Maxence Larrieu
email: maxence@larri.eu

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A Consideration of the Code of Computer Music as Writing, and Some Thinking on Analytical Theories

) and the answer of (Risset 2001). In the fourth section I will explain with Jean-Claude Risset and Horacio Vaggione the specificity of computer music; this music is written. Finally, I will look into a recent analysis theory, the Interactive Aural Analysis by Michael Clarke, which seems to fit with the latter specificity.

1 Therefore, computer music here is not restricted to a genre, it includes either 'live or tape' computer music, algorithmic or generative music, mixed and interactive music, and computer synthesis and treatment.

INTRODUCTION

My article deals with the analysis of computer music, and more precisely the one using at least one programming language, so that what the listener hears is the consequence of some computer codes 1 . This music is peculiar in that it does exist with a digital medium, which contains all the 'instructions' to be 'transformed into music'. Therefore, the medium of computer music has to be differentiated from the one of electroacoustic music. The former is made of 'code', which (i) contains some operations made by the composer and (ii) permits us to hear the work, whereas the latter is an audio signal, which allow us to hear the work. Considering this difference, one may wonder how this peculiar medium can be used to analyse music, which tools and theories can be helpful for the analyst ?

The question of code in musical analysis is not new. One pioneer is certainly Marco Stroppa, an author whom we will consider in detail later in this article. Another key author 2-17 is Jean-Claude Risset, who has shown that computer code should be considered as a key resource in computer music analysis:

It is only by integrating the musical and technical data that one can reach a structural description of the work which can make up for the lack of score. (Risset 2001: 157), my translation.

Over the last twenty years following the article of Risset, more analysis of computer music based on computer code has become available. For example let's consider [START_REF] Bonardi | Analyser l'orchestre électronique interactif dans les oeuvres de Manoury[END_REF] [START_REF] Bergsland | The six Fantasies Machine -an instrument modelling phrases from Paul Lansky's Six Fantasies[END_REF]) and [START_REF] Clarke | Wind Chimes and Interactive Aural Anlaysis[END_REF]). However, although there are some applications, I found there are not many theories which consider code, with all its richness, as Jean-Claude Risset pointed out. My article can be understood as a humble contribution to the analysis of computer music works based on their code. It is composed of the following four sections. In the first one I will present, with the help of a personal diagram, different theories for analysing electroacoustic music; in the second one I will shed light on the pioneering period where it was difficult to consider code of computer music to make analysis; in the third one, I will deal with the specificity of computer music; finally, in the last section I will discuss a recent analysis theory, the Interactive Aural Analysis by Michael Clarke, which seems to fit the specificity of computer music.

ANALYSING COMPUTER MUSIC

Since the middle of the twentieth century there have been a lot of theories to analyse electroacoustic music. In order to organise these theories, I suggest taking the following diagram as a starting point. musical scores drafts sketches working notes audio signal files codes scripts data …

As can be seen the diagram is based on the materials used in analysis. It allows the reader to clearly distinguish two general directions in analysis. The first one uses mainly the audio signal of the work, it is the key object. It should be mentioned here that audio signal has to be understood as something which permits us to hear the work: samples used in composition, or signal in technological meaning 2 are both excluded -here, audio signal is an object which allows us to hear the work. By contrast, 2 Indeed my definition go over the technological meaning: a signal without sound (e.g. very low/high frequency) is not considered here.

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the second direction in analysis can use a diversity of objects, and all of them are related to the composition and are situated 'before the signal'. One advantage of this organisation is that it is highly legible. For example, Pierre [START_REF] Couprie | Analyse Comparée des Trois rêves d'oiseau de François Bayle[END_REF])'s analysis of Trois rêves d'oiseau (1971) by François Bayle, is 'based on the audio signal', whereas John Chowning 's Stria (1977), analysed by Matteo (Menegheni 2007) is 'before the signal', since it used essentially sketches of Music N3 . This difference is also present in a recent work made by Pierre Couprie. After presenting a synthesis of the main theories that have been developed for 'the composition, the interpretation or the musical analysis', the author shows two methods of analysis. The first is 'based on the perception of the audio support' and the second uses 'the creation support which has been made available by the composer' (Couprie 2016: 160).

Lastly, I have to notice that this distinction is relevant on a theoretical level, so that one can organise theories without ambiguity. Indeed, even if the analysis is realised 'before the signal', it is obvious that at one point the analysts have to listen to the music. For example, in the famous analysis made the GRM 4 entitled L'envers d'une oeuvre : 'de natura sonorum' de Bernard Parmegiani, which is made 'before the signal', one can read 'Our first object of study was, chronologically, De Natura, it was the music' (Mion, Nattiez, and Thomas 1982 : 32), my translation.

Analysing 'from the signal'

The analysis from the signal fits electroacoustic music. Indeed, without a traditional score, the first object which comes into contact with the work is the sound of the work, the audio signal.

Analyses from the signal include (i) analyses where the signal is only read, i.e. the signal is used as a tool to proceed in the listening of the work, and (ii) analyses where the audio signal is examined by technological means. Consequently, this way of analysing music gathers a lot of practices and theories which are still in evolution. For example, the recent three year project New multimedia Tools for Electroacoustic Music Analysis considers analysis from the signal: 'Our research is primarily listener-focused -the composer's intentions, methods and approaches may influence but not define the experience of the music' (Emmerson and Landy 2012: 2).

Thinking musical analysis only in the act of perception has needed intellectual efforts. For example, one can think of Schaeffer's catchphrase 'music is made to be heard' (Schaeffer 1952: 117). One way to explain it is to focus on 'the embryogenesis' of electroacoustic music: musique concrète of Pierre Schaeffer. As we know, this music was born with the famous phenomenological concept of écoute réduite: sound has to be perceived without taking into account the causality ; it must be perceived only in the act of perception. This concept leads to a new field in musical analysis, which can be seen as the research of means to analysing music only in the act of listening, i.e. without searching for something from the creation process (the poietic). This situation was not self-evident, as François Delalande points out in the introduction of his recent collection [START_REF] Delalande | Analyser la musique, pourquoi, comment ?[END_REF]. After describing as a 'rich period' the moment when the 4-17 classical theories of musical analysis appeared -e.g. Generative Theory of Tonal Music [START_REF] Lerdahl | A generative theory of tonal music[END_REF], Schenkerian model [START_REF] Forte | Introduction to schenkenrian analysis[END_REF] and Paradigmatic analysis [START_REF] Ruwet | Langage, musique[END_REF]) -the author sees in the electroacoustic music 'a new paradigm [which] shows some disorder, if not revolution' (Delalande 2013: 7).

In this framework -no conventional notation ; with the concept of reduced listening ; and where listening is fundamental -the theoretical means of electroacoustic music analysis necessarily comes from the audio signal of the work :

The object to be analysed, that is to say a work or a given extract under its sound form, does have a certain materiality: it is a signal, fixed on a support, which becomes an acoustic wave. (Delalande 2013: 10), my translation.

My goal here was to show that those theories have something in common: all of them are based on audio signal. However, if we consider all the theories for analysing electroacoustic music from the signal, our latter diagram is no longer significant. Another step could be to clarify how the audio signal is used. For example, I can differentiate analysis where the signal is technically analysed (e.g. waveform, sonogram or sound descriptor), from the analysis where the signal is used just as a support for the listening. This is present in [START_REF] Marty | Introduction -Écoutes et analyses[END_REF], with a vertical dimension: at the bottom there is the sonogram and sound descriptors, and then, successively, the famous electroacoustic theories of typo-morphology [START_REF] Chion | Guide des Objets Sonores[END_REF]) and spectromorphology [START_REF] Smalley | Spectro-morphology and Structuring Processes[END_REF], and lastly music as a human experience, with the 'listening conduct' of François Delalande.

Analysing 'before the signal'

Even if there is no score as the musicologists knew from past centuries, it is possible to analyse electroacoustic works 'before the signal'. One historic example is the analysis made by the GRM (Mion, Thomas and Nattiez 1982). The documents used were an interview between the author and the composer, and secondly some sketches and listening scores made available by the composer at the time of the composition. In the following subtitles, I will give a preview of two theories for analysing 'before the signal', which are more related to computer music: genetic criticism applied to music and the faktura.

Genetic criticism applied to music

Genetic criticism applied to music is a main topic which is taking shape between philology and hermeneutics. Facing this density, my goal here is only to present the main idea, and then present the viewpoint of Laura Zattra. My point of view lies in a recent collective book, [START_REF] Donin | Introduction[END_REF] where genetic criticism is applied to music. Two key points can be found in the introduction. The first one is that genetic criticism is built with a main concept of writing, from which comes its 'love for the text', and the second one is that the theory is interested in the gestation of the artistic work. Of course, these two points are correlated: it is because of the real principle of writinglet's say that it freezes the time in which it takes place -that the genetic criticism can trace back to the gestation of the work :

In suspending the time, by offering some tools to organise and hierarchise the contents, by allowing the possibility to go back, the re-reading, the 5-17 repentance, the writing constitutes a cognitive technology particularly adapted to complex creation process. (Donin et al 2015 : 11), my translation.

So the genetic criticism applied to music is made with sketches coming from the composition. Of course traditionally those sketches were manuscripts, with musical notation, but nowadays, as we will see, it can also be computer code. Therefore, genetic criticism does not need to be restricted to traditional music. With a non trivial conceptual step, it can also be applied to computer music. However, possibly because of this step, the applications are not numerous5 ; one of the first authors is [START_REF] Zattra | Génétiques de la computer music[END_REF].

The conceptual step consists of seeing the act of writing within the computer, and in the same way conceiving the traces coming from the poietic process as 'text'.

The computer music works are generally realised in wide research and production centres, and show, after analysis, a predilection for the idea of 'crafts' which is realised through the writing, an action characterised by the presence of texts, in the wider sense of the term. [Les oeuvres sont en général réalisées dans de grands centre de recherche et production, et montrent, à l'analyse, une prédilection pour l'idée "d'artisanat" qui se réalise à travers l'écriture, une action caractérisée par la présence et l'élaboration de textes, au sens large du terme] (Zattra 2015: 213).

Indeed, to apply genetic criticism to computer music, the concept of text 'has to be widened ' (ibid. 220). Then the computer is seen as a 'source for the study of the process/translation between the musical universe of the composer and the digital one'. '[This process] produces some ''texts'' which schematise and operationalize the compositional choices'(ibid. 220). For the analyst, those 'texts' allow to shed light on the compositional choices, and to describe the 'construction process of the work'. (ibid. 220).

The faktura

In 2003, M. Battier developed a new theory for analysing electroacoustic music, with some ideas inspired from Constructivism, a Russian artistic movement at the beginning of the twentieth century. The theory is described in an article entitled A Constructivist approach to the analysis of electronic music and audio art -between instruments and faktura [START_REF] Battier | A Constructivist approach to the analysis of electronic music and audio artbetween instruments and faktura[END_REF]. It is a well-known article and most of the analysis based on the study of the sound production technologies make reference to it, e.g. [START_REF] Bonardi | Pérenniser pour transmettre, transmettre pour pérenniser -Destins de l'oeuvre mixte interactive -autour de en Echo, pièce de Philippe Manoury[END_REF], [START_REF] Dufeu | L'instrument numériqe comme objet d'analyse des musiques mixtes[END_REF] and (Baudoin 2007).

The faktura concept starts with a general idea: 'technology, artistic technique and aesthetic thoughts [are highly] intertwined' (Battier 2003 : 251). Constructivism is used to focus on the interactions between technology, technique and style, which took place during the creative process.

The faktura works with another concept, inherited from Constructivism, the tectonic. This one allows to distinguish the artist action from the materials that existed at the time of the creation.

The theoretical effort of the Constructivist was aimed at defining the relation between the act of creation and its materials. (ibid. 251)

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The definition of faktura relies on this distinction. It focuses on the moment when the artist -'either a sculptor, painter, poet, musician or audio artist' (ibid. 251) -acts on this collection of materials : 'Faktura is the category with which an artist transforms materials ' (ibid. 251). And of course this transformation is made with technology. The idea behind faktura is that once light has been brought on how the artist uses and transforms materials, the artist's style can then be described : 'in the Constructivist's approach, style becomes a topic which should be dealt with using a poietic method' (ibid. 254).

The music issues are tackled after dealing with the general definition of Constructivism. Battier's proposal is based on the sound-producing system and twowell known levels in musicology: the symbolic and signal ones -see [START_REF] Malloch | Towards a new conceptual framework for digital musical instruments[END_REF] for example. First, the sound-production system has to be deeply understood, and then the link between this system and (i) the composition (symbolic domain) and (ii) the 'actual realisation' (signal domain) have to be enlightened. (Battier 2003: 252).

Finally, it should be said that Battier's faktura is not dedicated to computer music. Indeed the concept aims to grasp all the sound technology of the twentieth century, i.e. sound recording, electrical, electroacoustic, electronic and lastly digital sound (ibid. 252). In the same way, Battier didn't speak about electroacoustic music or computer music, but targeted a larger view, speaking of 'audio art'.

In this first section I have given a diagram which permits the organisation of theories to analyse electroaoustic music. Those theories can be situated 'before' or 'from the audio signal' of the work. Next, I have given two examples of theories which aim to analyse 'before the signal': genetic criticism applied to music, which focuses on the concept of text, and the faktura, with the notions of style and materials. In the next section I will focus on computer music and the possibility to make analysis 'before the signal'.

ANALYSING COMPUTER MUSIC BEFORE THE SIGNAL

To my knowledge, the question of analysing computer music 'before the signal' was treated for the first time in 1984 by Marco Stroppa [START_REF] Stroppa | The analysis of electronic music[END_REF]. Ironically, Stroppa's article points out the difficulties of analysing this music in a non aural way : it results from the impossibility for the author to analyse the famous work Songes (1979) by Jean-Claude Risset, for recording instrumental sounds, computer synthesis and treatments.

In this section, I present three points which highlight this difficulty.

Code

First of all, and even if it is tautological, we have to consider that computer music produces some 'coded traces' (Risset 2001: 16).

According to this, we can then put forward two other considerations. Coded traces:  can only be read if the programming language is known;  come from an activity of design;

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 so can be seen as the programmer's own reflection: computer music traces contain some strong idiosyncrasies. This underlines the fact that reading coded traces is not easy. Firstly, one has to know, or even be a master in the programming language used -and those languages evolve with time -and secondly, these traces are idiosyncratic. Indeed we have to distinguish the reading of coded traces from that of natural language 6 . This difficulty is present in Jean-Claude Risset's writings when he speaks of 'cryptic traces':

Even for someone who is familiar with the language, the MUSIC V scores are cryptic, the process are described in an almost atomic way. Risset in (Lorrain 1980 : preface), my translation.

To take full advantage of these somewhat cryptic traces, those who undertake the analysis must be enlightened specialist, often composers themselves. Risset in (Licata 2002 : xvii) It is also present in an analysis made by Agostino Di Scipio on Contours (1982) by the same Risset. Di Scipio writes: 'I scrutinized many Music V code listings that the composer had written for this piece' (Di Scipio 2000: 2).

Lastly, this difficulty is glaringly present in [START_REF] Stroppa | The analysis of electronic music[END_REF]. One can think that Stroppa project of analysing Songes could have been accomplished if he had had, as Di Scipio did for Contours, the code of the work. Indeed, this is what (Risset 2001: 154) suggests. However, this would not work as at the time of writing (1983 -it is fundemental to notice that at that time the computer was not yet democratized) the 'cryptic traces' were seen by Stroppa as 'operational data', and were useless for musical analysis:

This is therefore linked to a specific machine and program. It is completely incomprehensible to most musicians, and directed simply towards specialists. (Stroppa 1984: 177)

Tape music ?

My second point concerns the words used to name this music. In the pioneering period, we used to refer to computer music as tape music. And obviously using the term tape, which refers to the magnetic tape on which audio signal is written, slows down, inevitably, the consideration of code.

For example this problem is present in [START_REF] Stroppa | The analysis of electronic music[END_REF]. Just after the introduction, he writes that 'To narrow the horizon further, I shall limit myself to consideration of tape music which has kept abreast of the evolution of thought in contemporary music' (ibid. 176). Then, in the second paragraph, which deals with the possibilities to analyse this music, the focus is now on computer music :

If we attempt to hold firm to these principles and apply them to computer music, we discover quite new problems and a new reality (ibid. 177) 6 Briefly one can notice that in the latter at least two words can be invented, whereas in the former all the variables names and the subroutines, for example, are invented

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When [START_REF] Risset | Problèmes posés par l'analyse d'oeuvres musicales dont la réalisation fait appel à l'informatique. Analyse et création musicales : actes du Troisième congrès européen d'Analyse Musicale[END_REF]) answers Stroppa, one key point he put forward is how we name this music. With Risset the issue is not about electronic music or tape music, but precisely, as the subtitle mentions 'musical works whose realisation involves computer' (my translation).

In a more astonishing way we can find this problem in [START_REF] Lorrain | Analyse de la bande magnétique de l'oeuvre de Jean-Claude Risset Inharmonique[END_REF]. Simply, the title of the document is 'Analysis of the tape of the work Inharmonique by Jean-Claude Risset' [Analyse de la bande magnétique de l'oeuvre de Jean-Claude Risset Inharmonique]. This is astonishing because if the work of Lorrain is historic and 'rare' as (Nucibella, Porcelluzia and Zattra 2005) pointed out, this is precisely because the medium used for the analysis is not the tape but the Music N score ... those very cryptic traces.

In the same way, we have to consider that the historic work of Lorrain was not meant to be an analysis. It is an Ircam report -which has never been published and is without pagination -ordered for teaching purpose, as Jean-Claude Risset said in the preface :

Denis Lorrain made this report at the request of Michel Decoust and for IRCAM internships for composers, but my experience with some similar documents (like my catalog of digital sound synthesis) reveals that they can be used in a broader scope. Risset in (Lorrain 1980: preface), my translation.

My comment does not impact the quality and the relevance of Lorrain's work, but it shows that in 1980, a musical analysis based on computer traces is not obvious ; it shows the difficulty of an analysis based on computer music code.

Ontology

I suggest that the ontology of computer works can deeply explain this lack of consideration of code. Substantially, in this pioneering period code of computer music wasn't considered because no link was made between code itself and the existence of the work. Once again, this is significant in (Stroppa 1984: 177):

One of the principal characteristics of [computer music] is that it exists and is performed more or less exclusively on tape, without any effective visual representation.

Indeed, in this period of time it was hard to conceive code as something related to the existence of musical works. First of all, the musicological community has been used for centuries to consider one material object, the graphical score, as the centre of the existence of musical works. For example in the famous Sémiologie musicale et musicologie générale one can read :

What results from the composer's creative gesture, is, in Western tradition, the score ; what makes the work feasible and recognisable as an entity, is the score ; what allows it to cross the centuries is again the score. (Nattiez 1987: 99, my translation) Secondly, at the time of Stroppa's article (80s), it should be said that code was not similar to the one of our digital age, it was low level (low abstraction) and some 9-17 languages belonged to specific hardware -cf. (Wang 2008: 14) for example. Consequently, code was of short duration and quickly became obsolete.

COMPUTER MUSIC CODE AS WRITING

In the last section I have shown that computer music has something specific, which makes it difficult to fit inside the tape music appellation. I found that this specificity has been highlighted by two famous composers and researchers, Jean-Claude Risset and Horacio Vaggione.

Jean-Claude Risset's 'exhaustive score'

Risset had a specific idiom to refer to computer music code: 'exhaustive score', 'complete score' are recurrent in his writings. Before getting into detail, I would like to make a link with the scheme of Music N. In (Larrieu 2018: 70) I argue that the scheme is made of two parts, working together: on one hand, from top to bottom, there is the concept of orchestra, Ugen and parameters, and on the other hand, from top to bottom, the concept of score, instruments and notes. First, one have to notice that Jean-Claude Risset is a pioneer of computer music, before the democratisation of computer. Second, I note that the author is coherent with the Music N scheme, so that, as we will see, the understanding of this scheme permits us to shed light on the thinking of Risset.

Risset recurrently used the words 'exhaustive' and 'complete' to qualify the score of computer music. For example, in chronological order, I can cite:

The program [Music V] requires a 'score' which defines the desired sound: so this score is like a recipe to obtain the sound as well as an exhaustive description of the physical sound structure. [Le programme demande une « partition » qui lui spécifie le son voulu : cette partition est donc une recette d'obtention du son en même temps qu'une description exhaustive de sa structure physique]. Risset in (Lorrain 1980 : preface) The programs giving the computer a synthesis recipe are the very scores of the sound structure, exhaustive and transmittable scores. [Les programmes stipulant à l'ordinateur la recette de synthèse sont de véritables partitions de la structure sonore, partitions exhaustives et transmissibles]. [START_REF] Risset | Timbre et synthèse des sons[END_REF] A Music V score is, as well as a production recipe, a complete score : it represents the compositional elaboration of the sound microstructure. [Une partition Music 5, en même temps qu'une recette de production, est aussi une partition intégrale : elle représente l'élaboration compositionnelle de la microstructure du son] [START_REF] Risset | Problèmes posés par l'analyse d'oeuvres musicales dont la réalisation fait appel à l'informatique. Analyse et création musicales : actes du Troisième congrès européen d'Analyse Musicale[END_REF].

I can understand those terms as an extension of the Music N scheme, where the Music N score (alphanumeric) is related to the traditional one (graphic). This is exactly what is done in (Mathews 1963: 554), where the conceptor of the language, Max Mathews, explains the scheme of Music N. And then, in the same way, while the traditional score needs a musician to be transformed into sound, the digital one does 10-17 not require it. But, as a result, the digital score does need an abundance of details, 'processes are described almost atomic ' Risset in (Lorrain 1980).

This last paragraph was useful to understand what Risset meant with these terms, but what is important in this article is the relationship the author enlightens between the digital score and sound. Indeed within a poietic view, one can easily see code as something powerful for studying the creative process, the composition, i.e. code as something resulting from a past process. But Risset invites us to see code also as something related to the sound heard: between code and sound there is a reciprocity. Indeed the reading of the former allows for a deeper listening of the latter. And this was already present in the introduction of the historic catalogue of Risset:

the computer data used for the synthesis of the sound affords a thorough description of the physical structure of these sounds (Risset 1969: 109).

Nearly half a century later, it can be said, more generally, that code and sound, or more precisely the computational process made by the composer and what we hear, are enriching each other.

Horacio Vaggione's 'direct writing'

I found this relation between code and sound in the writings of Horacio Vaggione, but in a different way. First, as I did with Jean-Claude Risset, I would like to make a link with the Music N scheme.

For Vaggione the note of the traditional musical theory, to which a graphic sign corresponds, is a 'syntactic brick' which is part of a 'reference level' (Vaggione 1998: 172). And, on each side of this level there are two domains, which are now well known :

 the macrotime level, above the note level, 'embracing all the possible scales going towards the ''tall'' ' (ibid.), my translation.  the microtime level, below the note level, 'embracing all the possible scales going towards the ''little'' (ibid.), my translation. In this respect, it goes without saying that Vaggione, unlike Risset, does not match with the Music N scheme. Indeed for Vaggione, musical score and notes cannot be assimilated to the score and note of Music N: the former are part of fixed operating level whereas the latter are not part of an operating level a priori : the denomination list of notes is nevertheless misleading because, even if it evokes some items 'list', those items do not correspond to the 'note' of the conventional notation, because here the item of the list can belong to any time level, and thereby can represent some duration belonging to the domain of microtime. (Vaggione 2010: 64), my translation.

The reciprocity between code and sound is not something which is at the centre of the thought of the author, it has not been substantially described. It is present as a key notion of the author, the one of object. One aspect of the notion is that object combines some sound and writing 'facts' :

[the object notion] no longer corresponds to the purely macroscopic entity of Schaeffer's (1966), but now underpin another type of compound entity in 11-17 which time structures (some process, some morphologies) are determined not only as sonic facts but also, and indissolubly, as writing facts. [ [la notion d'Objet] ne correspond plus à l'entité purement macroscopique définie par Schaeffer (1966), mais sous-tend désormais un autre type d'entité composée dans laquelle des structures temporelles (des process, des morphologies) se trouvent déterminées non seulement en tant que faits sonores mais aussi, et indissolublement, en tant que faits d'écriture. (Vaggione 1998: 188).

And moreover, in the same text this 'double determination' leads the author to qualify the object as 'transparent', and also to put forward the idea of 'direct writing': the digital sound is transparent, that is to say it can be open in order to provide access to its internal structure, and therefore allow direct writing of the sound material itself. (Vaggione 1998: 192), my translation.

I also found this idea in the previous article, but only by an exogenously means, that is to say with some reflections on the historic work of Risset :

In fact, that is the study of a morphological salience [that of brass sounds] that led Risset […] to state a model of dynamic character: an operative model, that you can translate in terms of parameters, thus inaugurating a 'direct writing' of the microtime phenomena forming part of the physical structure of sounds as well as of their perceptual identity. [En fait, c'est l'étude d'une saillance morphologique [celle des sons cuivrés] qui a conduit Risset […] à énoncer un modèle de ce caractère dynamique : un modèle opératoire, traduisible en termes paramétriques, inaugurant ainsi une "écriture directe" des phénomènes micro-temporels faisant partie de la structure physique des sons ainsi que de leur identité perceptive]. (Vaggione 2003: 92).

Writing

Through Risset and a fortiori Vaggione's writings, the specificity of computer music is now clear. The sound of this music is not evanescent, and cannot 'only be heard' as the one of acousmatic music. The sound of computer music can be enlightened with something that is written. Moreover, with Vaggione, this 'something else' of computer music -which is usually named code -is less understood as a 'new causality', or as a 'production process'. With [START_REF] Zattra | Génétiques de la computer music[END_REF], who speaks of 'text synthesis', and above all after reading Risset and Vaggione's writings, it can be seen that the specificity of computer music is well enlightened as far as writing is considered.

This feature can be surprising if I relate it with what is usually meant by 'music writing'. However it decreases if I compare two composition environments, the analog and the digital. In the former the composition is made with a mass of manipulations on electroacoustic devices, with some gestures, whereas in the latter, at the centre of the environment, there are two key devices: the screen and the alphanumeric keyboard.

In computer music the composition is made especially with a minimal unit, symbols, which lead, inevitably, to the idea of writing :

The computer has been to me the ideal tool because it brought the ability to work with discrete symbols on the level of the sound material and,

ANALYSING COMPUTER MUSIC

In the previous section I have enlighten the very spec-ificity of computer music when considering analysis: computer music is a written music. In this section I will now use this specificity to question computer music analysis. First, I will discuss the theories 'before the signal' previously mentioned, and second, I will dis-cuss of a recent analytical theory, the 'interactive aural analysis' conceived by Michael Clarke, which takes into account the written part of computer music.

Faktura and genetic criticism applied to music

Considering faktura first, since it is not dedicated to computer music, it does not take into account the specificity of computer music, the written code. Indeed, as we have seen, faktura can be applied to computer code as well as any electronic sound device used for composition.

By contrast, genetic criticism applied to music by Laura Zattra seems suitable. As we have seen, one key concept of the genetic criticism is writing, from which comes the original idea that 'synthesis data' can be seen as text. However, it has to be said that genetic criticism is fundamentally situated in the poietic: it did not take into account the sound actualisation of the musical work. As Zattra said, the goal of genetic criticism applied to computer music is to enlighten the 'forming process of the work' (Zattra 2015: 220). All the sounding and hearing aspect of those writings -which have been enlightened with Risset and Vaggione -are missing. One can say that genetic criticism applied to music is built on the idea that text is part of the genesis of the work, the creative process. If genetic criticism permits light to be shed on a musical work, it is thanks to its genesis. This focus on the creative process is for example present in a remark from Zattra on the work made by Lorrain on Inharmonique. To make his report, Lorrain rewrote the Music N score of Risset. In this operation, Zattra would appreciate 'a more precise explanation' of the 'interventions made on the original data' (ibid. 215), so that the reader could be as close as possible to the original 'text'. In this focus on the original text lies a second criticism I can make on genetic criticism applied to computer music. In my understanding the 'original text' is not essential, it is not 'the one' to absolutely get and study. One specificity of computer music is that, because of technology, one work can have many codes 7 . It is not only because of the obsolescence of digital technology, but also because in computer programming you can obtained the same result in different way. Analysts and musicologists have to deal with that specificity. It is not because one work can have many code versions, in that studying all these versions and comparing them, is the only way to analyse the work. This does not mean that comparing different versions of one work is not interesting. It means that another way for analysing computer music could exist, going beyond all the versions of one work, not focusing on the differences which occur between versions and codes, but focusing on the work itself 8 .

7 One historic example is Stria (1977), with the original version and at least the two versions of [START_REF] Dahan | Reconstructing Stria[END_REF] and (Baudoin 2007). 8 This point is truly important and must be developed in a substantial paper. At that time, one can think of the two concepts of early linguistic, synchrony and diachrony cf. [START_REF] Saussure | Cours de linguistique générale[END_REF].

13-17

Between reading and listening

Before concluding, I would like to present a recent analytical theory made by [START_REF] Clarke | Analysing Electroacoustic Music : an Interactive Aural Approach[END_REF], which takes into account the written part of computer music. Michael Clarke's theory was made during an analysis of Mortuos Plango, Vivo Voco (1980) a tape for eight channels 9 , composed by Jonathan Harvey and assisted by Stanley Haynes. The article [START_REF] Clarke | An interactive aural approach to the analysis of computer music[END_REF] is quite original because it is attached with a Max patch, which allows (i) to read the audio files of the work, (ii) to navigate within an aural pragmatic chart, and (iii) to compute some elements which enable to reproduce some extracts of the work.

The author has named his theory interactive aural analysis. But I am quite surprised with the term aural. Indeed, as we know, aural means that the analysis is based on hearing, that the only contact with the work is made through the ears. So, following my earlier diagram above, Clarke's theory is based on audio signal, on the listening of the signal. But, the computational elements present in his analysis testifies that his analysis is also made with an understanding 'before the signal' :

the software provides a series of interactive exercises illustrating the techniques used by Harvey in the work. In some cases synthesis is used to simulate the processes used by the composer. (Clarke 2006: 3) Synthesis and sound processing are also used to illustrate some of the techniques used in the composition and the reader can experiment with modifying the parameters. (Clarke 2005: 115) and more explicitly : in this analysis detailed attention has been given in particular to the poietic element […]. The composer also generously made his hand-written sketches available and these greatly aided the process of analysis. (Clarke 2006: 3) In our context the analysis made by Clarke is quite particular because it seems that the author has not worked with the code of the work. The programmes used for composition were Music V and Chant cf. (Clarke 2005: 115, 117). Maybe, 26 years after the composition, these codes were no longer available. In this context, Clarke has simulated the transformations and synthesis made by the composer and his assistant in a more recent programming language, Max.

What I want to point out, despite the fact that the code of the work has not been used, is that Clarke's analysis is situated both 'before' and 'from' the signal. 'Before', because the analyst has used Max to simulate some extract of the work, and 'from' because the analysis contains an aural paradigmatic chart.

It is precisely in this combination of ways of making analyses that I found Clarke's theory interesting. If the author focuses on interactivity and the aural, I would like to focus on the fact that the analysis is made both by reading code -in Clarke's analysis the code was made by the author but one can think to a more recent work where all the code needed is available -and by listening to the music -by producing an audio paradigmatic chart.

14-17

This way of analysing is the one that I argue in my PhD, (Larrieu 2018: 158). Since computer music is not a non-written music, the musicologist can both read and listen to the work. I support the idea of a circulation of meanings between the reading and the listening.

Thus, where we used to have only auditive feedback, we now have a confluence of the ears and eyes as well as the ability to store our actions in the form of codes. I think this is is a very positive aspect of computer music. (Vaggione interviewed in Budón 2013: 116) With Vaggione, I understand that a theory for analysing computer music is relevant if it does allow a circulation of meanings, between what is read and what is heard.

CONCLUSION

In this article I have explored different theories for analysing computer music. In the first section I have made a diagram which organise some theories without ambiguity: analysis can be made before or from the signal of the work. Then I have explained two theories situated before the signal, the famous faktura by Battier, and the recent genetic criticism applied to music by Zattra. Even if the latter theories make a pointsuch as (Menegheni 2007) -I feel they miss something which I find central to computer music, something which is related to the ontology of computer music works: they exist through writings, through what we used to call code. Tracing back the sparse history of computer music analysis, I have found in the second section that code was the object those theories do not apply to, because they did not manage to consider it. With the article [START_REF] Stroppa | The analysis of electronic music[END_REF]) used as a watermark, I have presented three points to illustrate this 'historical impermeability. In the third section I have considered 'the specificity of computer music', code as writing. The thinking of Risset and Vaggione helped me understand code not only as something coming from the poietic process, but also as something useful to understand music and sound. Finally, in the last section I have presented a recent theory by Clarke for analysing computer music, the Interactive Aural Analysis. Assuming the writing of computer music, I argue that this theory is relevant because it takes into account both the writing and the sounding part of computer music, even if it is not present in Clarke's demonstration.

I believe that if we understand computer music as a written one, it can open new directions for studying it. The work of Zattra, with genetic criticism, and of Clarke, with interactivity allowed with a Max patch, are two typical examples. However, to my knowledge, the writing of computer music hasn't been considered in a theory of analysis. One question which arises is: 'is a theory based on the writing of computer music relevant ?' For example, the Interactive Aural Analysis started with computer music [START_REF] Clarke | An interactive aural approach to the analysis of computer music[END_REF], and a few years later the author expanded it to the acousmatic repertoire [START_REF] Clarke | Extending Interactive Aural Analysis : Acousmatic Music[END_REF]. Moreover, in a recent article, [START_REF] Clarke | Analysing Electroacoustic Music : an Interactive Aural Approach[END_REF] writes 'it would be unfortunate to […] restrict the interactive aural approach to the electroacoustic repertoire' suggesting that '[in] principle, the interactive aural analysis approach can be applied to any piece, though each new work will inevitably lead to extension and adaptation of the approach'. I would like to keep the question open, but I see one direction to go further: making some analysis of computer music works with high consideration to code as writing, and then thinking about how the knowledge inside

  literally 'write' sounds.(Vaggione interviewed in Budón 2013: 102), my translation.

Music N refer to a family of programming languages which uses the scheme of Music V. See section 3.1 for one explanation of the Music V scheme ;(Mathews 1963: 555) for the historic presentation and(Manning 2004: 187) for a historical viewpoint.

For example in[START_REF] Zattra | Analysis and analyses of ecltroacoustic music[END_REF] the author made an overview on the existing methods to analyse electroacoustic music, and in the 'Genetic analysis' subtitles, except of her own works, only[START_REF] Lorrain | Analyse de la bande magnétique de l'oeuvre de Jean-Claude Risset Inharmonique[END_REF] appears.


Rheadel G Fulgencio
email: rheadel.fulgencio@univ-rouen.fr

Olivier Guibé
email: olivier.guibe@univ-rouen.fr

Quasilinear Elliptic Problem in a Two-Component Domain with L 1 data

In the present paper we consider the following class of quasi-linear equations:

The domain Ω is composed of two components, Ω1 and Ω2, with Γ being the interface between them. The given function f belongs to L 1 (Ω). We will first give a definition of a renormalized solution for this class of equations. The main result of this paper is the existence of such a solution.

Introduction

In the present paper, we study the existence of a solution u := (u 1 , u 2 ) of the following class of quasi-linear equations:

               -div(B(x, u 1 )∇u 1 ) = f in Ω 1 , -div(B(x, u 2 )∇u 2 ) = f
in Ω 2 u 1 = 0 on ∂Ω, (B(x, u 1 )∇u 1 )ν 1 = (B(x, u 2 )∇u 2 )ν 1 on Γ, (B(x, u 1 )∇u 1 )ν 1 = -h(x)(u 1 -u 2 ) on Γ.

(P)

Here, Ω is our two-component domain with ∂Ω as its boundary. The open sets Ω 1 and Ω 2 are the two disjoint components of Ω with Γ as the interface between them (see Figure 1) and the vector ν i is the unit outward normal to Ω i . The matrix field B is coercive but does not satisfy any growth condition, and the data f is an L 1 -function. On the boundary ∂Ω, we have a Dirichlet boundary condition, while on the interface Γ, we have a continuous flux and the jump of the solution is proportional to the flux. The existence and uniqueness of solution of (P) when f ∈ L 2 (Ω) was studied in [START_REF] Beltran | Homogenization of a quasilinear elliptic problem in a twocomponent domain with an imperfect interface[END_REF][START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF][START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF]. In [START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF][START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF] the equations are linear, that is, the matrix field B does not depend on the solution u, while in [START_REF] Beltran | Homogenization of a quasilinear elliptic problem in a twocomponent domain with an imperfect interface[END_REF], the equations are quasilinear, which is also the case in this study. The above mentioned papers are all motivated by homogenization, which is also our main goal (see [START_REF] Donato | Homogenization results for quasilinear elliptic problems in a two-component domain with L 1 data[END_REF]).

Since we consider in the present paper an L 1 -data, we need an appropriate notion of solution. Indeed, for the elliptic equation -div(A(x, u)∇u) = f with Dirichlet boundary condition if the matrix A is bounded, a solution in the sense of distribution exists (see [START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF]) but it is not unique in general (see the counterexamples in [START_REF] Prignet | Remarks on existence and uniqueness of solutions of elliptic problems with right hand side measures[END_REF][START_REF] Serrin | Pathological solutions of elliptic equations[END_REF]). If the matrix field is not bounded, then we cannot expect to have a solution in the sense of distribution since there is no reason to have A(x, u) ∈ L 1 loc . In the present paper, we use the notion of renormalized solution, which is first discussed in [START_REF] Diperna | On the Cauchy problem for Boltzmann equations: Global existence and weak stability[END_REF] by R.J. DiPerna and P.L. Lions for first order equations. This notion was then further developed by F. Murat in [START_REF] Murat | Soluciones renormalizadas de EDP elipticas no lineales[END_REF], by P.L. Lions and F. Murat in [START_REF] Lions | Sur les solutions renormalisées d'équations elliptiques[END_REF] for elliptic equations with Dirichlet boundary conditions and L 1 data, and by G. Dal Maso et al. in [START_REF] Maso | Renormalized solutions of elliptic equations with general measure data[END_REF] for elliptic equations with general measure data. There is a wide range of literature for elliptic equations with Dirichlet boundary condition and L 1 data, among them are [START_REF] Bénilan | An L 1 theory of existence and uniqueness of nonlinear elliptic equations[END_REF][START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF][START_REF] Maso | Renormalized solutions of elliptic equations with general measure data[END_REF][START_REF] Dall | Approximated solutions of equations with L 1 data. Application to the h-convergence of quasi-linear parabolic equations[END_REF][START_REF] Lions | Sur les solutions renormalisées d'équations elliptiques[END_REF][START_REF] Murat | Soluciones renormalizadas de EDP elipticas no lineales[END_REF]. Considering elliptic equations with Neumann or Fourier boundary conditions and L 1 data, which are connected to our problem, gives in general additional difficulties due the lack of Poincaré inequality or the low regularity of the solution (definition of the trace for e.g.). In the case of one-component domain (and L 1 data), using the framework of entropy solution existence results are given in [START_REF] Andreu | Quasilinear elliptic and parabolic equations in L 1 with nonlinear boundary conditions[END_REF][START_REF] Andreu | Existence and uniqueness for a degenerate parabolic equation with L 1 -data[END_REF][START_REF] Ouaro | Entropy solution to an elliptic problem with nonlinear boundary conditions[END_REF], convection-diffusion equations with mixed Neumann and Robin boundary conditions are studied in [START_REF] Droniou | Solving convection-diffusion equations with mixed, Neumann and Fourier boundary conditions and measures as data, by a duality method[END_REF] by a duality method, and in [START_REF] Guibé | Renormalized solutions of elliptic equations with Robin boundary conditions[END_REF] the authors prove existence and uniqueness results using the notion of renormalized solution for equations with Robin boundary conditions.

The main originality of the present paper is the jump of the solution which produces in the formulation a term in the interface Γ. Recalling that the regularity of the renormalized solution is given through the truncate, the first difficulty is to give a sense on the interface for functions (u 1 , u 2 ) whose truncates belong to H 1 . Following the ideas of [START_REF] Andreu | Quasilinear elliptic and parabolic equations in L 1 with nonlinear boundary conditions[END_REF][START_REF] Guibé | Renormalized solutions of elliptic equations with Robin boundary conditions[END_REF] (but in the case of one-component domain), we define an appropriate notion of trace (see Proposition 2.2). The second difficulty is the regularity of γ 1 (u 1 ) -γ 2 (u 2 ) (where γ 1 is the trace function for H 1 (Ω 1 )-functions and γ 2 is the trace function for H 1 (Ω 2 )-functions), since we have to deal with terms on the boundary like (γ 1 (u 1 ) -γ 2 (u 2 ))χ {|γ1(u1)|<k} (where χ A is the characteristic function of any set A) in the renormalized formulation. To have (γ 1 (u 1 ) -γ 2 (u 2 ))χ {|γ1(u1)|<k} belonging to L 1 (Γ) is then equivalent to have γ 2 (u 2 )χ {|γ1(u1)|<k} ∈ L 1 (Γ), which is unusual and is in some sense a coupled regularity on the boundary. It is worth noting that it is not a direct consequence of T k (u 1 ) ∈ H 1 (Ω 1 ) and T k (u 2 ) ∈ H 1 (Ω 2 ). Using the structure of the equation, we give an extra regularity (see (2.11b) in Definition 2.4) which allows one to complete our notion of renormalized solution for problem (P). We are then able to give a definition of renormalized solution for problem (P) for which we prove the existence (see Theorem 3.1).

This paper is organized as follows. The next section discusses the assumptions on our problem and some definitions including the definition of a renormalized solution of (P). Section 3 is devoted to the proof of the existence of a renormalized solution for (P).

Assumptions and Definitions

In this section, we present the assumptions and definitions necessary for our problem. We begin by introducing the two-component domain Ω. The domain Ω is a connected bounded open set in R N with its boundary ∂Ω. We can write Ω as the disjoint union Ω = Ω 1 ∪ Ω 2 ∪ Γ, where Ω 2 is an open set such that Ω 2 ⊂ Ω with a Lipschitz boundary Γ and Ω 1 = Ω \ Ω 2 . We denote by ν i the unit outward normal to Ω i . for some h 0 ∈ R + .

Ω Ω 2 Γ Ω 2 Γ Ω 1 ∂Ω
( and it has the following properties:

(A3.1) B(x, t)ξ • ξ ≥ α|ξ| 2 , for some α > 0, for a.e. x ∈ Ω, ∀t ∈ R, ∀ξ ∈ R N ; (A3.2) for any k > 0, B(x, t) ∈ L ∞ (Ω × (-k, k)) N ×N .
The space for this class of equations is not a usual L p -space or a Sobolev space due to the jump on the interface. We need the normed space V defined as follows. Let V 1 be the space defined by

V 1 = {v ∈ H 1 (Ω 1 ) : v = 0 on ∂Ω} with v V1 := ∇v L 2 (Ω1) . Define V := {v ≡ (v 1 , v 2 ) : v 1 ∈ V 1 and v 2 ∈ H 1 (Ω 2 )}, equipped with the norm v 2 V := ∇v 1 2 L 2 (Ω1) + ∇v 2 2 L 2 (Ω2) + v 1 -v 2 2 L 2 (Γ) .
(2.2)

Identifying ∇v := ∇v 1 + ∇v 2 we have that v 2 V = ∇v 2 L 2 (Ω\Γ) + v 1 -v 2 Proposition 2.1 ([16]). The norm given in (2.2) is equivalent to the norm of V 1 × H 1 (Ω 2 )
, that is, there exist two positive constants c 1 , c 2 such that

c 1 v V ≤ v V1×H 1 (Ω2) ≤ c 2 v V , ∀v ∈ V.
We now define the function T k , which is known as the truncation function at height ±k.The function T k : R -→ R is given by

T k (t) =      -k, if t ≤ k, t, if -k ≤ t ≤ k, k, if t ≥ k. (2.3)
This function will be crucial in the definition of a renormalized solution of (P).

In the case of L 1 -data, we cannot expect to have the solution u belonging to V . In general, in the framework of renormalized solution, the regularity of the solution is given through the regularity of the truncate. So it is necessary in our case to define the gradient and the trace of the solution u. For the gradient, we follow the definition given in [START_REF] Bénilan | An L 1 theory of existence and uniqueness of nonlinear elliptic equations[END_REF]. For the trace, we have to precise the trace of u 1 on Γ and the one of u 2 on Γ. With respect to [START_REF] Andreu | Quasilinear elliptic and parabolic equations in L 1 with nonlinear boundary conditions[END_REF][START_REF] Guibé | Renormalized solutions of elliptic equations with Robin boundary conditions[END_REF], we have the additional difficulty for u 2 since we do not have the Poincaré inequality.

Proposition 2.2. Let u = (u 1 , u 2 ) : Ω \ Γ -→ R be a measurable function such that T k (u) ∈ V for every k > 0. For i = 1, 2,

there exists a unique measurable function v

i : Ω i -→ R N such that for all k > 0, ∇T k (u i ) = v i χ {|ui|<k} a.e. in Ω i , (2.4)
where χ {|ui|<k} denotes the characteristic function of {x ∈ Ω i : |u i (x)| < k}. We define v i as the gradient of u i and write v i = ∇u i .

if

sup k≥1 1 k T k (u) 2 V < ∞, (2.5)
then there exists a unique measurable function w i : Γ -→ R, for i = 1, 2, such that for all k > 0,

γ i (T k (u i )) = T k (w i ) a.e. in Γ, (2.6)
where

γ i : H 1 (Ω i ) -→ L 2 (Γ)
is the trace operator. We define the function w i as the trace of u i on Γ and set

γ i (u i ) = w i , i = 1, 2.
Proof.

1. This is proved in [START_REF] Bénilan | An L 1 theory of existence and uniqueness of nonlinear elliptic equations[END_REF] (see Lemma 2.1).

2. The case i = 1, or more generally the truncates have a zero trace on a part of the boundary (which allows one to use Poincaré-kind inequality) is presented in [START_REF] Guibé | Renormalized solutions of elliptic equations with Robin boundary conditions[END_REF]. We just have to prove the result for i = 2.

The uniqueness is in the almost everywhere sense. Note that if we find a function that satisfies (2.6), then the uniqueness of w 2 is assured by the monotonicity of T k and the fact that w 2 is finite a.e. in Γ.

By Proposition 2.1, we know that

T k (u 2 ) H 1 (Ω2) ≤ c 1 T k (u) V ,
for some positive constant c 1 , independent of k. It follows from (2.5) that

T k (u 2 ) 2 H 1 (Ω2) ≤ M k, (2.7)
with M ∈ R + independent of k. Due to the regularity of Γ, γ 2 (T n (u 2 )) is well-defined and

k 2 meas Γ {|γ 2 (T k (u 2 ))| ≥ k} = Γ∩{|T k (u2)|≥k} (γ 2 (T k (u 2 ))) 2 dσ ≤ γ 2 (T k (u 2 )) 2 L 2 (Γ)
. Hence, by Trace Theorem and (2.7), we have

k 2 meas Γ {|γ 2 (T k (u 2 ))| ≥ k} ≤ γ 2 (T k (u 2 )) 2 L 2 (Γ) ≤ T k (u 2 ) 2 L 2 (Ω2) + ∇T k (u 2 ) 2 L 2 (Ω2) ≤ M k.
As a result,

meas Γ {|γ 2 (T k (u 2 ))| ≥ k} -→ 0 as k -→ 0. (2.8) Define Γ n = {x ∈ Γ : |γ 2 (T n (u 2 ))| < n} for n ∈ N. From (2.8), it follows that Γ = n≥1 Γ n ∪ A, (2.9)
where A is a subset of Γ with zero measure.

Note that for k < n, we have

T k (T n (u 2 )) = T k (u 2 ). Fix k > 0.
Then for every n ∈ N such that n > k, we have the following equality

T k (γ 2 (T n (u 2 ))) = γ 2 (T k (T n (u 2 ))) = γ 2 (T k (u 2 )) a.e. on Γ,
and then

γ 2 (T k (u 2 )) = γ 2 (T n (u 2 )) a.e. on Γ k . (2.10)
Since for every n 1 ≤ n, we have Γ n1 ⊆ Γ n , in view of (2.9) and (2.10), we can define w 2 in the following way:

w 2 = γ 2 (T n (u 2 )) on Γ n
and noting that Γ = n≥1 Γ n (up to measure zero set), we have for any

k > 0 γ 2 (T k (u 2 )) = T k (w 2 ) a.e. on Γ.
This concludes the proof.

Remark 2.3. In the following, we give an example of a measurable function u where T k (u) ∈ V but u 2 is not defined on a part of the interface. We consider

Ω = (-1, 2) with Ω 1 = (-1, 0) ∪ (1, 2)
and Ω 2 = (0, 1) (so Γ = {0, 1}), and u = (u 1 , u 2 ) is defined as

u(x) = u 1 (x) = (x + 1)(x -2) if x ∈ Ω 1 u 2 (x) = x -2 if x ∈ Ω 2 .
We have for some positive constants

C 1 , C 2 , ∇T k (u 1 ) 2 L 2 (Ω1) = |u1|<k (2x -1) 2 dx ≤ Ω1 (2x -1) 2 dx ≤ C 1 ,
and

∇T k (u 2 ) 2 L 2 (Ω2) = 1 k 1/2 (-2x -3 ) 2 dx = 4 - x 7 7 1 x=k 1/2 = 4 7 (k 7/2 -1).
Thus, we can see that

k 7/2 C ≤ T k (u) 2 V ≤ Ck 7/2 ,
for some C > 0 but clearly u 2 does not have a trace on {0} ⊂ Γ.

We are now in a position to give the definition of renormalized solution.

Definition 2.4. Let u = (u 1 , u 2 ) : Ω \ Γ -→ R be a measurable function. Then u is a renormalized solution of (P) if

T k (u) ∈ V, ∀k > 0; (2.11a) (u 1 -u 2 )(T k (u 1 ) -T k (u 2 )) ∈ L 1 (Γ), ∀k > 0; (2.11b) lim n→∞ 1 n {|u|<n} B(x, u)∇u • ∇u dx = 0; (2.12a) lim n→∞ 1 n Γ (u 1 -u 2 )(T n (u 1 ) -T n (u 2 )) dσ = 0; (2.12b)
and for any S ∈ C 1 (R) (or equivalently for any

S ∈ W 1,∞ (R)) with compact support, u satisfies Ω1 S(u 1 )B(x, u 1 )∇u 1 • ∇v 1 dx + Ω1 S (u 1 )B(x, u 1 )∇u 1 • ∇u 1 v 1 dx + Ω2 S(u 2 )B(x, u 2 )∇u 2 • ∇v 2 dx + Ω2 S (u 2 )B(x, u 2 )∇u 2 • ∇u 2 v 2 dx + Γ h(x)(u 1 -u 2 )(v 1 S(u 1 ) -v 2 S(u 2 )) dσ = Ω f vS(u) dx, (2.13
)

for all v ∈ V ∩ (L ∞ (Ω 1 ) × L ∞ (Ω 2 )).
Remark 2.5. Conditions (2.11a) (the regularity of the truncate) and (2.12a) (the decay of the "truncated energy") are standard in the framework of renormalized solutions. As mentioned in Introduction, the main originality of the present paper is the presence of the trace. In view of Proposition 2.2, γ(u 1 ) and γ(u 2 ) are well-defined. Condition (2.11b) is an extra regularity of (u

1 -u 2 )(T k (u 1 ) -T k (u 2 )). Indeed, (u 1 - u 2 )(T k (u 1 ) -T k (u 2 )
) cannot be written as

(u 1 -u 2 )(T k (u 1 ) -T k (u 2 ))χ {|u1|<n} χ {|u2|<n} , for any n ∈ R + , so that having (u 1 -u 2 )(T k (u 1 ) -T k (u 2 )) belonging to L 1 (Γ) is not a consequence of (2.11a).
Conditions (2.11a) and (2.11b) allow one to give a sense of all the terms in (2.13).

Let S ∈ C 1 (R) with compact support. Then for all v ∈ V ∩ (L ∞ (Ω 1 ) × L ∞ (Ω 2 )), i = 1, 2, we have if supp(h) ⊂ [-k, k] S(u i )B(x, u i )∇u i • ∇v i = S(u i )B(x, T k (u i ))∇T k (u i ) • ∇v i ∈ L 1 (Ω i ), S (u i )B(x, u i )∇u i • ∇u i v i = S (u i )B(x, T k (u i ))∇T k (u i ) • ∇T k (u i ) v i ∈ L 1 (Ω i ), f vS(u) ∈ L 1 (Ω).
For the boundary term, let us define S n : R -→ R by

S n (s) =                  0, if s ≤ -2n s n + 2, if -2n ≤ s ≤ -n 1, if -n ≤ s ≤ n - s n + 2, if n ≤ s ≤ 2n 0, if s ≥ 2n, (2.14)
then since S has a compact support, for some large enough n, we have

h(u 1 -u 2 )v 1 S(u 1 ) = hv 1 (u 1 -u 2 )(S(u 1 ) -S(u 2 ))S n (u 1 ) + hv 1 (u 1 -u 2 )S(u 2 )S n (u 1 ).
Since both S and S n have compact support, we have that hv 1 (u 1 -u 2 )S(u 2 )S n (u 1 ) is bounded and is therefore in L 1 (Γ). Moreover, since

S(u 1 ) -S(u 2 ) = S(T 2n (u 1 )) -S(T 2n (u 2 ))
and S is Lipschitz, we have

|hv 1 (u 1 -u 2 )(S(u 1 ) -S(u 2 ))S n (u 1 )| ≤ hv 1 L ∞ (Γ) S L ∞ (R) × |u 1 -u 2 ||T 2n (u 1 ) -T 2n (u 2 )|, a.e. in Γ. Thus, in view of (2.11b), h(u 1 -u 2 )v 1 S(u 1 ) ∈ L 1 (Γ). Similarly, h(u 1 -u 2 )v 2 S(u 2 ) ∈ L 1 (Γ).
It is worth noting that condition (2.11b) is equivalent to have

u 2 χ {|u1|<k} ∈ L 1 (Γ) and u 1 χ {|u2|<k} ∈ L 1 (Γ), (2.15)
for any k > 0. Indeed,

u 2 χ {|u1|<k} = (u 2 -u 1 )χ {|u1|<k} (S n (u 1 ) -S n (u 2 )) + u 2 S n (u 2 )χ {|u1|<k} + u 1 S n (u 1 )χ {|u1|<k} -u 1 S n (u 2 )χ {|u1|<k} ,
and by condition (2.11b), the first term on the right-hand side belongs to L 1 (Γ) while the next 3 terms are bounded and thus also belong to L 1 (Γ).

Finally, let us comment that conditions (2.12a) and (2.12b) which are crucial to obtain uniqueness results (see other paper [START_REF] Fulgencio | Uniqueness results for quasilinear elliptic problems in a two-component domain with L 1 data[END_REF]) and also to recover that formally, for any k > 0, T k (u) is an admissible function in (P), that is,

Ω1 B(x, u 1 )∇u 1 ∇T k (u 1 ) dx + Ω2 B(x, u 2 )∇u 2 ∇T k (u 2 ) dx + Γ h(x)(u 1 -u 2 )(T k (u 1 ) -T k (u 2 )) dσ = Ω f T k (u 1 ) dx.
To prove this, fix k > 0. Using S n (u)T k (u) which is an admissible test function for any n ∈ N, we have

Ω1 S n (u 1 )B(x, u 1 )∇u 1 • ∇T k (u 1 ) dx + Ω1 S n (u 1 )B(x, u 1 )∇u 1 • ∇u 1 T k (u 1 ) dx + Ω2 S n (u 2 )B(x, u 2 )∇u 2 • ∇T k (u 2 ) dx + Ω2 S n (u 2 )B(x, u 2 )∇u 2 • ∇u 2 T k (u 2 ) dx + Γ h(x)(u 1 -u 2 )(S n (u 1 )T k (u 1 ) -S n (u 2 )T k (u 2 )) dσ = Ω f T k (u)S n (u) dx.
(2.16) Condition (2.12a) allows one to pass to the limit of the second and fourth integral in (2.16) while condition (2.12b) is useful for passing to the limit of the integral on the boundary in (2.16).

Remark 2.6. As observed in the previous remark, we have the condition (2.11b) for the integral on the interface to make sense. We can avoid this problem by using the Boccardo-Gallouët estimates presented in [START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF]. However, these estimates are heavily dependent on the Sobolev constants. With the final aim of doing the homogenization process, we try as much as possible to refrain from using these estimates (see Remark 3.2).

Existence Results

In this section, we present the proof for the existence of a renormalized solution of (P). Proof. The proof is divided into 4 steps. In Step 1, we consider an approximate problem (see (P ε )) in which B is approximated and f ε is an L 2 -data. Using Schauder's fixed point theorem, the existence of at least a variational solution of (P ε ) can be shown. Step 2 is devoted to prove some a priori estimates and then extracting a subsequence. In Step 3, we prove that conditions (2.11a), (2.11b), (2.12a) and (2.12b) are satisfied. Finally, in Step 4, we pass to the limit and we show that the constructed function is a renormalized solution.

From this point until the end of the proof, we let i ∈ {1, 2}.

Step 1: Introducing the approximate problem and showing the existence of solution of the approximate problem Let ε > 0. Suppose {f ε } ⊂ L 2 (Ω) such that

f ε -→ f strongly in L 1 (Ω) as ε → 0. Define B ε (x, t) = B(x, T 1/ε (t))
. We now consider the following approximate problem

               -div(B ε (x, u ε 1 )∇u ε 1 ) = f ε in Ω 1 , -div(B ε (x, u ε 2 )∇u ε 2 ) = f ε in Ω 2 , u ε 1 = 0 on ∂Ω, (B ε (x, u ε 1 )∇u ε 1 )ν 1 = (B ε (x, u ε 2 )∇u ε 2 )ν 1 on Γ, (B ε (x, u ε 1 )∇u ε 1 )ν 1 = -h(x)(u ε 1 -u ε 2 ) on Γ, (P ε )
The variational formulation of problem (P ε ) is the following

           Find u ε ∈ V such that ∀ϕ ∈ V Ω1 B ε (x, u ε 1 )∇u ε 1 • ∇ϕ 1 dx + Ω2 B ε (x, u ε 2 )∇u ε 2 • ∇ϕ 2 dx + Γ h(x)(u ε 1 -u ε 2 )(ϕ 1 -ϕ 2 ) dσ = Ω f ε ϕ dx. (3.1)
Using Proposition 2.1 and Schauder's Fixed Point Theorem, the proof of the existence of solution for (3.1) is quite standard (see e.g. [START_REF] Beltran | Homogenization of a quasilinear elliptic problem in a twocomponent domain with an imperfect interface[END_REF]).

Step 2: Extracting subsequences and examining convergences Let u ε = (u ε 1 , u ε 2 ) be a solution to the approximate problem (P ε ). By Stampacchia's theorem, for k > 0, T k (u ε ) ∈ V since u ε ∈ V . Using T k (u ε ) as a test function in the variational formulation (3.1), we have

Ω1 B ε (x, u ε 1 )∇u ε 1 ∇T k (u ε 1 ) dx + Ω2 B ε (x, u ε 2 )∇u ε 2 ∇T k (u ε 2 ) dx + Γ h(x)(u ε 1 -u ε 2 )(T k (u ε 1 ) -T k (u ε 2 )) dσ = Ω f ε T k (u ε ) dx. (3.2)
By the definition of T k , the coercivity of B, and the assumption on h, we have

Ω1 B ε (x, u ε 1 )∇u ε 1 ∇T k (u ε 1 ) dx + Ω2 B ε (x, u ε 2 )∇u ε 2 ∇T k (u ε 2 ) dx + Γ h(x)(u ε 1 -u ε 2 )(T k (u ε 1 ) -T k (u ε 2 )) dσ ≥ α ∇T k (u ε 1 ) 2 L 2 (Ω1) + α ∇T k (u ε 2 ) 2 L 2 (Ω2) + h 0 T k (u ε 1 ) -T k (u ε 2 ) 2 L 2 (Γ) ≥ C 1 T k (u ε ) 2 V ,
for some positive constant C 1 . On the other hand, by Holdër inequality,

Ω f ε T k (u ε ) dx = Ω1 f ε T k (u ε 1 ) dx + Ω2 f ε T k (u ε 2 ) dx ≤ f ε L 1 (Ω) k ≤ M k,
for some positive constant M , which is independent of ε.

Thus,

T k (u ε ) 2 V ≤ M k C 1 , (3.3) that is, the sequence {T k (u ε )} is bounded in V for every k > 0.
By the Rellich theorem, the inclusions V → L 2 (Ω 1 )×L 2 (Ω 2 ) and H 1/2 (Γ) → L 2 (Γ) are compact. Consequently, since {T k (u ε )} is bounded in V for every k > 0 (countable), by the diagonal process, we can extract a subsequence of {T k (u ε )} such that for any k > 0 (k being a rational number), there is a v k ∈ V such that for

     T k (u ε i ) -→ v k,i strongly in L 2 (Ω i ), a.e. in Ω i , T k (u ε i ) v k,i weakly in V, γ i (T k (u ε i )) -→ γ i (v k,i
) strongly in L 2 (Γ), a.e. on Γ.

(3.4)

Now, we show that {u ε i } and {γ i (u ε i )} are Cauchy sequences in measure. For u ε i , we follow the arguments developed in [START_REF] Bénilan | An L 1 theory of existence and uniqueness of nonlinear elliptic equations[END_REF]. For γ i (u ε i ), we have additional difficulties which are overcome by using Proposition 2.1. Note that we have

T k (u ε i ) 2 L 2 (Ωi) = {|u ε i |≥k} |T k (u ε i )| 2 dx + {|u ε i |<k} |T k (u ε i )| 2 dx = {|u ε i |≥k} k 2 dx + {|u ε i |<k} |u ε i | 2 dx.
It follows by Poincaré inequality, Proposition 2.1, and (3.3), we have

k 2 meas{|u ε | ≥ k} = {|u ε 1 |≥k} k 2 dx + {|u ε 2 |≥k} k 2 dx ≤ T k (u ε 1 ) 2 L 2 (Ω1) + T k (u ε 2 ) 2 L 2 (Ω2) ≤ C 3 ∇T k (u ε 1 ) 2 L 2 (Ω1) + T k (u ε 2 ) 2 H 1 (Ω2) ≤ C 4 T k (u ε ) 2 V ≤ C 4 M k C 1 .
for some C 3 , C 4 ∈ R + . Thus, we can find a positive constant C independent of ε such that

meas{|u ε i | ≥ k} ≤ C k (3.5)
For γ 1 (u ε 1 ), observe that by Poincaré inequality and (3.3),

k 2 meas Γ {|γ 1 (u ε 1 )| ≥ k} = {|γ1(u ε 1 )|≥k} k 2 dσ = {|γ1(u ε 1 )|≥k} γ 1 (T k (u ε 1 )) 2 dσ ≤ γ 1 (T k (u ε 1 )) 2 L 2 (Γ) ≤ C 5 ∇T k (u ε 1 ) 2 L 2 (Ω2) ≤ C 6 k.
Consequently,

meas Γ {|γ 1 (u ε 1 )| ≥ k} ≤ C 6 k -→ 0 as k -→ ∞. (3.6)
For γ 2 (u ε 2 ), by the Trace Theorem, Proposition 2.1, and (3.3), we have

k 2 meas Γ {|γ 2 (u ε 2 )| ≥ k} = {|γ2(u ε 2 )|≥k} k 2 dσ = {|γ2(u ε 2 )|≥k} γ 2 (T k (u ε 2 )) 2 dσ ≤ γ 2 (T k (u ε 2 )) 2 L 2 (Γ) ≤ C 7 T k (u ε 2 ) 2 H 1 (Ω2) ≤ C 8 k. It follows that meas Γ {|γ 2 (u ε 2 )| ≥ k} ≤ C 8 k -→ 0 as k -→ ∞. (3.7)
By (3.6) and (3.7), for every η > 0, there exists k 0 such that for every k ≥ k 0 ,

meas Γ {x ∈ Γ; |γ i (u ε i )| ≥ k} < η. (3.8)
Let ω, η > 0. By (3.5) and (3.8), we can find k large enough such that

meas{|u ε i | ≥ k} ≤ η 3 (3.9) meas Γ {x ∈ Γ; |γ i (u ε i )| ≥ k} ≤ η 3 , (3.10)
for every ε > 0. Note that from (3.4), we can deduce that the sequences

{T k (u ε i )}, {γ i (T k (u ε i ))} are Cauchy in measure. Hence, there exists ε 0 > 0 such that meas{|T k (u ε i ) -T k (u ε i )| ≥ ω} < η 3 (3.11) meas Γ {|γ i (T k (u ε i )) -γ i (T k (u ε i ))| ≥ ω} < η 3 , (3.12)
for every 0 < ε , ε < ε 0 .

Observe that

{|u ε i -u ε i | ≥ ω} ⊂ {|u ε i | ≥ k} ∪ {|u ε i | ≥ k} ∪ {|T k (u ε i ) -T k (u ε i )| ≥ ω} and thus, meas{|u ε i -u ε i | ≥ ω} ≤ meas{|u ε i | ≥ k} + meas{|u ε i | ≥ k} + meas{|T k (u ε i ) -T k (u ε i )| ≥ ω}.
It follows from (3.9) and (3.11) that

meas{|u ε i -u ε i | ≥ ω} < η, that is, {u ε i } is indeed Cauchy in measure.
Using the inequalities (3.10) and (3.12), and similar arguments, it can be shown that {γ i (u ε i )} is Cauchy in measure.

Consequently, there is a subsequence of {u ε i } that is convergent a.e. to some measurable function u i : Ω i -→ R, that is

u ε i -→ u i a.e. in Ω i . (3.13)
It follows from (3.5) that u i is finite a.e. in Ω i . This u := (u 1 , u 2 ) is our candidate for a renormalized solution for problem (P). We now prove that u satisfies the conditions (2.11). Indeed, by the continuity of T k , we have

T k (u ε ) -→ T k (u) = v k ∈ V a.e. in Ω \ Γ. ( 3.14)
Moreover, we can deduce that {γ i (u ε i )} is convergent a.e. on Γ up to a subsequence. Hence there exists ω i : Γ -→ R such that

γ i (u ε i ) -→ ω i a.e. on Γ, (3.15)
with ω i finite a.e. on Γ by (3.8). We now identify w i and γ i (u i ). Using (3.3) and (3.4), we obtain

1 k T k (u) 2 V ≤ M C 1 ,
for any k > 0. By Proposition 2.2, γ i (u i ) (the trace in the truncate sense) is well defined. From (3.4), (3.14) and (3.15), we obtain that for any k > 0,

T k (ω i ) = γ i (v k,i ) = γ i (T k (u i )) = T k (γ i (u i )) a.e. on Γ.
Then we have ω i = γ i (u i ) a.e. on Γ. By Fatou's Lemma, T k being nondecreasing, we have for all k > 0,

Γ (u 1 -u 2 )(T k (u 1 )-T k (u 2 )) dσ ≤ lim inf ε →0 Γ (u ε 1 -u ε 2 )(T k (u ε 1 )-T k (u ε 2 )) dσ ≤ kM, which is (2.11b).
Furthermore, since T n (u ε ) converges to T n (u) in L ∞ weak- * and f ε converges to f in L 1 (Ω), we have

1 n Ω f ε T n (u ε ) dx -→ 1 n Ω f T n (u) dx as ε -→ 0. It follows that lim sup ε→0 1 n Ω\Γ B(x,T n (u ε ))∇T n (u ε )∇T n (u ε ) dx + Γ (u ε 1 -u ε 2 )(T n (u ε 1 ) -T n (u ε 2 )) dσ = 1 n Ω f T n (u) dx.
Observe that since u is finite a.e., 1 n T n (u) -→ 0 a.e. in Ω \ Γ.

In addition, for any n > 0, |T n (u)| ≤ n a.e. and thus,

1 n f T n (u) ≤ |f | ∈ L 1 (Ω).
By the Lebesgue Dominated Convergence Theorem, we obtain We need to show that for any v ∈ V ∩ (L ∞ (Ω 1 ) × L ∞ (Ω 2 )), u satisfies (2.13).

lim n→∞ 1 n Ω f T n (u) dx = 0,
We use the function S n defined in (2.14). Note that

S n (u ε i ) = S n (T 2n (u ε i )) ∈ H 1 (Ω i ) ∩ L ∞ (Ω i ) and thus, for any v ∈ V ∩ (L ∞ (Ω 1 ) × L ∞ (Ω 2 )), ψ = vS(u)S n (u ε ) ∈ V ∩ (L ∞ (Ω 1 ) × L ∞ (Ω 2 )).
Using ψ as a test function in (3.1), we have

I 11 + I 12 + I 21 + I 22 + I 31 + I 32 + I 4 = I 5 , (3.21)
where

I 1i = Ωi B ε (x, u ε i )∇u ε i • ∇v i S(u i )S n (u ε i ) dx I 2i = Ωi B ε (x, u ε i )∇u ε i • ∇u i S (u i )S n (u ε i ) dx I 3i = Ωi B ε (x, u ε i )∇u ε i • ∇u ε i S(u i )S n (u ε i ) dx I 4 = Γ h(x)(u ε 1 -u ε 2 )(v 1 S(u 1 )S n (u ε 1 ) -v 2 S(u 2 )S n (u ε 2 )) dσ I 5 = Ω f vS(u)S n (u ε ) dx.
We look at the behavior of each integral. In particular, we will pass to the limit as ε -→ 0 and then as n -→ ∞.

Note that for n ≥ k, we have S n (s)S(s) = S(s) and S n (s)S (s) = S (s), for a.e. s ∈ R. (3.22) We first look at I 1i . Observe that if ε is small enough, we have

B ε (x, u ε i )∇u ε i ∇v i S(u i )S n (u ε i ) = B(x, T 1/ε (u ε i ))∇T 2n (u ε i )S(u i )S n (u ε i ).
Choosing ε small enough, we have

B(x, T 1/ε (u ε i ))S n (u ε i ) = B(x, T 2n (u ε i ))S n (u ε i ) -→ S n (u i )B(x, T n (u i )),
a.e. in Ω i . Moreover, by the assumptions on B, we have

|B(x, T 1/ε (u ε i ))S n (u ε i )| ≤ sup Ωi×[-2n,2n] |B(x, t)|.
It follows from the Lebesgue Dominated Convergence Theorem that

B(x, T 1/ε (u ε i ))S n (u ε i ) = B(x, T 2n (u ε i ))S n (u ε i ) -→ S n (u i )B(x, T 2n (u i )).
a.e. in Ω i and in L ∞ (Ω i ) weak- * . This and (3.17) imply as ε -→ 0, We now observe the behavior of I 2i . For small enough ε, we have

I 1i -→ Ωi B(x, T 2n (u i ))∇T 2n (u i )∇v i S(u i )S n (u i ) dx = Ωi B(x, u i )∇u i ∇v i S(u i )S n (u i ) dx.
B ε (x, u ε i )∇u ε i ∇u i v i S (u i )S n (u ε i ) = B(x, T 2n (u ε i ))∇T 2n (u ε i )∇u i v i S (u i )S n (u ε i ), a.e. in Ω i . Since ∇u i v i S (u i ) = ∇T 2n (u i )vS (u i ) ∈ (L 2 (Ω i ))
N and by (3.17), we obtain as ε -→ 0, For the behavior of I 3i , we observe that

I 2i -→ Ω\Γ B(x, T 2n (u))∇T 2n (u)∇u vS (u)S n (u) dx = Ω\Γ B(x, u)∇u∇u vS (u)S n (u) dx.
|S n (s)| ≤ 1 n , for |s| ≤ 2n.
Consequently, For I 4 , we note that

|I 3i | ≤ v i L ∞ (Ωi) S L ∞ (R) n {|u ε i |<2n} B(x, u ε i )∇T 2n (u ε i )∇T 2n (u ε i ) dx.
h(x)(u ε 1 -u ε 2 )v i S(u i )S n (u ε i ) = h(x)(u ε 1 -u ε 2 )v i S(u i )S n (u ε i )S 2n (u ε i ).
Then we can write I 4 as

I 4 = I 41 + I 42 + I 43 -I 44 ,
where

I 41 = Γ h(x)(u ε 1 -u ε 2 )v 1 S(u 1 )S 2n (u ε 1 )(S n (u ε 1 ) -S n (u ε 2 )) dσ I 42 = Γ h(x)(u ε 1 -u ε 2 )v 1 S(u 1 )S n (u ε 2 )S 2n (u ε 1 ) dσ I 43 = Γ h(x)(u ε 1 -u ε 2 )v 2 S(u 2 )S 2n (u ε 2 )(S n (u ε 1 ) -S n (u ε 2 )) dσ I 44 = Γ h(x)(u ε 1 -u ε 2 )v 2 S(u 2 )S n (u ε 1 )S 2n (u ε 2 ) dσ.
Observe that S n is Lipschitz and S n (u ε i ) = S n (T 2n (u ε i )). This gives

|S n (u ε 1 ) -S n (u ε 2 )| = |S n (T 2n (u ε 1 )) -S n (T 2n(u ε 2 ) )| = 1 n |T 2n (u ε 1 ) -T 2n (u ε 2 )|.
Consequently, For large enough n, for j = 1, 2, i = j, we have S(u i )S 2n (u i )S n (u j ) = S(u i ).

|I 41 | ≤ h L ∞ (Γ) v 1 L ∞ (Γ) S L ∞ (Γ) S n L ∞ (Γ) n Γ |u ε 1 -u ε 2 ||T 2n (u ε 1 )-T 2n (u ε 2 )| dσ,
In view of (2.15) in Remark 2.5, (u 1 -u 2 )S(u 1 ) ∈ L 1 (Γ) so that by Lebesgue Dominated Convergence Theorem, This concludes the proof for the existence of a renormalized solution.

Remark 3.2. As explained in Introduction, it is possible to use the Boccardo-Gallouët estimates, that is, to show that u ε = (u ε 1 , u ε 2 ) is bounded in W 1,q (Ω 1 ) × W 1,q (Ω 2 ), and that leads to u = (u 1 , u 2 ) ∈ W 1,q (Ω 1 ) × W 1,q (Ω 2 ), for all q < N N -1

. Such a result may simplify the proof since it implies that γ 1 (u ε 1 ), γ 2 (u ε 2 ) ∈ W 1-1 q ,q (Γ) and in particular, γ 1 (u ε 1 ), γ 2 (u ε 2 ) are bounded in L 1+η (Γ), for some small enough η. It follows that we can give another definition of renormalized solution including u 1 -u 2 ∈ W 1-1 q ,q (Γ) instead of (2.11b) and then (2.12b) is not necessary since it is a direct consequence of the regularity u 1 -u 2 ∈ W 1-1 q ,q (Γ). However, dealing with homogenization does not allow to use the Boccardo-Gallouët estimates since they are strongly related to the Sobolev constant which may blow up in a varying domain. Moreover, our techniques allow to consider more general equations (with a nonlinear boundary terms) for which the Boccardo-Gallouët estimates are not useful.

Figure 1 :

 1 Figure 1: The two-component domain Ω If we have a function u defined on Ω \ Γ, then we denote u i = u Ωi the restriction of u in Ω i . Furthermore, we have the following assumptions: (A1) The data f belongs to L 1 (Ω). (A2) The function h satisfies h ∈ L ∞ (Γ) and 0 < h 0 < h(y) a.e. on Γ, (2.1)

Theorem 3 . 1 .

 31 Suppose the assumptions (A1)-(A3) hold. Then there exists a renormalized solution to (P) in the sense of Definition 2.4.

By ( 3

 3 , u i )∇u i ∇v i S(u i ) dx.(3.23)

By ( 3

 3 u)∇u • ∇u vS (u) dx.(3.24)

By ( 3

 3

For

  I 42 , we observe that|h(u ε 1 -u ε 2 )v 1 S(u 1 )S 2n (u 1 )S n (u ε 2 )| ≤ M,where the constant M depends only on the L ∞ -norms of h, S, S n and S 2n , and n. Also,h(x)(u ε 1 -u εa.e. on Γ as ε -→ 0. By Lebesgue Dominated Convergence Theorem, as ε -→ 0,I 42 -→ Γ h(x)(u 1 -u 2 )v 1 S(u 1 )S 2n (u 1 )S n (u 2 ) dσand similarly,I 44 -→ Γ h(x)(u 1 -u 2 )v2S(u 2 )S 2n (u 2 )S n (u 1 ) dσ.

1 - 1 -From

 11 u 2 )v 1 S(u 1 ) dσ (3.28) lim n→∞ lim ε→0 I 44 = Γ h(u 1 -u 2 )v 2 S(u 2 ) dσ. (3.29) Combining (3.26), (3.28), (3.27) and (3.29), we conclude thatlim u 2 )(v 1 S(u 1 ) -v 2 S(u 2 )) dσ.(3.30)Finally for I 5 , observing that S n (u ε ) weakly converges to S n (u) in L ∞ (Ω \ Γ) weak- * and a.e. in Ω \ Γ, f ε converges strongly to f in L 1 (Ω \ Γ), we haveI 5 = Ω\Γ f ε vS(u)S n (u ε ) dx -→ Ω\Γ f vS(u)S n (u) dx = Ωf vS(u)S n (u) dx. limit of (3.21) and using (3.23), (3.24), (3.25), (3.30), and (3.31), we have the desired conclusion.

L 2 (Γ) .

)v 1 S(u 1 )S 2n (u ε 1 )S n (u ε 2 ) -→ h(x)(u 1 -u 2 )v 1 S(u 1 )S 2n (u 1 )S n (u 2 ),

From this point, we just denote our sequence by ε. Rewriting all the results we got in terms of ε, we have the following: for all k > 0,

(3.16)

In addition, we have

(3.17)

Step 3: Showing conditions (2.12) of Definition 2.4.

From the continuity of B and (3.16), we have that for any fixed n > 0,

Due to (A3.1) and the lower semi-continuity of the weak convergence,

and by Fatou's Lemma,

We use 1 n T n (u ε ) as a test function in (3.1) to arrive at

Consequently, for ε small enough, we have

Remark 3.3 (Stability)

. By adapting the proof of Theorem 3.1, it is possible to derive a stability result. More precisely, let us consider u ε , a renormalized solution of

x ∈ Ω, for all t ∈ R, for any ξ ∈ R N , and

a.e. on Γ and h ε (y) < M (uniform), for some M > 0.

for every sequence r ε ∈ R such that r ε -→ r a.e. on R; h ε -→ h a.e. in Γ, then u ε converges to u a.e. where u is a renormalized solution of

The main point is to obtain the a priori estimates of Step 2. In view of (2.16) in Remark 2.5, T k (u ε ) is an "admissible" test function, so that

which gives all the necessary estimates. Then we can extract subsequences so that (3.16) hold true. In view of conditions on f ε , B ε , and h ε , we can perform

Step 3 of the proof of Theorem 3.1.