Jeanne Ropars

Ricardo C Rodríguez De La Vega

Manuela López-Villavicencio

Jérôme Gouzy

Joëlle Dupont

Dominique Swennen

Emilie Dumas

Tatiana Giraud
email: tatiana.giraud@u-psud.fr

Antoine Branca

P. decumbens, P. digitatum. P Expansum

Diversity and mechanisms of genomic adaptation in Penicillium

Keywords: convergence, parallel adaptation, domestication, mycotoxins, secondary metabolites, P. rubens

Penicillium is a diverse fungal genus with hundreds of species occurring worldwide in various substrates, from soil to food, and with various lifestyles, from necrotrophic pathogenicity to endophytic mutualism. Several species are important for human affairs, being widely used in industry, such as the penicillin-producer P. rubens, the two cheese starters P. camemberti and P. roqueforti, and the mold used for fermenting sausages, P. nalgiovense. Other species are food spoilers that produce harmful mycotoxins or cause damages in fruit crops. Currently, 30 genomes of Penicillium belonging to 18 species are available. In this chapter, we reconstruct a phylogenetic tree based on available Penicillium genomes and outline the main features of the genomes, such as gene and transposable element content. We then review the recent advances that the available genomic and transcriptomic resources in the Penicillium genus have allowed regarding our understanding of the genomic processes of adaptation, including changes in gene content, expression and strikingly frequent and recent horizontal gene transfers. In addition, we summarize recent studies using genetic markers on the level of genetic diversity, mode of reproduction and population structure within Penicillium species. Overall, the Penicillium genus appears highly suitable models for studying the mechanisms of adaptation.

Introduction

Penicillium is a diverse fungal genus with 354 accepted species today (Visagie et al., 2014b), occurring worldwide in various substrates, from soil to food. Their lifestyles also cover a broad range, from necrotrophic pathogenicity to endophytic mutualism, although most are saprotrophs. As a consequence of these ecological niches, many Penicillium species have important economic and social relevance for human populations. Several species are widely used in industry, such as the penicillin-producer P. rubens, the two cheese starters P. camemberti and P. roqueforti, and the mold used for fermenting sausages, P. nalgiovense [START_REF] Bernáldez | Effect of Penicillium nalgiovense as protective culture in processing of dryfermented sausage "salchichón[END_REF]. Other species cause damages and yield loss in fruit crops, e.g., P. digitatum and P. italicum, while others are a concern for food safety because of their production of mycotoxins, such as patulin [START_REF] Eckert | Postharvest disorders and diseases of citrus fruits. In "The Citrus Industry[END_REF][START_REF] Mccallum | Factors affecting patulin production by Penicillium expansum[END_REF].

In addition to their economic significance, Penicillium molds also provide a tractable model for understanding the genetic and genomic processes underlying adaptation, due to the diversity of their ecological niches, their small genomes, their long haploid phase, their short generation time, and their easy manipulations in laboratory. Therefore, they can help addressing the current key challenges in evolutionary biology, including the identification of the genes involved in ecologically relevant traits as well as the understanding of the nature, time course, and architecture of the genomic changes involved in the origin and processes of population adaptation and divergence [START_REF] Gladieux | Fungal evolutionary genomics provides insight into the mechanisms of adaptive divergence in eukaryotes[END_REF].

The Penicillium species used in industry (e.g., for cheese maturation, sausage fermentation or for penicillin production) represent particularly well-suited organisms for studying domestication, a selection-based process studied since the dawn of evolutionary thinking as a model of rapid adaptation and diversification [START_REF] Darwin | The variation of animals and plants under domestication[END_REF]). Yet, the process of domestication has been much less studied in eukaryote microorganisms than in plants or animals. Several traits are typically modified in domesticated fungi, such as color of colonies, growth rate, thallus density, length of conidiophores, rate and rapidity of spore germination [START_REF] Eichler | Beurteilung der Auskeirnfâhigkeit von Blauschimmelsporen[END_REF][START_REF] Moreau | Nomenclature des Penicillium utiles à la préparation du Camembert[END_REF]. In the baker's yeast Saccharomyces cerevisiae, the genomic processes associated with domestication include large-scale duplications leading to genome size expansions [START_REF] Liti | Yeast evolution and comparative genomics[END_REF][START_REF] Machida | Genome sequencing and analysis of Aspergillus oryzae[END_REF], the acquisition of new traits by horizontal gene transfers (HGT) [START_REF] Hall | Contribution of horizontal gene transfer to the evolution of Saccharomyces cerevisiae[END_REF]Khaldi and Wolfe, 2008;Khaldi et al., 2008;[START_REF] Novo | Eukaryote-to-eukaryote gene transfer events revealed by the genome sequence of the wine yeast Saccharomyces cerevisiae EC1118[END_REF] and hybridization [START_REF] Liti | Sequence diversity, reproductive isolation and species concepts in Saccharomyces[END_REF]. These mechanisms have been suggested as ways for fungi to increase their biochemical repertoire and their ability to adapt to new ecological niches, but its generality remains to be assessed [START_REF] Friesen | Emergence of a new disease as a result of interspecific virulence gene transfer[END_REF]Khaldi and Wolfe, 2008;Khaldi et al., 2008;[START_REF] Ropars | Adaptive horizontal gene transfers between cheese-associated fungi[END_REF][START_REF] Wisecaver | The Evolution of Fungal Metabolic Pathways[END_REF].

In this chapter, we aimed at summarizing the insights that the available genomic resources in the Penicillium genus have contributed to our understanding of the genomic processes of adaptation. In addition, we review recent studies using genetic markers on the level of genetic diversity, mode of reproduction and population structure within Penicillium species.

We start with a section describing the Penicillium species for which genomes are available, with special emphasis on their ecological niches and life history traits. Second, we reconstruct a phylogenetic tree based on available Penicillium genomes and outline the main features of the genomes, such as gene and transposable element content. Third, we summarize recent findings on strikingly frequent and recent horizontal gene transfers. Fourth, we review some of the recent transcriptomic studies performed in Penicillium fungi. We finally review recent investigations on genetic diversity and population structure within Penicillium species.

Ecological niches and life history traits

Currently, 30 genomes of Penicillium belonging to 18 species are available in public databases (Table 1); these species are necrotrophic plant pathogens (P. digitatum, P. expansum, P. italicum), common food spoilers (P. biforme, P. fuscoglaucum, P. carneum, P. paneum), or key industrial species for food production (P. camemberti, P. roqueforti, P. nalgiovense), pharmaceutic industry (P. rubens) or biorefinery (P. decumbens). The genome described as P. aurantiogriseum in the databases [START_REF] Yang | Genome sequencing and analysis of the paclitaxelproducing endophytic fungus Penicillium aurantiogriseum NRRL 62431[END_REF] most likely belongs to P. expansum [START_REF] Ballester | Genome, transcriptome, and functional analyses of Penicillium expansum provide new insights into secondary metabolism and pathogenicity[END_REF]. We detail below some of the specific traits and lifestyles of the Penicillium species with a sequenced genome.

The two species used for cheese production, P. camemberti and P. roqueforti, though sharing the same nutrient-rich ecological niche, are not closely related and have different domestication histories, thus providing ideal models to study parallel adaptation [START_REF] Elmer | Adaptation in the age of ecological genomics: insights from parallelism and convergence[END_REF]. The fungus P. camemberti, used for the maturation of soft cheeses like Brie and Camembert, is the result of selection programs aiming at improving the texture of the colony, the color of the conidia and physiological characteristics of mycelia. This human-created white species is thought to be derived from a single clone of P. commune [START_REF] Pitt | Penicillium commune, P. camemberti, the origin of white cheese moulds, and the production of cyclopiazonic acid[END_REF], a species complex split into P. biforme and P. fuscoglaucum [START_REF] Giraud | Microsatellite loci to recognize species for the cheese starter and contaminating strains associated with cheese manufacturing[END_REF]. Penicillium camemberti has never been isolated from other substrates than dairy products, in contrast to P. fuscoglaucum and P. biforme, considered as contaminants by stakeholders because of their blue-grey color [START_REF] Frisvad | Polyphasic taxonomy of Penicillium subgenus Penicillium. A guide to identification of food and air-borne terverticillate Penicillia and their mycotoxins[END_REF]. Penicillium roqueforti, used as a starter culture in the production of blue veined-cheeses, is in contrast widespread in food environments and has also been isolated from plant environments such as silage or wood [START_REF] Frisvad | Polyphasic taxonomy of Penicillium subgenus Penicillium. A guide to identification of food and air-borne terverticillate Penicillia and their mycotoxins[END_REF].

Penicillium carneum and P. paneum, two sister species of P. roqueforti, are common food spoilers responsible for the production of harmful mycotoxins [START_REF] O'brien | Mycotoxins and other secondary metabolites produced in vitro by Penicillium paneum Frisvad and Penicillium roqueforti Thom isolated from baled grass silage in Ireland[END_REF][START_REF] Petersson | Growth of Penicillium roqueforti, P. carneum, and P. paneum during malfunctioning airtight storage of high-moisture grain cultivars[END_REF]. Penicillium nalgiovense is used in the food industry as an inoculum on fermented dry sausages in Italy, Spain and France [START_REF] Bernáldez | Effect of Penicillium nalgiovense as protective culture in processing of dryfermented sausage "salchichón[END_REF]. It contributes to the taste of sausages and it helps preventing desiccation and protecting them from undesirable microorganisms through antibacterial and antifungal activities [START_REF] Lücke | Fermented sausages[END_REF][START_REF] Lücke | Starter cultures for dry sausages and raw ham composition and effect[END_REF].

The discovery by Alexander Fleming and subsequent mass production of β-lactam antibiotics (including penicillin) have revolutionized medicine and greatly contributed to reduce the mortality due to infectious bacterial diseases in the world [START_REF] Fleming | On the antibacterial action of cultures of a penicillium, with special reference to their use in the isolation of B. influenzae[END_REF][START_REF] Hersbach | The penicillins: Properties, biosynthesis, and fermentation[END_REF]. The production of antibiotics in large amounts has been made possible through strain improvement in P. rubens. Industrial strains are all derived from a single strain, NRRL1951, isolated from a spoiled cantaloupe during World War II [START_REF] Raper | Penicillin: II. Natural Variation and Penicillin Production in Penicillium notatum and Allied Species1, 2[END_REF]. Until recently, P. rubens was considered as a synonym of P. chrysogenum, but it is now accepted as a closely related but separated species (Houbraken et al., 2011). Penicillium solitum is a common food spoiler (in cheeses and dry meats such as sausages and salami), a pomaceous fruit pathogen [START_REF] Frisvad | Physiological criteria and mycotoxin production as aids in identification of common asymmetric penicillia[END_REF][START_REF] Pitt | Toxigenic Penicillium species[END_REF][START_REF] Sanderson | Postharvest decay of winter pear and apple fruit caused by species of Penicillium[END_REF] and it has also been isolated from apple orchards' soil and house dust [START_REF] Frisvad | Polyphasic taxonomy of Penicillium subgenus Penicillium. A guide to identification of food and air-borne terverticillate Penicillia and their mycotoxins[END_REF][START_REF] Papagianni | Mould growth on traditional Greek sausages and penicillin production by Penicillium isolates[END_REF]Visagie et al., 2014a). It is also used for production of compactin,a cholesterol-lowering agent with also an antifungal effect [START_REF] Frisvad | Polyphasic taxonomy of Penicillium subgenus Penicillium. A guide to identification of food and air-borne terverticillate Penicillia and their mycotoxins[END_REF].

Penicillium digitatum is a necrotrophic pathogen responsible for up to 90% post-harvest losses in citrus storage, particularly in arid and sub-tropical climates [START_REF] Eckert | Postharvest disorders and diseases of citrus fruits. In "The Citrus Industry[END_REF].

In contrast to most other necrotrophic fungi, P. digitatum seems highly specialized as it has never been collected in any other substrate than citrus [START_REF] Barkai-Golan | Postharvest diseases of fruits and vegetables: development and control[END_REF]. It is therefore a model for the study of specialization in the necrotrophic lifestyle. Penicillium italicum presents a very similar lifestyle on the same substrate [START_REF] Palou | Penicillium digitatum, Penicillium italicum (Green Mold, Blue Mold)[END_REF].

Penicillium expansum is famous for being the first described Penicillium species; it is an important post-harvest spoilage agent that can cause great losses in apple storage facilities [START_REF] Jurick | Identification of wild apple germplasm (Malus spp.) accessions with resistance to the postharvest decay pathogens Penicillium expansum and Colletotrichum acutatum[END_REF]. In contrast to P. digitatum, P. expansum can be found in a wide array of other substrates and is a concern for health care as it can produce patulin, a highly toxic mycotoxin [START_REF] Mccallum | Factors affecting patulin production by Penicillium expansum[END_REF]. The role of patulin in the capacity to colonize fruits and diverse ecological niches remains debated [START_REF] Ballester | Genome, transcriptome, and functional analyses of Penicillium expansum provide new insights into secondary metabolism and pathogenicity[END_REF].

Penicillium capsulatum and P. decumbens both produce secondary metabolites useful for the industry, in particular highly efficient enzymes for degrading cellulose; P. decumbens is also used in biorefinery as a renewable source for oil production [START_REF] Li | Development of a highly efficient gene targeting system allowing rapid genetic manipulations in Penicillium decumbens[END_REF]. Penicillium verrucosum and P. nordicum are both known for producing ochratoxin A, one of the most common mycotoxins in spoiled food [START_REF] Castella | Molecular characterization of ochratoxin A producing strains of the genus Penicillium[END_REF]. Penicillium paxilli is the species which the potent tremor-inducing blocker of calcium-activated potassium channels paxillin was originally isolated from [START_REF] Berry | Draft genome sequence of the filamentous fungus Penicillium paxilli (ATCC 26601)[END_REF].

Phylogenetic relationships, genome size and content, and changes in gene content

Despite the economically and ecologically important role of Penicillium fungi, their phylogenetic relationships still remain unclear, with only a few phylogenetic trees published, based on single genes (Houbraken and Samson, 2011;[START_REF] Samson | Phylogenetic analysis of Penicillium subgenus Penicillium using partial β-tubulin sequences[END_REF][START_REF] Seifert | Prospects for fungus identification using CO1 DNA barcodes, with Penicillium as a test case[END_REF].

We therefore constructed a phylogenetic tree, including all Penicillium strains with an available sequenced genome and four Aspergillus species as outgroups, based on 3,986 shared singlecopy orthologs, corresponding to a concatenated alignment of 1,198,500 bp (Figure 1). In the Penicillium clade, some internal nodes remained poorly supported; this may reflect a rapid radiation in this clade, leading to incomplete lineage sorting, and/or hybridizations or recurrent horizontal gene transfers [START_REF] Cheeseman | Multiple recent horizontal transfers of a large genomic region in cheese making fungi[END_REF][START_REF] Ropars | Adaptive horizontal gene transfers between cheese-associated fungi[END_REF].

Genomes in the Penicillium genus appear highly dynamic, with estimated genome sizes ranging from 25 Mb to 36 Mb (Table 1). A relationship between host range and genome size has been proposed for fruit pathogens [START_REF] Ballester | Genome, transcriptome, and functional analyses of Penicillium expansum provide new insights into secondary metabolism and pathogenicity[END_REF][START_REF] Marcet-Houben | Genome sequence of the necrotrophic fungus Penicillium digitatum, the main postharvest pathogen of citrus[END_REF]. For example, P. digitatum, which presents the smallest genome (25.7Mb in average), is only able to infect citrus fruits, whereas the generalist P. expansum (pathogen of pome and stone fruits) has the largest genome among fruit pathogens (ca 31 Mb); P. italicum presents intermediate host range (mainly citrus fruits) and genome size (ca 29 Mb) (Table 1).

We analysed the repeat content of all available genome using RepeatMasker [START_REF] Smit | RepeatMasker Open-4[END_REF]. Overall, Penicillium genomes have a low proportion of interspersed repeats, ranging from 0.32 to 1.71% of total genome assembly lengths, and yet all classes of transposable elements (TEs) are represented (Figure 2 Penicillium biforme, P. camemberti and P. fuscoglaucum, three closely related species occurring in cheese, show the largest genome sizes among available Penicillium genomes (ca.

35-36 Mb, Table 1), which is not associated with a particularly high TE content (Figure 2). Interestingly, a large expansion of the proteome seems to have occurred in the ancestor of this clade, indicating that the increase in genome size might be correlated with the acquisition of new genes (Rodríguez de la Vega et al., 2015). In contrast, P. digitatum is characterized by a much lower number of genes than other Penicillium lineages [START_REF] Marcet-Houben | Genome sequence of the necrotrophic fungus Penicillium digitatum, the main postharvest pathogen of citrus[END_REF][START_REF] Ropars | Adaptive horizontal gene transfers between cheese-associated fungi[END_REF], which probably relates to its highly specialized necrotrophic lifestyle.

Comparative genomic studies have revealed interesting patterns of gains or losses in genes involved in the production of secondary metabolites, such as penicillin, patulin or small secreted proteins acting as effectors in pathogens. Penicillium rubens for instance has acquired its capacity to produce high quantities of penicillin through the duplication of its penicillin biosynthetic gene cluster [START_REF] Fierro | The penicillin gene cluster is amplified in tandem repeats linked by conserved hexanucleotide sequences[END_REF] A comparison of two industrial P. rubens genomes, one of a high-penicillin producing strain and one of a low-penicillin producing strain, revealed in the high-penicillin producing strain an even higher number of copies of the penicillin biosynthetic gene cluster, as well as many genomic structural variations, such as translocations and gene gains/losses, likely related to an enhanced nitrogen and energy metabolism [START_REF] Wang | Genome sequencing of high-penicillin producing industrial strain of Penicillium chrysogenum[END_REF]. In the necrotrophic species P. expansum, a large number of secondary metabolism gene clusters were identified that were absent in other sequenced Penicillium genomes and may be involved in pathogenicity [START_REF] Ballester | Genome, transcriptome, and functional analyses of Penicillium expansum provide new insights into secondary metabolism and pathogenicity[END_REF]. Actually, despite a major genome contraction compared with other Penicillium species, P. expansum had the largest repertoire of secondary metabolites genes, indicating high numbers of gene gains and losses in this species. The patulin gene cluster was inferred to be present in the ancestor of P. expansum and P. roqueforti, its absence in other lineages of this clade implying gene losses [START_REF] Ballester | Genome, transcriptome, and functional analyses of Penicillium expansum provide new insights into secondary metabolism and pathogenicity[END_REF]. Overall, Penicillium genomes thus appear highly dynamic, with changes in gene content relating to genomic adaptations.

Horizontal gene transfers

The multiple available Penicillium genomes have further allowed detecting dozens of horizontal gene transfers (HGTs) that are transmissions of genetic material between species by other means than sexual reproduction. HGT events are most often detected by the existence of incongruences between gene genealogies and the species tree. Indeed, the finding of orthologs from distantly related species placed close together in a gene tree most likely indicates that this gene has recently been horizontally transferred between the two species instead of having followed vertical inheritance and divergence along the species tree. Despite long thought to be rare in eukaryotes, recent studies have shown that HGTs may play a major role in adaptation in this lineage, in particular in fungi [START_REF] Gladieux | Fungal evolutionary genomics provides insight into the mechanisms of adaptive divergence in eukaryotes[END_REF][START_REF] Keeling | Functional and ecological impacts of horizontal gene transfer in eukaryotes[END_REF][START_REF] Keeling | Horizontal gene transfer in eukaryotic evolution[END_REF][START_REF] Wisecaver | The Evolution of Fungal Metabolic Pathways[END_REF].

Fungi are the eukaryotic group for which the largest number of HGT events has been described so far. Most of described HGTs have a prokaryotic origin, likely reflecting the abundance of prokaryotes in all environments and the relative ease of detecting such HGT events compared to those from a eukaryotic origin [START_REF] Gladieux | Fungal evolutionary genomics provides insight into the mechanisms of adaptive divergence in eukaryotes[END_REF]. However, many HGTs between fungi have also been described, such as transfers of genes involved in secondary metabolite pathways in Aspergillus and Penicillium clades [START_REF] Wisecaver | The Evolution of Fungal Metabolic Pathways[END_REF].

Among Penicillium species, the penicillin-producer P. rubens may have acquired several important genes from bacteria by horizontal gene transfers, including some of the penicillin biosynthetic genes, pcbAB and pcbC, and the arsenate-resistance cluster (van den Berg et al., 2008). Genome analysis of the necrotrophic fungus P. digitatum revealed four putative genes that have been horizontally acquired from prokaryotes, including DEC1, likely playing a role in pathogenicity [START_REF] Marcet-Houben | Genome sequence of the necrotrophic fungus Penicillium digitatum, the main postharvest pathogen of citrus[END_REF]; indeed, it belongs to a gene family associated with virulence in maize infections, with homologs in plant pathogenic fungi and in bacteria, but without any homolog in non-pathogenic Penicillium species.

Several other horizontal gene transfers in Penicillium have occurred in the cheese environment, being striking by 1) the size of the transferred regions (i.e., several kilobases), 2) the eukaryotic origin of these transfers (likely among cheese-associated Penicillium species),

3) the number of species in which the same regions have been horizontally transferred, and 4) the very recent date of the transfers, likely associated with the human history of cheese production [START_REF] Cheeseman | Multiple recent horizontal transfers of a large genomic region in cheese making fungi[END_REF][START_REF] Ropars | Adaptive horizontal gene transfers between cheese-associated fungi[END_REF]. These horizontally transferred regions (HTRs) indeed occurred between Penicillium species from the cheese environment, and were completely identical at the nucleotide level between distant species (otherwise having a pairwise sequence identity of 85-90%) while lacking in other closely related species. One of these HTRs, Wallaby, is a 575 kb region that accounts for 2% of the P. roqueforti genome, and it can be in a single block or in a few fragments depending on the species [START_REF] Cheeseman | Multiple recent horizontal transfers of a large genomic region in cheese making fungi[END_REF]. Another HTR, CheesyTer, is 80 kb long, and is always found in a single block [START_REF] Ropars | Adaptive horizontal gene transfers between cheese-associated fungi[END_REF].

These two HTRs are flanked by copies of transposable elements (TEs) belonging to a specific family, the i non-LTR retrotransposons, that are rare elsewhere in the genomes [START_REF] Ropars | Adaptive horizontal gene transfers between cheese-associated fungi[END_REF]. This suggests that these TEs may be involved in the horizontal gene transfers. In fungi, the transfer of genetic material is thought to occur through conjugation, natural and agrobacterial transformation, viral transduction, or anastomosis [START_REF] Coelho | Extensive intrakingdom horizontal gene transfer converging on a fungal fructose transporter gene[END_REF][START_REF] Wisecaver | Fungal metabolic gene clusters -caravans traveling across genomes and environments[END_REF]. In Fusarium, for example, the transfer of an entire chromosome can occur by simple co-incubation of mycelial of two strains [START_REF] Ma | Comparative genomics reveals mobile pathogenicity chromosomes in Fusarium[END_REF].

The gene content in Wallaby and CheesyTer suggests that these transfers may play an important role in the adaptation of these fungi to the cheese environment. Among the 248 genes that Wallaby was predicted to contain, two genes, paf and Hce2, encoded proteins that may be involved in antagonistic interactions with other microorganisms [START_REF] Cheeseman | Multiple recent horizontal transfers of a large genomic region in cheese making fungi[END_REF].

CheesyTer carries 37 putative genes, including genes coding for lactose permease and betagalactosidase, which likely provide advantages in terms of use of the cheese substrate [START_REF] Ropars | Adaptive horizontal gene transfers between cheese-associated fungi[END_REF]. Actually, these two genes were found to be overexpressed in the first days of cheese maturation [START_REF] Lessard | Metatranscriptome analysis of fungal strains Penicillium camemberti and Geotrichum candidum reveal cheese matrix breakdown and potential development of sensory properties of ripened Camembert-type cheese[END_REF][START_REF] Ropars | Adaptive horizontal gene transfers between cheese-associated fungi[END_REF].

In P. roqueforti, all strains were found to carry either both or none of these two HTRs.

The two HTRs were only present in strains found in the dairy environment, while lacking in some strains from cheese and in all the strains isolated from other environments, such as silage or wood [START_REF] Cheeseman | Multiple recent horizontal transfers of a large genomic region in cheese making fungi[END_REF][START_REF] Ropars | Adaptive horizontal gene transfers between cheese-associated fungi[END_REF]. This further indicates an advantage conferred by these two HTRs in cheese.

Experiments of growth and competition on different media have further supported a role of the two HTRs in adaptation to cheese. Indeed, P. roqueforti strains carrying the two HTRs showed a significantly higher growth rate on cheese medium and a significantly lower growth rate on minimal medium [START_REF] Ropars | Adaptive horizontal gene transfers between cheese-associated fungi[END_REF]. Furthermore, P. roqueforti strains carrying the two HTRs showed a significant competitive advantage, both against P. roqueforti strains lacking the HTRs and against other Penicillium species also lacking the HTRs. Interestingly, this effect was only significant when strains were grown on cheese medium and not on minimal medium [START_REF] Ropars | Adaptive horizontal gene transfers between cheese-associated fungi[END_REF].

Transcriptomics

The availability of Penicillium genomes have also facilitated transcriptomic studies, i.e., investigations of mRNA expression in different conditions, which allows studying the regulation of genes and therefore detecting important genes involved in the adaptation to particular environments. Transcriptomics in Penicillium has focused so far on penicillin production in P. chrysogenum, on secondary metabolites production in plant pathogens, such as P. expansum, P. digitatum and P. italicum, and on cheese-making fungi.

The transcriptome analyses of the pathogenic P. expansum on apple revealed the induction of several metabolic pathways during infection and thus identified putative pathogenicity factors, such as proteases, cell-wall degrading enzymes and oxidoreductases [START_REF] Ballester | Genome, transcriptome, and functional analyses of Penicillium expansum provide new insights into secondary metabolism and pathogenicity[END_REF]. Putative effectors that are able to modulate host physiology were also identified [START_REF] Ballester | Genome, transcriptome, and functional analyses of Penicillium expansum provide new insights into secondary metabolism and pathogenicity[END_REF].

A metatranscriptome analysis of P. camemberti and Geotrichum candidum was performed in a camembert-type cheese matrix [START_REF] Lessard | Metatranscriptome analysis of fungal strains Penicillium camemberti and Geotrichum candidum reveal cheese matrix breakdown and potential development of sensory properties of ripened Camembert-type cheese[END_REF]. The functional annotation allowed the identification of the biological processes involved in cheese ripening. Globally, similar functions appeared involved in the use of the cheese substrate in both the yeast G. candidum and the mold P. camemberti [START_REF] Lessard | Metatranscriptome analysis of fungal strains Penicillium camemberti and Geotrichum candidum reveal cheese matrix breakdown and potential development of sensory properties of ripened Camembert-type cheese[END_REF].

A system biology approach, including transcriptomic but also metabolome and metabolic flux analyses, was used to understand the loss of penicillin production capacity by the high-producing P. chrysogenum strain during long-term ethanol-limited cultivation, a phenomenon called degeneration. The findings indicated that degeneration was due to the production of a lower quantity of the first two enzymes acting during penicillin biosynthesis, which may be due to a decrease of translation efficiency [START_REF] Douma | Degeneration of penicillin production in ethanol-limited chemostat cultivations of Penicillium chrysogenum: A systems biology approach[END_REF].

Population genetic diversity within Penicillium species and mode of reproduction

The genetic diversity has also been investigated within Penicillium species in some cases, although only with neutrally-evolving markers so far. The population genetic variability has been found to differ drastically among Penicillium species. In P. camemberti, no genetic polymorphism could be detected using either DNA fragments or microsatellites. This is consistent with this species being a clonal lineage originating from a white mutant in the cheesemaking mold formerly used for making Brie, P. commune [START_REF] Giraud | Microsatellite loci to recognize species for the cheese starter and contaminating strains associated with cheese manufacturing[END_REF]. In contrast, the genetic diversity in P. roqueforti was revealed to be substantial using microsatellites [START_REF] Ropars | Induction of sexual reproduction and genetic diversity in the cheese fungus Penicillium roqueforti[END_REF] and DNA fragment sequences [START_REF] Gillot | Fifty shades of blue: insights into Penicillium roqueforti morphological and genetic diversity[END_REF]. A strong population structure was found, with one population containing only cheese strains, most of which carried the HTRs described above, Wallaby and CheesyTer, and a second population containing cheese and noncheese strains, all lacking the HTRs. These two populations showed further, although weaker, subdivisions that corresponded to different morphologies and different cheese types [START_REF] Gillot | Fifty shades of blue: insights into Penicillium roqueforti morphological and genetic diversity[END_REF].

To date, genetic diversity has not been investigated within other Penicillium species, with the exception of P. chrysogenum, which has actually led to the identification of cryptic species and in the subsequent renaming of the penicillin-producing strain in P. rubens (Houbraken et al., 2011).

Although sexual stages have been described in many Penicillium species (Visagie et al 2014b), others has been historically considered to be exclusively asexual. This actually holds true for one fifth of fungal species [START_REF] Taylor | The evolution of asexual fungi: reproduction, specification and classification[END_REF], but was mainly due to the difficulty of observing sex in this phylum in nature. Indeed, direct or indirect evidence of sex have been observed in most cases when thoroughly investigated. Indirect evidence include 1) the presence of the complete meiotic toolbox (i.e., all the genes known to be necessary for meiosis and for mating-type determinism, with sequences apparently functional, i.e., under purifying selection), 2) footprints of recombination in populations, and 3) footprints of RIP (repeat-induced pointmutation), a defense mechanism of fungal genome inducing C/T transition mutations in repeated sequences during sexual reproduction in ascomycetes [START_REF] Galagan | RIP: The evolutionary cost of genome defense[END_REF].

Recent studies have improved our knowledge of the reproduction mode and breeding system in Penicillium. All species studied so far were shown to be heterothallic, as haploid genomes carried a single mating-type allele, either MAT1-1 or MAT1-2 [START_REF] Hoff | Eighty years after its discovery, Fleming's Penicillium strain discloses the secret of its sex[END_REF][START_REF] Ropars | Sex in cheese: evidence for sexuality in the fungus Penicillium roqueforti[END_REF].

After discoveries of indirect evidence of sex in populations of the penicillin-producer P. rubens, with occurrence of both mating-type alleles [START_REF] Hoff | Eighty years after its discovery, Fleming's Penicillium strain discloses the secret of its sex[END_REF] and of RIP footprints [START_REF] Braumann | Repeat induced point mutation in two asexual fungi, Aspergillus niger and Penicillium chrysogenum[END_REF], a sexual cycle could be induced in this species [START_REF] Böhm | Sexual reproduction and mating-typemediated strain development in the penicillin-producing fungus Penicillium chrysogenum[END_REF].

Similarly, in the cheese species P. roqueforti, mating types were shown to occur in balanced ratios in populations, RIP footprints were observed and purifying selection was inferred on genes involved in mating [START_REF] Ropars | Sex in cheese: evidence for sexuality in the fungus Penicillium roqueforti[END_REF]. Later, population analyses showed no linkage disequilibria among markers, suggesting recurrent recombination events and fruiting bodies and recombinant sexual ascospores could be successfully produced in vitro [START_REF] Ropars | Induction of sexual reproduction and genetic diversity in the cheese fungus Penicillium roqueforti[END_REF].

Conclusions and future prospects

Genomic and transcriptomic analyses have revealed several interesting genes and mechanisms likely involved in the adaptation of Penicillium species to various environments. In particular, the domesticated penicillin-producing and cheese-making Penicillium appear ideal model eukaryotes for studying the genomic processes of adaptation, given the recent and strong selection by humans. These genomic inferences now need to be validated using functional genetics, which will be allowed by the recent development of transformation and gene silencing methods [START_REF] Durand | Transformation of Penicillium roqueforti to phleomycin-and to hygromycin B-resistance[END_REF][START_REF] Gil-Durán | The pcz1 Gene, which Encodes a Zn (II) 2Cys6 Protein, Is Involved in the Control of Growth, Conidiation, and Conidial Germination in the Filamentous Fungus Penicillium roqueforti[END_REF][START_REF] Goarin | Gene replacement in Penicillium roqueforti[END_REF][START_REF] Kosalková | A natural short pathway synthesizes roquefortine C but not meleagrin in three different Penicillium roqueforti strains[END_REF][START_REF] Ullán | RNA-silencing in Penicillium chrysogenum and Acremonium chrysogenum: validation studies using β-lactam genes expression[END_REF]. It will also be very interesting to explore the population genomics of adaptation within species, in particular in the Penicillium fungi with high genetic diversity and a variety of ecological niches, such as P. roqueforti. Footprints of selective sweeps for instance may reveal selection having acting recently on other genes that the horizontally-transferred regions.

  ). The 10 most abundant types of TEs include four non-LTR i retroelements (I-1_AO, I-4_AO, I-5_AO and I-6_AO), three mariner DNA transposons (Mariner-6_AN, MarinerL-1_AO and Mariner-1_AF), one hAT DNA transposon (hAT-1_AN), one gypsy LTR retrotransposon (Gypsy1-I_AO) and one R1 non-LTR retroelement (RTAg4). Gypsy elements are by far the most abundant TE class found in Penicillium genomes as they account for 20% of all TEs; mariner elements constitute the second most abundant class (13%).

Figure 1 :

 1 Figure1: Maximum likelihood tree of genome sequenced Penicillium based on concatenated alignment of 3,986 single copy orthologs using RaxML[START_REF] Stamatakis | RAxML version 8: a tool for phylogenetic analysis and postanalysis of large phylogenies[END_REF]. Node labels correspond to the proportion of gene trees supporting the node.

Figure 2 :

 2 Figure 2: (A) Number and classification of transposable elements found in Penicillium genomes; (B) Percentage of the Penicillium genomes composed of interspersed repeats.

Acknowledgements

This work was supported by the ANR FROMA-GEN grant (ANR-12-PDOC-0030) awarded to AB, an "attractivité" grant from Paris-Sud University to AB, the ERC starting grant GenomeFun 309403 awarded to TG, a Marie Curie postdoctoral fellowship to RCRdlV [FP7 COFUND PRES-SUD No. 246556], and the ANR grant 'Food Microbiomes' (ANR-08-ALIA-007-02) awarded to JD.


Lorenzo Brasco

Chiara Leone

AND Giovanni Pisante

Anna Verde

SOBOLEV AND LIPSCHITZ REGULARITY FOR LOCAL MINIMIZERS OF WIDELY DEGENERATE ANISOTROPIC FUNCTIONALS

Keywords: 2010 Mathematics Subject Classification. 35J70, 35B65, 49K20 Anisotropic problems, degenerate elliptic equations, Besov-Nikol'skiȋ spaces. 1

come

Sobolev and Lipschitz regularity for local minimizers of widely degenerate anisotropic functionals

1. Introduction 1.1. Overview. In this paper we continue to investigate differentiability properties of local minimizers of convex functionals, exhibiting wide degeneracies and an orthotropic structure. The model case of functional we want to study is given by (1.1)

F(u; Ω ) = N i=1 1 p i ˆΩ (|u x i | -δ i ) p i + dx + ˆΩ f u dx, u ∈ W 1,p loc (Ω), Ω Ω,
with δ i ≥ 0 and p i ≥ 2. We denote p = (p 1 , p 2 , . . . , p N ) and W 1,p loc (Ω) = u ∈ W 1,1 loc (Ω) : u x i ∈ L p i loc (Ω), i = 1, . . . , N . The symbol ( • ) + above stands for the positive part.

For p 1 = • • • = p N , some results can be found in the recent papers [3] and [START_REF] Brasco | On certain anisotropic elliptic equations arising in congested optimal transport: local gradient bounds[END_REF]. We refer to the introduction of [START_REF] Brasco | On certain anisotropic elliptic equations arising in congested optimal transport: local gradient bounds[END_REF] for some motivations of this kind of functionals, arising from Optimal Transport problems with congestion effects.

Our scope is to generalize these results to the anisotropic case, i.e. to the case where at least one of the exponents p i is different from the others. These functionals pertain to the class of variational problems with non standard growth conditions, first introduced by Marcellini in [17,18].

Similar functionals have been considered in the past by the Russian school, see for example [12] and [29]. More recently, they have been considered by many authors also in western countries. Among others, we mention (in alphabetical order) Bildhauer, Fuchs and Zhong [START_REF] Bildhauer | A regularity theory for scalar local minimizers of splitting-type variational integrals[END_REF]2] (where the terminology splitting-type integrals is used), Esposito, Leonetti and Mingione [7], Leonetti [START_REF] Leonetti | Higher integrability for minimizers of integral functionals with nonstandard growth[END_REF], Liskevich and Skrypnik [16] and Pagano [START_REF] Bildhauer | Variational integrals with a wide range of anisotropy[END_REF]. However, we point out that the type of degeneracy admitted in (1.1) is heavier than those of the above mentioned references, due to the presence of the coefficients δ i ≥ 0 above.

We observe that local minimizers of the functional (1.1) are local weak solution of the degenerate elliptic equation

N i=1 (|u x i | -δ i ) p i -1 + u x i |u x i | x i = f.
The particular case

δ 1 = • • • = δ N = 0 and p 1 = • • • = p N = p corresponds to N i=1 |u x i | p-2 u x i x i = f,
which has been called pseudo p-Laplace equation in the recent literature. Here we prefer to use the terminology orthotropic p-Laplace equation, which seems more adapted and meaningful.

1.2. Main results. Our first result is the Sobolev regularity for some nonlinear functions of the gradient of a bounded local minimizer. For p = (p 1 , . . . , p N ), we will use the notation p := (p 1 , . . . , p N ), where p i is the Hölder conjugate of p i .

Theorem 1.1 (Sobolev regularity for bounded minimizers). Let ∈ {1, . . . , N -1} and 2 ≤ p ≤ q.

We set p = (p,

• • • , p , q, • • • , q N - ),
and let u ∈ W 1,p loc (Ω) ∩ L ∞ loc (Ω) be a local minimizer of

F(u; Ω ) = N i=1 ˆΩ g i (u x i ) dx + ˆΩ f u dx,
where f ∈ W 1,p loc (Ω) and g 1 , . . . , g N : R → R + are C 2 convex functions such that

1 C (|s| -δ) p-2 + ≤ g i (s) ≤ C (|s| p-2 + 1), s ∈ R, i = 1, . . . , , 1
C (|s| -δ) q-2 + ≤ g i (s) ≤ C (|s| q-2 + 1), s ∈ R, i = + 1, . . . , N, for some C ≥ 1 and δ ≥ 0. We set

V i = V i (u x i ),
where V i (s) = ˆs 0 g i (τ ) dτ, i = 1, . . . , N.

• If = N -1 and p, q satisfy p ≥ N -1 or

         p < (N -2) 2 N -1 , q < (N -2) p (N -2) -p , or            (N -2) 2 N -1 ≤ p < N -1, q < p √ N -1 - √ p 2 , (1.2)
then we have V i ∈ W 1,2 loc (Ω), for i = 1, . . . , N . • if 1 ≤ ≤ N -2 and p, q satisfy

(1.3) p ≥ N -2, or        p < N -2, q < (N -2) p (N -2) -p ,
then we have V i ∈ W 1,2 loc (Ω), for i = 1, . . . , N . Moreover, for every B r B R Ω, we have (1.4) ∇V i L 2 (Br) ≤ C = C N, p, q, C, δ, dist(B R , ∂Ω), u L ∞ (B R ) , f W 1,p (B R ) .

Remark 1.2. The previous result is proved under the additional assumption u ∈ L ∞ loc (Ω). Indeed, since the appearing of the celebrated counterexamples by Marcellini [19] and Giaquinta [START_REF] Giaquinta | Growth conditions and regularity, a counterexample[END_REF], it is well-known that local minimizers to this kind of functionals may be unbounded if p and q are too far apart (see also Hong's paper [START_REF] Min-Chun | Some remarks on the minimizers of variational integrals wtih non standard growth conditions[END_REF]). Sharp conditions in order to get u ∈ L ∞ loc can be found in [8, Theorem 3.1], see also the recent paper [START_REF] Cupini | Local boundedness of minimizers with limit growth conditions[END_REF].

Remark 1.3 (Comparison with previous results, part I). Theorem 1.1 contains as a particular instance the scalar case of [5, Theorem 2] by Canale, D'Ottavio, Leonetti and Longobardi, which still concerns bounded local minimizers. The latter corresponds to the particular case = N -1, p = 2 and δ = 0. However, even in this case, our result is stronger than [5, Theorem 2], since our conditions (1.2) are less restrictive for dimension N ∈ {2, 3, 4, 5}. In particular, we observe that for N ∈ {2, 3}, the first condition in (1.2) is always fulfilled. Thus in low dimension we have Sobolev regularity no matter how large q is, provided local minimizers are locally bounded.

In the model case (1.1), the result of Theorem 1.1 boils down to

(|u x i | -δ i ) p i 2 + u x i |u x i | ∈ W 1,2 loc (Ω), i = 1, . . . , N.
In particular, for local weak solutions of the anisotropic orthotropic p-Laplace equation (i.e. for δ i = 0), we get

|u x i | p i -2 2 u x i ∈ W 1,2 loc (Ω), i = 1, . . . , N.
This is the analog of the well-known result |∇u| p-2 2 ∇u ∈ W 1,2 loc (Ω), for local weak solution of the p-Laplace equation ∆ p u := div(|∇u| p-2 ∇u) = f, in the case p ≥ 2 (see [START_REF] Fusco | Some remarks on the regularity of minima of anisotropic integrals[END_REF]Lemma 3.1]).

We now restrict the discussion to the case of dimension N = 2 and consider for simplicity the model case presented at the beginning. We can prove the Lipschitz regularity of local minimizers. Namely, we obtain the following generalization of [3, Theorem A], the latter corresponding to the case

p 1 = p 2 . Theorem 1.4 (Lipschitz regularity in dimension 2). Let N = 2, 2 ≤ p 1 ≤ p 2 and δ 1 , δ 2 ≥ 0. Let f ∈ W 1,p loc (Ω), then every local minimizer u ∈ W 1,p loc (Ω) of the functional F(u; Ω ) = 2 i=1 1 p i ˆΩ (|u x i | -δ i ) p i + dx + ˆΩ f u dx, u ∈ W 1,p loc (Ω), Ω Ω,
is a locally Lipschitz continuous function.

Remark 1.5 (Comparison with previous results, part II). To the best of our knowledge, this result is new already in the simpler case of the functional

u → 2 i=1 1 p i ˆΩ |u x i | p i dx.
The only result of this type we are aware of is the pioneering one [29, Theorem 1] by Ural'tseva and Urdaletova. Though their result holds for every dimension N ≥ 2, this needs the additional assumptions p 1 ≥ 4 and p N < 2 p 1 . On the contrary, these restrictions are not needed in Theorem 1.4.

1.3. Some comments on the proofs. Let us spend some words on the proofs of our main results. As for Theorem 1.4, the proof is the same as that of [3, Theorem A], up to some technical modifications. This is based on a trick introduced in [3]: this permits to obtain Caccioppoli inequalities for convex functions of the gradient ∇u, by combining the linearized equation and the Sobolev regularity of Theorem 1.1. One can then build an iterative scheme of reverse Hölder inequalities and obtain the desired result by performing a Moser's iteration. The trick is a twodimensional one and does not seem possible to extend it to higher dimensional cases. On the other hand, we show here that the limitation p 1 = p 2 is not needed and the same proof works for p 1 < p 2 as well.

On the contrary, the proof of Theorem 1.1 contains a crucial novelty, which permits to improve the range of validity of Sobolev regularity, compared to previous results based on similar proofs. In order to neatly explain this point, we briefly resume the strategy for proving Theorem 1.1 in the model case

F(u; Ω ) = N i=1 1 p i ˆΩ (|u x i | -δ i ) p i + dx,
with the exponents p 1 ≤ p 2 ≤ . . . p N which could all differ. The starting point of the proof is differentiating the relevant Euler-Lagrange equation in a discrete sense, i.e. we use the Nirenberg's method of incremental ratios. This is very classical and permits to estimate integrated finite differences of the type (1.5)

ˆBr V i (• + h e j ) -V i |h| t 2 dx, i, j = 1, . . . , N,
by appealing to the monotonicity properties of the operator. The integrals (1.5) are estimated in terms of quantities like

(1.6) ˆBR u(• + h e j ) -u |h| t p i dx.
This is a (possibly fractional) derivative of u in the j-th direction, but raised to the power competing to the i-th direction. In the case

p 1 = • • • = p N = p ≥ 2,
one can take t = 1 and conclude directly that V i ∈ W 1,2 loc , thanks to the finite differences characterization of Sobolev spaces. On the contrary, the anisotropic case is subtler. We first observe that since by assumptions u x j ∈ L p j loc , when i ≤ j we could take again t = 1 in (1.5) and (1.6) and obtain full differentiability in these directions. For example, this is always the case if j = N , i.e. if we derive the equation in the x N direction, corresponding to the direction of maximal growth of the functional.

On the other hand, when j ≤ N -1 we have to pay attention to the "bad directions", corresponding to terms (1.6) with i ≥ j + 1. Indeed, in this case we do not know that u x j ∈ L p i loc . Rather, we choose 0 < t < 1 (depending on the ratio p j /p i ) and we use a L ∞ -W 1,p j interpolation in order to control this term (it is here that the assumption u ∈ L ∞ loc comes into play). By proceeding in this way, we get for every i = 1, . . . , N (1.7)

V i is weakly differentiable of order p j p N in the direction e j .

However, this is not the end of the story. Indeed, this information now entails that V i (and thus u x i ) enjoys better integrability properties, by fractional Sobolev embeddings. This in turn implies that we can re-initialize the previous scheme and exploit this new integrability in order to have a better control on (1.6). As a consequence, we can improve (1.7). The final outcome is thus obtained by a (finite) iteration of the scheme just described.

Up to now, the proof is very similar to that of [5]. The main difference is in the way we exploit (1.7) in order to improve the integrability of V i . In a nutshell, what usually one does is to extrapolate from (1.7) the weaker isotropic information

V i ∈ W p 1 p N
,2 loc (Ω), and then use the Sobolev embedding for usual fractional Sobolev-Slobodeckiȋ spaces. Then the algorithm runs as described above. Since in every direction we pass from p j /p N to p 1 /p N , each time p j > p 1 this gives rise to a loss of information which may be important.

In this paper, on the contrary, we take advantage of the full information contained in (1.7). The latter means that each V i is contained in an anisotropic Besov-Nikol'skiȋ space, where the anisotropy is now in the order of differentiability (we refer to Section 2.3 for the relevant definition). As one may expect, such a space has an improved Sobolev embedding, thus by proceeding in this alternative way the gain of integrability is strictly better at each step. This kind of anisotropic spaces and their embeddings seem to be completely overlooked or neglected by the recent literature on anisotropic problems, we refer to Nikol'skiȋ's monography [START_REF] Nikol | skiȋ, Approximation of functions of several variables and imbedding theorems[END_REF] for a comprehensive treatment of the subject (an alternative approach can be also find in Triebel's book [START_REF] Triebel | Theory of function spaces[END_REF]). We believe on the contrary this to be the natural setting for the problem and the natural tools to be exploited. These spaces are also briefly treated in the classical monography [START_REF] Kufner | Function spaces[END_REF] by Kufner, John and Fučík (see Sections 2 and 4 of [14, Chapter 8]).

Remark 1.6 (Why two exponents only?). After the previous description of the method of proof for the general case of p 1 ≤ p 2 ≤ • • • ≤ p N , the reader may be perplexed to see that in Theorem 1.1 we confine ourselves to the case of only two different exponents p ≤ q. The reason is easy to explain: the iterative scheme described above quickly becomes fairly intricate, in the general case

p 1 ≤ p 2 ≤ • • • ≤ p N .
In particular, when one tries to perform the iteration, at each step many subcases should be discussed by making the proof very difficult to be written (and read). For this reason, we preferred to confine our discussion to the case of two exponents.

At the same time, we believe our approach to be interesting and promising. Thus we explicitely write down the iterative step in the general case of N exponents, without running the scheme up to the end, see Propositions 3.2 and 3.4 below. These are valid under the assumption u ∈ L ∞ loc (Ω) and without restrictions on the spreadness of the exponents, thus they can be used in the general case p 1 ≤ p 2 ≤ • • • ≤ p N to obtain partial higher differentiability results.

1.4. Plan of the paper. In Section 2 we fix the notations and we set the preliminaries results needed throughout the paper, particularly focusing on embedding theorems for anisotropic Besov-Nikol'skiȋ spaces. In Section 3 we present, in a general form, the details of the scheme for improving differentiability roughly described above. Sections 4 and 5 are devoted to the proofs of Theorem 1.1 and Theorem 1.4 respectively. Some useful technical inequalities are finally collected in the Appendix.

Acknowledgements. The authors wish to thank Nicola Fusco for bringing reference [8] to their attention. The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Part of this work has been done during some visits of the first author to Napoli, as well as during the "XXVI Italian Workshop on Calculus of Variations" held in Levico Terme, in January 2016. Hosting institutions and organizers are gratefully acknowledged.

Preliminaries

2.1. Notation. Given h ∈ R N \ {0}, for a measurable function ψ : R N → R we introduce the notation

ψ h (x) := ψ(x + h) and δ h ψ(x) := ψ h (x) -ψ(x).
We recall that for every pair of functions ϕ, ψ we have

(2.1) δ h (ϕ ψ) = (δ h ϕ) ψ + ϕ h (δ h ψ).
We also use the notation

δ 2 h ψ(x) := δ h (δ h ψ(x)) = ψ(x + 2 h) + ψ(x) -2 ψ(x + h).
We indicate by {e 1 , . . . , e N } the canonical basis of R N .

Given 1 ≤ p 1 ≤ p 2 ≤ • • • ≤ p N , we denote p = (p 1 , p 2 , . . . , p N ). Let E ⊂ R N be
an open set, we define the anisotropic Sobolev spaces

W 1,p (E) = {u ∈ W 1,1 (E) : u x i ∈ L p i (E), i = 1, . . . , N }, and W 1,p 0 (E) = {u ∈ W 1,1 0 (E) : u x i ∈ L p i (E), i = 1, . . . , N }.
We define the harmonic mean p of the exponents

p 1 ≤ p 2 ≤ • • • ≤ p N 1 p = 1 N N i=1 1 p i ,
then the associated Sobolev-type exponent is defined by

p * =    N p N -p , if 1 ≤ p < N, +∞, if p > N.
Finally, for 0 < t < 1 and 1 ≤ p < ∞ we denote by W t,p (R N ) the Sobolev-Slobodeckiȋ space, i. e.

W t,p (R N ) = u ∈ L p (R N ) : [u] W t,p (R N ) < +∞ , where [u] p W t,p (R N ) = ˆRN ˆRN |u(x) -u(y)| p |x -y| N +t p dx dy.
Though we will not need this, we recall that W t,p (R) can be seen a particular instance of the larger class of Besov spaces.

2.2.

Embedding for anisotropic Sobolev spaces. We collect here a couple of embedding results that will be needed in the sequel. The first one is well-known 1 , a proof can be found for example in [27, Theorem 1 & Corollary 1].

Theorem 2.1 (Anisotropic Sobolev embeddings). Let Ω ⊂ R N be an open set, then for every u ∈ W 1,p 0 (Ω) we have:

(1) if p < N c u L p * (Ω) ≤ N i=1 ˆΩ |u x i | p i dx 1 p i , for a constant c = c(N, p) > 0; (2) if p = N and |Ω| < +∞, for every 1 ≤ χ < ∞ c u L χ (Ω) ≤ |Ω| 1 χ N i=1 ˆΩ |u x i | p i dx 1 p i . for a constant c = c(N, p, χ) > 0;
1 This result is usually attributed to Troisi in the literature of western countries, see [START_REF] Troisi | Teoremi di inclusione per spazi di Sobolev non isotropi[END_REF]. However, Trudinger in

[27] attributes the result for p = N to Nikol'skiȋ, whose paper [START_REF] Nikol | skiȋ, Imbedding theorems for functions with partial derivatives considered in various metrics[END_REF] appeared before [START_REF] Troisi | Teoremi di inclusione per spazi di Sobolev non isotropi[END_REF]. In any case, the methods of proof are different.

(3

) if p > N and |Ω| < +∞ c u L ∞ (Ω) ≤ |Ω| 1 N -1 p N i=1 ˆΩ |u x i | p i dx 1 p i , for a constant c = c(N, p) > 0.
The next embedding result is stated in [13, Theorem 1]. We provide a proof for the reader's convenience.

Proposition 2.2. Let Ω ⊂ R N be an open set and let p = (p 1 , . . . , p N ) be such that

(2.2) 1 < p 1 ≤ • • • ≤ p N < p * .
Then for every E Ω we have

W 1,p (Ω) → L p * (E), if p = N,
and

W 1,p (Ω) → L χ (E), for every 1 ≤ χ < ∞, if p = N. Proof. Let us fix two concentric balls B 0 B R 0 Ω. For every 0 ≤ < R ≤ R 0 we take a standard cut-off function η ∈ C ∞ 0 (B R ) such that η ≡ 1 on B , with ∇η L ∞ (R N ) ≤ C R - ,
for some universal constant C > 0. Let u ∈ W 1,p (Ω), then for every M > 0 we define

u M = min |u|, M ∈ W 1,p (Ω) ∩ L ∞ (Ω),
and finally take u M η ∈ W 1,p 0 (B R ) ⊂ W 1,p 0 (Ω). Let us suppose for simplicity that p < N , by Theorem 2.1 we have

N i=1 ˆΩ |(u M η) x i | p i dx 1 p i ≥ c ˆΩ |u M η| p * dx 1 p * ,
for some c = c(N, p) > 0. By using the properties of η, with simple manipulations we get

(2.3) N i=1 ˆΩ |(u M ) x i | p i dx 1 p i + N i=1 C R - ˆBR |u M | p i dx 1 p i ≥ c ˆB |u M | p * dx 1 p * ,
for a possibly different constant c > 0, still depending on N and p only. We now observe that by hypothesis (2.2) we have 1 < p i < p * , thus by interpolation in Lebesgue spaces

ˆBR |u M | p i dx 1 p i ≤ ˆBR |u M | dx (1-ϑ i ) ˆBR |u M | p * dx ϑ i p *
, where

ϑ i = p i -1 p i p * p * -1 ∈ (0, 1).
Thus by Young inequality we get

1 R - ˆBR |u M | p i dx N p i ≤ τ -ϑ i R - ˆBR |u M | dx (1-ϑ i ) 1 1-ϑ i + τ ˆBR |u M | p * dx 1 p *
, for every 0 < τ < 1. By choosing τ small enough and using the previous estimate in (2.3), we get

N i=1 ˆΩ |(u M ) x i | p i dx 1 p i + N i=1 C (R -) 1 1-ϑ i ˆΩ |u M | dx + c 2 ˆBR |u M | p * dx 1 p * ≥ c ˆB |u M | p * dx 1 p * .
The previous holds for every 0 ≤ < R ≤ R 0 , from [10, Lemma 6.1] we obtain

C N i=1 ˆΩ |(u M ) x i | p i dx 1 p i + N i=1 C (R 0 -0 ) 1 1-ϑ i ˆΩ |u M | dx ≥ ˆB 0 |u M | p * dx 1 p * , for some constant C = C(N, p) > 0. By arbitrariness of B r 0 B R 0 Ω, for every E Ω a standard covering argument leads to (2.4) C N i=1 ˆΩ |u x i | p i dx 1 p i + ˆΩ |u| dx ≥ ˆE |u M | p * dx 1 p * , for some constant C = C(N, p, dist(E, ∂Ω)) > 0.
In the previous inequality we also used that

|u x i | ≥ |(u M ) x i | and |u M | ≤ |u|,
almost eveywhere on Ω.

If we now take the limit as M goes to +∞ in (2.4), we get the desired result.

Remark 2.3 (Optimality of assumptions). In general we can not take E = Ω or p N ≥ p * in the previous result, see [13] for a counter-example. On the contrary, the hypothesis p 1 > 1 can be easily removed and we can relax it to p 1 ≥ 1. We leave the verification of this fact to the reader.

2.3. Anisotropic Besov-Nikol'skiȋ spaces. Let ψ ∈ L p (R N ), for p ≥ 1 and 0 < t ≤ 1 we define the quantities

(2.5) [ψ] n t,p ∞,i = sup |h|>0 δ he i ψ |h| t L p (R N ) , i = 1, . . . , N, and
(2.6) [ψ] b t,p ∞,i = sup |h|>0 δ 2 he i ψ |h| t L p (R N ) , i = 1, . . . , N.
Lemma 2.4. Let 0 < t < 1, then for every ψ ∈ L p (R N ) we have

(2.7) 1 2 [ψ] b t,p ∞,i ≤ [ψ] n t,p ∞,i ≤ C 1 -t [ψ] b t,p ∞,i + ψ L p (R N ) . For t = 1, for every ψ ∈ L p (R N ) we have 1 2 [ψ] b t,p ∞,i ≤ [ψ] n t,p ∞,i
, and there exists

ψ 0 ∈ L p (R N ) such that [ψ 0 ] b 1,p ∞,i < +∞ and [ψ 0 ] n 1,p ∞,i = +∞.
Proof. The first inequality in (2.7) is a plain consequence of triangle inequality and invariance by translations of L p norms. The second one can be proved by using a standard device, see [ If t = (t 1 , . . . , t N ) ∈ (0, 1] N , by following Nikol'skiȋ we define the corresponding anisotropic Besov-Nikol'skiȋ spaces as

N t,p ∞ (R N ) := ψ ∈ L p (R N ) : N i=1 [ψ] n t i ,p ∞,i < +∞ , and 2 B t,p ∞ (R N ) := ψ ∈ L p (R N ) : N i=1 [ψ] b t i ,p ∞,i < +∞ ,
see [20, pages 159-161]. We equip them with the norms

ψ N t,p ∞ (R N ) := ψ L p (R N ) + N i=1 [ψ] n t i ,p ∞,i and ψ B t,p ∞ (R N ) := ψ L p (R N ) + N i=1 [ψ] b t i ,p ∞,i
.

From now on we will always implicitly assume that t 1 ≤ t 2 ≤ • • • ≤ t N . Before going on, a couple of comments are in order.

Remark 2.5 (Comparison of the two spaces). By Lemma 2.4 we get that if 0

< t 1 ≤ • • • ≤ t N < 1, then N t,p ∞ (R N ) = B t,p ∞ (R N ). On the contrary, if t i = 1 for some i ∈ {1, . . . , N }, then N t,p ∞ (R N ) → B t,p ∞ (R N ) and N t,p ∞ (R N ) = B t,p ∞ (R N ). Moreover, we recall that if [ψ] n 1,p ∞,i < +∞,
for some i ∈ {1, . . . , N },

then its distributional derivative ψ x i belongs to L p (R N ), see [20, Theorem 4.8].
Remark 2.6. In the isotropic case

t 1 = • • • = t N = t with 0 < t < 1, we simply denote these spaces by N t,p ∞ (R N ) and B t,p ∞ (R N ). By Lemma 2.4 the seminorms ψ → N i=1 [ψ] n t,p ∞,i and ψ → N i=1 [ψ] b t,p ∞,i , 2
In [START_REF] Nikol | skiȋ, Approximation of functions of several variables and imbedding theorems[END_REF] this space is denoted by H t p and is seen to be a particular instance of a general class of anisotropic Besov spaces noted B t pθ , with 1 ≤ θ ≤ ∞.

are equivalent. Moreover, these in turn are equivalent to

ψ → sup |h|>0 δ h ψ |h| t L p (R N ) or ψ → sup |h|>0 δ 2 h ψ |h| t L p (R N )
.

The next very simple result asserts that N t,p ∞ (R N ) and B t,p ∞ (R N ) do not change, if in (2.5) and (2.6) the supremum is restricted to 0 < |h| < h 0 . The easy proof is left to the reader.

Lemma 2.7. Let 0 < t ≤ 1 and ψ ∈ L p (R N ), then for every h 0 > 0 and every i = 1, . . . , N we have

[ψ] n t,p ∞,i ≤ sup 0<|h|<h 0 δ he i ψ |h| t L p (R N ) + 2 h -t 0 ψ L p (R N ) ,
and

[ψ] b t,p ∞,i ≤ sup 0<|h|<h 0 δ 2 he i ψ |h| t L p (R N ) + 3 h -t 0 ψ L p (R N ) .
The following interpolation-type result as well is straightforward.

Lemma 2.8. Let 0 < t < s ≤ 1 and ψ ∈ L p (R N ), then for every i = 1, . . . , N we have

(2.8) [ψ] b t,p ∞,i ≤ s t -t s 3 s -t s-t s [ψ] t s b s,p ∞,i ψ s-t s L p (R N ) .
In particular, we have the continuous embedding

B t,p ∞ (R N ) → B t 1 ,p ∞ (R N ). Proof.
We can suppose that the right-hand side of (2.8) is finite, otherwise there is nothing to prove. For every i = 1, . . . , N and 0 < |h| < h 0 , we have

δ 2 he i ψ |h| t L p (R N ) ≤ h s-t 0 δ 2 he i ψ |h| s L p (R N )
.

By taking the supremum over 0 < |h| < h 0 and using Lemma 2.7 , we obtain

[ψ] b t,p ∞,i ≤ h s-t 0 [ψ] b s,p ∞,i + 3 h -t 0 ψ L p (R N ) .
If we now optimize in h 0 , we get the claimed inequality.

We need the following embedding property in standard Sobolev-Slobodeckiȋ spaces.

Lemma 2.9. Let t = (t 1 , . . . , t N ) ∈ (0, 1] N . Then we have the continuous embeddings

N t,p ∞ (R N ) → B t,p ∞ (R N ) → W κ,p (R N ), for every 0 < κ < t 1 .
Proof. The first embedding follows from Remark 2.5. Then it is sufficient to combine Lemma 2.8 with the well-known embedding

B t 1 ,p ∞ (R N ) → W κ,p (R N ), valid for every 0 < κ < t 1 (see [14, Section 8.2.5]).
Finally, the following embedding result in Lebesgue spaces will be important.

Theorem 2.10. Let 1 ≤ p ≤ N and let t = (t 1 , . . . , t N ) ∈ (0, 1] N be such that t 1 < 1. If we set

γ := N i=1 1 t i ,
then we have the continuous embeddings

N t,p ∞ (R N ) → B t,p ∞ (R N ) → L p χ (R N ), for every 1 ≤ χ < γ γ -p .
Proof. It is sufficient to prove the embedding for B t,p ∞ (R N ). We first observe that γ > N ≥ p, thus the condition on χ is well-posed. By [20, Chapter 6, Section 3] we have the embedding

B t,p ∞ (R N ) → B s,q
∞ (R N ), where s = (s 1 , . . . , s N ) and q > p are such that

s i = β t i and β = 1 - 1 p - 1 q γ > 0.
By Lemma 2.9 and Sobolev inequality for Sobolev-Slobodeckiȋ spaces (see for example [24, Theorem 1.73]), for every 0 < κ < s 1 = β t 1 we have

B s,q ∞ (R N ) → W κ,q (R N ) → L N q N -κ q (R N ).
We now observe that we can take β > 0 arbitrarily close to 0. Since we have

q = p γ γ + p β -p and N q N -κ q = N N -κ p γ γ + p β -p p γ γ + p β -p
this implies that the last exponent can be taken as close as desired to p γ/(γ -p) (observe that κ converges to 0 as β goes to 0). Remark 2.11. We observe that for the isotropic case t 1 = • • • = t N = t ∈ (0, 1] the exponent p γ/(γ -p) coincide with the usual Sobolev exponent N p/(N -t p) for the space W t,p (R N ) in the case t p < N .

We conclude this section by considering the localized versions of the spaces above.

If Ω ⊂ R N is an open set, for h ∈ R N \ {0} we denote Ω h = {x ∈ Ω : x + t h ∈ Ω for every t ∈ [0, 1]}.
For a function ψ ∈ L p (Ω), we define

(2.9) [ψ] n t,p ∞,i (Ω) = sup |h|>0 δ he i ψ |h| t L p (Ω he i ) , i = 1, . . . , N, and
(2.10) [ψ] b t,p ∞,i (Ω) = sup |h|>0 δ 2 he i ψ |h| t L p (Ω 2he i ) , i = 1, . . . , N.
Accordingly, we introduce the anisotropic Besov-Nikol'skiȋ spaces on Ω as

N t,p ∞ (Ω) := ψ ∈ L p (Ω) : N i=1 [ψ] n t i ,p ∞,i (Ω) < +∞ , and
B t,p ∞ (Ω) := ψ ∈ L p (Ω) : N i=1 [ψ] b t i ,p ∞,i (Ω) < +∞ .
Finally, we define

N t,p ∞,loc (Ω) := ψ ∈ L p loc (Ω) : ψ ∈ N t,p ∞ (E) for every E Ω , and
B t,p ∞,loc (Ω) := ψ ∈ L p loc (Ω) : ψ ∈ B t,p ∞ (E) for every E Ω .
Remark 2.12. As for the case of R N , the definitions of N t,p ∞ (Ω) and B t,p ∞ (Ω) do not change if we perform the supremum in (2.9) and (2.10) over 0 < |h| < h 0 for some h 0 > 0.

Corollary 2.13. Let Ω ⊂ R N be an open set. Under the assumptions of Theorem 2.10 and with the same notations, we have

N t,p ∞,loc (Ω) ⊂ B t,p ∞,loc (Ω) ⊂ L p χ loc (Ω), for every 1 ≤ χ < γ γ -p .
Proof. Let ψ ∈ N t,p ∞,loc (Ω) and let E Ω, we prove first that ψ ∈ B t,p ∞ (E). By triangle inequality

δ 2 he i ψ |h| t L p (E 2he i ) ≤ δ he i ψ he i |h| t L p (E 2he i ) + δ he i ψ |h| t L p (E 2he i ) ≤ δ he i ψ |h| t L p (E he i ) + δ he i ψ |h| t L p (E he i )
,

where we used a simple change a variable and the inclusion E 2he i ⊂ E he i . By taking the supremum over h, we get the first conclusion.

Let ψ ∈ B t,p ∞,loc (Ω) and let E Ω, we prove that ψ ∈ L p χ (E). We set d = dist(E, ∂Ω) > 0, then there exist x 1 , . . . , x k ∈ E such that

E ⊂ k j=1 B d 8 (x j ).
It is sufficient to prove that ψ ∈ L p χ (B d/8 (x j )) for every j = 1, . . . , k. We fix one of these balls and omit to indicate the center x j for simplicity. We then take a standard cut-off function

η ∈ C ∞ 0 (B d/4 ) ⊂ C ∞ 0 (Ω) such that η ≡ 1 on B d/8
. Then we observe that ψ η ∈ B t,p ∞ (R N ): indeed, by triangle inequality and (2.1) for every h = 0 such that |h| < d/8 we have

δ 2 he i (ψ η) |h| t i L p (R N ) ≤ δ 2 he i η |h| t i ψ L p (R N ) + 2 δ he i η he i |h| t i δ he i ψ L p (R N ) + η 2he i δ 2 he i ψ |h| t i L p (R N ) ≤ 4 d 8 1-t i ∇η L ∞ ψ L p (B d 2 ) + δ 2 he i ψ |h| t i L p (B d 2 ) , i = 1, . . . , N,
and the supremum of the latter over 0 < |h| < d/8 is finite, since B d/2 Ω by construction. By appealing to Lemma 2.7, we thus get ψ η ∈ B t,p ∞ (R N ). We can use Theorem 2.10 and get ψ η ∈ L p χ (R N ). Since η ≡ 1 on B d/8 , this gives the desired result.

A general scheme for improving differentiability

In this section we consider a slightly more general framework, with respect to that of Theorem 1.1. Namely, we consider a set of C 2 convex functions g i : R → R + such that

(3.1) 1 C (|s| -δ) p i -2 + ≤ g i (s) ≤ C (|s| p i -2 + 1), i = 1, . . . , N, for some C ≥ 1, δ ≥ 0 and 2 ≤ p 1 ≤ • • • ≤ p N -1 ≤ p N .
Remark 3.1. Let us point out the following simple inequality that will be used in what follows: for every a ≤ s ≤ b, we have

(3.2) g i (s) ≤ C i g i (a) + g i (b) + 1 ,
for some

C i = C i (p i , δ) ≥ 1.
This follows with elementary manipulations, by exploiting (3.1). We leave the details to the reader.

We then consider u ∈ W 1,p loc (Ω) a local minimizer of

F(u; Ω ) = N i=1 ˆΩ g i (u x i ) dx + ˆΩ f u dx.
In particular, u solves

(3.3) N i=1 ˆg i (u x i ) ϕ x i dx + ˆf ϕ dx = 0,
for every ϕ ∈ W 1,p 0 (Ω ) and every Ω Ω. For every i = 1, . . . , N , we define

V i = V i (u x i ), where V i (t) = ˆt 0 g i (τ ) dτ.
Our aim is to prove that every V i enjoys some weak differentiability properties. We start with the following result.

Proposition 3.2 (Initial gain). Let 2 ≤ p 1 ≤ • • • ≤ p N -1 ≤ p N and let f ∈ W 1,p loc (Ω). We suppose that u ∈ L ∞ loc (Ω).
Then for every i = 1, . . . , N we have

V i ∈ N t,2 ∞,loc (Ω), where t = p 1 p N , . . . , p N -1 p N , 1 .
Proof. We take B r 0 B R 0 Ω a pair of concentric balls centered at x 0 and set

h 0 = (R 0 -r 0 )/4 and R = R 0 + r 0 2 .
Then we pick ϕ ∈ W 1,p 0 (B R ) that we extend it to zero on R N \ B R . For every 0 < |h| < h 0 we can insert the test function ϕ -he j (x) in (3.3). With a simple change of variables we get

(3.4) N i=1 ˆΩ g i (u x i ) he j ϕ x i dx = ˆΩ f he j ϕ dx.
By subtracting (3.3) and (3.4) and dividing by |h|, we thus get

N i=1 ˆΩ g i (u x i ) he j -g i (u x i ) |h| ϕ x i dx = ˆΩ δ he j f |h| ϕ dx.
We now make the following particular choice

ϕ = ζ 2 δ he j u |h| s j ,
where s j ∈ (-1, 1] will be chosen below and ζ is the standard cut-off function

ζ(x) = min 1, R -|x -x 0 | R -r 0 + .
We obtain

N i=1 ˆ δ he j g i (u x i ) |h| δ he j u x i |h| s j ζ 2 dx ≤ 2 N i=1 ˆ δ he j g i (u x i ) h |ζ x i | ζ δ he j u |h| s j dx + ˆ δ he j f h δ he j u |h| s j ζ 2 dx.
Recalling the definition of V i , using (A.1) in the left-hand side and (A.2) (in combination with (3.2)) in the right-hand side, we obtain

N i=1 ˆ δ he j V i |h| s j +1 2 2 ζ 2 dx ≤ C N i=1 ˆ δ he j V i |h| s j +1 2 g i (u x i ) he j + g i (u x i ) + 1 |ζ x i | ζ δ he j u |h| s j +1 2 dx + ˆ δ he j f h δ he j u |h| s j ζ 2 dx.
If we use Hölder and Young inequalities in the right-hand side, we can absorb the higher-order term. Namely, since we have

N i=1 ˆ δ he j V i |h| s j +1 2 g i (u x i ) he j + g i (u x i ) + 1 |ζ x i | ζ δ he j u |h| s j +1 2 dx ≤ C τ N i=1 ˆ δ he j V i |h| s j +1 2 2 ζ 2 dx + C τ N i=1 ˆ g i (u x i ) he j + g i (u x i ) + 1 |ζ x i | 2 δ he j u |h| s j +1 2 2 dx,
where 0 < τ < 1, by choosing τ small enough, we thus get

N i=1 ˆ δ he j V i |h| s j +1 2 2 ζ 2 dx ≤ C N i=1 ˆ g i (u x i ) he j + g i (u x i ) + 1 |ζ x i | 2 δ he j u |h| s j +1 2 2 dx + C   ˆBR δ he j f |h| s j +1 2 p j dx   1 p j ˆBR δ he j u |h| s j +1 2 p j dx 1 p j
.

By basic properties of differential quotients, we get for 0

< |h| < h 0 ˆBR δ he j u |h| s j +1 2 p j dx ≤ C h 1-s j 2 p j 0 ˆBR 0 |u x j | p j dx,
and similarly

ˆBR δ he j f |h| s j +1 2 p j dx ≤ C h 1-s j 2 p j 0 ˆBR 0 |f x j | p j dx.
This yields

N i=1 ˆ δ he j V i |h| s j +1 2 2 ζ 2 dx ≤ C (R 0 -r 0 ) 2 N i=1 ˆ g i (u x i ) he j + g i (u x i ) + 1 δ he j u |h| s j +1 2 2 dx + C h 1-s j 0 ˆBR 0 f x j p j dx 1 p j ˆBR 0 u x j p j dx 1 p j . (3.5)
We use again Hölder inequality in the first term in the right-hand side, so that

N i=1 ˆ g i (u x i ) he j + g i (u x i ) + 1 δ he j u |h| s j +1 2 2 dx ≤ N i=1 ˆBR g i (u x i ) he j + g i (u x i ) + 1 p i p i -2 dx p i -2 p i ˆBR δ he j u |h| s j +1 2 p i dx 2 p i
.

We now observe that with simple manipulations we have

ˆBR g i (u x i ) he j + g i (u x i ) + 1 p i p i -2 dx p i -2 p i ≤ C ˆBR 0 |g i (u x i ) + 1| p i p i -2 dx p i -2 p i
, since for every 0 < |h| < h 0 we have B R + he j ⊂ B R 0 , by construction. Thus from (3.5) we obtain

N i=1 ˆ δ he j V i |h| s j +1 2 2 ζ 2 dx ≤ C (R 0 -r 0 ) 2 N i=1 g i (u x i ) + 1 L p i p i -2 (B R 0 ) ˆBR δ he j u |h| s j +1 2 p i dx 2 p i + C h 1-s j 0 ˆBR 0 f x j p j dx 1 p j ˆBR 0 u x j p j dx 1 p j . (3.6)
The first term in the right-hand side is more delicate and we have to distinguish between two cases.

Case A: j = N . By hypothesis we have p i ≤ p N for every 1 ≤ i ≤ N . Thus we get

ˆBR δ he N u |h| s N +1 2 p i dx ≤ C R N p N -p i p N ˆBR δ he N u |h| s N +1 2 p N dx p i p N , i = 1, . . . , N.
We can then choose s N = 1 so that (s N + 1)/2 = 1 as well. Then from (3.6) we get

N i=1 ˆ δ he N (V i ) h 2 ζ 2 dx ≤ C (R 0 -r 0 ) 2 N i=1 R 2 N p N -p i p N p i 0 g i (u x i ) + 1 L p i p i -2 (B R 0 ) × ˆBR 0 |u x N | p N dx 2 p N + C ˆBR 0 |f x N | p N dx 1 p N ˆBR 0 |u x N | p N 1 p N . Case B: 1 ≤ j ≤ N -1.
This in turn has to be divided in two sub-cases.

Case B.1: 1 ≤ i ≤ j. This is similar to Case A, since by hypothesis we have p i ≤ p j . Then for 0 < |h| < h 0 we simply have

ˆBR δ he j u |h| s j +1 2 p i dx ≤ C R N p j -p i p j ˆBR δ he j u |h| s j +1 2 p j dx p i p j ≤ C h 1-s j 2 p i 0 R N p j -p i p j ˆBR 0 |u x j | p j dx p i p j . Case B.2: j + 1 ≤ i ≤ N .
Here we should be more careful. The order of maximal differentiability t j = (s j + 1)/2 is determined here. We set t j = p j /p N as in the statement, we thus get

ˆBR δ he j u |h| t j p i dx ≤ ˆBR δ he j u p j |h| t j p i dx δ he j u p i -p j L ∞ (B R ) .
Since p j -t j p i ≥ 0, we further observe that for every 0 < |h| < h 0 we have ˆBR δ he j u

p j |h| t j p i dx ≤ h p j -t j p i 0 ˆBR δ he j u h p j dx ≤ C h p j -t j p i 0 ˆBR 0 |u x j | p j dx. Moreover δ he j u L ∞ (B R ) ≤ 2 u L ∞ (B R 0 ) .
By using the previous estimates in (3.6) we thus obtain3

N i=1 ˆ δ he j V i |h| t j 2 ζ 2 dx ≤ C h 2 (1-t j ) 0 (R 0 -r 0 ) 2 j i=1 R 2 N p j -p i p j p i 0 g i (u x i ) + 1 L p i p i -2 (B R 0 ) u x j 2 L p j (B R 0 ) + C (R 0 -r 0 ) 2   N i=j+1 h 2 p j p i - p j p N 0 g i (u x i ) + 1 L p i p i -2 (B R 0 ) u 2 1- p j p i L ∞ (B R 0 )   u x j 2 p j p i L p j (B R 0 ) + C h 2 (1-t j ) 0 f x j L p j (B R 0 ) u x j L p j (B R 0 ) ,
for a constant C = C(N, p 1 , . . . , p N ) > 0. By taking the supremum over 0 < |h| < h 0 , summing over j = 1, . . . , N and recalling that ζ ≡ 1 on B r , we finally conclude that

N j=1 sup 0<|h|<h 0 δ he j V i |h| t j L 2 (Br)
< +∞, i = 1, . . . , N.

We now take E Ω such that d = dist(E, ∂Ω) > 0. There exist J ∈ N and x 1 , . . . , x J ∈ E such that

E ⊂ J k=1 B d 4 (x k ).
By observing that each set E he j is still covered by this family of balls, we thus obtain

N j=1 sup 0<|h|< d 4 δ he j V i |h| t j L 2 (E he j ) ≤ N j=1 J k=1 sup 0<|h|< d 4 δ he j V i |h| t j L 2 (B d 4 (x k ))
< +∞, i = 1, . . . , N.

By taking into account Remark 2.12, this gives V i ∈ N t,2 ∞,loc (Ω), as desired.

By using Corollary 2.13, we also get the following higher integrability result.

Corollary 3.3. Under the previous assumptions, for every i = 1, . . . , N we have

V i ∈ L 2 χ loc (Ω), for every 1 ≤ χ < γ γ -2
, where γ = N j=1

1 t j = p N N p .
The next result shows that each time V 1 , . . . , V N gain integrability, then we can improve their differentiability as well.

Proposition 3.4 (Improvement of differentiability). Let us suppose u ∈ L ∞ loc (Ω) and V 1 , . . . , V N ∈ L 2 χ loc (Ω)
, for some χ > 1. Then we have

V i ∈ N r,2 ∞,loc (Ω), i = 1, . . . , N,
where the vector r = (r 1 , . . . , r N ) is given by

(3.7) r j = min p j p N + p j 2 (χ -1), 1 , j = 1, . . . , N.
Proof. We first observe that the hypothesis on V i implies that u x i ∈ L p i χ loc (Ω), thanks to (3.1). Moreover, for j = N by Proposition 3.2 we already know that we have maximal differentiability, i.e. r N = 1.

Let us fix 1 ≤ j ≤ N -1, we go back to (3.5) and we use Hölder inequality in the right-hand side for the terms i ≥ j + 1, with exponents

p i χ p i -2 and p i χ p i (χ -1) + 2 .
This gives

N i=1 ˆ δ he j V i |h| s j +1 2 2 ζ 2 dx ≤ C (R 0 -r 0 ) 2 j i=1 ˆBR g i (u x i ) he j + g i (u x i ) + 1 p i p i -2 dx p i -2 p i × ˆBR δ he j u |h| s j +1 2 p i dx 2 p i + C (R 0 -r 0 ) 2 N i=j+1 ˆBR g i ((u he j ) x i ) + g i (u x i ) + 1 χ p i p i -2 dx p i -2 χ p i ×   ˆBR δ he j u |h| s j +1 2 2 χ p i p i (χ-1)+2 dx   p i (χ-1)+2 χ p i + C h 1-s j 0 f x j L p j (B R 0 ) u x j L p j (B R 0 ) . (3.8)
The first sum on the right-hand side is estimated as in Proposition 3.2. For the second one, we have to make two separate discussione, depending on whether

• χ is such that (3.9) χ ≥ 1 + 2 1 p j - 1 p N ;
• or χ is such that

(3.10) χ < 1 + 2 1 p j - 1 p N .
If we assume that (3.9) is satisfied, then we have as well

χ ≥ 1 + 2 1 p j - 1 p i , for every i = 1, . . . , N.
that is

2 p i p i (χ -1) + 2 ≤ p j ,
for every i = 1, . . . , N.

Back to (3.8), we can choose s j = 1 and we simply have

ˆBR δ he j u h 2 χ p i p i (χ-1)+2 dx ≤ C R N 1- 2 p i p j [p i (χ-1)+2] ˆBR 0 |u x j | χ p j dx 2 p i p j [p i (χ-1)+2] ,
thus with the usual manipulations we obtain

N i=1 ˆ δ he j V i h 2 ζ 2 dx ≤ C (R 0 -r 0 ) 2 j i=1 R 2 N p j -p i p j p i 0 g i (u x i ) + 1 L p i p i -2 (B R 0 ) u x j 2 L p j (B R 0 ) + C (R 0 -r 0 ) 2   N i=j+1 R N χ-1 χ + 2 (p j -p i ) χ p j p i 0 g i (u x i ) + 1 L χ p i p i -2 (B R 0 )   u x j 2 L χ p j (B R 0 ) + C f x j L p j (B R 0 ) u x j L p j (B R 0 ) .
Let us now consider the case where (3.10) is verified. In this case, by using that u ∈ L ∞ loc , if we set

r j = 1 + s j 2 = p j p N + p j 2 (χ -1) < 1, we obtain ˆBR δ he j u |h| r j 2 χ p i p i (χ-1)+2 dx ≤ δ he j u 2 χ p i p i (χ-1)+2 -χ p j L ∞ (B R ) ˆBR |δ he j u| p j χ |h| r j 2 χ p i p i (χ-1)+2 dx.
We observe that by construction

r j 2 χ p i p i (χ -1) + 2 ≤ χ p j .
Then as before, we obtain for 0

< |h| < h 0 , N i=1 ˆ δ he j V i |h| r j 2 ζ 2 dx ≤ C h 2 (1-r j ) 0 (R 0 -r 0 ) 2 j i=1 R 2 N p j -p i p j p i 0 g i (u x i ) + 1 L p i p i -2 (B R 0 ) u x j 2 L p j (B R 0 ) + C (R 0 -r 0 ) 2 N i=1 g i (u x i ) + 1 L χ p i p i -2 (B R 0 ) h 2 p j p i - p j p N 0 u 2 1- p j p i -p j (χ-1) L ∞ (B R 0 ) u x j 2 p j p i +p j (χ-1) L χ p j (B R 0 ) + C h 2 (1-r j ) 0 f x j L p j (B R 0 ) u x j L p j (B R 0 ) .
Thus, from the previous estimate, we get V i ∈ N r,2 ∞,loc (Ω) by proceeding as in the final part of Proposition 3.2.

Again by Corollary 2.13, we also get the following.

Corollary 3.5. Under the previous assumptions, for every i = 1, . . . , N we have

V N ∈ L 2 χ loc (Ω) =⇒ V i ∈ L 2 ϑ loc (Ω), for every 1 ≤ ϑ < γ γ -2
, where γ = N j=1 1 r j , and r j is defined in (3.7).

Local Sobolev estimate in a particular case

We now specialize the discussion to the situation where we just have two growth exponents 2 ≤ p < q. Namely, let ∈ {1, 2, • • • , N -1} and consider

p 1 = • • • = p = p < p +1 = • • • = p N = q, with p ≥ 2,
as in the statement of Theorem 1.1.

Proof of Theorem 1.1. Let us set (4.1)

τ 0 := 1 - 1 N -1 p q ,
and observe that 0 < τ 0 < 1. We take {α k } an increasing sequence of positive numbers with

1 > α k > τ 0 , for every k ∈ N, lim k→∞ α k = 1. Let i = 1, • • • , N , by Proposition 3.2 we have V i ∈ N t 0 ,2 ∞,loc (Ω)
, where

t 0 = (t 0 , • • • , t 0 , 1, • • • , 1) =   p/q, . . . , p/q , 1, • • • , 1   .
Moreover, if we set

γ 0 = q p + N - and χ 0 = 1 + α 0 2 γ 0 -2 ,
we have V i ∈ L 2 χ 0 loc (Ω) by Corollary 3.3. We now repeatedly apply Proposition 3.4 and Corollary 3.5: after k + 1 steps, we get

V i ∈ N t k ,2
∞,loc (Ω) where

t k = (t k , . . . , t k , 1, • • • , 1) with t k = min p q + p 2 (χ k-1 -1), 1 , and (4.2)
χ k-1 = 1 + α k-1 2 γ k-1 -2 , γ k-1 = t k-1 + N -.
We want to prove that under the standing assumptions (1.2) or (1.3), there exists

k 0 ∈ N such that p q + p 2 (χ k 0 -1 -1) ≥ 1.
By using the relations (4.2), this is the same as

(4.3) p q + α k 0 -1 p t k 0 -1 + N --2 ≥ 1.
Until this does not occur, we thus have that {t k } k∈N coincides with the recursive sequence defined by (4.4)

       t 0 = p q t k+1 = p q + α k b(t k ),
where the function t → b(t) is defined by

b(t) = p t + N -2 - , for t > 0 and t = -(N -2) .
We observe that, for any ∈ {1, 2, • • • , N -1}, b(t) is an increasing function on its domain and it is positive for t in the interval (0, N -1) 4 . In order to obtain (4.3) and conclude the proof, we consider two possibilities for the sequence (4.4):

Alternative I) either there exists k 0 such that t k 0 ≥ N -1;

Alternative II) or t k < N -1 for every k ∈ N.

If Alternative I) occurs the proof ends, since we automatically get (4.3) and we can stop the process at t k 0 .

In Alternative II), using the monotone behaviour of b and {α k } k∈N , we get that {t k } k∈N is an increasing sequence, thus it admits a limit L with (4.5)

p q < L ≤ N -1.
In order to obtain (4.3) and conclude the proof, it would be sufficient to show that L > 1. By recalling that {α k } k∈N converges to 1 by construction, the possible limits L of {t k } k∈N can be found among the solutions of the equation

(4.6) L = p q + p L + L (N --2)
.

Case = N -2 In this case (which can happen only for N ≥ 3), the equation (4.6) is linear and we immediately get

L = p q N -2 N -2 -p .
This implies that if N -2 ≤ p we are indeed in Alternative I), since we violate 5 (4.5). If on the other hand N -2 > p, then L > 1 thanks to hypothesis (1.3).

Case 1 ≤ ≤ N -3 Observe that this can happen only for N ≥ 4. From (4.6) we get that the possible limits of t k are determined by the roots of the polynomial: (4.7)

P (t) = t 2 (N -2 -) -t (N -2 -) p q + p -- p q .
By a simple computation, we see that P has real roots L 1 ≤ L 2 if and only if

(4.8) (N -2 -) p q + ( √ - √ p) 2 ≥ 0.
Since ≤ N -3 the previous condition is always satisfied (with strict inequality sign, indeed). We have

P (t) < 0 ⇐⇒ L 1 < t < L 2 .
If we observe that P (p/q) = -p 2 /q < 0, we thus get

L 1 < p q < L 2 .
Since {t k } k∈N is increasing and t 0 = p/q, this implies

lim k→∞ t k = L 2 .
We now observe that we have

L 2 > 1 ⇐⇒ P (1) < 0 ⇐⇒ hypothesis (1.3).
and we are done.

Case = N -1 . This case is subtler. Let us start by looking at the subcase p ≥ N -1.

Case p ≥ N -1. We first recall that

t k+1 -t k = p q + α k b(t k ) -t k .
Then observe that the function (recall the definition (4.1) of τ 0 )

ϕ(t) = τ 2 0 b(t) -t, t ∈ p q , N -1 , is such that ϕ (t) = τ 2 0 p (N -1) (N -1 -t) 2 -1 ≥ 0 ⇐⇒ N -1 > t ≥ N -1 -τ 0 (N -1) p =: t.
Since we are supposing p ≥ N -1, the choice of τ 0 entails t ≤ N -1 -τ 0 (N -1) = p q .

This implies that if p ≥ N -1, then ϕ is strictly increasing on [p/q, N -1). By recalling that α k > τ 0 > τ 2 0 and t 0 = p/q we get

t k+1 -t k > p q + τ 2 0 b(t 0 ) -t 0 = τ 2 0 b p q > 0,
thus the sequence can not converge to a finite value. This means that in this case we are indeed in Alternative I) and thus we are done.

Observe in particular that since by assumption p ≥ 2, the previous discussion implies that for N = 2 and N = 3 we finished the proof.

Case 2 ≤ p < N -1 and N ≥ 4. Again, the possible limits of {t k } n∈N are given by the roots of the polynomial P defined in (4.7). We first observe that condition (4.8) now reads (4.9)

p √ N -1 - √ p 2 ≤ q.
When this is fulfilled, P admits real roots. We can thus observe that if p and q satisfy the third block of assumptions in (1.2), P has not real roots which implies that in this case we are in the situation I) and the proof is over.

We assume that (4.9) is verified. In this case we have

P (t) < 0 ⇐⇒ t < L 1 or t > L 2 .
We still have P (p/q) < 0, so that (4.10)

p q < L 1 < L 2 or L 1 < L 2 < p q .
Since {t k } k∈N is increasing and t 0 = p/q, this implies

lim k→∞ t k = L 1 ,
and thus the second alternative in (4.10) is ruled out. We compute L 1 , this is given by

L 1 = N -1 - p q - N -1 - p q 2 -4 (N -1) p q 2 .
Observe that

L 1 > 1 ⇐⇒ N -3 - p q > N -1 - p q 2 -4 (N -1) p q .
A necessary condition for this to happen is that

N -3 > p q ⇐⇒ p ≤ N -3 ∪      p > N -3, q < p p -(N -3)
.

When these conditions are in force, then we obtain

N -1 - p q 2 + 4 -4 N -1 - p q > N -1 - p q 2 -4 (N -1) p q ,
which is the same as

(4.11) N -2 - p q < (N -1) p q ⇐⇒ N -2 -p < (N -2) p q .
By recalling that we are in the case p < N -1 and we are assuming (1.2) and (4.9), we need to consider the two possibilities:

A) p ≤ N -3; B) N -3 < p < (N -2) 2 N -1 .
In case A), the second set of assumptions in (1.2) implies that (4.11) is verified and thus we are done. Observe that the bound

q < (N -2) p N -2 -p .
is compatible with p ≤ N -3 and (4.9), since for p ≤ N -2 we have

(N -2) p N -2 -p > p √ N -1 - √ p 2 ⇐⇒ p = (N -2) 2 N -1 ,
and the latter is strictly greater than N -3.

In case B), in order to verify (4.11) we would need

         q < (N -2) p N -2 -p , q < p p -(N -3)
.

Observe that

N -3 < p < (N -2) 2 N -1 =⇒ (N -2) p N -2 -p < p p -(N -3)
.

Thus the condition becomes

         N -3 < p < (N -2) 2 N -1 , q < (N -2) p N -2 -p ,
which is again covered by our assumptions (1.2). This concludes the proof.

Local Lipschitz estimate in dimension two

5.1. Proof of Theorem 1.4. We now restrict the discussion to the case of dimension N = 2 and consider the model case

F(u; Ω ) = 2 i=1 1 p i ˆΩ (|u x i | -δ i ) p i + dx + ˆΩ f u dx, u ∈ W 1,p loc (Ω), Ω Ω.
We can suppose that p 1 < p 2 , since for p 1 = p 2 the result has already proved in [3]. Under the standing assumption, we take U ∈ W 1,p loc (Ω) to be a local minimizer. Then we proceed as in [3]. We take Ω Ω and set d = dist(Ω , ∂Ω). Since Ω can be covered by a finite number of balls centered at Ω and with radius r 0 ≤ d/100, it is sufficient to show that ∇U L ∞ (Br 0 (x 0 )) < +∞, where B r 0 (x 0 ) is one of these balls. To this aim, we set B = B 4 r 0 (x 0 ) and solve the regularized problem for 0 < ε 1

(5.1) min{F ε (u; B) : u -U ε ∈ W 1,p 0 (B)}, where:

• the regularized functional F ε is defined by

F ε (u; B) = 2 i=1 ˆB g i,ε (u x i ) dx + ˆB f ε u dx;
• the functions g i,ε are given by

g i,ε (t) = (|t| -δ i ) p i + p i + ε t 2 2 , i =
1, 2;

• U ε and f ε are regularizations of U and f . By [3, Theorem 2.4], we know that (5.1) admits a unique solution u ε , which is smooth by proceeding as in [3, Lemma 2.8]. In order to conclude, it is sufficient to prove the uniform estimate

(5.2) ∇u ε L ∞ (Br 0 ) ≤ C,
with C > 0 independent of ε and depending only on p 1 , p 2 , δ 1 , δ 2 , r 0 , f W 1,p (2 B) and U W 1,p (2 B) . This is proved in the next subsection. As in [3] (to which we refer for the missing details), this gives the estimate on ∇U and thus the conclusion.

5.2. Uniform Lipschitz estimate. The proof of (5.2) is the same as that of [3, Proposition 4.1], up to a couple of crucial modifications needed. We give the details of the latter and sketch the rest of the proof, by referring the reader to [3]. For notational simplicity, we write u in place of u ε . We introduce the quantity

δ = 1 + max{δ 1 , δ 2 }.
then in what follows we set

W i = δ 2 + (|u x i | -δ) 2 + , i = 1, 2.
First of all, we need the following Caccioppoli-type inequality. The proof is a slight variation of [3, Lemma 3.6 & Corollary 3.7], we omit it.

Lemma 5.1. There exists a constant C = C(p 1 , p 2 ) > 0 such that for every s ≥ 0, every Lipschitz function η with compact support in B and j = 1, 2, we have

ˆ W p j 4 + s 2 j x j 2 η 2 dx ≤ Cδ p j -2 2 i=1 ˆW p i -2 2 i W s+1 j |∇η| 2 dx + (s + 1) 2 ˆ|f ε | 2 W s j η 2 dx . (5.3)
We can now start the proof of the estimate (5.2) for the gradient of u. We may consider the case of the first component u x 1 only, the other one being similar. With standard manipulations, from (5.3) we get

ˆ W p 1 4 + s 2 1 η x 1 2 dx ≤ C δ p 1 -2 2 i=1 ˆW p i -2 2 i W s+1 1 |∇η| 2 dx + C δ p 1 -2 (s + 1) 2 ˆ|f ε | 2 W s 1 η 2 dx, (5.4)
with C = C(p 1 , p 2 ) > 0, where we used that δ ≥ 1. In order to reconstruct the full gradient

∇W p 1 4 + s 2 1
on the left-hand side, we observe that

W p 1 4 + s 2 1 x 2 = p 1 + 2 s p 1 W p 1 4 1 x 2 W s 2
1 .

Then if we fix 1 < q < 2, by Hölder's inequality with exponents 2/q and 2/(2 -q), we have

ˆ W p 1 4 + s 2 1 x 2 q η q dx 2 q ≤ p 1 + 2 s p 1 2   ˆ W p 1 4 1 x 2 2 η 2 dx   ˆspt(η) W q 2-q s 1 dx 2-q q .
By using the same manipulations as in [3], we thus get

ˆ W p 1 4 + s 2 1 η x 2 q dx 2 q ≤ C (1 + s) 2   ˆ W p 1 4 1 x 2 2 η 2 dx   ˆspt(η) W q 2-q s 1 dx 2-q q + C ˆW p 1 q 4 + s q 2 1 |η x 2 | q dx 2 q , ( 5.5)
with C = C(p 1 , p 2 ) > 0. We assume for simplicity that all the balls are centered at the origin. We then fix the radius r 0 > 0 as above and define

R 0 = 2 r 0 R 1 := 3 2 r 0 . For r 0 < r < R < R 1 , we take η ∈ W 1,∞ 0 (B R ) to be the standard cut-off function η(x) = min 1, (R -|x|) + R -r .
By multiplying (5.4) and (5.5) we get

  ˆ W p 1 4 + s 2 1 η x 1 2 dx   ˆ W p 1 4 + s 2 1 η x 2 q dx 2 q ≤ C δ p 1 -2 1 (R -r) 2 2 i=1 ˆBR W p i -2 2 i W s+1 1 dx + (s + 1) 2 ˆBR |f ε | 2 W s 1 dx ×   (s + 1) 2   ˆBR W p 1 4 1 x 2 2 dx   ˆBR W q 2-q s 1 dx 2-q q + 1 (R -r) 2 ˆBR W p 1 q 4 + s q 2 1 dx 2 q .
Then we apply the anisotropic Sobolev inequality of Theorem 2.1 to the compactly supported function W

(p 1 +2 s)/4 1 η. This yields

T q ˆ W p 1 4 + s 2 1 η q * dx 4 q * ≤ C δ p 1 -2 1 (R -r) 2 2 i=1 ˆBR W p i -2 2 i W s+1 1 dx + (s + 1) 2 ˆBR |f ε | 2 W s 1 dx × (s + 1) 2 ˆBR W p 1 4 1 x 2 2 dx ˆBR W q 2-q s 1 dx 2-q q + 1 (R -r) 2 ˆBR W p 1 q 4 + s q 2 1 dx 2 q .
(5.6)

The exponents q and q * are given by q = 4 q 2 + q and q * = 4 q 2 -q , the constant T q only depends on q and it degenerates to 0 as q goes to 2. The idea is to use the previous fundamental estimate (5.6) to produce an iterative scheme of reverse Hölder inequalities on shrinking balls. Then we perform a Moser's iteration in order to conclude. We need to estimate the terms appearing in the right-hand side of (5.6). The crucial difference with respect to [3] is in the first term on the right-hand side of (5.6), i.e.

(5.7)

2 i=1 ˆBR W p i -2 2 i W s+1 1 dx = ˆBR W p 1 2 1 W s 1 dx + ˆBR W p 2 -2 2 2 W 1 W s 1 dx.
On the contrary, all the other terms are estimated exactly as in [3], thus we omit the details. Let us now focus on the term above, it is useful to introduce the quantity

I(W 1 , W 2 , f ε ; R 0 , R 1 ) = 2 i=1 R 0 R 1 2 B R 0 W p i 2 i dx + ˆBR 1 ∇W p i 4 i 2 dx p i -2 p i p 1 p 1 -2 + R 2 p 1 0 ˆBR 1 |f ε | 2 p 1 dx 1 p 1
.

First we claim that I(W 1 , W 2 , f ε ; R 0 , R 1 ) is uniformly bounded, independently of ε. To this aim, as for the term containing f ε , we observe that by Proposition 2.2 we have the continuous embedding (recall that R 1 < R 0 ) W

|Φ i,ε (t)| = f i (V -1 i,ε (t)) 1 V i,ε (V -1 i,ε (t)) ≤ C i , t ∈ R.
By using this observation, we thus obtain ˆBR 1 ∇W

p i 4 i 2 dx ≤ L i ˆBR 1 |∇V i,ε | 2 dx,
with L i = L i (p i , δ) > 0. We can now invoke (1.4), in order to bound uniformly the last term. It is only left to observe that the bound in (1.4) also depends on the local L ∞ norm of u ε . This can be uniformly bounded by appealing to [8, Theorem 3.1], proving the claim. We now come back to estimate the quantities in (5.7). Let us recall that, since we are in dimension N = 2, we have the continuous embedding W 1,2 (B R 1 ) → L ϑ (B R 1 ) for every 1 ≤ ϑ < +∞. Then by Hölder's inequality and Sobolev-Poincaré inequality, exactly as in [3] we get

ˆBR W p 1 2 1 W s 1 dx ≤ C I(W 1 , W 2 , f ε ; R 0 , R 1 ) R 2 p 1 0 ˆBR W s p 1 1 dx 1 p 1
.

For the second term we have to be more careful. By using Hölder inequality with exponents Proof. For ε > 0, let us consider the function g ε (t) = g(t) + ε t 2 . We set

p
V ε (t) = ˆt 0 g ε (τ ) dτ,
then we observe that this is a strictly increasing function, thus invertible. Finally, we define

F ε (t) = g ε V -1 ε (t)
, which is an increasing function. Indeed, we have

F ε (t) = g ε (V -1 ε (t)) 1 V ε (V -1 ε (t)) = g ε (V -1 ε (t)) > 0.
By basic Calculus, this yields

|g ε (a) -g ε (b)| = |F ε (V ε (a)) -F ε (V ε (b))| ≤ sup s∈[a,b] F ε (V ε (s)) |V ε (a) -V ε (b)| = sup s∈[a,b] g ε (s) |V ε (a) -V ε (b)|.
By taking the limit as ε goes to 0, we get the desired conclusion.

W 1 p 1 0 ˆBR W s p 1 1 dx 1 p 1 ,

 10111 W s 1 dx ≤ C I(W 1 , W 2 , f ε ; R 0 , R 1 ) R 2as well, where we used again that R 1 < R 0 . Lemma A.2. Let g : R → R + be a C 1,1 convex increasing function. For every a, b ∈ R we have (A.2) g (a) -g (b) ≤ sup s∈[a,b] g (s) |V (a) -V (b)|.

Remark A. 3 .

 3 When g(t) = |t| p /p, the previous inequalities imply the familiar estimates |a| p-2 a -|b| p-2 b (a -b) ≥ (p -

,

  1,p (B R 0 ) → L 2 p 1 (B R 1 ), since p 2 < p 1 ≤ 2 and 2 p 1 < p * = 2 p 1 p 2 p 1 + p 2 -p 1 p 2 , can be uniformly bounded in terms of the W 1,p norm of f on B R 0 . The terms containing the gradients of W are more delicate, for them we need Theorem 1.1. Indeed, let us defineV i,ε (t) = ˆt 0 g i,ε (s) ds and V i,ε = V i,ε ((u ε ) x i ), i = 1, 2.We observe that V i,ε : R → R is a locally Lipschitz omeomorphism, with V i,ε > 0. If we setf i (t) = δ 2 + (|t| -δ) 2 It is not difficult to see that Φ i,ε is a Lipschitz function, with Lipschitz constant independent of ε. i,ε (t) = g i,ε (t) ≥ 1 √ C i |t| p i -2 2 , for |t| ≥ δ,for some C i = C i (p i , δ) ≥ 1. Thus we get

	thus the term		ˆBR 1	|f ε | 2 p 1 dx	1 p 1
	p 1 /4 1	p 2 /4 2 and W
					p i
				+	4 ,	t ∈ R,
	then we obtain that W i p i /4	= Φ i,ε (V i,ε ), where
			Φ i,ε (t) = f i (V -1 i,ε (t)),	t ∈ R.

Indeed, we have

f i (t) = 0, for |t| < δ and |f i (t)| ≤ C i |t| p i -2

2 , for |t| ≥ δ.

V

,

  where we further used Young's inequality and the constant C = C(p 1 ) > 0 depends only on p 1 . To treat the term into square brakets, we use again Sobolev-Poincaré inequalities. Namely, we have ˆBR 1 W

												1	p 1 p 1 -2	,		p 1	p 1 2	,	p 1 ,
	we get
	ˆBR	W 2 p 2 -2 2	W 1 W s 1 dx ≤ C	  	ˆBR 1		W 2 p 2 4	2 p 1	p 2 -2 p 1 -2	p 1 p 2	dx	1 p 1	+	ˆBR 1	p 1 4 1 W	2 p 1	dx	1 p 1	  
							×	ˆBR	W s p 1 1	dx		1 p 1
						p 1 4 1	2 p 1	dx	1 p 1	≤ C R	2 p 1 1	B R 1	W 1 dx + p 1 2	ˆBR 1	∇W	p 1 4 1	2	dx ,
	and
	ˆBR 1	W 2 p 2 4	2 p 1	p 2 -2 p 1 -2	p 1 p 2	dx	1 p 1	p 1 -2 p 2 -2		p 2 p 1	≤ C R	2 p 1 1	p 1 -2 p 2 -2	p 2 p 1	B R 1	W 2 dx + p 2 2	ˆBR 1	∇W	p 2 4

It is intended that the second term in the right-hand side is 0 for j = N .

Indeed, for ≤ N -2, b(t) is positive increasing for t > 0.

By using these estimates in (5.6) and proceeding as in [3] for all the other terms, we obtain

for a constant C = C(p 1 , q) > 0. The exponent 1 < q < 2 is chosen as

By further observing that W 1 ≥ 1, from (5.8) we gain ˆBr

, for s ≥ 0. This is an iterative scheme of reverse Hölder inequalities, we can now iterate infinitely many times this estimate, as in [3].

Appendix A. Pointwise inequalities

Lemma A.1. Let g : R → R + be a C 1,1 convex function. Let us set

For every a, b ∈ R we have

Proof. Without loss of generality, we can assume that a ≥ b. Indeed, g (a) -g (b) and a -b have the same sign, thanks to the monotonicity of g . For a = b there is nothing to prove, so we take a > b. By using Jensen inequality, we have

as desired.