Pierre-Carl Langlais

Jean-Baptiste Camps

Nicolas Baumard

Olivier Morin

From Roland to Conan: First results on the corpus of French literary fictions (1050-1920)

A corpus of French literary fictions: 1050-1920

A decade ago, when he set out to explore the evolution of English novels in the period 1740-1850, F. Moretti had to focus on the titles as the main available data, yet noting that: in a few years, we will have a digital archive with the full texts of (almost) all novels ever published [START_REF] Moretti | Style, Inc. Reflections on Seven Thousand Titles (British Novels, 1740-1850)[END_REF].

Today, we propose to embark on a journey to model the evolution of French literary fictions, from their epic and chivalric origins up to the more modern productions of genres such as heroïc fantasy or historical novels. We offer to do that not from the titles, but by analysing a corpus, currently in construction, whose aim is to cover all French literary fiction digitized from the earliest sample of French literature to the 20th century (or more precisely, to the point were a significant share of published literature is not yet in the public domain).

This project is at the intersection of two major trends in computational literary analysis: the creation and documentation of large literary corpora and the analysis of literary genre and discourse through machine learning classification. In comparison with previous examples of French literary corpus (like théâtre classique [START_REF] Fièvre | Théâtre classique[END_REF]) or the French corpus of the European literary text collection [START_REF] Odebrecht | European literary text collection (eltec)[END_REF], French novels make up a massive amount of texts (80,000 registered work in the French National Library before the 20th century), a large share of which is non-canonical and little documented. Text mining techniques make it possible to explore and document large digitized corpora with little editorial work. Classification is not simply used as cataloguing tool: its limitations can in fact inform in a more complex way the development of genres and the intertextual interplay between one genre and another.

The French fiction corpus initially results from the collocation of three different collections:

1. A collection of medieval fictions and chansons de gestes (1050-1450) -(cf. [START_REF] Camps | Geste: un corpus de chansons de geste, 2016[END_REF] 2. A collection of printed fictions of Gallica from the modern period and the 19th century (Langlais, 2021b).

3. A new collection comprising most of the digitized fictions from the early modern period (1450-1700).

The second corpus relied on one of the oldest bibliographic database: the catalog of the French National Library. The classification scheme used by the library until 1996 was developed between 1684 and 1688 by the librarian of Louis XIV, Nicolas Clément. A specific category for novels (Y2) has been existing since 1730 as a duplication of the category for poetry (Y). The novels are now classified in Y2 or Y Bis. For cultural history, the BNF catalog presents a major interest: the categories are often (nearly) contemporary of the documents they aim to classify.

This corpus was made possible thanks to the policy of open data and open content which the BNF has been engaged in for several years. While the Clément classification ceased to be used for the classification of physical collections in 1996, it has become available in 2017 for data analysis when the BNF opened a new catalog access service, the SRU 1 . It was originally limited to novels digitized by the French National Library with a strong focus on the 19th century.

The third corpus was created specifically for the project: it aims to cover all the digitized versions of early modern novels published in France between 1473 (publication date of our earliest entry, Le Roman de Jason et Médée) and 1700 (our arbitrary cutoff, for now). This collection originally aimed to bridge the two previous corpora which were focused on two extreme temporal points of the history of French literature (the medieval period and the post-1800 period). It became a more ambitious experiment: a systematic harvesting of all available digitizations. The collections of digital documents already available in Fictions littéraires de Gallica have been significantly expanded thanks to the semi-automatical retrieval of documents digitized by Google Books and other online digital library. The creation of this composite corpus underline that digitized collections may be already more representative than expected, although nobody knew it or could measure the scale of it.

Using the combined collections of several digital libraries made it possible to cover a large amount of the novels registered in the catalog of the French National Library. For the 1470-1600 period, we have retrieved 275 novels out 349, that is 78.8% of the corpus. For the 1600-1700 period, we have 724 novels out of 1058, that is 68.4% of the corpus. Representativeness ratio is not only high but consistent on the entire time period as shown in figure 2 as the size of the available digitized corpus remains proportional to the total amount of novels identified by the French National Library.

Such high coverage seems to alleviate most concerns related to the representativeness of digital corpora: it becomes less likely that important genres or themas are neglected. Yet the novels registered on the French National Library catalog do not encompass the total sum of literary fiction published or circulated on the period. The comparison with Google Books showed that numerous editions were not recorded by Gallica. In the context of the project, we only checked novels recorded on the catalogue, but this discrepancy shows that some published novels could definitely have been overlooked, although by definition these documents are not necessarily expected to belong to those with the largest impact, as they failed to be noticed by bibliographic records. After the 19th century, nearly all published monographs are expected to be indexed in the catalogue of the French National Library due to progress in the implementation of the policy of legal deposit (dépôt légal, established in 1537). Yet, at this time, a large amount of literature was increasingly being published in periodicals and newspapers.

In short as representativeness of existing library catalogues becomes less of a concern, digital collections are bound to raise more complex issues and address a more critical perspective on existing bibliographic resources: what is a "novel"? what is a "publication"? what about piracy edition or periodical fictions?

The creation of the third corpus also aims to benefit from unprecedented progresses in the detection of historical OCR. The OCR17+ model trained by Claire Jahan and Simon Gabay on a collection of 17th century prints already yields a usable text for most of the 16th to 18th century 1 Search/Retrieve via URL, https://api.bnf.fr/fr/api-sru-catalogue-general.

Modelling literary genre: a back and forth approach

Chivalric romance is one of oldest and most enduring genre of European literature. It finds its origin in Old French epics, known as chansons de geste, whose first attestations go back to the 11th century, and in the genre of roman that emerges in the second half of the 12the century. Continuously modified, adapted and rewritten during the Middle Ages and the Early Modern times, this repertoire of fictions provided numerous tropes and patterns that arguably live on in more modern and even contemporary literature, especially in popular literature and even fantasy novels.

Preliminary exploration of our large corpus of French literary fictions suggests that the chivalric novel, rather than ceasing to exist, has been continually evolving and gradually morphed into forms close to the contemporary fantasy novels. This structural transformation has been largely overlooked as forms of chilvalric romances have largely disappeared from the high literary canon 2 [START_REF] Jahan | Ocr17+ -layout analysis and text recognition for 17th c. french prints[END_REF]. This model will be applied on the entire corpus. Preliminary results on a random selection of 60 pages from the corpus show that OCR quality is already high for the 17th century (nearly 99% character accuracy). The 16th century corpus is more challenging due to specific issues with Gothic fonts and segmentation, although we intend to solve these limitations by fine tuning a model for Gothic fonts in incunabula and 16th century prints. In comparison, the mean OCR quality in the collections held by Gallica is at roughly 80% for the 16th and 17th centuries (according to available metadata) and a large part is currently unavailable for full-text search, possibly because OCR quality was too low. The project will also benefit from the development made by the ongoing Gallic(orpor)a project, that aims to provide a fully reusable pipeline for Gallica documents in French from the Middle Ages to the Revolution. By the end of this project, we plan to create a small search engines of the entire collected corpus of 15th to 17th century documents. This would be the first resource to give access to a large share of the published literature in France in the early modern period. An initial version of this search engine may be available by the time of the conference and serve as a demonstration of our methodology of systematic collection and enhancement of available digitized collections.

after the Renaissance and continued to evolve in the production of lesser known and more obscure authors.

Our analysis relies on a anachronistic use of classification recently pioneered in cultural analytics [START_REF] José | The Novel in the Spanish Silver Age: A Digital Analysis of Genre Using Machine Learning[END_REF], Underwood, 2019). Probability rates, cross-classifications of the same text and, even, classification failures are reinterpreted as a way to measure the complex evolution of literary genres.

To investigate the transformation of chivalric novels into contemporary genres like fantasy, we created two historical models of literary genres: a 21st century model and an early modern "Fresnoy" model (from a 1731 catalog). The combined use of anachronistic classification aims to locate the two parallel processes of genre survival/transformation (for chivalric romance) and genre emergence/coagulation (for fantasy).

Classification was made with SVM using an R library initially developed for the classification of newspaper genres in the 19th century, TidySupervise (Langlais, 2021a).

The 21st century model is created using nine genre categories of the social cataloguing website Babelio3 . User-generated tags have recently emerged as an important source of information in computational analysis of contemporary literature and literary reception [START_REF] Walsh | The goodreads "Classics": A computational study of readers, amazon, and crowdsourced amateur criticism[END_REF]. A French counterpart to Goodreads, Babelio has a significant impact among French literary readers with nearly 1 million visits per month. In this project, we focused on a subset of generic tags that were either major acknowledged genres in contemporary literature or relevant for our ongoing projects: romance, fantasy, detective fiction, science fiction, historical novel, adventure novel, social novel, fantastique novel and erotica. Obviously, this is not a straightforward classification, since one novel could belong to several categories. We aimed rather to reconstruct the fuzzy space of contemporary literary genres in France with all its underlying uncertainties and overlaps.

The model was trained on 4,081 segments of 1000 words extracted from 1,346 novels. We applied a random selection of three 1000 words segments by work. This selection aims to limit over-fitting and ensure that the model will be correctly trained on generic features and not on the style of specific novels. Bootstrap evaluation of the model yield a 75% accuracy, yet with significant variations among the genres (fantasy being the highest rated with science fiction and erotica).

Preliminary results from the 21st century model have revealed significant examples of "missing links" between the early modern chivalric romances and the fantasy novels (Table 1). Of special interest is La Mort de Roland a 1858 rewrite of the Chanson de Roland by Alfred Assolant that explicitly claims to be an epic fantasy (Fantaisie épique).

Most of these works are poorly attested in literary history. In comparison, the results from Science-fiction yields much more expected and "canonic" works (especially from Jules Verne).

The early modern model (or "Fresnoy" model) has been made possible by an exceptional historical source on literary genre classification: the second volumes of De l'usage des romans, où l'on fait voir leur utilité & leurs différens caractères by Nicolas Lenglet du Fresnoy (first published in 1731 under the pseudonym of Gordon de Percel). It is a catalog of a large among of French, Spanish and Italian novels published since 1731 broken down by genres according to the prevalent taxonomy of the time: Roman de chevalerie, Roman d'amour, Roman historique, Roman comique, Roman politique, etc. Ongoing work aims to reconcile the classification of Lenglet du Fresnoy with our corpus of digitized novels. Preliminary results suggest that the generic identity in the catalogue of Fresnoy is much stronger than in the Babelio dataset, with as much as 93% accuracy on four genres (Roman d'amour, Roman historique, Roman de chevalerie and Roman comique & satirique) in our initial run. While this high accuracy may be caused by overfitting on a limited samples of novels, it seems also consistent with the significance of genre classification in the Title

Figure 1 :

 1 Figure 1: Composition of the corpus by decade

https://www.babelio.com/.

A this stage, the expansion of the corpus and the re-digitization of available documents with historical models for OCR will be crucial to move beyond an exploratory phase and design a more systematic examination of the metamorphosis of literary genre.


BAYESIAN MULTI-OBJECTIVE OPTIMIZATION FOR QUANTITATIVE RISK ASSESSMENT IN MICROBIOLOGY

Subhasish Basak 1,2 , Julien Bect 2 , Laurent Guillier 1 , Fanny Tenenhaus-Aziza 3 , Janushan Christy 4 , Emmanuel Vazquez 2 1 Agence nationale de sécurité sanitaire (ANSES), 94700, Maisons-Alfort, France 2 Université Paris-Saclay, CNRS, CentraleSupélec, L2S, 91190, Gif-sur-yvette, France 3 Centre national interprofessionnel de l'économie laitière (CNIEL), 75009, Paris, France 4 Centre technique d'expertise agroalimentaire (ACTALIA), 74800, La-Roche-sur-Foron, France BAYESIAN MULTI-OBJECTIVE OPTIMIZATION FOR QUANTITATIVE RISK ASSESSMENT IN MICROBIOLOGY Subhasish Basak 1,2 , Julien Bect 2 , Laurent Guillier 1 , Fanny Tenenhaus-Aziza 3 , Janushan Christy 4 , Emmanuel Vazquez 2 1 Agence nationale de sécurité sanitaire (ANSES), 94700, Maisons-Alfort, France 2 Université Paris-Saclay, CNRS, CentraleSupélec, L2S, 91190, Gif-sur-yvette, France 3 Centre national interprofessionnel de l'économie laitière (CNIEL), 75009, Paris, France 4 Centre technique d'expertise agroalimentaire (ACTALIA), 74800, La-Roche-sur-Foron, France This work is part of the ArtiSaneFood project (grant number : ANR-18-PRIM-0015) which is part of the PRIMA program supported by the European Union.

Context

As a part of the European project ArtiSaneFood, the primary goal of this collaborative work between ANSES, CNIEL and L2S is to establish efficient bio-intervention strategies for cheese producers in France, in order to "economically" reduce the risk of Haemolytic Uremic Syndrome (HUS) caused by Shiga-Toxin producing Escherichia Coli (STEC) present in raw-milk soft cheese. This translates into a multiobjective optimization problem of a stochastic simulator based on a quantitative risk assessment (QRA) model proposed by [START_REF] Perrin | Quantitative risk assessment of haemolytic and uremic syndrome linked to O157:H7 and non-O157:H7 shiga-toxin producing escherichia coli strains in raw milk soft cheeses[END_REF], to estimate the Pareto optimal solutions for the process intervention parameters.

QRA simulator

farm

P batch R batch cheese postharvest consumer Y STEC milk N colony s Y colony s Batch level stochastic simulator
The farm-to-fork continuum for one batch

P batch = P [Γ > 0] = 1 -exp(-K • n sample )
Quantities of Interest (QoI)

Several batches are simulated to estimate

• R avg = E[R batch • (1 -P batch • p cheese test )] • P avg = E[P batch • p cheese test ] QoIs are • Relative risk: f 1 = R avg (1-P avg )•K 1 , (K 1 : baseline risk)
• Batch rejection rate: f 2 = P avg

Multi-objective optimization

We consider the multi-objective optimization problem

min x∈X f (x)
where, f = (f 1 , f 2 , . . .)

• The solution set P contains Pareto optimal points

P = {x ∈ X : x ∈ X, f (x ) ≺ f (x)} where f ≺ f =⇒ f i ≤ f i , ∀i, with at least one of the inequalities being strict z 1 z 2 z 3 z 4 z 5 f 1 f 2
Pareto optimal points z 1 and z 2

• In a stochastic setting, we assume additive noise: for a given x i ∈ X, we observe

Z i = f (x i ) + ε i , ε i ∼ N (0, Σ)

PALS

Proposed by [START_REF] Zuluaga | Active learning for multi-objective optimization[END_REF] and extended for the stochastic case by [START_REF] Barracosa | Extension of the Pareto Active Learning method to multi-objective optimization for stochastic simulators[END_REF].

• Initial design I 0 , n = 0, β > 0

• Step I -∀x ∈ X, construct confidence rectangles R n (x) = {z ∈ R 2 + : R - n (x) ≺ z ≺ R + n (
x)} using posterior means µ n,j (x) and variances σ 2 n,j (x) R ± n,j (x) = µ n,j (x) ± β 1/2 σ n,j (x) based on I n (indep. GP models for f j , j ∈ {1, 2})

• Step II -Classify each x ∈ X into one of three classes -Deemed Pareto optimal P n :

P n = {x ∈ X| x ∈ X \ {x}, R - n (x ) ≺ R + n (x)} -Non Pareto optimal N n : N n = {x ∈ X|∃x ∈ X \ {x}, R + n (x ) ≺ R - n (x)} -Unclassified U n = X \ (P n ∪ N n ) • Step III -Sample the next point of evaluation X n+1 = arg max x∈(U n ∪P n ) R - n (x) -R + n (x) 2 I n+1 = I n ∪ (X n+1 , Z n+1 ), Z n+1 = f (X n+1 ) + ε n+1 • Increase n and Repeat I-III until n ≤ n max

PALS with quantiles

PALS is not directly suitable for our application:

• f 1 is not the expectation of a simulator output,

• we build a GP model for R avg instead. So, we propose using quantiles of QoI to construct confidence rectangles of coverage probability α R - n,j (x) = q n,j (α/2) and R + n,j (x) = q n,j (1 -α/2) with q n,j (.) being a posterior quantile of f j (x)

Experimental results

• Minimizing f , given p milk test = 1, p cheese test = 0.5, w.r.t.

n sample ∈ {1, 5, 10, 15, , 12, 14, 16, 18, 20, 25, 30, 40

• • • , 50} -l sort ∈ {10

  front (stripes corresponding diff. values of l sort ) -Pareto optimal (green) and dominated points (red) • Pareto front estimated using PALS -|I 0 | = 10, n max = 30, samples/evaluation = 200 using significantly less (40 × 200) budget, the user can provide the following insights -Points with l sort > 12 are dominated (red) l sort ≤ 12 remain unclassified (blue) Future work • Integrate milk loss as objective and p milk test , p cheese test as design variables • Take the correlation between outputs into account


Omar Anza

Jean-Philippe Mandallena
email: jean-philippe.mandallena@unimes.fr.

Γ Γ Γ-CONVERGENCE OF NONCONVEX UNBOUNDED INTEGRALS IN STRONGLY CONNECTED SETS

Keywords: convergence, strong connectedness, perforated sets, nonconvex unbounded integrals, homogenization

We study Γ-convergence of nonconvex integrals of the calculus of variations in strongly connected sets when the integrands have not polynomial growth and can take infinite values. Application to homogenization of unbounded integrals in strongly perforated sets is also developed.

Introduction

Asymptotic analysis for boundary value problems in perforated sets was studied for the first time by Cioranescu and Saint Jean Paulin (see [START_REF] Cioranescu | Homogenization of reticulated structures[END_REF] and the reference therein) and Khruslov and Marchenko (see [START_REF] Marchenko | Homogenization of partial differential equations[END_REF] and the reference therein). The approach of Cioranescu and Saint Jean Paulin is based upon multiscale methods like formal two-scale asymptotic expansions, compensated compactness and oscillating test functions and twoscale convergence (see [START_REF] Jikov | Homogenization of differential operators and integral functionals[END_REF][START_REF] Cioranescu | An introduction to homogenization[END_REF][START_REF] Chechkin | Homogenization[END_REF]) while the one of Khruslov and Marchenko uses variational analysis like Γ-convergence (see [START_REF] Dal | An introduction to Γ-convergence[END_REF][START_REF] Braides | Homogenization of multiple integrals[END_REF]). The common point of the two approaches is the use of extension theorems for passing from perforated to non-perforated sets (see [START_REF] Cioranescu | Homogenization in open sets with holes[END_REF][START_REF] Acerbi | An extension theorem from connected sets, and homogenization in general periodic domains[END_REF]). In this paper we consider the variational approach which consists of computing the Γ-limit as ε 0 of integral functionals of type ż O W ε px, ∇upxqq1 Oε pxqdx,

(1.1)

where u P W 1,p pO; R m q with p ą 1 and, for each ε ą 0, W ε : O ˆM r0, 8s is a not necessarily convex Borel measurable function with M denoting the space of m ˆN matrices and OzO ε represents the holes at the scale ε in the bounded open set O Ă R N . Following Khruslov and Marchenko, the Γ-convergence is taken with respect to the L p pO ε , Oqconvergence, i.e. }u ε ´u} L p pOε;R m q " 0, and our object is to deal with the problem of finding a ΓpL p pO ε , Oqq-limit of type ż

O W lim px, ∇upxqqdx (1.2)
with W lim : O ˆM r0, 8s an integrand for which we wish to give a formula depending on W ε . When W ε has p-growth, i.e. there exist α, β ą 0, which does not depend on ε, such that for every px, ξq P O ˆM, α|ξ| p 1 Oε pxq ď W ε px, ξq1 Oε pxq ď βp1 `|ξ| p q1 Oε pxq, the problem was treated (for general periodic perforated sets) by Acerbi, Chiadò Piat, Dal Maso and Percivale in the scalar case (see [START_REF] Acerbi | An extension theorem from connected sets, and homogenization in general periodic domains[END_REF]) and by Braides and Chiadò Piat in the vector-valued case (see [START_REF] Braides | Remarks on the homogenization of connected media[END_REF]). In the present paper we consider the case where p ą N and W ε has G-growth, i.e. there exists a Borel measurable and p-coercive function G : M r0, 8s and there exist α, β ą 0, which does not depend on ε, such that for every px, ξq P O ˆM, αGpξq1 Oε pxq ď W ε px, ξq1 Oε pxq ď βp1 `Gpξqq1 Oε pxq, which allows to W ε to take infinite values. Note that, as in the p-growth case, since G is pcoercive, there exists C ą 0, which does not depend on ε, such that for every px, ξq P O ˆM, W ε px, ξq1 Oε pxq ě C|ξ| p 1 Oε pxq.

(1.3) Such a unbounded case is of interest in nonlinear elasticity where a fundamental open problem is to develop variational techniques to deal with energy densities that can take infinite values and verify the two basic conditions of nonlinear elasticity, namely the noninterpenetration of the matter and the neccessity of an infinite amount of energy to compress a finite volume into zero volume. The results of this paper give some improvments in this direction in the framework of perforated nonlinear elastic materials.

The plan of the paper is as follows. In §2.1 we recall the definition of L p pO ε , Oq-convergence (see Definition 2.1) and the one of strongly connected set (see Definition 2.3). Note that a weaker notion of connected set exists (see [START_REF] Cioranescu | Homogenization of reticulated structures[END_REF] Chapter 1, §2.5 pp. 40] and [BD98, Chapter 19 pp. 167] for more details). This weak notion allows to consider more general perforated sets but it is not considered here due to the fact that our Γ-convergence method does not apply in such a situation. In fact, the Γ-convergence of unbounded integrals in weakly connected sets is an open problem. When O ε is strongly connected, bounded sequences in W 1,p pO ε ; R m q are relatively compact with respect to the L p pO ε , Oq-convergence (see Theorem 2.4). This makes that ΓpL p pO ε , Oqq-convergence (see Definition 2.5 in §2.2) is well adapted to deal with variational problems involving integral functionals of type (1.1) satisfying (1.3) (see Proposition 2.7 in §2.2). Our main result, which establishes the ΓpL p pO ε , Oqq-convergence of (1.1) to (1.2), is stated in §3.1 and proved in §5.3, see Theorem 3.6 and also Proposition 3.7 which makes more precise the formula of the limit integrand W lim in (1.2). Classically, its proof is a consequence of Proposition 3.4 (the lower bound) and Proposition 3.5 (the upper bound). The proofs of Propositions 3.4 and 3.5 are given in §5.1 and §5.2 respectively. In §2.3 we recall the concept of (family of) ru-usc 1 integrand(s) and its main properties which are used in the proof of both Propositions 3.4 and 3.5 (and in Proposition 3.7). The proof of Proposition 3.5 also needs the use of the Vitali envelope of a set function which is recalled in §4.1. Finally, application to homogenization of unbounded integrals in strongly perforated sets is developed in §3.2, see Theorem 3.11. This homogenization theorem is proved in §5.4 by using an extension theorem (see Theorem 3.8) and a subadditive theorem (see Theorem 4.4 in §4.2).

Notation. Let M be the space of m ˆN matrices, for A Ă M we denote the interior and the closure of A by intpAq and A respectively. The symbol ş stands for the mean-value integral with respect to the Lebesgue measure L N on R N , i.e. şQ " 1 L N pQq şQ .

Preliminaries

In what follows, m, N ě 1 are two integers and p ą 1 is a real number. L N pOzO ε q " 0;

1 The abbreviation ru-usc means radially uniformly upper semicontinuous.

(C 2 ) there exists C ą 0 such that for all tu ε u εą0 Ă W 1,p pO; R m q there exists tp u ε u εą0 Ă W 1,p pO; R m q such that for all ε ą 0,

$ & % p u ε " u ε on O ε }p u ε } L p pO;R m q ď C}u ε } L p pOε;R m q }∇p u ε } L p pO;R m q ď C}∇u ε } L p pOε;R m q .
The following two definitions are due to Khruslov and Marchenko (see [MK06, Chapter 4, Definition 4.5 pp. 114 and Definition 4.7 pp. 116]).

Definition 2.1. We say that tu ε u εą0 Ă L p pO; R m q is L p pO ε , Oq-convergent if there exists u P L p pO; R m q such that lim

ε 0 }u ε ´u} L p pOε;R m q " 0, i.e. lim ε 0 ż Oε |u ε ´u| p dx " 0.
We then write u ε L p pOε,Oq u.

Remark 2.2. Under (C 1 ) if tu ε u εą0 Ă L p pO; R m q is L p pO ε , Oq-convergent then its limit is unique.

Definition 2.3. When (C 1 )-(C 2 ) hold we say that tO ε u εą0 is p-strongly connected.

Khruslov and Marchenko have also proved the following compactness result (see [MK06, Chapter 4, Theorem 4.8 pp. 116]).

Theorem 2.4. Assume that tO ε u εą0 is p-strongly connected. If tu ε u εą0 Ă W 1,p pO; R m q and if sup εą0 }u ε } W 1,p pOε;R m q ă 8 then, up to a subsequence, there exists u P W 1,p pO; R m q such that u ε L p pOε,Oq u.

2.2. ΓpL p pO ε , Oqq-convergence. We begin with the definition of ΓpL p pO ε , Oqq-convergence.

(For more details on the theory of Γ-convergence we refer to [START_REF] Dal | An introduction to Γ-convergence[END_REF].)

Definition 2.5. By the ΓpL p pO ε , Oqq-limit of I ε : W 1,p pO; R m q r0, 8s as ε 0 we mean a functional I lim : W 1,p pO; R m q r0, 8s such that: Γ-lim: for every u P W 1,p pO; R m q, ΓpL p pO ε , Oqq-lim ε 0 I ε puq ě I lim puq with ΓpL p pO ε , Oqq-lim

ε 0 I ε puq :" inf " lim ε 0 I ε pu ε q : u ε L p pOε,Oq - u * ,
or equivalently, for every u P W 1,p pO; R m q and every tu

ε u εą0 Ă W 1,p pO; R m q such that u ε L p pOε,Oq - u, lim ε 0 I ε pu ε q ě I lim puq;
Γ-lim: for every u P W 1,p pO; R m q, ΓpL p pO ε , Oqq-lim ε 0 I ε puq ď I lim puq with ΓpL p pO ε , Oqq-lim

ε 0 I ε puq :" inf ! lim ε 0 I ε pu ε q : u ε L p pOε,Oq - u ) ,
or equivalently, for every u P W 1,p pO; R m q there exists tu ε u εą0 Ă W 1,p pO; R m q such that u ε L p pOε,Oq u and lim ε 0

I ε pu ε q ď I lim puq.
We then write I lim " ΓpL p pO ε , Oqq-lim

ε 0 I ε .
Remark 2.6. It is easy to see that ΓpL p pO ε , Oqq-lim ε 0 I ε and ΓpL p pO ε , Oqq-lim ε 0 I ε are lsc 2 with respect to the L p pO; R m q-convergence.

Theorem 2.4 makes that ΓpL p pO ε , Oqq-convergence is well adapted to deal with variational problems involving integral functionals of type (1.1) satisfying (1.3). This is summarized in the following proposition whose proof is left to the reader.

Proposition 2.7. Assume that tO ε u εą0 is p-strongly connected, ΓpL p pO ε , Oqq-lim ε 0 I ε " I lim and there exists C ą 0 such that for every ε ą 0 and u P W 1,p pO; R m q,

I ε puq ě C ż Oε |∇upxq| p dx.
Let f P L q pO; R m q with 1 p `1 q " 1 and, for each ε ą 0, set

θ ε :" inf " I ε puq ´żOε f pxqupxqdx : u P W 1,p pO; R m q * .
Then, every minimizing sequence tu ε u εą0 for the variational problems θ ε is relatively compact with respect to the L p pO ε , Oq-convergence, and every L p pO ε , Oq-cluster point u of tu ε u εą0 is such that

lim ε 0 θ ε " I lim puq ´żO f pxqupxqdx " inf " I lim puq ´żO f pxqupxqdx : u P W 1,p pO; R m q * .
2.3. Ru-usc property. We begin by recalling the concept of ru-usc function which was introduced in [AH10] (see also [START_REF] Anza | Radial representation of lower semicontinuous envelope[END_REF] and [AHM11, §3.1]). Definition 2.10. For each ε ą 0, let L ε : O ˆM r0, 8s be a Borel measurable function. We say that the family tL ε u εą0 is ru-usc if there exist ta ε u εą0 Ă L 1 pO; s0, 8sq and a 0 P L 1 pO; s0, 8sq such that:

2.

lim ε 0 ż A a ε pxqdx " ż A a 0 pxqdx for all A P OpOq; lim t 1 ´sup εą0 ∆ aε
Lε ptq ď 0.

The following lemma will be useful for dealing with ΓpL p pO ε , Oqq-convergence. (For a proof we refer to [AHM21, Lemma 2.21].)

Lemma 2.11. For each ε ą 0, let L ε : O ˆM r0, 8s be a Borel measurable function and, for each ρ ą 0, let H ρ rL ε s : O ˆM r0, 8s be defined by

H ρ rL ε spx, ξq :" inf # ż Qρpxq L ε py, ξ `∇vpyqqdy : v P W 1,p 0 pQ ρ pxq; R m q + (2.4)
with Q ρ pxq :" x`s ´ρ 2 , ρ 2 r N . If tL ε u εą0 is ru-usc with ta ε u εą0 Ă L 1 pO; s0, 8sq and a 0 P L 8 pO; s0, 8sq then L 8 :" lim

ρ 0 lim ε 0 H ρ rL ε s : O ˆM r0, 8s
is ru-usc with the constant function }a 0 } L 8 .

For application to homogenization (see §3.2) we will need the following result. (For a proof we refer to [AHM21, Lemma 2.24].)

Lemma 2.12. Let L : R N ˆM r0, 8s be Borel measurable function such that Lp¨, ξq is 1-periodic for all ξ P M, i.e. for every px, zq P R N ˆZN , Lpx `z, ξq " Lpx, ξq, and, for each ε ą 0, let L ε : O ˆM r0, 8s be defined by

L ε px, ξq :" L ´x ε , ξ ¯.
Let a P L 1 loc pR N ; s0, 8sq be a 1-periodic function and, for each ε ą 0, let a ε P L 1 loc pR N ; s0, 8sq be defined by

a ε pxq :" a ´x ε ¯.
If L is ru-usc with the function a then tL ε u εą0 is ru-usc with the family of functions ta ε u εą0 and the constant function xay :" ş s0,1r N apyqdy.

Main results

3

Q Ă O, if L N pO ε X Qq ą 0 then H N ´1pO ε X BQq ą 0;
Let M denote the space of m ˆN matrices and let G : M r0, 8s be a Borel measurable function satisfying the following conditions: (A 0 ) 0 P intpGq, where G denotes the effective domain of G, i.e. G :" tξ P M : Gpξq ă 8u; (A 1 ) there exists γ ą 0 such that for every ξ, ζ P M and every t Ps0, 1r, Gptξ `p1 ´tqζq ď γp1 `Gpξq `Gpζqq;

(A 2 ) G is p-coercive, i.e. there exists c ą 0 such that for every ξ P M, Gpξq ě c|ξ| p . Remark 3.1. If (A 1 ) is satisfied then G is convex, but G is not necessarily convex (see [AHMZ15, Sect. 9]). So, if moreover (A 0 ) holds then tG Ă intpGq for all t Ps0, 1r, and there exists r ą 0 such that sup

|ξ|ďr Gpξq ă 8, see [AHM12, Lemma 4.1].
Let G 8 : O ˆM r0, 8s be defined by

G 8 px, ξq :" lim ρ 0 lim ε 0 H ρ rG1 Oε spx, ξq (3.1)
with H ρ rG1 Oε s given by (2.4) with L ε " G1 Oε where G1 Oε : O ˆM r0, 8s is defined by G1 Oε px, ξq " Gpξq1 Oε pxq. Denote the effective domain of G 8 px, ¨q by G 8,x . We further suppose that:

(A 3 ) for every u P W 1,p pO; R m q, if ş O G 8 px, ∇upxqqdx ă 8 and if ∇upxq P intpG 8,x q for L N -a.a. x P O, then ş O Gp∇upxqqdx ă 8; (A 4 ) for every x P O, G 8 px, ¨q is lsc on intpG 8,x q.
Remark 3.2.

(i) For every px, ξq P O ˆM, G 8 px, ξq ď Gpξq, and so G Ă G 8,x for all x P O.

(ii) Defining G, G 8 : W 1,p pO; R m q r0, 8s by Gpuq :" ş O Gp∇upxqqdx and G 8 puq :" ş O G 8 px, ∇upxqqdx and denoting their effective domains by dompGq and dompG 8 q, we see that (A 3 ) means that ! u P dompG 8 q : ∇upxq P intpG 8,x q for L N -a.a. x P O ) Ă dompGq.

(iii) If either dompG 8 q " dompGq or Gpuq ă 8 for all u P W 1,p pO; R m q such that ∇upxq P intpG 8,x q for L N -a.a. x P O, then (A 3 ) can be dropped. (iv) If G satisfies (A 1 ) then G 8 verifies the same condition, i.e. for every x P O, every ξ, ζ P M and every t Ps0, 1r,

G 8 px, tξ `p1 ´tqζq ď γp1 `G8 px, ξq `G8 px, ζqq,
and so G 8,x is convex for all x P O. Hence, under (A 0 )-(A 1 ), for every x P O, tG 8,x Ă intpG 8,x q for all t Ps0, 1r.

(v) If G is convex then G 8 px, ¨q is convex for all x P O, and so (A 4 ) can be dropped. More Generally, if for every x P O, G 8 px, ¨q is quasiconvex, i.e. for every ξ P M,

G 8 px, ξq " inf ! ş s0,1r N G 8 px, ξ `∇ϕpyqqdy : ϕ P W 1,8 0 ps0, 1r N ; R m q )
, then (A 4 ) can be dropped (see [START_REF] Fonseca | The lower quasiconvex envelope of the stored energy function for an elastic crystal[END_REF]).

For each ε ą 0, let W ε : O ˆM r0, 8s be a Borel measurable function with the following G-growth condition:

(A 5 ) there exist α, β ą 0 such that for every ε ą 0 and every px, ξq P O ˆM, αGpξq1 Oε pxq ď W ε px, ξq1 Oε pxq ď βp1 `Gpξqq1 Oε pxq.

We further assume that (A 6 ) tW ε 1 Oε u εą0 is ru-usc with ta ε u εą0 Ă L 1 pO; s0, 8sq and a 0 P L 8 pO; s0, 8sq, where W ε 1 Oε : O ˆM r0, 8s is defined by pW ε 1 Oε qpx, ξq :" W ε px, ξq1 Oε pxq.

Remark 3.3. As ∆ aε Wε1 Oε ptq ď maxt0, ∆ aε Wε ptqu for all ε ą 0 and all t P r0, 1s, if tW ε u εą0 is ru-usc with ta ε u εą0 Ă L 1 pO; s0, 8sq and a 0 P L 8 pO; s0, 8sq, then also is tW ε 1 Oε u εą0 .

For each ε ą 0 and each ρ ą 0, let H ρ rW ε 1 Oε s : O ˆM r0, 8s be defined by (2.4) with

L ε " W ε 1 Oε , i.e. H ρ rW ε 1 Oε spx, ξq :" inf # ż Qρpxq W ε py, ξ `∇vpyqq1 Oε pyqdy : v P W 1,p 0 pQ ρ pxq; R m q + (3.2)
and consider the following assumption: (A 7 ) for every x P O and every ξ P intpG 8,x q,

lim ρ 0 lim ε 0 H ρ rW ε 1 Oε spx, ξq ě lim ρ 0 lim ε 0 H ρ rW ε 1 Oε spx, ξq.
For each ε ą 0, let I ε : W 1,p pO; R m q r0, 8s be defined by

I ε puq :" ż Oε W ε px, ∇upxqqdx " ż O W ε px, ∇upxqq1 Oε pxqdx.
Here are the main results of the paper.

Proposition 3.4. Assume that p ą N . Under (C 1 )-(C 2 ) and (A 0 )-(A 6 ) we have

ΓpL p pO ε , Oqq-lim ε 0 I ε puq ě ż O lim t 1 ´lim ρ 0 lim ε 0 H ρ rW ε 1 Oε spx, t∇upxqqdx
for all u P W 1,p pO; R m q.

Proposition 3.5. Assume that p ą N . Under (C 0 )-(C 2 ) and (A 0 )-(A 6 ) we have

ΓpL p pO ε , Oqq-lim ε 0 I ε puq ď ż O lim t 1 ´lim ρ 0 lim ε 0 H ρ rW ε 1 Oε spx, t∇upxqqdx for all u P W 1,p pO; R m q.
As a consequence of Propositions 3.4 and 3.5 we have Theorem 3.6. Assume that p ą N . Under (C 0 )-(C 2 ) and (A 0 )-(A 7 ) we have

ΓpL p pO ε , Oqq-lim ε 0 I ε puq " ż O W lim px, ∇upxqqdx
for all u P W 1,p pO; R m q with W lim : O ˆM r0, 8s given by

W lim px, ξq :" lim t 1 ´lim ρ 0 lim ε 0 H ρ rW ε 1 Oε spx, tξq.
Let W 8 : O ˆM r0, 8s be defined by

W 8 px, ξq :" lim ρ 0 lim ε 0 H ρ rW ε 1 Oε spx, ξq.
Let x W 8 : O ˆM r0, 8s be given by x W 8 px, ξq :" lim t 1

´W8 px, tξq and, for each x P O, let W 8 px, ¨q denotes the lsc envelope of W 8 px, ¨q. The following proposition makes more precise the formula of the limit integrand W lim in Theorem 3.6. Proposition 3.7. Assume that (A 0 )-(A 1 ) and (A 5 )-(A 6 ) hold.

(i) For every x P O,

x W 8 px, ξq " lim t 1 ´W8 px, tξq " # lim t 1 ´W8 px, tξq if ξ P G 8,x 8 otherwise.
So, in Theorem 3.6 we have W lim " x W 8 . (ii) Suppose furthermore that for each x P O, W 8 px, ¨q is lsc on intpG 8,x q. Then x W 8 px, ξq " W 8 px, ξq "

$ & % W 8 px, ξq if ξ P intpG 8,x q lim t 1 ´W8 px, tξq if ξ P BG 8,x 8
otherwise.

(3.3)

In such a case, in Theorem 3.6, W lim is given by (3.3).

Proof of Proposition 3.7. From (A 6 ) and Lemma 2.11, we can assert that W 8 is ru-usc. Moreover, by (A 5 ) it is easily seen that for every x P O, the effective domain of W 8 px, ¨q is equal to G 8,x . So, taking (A 0 )-(A 1 ) (see Remark 3.2(iii)) into account, Proposition 3.7 follows from Theorem 2.9.

! x P R N : x ε P E ) ,
and we set O ε :" O X E ε . (Note that E is 1-periodic, i.e. for every px, zq P R N ˆZN , 1 E px `zq " 1 E pxq.) We further assume that (H 1 ) for every ε ą 0, pR N zE ε q X BO " H, i.e. the holes do not intersect the boundary BO. In this framework, it is clear that (C 1 ) in §2.1 and (C 0 ) in §3.1 are satisfied. Moreover, we have the following extension result due to Cioranescu and Saint Jean Paulin [START_REF] Cioranescu | Homogenization in open sets with holes[END_REF] (see also [CSJP99, §2.3 pp. 25]).

Theorem 3.8. If (H 0 )-(H 1 ) hold then (C 2 ) in §2.1 holds.
Let M denote the space of m ˆN matrices and let G : M r0, 8s be a p-coercive convex function and let G 8 : O ˆM r0, 8s be defined by (3.1). We suppose that: (H 2 ) for every x P O, intpG 8,x q Ă G; (H 3 ) for every u P W 1,p pO; R m q, if ş O G 8 px, ∇upxqqdx ă 8 and if ∇upxq P intpGq for L N -a.a. x P O, then ş O Gp∇upxqqdx ă 8. Remark 3.9. Under (H 2 ) we have intpG 8,x q " intpGq and G 8,x " G for all x P O.

Let W : R N ˆM r0, 8s be a Borel measurable function satisfying the following conditions: (H 4 ) there exist α, β ą 0 such that for every px, ξq P R N ˆM,

αGpξq1 E pxq ď W px, ξq1 E pxq ď βp1 `Gpξqq1 E pxq;
(H 5 ) for every ξ P M, W p¨, ξq is 1-periodic, i.e. for every px, zq P R N ˆZN , W px `z, ξq " W px, ξq;

(H 6 ) W 1 E is ru-usc with a 1-periodic function a P L 1 loc pR N ; s0, 8sq, where W 1 E : R N M r0, 8s is defined by pW 1 E qpx, ξq :" W px, ξq1 E pxq.

Remark 3.10. As for a family of functions (see Remark 3.3), if W is ru-usc with a 1-periodic function a P L 1 loc pR N ; s0, 8sq, then also is W 1 E .

For each ε ą 0, let J ε : W 1,p pO; R m q r0, 8s be defined by

J ε puq :" ż Oε W ´x ε , ∇upxq ¯dx " ż O W ´x ε , ∇upxq ¯1Oε pxqdx.
As a consequence of Theorem 3.6 and Proposition 3.7(i) we have the following homogenization result.

Theorem 3.11. Assume that p ą N . Under (H 0 )-(H 6 ) we have

ΓpL p pO ε , Oqq-lim ε 0 J ε puq " ż O x W hom p∇upxqqdx
for all u P W 1,p pO; R m q with x W hom : M r0, 8s given by x W hom pξq "

# lim t 1 ´Whom ptξq if ξ P G 8 otherwise,
where W hom : M r0, 8s is defined by

W hom pξq :" inf kPN ˚1 k N inf "ż s0,kr N XE W py, ξ `∇ϕpyqqdy : ϕ P W 1,p 0 ps0, kr N ; R m q * .

Auxiliary results

Integral representation of the

i " Q ρ i px i q :" x i `s ´ρi 2 , ρ i 2 r N u iPI of disjoint
open cubes of A with x i P A and ρ i Ps0, δr and such that L N pAz Y iPI Q i q " 0 by V δ pAq. Let CubpR N q be the class of all open cubes in R N . The following theorem is due to Akcoglu and Krengel (see [START_REF] Akcoglu | Ergodic theorems for superadditive processes[END_REF] and also [START_REF] Licht | Global-local subadditive ergodic theorems and application to homogenization in elasticity[END_REF] and [AHM11, Theorem 3.11]).

Theorem 4.4. Let S : O b pR N q r0, 8s be a subadditive and Z N -invariant set function for which there exists C Ps0, 8r such that for every A P O b pR N q, SpAq ď CL N pAq.

Then, for every

Q P CubpR N q, lim ε 0 S `1 ε Q LN `1 ε Q ˘" inf kě1 Sps0, kr N q k N .

Proofs

5.1. Proof of the lower bound. Here we prove Proposition 3.4.

Proof of Proposition 3.4. Let u P W 1,p pO; R m q and let tu ε u εą0 Ă W 1,p pO; R m q be such that u ε L p pOε,Oq u, i.e. lim ε 0 }u ε ´u} L p pOε;R m q " 0.

(5.1)

We have to prove that lim ε 0

I ε pu ε q ě ż O lim t 1 ´lim ρ 0 lim ε 0 H ρ rW ε 1 Oε spx, t∇upxqqdx.
(5.2)

Without loss of generality we can assume that lim ε 0 I ε pu ε q " lim ε 0 I ε pu ε q ă 8, and so sup εą0 I ε pu ε q ă 8.

(5.3)

From (5.3) it follows that sup εą0 ż O W ε px, ∇u ε pxqq1 Oε pxqdx ă 8.
(5.4) From (5.1) and (5.4) together with (A 2 ) and the left inequality in (A 3 ), we have

$ & % sup εą0 }u ε } L p pOε;R m q ă 8 sup εą0 }∇u ε } L p pOε;R m q ă 8.
(5.5) By (C 2 ) there exist C ą 0 and tp u ε u εą0 Ă W 1,p pO; R m q such that for every ε ą 0,

p u ε " u ε on O ε (5.6) and " }p u ε } L p pO;R m q ď C}u ε } L p pOε;R m q }∇p u ε } L p pO;R m q ď C}∇u ε } L p pOε;R m q .
(5.7) From (5.5) and (5.7) we deduce that sup εą0 }p u ε } W 1,p pO;R m q ă 8, and so, up to a subsequence, there exists p u P W 1,p pO; R m q such that: lim ε 0 }p u ε ´p u} L p pO;R m q " 0;

(5.8)

∇p u ε á ∇p u in L p pO; R m q.
But (5.1), (5.6) and (5.8) implies that p u " u, and consequently lim ε 0 }p u ε ´u} L p pO;R m q " 0;

(5.9)

∇p u ε á ∇u in L p pO; R m q.
(5.10)

As p ą N , from (5.9) and (5.10) we can assert that, up to a subsequence,

lim ε 0 }p u ε ´u} L 8 pO;R m q " 0.
On the other hand, taking (5.6) into account, from the left inequality in (A 3 ) we deduce that (5.12)

Step 1: localization. For every ε ą 0, we define the (positive) radon measure µ ε on O by

µ ε :" W ε p¨, ∇p u ε p¨qq1 Oε p¨qL N .
(5.13) From (5.4) we see that sup εą0 µ ε pOq ă 8, and so there exists a (positive) Radon measure µ on O such that, up to a subsequence, µ ε á µ weakly. By Lebesgue's decomposition theorem, we have µ " µ a `µs where µ a and µ s are (positive) Radon measures on O such that µ a ! L N and µ s K L N . Thus, to prove (5.2) it suffices to show that

µ a ě lim t 1 ´lim ρ 0 lim ε 0 H ρ rW ε 1 Oε sp¨, t∇up¨qqL N .
(5.14)

From Radon-Nikodym's theorem we have µ a " f p¨qL N with f p¨q :" lim ρ 0 µpQ ρ p¨qq L N pQ ρ p¨qq P L 1 pO; r0, 8rq, (5.15) and so to prove (5.14) it is sufficient to establish that for L N -a.e. x 0 P O,

f px 0 q " lim ρ 0 µpQ ρ px 0 qq L N pQ ρ px 0 qq ě lim t 1 ´lim ρ 0 lim ε 0 H ρ rW ε 1 Oε spx 0 , t∇upx 0 qq.
(5.16) Fix x 0 P OzN where N Ă O is a suitable set such that L N pN q " 0. As µpOq ă 8, without loss of generality we can assume that µpBQ ρ px 0 qq " 0 for all ρ ą 0, which implies, by Alexandrov's theorem, that µpQ ρ px 0 qq " lim ε 0 µ ε pQ ρ px 0 qq. Consequently, to prove (5.16) it suffices to show that lim ρ 0

lim ε 0 ż Qρpx 0 q W ε px, ∇p u ε pxqq1 Oε pxqdx ě lim t 1 ´lim ρ 0 lim ε 0
H ρ rW ε 1 Oε spx 0 , t∇upx 0 qq. (5.17)

On the other hand, as G is convex, see (A 1 ) and Remark 3.1, and 0 P intpGq, see (A 0 ), from (5.11) we can assert for every t Ps0, 1r, t∇p u ε pxq1 Oε pxq P G for all ε ą 0 and for L N -a.a. x P O.

Hence, given any t Ps0, 1r, we see that for every ε ą 0 and every ρ ą 0,

ż Qρpx 0 q W ε px, t∇p u ε pxqq1 Oε pxqdx ď `1 `∆ptq ˘ż Qρpx 0 q W ε px, ∇p u ε pxqq1 Oε pxqdx `∆ptqż
Qρpx 0 q a ε pxqdx with ∆ptq :" sup εą0 ∆ aε Wε1 Oε ptq, where ∆ aε Wε1 Oε ptq is given by (2.1) with Lpx, ξq " W ε px, ξq1 Oεpxq and a " a ε . Letting ε 0 and ρ 0 we obtain

lim ρ 0 lim ε 0 ż Qρpx 0 q W ε px, t∇p u ε pxqq1 Oε pxqdx ď `1 `∆ptq ˘lim ρ 0 lim ε 0 ż Qρpx 0 q W ε px, ∇p u ε pxqq1 Oε pxqdx `lim ρ 0 lim ε 0 ∆ptqż Qρpx 0 q a ε pxqdx.
But, from (A 6 ) (see also Definition 2.10) we have: lim ρ 0 lim ε 0 ż Qρpx 0 q a ε pxqdx " lim ρ 0 ż Qρpx 0 q a 0 pxqdx " a 0 px 0 q P r0, 8r;

lim t 1 ´∆ptq ď 0, hence lim t 1 ´lim ρ 0 lim ε 0 ∆ptqż Qρpx 0 q a ε pxqdx ď 0, and consequently lim t 1 ´lim ρ 0 lim ε 0 ż Qρpx 0 q W ε px, t∇p u ε pxqq1 Oε pxqdx ď lim ρ 0 lim ε 0 ż Qρpx 0 q W ε px, ∇p u ε pxqq1 Oε pxqdx.
Thus, to prove (5.17) it is sufficient to show that lim

t 1 ´lim ρ 0 lim ε 0 ż Qρpx 0 q W ε px, t∇p u ε pxqq1 Oε pxqdx ě lim t 1 ´lim ρ 0 lim ε 0 H ρ rW ε 1 Oε spx 0 , t∇upx 0 qq. (5.18)
Step 2: cut-off method. Fix any ε ą 0, any t Ps0, 1r, any σ Pst, 1r, any λ Ps0, 1r and any ρ ą 0. Let ϕ P C 8 pOq be a cut-off function for the pair pOzQ ρ px 0 q, Q λρ px 0 qq, i.e. ϕpxq P r0, 1s for all x P O, ϕpxq " 0 for all x P OzQ ρ px 0 q and ϕpxq " 1 for all x P Q λρ px 0 q, such that }∇ϕ} L 8 ď θ ρp1´λq for some θ ą 0 (which does not depend on ρ and λ). Define v ε P W 1,p pO; R m q by v ε :" ϕp u ε `p1 ´ϕqu x 0 " ϕpp u ε ´ux 0 q `ux 0 with u x 0 p¨q :" upx 0 q `∇upx 0 qp¨´x 0 q. Then tv ε ´tu x 0 P W 1,p 0 pQ ρ px 0 q; R m q (5.19) and t∇v ε "

" t∇p u ε in Q λρ px 0 q t σ `ϕσ∇p u ε `p1 ´ϕqσ∇upx 0 q ˘``1 ´t σ ˘Ψε,ρ in Q ρ px 0 qzQ λρ px 0 q (5.20)
with Ψ ε,ρ :" t 1´t σ ∇ϕ b pp u ε ´ux 0 q. Using the right inequality in (A 5 ) it follows that

ż Qρpx 0 q W ε px, t∇v ε q1 Oε dx " 1 L N pQ ρ px 0 qq ż Q λρ px 0 q W ε px, t∇p u ε q1 Oε dx `1 L N pQ ρ px 0 qq ż Qρpx 0 qzQ λρ px 0 q W ε px, t∇v ε q1 Oε dx ď ż Qρpx 0 q W ε px, t∇p u ε q1 Oε dx `βp1 ´λN q `β L N pQ ρ px 0 qq ż Qρpx 0 qzQ λρ px 0 q
Gpt∇v ε q1 Oε dx.

(5.21)

On the other hand, taking (5.20) into account and using (C 1 ) and the left inequality in (A 5 ), we have

Gpt∇v ε q1 Oε ď c 1 p1 `Gpσ∇p u ε q `Gpσ∇upx 0 qq `GpΨ ε,ρ qq 1 Oε ď c 1 ˆ1 `1 α W ε px, σ∇p u ε q1 Oε `Gpσ∇upx 0 qq `GpΨ ε,ρ q ˙(5.22)
with c 1 :" 2pγ `γ2 q ą 0. Note that from (A 0 ) and (5.12) we can assert that σ∇upx 0 q P G, and so Gpσ∇upx 0 qq ă 8. Moreover, it is easy to see that

}Ψ ε,ρ } L 8 pQρpx 0 q;Mq ď θt p1 ´t σ qp1 ´λq 1 ρ }u ´ux 0 } L 8 pQρpx 0 q;R m q `θt ρp1 ´t σ qp1 ´λq }p u ε ´u} L 8 pO;R m q ,
where lim ρ 0 θt p1 ´t σ qp1 ´λq 1 ρ }u ´ux 0 } L 8 pQρpx 0 q;R m q " 0 (5.23) because lim ρ 0 1 ρ }u ´ux 0 } L 8 pQρpx 0 q;R m q " 0 since p ą N , and lim ε 0 θt ρp1 ´t σ qp1 ´λq }p u ε ´u} L 8 pO;R m q " 0 (5.24) by (5.9). From (A 0 ) and (A 1 ) there exists r ą 0 such that c 2 :" sup |ξ|ďr Gpξq ă 8

(see Remark 3.1). By (5.23) there exists ρ ą 0 such that θt p1´t σ qp1´λq 1 ρ }u´u x 0 } L 8 pQρpx 0 q;R m q ă r 2 for all ρ Ps0, ρr. Fix any ρ Ps0, ρr. Taking (5.24) into account we can assert that there exists ε ρ ą 0 such that GpΨ ε,ρ q ď c 2 for all ε Ps0, ε ρ r.

(5.25) Thus, from (5.21), (5.22) and (5.25) we deduce that

ż Qρpx 0 q W ε px, t∇v ε q1 Oε dx ď ż Qρpx 0 q W ε px, t∇p u ε q1 Oε dx `c3 pσqp1 ´λN q `βc 1 α Γ ε,ρ,λ,σ
for all t Ps0, t ρ r with:

c 3 pσq :" βc 1 ˆ1 `1 c 1 `Gpσ∇upx 0 qq `c2 ˙Ps0, 8r ; Γ ε,ρ,λ,σ :" 1 L N pQ ρ px 0 qq ż Qρpx 0 qzQ λρ px 0 qq W ε px, σ∇p u ε q1 Oε dx.
But, taking (5.19) into account, we see that

H ρ rW ε 1 Oε spx 0 , t∇upx 0 qq ď ż Qρpx 0 q W ε px, t∇v ε q1 Oε dx,
hence, for every ρ ą 0, every ε Ps0, ε ρ r, every λ Ps0, 1r, every t Ps0, 1r and every σ Pst, 1r, we have

H ρ rW ε 1 Oε spx 0 , t∇upx 0 qq ď ż Qρpx 0 q W ε px, t∇p u ε q1 Oε dx `c3 pσqp1 ´λN q `βc 1 α Γ ε,ρ,λ,σ .
(5.26)

Step 3: passing to the limit. Letting ε 0, ρ 0, λ 1 ´, σ 1 ´and t 1 ´in (5.26), we obtain lim

t 1 ´lim ρ 0 lim ε 0 H ρ rW ε 1 Oε spx 0 , t∇upx 0 qq ď lim t 1 ´lim ρ 0 lim ε 0 ż Qρpx 0 q W ε px, t∇p u ε q1 Oε dx `βc 1 α lim σ 1 ´lim λ 1 ´lim ρ 0 lim ε 0 Γ ε,ρ,λ,σ .
(5.27)

Substep 3-1: proving that lim σ 1 ´lim λ 1 ´lim ρ 0 lim ε 0 Γ ε,ρ,λ,σ " 0 lim σ 1 ´lim λ 1 ´lim ρ 0 lim ε 0 Γ ε,ρ,λ,σ " 0 lim σ 1 ´lim λ 1 ´lim ρ 0 lim ε 0 Γ ε,ρ,λ,σ " 0. For every ε Ps0, ε ρ r, we have Γ ε,ρ,λ,σ ď `1 `∆pσq ˘µε `Qρ px 0 qzQ λρ px 0 qq LN pQ ρ px 0 qq `∆pσq 1 L N pQ ρ px 0 qq ż Qρpx 0 qzQ λρ px 0 q a ε pxqdx.
(5.28) But from (A 6 ) (see also Definition 2.10) we have lim σ 1 ´∆pσq ď 0 and lim

λ 1 ´lim ρ 0 lim ε 0 1 L N pQ ρ px 0 qq ż Qρpx 0 qzQ λρ px 0 q a ε pxqdx ď lim ρ 0 lim ε 0 ż Qρpx 0 q a ε pxqdx
" lim ρ 0 ż Qρpx 0 q a 0 pxqdx " a 0 px 0 q P r0, 8r

with 1 L N pQρpx 0 qq ş Qρpx 0 qzQ λρ px 0 q a ε pxqdx ě 0, hence lim σ 1 ´lim λ 1 ´lim ρ 0 lim ε 0 ∆pσq 1 L N pQ ρ px 0 qq ż Qρpx 0 qzQ λρ px 0 q a ε pxqdx ď 0.
(5.29)

As µ ε á µ weakly and Q ρ px 0 qzQ λρ px 0 q is compact, by Alexandrov's theorem, we have lim ε 0 µ ε `Qρ px 0 qzQ λρ px 0 q ˘ď µ `Qρ px 0 qzQ λρ px 0 q ˘, hence lim ε 0 µ ε `Qρ px 0 qzQ λρ px 0 q ˘ď µ `Qρ px 0 q ˘´µ pQ λρ px 0 qq , and consequently lim ε 0 µ ε `Qρ px 0 qzQ λρ px 0 q LN pQ ρ px 0 qq ď µ `Qρ px 0 q LN pQ ρ px 0 qq ´λN µ pQ λρ px 0 qq L N pQ λρ px 0 qq .

It follows that lim

ρ 0 lim ε 0 µ ε `Qρ px 0 qzQ λρ px 0 q LN pQ ρ px 0 qq ď `1 ´λN ˘f px 0 q with f P L 1 pO; r0, 8rq given by (5.15), and so m ε u pAq. with, for each ε ą 0, m ε u : OpOq r0, 8s given by

lim λ 1 ´lim ρ 0 lim ε 0 µ ε `Qρ px 0 qzQ λρ px 0 q LN pQ ρ px 0 qq " 0.
m ε u pAq " inf "ż OεXA W ε px, ∇vpxqqdx : v ´u P W 1,p 0 pA; R m q * " inf "ż A W ε px, ∇vpxqq1 Oε pxqdx : v ´u P W 1,p 0 pA; R m q * .
(5.32)

For each δ ą 0 and each A P OpOq, we denote the class of countable families tQ i :"

Q ρ i px i qu iPI of disjoint
open cubes of A with x i P A and ρ i Ps0, δr such that L N pAzY iPI Q i q " 0 by V δ pAq, and we consider m δ u : OpOq r0, 8s given by m δ u pAq :" inf

# ÿ iPI m u pB i q : tB i u iPI P V δ pAq + ,
and we define m ů : OpOq r0, 8s by

m ůpAq :" sup δą0 m δ u pAq " lim δ 0 m δ u pAq.
The set function m ů is called the Vitali envelope of m u (see §4.1).

Step 1: link between Γ-lim Γ-lim Γ-lim and Vitali envelope. Let u P W 1,p pO; R m q. We are going to prove that ΓpL p pO ε , Oqq-lim ε 0 I ε puq ď m ůpOq.

(5.33)

Without loss of generality we can assume that m ůpOq ă 8. Fix any δ ą 0. By definition of m δ u pOq there exists tQ i u iPI P V δ pOq such that ÿ iPI m u pQ i q ď m δ u pOq `δ 2 .

(5.34)

Fix any ε ą 0. For each i P I, by definition of m ε u pQ i q there exists v i ε P W 1,p pO; R m q such that v i ε ´u P W 1,p 0 pQ i ; R m q and ż

OεXQ i W ε `x, ∇v i ε pxq ˘dx ď m ε u pQ i q `δL N pQ i q 2L N pOq . (5.35) Define u δ ε : O R m by u δ ε :" # u in Oz Y iPI Q i v i ε in Q i . Then u δ ε ´u P W 1,p 0 pO; R m q.
Moreover, we have ∇u δ ε pxq " ∇v i ε pxq for L N -a.a. x P Q i . From (5.35) we see that

I ε pu δ ε q ď ÿ iPI m ε u pQ i q `δ 2 ,
hence lim ε 0 I ε pu δ ε q ď m δ u pOq `δ by using (5.34), and consequently lim

δ 0 lim ε 0 I ε pu δ ε q ď m ůpOq.
(5.36)

On the other hand, we have

}u δ ε ´u} p L p pOε;R m q " ż Oε ˇˇu δ ε ´uˇˇp dx " ÿ iPI ż OεXQ i ˇˇv i ε ´uˇˇp dx.
Taking (C 0 ) into account and since ρ i Ps0, δr for all i P I, by Poincaré's inequality (see [DD12, Exercise 2.9 pp. 106]) we have

ÿ iPI ż OεXQ i ˇˇv i ε ´uˇˇp dx ď Cδ p ÿ iPI ż OεXQ i ˇˇ∇v i ε ´∇u ˇˇp dx
with C ą 0 (which only depends on p) and so

}u δ ε ´u} p L p pOε;R m q ď 2 p´1 Cδ p ˜ÿ iPI ż OεXQ i |∇v i ε | p dx `żOεXQ i |∇u| p dx ¸.
(5.37)

Taking the left inequality in (A 3 ), (5.35) and (5.34) into account, from (5.37) we deduce that lim

ε 0 }u δ ε ´u} p L p pO;R m q ď 2 p´1 Cδ p ˆ1 α `mδ u pOq `δ˘`ż O |∇u| p dx ẇith α ą 0 given by (A 3 ), which gives lim δ 0 lim ε 0 }u δ ε ´u} p L p pOε;R m q " 0 (5.38)
because lim δ 0 m δ u pOq " m ůpOq ă 8 and u P W 1,p pO; R m q. According to (5.36) and (5.38), by diagonalization there exists a mapping ε δ ε , with δ ε 0 as ε 0, such that: lim ε 0 }w ε ´u} p L p pOε;R m q " 0;

(5.39) lim ε 0

I ε pw ε q ď m ůpOq (5.40)
with w ε :" u δε ε . By (5.39) we have ΓpL p pO ε , Oqq-lim ε 0 I ε puq ď lim ε 0 I ε pw ε q, and (5.33) follows from (5.40).

Step 2: differentiation with respect to L N L N L N . Let u P W 1,p pO; R m q be such that Gpuq :" ş O Gp∇upxqqdx ă 8. We are going to prove that

m ůpOq " ż O lim ρ 0 m u pQ ρ pxqq L N pQ ρ pxqq dx.
(5.41)

According to Theorem 4.2, to prove (5.41) it suffices to establish that m u is subadditive and there exists a finite Radon measure ν on O which is absolutely continuous with respect to L N such that m u pAq ď νpAq (5.42) for all A P OpOq. For each ε ą 0, from the definition of m ε u in (5.32), it is easy to see that for every A, B, C P OpOq with B, C Ă A, B X C " H and L N pAzpB Y Cqq " 0, one has m ε u pAq ď m ε u pBq `mε u pCq, and so lim

ε 0 m ε u pAq ď lim ε 0 m ε u pBq `lim ε 0 m ε u pCq, i.e.
m u pAq ď m u pBq `mu pCq, which shows the subadditivity of m u . On the other hand, given any ε ą 0, by using the right inequality in (A 5 ) we have

m ε u pAq ď ż A βp1 `Gp∇upxqqqdx
for all A P OpOq. Thus (5.42) holds with the Radon measure ν :" βp1 `Gp∇up¨qqqL N which is necessarily finite since Gpuq ă 8.

Step 3: cut-off method. Let t Ps0, 1r, let σ Pst, 1r and let u P W 1,p pO; R m q be such that Gpσuq ă 8. We are going to prove that for L N -a.e. Fix any ε ą 0, any λ Ps0, 1r, any ρ ą 0 and any δ ą 0. By definition of m ε tux pQ λρ pxqq in (5.32), there exists w P W 1,p pO; R m q such that tw ´tu x P W 1,p 0 pQ λρ pxq; R m q (5.44) and ż

Q λρ pxq
W ε py, t∇wpyqqdy ď m ε tux pQ λρ pxqq `δL N pQ λρ pxqq.

(5.45)

Let ϕ P C 8 pOq be a cut-off function for the pair pOzQ ρ pxq, Q λρ pxqq, i.e. ϕpyq P r0, 1s for all y P O, ϕpyq " 0 for all y P OzQ ρ pxq and ϕpyq " 1 for all y P Q λρ pxq, such that }∇ϕ} L 8 ď θ ρp1 ´λq for some θ ą 0 (which does not depend on ρ and λ). Define v P W 1,p pO; R m q by v :" ϕu x `p1 ´ϕqu " ϕpu x ´uq `u.

Then

tv ´tu P W 1,p 0 pQ ρ pxq; R m q (5.46) and t∇v " " t∇upxq in Q λρ px 0 q t σ `ϕσ∇upxq `p1 ´ϕqσ∇u ˘``1 ´t σ ˘Ψρ in Q ρ px 0 qzQ λρ px 0 q (5.47) with Ψ ρ :" t 1´τ σ ∇ϕ b pu x ´uq. From (5.44) and (5.46) we have tv `ptw ´tu x q ´tu P W 1,p 0 pQ ρ pxq; R m q, and so, noticing that ∇ptw ´tu x qpyq " t∇w ´t∇u x " 0 for L N -a.a. y P Q ρ pxqzQ λρ pxq, Gpt∇vqdy.

m ε tu pQ ρ pxqq L N pQ λρ pxqq ď 1 L N pQ λρ pxqq ż Qρpxq W ε py, t∇v `t∇w ´t∇u x q 1 Oε pyqdy " 1 L N pQ λρ pxqq ż Q λρ pxq W ε py, t∇upxq `t∇w ´t∇upxqq 1 Oε pyqdy `1 L N pQ λρ pxqq ż QρpxqzQ λρ pxq W ε py, t∇vq1 Oε pyqdy " 1 L N pQ λρ pxqq ż Q λρ pxq
(5.48)

On the other hand, taking (5.47) into account and using (A 1 ), we have Gpt∇vq ď c 1 p1 `Gpσ∇upxqq `Gpσ∇uq `GpΨ ρ qq (5.49) with c 1 :" 2pγ `γ2 q ą 0. Moreover, it is easy to see that

}Ψ ρ } L 8 pQρpxq;Mq ď θt p1 ´t σ qp1 ´λq 1 ρ }u ´ux } L 8 pQρpxq;R m q with lim ρ 0 θt p1 ´t σ qp1 ´λq 1 ρ
}u ´ux } L 8 pQρpxq;R m q " 0 (5.50) because lim ρ 0 1 ρ }u ´ux } L 8 pQρpxq;R m q " 0 since p ą N . From (A 0 ) and (A 1 ) there exists r ą 0 such that c 2 :" sup |ξ|ďr Gpξq ă 8

(see Remark 3.1). By (5.50) there exists ρ ą 0 such that θt p1´t σ qp1´λq 1 ρ }u ´ux } L 8 pQρpxq;R m q ă r for all ρ Ps0, ρr. Fix any ρ Ps0, ρr. We then have GpΨ ρ q ď c 2 .

( Conclusion of the steps 1, 2 and 3. As a direct consequence of (5.33), (5.41) and (5.43) together with Remarks 5.1 and 5.2, we have the following lemma.

Lemma 5.3. For every t Ps0, 1r and every u P W 1,p pO; R m q such that Gptuq ă 8 and Gpσuq ă 8 for some σ Pst, 1r, one has

ΓpL p pO ε , Oqq-lim ε 0 I ε ptuq ď ż O lim ρ 0 lim ε 0 H ρ rW ε 1 Oε spx, t∇upxqqdx.
Step 4: end of the proof. Let u P W 1,p pO; R m q. We have to prove that ΓpL p pO ε , Oqq-lim

ε 0 I ε puq ď ż O lim t 1 ´lim ρ 0 lim ε 0 H ρ rW ε 1 Oε spx, t∇upxqqdx.
(5.58)

Without loss of generality we can assume that (5.61)

Substep 4-1: proving (5.58) under the constraint ∇upxq P intpG 8,x q ∇upxq P intpG 8,x q ∇upxq P intpG 8,x q for L N L N L N -a.a.

H ρ rW ε 1 Oε spx, t∇upxqqdx " ż O lim t 1 ´lim ρ 0 lim ε 0 H ρ rW ε 1 Oε spx, t∇upxqqdx.
From (5.65) we conclude that lim

t 1 ´ΓpL p pO ε , Oqq-lim ε 0 I ε ptuq ď ż O lim t 1 ´lim ρ 0 lim ε 0 H ρ rW ε 1 Oε spx, t∇upxqqdx,
and (5.58) follows because ΓpL p pO ε , Oqq-lim t 8 I ε is lsc with respect to the L p pO; R m qconvergence and tu u in L p pO; R m q as t 1

´.

Substep 4-2: proof of (5.58). First of all, from (A 6 ) and Lemma 2.11 we can assert that W 8 :" lim ρ 0 lim ε 0 H ρ rW ε 1 Oε s is ru-usc with the constant function }a 0 } L 8 . Moreover, by I ε ptuq ď I lim puq, and (5.58) follows because ΓpL p pO ε , Oqq-lim ε 0 I ε is lsc with respect to the L p pO; R m qconvergence and tu u in L p pO; R m q as t 1 ´. where dom `lim ρ 0 lim ε 0 H ρ rW ε 1 Oε spx, ¨q˘a nd dom `lim ρ 0 lim ε 0 H ρ rW ε 1 Oε spx, ¨q˘d enotes the effective domain of lim ρ 0 lim ε 0 H ρ rW ε 1 Oε spx, ¨q and lim ρ 0 lim ε 0 H ρ rW ε 1 Oε spx, ¨q respectively. Let px, ξq P O ˆM. If ξ R G 8,x then there exists t ξ Ps0, 1r such that tξ R G 8,x for all t P rt ξ , 1r. Hence: From (H 4 ) we can assert that for every ε ą 0 and every px, ξq P R N ˆM,

Proof of the

' if ξ R G 8,
αGpξq1 E ´x ε ¯ď W ´x ε , ξ ¯1E ´x ε ¯ď βp1 `Gpξqq1 E ´x ε ¯,
i.e. αGpξq1 Eε pxq ď W ε px, ξq1 Eε pxq ď βp1 `Gpξqq1 Eε pxq, and so (A 5 ) is verified. By (H 5 ) we see that for each ξ P M, x W px, ξq1 E pxq is 1-periodic because, by assumption, 1 E is 1-periodic. Hence, taking (H 6 ) into account, from Lemma 2.12 we deduce that (A 6 ) holds. So, it remains to prove that (A 7 ) is satisfied, i.e. according to (H 2 ), for every x P O and every ξ P intpGq, First of all, it is clear that S ξ is subadditive. Then, by using (H 5 ) and the fact that 1 E is 1-periodic, it easy to see that S ξ is Z N -invariant. Finally, from the right inequality in (H 4 ) we deduce that for every A P O b pR N q, S ξ pAq ď C ξ L N pAq with C ξ :" βp1 `Gpξqq ă 8 because ξ P intpGq. From Theorem 4.4 it follows that for every ρ ą 0, lim

lim ρ 0 lim ε 0 S ξ `1 ε Q ρ pxq LN `1 ε Q ρ pxq ˘ě lim ρ 0 lim ε 0 S ξ `1 ε Q ρ pxq LN `1 ε Q ρ pxq ˘(5.
ε 0 S ξ `1 ε Q ρ pxq LN `1 ε Q ρ pxq ˘" inf kě1
S ξ ps0, kr N q k N , which implies (5.72). Thus (C 0 )-(C 2 ) and (A 0 )-(A 7 ) are satisfied, and the theorem follows from Theorem 3.6 and Proposition 3.7(i) (with Remark 3.9) in noticing that inf kě1 S ξ ps0,kr N q k N " W hom pξq.

2. 1 .

 1 L p pO ε , Oq-convergence and strong connectedness. Let O Ă R N be a bounded open set and, for each ε ą 0, let O ε Ă O be an open set. In what follows we consider the following two conditions: (C 1 ) lim ε 0

. 1 .

 1 The ΓpL p pO ε , Oqq-convergence result. Let O Ă R N be a bounded open set and, for each ε ą 0, let O ε Ă O be an open set and consider the following condition: (C 0 ) for all ε ą 0 and all open cube

  Vitali envelope of a set function. Let O Ă R N be a bounded open set and let OpOq be the class of open subsets of O. For each δ ą 0 and each A P OpOq, we denote the class of countable families tQ

lim ρ 0

 0 Definition 4.1. Given S : OpOq r0, 8s, for each δ ą 0 we define S δ : OpOq r0, 8s by S δ pAq :" inf # ÿ iPI SpQ i q : tQ i u iPI P V δ pAq + . By the Vitali envelope of S we call the set function S ˚: OpOq r´8, 8s defined by S ˚pAq :" sup δą0 S δ pAq " lim δ 0 S δ pAq. The interest of Definition 4.1 comes from the following integral representation result. (For a proof we refer to [AHM18, §3.3] or [AHCM17, §A.4].) Theorem 4.2. Let S : OpOq r0, 8s be a set function satisfying the following two conditions: (i) there exists a finite Radon measure ν on O which is absolutely continuous with respect to L N such that SpAq ď νpAq for all A P OpOq; (ii) S is subadditive, i.e. SpAq ď SpBq `SpCq for all A, B, C P OpOq with B, C Ă A, B X C " H and L N pAzpB Y Cqq " 0. Then lim ρ 0 SpQρp¨qq ρ N P L 1 pOq and for every A P OpOq, one has S ˚pAq " ż A SpQ ρ pxqq ρ N dx. 4.2. A subadditive theorem. Let O b pR N q be the class of all bounded open subsets of R N . We begin with the following definition. Definition 4.3. Let S : O b pR N q r0, 8s be a set function. (i) We say that S is subadditive if SpAq ď SpBq `SpCq for all A, B, C P O b pR N q with B, C Ă A, B X C " H and L N pAzpB Y Cqq " 0. (ii) We say that S is Z N -invariant if SpA `zq " SpAq for all A P O b pR N q and all z P Z N .

sup εą0 ż O

 ż Gp∇p u ε pxqq1 Oε pxqdx ă 8, hence Gp∇p u ε pxqq1 Oε pxq ă 8 for all ε ą 0 and L N -a.a. x P O, and so, taking (A 0 ) into account, ∇p u ε pxq1 Oε pxq P G for all ε ą 0 and L N -a.a. x P O.(5.11) As G is convex, see (A 1 ) and Remark 3.1, from (C 1 ), (5.10) and (5.11) it follows that ∇upxq P G for L N -a.a. x P O.

with µεpQρpx 0 Γ

 0 qzQ λρ px 0 qq L N pQρpx 0 qq ě 0. Consequently, by using (A 6 ), ˘µε `Qρ px 0 qzQ λρ px 0 q LN pQ ρ px 0 qq ď ε,ρ,λ,σ " 0.(5.31)Substep 3-3: end of the proof. Combining (5.31) with (5.27) we obtain (5.18), and the proof of the lower bound is complete.5.2. Proof of the upper bound.Here we prove Proposition 3.5.Proof of Proposition 3.5. In what follows, OpOq denotes the class of all open subsets of O. For each u P W 1,p pO; R m q, let m u : OpOq r0, 8s be defined by m u pAq :" lim ε 0

x P O, lim ρ 0 m 0 m tux pQ ρ pxqq L N pQ ρ pxqq " lim ρ 0 lim ε 0 H ρ rW ε 1 0 m tu pQ ρ pxqq L N pQ ρ pxqq " lim ρ 0 m tu pQ ρ pxqq L N pQ ρ pxqq and lim ρ 0 m tux pQ ρ pxqq L N pQ ρ pxqq " lim ρ 0 m

 00010000 tu pQ ρ pxqq L N pQ ρ pxqq ď lim ρ 0 m tux pQ ρ pxqq L N pQ ρ pxqq (5.43) with u x p¨q :" upxq `∇upxqp¨´xq. Remark 5.1. For L N -a.e. x P O, one has lim ρ Oε spx, t∇upxqq.Remark 5.2. If Gptuq ă 8 then Gptu x q ă 8 for L N -a.a. x P O, and so, by the step 2, lim ρ tux pQ ρ pxqq L N pQ ρ pxqq .

W

  ε py, t∇wq 1 Oε pyqdy `1 L N pQ λρ pxqq ż QρpxqzQ λρ pxq W ε py, t∇vq1 Oε pyqdy. From (5.45) and the right inequality in (A 5 ) it follows that m ε tu pQ ρ pxqq L N pQ ρ pxqq ď m ε tu pQ ρ pxqq L N pQ λρ pxqq ď m ε tux pQ λρ pxqq L N pQ λρ pxqq `δ `βp1 ´λN q λ N `β L N pQ λρ pxqq ż QρpxqzQ λρ pxq

H ρ rW ε 1 H ρ rW ε 1 H ρ rW ε 1

 111 Oε spx, t∇upxqqdx ": I lim puq ă 8.(5.59) Then, by Proposition 3.7(i) we have ∇upxq P G 8,x for L N -a.a. x P O (5.60) and, for L N -a.e. x P O, Oε spx, t∇upxqq " lim Oε spx, t∇upxqq.

0 lim ε 0 H ρ rW ε 1 0 lim ε 0 H ρ rW ε 1 ˆlim ρ 0 lim ε 0 H ρ rW ε 1 H ρ rW ε 1

 0101011 ΓpL p pO ε , Oqq-convergence theorem. Here we prove Theorem 3.6. Proof of Theorem 3.6. By (A 5 ) we see that αG 8 px, ξq ď lim ρ Oε spx, ξq ď lim ρ Oε spx, ξq ď β `1 `G8 px, ξq for all px, ξq P O ˆM. So, for every x P O, one has dom Oε spx, Oε spx, ¨q˙" G 8,x , (5.71)

  72)with S ξ : O b pR N q r0, 8s defined byS ξ pAq :" inf "ż A W px, ξ `∇ϕpxqq1 E pxqdx : ϕ P W 1,p 0 pA; R m q * ,where O b pR N q denotes the class of all bounded open subsets of R N . Fix x P O and ξ P intpGq.

where M denotes the space of m ˆN matrices. For each x P O, we denote the effective domain 3 of Lpx, ¨q by L x and, for each

  X r0, 8s, by the effective domain of f we mean the set of z P X such that f pzq ă 8.

	Theorem 2.9. If L : O ˆM	r0, 8s is ru-usc and if for every x P O,
				tL x Ă intpL x q for all t Ps0, 1r,	(2.3)
	then:
	(a) p L is ru-usc;
	(b) p Lpx, ξq " lim t 1 ´Lpx, tξq for all px, ξq P O ˆM.
	If moreover, for every x P O, Lpx, ¨q is lsc on intpL x q then:
	(c) p Lpx, ξq "	$ &	Lpx, ξq lim t 1 ´Lpx, tξq if ξ P BL x if ξ P intpL x q
		%	8	otherwise;
	(d) for every x P O, p Lpx, ¨q is the lsc envelope of Lpx, ¨q.
	2.3.2. Family of ru-usc functions. The following definition generalizes Definition 2.8 to the
	case of a family of functions.
	3.1. Ru-usc function. Let O Ă R N be an open set and L : O ˆM	r0, 8s be a Borel
	measurable function, a P L 1 loc pO; s0, 8sq, we consider ∆ a L : r0, 1s s ´8, 8s defined by
			∆ a L ptq :" sup xPO	sup ξPLx	Lpx, tξq ´Lpx, ξq apxq `Lpx, ξq	.	(2.1)
	Definition 2.8. We say that L : O ˆM	r0, 8s is ru-usc if there exists a P L 1 loc pO; s0, 8sq
	such that
				lim t 1 ´∆a L ptq ď 0.	(2.2)
	The interest of Definition 2.8 comes from the following theorem. (For a proof we refer to
	[AHM11, Theorem 3.5] and also [AHM12, §4.2]) Let p L : O ˆM	r0, 8s be defined by
				p Lpx, ξq :" lim t 1 ´Lpx, tξq.

2

The abbreviation lsc means lower semicontinuous.

3 Given a set X and a function f :

  3.2. Homogenization of unbounded integrals in strongly perforated sets. Let Y :" r0, 1r N , let H Ă Y be an open set with Lipschitz boundary, which represents the holes in Y , such that (H 0 ) H XBY " H, i.e. the holes do not intersect the boundary BY , and Y zH is connected. Let E :" Y zH `ZN and, for each ε ą 0, we denote the ε-homothetic set of E by E ε , i.e.

	E ε :"

  QρpxqzQ λρ pxqGpσ∇upyqqdy ď L N pQ ρ pxqqż

	Letting ε	0, ρ			0 and λ			1 ´in (5.53) and using (5.54), (5.55), (5.56) and (5.57) we
	conclude that
							lim ρ 0	m tu pQ ρ pxqq L N pQ ρ pxqq	ď lim ρ 0	m tux pQ ρ pxqq L N pQ ρ pxqq	`δ,	5.51)
	From (5.49) and (5.51) it follows that and (5.43) follows by letting δ 0.
	β L N pQ λρ pxqq	ż	QρpxqzQ λρ pxq	Gpt∇vqdy ď βc 1 `1 `Gpσ∇upxqq `c2	˘1 ´λN λ N
											ż
											`βc 1 L N pQ λρ pxqq	QρpxqzQ λρ pxq	Gpσ∇upyqqdy.
	But
	ż
											|Gpσ∇upyqq ´Gpσ∇upxqq|dy
											Qρpxq
											`LN pQ ρ pxqzQ λρ pxqqGpσ∇upxqq,
	hence
	β L N pQ λρ pxqq	ż	QρpxqzQ λρ pxq Gpt∇vqdy ď βc 1 `1 `2Gpσ∇upxqq `c2	˘1 ´λN λ N
											`βc 1 λ N	ż Qρpxq |Gpσ∇upyqq ´Gpσ∇upxqq|dy. (5.52)
	From (5.48) and (5.52) we deduce that
		m ε tu pQ ρ pxqq L N pQ ρ pxqq	ď	m ε tux pQ λρ pxqq L N pQ λρ pxqq	`δ
										`βc 1 ˆ1	`1 c 1	`2Gpσ∇upxqq `c2	˙1 ´λN λ N
										`βc 1 λ N	ż Qρpxq	|Gpσ∇upyqq ´Gpσ∇upxqq|dy.	(5.53)
	As Gpσuq ă 8, i.e. Gpσ∇up¨qq P L 1 pOq, we can assert that:
							Gpσ∇upxqq ă 8;	(5.54)
							lim ρ 0 ż Qρpxq	|Gpσ∇upyqq ´Gpσ∇upxqq|dy " 0.	(5.55)
	Moreover, we have:
		lim ρ 0	lim ε 0	m ε tu pQ ρ pxqq L N pQ ρ pxqq	" lim ρ 0	m tu pQ ρ pxqq L N pQ ρ pxqq	;	(5.56)
		lim ρ 0	lim ε 0	m ε tux pQ λρ pxqq L N pQ λρ pxqq	ď lim ρ 0	lim ε 0	m ε tux pQ ρ pxqq L N pQ ρ pxqq	" lim ρ 0	m tux pQ ρ pxqq L N pQ ρ pxqq	.	(5.57)

  ) we see that for every t Ps0, 1r, Gptuq ď γL N pOqp1 `Gp0qq `γGpuq, hence, by (A 0 ) and (5.64), Gptuq ă 8 for all t Ps0, 1r, and so, by Lemma 5.3 we have ΓpL p pO ε , Oqq-lim Oε spx, t∇up¨qq ď β `1 `Gp0q `Gp∇up¨q ˘:" f p¨q for all t Ps0, 1r with f P L 1 pOq by (A 0 ) and (5.64).

	But, by (A 1 ε 0	I ε ptuq ď	ż O	lim ρ 0	lim ε 0	H ρ rW ε 1 Oε spx, t∇upxqqdx for all t Ps0, 1r. (5.65)
	On the other hand, from the right inequality in (A 5 ) we see that for every t Ps0, 1r,
						lim ρ 0	lim ε 0	H ρ rW ε 1 Oε spx, t∇up¨qq ď β `1 `G8 px, t∇up¨qq	ď
							β `1 `Gpt∇up¨qq ˘,
	and consequently, by using (A 1 ),
	lim ρ 0	lim ε 0	H ρ rW ε 1 Taking (5.61) into account, from Lebesgue's dominated
	convergence theorem we deduce that
	lim t 1 ´żO	lim ρ 0	lim ε 0
	x P O x P O x P O. Assume that
							∇upxq P intpG 8,x q for L N -a.a. x P O.	(5.62)
	Then, since (A 0 )-(A 1 ) (see Remark 3.2(iii)) implies that t∇upxq P intpG 8,x q for all t Ps0, 1r
	and for L N -a.a. x P O, by (A 4 ) we have
						lim	(5.63)
						t 1
	Using (5.63) and the left inequality in (A 5 ) we see that
			α 1	I lim puq ě	O ż	lim t 1 ´G8 px, t∇upxqqdx ě	ż
							Gpuq ă 8.	(5.64)

´G8 px, t∇upxqq ě G 8 px, ∇upxqq for L N -a.a. x P O. O G 8 px, ∇upxqqdx ": G 8 puq, hence, by (5.59), G 8 puq ă 8, and so taking (A 3 ) into account, from (5.62) it follows that

  Oε s is ru-usc with the constant function }a 0 } L 8 .On the other hand, from (5.60) and (A 0 )-(A 1 ) (see Remark 3.2(iii)) we deduce that ∇ptuqpxq P intpG 8,x q for all t Ps0, 1r and L N -a.a. x P O. Ps0, 1r. Moreover, by (5.66) we have lim t 1

								(5.69)
	According to (5.69) and (5.68), from the substep 4-1 we can assert that
								ΓpL p pO ε , Oqq-lim ε 0	I ε ptuq ď I lim ptuq
	for all t Ps0, 1r, and so, taking (5.67) into account,
	ΓpL p pO ε , Oqq-lim ε 0	I ε ptuq ď `1	`∆}a 0 } L 8 x W8	ptq ˘Ilim puq	`∆}a 0 } L 8 x W8	ptq}a 0 } L 8 L N pOq	(5.70)
	for all t ´∆}a 0 } L 8 x W8	ptq ď 0. Hence, letting t	1 ín
	(5.70) we conclude that
								lim
								t 1
			t 1	´lim ρ 0	lim ε 0	H (5.66)
	From (5.59) we see that ∇upxq P x W 8,x for L N -a.a. x P O, where x W 8,x denotes the effective
	domain of x W 8 px, ¨q. Hence, for every t Ps0, 1r,
			ż
				O	x W 8 px, t∇upxqqdx ď `1	`∆}a 0 } L 8 x W8	ptq	˘żO	x W 8 px, ∇upxqqdx
								`∆}a 0 } L 8 x W8	ptq}a 0 } L 8 L N pOq
	with ∆	}a 0 } L 8 x W8	ptq :" sup xPO sup ξP x W8,x	x W8px,tξq´x W8px,ξq }a 0 } L 8 `x W8px,ξq , i.e.
				I lim ptuq ď `1	`∆}a 0 } L 8 x W8	ptq ˘Ilim puq	`∆}a 0 } L 8 x W8	ptq}a 0 } L 8 L N pOq	(5.67)
	for all t Ps0, 1r. Using (5.59) we see that
								I lim ptuq ă 8 for all t Ps0, 1r.	(5.68)

(A 5 ) we see that for every x P O, the effective domain of W 8 px, ¨q is equal to G 8,x . Taking (A 0 )-(A 1 ) (see Remark 3.2(iii)) into account, from Theorem 2.9(ii) it follows that x W 8 :" lim ρ rW ε 1 ´ΓpL p pO ε , Oqq-lim ε 0

  x then, by (5.71), H ρ rW ε 1 Oε spx, tξq " lim H ρ rW ε 1 Oε spx, tξq " 8 for all t P rt ξ , 1r; ' if ξ P G 8,x then, from (A 0 )-(A 1 ) (see Remark 3.2(iv)), we have tξ P intpG 8,x q for all t Ps0, 1r, and so, by (A 7 ),H ρ rW ε 1 Oε spx, tξq ě lim H ρ rW ε 1 Oε spx,tξq for all t Ps0, 1r. Oε spx, tξq for all px, ξq P O ˆM. From Propositions 3.4 and 3.5 we deduce that ΓpL p pO ε , Oqq-lim H ρ rW ε 1 Oε spx, t∇upxqqdx H ρ rW ε 1 Oε spx, t∇upxqqdx ě ΓpL p pO ε , Oqq-lim ε 0I ε puq.for all u P W 1,p pO; R m q. Hence ΓpL p pO ε , Oqq-lim lim px, ∇upxqqdx for all u P W 1,p pO; R m q.5.4. Proof of the homogenization theorem. Here we prove Theorem 3.11. Proof of Theorem 3.11. Taking (H 0 )-(H 1 ) and Theorem 3.8 into account, we see that (C 0 )-(C 2 ) hold. Since G is p-coercive convex function satisfying (H 2 )-(H 3 ), it is clear that (A 0 )-(A 4 ) hold. For each ε ą 0, let W

			ż
	lim ρ 0 lim lim ε 0 ρ 0 lim ε 0 It follows that lim t 1 ´lim ρ 0	ε 0 lim H ε 0	ρ 0 ρ 0 lim ε 0 lim ε 0 I ε puq ě O lim t 1 ´lim ρ 0 ě ż O lim t 1 ´lim ρ 0 ε 0 I ε puq " ż	lim ε 0 lim ε 0

ρ rW ε 1 Oε spx, tξq ě lim t 1 ´lim ρ 0 lim ε 0 H ρ rW ε 1 O W ε : O ˆM r0, 8s be defined by W ε px, ξq :" W ´x ε , ξ ¯.