealing with risks • How ontologies will be used to bring interoperability from a syntactical to a semantical level.

The paper ends with two examples demonstrating the benefits of the Orchestra Architecture. One is in the area of coastal zone management, and the other is related with managing earthquake risks.

The current situation in risk management

Increasing numbers of natural disasters have demonstrated to the European Commission and the Member States of the European Union the paramount importance of the natural hazards subject for the protection of the environment and the citizens. The flooding experienced throughout central Europe in August 2002 is the most recent example of the damage caused by unforeseen weather driven natural hazards. The summer of 2003 clearly showed the growing problem of droughts in Europe including the Forest Fires in Portugal with more than 90,000 ha of burnt areas. There is strong scientific evidence of an increase in mean precipitation and extreme precipitation events on the one hand and water shortages for certain regions on the other hand which implies that weather driven natural hazards may become more frequent.

The European Spatial Development Perspective and new EC Regional Policy regulations emphasize the need of a better spatial planning and require new tools for the impact assessment of regional developments on natural and technological risks and vice versa. Now the new regulations for renewed Structural Funds and instruments for the period 2007-2013, adopted by the EC on 14 July 2004, foresee specific measures for "developing plans and measures to prevent and cope with natural risks".

The different types of risks (affecting the territory of the EU) need to be better addressed by an integrated approach to risk management. Prevention, preparedness and response, three major phases of risk management, usually involve a vast range of sectoral institutions and organisations at various administrative levels with different systems (monitoring, forecasting, warning, information, etc.) and services. Unfortunately, exchange of relevant information needed for dealing with risks is often limited to a raw data exchange level and true efficiency, in most cases, is hindered by administrative and legal boundaries as well as a lack of interoperability on the technical side.

Because natural and man-made disasters are not limited by administrative boundaries, cross-border aspects also need to be carefully considered and as a consequence major efforts are required to harmonise data and making services interoperable.

The application of different policies, procedures, standards and the lack of interoperability of systems, result in problems related to a efficient data management and information delivery, all critical elements of Risk Management. Interoperability based on standardisation will help close the gap between the different actors and is also expected to stimulate the development of an operational European service market supported by the sharing of procedures, interfaces and resources.

A substantial portion of IT expenditure in Europe supports the maintenance of thousands of legacy systems, the vast majority of which were not designed to work together. Recent events have underscored the need to be able to consolidate information from disparate systems to support citizen protection and security, disaster management, criminal justice, and other missions, crossing pan-European agency boundaries and extending into national, state and local government areas. One of the most urgent and important challenges currently facing governments is to get these systems to interoperate and share information.

Alessandro Annoni1, Lars Bernard1, John Douglas2, Joseph Greenwood3, Irene Laiz4, Michael Lloyd5, Zoheir Sabeur4, Anne-Marie Sassen6, Jean-Jacques Serrano2, Thomas Usländer7

The Orchestra project

In an attempt to solve some of these problems, the European Commission has made "Improving risk management" one of its strategic objectives of the IST programme. The integrated project Orchestra is one of the projects that recently started in this area. The main goal of Orchestra is to design and implement an open service oriented software architecture that will improve the interoperability among actors involved in multi-risk management.

In order to realise this goal, the key objectives for the project are the following:

• To design an open service-oriented architecture for risk management that links spatial and non-spatial information services. In this context Orchestra will provide input to INSPIRE and GMES (see below). • To develop the service infrastructure for deploying risk management services. • To develop thematic services that are useful for various multi-risk management applications based on the architecture. • To validate the Orchestra architecture and thematic services in a multirisk scenario. • To provide software standards for risk management applications. In particular, the de facto standard of OGC and the de jure standards of ISO and CEN are envisaged to be influenced.

The Orchestra project started in September 2004. Currently the focus of the work is on understanding user needs, system requirements and an assessment of useful technologies. This is considered the necessary for input for design decisions for the Orchestra architecture. In the following we will first explain the process used to create the architecture, the so-called Orchestra Reference Model.

The Orchestra Reference Model

The architectural process of Orchestra is based on the principles of the following international standards:

• The Reference Model for Open Distributed Processing (ISO/IEC 10746 RM-ODP) is used for the structuring of ideas and documentation.

• The OpenGIS Service Architecture (especially ISO/DIS 19119) is used for the taxonomy of the Orchestra services.

Mapping of RM-ODP Viewpoints

RM-ODP is an international standard for creating open, distributed processing systems. It provides an overall conceptual framework for building distributed systems in an incremental manner. The Orchestra architectural process uses the RM-ODP viewpoints for the structuring of ideas and their documentation. The mapping of the viewpoints to Orchestra is indicated in Table 1. As the Orchestra deployment will have the nature of a looselycoupled distributed system based on operational services rather than a distributed application based on computational objects, in Orchestra the "computational viewpoint" is referred to as the "service viewpoint". The Orchestra Reference Model covers all five viewpoints in the following manner: Alessandro Annoni1, Lars Bernard1, John Douglas2, Joseph Greenwood3, Irene Laiz4, Michael Lloyd5, Zoheir Sabeur4, Anne-Marie Sassen6, Jean-Jacques Serrano2, Thomas Usländer7

• The analysis phase is described as part of the Enterprise Viewpoint.

• The design phase encompasses the harmonised specification of the Information and Service viewpoint resulting from requirements of the Enterprise viewpoint. The result is the Orchestra architecture that is, by definition, a platform-neutral specification according to the requirements of ISO/DIS 19119 (i.e. specification in UML). • The Orchestra architecture does not cover the Engineering and Technology viewpoints. • The aspects of the Engineering and Technology viewpoints are combined in one or more process steps. Each step represents one mapping to a specific service infrastructure (e.g. W3C Web Services) and leads to a platform-specific Orchestra Implementation Specification.

Compliance with the OpenGIS Service Architecture

The Orchestra architecture is a "simple service architecture" and shall use the service taxonomy of the Open GIS Architecture (ISO/DIS 19119). Thus, Orchestra will provide human interaction services (e.g. catalogue or map viewers), model/information management services (e.g. feature and map access services, query support services), workflow/task services (e.g. service chaining support), processing services (e.g. statistical calculation), communication services (e.g. web services) and system management services (e.g. authorisation support).

Analysis and Design Process

The Orchestra architecture is being designed in an iterative way recognising the fact that both the requirements of the system and end users and the technological progress in the IT market and in IT standardisation have a dynamic nature and cannot be completely caught in a one-shot design. Thus, a global iteration cycle between the analysis and the design phase of the architecture is foreseen (see Figure 1).

A consolidation process in-between ensures that, at a defined point in time, there is a common understanding of the system requirements, the user requirements and an assessment of the current technology as a foundation to design the Orchestra architecture.

System requirements encompass all aspects that today prevent interoperability between systems. They are expressed in terms of architectural properties that a system should follow in order to improve the exchange, sharing and using of information across system boundaries. User requirements, on the one hand, represent the view of the end user of the Orchestra system that is being specified independent of the technological foundation of the system. It is expressed in terms of end-user services, information presentation and availability requirements and nonfunctional aspects such as trustworthiness of information, response time or quality of service. On the other hand, they comprise the view of the system engineer that will build thematic services (e.g. for flood or fire forecast). Here, user requirements are expressed in terms of the domain-specific information model (ontology) that has to be supported by the principal information management capabilities of the Orchestra platform.

Both the system and the user requirements are dynamic in the sense that they will be prioritised and adapted in local iteration cycles due to the result of the consolidation process.

Technology assessment is also a continuous process. Orchestra aims at building the architecture on top of technologies, tools and products that are either standard approaches or have proven to be successful in solving interoperability problems in deployed use-cases.

The dynamic nature of the input factors of the Orchestra architecture naturally leads to an iterative architectural design process. Various but Alessandro Annoni1, Lars Bernard1, John Douglas2, Joseph Greenwood3, Irene Laiz4, Michael Lloyd5, Zoheir Sabeur4, Anne-Marie Sassen6, Jean-Jacques Serrano2, Thomas Usländer7 controlled upgrades of the Orchestra architecture will be required to adapt the architecture to the changing needs.

In the following two sections of this paper we will highlight two important technologies for Orchestra: geographic information and ontologies for improvement of semantic interoperability. We will explain what Orchestra will contribute to the state of the art in these fields.

Geographic information services for risk management

Geographic information is involved in all phases of risk management processes -from prevention to immediate reaction. The geospatial aspects may be explicit (e.g. topographic maps providing background information) or hidden (tables about population distribution in an affected area). In the same way either dedicated tools are used to analyse or incorporate geospatial aspects (e.g. the usage of a GIS by a GI expert) or the information is integrated via interoperable GI components or GI services in a specific risk management application. Orchestra focuses on the latter case.

The transition from closed and monolithic GIS to open and interoperable GIS introduces a paradigm shift in the development of GIS-standards. Standards do not any more focus on formats for the file exchange of geodata but are directed to the specification of service interfaces. These services provide geoinformation that are processed according the user's query and their specific needs, e.g. a specific part of the cadastre, a fastest route from A to B, a thematic map on the current ozone pollution etc.

Interoperable GI Services

On-going activities to realise interoperable GI services currently mainly focus on encoding and accessing geographic information in an interoperable manner. The existing (partly draft) ISO standards and OGC specifications cover encoding and accessing geographic information to achieve interoperable GI services. Thus following the OGC service taxonomy this work falls mainly into the category model/information management services. However, specifications to define interoperable GI services to process spatio-temporal information (processing services) need still to be researched and there is an urgent need to define the respective specifications to support risk management applications as intended by Orchestra. Exam-Orchestra: Developing a Unified Open Architecture for Risk Management Applications 9

ples of functionalities that need to be covered by these geoprocessing services are:

• Weighted combination of different geoinformation layers to support assessment and spatial decisions • Spatial and topological operators (buffer, generalisation, etc.)

• Feature extraction services, e.g. to operate on remote sensing data • Geostatistical operators to extract key values or aggregate spatial information.

Moreover Orchestra will investigate on interoperability of GI services with spatio-temporal simulation models to ease the incorporation of e.g. flooding or weather forecast simulations in risk management applications.

Interoperable GI services are one cornerstone for the successful implementation of Spatial Data Infrastructures (SDI). SDI evolve all over Europe on institutional, regional and national levels. In 2001 the European Commission started the INSPIRE initiative (INfrastructure for SPatial In-foRmation in Europe; http://inspire.jrc.it/home.html) to streamline national SDI developments with respect on supporting European environmental policies. The initiative intends to trigger the creation of a European Spatial Data Infrastructure (ESDI) that delivers to the users integrated geospatial information services from various distributed sources, crossing institutional and political boundaries. The target users of INSPIRE include policymakers, planners and managers at European, national and local level and the citizens and their organisations.

European Directive on Infrastructure for Spatial Information in Europe (INSPIRE)

Following 3 years of intensive collaboration with Member States experts and stakeholder consultation, the European Commission has adopted on the July 2004 a proposal for a Directive establishing an infrastructure for spatial information in the Community (INSPIRE) (COM( 2004) 516 final).

The adoption of the proposal marks an important step on the way forward to a European-wide legislative framework that helps in achieving an ESDI. This proposal does not only address policy related issues concerning the development of an ESDI but also dedicates three chapters to the technical requirements that have to be fulfilled by the member states to establish the ESDI. These three chapters are on Metadata, Interoperability of spatial data sets and services, and Network services. Under these chapters Alessandro Annoni1, Lars Bernard1, John Douglas2, Joseph Greenwood3, Irene Laiz4, Michael Lloyd5, Zoheir Sabeur4, Anne-Marie Sassen6, Jean-Jacques Serrano2, Thomas Usländer7 the proposal list general requirements on these issues as well as it formulates the requirement to adopt appropriate implementing rules.

During the INSPIRE preparatory phase (2005)(2006) the ORCHESTRA project will provide input towards the drafting as well as the piloting of the INSPIRE implementing rules in the risk management domain. The first input can be expected on the topic of the INSPIRE network services.

The establishment of an ESDI will represent significant added value for -and will also benefit from -other Community initiatives such as Council Regulation (EC) No 876/2002 of 21 May 2002 setting up the Galileo Joint Undertaking (OJ L 138, 28.5.2002) and Global Monitoring for Environment and Security (GMES): Establishing a GMES capacity by 2008 (COM(2004) 65 final). In order to exploit the synergies between these initiatives, Member States should consider using the data and services resulting from Galileo and GMES as they become available, in particular those related to the time and space references from Galileo.

The use of ontologies for improved semantic interoperability

Where there are many distributed and heterogeneous sources of information and application services to be used then the key is interoperability between them. This needs to work on at least three levels; the syntax level, the structural level and the semantic level. To date most standard work has focused on the syntax of communicating between disparate software services, for example the Web Map Service standard of the OGC, which specifies the syntax for communicating between the mapping client and the service. Work has also concentrated on the structure of the data used in those interactions such as Simple Feature Specification of the OGC which allows structured data to be shared between systems. These two aspects make the physical connectivity and exchange of data possible and enable a distributed architecture to be realized.

However the content of the information in the distributed systems being connected may be referring to substantially different things and consequently not interoperate at the level of its semantics. For example in the risk management domain the term 'bank' in different data sources could mean 'a steep natural incline' or 'a business establishment in which money is kept'. One set of data is clearly suitable for use in a flood modeling application and the other is not! This is further complicated in multiple risk Orchestra: Developing a Unified Open Architecture for Risk Management Applications 11 management scenarios where the information with different terminology from multiple information communities must be interpreted together.

In order to resolve this semantic interoperability when sharing data between systems the meaning of the terms must be stated so that the differences can be resolved by the computer. This can be done in an ontology which in a computer science setting is a 'specification of a conceptualization'. It formally describes a set of concepts and the relationships that hold between them in a given context in a logical manner so the semantics can be interpreted by machines. The W3C have defined the OWL (Web Ontology Language) standard for the representation of ontologies to describe the semantics of disparate resources on the network.

The approach in Orchestra will be to define ontologies for common domains of risk management such as flooding or forest fire and the common concepts of risk management that bind these domains. The rich high level formal semantics within these ontologies should be capable of representing the abstract concepts and relations such as the causes, propagators and effects of risks. The simpler semantics of individual data sources and applications are represented in their own ontologies and are then mapped to the rich high level semantics of these ontologies This allows heterogeneous and abstract semantics of different data sources and applications to be interpreted through the richer semantics of the domain. By having a common semantics to which data and services are ultimately mapped it improves the selection and comparison of data sources based on their content and it becomes possible for the machine to infer whether two data sources are comparable or suitable for use in an application service. The formal representation of the common concepts of risk management also provide a common semantics to support cross domain query in multi risk management. For example the data about the results of forest fire can act as an input to understanding the increased risk to flooding due to mass vegetation changes.

To allow the ontologies to be used to bind together the different data sources and services of the Orchestra architecture they need to be mapped to the common meta information model and be processed by the core services. The project will define the mapping of the ontologies to the meta information model of the Orchestra architecture to which the different services and data sources will also be referenced. This common structure for using meta information will be used to combine the different services by providing the relationship to the syntax and structural description of in-Alessandro Annoni1, Lars Bernard1, John Douglas2, Joseph Greenwood3, Irene Laiz4, Michael Lloyd5, Zoheir Sabeur4, Anne-Marie Sassen6, Jean-Jacques Serrano2, Thomas Usländer7 formation. Semantic web-services will need to be incorporated into the architecture as part of the core services to expose and manipulate the ontologies. These at a minimum include ontology management and storage services, inference services to infer the semantic equivalence between different data sources and services and query brokering services. These are required to transform the semantics into syntax and structural connectivity to enable access to the different data sources.

Through meta information model and the services of the Orchestra architecture Orchestra will seek to provide a framework in which semantic interoperability can start to be achieved in addition to and integrated with the syntactical and structural interoperability. The key benefits to using formal and explicit semantics lie in the area of selecting and combining disparate information from a variety of sources to be used for a particular risk management task. With a functional semantic layer the differing uses of terminology across different risk management systems is no longer a barrier to the productive sharing of information and approaches. The appropriate use of data for a given application service or function can also be matched to ensure the end results are consistent. In an open architecture this allows multiple organization to share information and collaborate as the formal semantic enables plurality of views particularly in multi-risk scenarios where different information communities must work together.

Examples of the use of Orchestra

The Orchestra architecture will not take the form of a ready to use application for risk management. To the contrary, it should be considered as a collection of services, tools and methodologies that so-called system users can use to develop risk management applications for end users. It is our vision that chains of cooperating services will become reality and that system users can add value to already existing services. They can for instance combine services for one risk with services for another risk, and that way develop a multi-risk management application. There will be less need for reinvention of the wheel, and more effort can be spend on creative solutions for specific problems.

The Orchestra architecture will be generic in the sense that it should provide basic functionality that is useful in applications for any kind of risk (natural or technological), also in cross border situations. Based on this architecture, the project will develop a few thematic services that are Orchestra: Developing a Unified Open Architecture for Risk Management Applications 13 useful in various risk management applications, such as a cadastre service, a weather service, etc. A large part of the project will be dedicated to the validation of the Orchestra architecture and thematic services. We will now describe two possible scenarios for the use of Orchestra. The first one is related to coastal zone management, and the second one to earth quake risk management.

An example of the use of Orchestra for risk management at coastal zone

The study of the environmental risks which may be generated from maritime transport activity in coastal European waters requires the establishment of advanced modelling techniques and evaluation benchmarks. The support of a generic evaluation of the environmental risks within an open information system infrastructure is key for improving the risk management methodology.

In Orchestra, various numerical modelling applications could be deployed to assess the environmental risks induced by ship traffic activity at coastal zones. This can be efficiently achieved through a generic sharing of common toxicity, traffic networks, coastal zone environmental databases and numerical modelling kernels. The generic sharing of data information is formalised through advanced knowledge modelling at coastal zone, including standard data and meta-data access methods. Maritime transport activity could induce the multiple risk of introducing anti-foulants, ballast waters, oil and chemical spills, and atmospheric emissions into coastal waters and coastal zones. Ship traffic spatial networks in European coastal waters can be geographically identified, updated in time and shared within common databases (Figure 2). The generic access to such geospatial information within Orchestra will lead to rapid risk management of ship traffic in European coastal waters.

The engendered multiple risks can be predicted through multi-modelling the fate of anti-foultants such as TBT, alien organisms in ballast waters and SO x /NO x gases in the atmosphere following on ship traffic network within a region ship traffic network of interest. The various discharges of ballast waters, anti-foulants and gas emmissions could lead to quantifiable risks of multiple orders around coastal zone environments, including health risks. For instance, TBT anti-foulants exposure may lead to marine species genetic disorders; ballast waters could introduce alien marine organisms and diseases and; gas emissions cause poor air quality and respiratory problems around the same zone of interest. Oil spills could not only Alessandro Annoni1, Lars Bernard1, John Douglas2, Joseph Greenwood3, Irene Laiz4, Michael Lloyd5, Zoheir Sabeur4, Anne-Marie Sassen6, Jean-Jacques Serrano2, Thomas Usländer7 damage marine organisms, but also damage commercial activities such as fishing and tourism, and hence contribute to raising unemployment levels in coastal working communities.

The above mentioned risks could themselves induce secondary risks on local economies at coastal zone, i.e. fishing and leisure industries, and may contribute to unemployment levels within coastal working communities.

The various types of risks are established as probability of occurrences of the so-called Predicted Environmental Concentration (PEC) and compared to critical exposures; the so-called No Effect Concentration (NEC) value. Spatial hazard maps can therefore be predicted within coastal zone and updated according to changes in environmental conditions and ship traffic activity with time (Figure 3). Hazards maps will lead to the dynamic evaluation of multi-risks and their probability of exceedence with respect to EU and international statutory criteria. These are with respect to air and water quality standards but also ecosystem toxicity thresholds. The economic indicator thresholds are much more challenging during the evaluation exercise but can still be implemented as additional information layers within the Orchestra model applications if historical economic indicators are available for the regions of study.

An example of the use of Orchestra for earthquake risk management

Earthquakes can occur in border regions (both national and international), e.g. the 1976 Friuli earthquake sequence that caused damage in Italy and in Slovenia. They can induce other potentially damaging events, such as landslides (e.g. the 2001 El Salvador earthquake where the biggest single lost of life was caused by a landslide triggered by the earthquake) or fires (e.g. the 1923 Kanto Plain earthquake where the fires following the earthquake caused many more causalities than the actual earthquake shaking).

Because of the potentially catastrophic nature of large earthquakes, which can affect all aspects of society, their correct management relies on close cooperation between different organisations at different levels. For these reasons earthquake risk management is a good example of the type of risk that is of particular interest to Orchestra. Earthquake risk management is an interesting contrast to coastal zone management because earthquakes are a rapid-onset hazard (the strong shaking that causes damage during an Alessandro Annoni1, Lars Bernard1, John Douglas2, Joseph Greenwood3, Irene Laiz4, Michael Lloyd5, Zoheir Sabeur4, Anne-Marie Sassen6, Jean-Jacques Serrano2, Thomas Usländer7 earthquake usually lasts for less than one minute), although the effects can last for decades, whereas coastal zone hazards are usually slow-onset hazards. In addition, the main technique for managing earthquake risk is to reduce the vulnerability of the elements at risk by using better construction techniques whereas coastal zone risk management also includes efforts to reduce the actual hazard (a step that is impossible for earthquakes). For these and other reasons it is a big challenge to develop an architecture that is appropriate for both these risks (and others).

Orchestra will help to improve the management of earthquake risk in Europe by facilitating easier data exchange between involved parties. For example, the estimated level of seismic hazard often does not exactly match at national and international borders due to differences in input data and methodology, whereas in reality earthquakes do not respect human borders. Similar problems can occur for building vulnerability assessments in different countries due to differences in the way this is performed and because of the difficulty in accessing vital information. Therefore the assessed earthquake risk across borders is difficult to compare and consequently it is difficult assign priorities to its management. This leads to a non-optimal allocation of resources and therefore waste and higher real risks in some regions than expected.

Figure 1 :

 1 Figure 1: Dynamic Orchestra Architectural Process

Figure 2 Figure 3

 23 Figure 2 Illustrative ship traffic network in the German Bight

Table 1 :

 1 Mapping of the RM-ODP Viewpoints to Orchestra

	Viewpoints	Mapping to Orchestra	Usage example
	Enterprise	Reflects the analysis phase in	Use case description
		terms of the system and the user	of a geoprocessing
		requirements as well as the	service
		technology assessment
	Information	Covers the conceptual model of	UML class diagram
		all kinds of information with	defining the infor-
		their thematic, spatial, temporal	mation elements that
		characteristics as well as their	are used by the geo-
		meta-data.	processing service
	Computation-	Covers the Orchestra services	UML specification of
	al (referred to	that enable syntactical and se-	the	geoprocessing
	as Service	mantic interoperability and ad-	service
	Viewpoint)	ministration across system
		boundaries
	Engineering	Covers the mapping of the Or-	Mapping of the UML
		chestra service specifications to	specification	to
		the chosen service infrastruc-	WSDL
		ture
	Technology	Covers the technological choic-	Usage of W3C Web
		es of the service infrastructure	Services and UDDI
		and the operational issues of the
		infrastructure.

Concluding remarks

The Orchestra project (www.eu-orchestra.org) started in September 2004 and will run until August 2007. The following organisations are involved in the project:

• Atos Origin, Spain • BMT Cordah Limited, United Kingdom • The Alliance of Maritime Regional Interests in Europe, Belgium

The project will work together closely with two other Integrated Projects in the field called WIN (http://www.win-eu.org) and OASIS (www.oasis-fp6.org). WIN will concentrate more on organisational issues relevant for improved interoperability in risk management and OASIS focuses on crisis management. The three projects will use the same architectural principles and make their results interoperable. These results will be provided as input to INSPIRE and to GMES (Global Monitoring for Environment and Security, www.gmes.info).

• Craglia, M., A.Annoni, R.S.


Marc Briane
email: mbriane@insa-rennes.fr

Laurent Pater
email: laurent.pater@ens-cachan.org

Magneto-resistance in three-dimensional composites

Keywords: Hall effect, homogenization, magneto-resistance, magneto-transport AMS classification: 35B27, 74Q15

published or not. The documents may come

Magneto-resistance in three-dimensional composites.

Introduction

In a conductor with a matrix-valued resistivity ρ, a low magnetic field h ∈ R 3 induces a perturbed resistivity ρ(h). Due to Onsager relations (see [START_REF] Landau | Électrodynamique des Milieux Continus[END_REF][START_REF] Omar | Elementary Solid State Physics: Principles and Applications[END_REF]), the perturbed resistivity satisfies ρ(h) = ρ(-h) T .

(1.1)

As a consequence, the perturbed resistivity admits the following second-order expansion (see Section 2):

ρ(h) = ρ(0) + R(h) + M (h, h) + o(|h| 2 ), (1.2)
where ρ(0), M (h, h) are symmetric matrices and R(h) is an antisymmetric matrix. On the one hand, according to the Hall effect (see, e.g., [START_REF] Landau | Électrodynamique des Milieux Continus[END_REF]), the magnetic field induces a transversal electric field E t (h) which balances the magnetic force acting on the charge carrier and is perpendicular to the current j. It is given by

E t (h) = R(h) j ⊥ j, (1.3)
where R(h) is the Hall tensor which reduces to r(j × h) in the isotropic case. On the other hand, the so-called magneto-resistance M (h, h) measures the difference between the perturbed dissipation energy and the unperturbed one, namely

ρ(h)j • j -ρ(0)j • j = M (h, h)j • j + o(|h| 2 ), (1.4)
in which the Hall term plays no role (due to the antisymmetry of R(h)). Expansion (1.4) has to be regarded in connection to the Kohler law [START_REF] Kohler | Zur magnetischen widerstandsnderung reiner metalle[END_REF] which states that the symmetrized resistivity (without the Hall term) ρ s (h) of a homogeneous conductor satisfies the asymptotic ρ s (h) -ρ(0) ≈ h→0 m |h| 2 with m > 0, (1.5) which corresponds to an increase of the magneto-resistance.

When the conductor has a microstructure characterized by a scale ε, the resistivity ρ ε (h) depends on the two parameters ε, h. In the framework of the Murat Tartar H-convergence theory (see Section 2 and [START_REF] Murat | Compacité par compensation[END_REF][START_REF] Murat | H-convergence[END_REF]), the conductivity σ ε (h) = ρ ε (h) -1 H-converges to the effective (or homogenized) conductivity σ * (h). Under appropriate regularity conditions for σ ε (h) (see (2.2)), the effective resistivity ρ * (h) = σ * (h) -1 also satisfies equality (1.1) and the second-order expansion

ρ * (h) = ρ * (0) + R * (h) + M * (h, h) + o(|h| 2 ), (1.6)
where R * is the effective Hall tensor and M * is the effective magneto-resistance tensor.

In his seminal work [START_REF] Bergman | Self-duality and the low field Hall effect in 2D and 3D metal-insulator composites. Percolation Structures and Processes[END_REF], Bergman gave for a periodic composite material an expression of the effective Hall matrix in terms of the local Hall matrix and the local current fields obtained in the absence of a magnetic field. Bergman's approach was extended in dimension two [START_REF] Briane | Homogenization of the two-dimensional hall effect[END_REF] and in dimension three [START_REF] Briane | Homogenization of the three-dimensional Hall effect and change of sign of the Hall coefficient[END_REF] in the non-periodic framework of H-convergence.

In dimension two, the conductor lies in a plane (e 1 , e 2 ) embedded in a transversal magnetic field h e 3 , so that the local/effective Hall coefficient r ε/ * and the local/effective magneto-resistance matrix M ε/ * are defined by

R ε/ * (h) = r ε/ * h 0 -1 1 0 and M ε/ * (h, h) = h 2 M ε/ * . (1.7)
In the periodic case, i.e. when r ε (x) = r(x/ε) and M ε (x) = M (x/ε) are oscillating functions of the fast variable x/ε, it was proved in [START_REF] Briane | Homogenization of the magneto-resistance in dimension two[END_REF] that

M * j • j -M j • j ≥ 0, (1.8)
for any unperturbed current j, and that (1.8) is an equality if and only if the Hall coefficient is a constant. By the Kohler law (1.5), the magneto-resistance in each phase satisfies M = µ I 2 with µ > 0, which implies the positivity of M * by (1.8). Then, the positivity of m in (1.5) corresponds to the positivity of M * in (1.8). Therefore, the inequality (1.8) extends the classical Kohler law (1.5) to anisotropic two-dimensional composites (see [START_REF] Briane | Homogenization of the magneto-resistance in dimension two[END_REF], Remark 2.6).

This paper extends the results of [START_REF] Briane | Homogenization of the magneto-resistance in dimension two[END_REF] to three-dimensional composites. In dimension three, the local/effective Hall tensor reads as

R ε/ * • h = E (R ε/ * h), with E (η) := 0 -η 1 η 2 η 1 0 -η 3 -η 2 η 3 0 , (1.9)
where R ε/ * is called the local/effective Hall matrix. First, we obtain a general expression (see Theorem 2.1) for the difference between the effective dissipation energy due to the magneto-resistance and the average of the local dissipation energy. Then, extending a classical duality principle (see Lemma 3.1), we prove that this difference is non-negative (see Theorem 3.1), and equal to zero if and only if the Hall matrix satisfies some compactness condition. In the periodic case, this reads as (see Corollary 3.1)

D(h, h) := M * (h, h) j • j -M (h, h) j • j ≥ 0, (1.10)
for any unperturbed current field j. Moreover, (1.10) is an equality if and only if

Curl (E (R h)j) = 0 in D ′ (R 3 ). (1.11) We also investigate the behaviour of higher even-order terms. We show that inequality (1.10) reverses when the magneto-resistance is replaced by the fourth-order term and the Hall matrix is assumed to be zero (see Proposition (3.1)). However, the equivalent of D(h, h) for even-order term higher or equal to 4 may have both a positive and a negative eigenvalue, so that (1.10) cannot be extended (see Proposition 3.2).

Then, the condition of equality (1.11) is discussed in the case of columnar composites. First, an explicit formula for the difference of dissipation energies D(h, h) (1.10) is given (see Proposition 4.1) for a periodic material which is layered in some direction ξ. Second, for a general columnar structure in the direction e 3 , the equality D(h, h) = 0 is shown to be very sensitive to the orientation of the magnetic field (see Proposition 4.2). More precisely, the equality D(h, h) = 0 implies that σ(y 1 , y 2 ) is a tensor product of type f (h 1 y 1 + h 2 y 2 ) g(h 2 y 1 -h 1 y 2 ). For example, in the case of a fourphase checkerboard α 1 , α 2 , α 3 , α 4 (see figure 4.1 below), we obtain that for any magnetic field h = 0

perpendicular to e 3 , D(h, h) = 0 if α 1 α 3 = α 2 α 4 (see Proposition 4.3).
The paper is organized as follows. In Section 2, we recall results on the homogenization of the Hall effect and the magneto-resistance in order to establish an asymptotic formula for the effective magneto-resistance. In Section 3, we prove inequality (1.10) and deal with the case of higher-order terms. Section 4 is devoted to the case of equality for layered and columnar composites. • × denotes the cross product and ⊗ the tensor product in R 3 .

• R d×d denotes the set of the real-valued (d× d) matrices and I d denotes the unit matrix of R d×d .

• R d×d a (resp. R d×d s ) is the set of the real-valued (d× d) antisymmetric matrices (resp. symmetric matrices).

• A s denotes the symmetric part of A, A T its transposed matrix, and Cof(A) its cofactors matrix.

• Ω denotes a bounded open set of R d .

• For α, β > 0, M(α, β; Ω) denotes the set of the invertible matrix-valued functions A measurable in Ω and such that

∀ ξ ∈ R d , A(x)ξ • ξ ≥ α|ξ| 2 and A(x) -1 ξ • ξ ≥ β -1 |ξ| 2 , a.e. x ∈ Ω.
(1.12)

• For a vector-valued function

U : Ω -→ R d , DU := ∂U j ∂x i 1≤i,j≤d , div (U ) = d i=1 ∂U i ∂x i and curl (U ) = ∂U i ∂x j - ∂U j ∂x i 1≤i,j≤d .
• For a matrix-valued function

Σ : Ω -→ R d×d , Div (Σ) = d i=1 ∂Σ i,j ∂x i 1≤j≤d and Curl (Σ) = ∂Σ i,k ∂x j - ∂Σ j,k ∂x i 1≤i,j,k≤d .
• If H is a vector space endowed with a norm • , the equality

g ε (h) = o H (|h| n ), for n ∈ N, means that lim h→0 1 |h| n sup ε>0 g ε (h) = 0. (1.13) • For k ∈ N, C k c (Ω)
denotes the space of k-continuously derivable functions with compact support in Ω.

• For ν = (ν 1 , . . . , ν d ) ∈ N d , we denote |ν| = ν 1 + • • • + ν d and ∂ |ν| ∂x ν = ∂ ν 1 ∂x ν 1 1 • • • ∂ ν d ∂x ν d d .
• Y := (0, 1) 3 , and the Y -average is denoted • .

• H ♯ (Y ; Z) denotes the space of the Y -periodic functions from R 3 to Z which belong to H loc (R d ) for a generic function space H.

Remark 1.1. Consider a sequence g ε (h) in H which satisfies the expansion of order n ∈ N,

g ε (h) = g 0 ε + g 1 ε (h) + • • • + g n ε (h, . . . , h) + o H (|h| n ), (1.14)
where for any k ≤ n, h → g k ε (h, . . . , h) are bounded sequences of k-linear symmetric forms in H. In view of (1.13) each term g k ε (h, . . . , h) of the expansion inherits of the same (weak or strong) convergence of g ε (h) in H.

Let us recall the definition of the H-convergence due to Murat, Tartar [START_REF] Murat | H-convergence[END_REF]:

Definition 1.1 (Murat, Tartar [18]). A sequence A ε in M(α, β; Ω) is said to H-converge to the matrix-valued function A * if for any distribution f ∈ H -1 (Ω), the solution u ε ∈ H 1 0 (Ω) of the equation div(A ε ∇u ε ) = f satisfies the convergences u ε ⇀ u * weakly in H 1 0 (Ω) and A ε ∇u ε ⇀ A * ∇u * weakly in L 2 (Ω) 2 , (1.15)
where u * solves in H 1 0 (Ω) the homogenized equation div(A * ∇u * ) = f . Murat and Tartar [START_REF] Murat | H-convergence[END_REF] proved that for any sequence A ε in M(α, β; Ω), there exist A * in M(α, β; Ω) and a subsequence of A ε which H-converges to A * .

2 The three-dimensional effective magneto-resistance 2.1 The three-dimensional Hall effect and magneto-resistance Let α, β > 0, and let Ω be a regular bounded domain of R 3 . Consider a heterogeneous conductor in Ω, with a symmetric matrix-valued conductivity σ ε ∈ M(α, β; Ω) (see (1.12)), associated with the resistivity ρ ε := (σ ε ) -1 . Here, ε is a small positive parameter which represents the scale of the microstructure. In the presence of a magnetic field h ∈ R 3 , it is known (see, e.g., [START_REF] Landau | Électrodynamique des Milieux Continus[END_REF]) that the perturbed resistivity satisfies the property

ρ ε (-h) = ρ ε (h) T .
(2.1) Also assume that the conductivity satisfies the following regularity properties: there exist an open ball O in R 3 centered at 0 and b ∈ L ∞ (Ω) such that for any ε > 0 and any multi-

index |ν| ≤ 2,              σ ε (h) ∈ M(α, β; Ω), ∀ h ∈ O, h → σ ε (h)(x) is of class C |ν| on O, ∀ x ∈ Ω, ∂ |ν| σ ε ∂h ν (h)(x) - ∂ |ν| σ ε ∂h ν (k)(x) ≤ b(x) |h -k|, ∀ h, k ∈ O, a.e. x ∈ Ω. (2.2)
As a consequence of (2.2), the resistivity ρ ε (h) satisfies the second-order expansion

ρ ε (h) = ρ ε + R ε (h) + M ε (h, h) + o L ∞ (Ω) 3×3 (|h| 2 ), (2.3)
where R ε : R 3 → L ∞ (Ω) 3×3 and M ε : R 3 × R 3 → L ∞ (Ω) 3×3 are sequences of linear operators uniformly bounded with respect to ε. By virtue of (2.1), for any h in R 3 , ρ ε (0) and M ε (h, h) are symmetric matrix-valued functions, while R ε (h) is an antisymmetric matrix-valued function. The matrix-valued function defined by (see (1.9) and [START_REF] Briane | Homogenization of the three-dimensional Hall effect and change of sign of the Hall coefficient[END_REF] for more details)

R ε h := E -1 R ε (h) , (2.4)
and the second-order term M ε (h, h) are respectively called the Hall matrix and the magnetoresistance associated with the perturbed resistivity ρ ε (h), so that

ρ ε (h) = ρ ε + E (R ε h) + M ε (h, h) + o L ∞ (Ω) 3×3 (|h| 2 ). (2.5) Remark 2.1. Since by assumption R ε : R 3 → L ∞ (Ω) 3×3 is uniformly bounded with respect to ε, R ε is a bounded sequence in L ∞ (Ω) 3×3 .
Similarly, we define the linear operators S ε : R 3 → L ∞ (Ω) 3×3 and N ε :

R 3 × R 3 → L ∞ (Ω) 3×3 by σ ε (h) = ρ ε (h) -1 = σ ε + E (S ε h) + N ε (h, h) + o L ∞ (Ω) 3×3 (|h| 2 ), (2.6)
which are uniformly bounded with respect to ε.

Homogenization of the magneto-resistance

Assume that σ ε (h) H-converges (see definition 1.1) to σ * (h) for any h ∈ O. In fact due to the compactness of H-convergence [START_REF] Murat | H-convergence[END_REF] this holds true for a subsequence of ε and a countable set of h. Then, by [START_REF] Colombini | Sur la convergence de solutions d'équations paraboliques[END_REF] (Theorem 2.5 in the symmetric case) and [START_REF] Boccardo | Homogénéisation de problèmes quasi-linéaires[END_REF] (Theorem 3.1 in the non symmetric case), the effective (or homogenized) conductivity σ * (h) satisfies σ * (-h) = σ * (h) T . By the regularity conditions (2.2) (see [START_REF] Briane | Homogenization of the magneto-resistance in dimension two[END_REF] and [START_REF] Colombini | Sur la convergence de solutions d'équations paraboliques[END_REF] for more details), as in (2.6), we have the second-order expansion

σ * (h) = σ * + E (S * h) + N * (h, h) + o(|h| 2 ).
(2.7)

Moreover, by taking the inverse of (2.7), the effective resistivity ρ * (h) := σ * (h) -1 also expands as

ρ * (h) = ρ * + E (R * h) + M * (h, h) + o(|h| 2 ), (2.8)
where R * ∈ L ∞ (Ω) 3×3 is the effective Hall matrix and M * : R 3 × R 3 → L ∞ (Ω) 3×3 is the effective magneto-resistance tensor of the composite. We have the following result:

Proposition 2.1. The following relations hold for any h ∈ O,

S ε = -Cof(σ ε ) R ε and S * = -Cof(σ * ) R * , (2.9
)

N ε (h, h) = -σ ε M ε (h, h) σ ε + E (S ε h) σ -1 ε E (S ε h), (2.10
)

N * (h, h) = -σ * M * (h, h) σ * + E (S * h) σ -1 * E (S * h). (2.11)
Proof. By the first-order expansions (2.5) and (2.6) we have, for any h ∈ O and any t > 0 small enough,

0 = -I 3 + σ ε + E (tS ε h) + N ε (th, th) ρ ε + E (tR ε h) + M ε (th, th) + o L ∞ (Ω) 3×3 (t 2 |h| 2 ) = t E (S ε h)σ -1 ε + σ ε E (R ε h) + t 2 N ε (h, h)σ -1 ε + E (S ε h)E (R ε h) + σ ε M ε (h, h) + o L ∞ (Ω) 3×3 (t 2 |h| 2
). Then, dividing by t the previous equality and letting t tend to zero we obtain

E (R ε h) = -σ -1 ε E (S ε h) σ -1 ε .
(2.12)

Hence, we get that

0 = t 2 N ε (h, h) σ -1 ε -E (S ε h) σ -1 ε E (S ε h) σ -1 ε + σ ε M ε (h, h) + o L ∞ (Ω) 3×3 (t 2 |h| 2 ). (2.13)
We divide this equality by t 2 and let t tend to 0 to get (2.10).

The first equality of (2.9) is a straightforward consequence of the following algebraic lemma which is proved in [START_REF] Briane | Homogenization of the three-dimensional Hall effect and change of sign of the Hall coefficient[END_REF] (Lemma 1):

∀ P ∈ R 3×3 , P T E (ξ)P = E Cof(P ) T ξ .
(2.14)

Applying (2.14) to the equality (2.30), and using that σ ε is symmetric, we get that for any h ∈ O,

E (S ε h) = -σ ε E (R ε h)σ ε = -σ T ε E (R ε h)σ ε = E -Cof(σ ε ) T R ε h = E -Cof(σ ε )R ε h , (2.15)
which shows the first part of (2.9) due to the invertibility of E . The proof for the homogenized quantities in (2.9) and (2.11) is quite similar.

The general case

We now give an analogous in dimension three of Theorem 2.2 of [START_REF] Briane | Homogenization of the magneto-resistance in dimension two[END_REF] in order to give weak convergences of the effective Hall matrix and the magneto-resistance. All the subsequences parametrized by h converge up to a subsequence of ε. Due to (2.2), the linearity or the quadratic dependence in h, the convergences hold for any h. From now on, we consider a subsequence still denoted by ε such that all the sequences converge as ε tends to 0 and for any h in O.

First of all, we need to introduce a corrector P ε (h) (or electric field) in the sense of Murat-Tartar (see [START_REF] Murat | H-convergence[END_REF]), which is the gradient of a vector-valued U ε (h) associated with the unperturbed conductivity σ ε in M(α, β; Ω). To this end consider the solution

U ε (h) in H 1 (Ω) 3 of the problem Div σ ε (h)DU ε (h) = Div σ * (h) in D ′ (Ω), U ε (h)(x) -x = 0 on ∂Ω. (2.16)
Thanks to H-convergence and the Meyers estimate of [START_REF] Meyers | An L p -estimate for the gradient of solutions of second-order elliptic divergence equations[END_REF], there exists a number p > 2 which only depends on α, β, Ω, such that the corrector P ε (h) := DU ε (h) satisfies, for any h ∈ O,

P ε (h) ⇀ I 3 weakly in L p (Ω) 3×3 .
(2.17)

The knowledge of such a corrector combined with the div-curl lemma (see [START_REF] Murat | Compacité par compensation[END_REF] and [START_REF] Tartar | Compensated compactness and applications to partial differential equations[END_REF]) permits to derive the effective perturbed effective conductivity by the following convergence

σ ε (h)P ε (h) ⇀ σ * (h) weakly in L p (Ω) 3×3 . (2.18)
By the regularity condition (2.2), the coercivity of σ ε (h) and the Meyers estimate [START_REF] Meyers | An L p -estimate for the gradient of solutions of second-order elliptic divergence equations[END_REF], the potential U ε (h) and the corrector P ε (h) admit the following second-order expansions in h:

U ε (h) = U 0 ε + U 1 ε (h) + U 2 ε (h, h) + o W 1,p (Ω) 3 (|h| 2 ), (2.19)
P ε (h) = P 0 ε + P 1 ε (h) + P 2 ε (h, h) + o L p (Ω) 3×3 (|h| 2 ). (2.20)
We can now state the following result:

Theorem 2.1. Assume that the conditions (2.1)-(2.8) are satisfied, and that the norms of the Hall matrix R ε and the local magneto-resistance tensor M ε are bounded in L ∞ (Ω). Then, the effective Hall matrix R * , the effective S-matrix S * and the effective magneto-resistance are given by the following limits, for any h ∈ O,

S * = lim w-L 1 (Ω) Cof(P 0 ε ) T S ε , Cof(σ * ) R * = lim w-L 1 (Ω) Cof(σ ε P 0 ε ) T R ε , (2.21)
and

σ * M * (h, h) σ * = lim w-L 1 (Ω) σ ε P 0 ε T M ε (h, h) σ ε P 0 ε -E (S * h) T σ -1 * E (S * h) + lim w-L 1 (Ω) E (S ε h)P 0 ε + σ ε P 1 ε (h) T σ -1 ε E (S ε h)P 0 ε + σ ε P 1 ε (h) , (2.22)
where w -L 1 (Ω) means that the convergence holds weakly in L 1 (Ω) and P 0 ε , P 1 ε (h) are the matrixvalued gradient which satisfy (2.20).

Proof. The proof uses similar expansions as in [START_REF] Briane | Homogenization of the magneto-resistance in dimension two[END_REF] combined with algebraic specificities of dimension 3. Taking into account the expansions (2.6) and (2.20), we have:

σ ε (h)P ε (h) = σ ε P 0 ε + σ ε P 1 ε (h) + E (S ε h)P 0 ε + σ ε P 2 ε (h, h) + E (S ε h)P 1 ε (h) + N ε (h, h)P 0 ε + o L 2 (Ω) 3×3 (|h| 2 ). ( 2
       P 0 ε -⇀ I 3 weakly in L p (Ω) 3×3 , P 1 ε (h) -⇀ 0 weakly in L p (Ω) 3×3 , P 2 ε (h, h) -⇀ 0 weakly in L p (Ω) 3×3 , (2.24)
and Div

σ ε P 0 ε = Div σ * , Div σ ε P 1 ε (h) + E (S ε h)P 0 ε = Div E (S * h) in D ′ (Ω) 3×3 . (2.25)
Moreover, from the expansions (2.20), (2.23) and the symmetry of σ ε , we deduce that

P ε (h) T σ ε (h)P ε (h) = P 0 ε ) T σ ε P 0 ε + P 0 ε ) T E (S ε h)P 0 ε + P 0 ε ) T E (S ε h)P 1 ε (h) + N ε (h, h)P 0 ε + σ ε P 0 ε ) T P 2 ε (h, h) + σ ε P 0 ε ) T P 1 ε (h) + P 1 ε (h) T σ ε (h)P ε (h) + P 2 ε (h, h) T σ ε (h)P ε (h) + o L p/2 (Ω) 3×3 (|h| 2 ). (2.26)
Then, taking into account (2.17), (2.24), (2.25), the div-curl lemma implies that

P ε (h) T σ ε (h)P ε (h) converges to σ * (h) in L p/2 (Ω) 3×3 , and P 1 ε (h) T σ ε (h)P ε (h), P 2 ε (h, h) T σ ε (h)P ε (h), σ ε P 0 ε ) T P 2 ε (h, h), σ ε P 0 ε ) T P 1 ε (h) converges to 0 in L p/2 (Ω) 3×3 . Noting that P 0 ε T E (S ε h)P 0 ε = E Cof(P 0 ε ) T S ε h by (2.
14) and passing to the limit in (2.26), we obtain

σ * (h) = σ * + lim w-L 1 (Ω) E Cof(P 0 ε ) T S ε h + lim w-L 1 (Ω) P 0 ε T E (S ε h)P 1 ε (h) + P 0 ε T N ε (h, h)P 0 ε + o L 1 (Ω) 3×3 (|h| 2 ). (2.27)
Equating this expression with (2.7) it follows that

E S * h = lim w-L 1 (Ω) E Cof(P 0 ε ) T S ε h , (2.28)
and

N * (h, h) = lim w-L 1 (Ω) P 0 ε T E (S ε h)P 1 ε (h) + P 0 ε T N ε (h, h)P 0 ε . (2.29)
As E is an invertible linear mapping, we deduce from (2.28) and (2.9) the convergences (2.21).

We have, as E (S ε h) is antisymmetric and σ ε symmetric,

σ ε P 1 ε (h) + E (S ε h)P 0 ε T σ ε -1 σ ε P 1 ε (h) + E (S ε h)P 0 ε = -P 0 ε T E (S ε h) σ ε -1 E (S ε h)P 0 ε -P 0 ε T E (S ε h)P 1 ε (h) + P 1 ε (h) T σ ε P 1 ε (h) + E (S ε h)P 0 ε .
(2.30) Again, taking into account (2.17), (2.24), (2.25), the div-curl lemma implies that

P 1 ε (h) T σ ε P 1 ε (h) + E (S ε h)P 0 ε -⇀ 0 weakly in L p (Ω) 3×3 . (2.31)
Hence, (2.30) implies that

lim w-L 1 (Ω) σ ε P 1 ε (h) + E (S ε h)P 0 ε T σ -1 ε σ ε P 1 ε (h) + E (S ε h)P 0 ε = -lim w-L 1 (Ω) P 0 ε T E (S ε h)σ -1 ε E (S ε h)P 0 ε + P 0 ε T E (S ε h)P 1 ε (h) . ( 2
σ * M * (h, h) σ * -lim w-L 1 (Ω) σ ε P 0 ε T M ε (h, h) σ ε P 0 ε = lim w-L 1 (Ω) P 0 ε T N ε (h, h)P 0 ε -P 0 ε T E (S ε h)σ -1 ε E (S ε h)P 0 ε -N * (h, h) + E (S * h) σ -1 * E (S * h) = -lim w-L 1 (Ω) P 0 ε T E (S ε h)σ -1 ε E (S ε h)P 0 ε + P 0 ε T E (S ε h)P 1 ε (h) -E (S * h) T σ -1 * E (S * h) = lim w-L 1 (Ω) σ ε P 1 ε (h) + E (S ε h)P 0 ε T σ ε -1 σ ε P 1 ε (h) + E (S ε h)P 0 ε -E (S * h) T σ -1 * E (S * h).
In fact, the convergences (2.21) and (2.22) hold in L p/2 (Ω) 3×3 .

The periodic case

We now give a corollary of Theorem 2.1 for periodic media. To this end, set Y := (0, 1) 3 , and consider the εY -periodic conductivity

σ ε (h)(x) := σ(h) x ε , (2.33)
where σ(h) is a Y -periodic matrix-valued function. We assume (2.1) and analogous regularity conditions to (2.2): there exists an open ball O in R 3 centered at 0 such that

σ(h) ∈ M(α, β; Ω), ∀ h ∈ O, h → σ(h)(y) is of class C 2 on O, ∀ y ∈ Y. (2.34)
These conditions gives the expansions, like in (2.6) and (2.3)

σ(h) = σ + E (Sh) + N (h, h) + o L ∞ (Ω) 3×3 (|h| 2 ), (2.35) ρ(h) = σ(h) -1 = ρ + E (Rh) + M (h, h) + o L ∞ (Ω) 3×3 (|h| 2 ), ( 2
(h) = 0 in D ′ (R 3 ), y → U (h)(y) -y = 0 is Y -periodic.
(2.37) and P (h) := DU (h) with

P (h) = I 3 . (2.38)
We have the classical periodic homogenization formula (see, e.g., [START_REF] Murat | H-convergence[END_REF] for more details) σ * (h) = σ(h)DU (h) . By virtue of (2.34) we have expansions similar to (2.7), (2.8) and (2.20)

σ * (h) = σ * + E (S * h) + N * (h, h) + o(|h| 2 ), (2.39) ρ * (h) = σ * (h) -1 = ρ * + E (R * h) + M * (h, h) + o(|h| 2 ), (2.40)
and

P (h) = P 0 + P 1 (h) + P 2 (h, h) + o L 2 (Ω) 3×3 (|h| 2 ). (2.41)
We can state a corollary to Theorem 2.1:

Corollary 2.1. For a periodic conductor, the effective Hall matrix R * , the effective S-matrix S * and the effective magneto-resistance tensor M * are given by the following relations, for any h ∈ O,

S * = Cof(P 0 ) T S , Cof(σ * ) R * = Cof(σ ε P 0 ) T R , (2.42)
and

σ * M * (h, h) σ * = σP 0 T M (h, h) σP 0 -E (S * h) T σ -1 * E (S * h) + E (Sh)P 0 + σP 1 (h) T σ -1 E (Sh)P 0 + σP 1 (h) , (2.43)
where P 0 , P 1 (h) are the matrix-valued gradient which satisfy (2.41).

3 Comparison between the effective magneto-resistance and the local magneto-resistance.

The main result

We now give a generalization of the two-dimensional Theorem 2.4 of [START_REF] Briane | Homogenization of the magneto-resistance in dimension two[END_REF], and a corollary in the periodic case with the notations of Section 2.2.2. Theorem 3.1. Assume that the conditions (2.1)-(2.8) are satisfied, and that the norm of the local Hall matrix R ε and the norm of the local magneto-resistance tensor M ε are bounded in L ∞ (Ω). Then, for any h ∈ O, we have

σ * M * (h, h) σ * ≥ lim w-L 1 (Ω) σ ε P 0 ε T M ε (h, h) σ ε P 0 ε . (3.1)
Moreover, (3.1) is an equality if and only if

Curl E (R ε h)σ ε P 0 ε lies in a compact subset of H -1 (Ω) 3×3×3 . (3.2)
Corollary 3.1. In the periodic case, the constant effective magneto-resistance tensor M * and the constant effective conductivity σ * satisfy the inequality for any h ∈ O,

σ * M * (h, h) σ * ≥ σP 0 T M (h, h) σP 0 , with σ * = σP 0 , (3.3)
where σ(y) is the local conductivity and M (h, h)(y) is the local magneto-resistance. Moreover, (3.3) is an equality if and only if

Curl E (Rh) σ P 0 = 0 in D ′ (R 3 ) 3×3×3 . (3.4)
Remark 3.1. Then the inequality (3.3) can be written, for any

h ∈ R 3 , M * (h, h) j • j ≥ M (h, h)j • j , with j = σ * e , (3.5)
where e(y) = P 0 (y) e is the local electric field and j(y) = σ(y)e(y) is the local current field. Inequality (3.5) means that the dissipation energy in a composite is greater than or equal to the average of the dissipation energy in each of its phases.

The proof of Theorem 3.1 is based on the following result:

Lemma 3.1.
Let Ω be a bounded open subset of R d . Consider a sequence A ε of symmetric matrixvalued functions in M(α, β; Ω) which H-converges to A * , and a sequence ξ ε of L 2 (Ω) d which satisfies

ξ ε -⇀ ξ weakly in L 2 (Ω) d and div (ξ ε ) -→ div (ξ) strongly in H -1 (Ω). (3.6) Also assume that A -1 ε ξ ε • ξ ε -⇀ ζ weakly- * in M(Ω). (3.7)
Then, we have the inequality

ζ ≥ A -1 * ξ • ξ in M(Ω). (3.8)
Moreover, the inequality (3.8) is an equality if and only if

curl A -1 ε ξ ε lies in a compact subset of H -1 loc (Ω) d×d . (3.9)
Remark 3.2. Inequality (3.8) is a classical duality inequality in the periodic case (see [START_REF] Jikov | Homogenization of Differential Operators and Integral Functionals[END_REF] pp.160-200). However, up our knowledge the non-periodic case and the condition (3.9) of equality are less classical and deserve a proof.

Proof of Theorem 3.1. Taking into account the expansions (2.6) and (2.20), we have as in (2.23):

σ ε (h)P ε (h) = σ ε P 0 ε + σ ε P 1 ε (h) + E (S ε h)P 0 ε + o L p (Ω) 3×3 (h). ( 3
       P 0 ε -⇀ I 3 weakly in L p (Ω) 3×3 , P 1 ε (h) -⇀ 0 weakly in L p (Ω) 3×3 , σ ε P 1 ε (h) + E (S ε h)P 0 ε -⇀ E (S * h) weakly in L p (Ω) 3×3 . (3.11) Let λ ∈ R 3 . We apply Lemma 3.1 with A ε := σ ε , ξ ε := σ ε P 1 ε (h) + E (S ε h)P 0 ε λ, ξ := E (S * h)λ and ζ := lim w-L 1 (Ω) E (S ε h)P 0 ε + σ ε P 1 ε (h) T σ -1 ε E (S ε h)P 0 ε + σ ε P 1 ε (h) λ • λ. (3.12) It follows that for any λ ∈ R 3 , lim w-L 1 (Ω) E (S ε h)P 0 ε + σ ε P 1 ε (h) T σ -1 ε E (S ε h)P 0 ε + σ ε P 1 ε (h) λ • λ ≥ E (S * h) T σ -1 * E (S * h)λ • λ. (3.13)
Using the fact that P 1 ε (h) is a gradient and (2.12), (3.13) is an equality if and only if

Curl A -1 ε ξ ε = Curl σ -1 ε E (S ε h)P 0 ε λ = -Curl E (R ε h)σ ε P 0 ε λ (3.14)
lies in a compact subset of H -1 loc (Ω) 3×3 . Due to the arbitrariness of λ, this can be rewritten

lim w-L 1 (Ω) E (S ε h)P 0 ε + σ ε P 1 ε (h) T σ -1 ε E (S ε h)P 0 ε + σ ε P 1 ε (h) ≥ E (S * h) T σ -1 * E (S * h), (3.15)
which is an equality if and only if

Curl E (R ε h)σ ε P 0 ε lies in a compact subset of H -1 loc (Ω) 3×3×3 . (3.16)
We conclude to (3.1) by (2.22) and (3.15), and to (3.2) by (3.16).

Proof of Lemma 3.1. Proof of inequality (3.8): Let ϕ be a non-negative function in C 0 c (Ω). Let δ > 0, and for i = 1, . . . , k, let λ i ∈ R d and let ω i be balls in Ω such that

supp ϕ ⊂ k i=1 ω i and k i=1 ˆωi |A -1 * ξ -λ i | 2 dx ≤ δ. (3.17)
We consider a partition of unity (ψ i ) 1≤i≤k such that

∀ i = 1, . . . , k, ψ i ∈ C ∞ c (ω i ), 1 ≥ ψ i ≥ 0, k i=1 ψ i ≡ 1 in supp ϕ, (3.18)
and a sequence of functions ( ψ i ) 1≤i≤k such that

∀ i = 1, . . . , k, ψ i ∈ C ∞ c (ω i ), ψ i ≡ 1 in supp ψ i . (3.19)
For i = 1, . . . , k, let v i ε be the unique solution of the problem

div A ε ∇v i ε = div A * ∇( ψ i λ i • x) in D ′ (Ω) v i ε = 0 on ∂Ω. (3.20)
Thanks to the H-convergence of A ε (see Definition 1.1) we have the convergence

∀ i = 1, . . . , k, v i ε -⇀ v i = ψ i λ i • x weakly in H 1 0 (Ω), ∇v i ≡ λ i in supp ψ i . (3.21)
More generally, for any λ ∈ R d , we consider the unique solution v λ ε of the problem

div A ε ∇v ε = div A * λ in D ′ (Ω), v ε = λ • x on ∂Ω. (3.22)
Again, by the H-convergence of A ε , we have the convergences

v ε -⇀ λ • x weakly in H 1 0 (Ω), A ε ∇v ε -⇀ A * λ weakly in L 2 (Ω) d . (3.23)
We have by (3.18)

ˆΩ ζϕ dx = lim ε→0 ˆΩ A -1 ε ξ ε • ξ ε ϕ dx = k i=1 lim ε→0 ˆωi A -1 ε ξ ε • ξ ε ψ i ϕ dx. (3.24)
Combining this inequality with, for i = 1, . . . , k,

A -1 ε ξ ε • ξ ε -2ξ ε • ∇v i ε + A ε ∇v i ε • ∇v i ε = A -1 ε ξ ε -A ε ∇v i ε • ξ ε -A ε ∇v i ε ≥ 0, (3.25)
we obtain that

ˆΩ ζϕ dx ≥ k i=1 lim inf ε→0 ˆωi 2ξ ε • ∇v i ε -A ε ∇v i ε • ∇v i ε ψ i ϕ dx. (3.26)
By (3.20), (3.6) and (3.21), and by the classical div-curl lemma of [START_REF] Murat | Compacité par compensation[END_REF][START_REF] Murat | H-convergence[END_REF] we have

ξ ε • ∇v i ε -⇀ ξ • ∇v i = ξ • λ i and A ε ∇v i ε • ∇v i ε -⇀ A * ∇v i • ∇v i = A * λ i • λ i (3.27)
weakly- * in M(ω i ). This combined with (3.26) and (3.18) yields

ˆΩ ζϕ dx ≥ k i=1 ˆωi 2ξ • λ i -A * λ i • λ i ψ i ϕ dx = k i=1 ˆωi A -1 * ξ • ξ ψ i ϕ dx - k i=1 ˆωi A * A -1 * ξ -λ i • A -1 * ξ -λ i ψ i ϕ dx = ˆΩ A -1 * ξ • ξ ϕ dx - k i=1 ˆωi A * A -1 * ξ -λ i • A -1 * ξ -λ i ψ i ϕ dx. (3.28)
Moreover, by the Cauchy-Schwarz inequality and A * ∈ M(α, β; Ω), we have for i = 1, . . . , k,

ˆωi A * A -1 * ξ -λ i • A -1 * ξ -λ i ψ i ϕ dx ≤ β ϕ ∞ ˆωi |A -1 * ξ -λ i | 2 dx. (3.29)
Summing these inequalities on i together with (3.28) and (3.17), we finally get that

ˆΩ ζϕ dx ≥ ˆΩ A -1 * ξ • ξϕ dx -β δ ϕ ∞ . (3.30)
We conclude to (3.1) since δ is arbitrary.

Proof of the case of equality: Let us now prove that the equality in (3.8) implies (3.9). Consider a compact subset K of Ω, and a sequence

Φ ε ∈ H 1 0 (Ω) d×d such that Φ ε -⇀ 0 weakly in H 1 0 (Ω) d×d and supp Φ ε ⊂ K. (3.31)
By the definitions of the curl and the divergence, and by (3.31) we have

curl A -1 ε ξ ε , Φ ε H -1 (Ω) d×d ,H 1 0 (Ω) d×d = ˆK A -1 ε ξ ε • Div Φ ε -Φ T ε dx. (3.32)
Consider a partition of unity (ψ i ) 1≤i≤k such that

∀ i = 1, . . . , k, ψ i ∈ C ∞ c (ω i ), 0 ≤ ψ i ≤ 1, k i=1 ψ i ≡ 1 in K, (3.33)
the functions ψ i defined by (3.19), and the function v i ε by (3.20). We decompose the equality (3.32) in two parts curl

A -1 ε ξ ε , Φ ε H -1 (Ω) d×d ,H 1 0 (Ω) d×d = I ε + J ε , (3.34)
where

I ε := k i=1 ˆωi A -1 ε ξ ε -∇v i ε • Div Φ ε -Φ T ε ψ i dx, (3.35)
and

J ε := k i=1 ˆωi ∇v i ε • Div Φ ε -Φ T ε ψ i dx. (3.36)
On the one hand, by the Cauchy-Schwarz inequality we have

|I ε | 2 ≤ k i=1 ˆωi |A -1 ε ξ ε -∇v i ε | 2 ψ i dx k i=1 ˆωi Div Φ ε -Φ T ε 2 ψ i dx , (3.37)
that is

|I ε | 2 ≤ Div Φ ε -Φ T ε 2 L 2 (Ω) d k i=1 ˆωi |A -1 ε ξ ε -∇v i ε | 2 ψ i dx. ( 3
A -1 ε ξ ε -∇v i ε 2 ψ i dx = lim sup ε→0 k i=1 ˆωi A -1 ε ξ ε • ξ ε -2ξ ε • ∇v i ε + A ε ∇v i ε • ∇v i ε ψ i dx = k i=1 ˆωi A -1 * ξ • ξ -2ξ • ∇v i + A * ∇v i • ∇v i ψ i dx = k i=1 ˆωi A * A -1 * ξ -λ i • A -1 * ξ -λ i ψ i dx ≤ β k i=1 ˆωi |A -1 * ξ -λ i | 2 dx ≤ βδ. (3.39)
This combined with (3.31) and (3.38) implies that

|I ε | = O( √ δ). (3.40)
On the other hand, an integration by parts gives

J ε = k i=1 ˆωi ∇v i ε • Div Φ ε -Φ T ε ψ i dx = k i=1 ˆωi ∇v i ε • Div ψ i (Φ ε -Φ T ε ) dx - k i=1 ˆωi ∇v i ε • Φ T ε -Φ ε ∇ψ i dx. (3.41) Since ψ i (Φ ε -Φ T ε
) is an antisymmetric matrix, we have for any i = 1, . . . , k,

ˆωi ∇v i ε • Div ψ i (Φ ε -Φ T ε ) dx = - d k,l=1 ∂ 2 v i ε ∂x k ∂x l , ψ i (Φ ε -Φ T ε ) k,l C ∞ c (ω i ),D(ω i ) = 0, (3.42)
hence

J ε = k i=1 ˆωi ∇v i ε • Φ ε -Φ T ε ∇ψ i dx. (3.43)
The Cauchy-Schwarz inequality gives

|J ε | 2 ≤ Φ ε -Φ T ε 2 L 2 (Ω) d×d sup 1≤i≤k ∇ψ i 2 ∞ k i=1 ∇v i ε 2 L 2 (ω i ) . (3.44)
Then, since k and (ψ i ) 1≤i≤k are independent of ε, there exists C δ > 0 such that

|J ε | ≤ C δ Φ ε -Φ T ε L 2 (Ω) d×d . ( 3
curl A -1 ε ξ ε , Φ ε H -1 (Ω) d×d ,H 1 0 (Ω) d×d ≤ C δ Φ ε -Φ T ε L 2 (Ω) d×d + O( √ δ). (3.46)
Moreover, by the definition of Φ ε in (3.31) and Rellich's theorem, Φ ε -Φ T ε converges strongly to 0 in L 2 (Ω) d×d . Therefore, we get that for any δ > 0, lim sup

ε→0 curl A -1 ε ξ ε , Φ ε H -1 (Ω) d×d ,H 1 0 (Ω) d×d ≤ C √ δ, (3.47)
which implies (3.9) due to the arbitrariness of δ > 0 and (3.31).

Finally, let us prove that (3.9) implies the equality in (3.8). As |A -1 ε ξ ε | ≤ β|ξ ε | is a bounded sequence in L 2 (Ω), the following convergence holds up to a subsequence

η ε := A -1 ε ξ ε -⇀ η weakly in L 2 (Ω) d . (3.48)
By the div-curl lemma and (3.9), we have

η ε • ξ ε = A -1 ε ξ ε • ξ ε -⇀ η • ξ in D ′ (Ω), (3.49)
η ε • A ε ∇v λ ε -⇀ η • A * λ in D ′ (Ω). (3.50)
Moreover, since A ε is symmetric, we have

η ε • A ε ∇v λ ε = ξ ε • ∇v λ ε -⇀ ξ • λ in D ′ (Ω). ( 3

Higher-order terms

In this section, we try to extend the inequality (3.1) when the magneto-resistance is replaced by any term of even-order in the expansion of the perturbed resistivity. We first establish an inequality (opposite to (3.1)) satisfied by the fourth-order term of the resistivity assuming that the Hall matrix is zero. Then, we prove that the positivity is not conserved for even-orders greater than two.

For the sake of simplicity, we lighten the notation of Remark 1.1: for any functional space H, any integer k, any k-linear form g k : R 3 k → H and any h ∈ R 3 , the k th -order g k (h, . . . , h) is symply denoted g k (h). Let n ≥ 4 be an integer. Assume that the conductivity satisfies the regularity condition (2.2) for any multi-index |ν| ≤ n. As a consequence, the conductivity σ ε (h), the resistivity ρ ε (h) = σ ε (h) -1 and the associated homogenized quantities σ * (h), ρ * (h) satisfy the n th -order expansions in h

σ ε (h) = σ ε + • • • + σ n ε (h) + o L ∞ (Ω) 3×3 (|h| n ), ρ ε (h) = ρ ε + • • • + ρ n ε (h) + o L ∞ (Ω) 3×3 (|h| n ), σ * (h) = σ * + • • • + σ n * (h) + o L ∞ (Ω) 3×3 (|h| n ), ρ * (h) = ρ * + • • • + ρ n * (h) + o L ∞ (Ω) 3×3 (|h| n ), (3.54)
where for any h ∈ R 3 , the matrices σ k ε/ * (h), ρ k ε/ * (h) are symmetric for even k and antisymmetric for odd k. Note that with the notations of Section 2 we have

σ 1 ε/ * (h) = E (S ε/ * h), ρ 1 ε/ * (h) = E (R ε/ * h), σ 2 ε/ * (h) = N ε/ * (h, h), ρ 2 ε/ * (h) = M ε/ * (h, h). (3.55)
Similarly to (2. [START_REF] Omar | Elementary Solid State Physics: Principles and Applications[END_REF]) and (2.20), by the above regularity condition, the coercivity of σ ε (h) and the Meyers estimate [START_REF] Meyers | An L p -estimate for the gradient of solutions of second-order elliptic divergence equations[END_REF], the potential U ε (h) and the corrector P ε (h) admit the following n th -order expansions in h,

U ε (h) = U 0 ε + U 1 ε (h) + • • • + U n ε (h) + o W 1,p (Ω) 3 (|h| n ), (3.56)
P ε (h) = P 0 ε + P 1 ε (h) + • • • + P n ε (h) + o L p (Ω) 3×3 (|h| n ). ( 3
σ ε P 0 ε T ρ 4 ε (h) σ ε P 0 ε . ( 3
σ 1 * = ρ 1 ε = ρ 1 * = 0. (3.60)
Considering the expansion at the fourth-order of σ ε (h)ρ ε (h) = I 3 , we obtain similarly to the proof of Proposition 2.1, for any h ∈ O,

σ ε ρ 4 ε (h) + σ 1 ε (h) ρ 3 ε (h) + σ 2 ε (h) ρ 2 ε (h) + σ 3 ε (h) ρ 1 ε (h) + σ 4 ε (h) ρ ε = 0, (3.61)
which gives, by (3.60),

σ ε ρ 4 ε (h) σ ε = -σ 2 ε (h) ρ 2 ε (h) σ ε -σ 4 ε (h). ( 3
σ ε ρ 4 ε (h) σ ε = σ 2 ε (h) σ -1 ε σ 2 ε (h) -σ 4 ε (h). (3.64)
Similarly, we have for the homogenized quantities

σ * ρ 4 * (h) σ * = σ 2 * (h) σ -1 * σ 2 * (h) -σ 4 * (h). (3.65)
Taking into account the expansions (3.54) and (3.57), we have (writing only the second and the fourth-order terms) like in (2.23)

σ ε (h)P ε (h) = • • • + σ ε P 2 ε (h) + σ 1 ε P 1 ε (h) + σ 2 ε (h)P 0 ε + • • • + σ ε P 4 ε (h) + σ 1 ε (h)P 3 ε (h) + σ 2 ε (h)P 2 ε (h) + σ 3 ε (h)P 1 ε (h) + σ 4 ε (h)P 0 ε + o L p (Ω) 3×3 (|h| 4 ), (3.66)
that is by (3.60)

σ ε (h)P ε (h) = • • • + σ ε P 2 ε (h) + σ 2 ε (h)P 0 ε + • • • + σ ε P 4 ε (h) + σ 2 ε (h)P 2 ε (h) + σ 3 ε (h)P 1 ε (h) + σ 4 ε (h)P 0 ε + o L p (Ω) 3×3 (|h| 4
       P 4 ε (h) -⇀ 0 weakly in L p (Ω) 3×3 , σ ε P 2 ε (h) + σ 2 ε (h)P 0 ε -⇀ σ 2 * (h) weakly in L p (Ω) 3×3 , σ ε P 4 ε (h) + σ 2 ε (h)P 2 ε (h) + σ 3 ε (h)P 1 ε (h) + σ 4 ε (h)P 0 ε -⇀ σ 4 * (h) weakly in L p (Ω) 3×3 .
(3.68) and Div

σ ε P 2 ε (h) + σ 2 ε (h)P 0 ε = Div σ 2 * (h) , Div σ ε P 4 ε (h) + σ 2 ε (h)P 2 ε (h) + σ 3 ε (h)P 1 ε (h) + σ 4 ε (h)P 0 ε = Div σ 4 * (h) in D ′ (Ω) 3×3 . (3.69)
Moreover, from σ 1 ε = 0, (3.60) and (2.25) σ ε P 1 ε (h) is a divergence free function. Then the divcurl lemma implies that σ ε P 1 ε (h) T P 1 ε (h) converges to 0 in L p/2 (Ω) 3×3 . This combined with σ ε ∈ M(α, β; Ω), we get that

P 1 ε (h) -→ 0 strongly in L p (Ω) 3×3 . (3.70)
Taking into account (2.24) and (3.69), the div-curl lemma implies the convergence

P 0 ε T σ ε P 4 ε (h) + σ 2 ε (h)P 2 ε (h) + σ 3 ε (h)P 1 ε (h) + σ 4 ε (h)P 0 ε -⇀ σ 4 * (h) weakly in L p/2 (Ω) 3×3 . ( 3
.71) Moreover by (2.24), (3.68), (2.25) and the symmetry of σ ε the div-curl lemma yields

P 0 ε T σ ε P 4 ε (h) = σ ε P 0 ε T P 4 ε (h) -⇀ 0 weakly in L p (Ω) 3×3 . ( 3
σ 4 * (h) = lim w-L 1 (Ω) P 0 ε T σ 2 ε (h)P 2 ε (h) + P 0 ε T σ 4 ε (h)P 0 ε . (3.73)
Taking into account (2.17), and (3.69) the div-curl lemma implies that

P 2 ε (h) T σ ε P 2 ε (h) + σ 2 ε (h)P 0 ε -⇀ 0 weakly in L p/2 (Ω) 3×3 , (3.74) hence lim w-L 1 (Ω) σ ε P 2 ε (h) + σ 2 ε (h)P 0 ε T σ -1 ε σ ε P 2 ε (h) + σ 2 ε (h)P 0 ε = lim w-L 1 (Ω) P 0 ε T σ 2 ε (h)P 2 ε (h) + P 0 ε T σ 2 ε (h)σ -1 ε σ 2 ε (h)P 0 ε . ( 3
σ ε P 0 ε T ρ 4 ε (h) σ ε P 0 ε = σ 2 * (h) (σ * ) -1 σ 2 * (h) -σ 4 * (h) -lim w-L 1 (Ω) P 0 ε T σ 2 ε (h)σ -1 ε σ 2 ε (h)P 0 ε -P 0 ε T σ 4 ε (h)P 0 ε = σ 2 * (h) (σ * ) -1 σ 2 * (h) -lim w-L 1 (Ω) P 0 ε T σ 2 ε (h)σ -1 ε σ 2 ε (h)P 0 ε + P 0 ε T σ 2 ε (h)P 2 ε (h) = σ 2 * (h) (σ * ) -1 σ 2 * (h) -lim w-L 1 (Ω) σ ε P 2 ε (h) + σ 2 ε (h)P 0 ε T σ -1 ε σ ε P 2 ε (h) + σ 2 ε (h)P 0 ε . (3.76) Let λ ∈ R 3 . We apply Lemma 3.1 with A ε := σ ε , ξ ε := σ ε P 2 ε (h) + σ 2 ε (h)P 0
ε λ which has a compact divergence by (3.69) and converges weakly in L p (Ω) 3×3 to ξ := σ 2 * (h)λ by (3.68). Thus, with the notation of Lemma 3.1, we have

ζ = lim w-L 1 (Ω) σ 2 ε (h)P 0 ε + σ ε P 2 ε (h) T σ -1 ε σ 2 ε (h)P 0 ε + σ ε P 2 ε (h) λ • λ. (3.77) As σ 2 * (h) is symmetric, it follows that for any λ ∈ R 3 , lim w-L 1 (Ω) σ 2 ε (h)P 0 ε + σ ε P 2 ε (h) T σ -1 ε σ 2 ε (h)P 0 ε + σ ε P 2 ε (h) λ • λ ≥ σ 2 * (h)σ -1 * σ 2 * (h)λ • λ. (3.78)
Using the fact that P 2 ε (h) is a gradient and the equality (3.63), (3.78) is an equality if and only if curl

A -1 ε ξ ε = curl σ -1 ε σ 2 ε (h)P 0 ε λ = -curl ρ 2 ε (h)σ ε P 0 ε λ (3.79)
lies in a compact subset of H -1 (Ω) 3×3 . This concludes the proof due to the arbitrariness of λ.

An example with changes of sign

In this section we build a rank-one laminate which shows that the inequality (3.1) (or its inverse) cannot be extended to higher even-order terms. Let p ∈ N * . Define the perturbed conductivity

σ(h) := χ σ 1 (h) + (1 -χ) σ 2 (h), (3.80)
where χ is a characteristic function. For i = 1, 2, the conductivities σ i (h) belong to M (α, β; Ω) and the resistivities ρ i (h) = σ i (h) -1 satisfy the n th -order expansions in h

σ i (h) = σ i + • • • + σ 2p i (h) + o(|h| 2p ), ρ i (h) = ρ i + • • • + ρ 2p i (h) + o(|h| 2p ), (3.81)
where for any h ∈ R 3 , the matrices σ k i (h), ρ k i (h) are symmetric for even k and antisymmetric for odd k. We have the following result: Proposition 3.2. For any even integer p ≥ 2, there exists a rank-one laminate such that for any h the matrix

D (2p) (h) := σ * ρ (2p) * (h)σ * -σP 0 T ρ (2p) (h) σP 0 (3.82)
is neither non-positive, nor non-negative.

Proof. We consider the particular case of (3.80) where the magnetic field is h = h 3 e 3 , χ is a 1-periodic function only depending on x 1 , and

σ 1 (h) = θ -1 I 3 + θ -1 E (h), σ 2 (h) = α 2 I 3 + E (h), with θ := χ ∈ (0, 1), α 2 > 0. (3.83)
The laminate corrector P (h) is explicitly given by (see, e.g., [START_REF] Briane | Corrector for the homogenization of a laminate[END_REF])

P (h) = χ P 1 (h) + (1 -χ) P 2 (h), (3.84)
where for i = 1, 2,

P i (h) = I 3 + (1 -θ) (2-i) (-θ) i-1 1 -θ + θ 2 α 2   θ α 2 -1 (1 -θ) h 3 0 0 0 0 0 0 0   . (3.85)
The homogenized conductivity is defined by

σ * (h) = χ σ 1 (h)P 1 (h) + (1 -χ) σ 2 (h)P 2 (h) , (3.86)
which yields

σ * (h) =           α 2 1 -θ + θ 2 α 2 - 1 -θ + θ α 2 1 -θ + θ 2 α 2 h 3 0 1 -θ + θ α 2 1 -θ + θ 2 α 2 h 3 1 + (1 -θ) α 2 + (1 -θ) 3 1 -θ + θ 2 α 2 h 2 3 0 0 0 1 + (1 -θ) α 2           . (3.87)
Inverting this matrix, we obtain the homogenized resistivity

ρ * (h) =          b (1 -θ + θ 2 α 2 ) 2 + (1 -θ) 3 h 2 3 b α 2 + c h 2 3 1 -θ + θ α 2 b α 2 + c h 2 3 0 - 1 -θ + θ α 2 b α 2 + c h 2 3 α 2 b α 2 + c h 2 3 0 0 0 b -1         
, where b

:= 1 + (1 -θ), c := 1 -θ + α 2 . (3.88)
Expanding the quantities ρ i (h) = σ i (h) -1 , we obtain the expressions

ρ (2p) 1 (h) = (-1) p θ h 2p 3 K and ρ (2p) 2 (h) = (-1) p h (2p) 3 α 2p+1 2 K, where K = 1 0 0 0 1 0 0 0 0 . (3.89) Using (3.83)-(3.89)
, we can compute D (2p) (h). All the coefficients in the matrix D (2p) (h) are zero except the entries

             D (2p) (h) 1,1 = (-1) p (1 -θ + θ 2 α 2 ) 2 (1 -θ + θα 2 ) 2 (1 -θ + α 2 ) p-1 α p-1 2 1 + (1 -θ)α 2 p - 1 -θ α 2p-1 2 -(θ α 2 ) 2 h 2p 3 , D (2p) (h) 2,2 := (-1) p (1 -θ + α 2 ) p α p 2 1 + (1 -θ)α 2 p-1 -1 - 1 -θ α 2p-1 2 h 2p 3 .
(3.90)

As p ≥ 2, passing to the limit in (3.90) successively when θ → 0 and α 2 → ∞, we obtain that

lim θ→0 D (2p) (h) ∼ α 2 →∞ (-1) p h 2p 3     1 α p 2 0 0 0 -1 0 0 0 0     . (3.91)
Finally, when θ is small enough and α 2 large enough, the matrix D (2p) (h) has a positive eigenvalue, and a negative eigenvalue, which concludes the proof.

Remark 3.3. The case p = 1 confirms Theorem 3.1. Indeed, we have

D (2) (h) = (1 + (1 -θ)α 2 ) -1 (1 -θ + θ 2 α 2 ) 2 1 -θ α 2 + θ θα 2 2 (1 -θ)α 2 + θ θ -1 -(1 -θ + θα 2 ) 2 h 2 3 (e 1 ⊗e 1 ),
which is positive by the Cauchy-Schwarz inequality. This formula is a particular case of formula (4.6) below.

4 Case of equality for a few periodic structures

In this section we consider various periodic microstructures in the case of equality for (3.3). On the one hand, Section 4.1 provides an explicit expression of the difference between the two terms of (3.3) for layered structures and thus the different cases of equality. On the other hand, Section 4.2 only provides the cases of equality for columnar structures: condition (3.4) would have consequences on the Hall matrix and the conductivity of the microstructure. We use the notations of Section 2.2.2.

More precisely, we study the consequences of D(h, h) = 0 in (1.10). For a given averaged-value λ ∈ R 3 of the electric field e = P 0 λ in a composite conductor, we have the relations for the local current and the averaged-value of the current (see Remark 3.1)

j = σe = σP 0 λ, j = σ * e = σ * λ. (4.1)
We set

D(h, h) := σ * M * (h, h) σ * -σP 0 T M (h, h) σP 0 , (4.2)
so that, by the symmetry of σ * , it follows that

D(h, h) = D(h, h)λ • λ. (4.3)

Periodic layered structures

In this section, we establish for a periodic layered structure depending on a direction ξ ∈ R 3 , |ξ| = 1, an exact formula for the difference between the effective magneto-resistance and the averaged local magneto-resistance.

Let σ(h) be a perturbed conductivity in M(α, β; Ω) only depending on ξ • y, and satisfying the expansion

σ(h)(y) = a(ξ • y) I 3 + s(ξ • y) E (h) + N (h, h)(ξ • y) + o(|h| 2 ), for a.e. y ∈ Y, (4.4)
where a : R → [α, β] and s : R → R are 1-periodic functions. By Proposition 2.1, we have

R = r I 3 = - s a 2 I 3 . (4.5)
Considering the expansions (2.39)-(2.41), we can state a result precising Corollary 3.1:

Proposition 4.1. Consider a conductivity σ(h) satisfying (4.4) and the matrix-valued function D(h, h) defined by (4.2).

• When h is not parallel to ξ, we have

O(h) T D(h, h) O(h) =     d 1 |h × ξ| 2 d 3 (h • ξ)|h × ξ| 0 d 3 (h • ξ)|h × ξ| d 2 (h • ξ) 2 0 0 0 d 2 (h • ξ) 2     , (4.6)
where O(h) is the change-of-basis matrix from the canonical basis to

B = ξ, ξ × (ξ × h) |h × ξ| , h × ξ |h × ξ| , and
         d 1 := a -1 -2 a r 2 -a -1 a r 2 , d 2 := a 3 r 2 -a -1 a 2 r 2 , d 3 := a -1 -1 a 2 r 2 -a -1 a r a 2 r . (4.7)
• When h is parallel to ξ, we have

O T D(h, h) O = d 2 (h • ξ) 2   0 0 0 0 1 0 0 0 1   , (4.8)
where O is the change-of-basis matrix from the canonical basis to an orthonormal basis B = (ξ, u, v), for suitable u, v ∈ R 3 .

Moreover, D(h, h) = 0 if and only if one of the following conditions holds:

• h = 0;

• h = 0 is orthogonal to ξ, and r is a constant;

• h = 0 is parallel to ξ, and ar is a constant;

• h = 0 is neither parallel to ξ, nor orthogonal to ξ, and ar, r are constant.

Proof. t

First case: h = 0 is not parallel to ξ. By Corollary 2.1 we have

D(h, h) = E (Sh)P 0 + σP 1 (h) T σ -1 E (Sh)P 0 + σP 1 (h) -E (S * T σ -1 * E (S * h). (4.9)
For the sake of simplicity denote O := O(h). Denoting by • the quantities with respect to the new basis B, we have the following change-of-basis formulas respectively for the system of coordinates, the local conductivity, the zero and first-order terms in the expansion of the corrector and the local S-matrix defined by (2.35):

             ŷ = O T y, ĥ = O T h, σ ĥ = O T σ(h) O, σ = O T σ O, Ŝ = O T S O
(as a consequence of (2.14)),

P 0 = O T P 0 O, P 1 ĥ = O T P 1 (h) O, (4.10)
where the last equality is a consequence of the relation

∀ λ ∈ R 3 , P ĥ λ = O T P (h)λ = O T P (h) O O T λ = O T P (h) O λ, with λ = O T λ. (4.11)
Due to Lemma 38 of [START_REF] Tartar | An introduction to the homogenization method in optimal design[END_REF] for the homogenized conductivity and to (2.14) for the homogenized Smatrix defined by (2.39), we have

σ * ĥ = O T σ * (h) O, σ * = O T σ * O and Ŝ * = O T S * O. (4.12)
From these relations and (4.9) it is easy to check that the difference term D(h, h) defined by (4.2) satisfies the relation D ĥ, ĥ = O T D(h, h) O, (

where D ĥ, ĥ := E Ŝ ĥ P 0 + σ P 1 ĥ) T σ-1 E Ŝ ĥ P 0 + σ P 1 ĥ -E Ŝ * ĥ) T σ-1 * E Ŝ * ĥ . (4.14)

Let us now compute D ĥ, ĥ in (4.14). We have

h = (h • ξ, -|h × ξ|, 0) T = ( ĥ1 , ĥ2 , 0) T . (4.15)
By isotropy, we have σ = a I 3 and Ŝ = s I 3 . By the uniqueness of the solution of problem (2.37)-(2.38), we have

P 0 =     a -1 -1 a 0 0 0 1 0 0 0 1     and P 1 ĥ =      0 0 - s -a -1 -1 s a a ĥ2 0 0 0 0 0 0      . (4.16)
Hence, combining the equality (4.16) with the classical periodic homogenization formula (see, e.g., [START_REF] Murat | H-convergence[END_REF] for more details)

σ * = σ P 0 =   a -1 -1 0 0 0 a 0 0 0 a   . (4.17)
Moreover, we have E Ŝ ĥ P 0 + σ P 1 ĥ = s E ĥ P 0 + a P 1 ĥ =

       0 0 σ -1 -1 s a ĥ2 0 0 -s ĥ1 -σ -1 -1 s a ĥ2 s ĥ1 0        . (4.18)
Also by Corollary 2.1, we obtain that

Ŝ * = Cof P 0 Ŝ =     s 0 0 0 a -1 -1 s a 0 0 0 a -1 -1 s a     , (4.19)
and

E Ŝ * ĥ =        0 0 a -1 -1 s a ĥ2 0 0 -s ĥ1 -a -1 -1 s a ĥ2 s ĥ1 0        . ( 4.20)
Putting (4.18)-(4.20) in (4.14), we get that D ĥ, ĥ =

     d 1 ĥ2 2 -d 3 ĥ1 ĥ2 0 -d 3 ĥ1 ĥ2 d 2 ĥ2 1 0 0 0 d 2 ĥ2 1      , (4.21)
where

                   d 1 := a -1 -2 s 2 a 3 -a -1 s a 2 , d 2 := s 2 a -a -1 s 2 , d 3 := a -1 -1 s 2 a 2 -a -1 s a s . (4.22)
We deduce (4.6) from (4.5) and (4.15). Second case: h = 0 is parallel to ξ. Then, we have P 1 h = 0, and the computations are quite similar.

Cases of equality. When h = 0 is not parallel to ξ but orthogonal to ξ, by (4.6) D(h, h) = 0 implies that d 1 = 0. Thus, the equality a r 2 = a -1 a r In this section, we consider columnar isotropic structures in the direction y 3 . More precisely, the Y -periodic conductivity σ(h) of (2.35) only depends on

y ′ = (y 1 , y 2 ) with        σ(0) := σ(y ′ ) I 3 , σ ∈ L ∞ ♯ (0, 1) 2 ; [α, β] , S := s(y ′ ) I 3 , s ∈ L ∞ ♯ (0, 1) 2 ; R , N (h, h) := N (h, h)(y ′ ), N (h, h) ∈ L ∞ ♯ (0, 1) 2 ; R 3×3 s . (4.24)
Consequently, by Proposition 2.1 the expansion (2.36) of ρ(h) satisfies

       ρ = σ(y ′ ) -1 I 3 , R = r(y ′ ) I 3 , with r = -σ -2 s, M (h, h) := M (h, h)(y ′ ), M (h, h) ∈ L ∞ ♯ (0, 1) 2 ; R 3×3 s . (4.25)
We have the following result: • h ′ = 0, h 3 = 0, and the Hall coefficient r is constant;

• h ′ = 0, h 3 = 0 and there exist two positive functions f , g in L ∞ (R), with f -1 , g -1 in L ∞ (R), which are h i -periodic for i = 1, 2, and a constant C such that

σ(y ′ ) = f (h ′ • y ′ ) g(Jh ′ • y ′ ) and r(y ′ ) = C f (h ′ • y ′ )
a.e. y ′ ∈ (0, 1) 2 , (4.27)

where J := 0 -1 1 0 ;

• h ′ = 0, h 3 = 0, the Hall coefficient r is constant, and there exists a function

g in L ∞ (R) with g -1 in L ∞ (R) which is h i -periodic for i = 1, 2, such that σ(y ′ ) = g(Jh ′ • y ′ ) a.e. y ′ ∈ (0, 1) 2 . ( 4

.28)

Moreover, when h 1 h 2 = 0 and h 1 /h 2 / ∈ Q, σ and r are constant.

Remark 4.1. The case (h 1 , h 2 ) = 0 corresponds to the two-dimensional case in [START_REF] Briane | Homogenization of the magneto-resistance in dimension two[END_REF] (Theorem 2.4).

In the case (h 1 , h 2 ) = (0, 0), h 3 = 0, f and g are not unique. For example f and g can be chosen such that f -1 = 1 to ensure the uniqueness.

Proof of Proposition 4.2. We work in the orthonormal basis (f 1 , f 2 , e 3 ) defined by

(f 1 , f 2 ) :=      h ′ |h ′ | , Jh ′ |h ′ | if h ′ = (0, 0), (e 1 , e 2 ) if h ′ = (0, 0). (4.29)
In the new basis, we have h = |h ′ |f 1 + h 3 e 3 . The associated system of coordinates is given by

             z 1 := h 1 y 1 + h 2 y 2 |h ′ | , z 2 := h 1 y 2 -h 2 y 1 |h ′ | , z 3 := y 3 , if h ′ = 0, and z = y, if h ′ = 0. (4.30)
We denote for i = 1, 2, P 0 f i := ∇u i , P 0 e 3 := ∇u 3 and for i = 1, 2, 3, v i (z) = u i (y).

Since the gradient, the divergence and the curl are invariant by a change of orthonormal righthanded basis, by (3.4) we have for any i = 1, 2, 3,

0 = curl (σr)E (h)∇u i =     |h ′ | ∂ z 2 (σr) ∂ z 2 v i -∂ z 3 (σr) (h 3 ∂ z 1 v i -|h ′ |∂ z 3 v i ) -h 3 ∂ z 3 (σr) ∂ z 2 v i -|h ′ | ∂ z 1 (σr) ∂ z 2 v i ∂ z 1 (σr) (h 3 ∂ z 1 v i -|h ′ |∂ z 3 v i ) + h 3 ∂ z 2 (σr) ∂ z 2 v i     . ( 4

.31)

As v i , σ, r are independent of z 3 , (4.31) reads as

    |h ′ | ∂ z 2 (σr) ∂ z 2 v i |h ′ | ∂ z 1 (σr) ∂ z 2 v i h 3 ∂ z 1 (σr) ∂ z 1 v i + h 3 ∂ z 2 (σr) ∂ z 2 v i     = 0, for i = 1, 2. (4.32)
First case: h ′ = 0 and h 3 = 0. We are led to the two-dimensional case of [START_REF] Briane | Homogenization of the magneto-resistance in dimension two[END_REF] with h = h 3 e 3 . The key ingredient is the positivity of the determinant of the corrector P 0 due to Alessandrini and Nesi [START_REF] Alessandrini | Univalent σ-harmonic mappings[END_REF].

Second case: h ′ = 0 and h 3 = 0. Without loss of generality, we can assume that |h ′ | = 1. The two first equalities of (4.32) give the existence of a constant C such that hence, C = 0, which implies that v 1 is a function of z 1 . On the other hand, the Alessandrini, Nesi [START_REF] Alessandrini | Univalent σ-harmonic mappings[END_REF] result combined with v 1 = v 1 (z 1 ) yields

(σr) ∂ z 2 v 1 = C. (4.33) Since ∇u 1 = ∂ z 1 v 1 f 1 + ∂ z 2 v 1 f 2 + ∂ z 3 v 1 e 3 , we have by (2.38) ∂ z 2 v 1 = ∇u 1 • f 2 = f 1 • f 2 = 0. ( 4
det(P 0 ) = ∂ z 1 v 1 ∂ z 2 v 2 -∂ z 1 v 2 ∂ z 2 v 1 =0 = ∂ z 1 v 1 ∂ z 2 v 2 > 0 a.e. in Y. (4.36)
Moreover, by (2.37) we have

0 = div σ∇v 1 = ∂ z 1 σ ∂ z 1 v 1 in D ′ (R 3 ), (4.37)
which implies that σ ∂ z 1 v 1 is a function of z 2 . By (4.36), we may define the two measurable functions f , g by f (z 1 ) :

= ∂ z 1 v 1 -1 and g(z 2 ) := σ ∂ z 1 v 1 . (4.38)
Therefore, we get that σ(y ′ ) = f (z 1 ) g(z 2 ). In particular, f , g are h i -periodic for i = 1, 2. Let us show that f , g, f -1 , g -1 are bounded functions. Denote δ := max(|h 1 |, |h 2 |) > 0. As α ≤ σ ≤ β, we have by (4.38),

β 2 f -2 (z 1 ) ≥ σ 2 f -2 (z 1 ) = g 2 (z 2 ) ≥ α 2 f -2 (z 1 ) > 0 a.e. in (z 1 , z 2 ) ∈ (0, δ) 2 . (4.39)
Integrating (4.39) successively with respect to z 1 and z 2 on (0, δ), we get that

α -2 C 1 ≥ f -2 (z 1 ) ≥ β -2 C 1 , β 2 α -2 C 1 ≥ g 2 (z 2 ) ≥ α 2 β -2 C 1 , with C 1 := δ 0 g 2 (z 2 ) dz 2 > 0 (4.40) that is f, g, f -1 , g -1 are L ∞ (R) functions. Integrating the inequality σ(y ′ ) = f (z 1 ) g(z 2
) ≥ α with respect to z 2 on (0, δ), we obtain that

f (z 1 ) ˆδ 0 g(z 2 ) dz 2 ≥ δ α > 0 a.e. z 1 ∈ R, (4.41)
that is f has a constant sign. Moreover, like in (4.34) we have

f -1 = ∂ z 1 v 1 = ∇u 1 • f 1 = f 1 • f 1 = 1. (4.42)
Hence f is a positive function, so is g by (4.38). Then, by a uniqueness argument the expression of σ implies that the potentials v i , i = 1, 2, 3, are given by

       ∂ z 1 v 1 = f -1 (z 1 ), ∂ z 2 v 1 = ∂ z 3 v 1 = 0, ∂ z 2 v 2 = g -1 -1 g -1 (z 2 ), ∂ z 1 v 2 = ∂ z 3 v 2 = 0, v 3 = z 3 .
(4.43)

The conditions (4.32) and (4.36) give the existence of a constant C such that Conversely, if the conductivity and the Hall coefficient satisfy (4.27) with f -1 = 1 (see Remark 4.1), the potentials v i , for i = 1, 2, 3, are given by (4.43). Hence, it follows immediately (4.31) and thus the case of equality.

r(y ′ ) = C σ ∂ z 2 v 2 = C g -1 f (z 1 ) . ( 4
Third case: h ′ = 0 and h 3 = 0. Considering the third equality of (4.32), the first case shows that r is constant in Y . Moreover, taking into account the first and second components of (4.32), σ, r takes the form (4.27) by the second case. Hence, f is constant which gives (4.28). The converse is similar to the second case.

Case where h 1 h 2 = 0 and h 1 /h 2 / ∈ Q. As h 1 , h 2 = 0, we are in the second or third case of the proof. Let i ∈ {1, 2}. We have proved that u i (y) = v i (z) is a function of z i . Moreover, ϕ : y → u i (y) -f i • y = v i (z) -z i is a function in H 1 ♯ (Y ; R). The function ϕ has a continuous representative and is h j -periodic for j = 1, 2. As h 1 /h 2 / ∈ Q, ϕ is constant so is ∇v i . Therefore, by (4.43), f, g are also constant. Finally by (4.27), σ and r are constant.

Four-phase checkerboard

In the section, we consider a four-phase checkerboard columnar structure. Let α 1 , α 2 , α 3 , α 4 be positive numbers. Consider the Y -periodic conductivity only depending on y ′ = (y 1 , y 2 ), defined on the unit square (-1/2, 1/2) 2 by (see figure 4 • h = 0 is not parallel to e i for i = 1, 2, 3, and σ, r are constant.

• h = h 3 e 3 = 0, and the Hall coefficient r is constant;

• h = h i e i = 0 for i = 1, 2, and there exists a constant C such that α 1 α 3 = α 2 α 4 , and r = C α 6-2i α 1 1 {y i >0} + 1 {y i <0} . (4.46)

Remark 4.2. The case equality α 1 α 3 = α 2 α 4 corresponds to the case where the conductivity of the four-phase checkerboard is a tensor product of functions (see [START_REF] Marino | Un tipo di approssimazione dell'operatore n i,j=1 d i (a ij (x)d j ) con operatori n j=1 d j (β(x)d j )[END_REF]). When α 1 α 3 = α 2 α 4 , Craster and Obnosov [START_REF] Craster | On effective resistivity and related parameters for periodic checkerboard composites[END_REF][START_REF] Craster | Four-phase checkerboard composites[END_REF] proved an intricate formula for the corrector P 0 . In this case, σ is not a tensor product of functions which is consistent with Proposition 4.2.

Proof of Proposition 4.3. The case (h 1 , h 2 ) = (0, 0) and h 3 = 0 is a direct consequence of Proposition 4.2. Set, like in Proposition 4.2, h ′ = (h 1 , h 2 ).

First case: h is not parallel to e i for i = 1, 2, 3. Assume that, without loss of generality, |h ′ | = 1, h i > 0 for i = 1, 2. We apply Proposition 4.2. There exist two positive functions f and g in L ∞ (R) which are h i -periodic for i = 1, 2, and a constant C such that (4.27) holds. Since σ(y ′ ) = f (z 1 ) g(z 2 ) (with the new variables (4.30)) is a piecewise constant function, f (z 1 ) and g(z 2 ) are constant in each open square Q i for i = 1, 2, 3, 4. Considering the particular case of Q 1 and, for δ small enough, the rectangle

Q 1,δ := y ′ ∈ Q 1 : z 1 ∈ h 1 h 2 δ, h 1 + h 2 2 , z 2 ∈ (-δ, δ) ⊂ Q 1 , (4.47)
we get successively that          f is constant on I := 0,

h 1 + h 2 2 , g is constant on J := - h 1 2 , h 2 2 .
(4.48)

Hence, σ is constant in the rectangle (see figure 4.2)

Q := y ′ ∈ - 1 2 , 1 2 : (z 1 , z 2 ) ∈ I × J . (4.49) h ′ Q 1 Q 2 Q 3 Q 4 y 2 y 1 z 1 z 2 Q Q 1,δ Q Figure 4
.2: Cross-section of the period cell of the checkerboard in the (y 1 , y 2 )-plane

As Q intersects Q 1 , Q 2 , Q 4 , σ is constant in Q 1 ∪ Q 2 ∪ Q 4 , hence α 1 = α 2 = α 4 .
Repeating the same argument with Q 3 in (4.48), we obtain that α 1 = α 2 = α 3 = α 4 . Therefore, σ, f , g are constant, so is r by (4.27).

Second case: h = h 1 e 1 = 0. Without loss of generality, assume that h 1 = 1. We again apply Proposition 4.2. There exist two positive functions f and g in L ∞ (R) which are h i -periodic for i = 1, 2, and a constant C ′ such that σ(y ′ ) = f (y 1 ) g(y 2 ), and r(y

′ ) = C ′ f (y 1 )
a.e. y ′ ∈ (-1/2, 1/2)

  Notations • | • | denotes the euclidean norm in R d for any positive integer d and (e 1 , . . . , e d ) the canonic basis of R d .

  .45) Combining (3.40) and (3.45) with (3.34), we have, for any δ > 0,

  .51) From (3.50) and (3.51), we deduce that for any λ ∈ R d ,η • A * λ = A * η • ξ = ξ • λ a.e.in Ω, (3.52) which implies that η = A -1 * ξ a.e. in Ω. (3.53) We conclude to the equality in (3.8) combining (3.53) with (3.49).

  .72) Hence, combining (3.70) and (3.72) in (3.71) we obtain

  .75) Combining the equalities (3.64), (3.65) with (3.73), (3.75) and the symmetry of σ ε and σ 2 ε (h), we obtain that σ * ρ 4 * (h) σ * -lim w-L 1 (Ω)

Figure 3 . 1 :

 31 Figure 3.1: A three-dimensional rank-one laminate

Proposition 4 . 2 .

 42 Consider a conductivity σ(h) satisfying (4.24) and set h ′ = (h 1 , h 2 ). Assume that r -1 ∈ L 1 (Y ) and (σ r) -1 = 0.(4.26)Then, D(h, h) = 0 (see (4.2)) is an equality if and only if one of the following conditions holds• h = 0;

  .34) By (4.26), since 0 < α ≤ σ ≤ β and r -1 ∈ L 1 (Y ), (σr) -1 = 0 almost everywhere in R 2 . Combining (4.34) with (4.33), we get that C (σr) -1 = 0, (4.35)

  .44) Using the expressions (4.30) and |h ′ | = 1, we obtain (4.27).

α 1

 1 in Q 1 := (0, 1/2) 2 , α 2 in Q 2 := (0, 1/2) × (-1/2, 0), α 3 in Q 3 := (-1/2, 0) 2 , α 4 in Q 4 := (-1/2, 0) × (0, 1/2).

Figure 4 . 1 :Proposition 4 . 3 .

 4143 Figure 4.1: Period cell of the four-phase checkerboard columnar structure

  .57)3.2.1 Fourth-order term with zero Hall matrixWe can now state the following result for the fourth-order term: Proposition 3.1. Assume that (2.1) and (2.2) for |ν| ≤ 4 are satisfied and that the norms of σ 2 ε , σ 3 ε and σ 4 ε are bounded in L ∞ (Ω). Then, in the absence of Hall effect (i.e. ρ 1 ε = 0), we have, for any h ∈ O, σ

* ρ 4 * (h) σ * ≤ lim w-L 1 (Ω)

  2 (4.23) can be regarded as the case of equality in the Cauchy-Schwarz inequality satisfied by the functions √ a and √ a r in L 2 [0, 1] . Therefore, √ a is proportional to √ a r, hence r is constant. The converse is immediate. The other cases are similar.

	4.2 Periodic columnar structures
	4.2.1 The general case

  2 . (4.50) Since σ is piecewise constant in the four-phase checkerboard and f , g are respectively functions of the independent variables y 1 , y 2 , f , g are constant in each open square Q i , for i = 1, 2, 3, 4. It follows immediately that there exist two positive constantsC ′ 1 and C ′ 2 such that C ′ 1 C ′ 2 = α -1 1 and f (y 1 ) = C ′ 1 α 1 1 {y 1 >0} + α 4 1 {y 1 ≤0} , g(y 2 ) = C ′ 2 α 1 1 {y 2 >0} + α 2 1 {y 2 ≤0} , a.e. y ′ ∈ (-1/2, 1/2) 2 . (4.51) Finally, (4.51) and (4.50) imply that there exists a constant C such that α 1 α 3 = α 2 α 4 , and r(y ′ ) = C α 4 α 1 1 {y 1 >0} + 1 {y 1 <0} . (4.52)

Conversely, if (4.52) is satisfied, we can define f and g as in (4.51) with

1 . The case h = h 2 e 2 = 0 is quite similar.