Akira F Peters
email: akirapeters@gmail.com

Aaron D Mann

César A Córdova

Juliet Brodie

Juan A Correa

Declan C Schroeder

J Mark Cock

Genetic diversity of Ectocarpus (Ectocarpales, Phaeophyceae) in Peru and northern Chile, the area of origin of the genome-sequenced strain

Keywords: Genetic diversity, Ectocarpus, Kuckuckia, Chile, Peru

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The Ectocarpus strain chosen for genome sequencing (henceforth referred to as the "genome strain") is a male gametophyte which has the designation "Ec32" in the Ectocarpus strain collection at Roscoff. It was selected because it displays an alternation of two morphologically distinguishable generations [START_REF] Peters | Life-cycle-generation-specific developmental processes are modified in the 1512[END_REF] and produces unilocular sporangia on parthenogenetic sporophytes, both of which facilitate studies on the complex brown algal life history (for more reasons and terminology of morphology and life history see Peters et al., 2004a;[START_REF] Coelho | Complex life cycles of multicellular eukaryotes: new approaches based on the use of model organisms[END_REF][START_REF] Peters | Life-cycle-generation-specific developmental processes are modified in the 1512[END_REF]. The strain was obtained from a meiospore of a field sporophyte collected by the first author in November 1988 at San Juan de Marcona, Peru (Site 3 in Fig. 1). Identified as E. siliculosus (Dillwyn) Lyngbye in a first molecular phylogeny (Stache-Crain et al., 1997, designation "SAm120h";lineage 1c), it later showed post-zygotic incompatibility (Peters et al., 2004b) when crossed with strains from Europe regarded as genuine E. siliculosus (i.e. genetically close to those used by [START_REF] Müller | Generationswechsel, Kernphasenwechsel und Sexualität der Braunalge Ectocarpus siliculosus im Kulturversuch[END_REF] for the description of the life history; lineage 1a in [START_REF] Stache-Crain | Molecular systematics of Ectocarpus and Kuckuckia (Ectocarpales, Phaeophyceae) inferred from phylogenetic analysis of nuclear and plastid-encoded DNA sequences[END_REF] or with a strain of Ectocarpus from New Zealand previously used in research on virus inheritance [START_REF] Müller | Mendelian segregation of a virus genome during host meiosis in the marine brown alga Ectocarpus siliculosus[END_REF]; lineage 4 in [START_REF] Stache-Crain | Molecular systematics of Ectocarpus and Kuckuckia (Ectocarpales, Phaeophyceae) inferred from phylogenetic analysis of nuclear and plastid-encoded DNA sequences[END_REF]: hybrids possessed a normal morphology, but were unable to form meiospores. Further crosses of the genome strain with an Ectocarpus from northern Chile, as well as with the recently reinstated E. crouaniorum Thuret in Le Jolis from Europe [START_REF] Peters | Reinstatement of E. crouaniorum Thuret in Le Jolis as a third common species of Ectocarpus (Ectocarpales, Phaeophyceae) in western Europe, and its phenology at Roscoff, Brittany[END_REF], gave similar results (unpublished data). Although sisters of the genome strain (female gametophytes from the same field sporophyte from Peru) were available for genetic experiments, for instance to test sexlinkage of a mutation [START_REF] Peters | Life-cycle-generation-specific developmental processes are modified in the 1512[END_REF], no female gametophyte of sufficient genetic distance was available for experiments requiring outcrossing (e.g. to produce a genetic map, [START_REF] Heesch | A sequence-tagged genetic map for the brown alga Ectocarpus siliculosus provides large-scale assembly of the genome sequence[END_REF]. Apart from the genome strain there was no unambiguous published record of Ectocarpus from Peru. Howe (1914: p. 50) mentioned "small specimens of Ectocarpus" on Desmarestia peruviana Montagne at Ancón (13 February 1907) and on Lessonia nigrescens Bory de Saint-Vincent at Chincha Island (18 June 1907); however, he regarded the material as "too meager to justify an attempt at determination or description". The nearest published record of Ectocarpus was from Iquique, northern Chile, at ca 800 km distance from Site 3 [START_REF] Ramírez | Catálogo de las algas marinas bentónicas de la costa temperada del Pacífico de Sudamérica[END_REF].

To isolate additional strains of Ectocarpus showing full compatibility with the genome strain, we collected more individuals from localities along the South American Pacific coast from central Peru to central Chile. They were identified based on comparisons with nuclear ribosomal internal transcribed spacer 1 (ITS1) sequences available for 43 strains of Ectocarpus and seven of Kuckuckia (the sister genus of Ectocarpus) from all continents except Antarctica [START_REF] Stache-Crain | Molecular systematics of Ectocarpus and Kuckuckia (Ectocarpales, Phaeophyceae) inferred from phylogenetic analysis of nuclear and plastid-encoded DNA sequences[END_REF]. In the studied area, Ectocarpus presented a surprisingly high genetic diversity described in the present paper. Isolates of one genotype were genetically similar to the genome strain and proved to be fully interfertile with it.

Materials and Methods

Field collection and isolation of cultures. Nine localities (four in Peru, five in Chile) spanning a distance of approximately 3000 km were visited for collection, most during one visit in February-March 2006 (Fig. 1, Table 1). Sampling at Site 8 was done for studies on the response of Ectocarpus to copper pollution (Mann et al., unpublished). From each field thallus macroscopically resembling Ectocarpus, filaments were inoculated in a 2 ml Eppendorff tube containing autoclaved sea water. After transfer to the Biological Station at Roscoff, isolation and cultivation of clean unilalgal clonal cultures were undertaken as described previously (Peters et al., 2010). A number of thalli from Site 8 were isolated by filtering nearshore surface seawater and cultivating filaments that developed on filter paper, or by inoculating field macrothalli of Scytosiphon and isolating all Ectocarpus thalli developing on them in culture. Only thalli possessing ribbon-shaped plastids were retained for further study.

Strain designations beginning with "CCAP1310/" are from the Culture Collection of Algae and Protozoa; numbers or numbers preceded by "Ec" are isolates housed in the Ectocarpus strain collection at Roscoff, and are maintained by the first author. Molecular methods. DNA was extracted from living cultures as described (Peters et al., 2004b). Initial identification was based on ITS1 length (see [START_REF] Peters | Reinstatement of E. crouaniorum Thuret in Le Jolis as a third common species of Ectocarpus (Ectocarpales, Phaeophyceae) in western Europe, and its phenology at Roscoff, Brittany[END_REF] for rationale), followed by sequencing of ITS1 or by a diagnostic PCR using a primer specific for the ITS1 sequence of the genome strain (for details and oligonucleotide primers used see supporting information). Sequences were aligned manually using Se-Al v. 2.0a11 [START_REF] Rambaut | Se-Al v2.0a11[END_REF].

Sequences similar over the full length of ITS1 and not requiring indels of more than 20bp were considered to belong to the same ITS1 genotype (GT). For at least one member of each ITS1 genotype discovered, three additional markers (ITS2, 3'-rbcL+Rubisco spacer, cox3) were sequenced to provide data for phylogenetic analyses. For the genome strain, the corresponding data were taken from the genome. Sequences were deposited in the EMBL/Genbank/DDBJ database (Table 2).

Data analyses. The ITS1 sequences obtained were compared to Ectocarpus and Kuckuckia sequences available in GenBank using blast [START_REF] Altschul | Gapped BLAST and PSI-BLAST: a new generation of protein database search programs[END_REF]. ITS1 was preferred over Rubisco spacer sequences because ITS1 is more variable, particularly in its first part [START_REF] Stache-Crain | Molecular systematics of Ectocarpus and Kuckuckia (Ectocarpales, Phaeophyceae) inferred from phylogenetic analysis of nuclear and plastid-encoded DNA sequences[END_REF].

Alignable sequences of the four markers were concatenated, sequences of reference strains (nine from Europe, one each from South America and New Zealand; Table 2) were then added, and phylogenetic analyses undertaken. Phylogenetic trees were constructed using the various programs in PHYLIP version 3.69 [START_REF] Felsenstein | PHYLIP (Phylogeny Inference Package) Version 3.57c[END_REF], and the robustness of the alignments was tested with the bootstrapping option (SeqBoot). Genetic distances, applicable for distance matrix phylogenetic inference, were calculated using the Dnadist program in the PHYLIP package.

Phylogenetic inferences based on the distance matrix (Neighbor), maximum likelihood (Dnaml) and parsimony (Dnapars) algorithms were applied to the alignments. In all cases, the best tree or majority rule consensus tree was selected using the consensus program (Consense). The trees were visualized and drawn using the TREEVIEW software version 2.1 [START_REF] Page | TREEVIEW software version 2[END_REF].

Results

In the Museo de Historia Natural, Lima, seven herbarium specimens were thalli of Ectocarpus collected by César Acleto in central Peru around Lima between September and February from 1964 to 1976. In the field, Ectocarpus occurred at eight out of the nine localities visited, usually as an epiphyte on other macroalgae such as Macrocystis, Lessonia, Desmarestia and Gracilaria, and only occasionally on rocks. It was present from sheltered to exposed conditions but was not found at a highly exposed locality (Table 1). Although some individuals were up to 14 cm in size others were minute felts or dark spots; they appeared in culture after inoculation with their substratum or they developed on paper that was used to filter sea water (Figs 2-4; Table 3). A total of 120 strains, including strain CCAP1310/40 isolated previously at Site 9, were available for molecular identification.

Determination of the length of the nuclear ribosomal ITS1 in all strains identified 5-6 ITS1 length types which were distinguishable on agarose gels, and one putative hybrid (Fig. 5).

Sequencing of 36 individuals revealed that two of the length types were heterogeneous and contained more than one ITS1 genotype (GT). In total we obtained nine different GTs representing seven Ectocarpus, one Kuckuckia and one hybrid between two of the Ectocarpus genotypes (Table 3).

Individuals with the same GT as the genome strain (GT4) occurred from Site 1 where they were the only Ectocarpus collected, to Site 5 at the northern border of Chile (Tables 1,2). The ITS1 regions of the seven sequenced strains of this GT were identical to that of the genome strain (n=2, Sites 2 and 3), differed by a single nucleotide substitution (n=4, three from Site 1 and one 2 for sequence accession of the latter strain), or had an ambiguity at the same position (n=1, Site 5).

There were eight additional genotypes in the study area (Table 3). GT1, which has not been sequenced previously (no blast hit with 100% coverage), was collected at Sites 2, 5 and 8. With 302-306 bp it had the shortest ITS1 so far found in any Ectocarpus. GT2 was common at Site 3, but also present at the most southern Site 10. It showed no perfect match with any published ITS1; the best blast hit was a previous isolate from Puerto Deseado, Patagonia, Argentina (93% identity). However, when we sequenced ITS1 of CCAP1310/40 from Site 9 (for which only the Rubisco spacer sequence had hitherto been available; U38736) its ITS1 closely resembled that of our isolates of GT2; CCAP1310/40 only differed by nine substitutions and four small indels of, in total, 10bp length. GT3 was collected once (at Site 6); its ITS1 was alignable with that of GT2 but differed by 22 substitutions and four indels of in total seven bp length. It was considered just different enough to justify recognition as a separate GT. It more strongly (95% identity) resembled that of the isolate from Argentina mentioned above which again gave the best blast hit.

GT5 (E. fasciculatus) was present from Site 3 to Site 10. It strongly resembled previous sequences from Isla Robinson Crusoe (off central Chile) and from Europe. GT6 (E. siliculosus) was the only Ectocarpus collected at Site 7, and was also common at Site 8. GT7, obtained in a single isolate from Site 5, was a genotype that had not been sequenced previously; the most similar sequence available was from a Kuckuckia, but it showed only 77% identity. In culture the isolate of GT7 developed phaeophycean hairs which are characteristic for Kuckuckia and absent in Ectocarpus [START_REF] Hamel | Sur la classification des Ectocarpales[END_REF]. As the ITS1 of GT7 had the same length as that of E. siliculosus, all isolates of this ITS1 length were examined for the presence of hairs, but they were only observed in the isolate from Site 5. ITS1 of two isolates of this ITS1 length from Site 7 and three from Site 8 (including strain CCAP1310/333 from Site 8 which showed high copper tolerance; F o r P e e r R e v i e w [START_REF] Ritter | Copper stress proteomics highlights local adaptation of two strains of the model brown alga Ectocarpus siliculosus[END_REF][START_REF] Dittami | Comparative genome hybridizations of Ectocarpus strains and ecotypes[END_REF] were sequenced and their ITS1 matched that of European E. siliculosus, e.g. of well-studied strains from Naples and Roscoff. We concluded that all isolates with an ITS1 length of 714-716 bp, apart from the one from Site 5, belonged to E. siliculosus. GT8 was present only at Site 7; it was identified as E. crouaniorum because its long ITS1 was similar (>90% identity) with strains of this species from Europe. The best blast hits were isolates from the salt-polluted German Werra river and from Niebla, Valdivia (southern Chile) (96 and 95% identity, respectively). GT9 presented a double band with lengths that corresponded to E. siliculosus and E. crouaniorum, two species which in Europe are known to occasionally produce viable hybrids; these are, however, unable to form meiospores [START_REF] Peters | Reinstatement of E. crouaniorum Thuret in Le Jolis as a third common species of Ectocarpus (Ectocarpales, Phaeophyceae) in western Europe, and its phenology at Roscoff, Brittany[END_REF]. The two strains of this putative hybrid genotype were isolated from Site 7 where they cooccurred with E. siliculosus and E. crouaniorum.

The concatenated alignable sequences from ITS1, ITS2, Rubisco spacer and cox3 (Table 2) had a length of 1533 bp. The different calculation methods of phylogenetic trees gave basically the same topology, and only one is shown (Fig. 6). The three isolates of Kuckuckia, including the isolate of GT7 from Site 5, formed the sister group to Ectocarpus which itself was split into two major lineages. The first consisted of E. fasciculatus and an additional European genotype, and the second comprised the remaining isolates. Within this latter clade, there were four sub-clades with strong and one sub-clade with moderate (>80%) statistical support but the hierarchy among the sub-clades varied and none of the hierarchies had strong statistic support in any analysis. In all trees, the genome strain clustered strongly with GT1, but was separated from genuine E. siliculosus, E. crouaniorum, the strain from New Zealand, and a lineage formed by the other South American endemic genotypes 2 and 3.

Discussion

Our study revealed a diverse Ectocarpus flora in Peru and northern Chile (Fig. 7). Three species recognised for Western Europe, including the recently reinstated E. crouaniorum [START_REF] Peters | Reinstatement of E. crouaniorum Thuret in Le Jolis as a third common species of Ectocarpus (Ectocarpales, Phaeophyceae) in western Europe, and its phenology at Roscoff, Brittany[END_REF], were present. In addition there was a Kuckuckia and four genotypes of Ectocarpus which have not been recorded outside South America, and for which we can only propose provisional designations, i.e. genotypes (GT) 1-4. Kuckuckia had been reported previously from Isla Robinson Crusoe, but according to sequences, our isolate was significantly different from both that strain and from six other previously sequenced isolates from Europe and South Africa which form two lineages within Kuckuckia [START_REF] Stache-Crain | Molecular systematics of Ectocarpus and Kuckuckia (Ectocarpales, Phaeophyceae) inferred from phylogenetic analysis of nuclear and plastid-encoded DNA sequences[END_REF]. Our new strain from northern Chile adds a third lineage. Based on culture studies, all Kuckuckia were merged into a single species, K. spinosa (Kützing) Kuckuck (Pedersen, 1979); the molecular data and our new isolate suggest the systematics of the genus may need to be re-assessed.

Our data suggest a separation of the study area into a northern half from Ancón, central Peru to Pisagua, northern Chile (Sites 1-6) dominated by the genotype of the genome strain (GT4; 61% of the strains isolated in the northern half but absent in the South) and a southern half from Antofagasta to Quintay (Sites 7-10) dominated by E. siliculosus, E. crouaniorum and their hybrids (together representing 65% of the samples in the southern half but missing in the North). This apparent split may reflect adaptation to different oceanographic environments (see below), but we cannot exclude that it is an artefact due to the different sampling methods: in the North we collected at the end of summer from drift or subtidal material, in the South we collected in spring and summer in the intertidal and from spores in the water column. The sampling zone and season certainly influence which species of Ectocarpus are encountered; for instance, in Western Europe, E. crouaniorum occurs only in the high intertidal and macrothalli are seasonal in spring and early summer [START_REF] Peters | Reinstatement of E. crouaniorum Thuret in Le Jolis as a third common species of Ectocarpus (Ectocarpales, Phaeophyceae) in western Europe, and its phenology at Roscoff, Brittany[END_REF]. More complete sampling would be required for a genuine quantification of the occurrence of the different genotypes in the area.

Isolates similar to the genome strain were the northernmost samples (Table 1) and represented two thirds of the isolates from Peru. The distribution area of this genotype is characterised by continuous upwelling of cool water (16-20ºC at the sea surface) irregularly interrupted by El Niño events which may result in several weeks of >10ºC higher sea surface temperatures due to the southward incursion of warm waters [START_REF] Peters | Temperature tolerance and latitudinal range of brown algae from temperate Pacific South America[END_REF]. So far no Ectocarpus has been collected from similar low latitudes. It remains to be examined whether it extends even further north, perhaps up to the deep-water kelp forests around Galápagos where its frequent host, Desmarestia, is present [START_REF] Graham | Deep-water kelp refugia as potential hotspots of tropical marine diversity and productivity[END_REF]. We have not found GT4 south of Arica. Apart from the two individuals from Arica, which were collected in situ at 3-5m depth, all samples of this genotype were from drift material, and their original habitat and depth distribution are not known. There are very limited data on seasonality of this genotype. All our new collections were from late summer, and it was present as a minute epiphyte or endophyte in Desmarestia peruviana in November 1988. The Museo de Historia Natural, Lima, houses seven herbarium specimens which are thalli of Ectocarpus, collected by César Acleto around Lima between September and February from 1964 to 1976. If it is assumed they belong to the same genotype, macroscopic thalli appear to be present from September to March. GT1 and GT2 were present in both parts of the study area. Whereas GT1 formed minute dark spots on Macrocystis and Lessonia, thalli of GT2 were large (Fig. 4) and grew on rock or on different red and brown algae, such as Gracilaria and Desmarestia. Both genotypes are endemic to the Pacific coast of South America. GT3, found once at an exposed site, was previously known from Atlantic South America; it also resembles Ectocarpus strains collected in New Zealand (U38761) and at Isla Robinson Crusoe (U38763) [START_REF] Stache-Crain | Molecular systematics of Ectocarpus and Kuckuckia (Ectocarpales, Phaeophyceae) inferred from phylogenetic analysis of nuclear and plastid-encoded DNA sequences[END_REF]. This genotype may be widely distributed in the southern hemisphere.

Our phylogenetic analyses, based on the genetic markers used in [START_REF] Stache-Crain | Molecular systematics of Ectocarpus and Kuckuckia (Ectocarpales, Phaeophyceae) inferred from phylogenetic analysis of nuclear and plastid-encoded DNA sequences[END_REF] plus cox3, confirmed most lineages obtained in that pioneer study. However, the lineages 2d and 3 in [START_REF] Stache-Crain | Molecular systematics of Ectocarpus and Kuckuckia (Ectocarpales, Phaeophyceae) inferred from phylogenetic analysis of nuclear and plastid-encoded DNA sequences[END_REF] were not represented in our taxon set. Our analyses confirmed that GT4 groups within the Ectocarpus subclade "siliculosi" (Fig. 6). In a previous analysis involving fewer taxa [START_REF] Peters | Reinstatement of E. crouaniorum Thuret in Le Jolis as a third common species of Ectocarpus (Ectocarpales, Phaeophyceae) in western Europe, and its phenology at Roscoff, Brittany[END_REF], it had clustered with Stache-Crain's lineage 1a which is genuine E. siliculosus. The addition of more South American strains led to its separation from that lineage. A post-zygotic sterility barrier between the genome strain and genuine E. siliculosus has been reported (Peters et al., 2004b). The genome strain was also genetically distant from E. crouaniorum and from an Ectocarpus from New Zealand referred to as E. siliculosus in previous studies [START_REF] Stache | Sexual compatibility and species concept in Ectocarpus siliculosus (Ectocarpales, Phaeophyceae) from Italy, North Carolina, Chile, and New Zealand[END_REF][START_REF] Müller | Mendelian segregation of a virus genome during host meiosis in the marine brown alga Ectocarpus siliculosus[END_REF]Peters et al., 2004b). It was likewise separated from the lineage formed by GT2 and GT3. GT2 and GT4 co-occurred at Site 3; a cross between the genome strain and a member of GT2 from Site 8 produced viable hybrids with weaklygrowing erect thalli which did not form meiosporangia (unpublished results). GT1 and GT3 were not yet involved in crossing studies.

In summary, the genome-sequenced Ectocarpus appears to be genetically separated from all genotypes with recognised names and also from all but one (GT1) of the genotypes endemic to South America or the southern hemisphere; it could therefore be regarded as a further species of Ectocarpus. The problem is to find its correct name. A proportion of the seaweeds that are reported from Peru and Chile also occur in the NE Pacific [START_REF] Santelices | Algas marinas de Chile[END_REF]; this is the case for Ectocarpus acutus Setchell et Gardner reported in Chile from Coquimbo and Cartagena [START_REF] Santelices | Algas marinas de Chile[END_REF][START_REF] Ramírez | Catálogo de las algas marinas bentónicas de la costa temperada del Pacífico de Sudamérica[END_REF]. It is possible that the lineages GT1+4 and 2+3 belong to this or another one of the species of Ectocarpus described from the Pacific coast of North America [START_REF] Setchell | Phycological contributions. VI. New species of Ectocarpus[END_REF][START_REF] Setchell | The marine algae of the Pacific coast of North America. Part III. Melanophyceae[END_REF] or NE Asia (Yoshida et al., 1995;Yoshida, 1998). So far four isolates of Ectocarpus from this vast region have been sequenced. Strains from San Francisco and Santa Barbara (California, USA) and from Kanagawa, Enoshima Prefecture, central Japan, were genuine E. siliculosus, a strain from Akkeshi, NE Hokkaido, Northern Japan, in contrast, was more closely related to GT4 [START_REF] Stache-Crain | Molecular systematics of Ectocarpus and Kuckuckia (Ectocarpales, Phaeophyceae) inferred from phylogenetic analysis of nuclear and plastid-encoded DNA sequences[END_REF]Tanaka et al., unpublished). The Akkeshi strain was nevertheless in crosses fully compatible with genuine E. siliculosus, but it has not been tested against GT4 [START_REF] Müller | Sexual reproduction of Ectocarpus siliculosus (Ectocarpales Phaeophyceae) in Japan[END_REF]. More comprehensive sampling in the North Pacific may be helpful for the taxonomic revision of the genus and the nomenclature of South American Ectocarpus.

Our work also aimed at finding strains that were sexually compatible with the genome strain.

Seven out of our 36 new strains of GT4 produced unilocular sporangia in culture; female gametophytes obtained from meiospores of two sporophytes from Sites 2 and 5 were used for crosses with the genome strain. In both combinations the zygotes developed into sporophytes capable of meiosis, and one of these sporophytes was selected for further experiments leading to the production of the genetic map of the Ectocarpus genome [START_REF] Heesch | A sequence-tagged genetic map for the brown alga Ectocarpus siliculosus provides large-scale assembly of the genome sequence[END_REF]. 1000, 1500 bp from the bottom). Lanes 1-9: genotypes (GT). 1. GT1, 2. GT2, 3. GT3, 4. GT4 (=genome-sequenced strain), 5. E. fasciculatus,6. E. siliculosus,7. Kuckuckia sp.,8. E. crouaniorum,9. Putative hybrid between E. siliculosus and E. crouaniorum; the middle band is a PCR artefact. Note that lanes 2-4 and 6-7 have lengths that are hardly (if at all) distinguishable in an ordinary 1% agarose gel. See Table 3 for precise ITS1 lengths. Brittany, France [START_REF] Peters | Reinstatement of E. crouaniorum Thuret in Le Jolis as a third common species of Ectocarpus (Ectocarpales, Phaeophyceae) in western Europe, and its phenology at Roscoff, Brittany[END_REF]. Names starting with "1310" are CCAP strain numbers, those with "Ec" are accessions at the Roscoff Ectocarpus strain collection, those with "Kck" are strain designations in [START_REF] Stache-Crain | Molecular systematics of Ectocarpus and Kuckuckia (Ectocarpales, Phaeophyceae) inferred from phylogenetic analysis of nuclear and plastid-encoded DNA sequences[END_REF]. Genotypes obtained in the present study (GT1-8) are provided to the right of the tree, as well as the lineage numbers (1a-6b) of strains included in [START_REF] Stache-Crain | Molecular systematics of Ectocarpus and Kuckuckia (Ectocarpales, Phaeophyceae) inferred from phylogenetic analysis of nuclear and plastid-encoded DNA sequences[END_REF]. Genuine E. siliculosus was in our analyses only represented by European isolates because individuals from Chile had highly similar ITS1 sequences (99% identity in blast; Table 3) and sequences of additional markers were not generated for them.

Ec319 represents Stache-Crain lineage 5a instead of strain CCAP1310/100 originally employed in [START_REF] Stache-Crain | Molecular systematics of Ectocarpus and Kuckuckia (Ectocarpales, Phaeophyceae) inferred from phylogenetic analysis of nuclear and plastid-encoded DNA sequences[END_REF] for which cox3 is not yet available. Accessions of sequences utilised are provided in Table 2. that of the genome strain (GT4). For details see Table 3. [START_REF] Peters | Reinstatement of E. crouaniorum Thuret in Le Jolis as a third common species of Ectocarpus (Ectocarpales, Phaeophyceae) in western Europe, and its phenology at Roscoff, Brittany[END_REF]. Names starting with "1310" are CCAP strain numbers, those with "Ec" are accessions at the Roscoff Ectocarpus strain collection, those with "Kck" are strain designations in [START_REF] Stache-Crain | Molecular systematics of Ectocarpus and Kuckuckia (Ectocarpales, Phaeophyceae) inferred from phylogenetic analysis of nuclear and plastid-encoded DNA sequences[END_REF]. Genotypes obtained in the present study (GT1-8) are provided to the right of the tree, as well as the lineage numbers (1a-6b) of strains included in [START_REF] Stache-Crain | Molecular systematics of Ectocarpus and Kuckuckia (Ectocarpales, Phaeophyceae) inferred from phylogenetic analysis of nuclear and plastid-encoded DNA sequences[END_REF]. Genuine E. siliculosus was in our analyses only represented by European isolates because individuals from Chile had highly similar ITS1 sequences (99% identity in blast; Table 3) and sequences of additional markers were not generated for them. Ec319 represents Stache-Crain lineage 5a instead of strain CCAP1310/100 originally employed in [START_REF] Stache-Crain | Molecular systematics of Ectocarpus and Kuckuckia (Ectocarpales, Phaeophyceae) inferred from phylogenetic analysis of nuclear and plastid-encoded DNA sequences[END_REF] 3. 130x99mm (600 x 600 DPI)

F o r P e e r R e v i e w

  mutant of the brown alga Ectocarpus siliculosus. Development 135: 1503-Marine algae of Japan. Tokyo: Uchida Rokakahu Publishing Co. Yoshida T, Yoshinaga K, Nakajima Y. 1995. Check list of marine algae of Japan (revised in 1995). Japanese Journal of Phycology 43: 115-171.

Fig. 1 .

 1 Fig.1. Origin of isolates for the present study. See Table1for details on collecting sites.

Fig. 5 .

 5 Fig. 5. ITS1 lengths in Ectocarpus and Kuckuckia from Peru and Chile. 2% agarose gel run at

Fig. 6 .

 6 Fig. 6. Molecular phylogeny of Ectocarpus and Kuckuckia inferred from genetic distance

Fig. 7 .

 7 Fig. 7. Diversity of Ectocarpus and Kuckuckia in Peru and northern Chile (n=120 samples). sil =

F

Fig. 5 .Fig. 6 .

 56 Fig.5. ITS1 lengths in Ectocarpus and Kuckuckia from Peru and Chile. 2% agarose gel run at low voltage, stained with ethidium bromide. Flanking lanes contain length standards (600, 800, 1000, 1500 bp from the bottom). Lanes 1-9: genotypes (GT). 1. GT1, 2. GT2, 3. GT3, 4. GT4 (=genomesequenced strain), 5. E. fasciculatus, 6. E. siliculosus, 7. Kuckuckia sp., 8. E. crouaniorum, 9. Putative hybrid between E. siliculosus and E. crouaniorum; the middle band is a PCR artefact. Note that lanes 2-4 and 6-7 have lengths that are hardly (if at all) distinguishable in an ordinary 1% agarose gel. See Table3for precise ITS1 lengths. 182x93mm (98 x 98 DPI)

Fig. 7 .

 7 Fig. 7. Diversity of Ectocarpus and Kuckuckia in Peru and northern Chile (n=120 samples). sil = E.siliculosus, fas = E. fasciculatus, cro = E. crouaniorum, sil+cro = putative hybrid between E. siliculosus and E. crouaniorum, Kuck = Kuckuckia sp. GT1-4 = other genotypes, including that of the genome strain (GT4). For details see Table3. 130x99mm (600 x 600 DPI)

Table 1 .

 1 Origin and number of isolates. Kuck = Kuckuckia; sil = E. siliculosus; fas = E. fasciculatus; cro = E. crouaniorum; ND = no data Ectocarpus strain collection at Roscoff or CCAP §For a description of the site see[START_REF] Andrade | Distribution of dissolved species and suspended particulate copper in an intertidal ecosystem affected by copper mine tailings in Northern Chile[END_REF] ¶Strain collected by Mariela Gonzales and isolated by Dieter G. Müller

					Genotypes
					present	ITS1
	Site			No of	(number of	sequenced
	no Locality	Date	Coordinates	isolates	isolates)	(n)	Site details †		Collection details	Strains ‡
							Playa de Ancón	Sheltered
	Ancón	07-03-06 -11.7 -77.2	4	GT4 (4)	3	Playa San Francisco	Sheltered	Drift, on Gracilaria	311-314
										201-202, 246-247,
										266-267, 269-271,
	Bahía Mendieta									274-285, 289-293,
	(South of Paracas)	08-03-06 -14.1 -76.3		GT4 (28)	1 Playa Mendieta	Sheltered	Drift, on various macroalgae	315-316
									Upper subtidal, on phylloid of
				31	GT1 (3)	2 Cueva de la Zorra	Mid-exposed	Macrocystis	286-288
	San Juan de Marcona 09-03-06 -15.4 -75.2		GT4 (2)	1 Playa Hermosa	Sheltered	Drift, on Desmarestia peruviana 301, 303
										294-300, 302, 304-
					GT2 (12)	7 Playa Hermosa	Sheltered	Drift, on various macroalgae	307
									Upper subtidal, on phylloid of
				16	fas (2)	1 Reserva Punta San Juan	Mid-exposed	Macrocystis	308-309
	Ilo	10-03-06 -17.6 -71.3		none	---	Playa Boca del Río	Highly	---	---
				0	none	---	Punta Coles	exposed	---	---
									Subtidal 3-5m, on Lessonia
	Arica	01-03-06 -18.5 -70.3		GT4 (2)	2 South end of town	Mid-exposed	trabeculata and red macroalga	161, 721
									Subtidal 3-5m, on Lessonia
					fas (1)	1 South end of town	Mid-exposed	trabeculata	310
									Upper subtidal, on Lessonia
					fas (1)	1 Yacht harbour	Sheltered	trabeculata	165
				5	Kuck (1)	1 Yacht harbour	Sheltered	Upper subtidal, on rope	160

Table 2 .

 2 DNA sequences utilised in the present study; newly generated sequences in bold. Acc = GenBank/EBI/DDBJ accession; bp=base pairs (length); -=not applicable ITS1: start atcattaCCGA, end GTTGTAaaacttt; ITS2: start gtctgttGACACC, end TTTCGTTcggacct[START_REF] Stache-Crain | Molecular systematics of Ectocarpus and Kuckuckia (Ectocarpales, Phaeophyceae) inferred from phylogenetic analysis of nuclear and plastid-encoded DNA sequences[END_REF] ‡ Including flanking rbcL and rbcS gene sequences § Strain used in[START_REF] Müller | Mendelian segregation of a virus genome during host meiosis in the marine brown alga Ectocarpus siliculosus[END_REF] andPeters et al. 2004b; lineage 4 in[START_REF] Stache-Crain | Molecular systematics of Ectocarpus and Kuckuckia (Ectocarpales, Phaeophyceae) inferred from phylogenetic analysis of nuclear and plastid-encoded DNA sequences[END_REF] ¶ Strain representing lineage 5a of[START_REF] Stache-Crain | Molecular systematics of Ectocarpus and Kuckuckia (Ectocarpales, Phaeophyceae) inferred from phylogenetic analysis of nuclear and plastid-encoded DNA sequences[END_REF]

								Rubisco
								spacer ‡
		Species/genotype	Strain	Origin	ITS1+2	ITS1 † ITS2 †	(515bp)	cox3 (665 bp)
					Acc	bp	bp	Acc	Acc
	1 GT1	Ec157	Site 5	FN564453	306	247 FN564475	FN564526
	2 GT1	Ec286	Site 2	FN564454	302	247 FN564476	FN564527
	3 GT2	Ec298	Site 3	FN564456	344	293 FN564477	FN564528
	4 GT2	CCAP1310/40	Site 9	FN564455	352	292 U38736	FN564529
	5 GT3	Ec156	Site 6	FN564457	347	264 FN564478	FN564530
					AJ550048
					and
			Genome strain		genome				from genome
	6 GT4	Ec32=CCAP1310/4	Site 3	sequence	362	252 AJ550050	project
	7 GT4	Ec721	Site 5	FN564446∫	362	-	-	-
		GT5 =
	8	E. fasciculatus	Ec310	Site 3	FN564458	433	292 FN564479	FN564531
	9 GT6 = E. siliculosus Ec147	Site 7	FN564466∫	716	-	-	-
		GT7 = Kuckuckia
	10	sp.	Ec160	Site 5	FN564460	714	290 FN564480	FN564532
		GT8 =
	11	E. crouaniorum	Ec608	Site 8	FN564459	855	253 FN564481	FN564533
				Plouescat,
				Brittany,
	12 E. fasciculatus	1310/12	France	U38824	429	286 U38711	FN564521
	13 E. fasciculatus	fas R1 = Ec395	Santec,	FN564441	428	282 FN564468	FN564513

F o r P e e r R e v i e w

Table 3 .

 3 Genotypes (GT) and ITS1 length types (LT) of Ectocarpus/Kuckuckia from Peru and northern Chile; sil = E. siliculosus; cro = E. crouaniorum; ND = no data; -=not applicable.Figures in bold face in last two columns indicate that there was no close blast hit ¶Note that for strains isolated from filtered water or developed in culture on inoculated substratum the field morphology is unknown †ITS1 start atcattaCCGA, end GTTGTAaaacttt[START_REF] Stache-Crain | Molecular systematics of Ectocarpus and Kuckuckia (Ectocarpales, Phaeophyceae) inferred from phylogenetic analysis of nuclear and plastid-encoded DNA sequences[END_REF] ‡Compare with Fig.5§Origin of strains giving best blast hits Fig. 1. Origin of isolates for the present study. See Table 1 for details on collecting sites. Diameters of closed circles represent the number of isolates from each site. No Ectocarpus was found at the highly exposed Site 4 (open circle). The genome strain was collected in 1988 at Site 3. The isolate from Site 9 was available from CCAP. Map generated using www.aquarius.ifm-geomar.de/omc/.

				ITS1	Length		Sites	Sites		Best hits in
				length	of PCR		in	in		blast of ITS1
	GT Designation/ species	Field morphology ¶	(bp) †	product ‡ LT ‡ Examples	Peru	Chile Elsewhere §	sequence	Coverage Identity
									Isle of Man
									(United
	1 GT1	Minute dark spots on hosts	302-6	526-30 1 Ec157	2 6, 8	Kingdom)	U38771	75%	93%
									Pto. Deseado
	2 GT2	Large thalli (Fig. 4)	344-52	568-76 2 Ec298	3 9, 10	(Argentina)	U38762	100%	93%
									Pto. Deseado
	3 GT3	Minute dark spot on host	347	571 2 Ec156	-	6	(Argentina)	U38762	100%	95%
		GT4 (the genome-
	4	sequenced Ectocarpus)	Small to large thalli (Figs 2-3)	362	586 2 Ec721	1-3	5 -	AJ550048	100%	99%
									Isla Robinson
								5, 8,	Crusoe (Chile);	U38780		98%
	5 E. fasciculatus Harvey	Minute dark spots on hosts or	431-3	655-7 3 Ec310	3	10	Europe	U38781	100%	97%
		E. siliculosus (Dillwyn)	From thin felt on rock							U38759		99%
	6	Lyngbye	or invertebrates to large drifting thalli	714-6	938 4 Ec147	-	7, 8 Cosmopolitan	U38760	100%	99%
									Isla Robinson
	7 Kuckuckia sp.	Minute thallus on rope	714	938 4 Ec160	-	5	Crusoe (Chile	U38825	100%	77%
									Europe (Werra
									river); Valdivia	U38774,
	8 E. crouaniorum Thuret	Minute thalli on rock and invertebrates	855	1079 5 Ec608	-	8	(Chile)	U38772	98%	96%
									Brittany
	9 Putative hybrid sil+cro	Minute thalli on rock and invertebrates	ND	ND	4+5 Ec513	-	8	(France))	-	-	-

F o r P e e r R e v i e w

Figs 2-4. Herbarium specimens of Ectocarpus from Peru, all collected from drift material. 2. GT4, Site 1. 3. Several thalli of GT4 epiphytic on Desmarestia firma Skottsberg, Site 2. 4. GT2, Site 3.

  for which cox3 is not yet available. Accessions of sequences utilised are provided in Table 2. 135x179mm (600 x 600 DPI)

	F o
	r
	P
	e e r
	R e v i e w

Manuscript submitted to New Phytologist for review

Acknowledgements

We thank César Acleto for providing access to the phycological herbarium at the Museo de Historia Natural, Lima, and Matt Hall for advice during the molecular experiments. AFP was supported by a travel grant from Université Pierre et Marie Curie for the collections and by a Ray Lankester Investigatorship by the Marine Biological Association of the United Kingdom for the molecular work. Collections and isolations in Chile were supported by FONDAP 1501 0001 funded by CONICYT, to the Center for Advanced Studies in Ecology & Biodiversity (CASEB) Program 7. Additional support was provided by ICA grant (to JAC).


Mourad Choulli
email: mourad.choulli@univ-lorraine.fr

Masahiro Yamamoto

LOGARITHMIC STABILITY OF PARABOLIC CAUCHY PROBLEMS

Keywords: Mathematics subject classification : 35R25, 35K99, 58J35 Parabolic Cauchy problems, logarithmic stability, Carleman inequality, Hardy inequality

The uniqueness of parabolic Cauchy problems is nowadays a classical problem and since Hadamard [8] these kind of problems are known to be ill-posed and even severely ill-posed. Until now there are only few partial results concerning the quantification of the stability for parabolic Cauchy problems. We bring in the present work an answer to this issue for smooth solutions under the minimal condition that the domain is Lipschitz.

Introduction

Throughout this article Ω is a bounded domain of R n with Lipschitz boundary Γ. Consider the parabolic operator

L = div(A∇ •) -∂ t .
Here A = (a ij ) is a symmetric matrix whose coefficients belong to W 1,∞ (Ω). Assume furthermore that there exists a constant 0 < κ ≤ 1 so that (1.1)

A(x)ξ • ξ ≥ κ|ξ| 2 , x ∈ Ω, ξ ∈ R n ,

and

(1.2)

a ij W 1,∞ (Ω) ≤ κ -1 , 1 ≤ i, j ≤ n.
Let t 0 < t 1 so that t 1 -t 0 ≤ T 0 , for some given T 0 > 0 and set Q = Ω × (t 0 , t 1 ).

Recall the notation

H 2,1 (Q) = L 2 ((t 0 , t 1 ), H 2 (Ω)) ∩ H 1 ((t 0 , t 1 ), L 2 (Ω)).
If Γ 0 is a nonempty open subset of Γ then a classical result says that any u ∈ H 2,1 (Q) satisfying Lu = 0 in Q and u = ∇u = 0 on Γ 0 × (t 0 , t 1 ) must be identically equal to zero (see [START_REF] Choulli | Une introduction aux problèmes inverses elliptiques et paraboliques[END_REF] and references therein). This result is known as the uniqueness of the Cauchy problem for the equation Lu = 0. Quantifying this uniqueness result consists in controlling a norm of a solution of Lu = 0 by a suitable function of the norm of (u, ∇u) |Γ0×(t0,t1) in some space.

In the rest of this paper, 0 < α < 1 is fixed and, for simplicity's sake, we use the notations

X (Q) = C 1+α,(1+α)/2 (Q) ∩ H 1 ((t 0 , t 1 ), H 2 (Ω)), Y (Q) = {u ∈ X (Q); ∂ t u ∈ X (Q)}, Z (Q) = Y (Q) ∩ H 3 ((t 0 , t 1 ), H 2 (Ω)).
We endow X (Q), Y (Q) and Z (Q) with their natural norms

u X (Q) = u C 1+α,(1+α)/2 (Q) + u H 1 ((t0,t1),H 2 (Ω)) u Y (Q) = u X (Q) + ∂ t u X (Q) . u Z (Q) = u Y (Q) + u H 3 ((t0,t1),H 2 (Ω))
Let * = e -e e . Set then, for µ > 0 and 0 ≤ * ,

Ψ 0,µ ( ) =    0 if = 0, (ln ln ln | ln |) -µ if 0 < ≤ 0 , if ≥ 0 .
We are mainly concerned in the present work with the stability issue for the Cauchy problem associated to the parabolic operator L. Precisely, we are going to prove the following result.

Theorem 1.1. Let Γ 0 be a nonempty open subset of Γ and s ∈ (0, 1/2). Then there exist two constants C > 0 and 0 < 0 ≤ * , depending on Ω, κ, T 0 , α, s and Γ 0 , so that, for any u ∈ Z (Q) satisfying Lu = 0 in Q, we have

C u L 2 ((t0,t1),H 1 (Ω)) ≤ u Z (Q) Ψ 0,µ C(u, Γ 0 ) u Z (Q) ,
with µ = min(α, s)/4 and C(u, Γ 0 ) = u H 3 ((t0,t1),L 2 (Γ0)) + ∇u H 2 ((t0,t1),L 2 (Γ0)) .

It is straightforward to check that C(u, Γ 0 ) in the preceding theorem can be substituted by C(u, Γ 0 ) = u H 3 ((t0,t1),L 2 (Γ0))∩H 2 ((t0,t1),H 1 (Γ0)) + ∂ n u H 2 ((t0,t1),L 2 (Γ0)) .

Here n is the unit exterior normal field on Γ and ∂ n u = ∇u • n.

We observe that Theorem 1.1 remains valid if L is substituted by L plus an operator of first order in space variable whose coefficients are bounded.

Since the proofs are quite complicated we limited ourselves to the case Lu = 0. We believe that one can remove this condition by adding to C(u, Γ 0 ) the norm of Lu is a suitable space.

The second author [14, Theorem 5.1, page 24] proved a Hölder stability in a proper subdomain of Q depending on the part of the lateral boundary where the Cauchy data is given. In [13, Theorem 3.5.1, pages 45 and 46], Vessella establishes a local Hölder stability corresponding to the continuation of Cauchy data to an interior subdomain for solutions vanishing at the initial time. Recently, Bourgeois [1, Main theorem, page 2] proved a result similar to the one in Theorem 1.1, with a single logarithm function, in the case where L = ∆ -∂ t , Ω = D \ O, D and O are two domains of class C 2 , O D, and Γ 0 is either ∂D or ∂O. His result is based on a global Carleman estimate in which the weight function is built from the distance to the boundary of the space variable.

We discuss in the present work the Cauchy problem in all of its generality, that is without any restriction on the part of the boundary where the Cauchy data is given.

The proof of the main result is inspired by that used in the elliptic case by the first author in [START_REF] Choulli | Applications of elliptic Carleman inequalities to Cauchy and inverse problems[END_REF] (a substantial improvement of this result will appear in [START_REF] Choulli | An introduction to the analysis of elliptic partial differential equations[END_REF]). Note however that there is a great difference between the elliptic case and the parabolic case. The main difficulty in the parabolic case is due to the fact that the initial time and the final time data are wanting. So the proofs are more technical. The idea to overcome the fact that the initial time and the final time data are not known is to use a Hardy inequality with respect to time variable. This explains partially why we need to work with sufficiently smooth solutions. Another difference between the parabolic case and the elliptic case relies on the fact that in the elliptic case the main tool is a three-ball inequality with arbitrary radius. While in the parabolic case the method is based on a three-cylinder inequality (see Theorem 2.1) in which the radius depends on the distance to the boundary of the time variable. Roughly speaking, the radius becomes smaller and smaller as the time variable approaches the boundary. For this reason, contrary to the elliptic case where the stability is only of single logarithmic type, the stability is of multiple logarithmic type.

The three-cylinder inequality appears to be the right tool for continuing a solution of a parabolic equation. For this reason, we are not convinced that Theorem 1.1 can be improved by using a global method.

Although we used classical tools to establish our main result, the result itself is completely new and our proof is entirely self-contained. This is our modest contribution to the stability issue for parabolic Cauchy problems.

The most part in our analysis is build on a Carleman inequality (Theorem 2.2 below). We observe that Carleman inequalities are very useful tool in control theory and for establishing the unique continuation property for elliptic and parabolic partial differential equations. There is wide literature on this subject. We just quote here the few references [START_REF] Bourgeois | Quantification of the unique continuation property for the heat equation[END_REF][START_REF] Fernàndez-Cara | Global Carleman inequalities for parabolic systems and applications to controllability[END_REF][START_REF] Fursikov | Controllability of evolution equations[END_REF][START_REF] Lebeau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF].

The rest of this article is organized as follows. Section 2 is devoted to a threecylinder interpolation inequality for the L 2 t (H 1 x )-norm. This inequality will be very useful for continuing the data on an interior subdomain to the lateral boundary data, and to continue the data from one subdomain to another subdomain. This is what we show in Section 3 and, as byproduct, we prove a stability estimate corresponding to the unique continuation from an interior data. The proof of Theorem 1.1 is completed in Section 4 by beforehand establishing a result that quantifies the stability from the Cauchy data to an interior subdomain.

Three-cylinder interpolation inequality

We prove in this section Theorem 2.1. There exist C > 0 and 0 < ϑ < 1, only depending on κ, Ω and T 0 , so that, for any 0 < < (t

1 -t 0 )/2, u ∈ H 1 ((t 0 , t 1 ), H 2 (Ω)) satisfying Lu = 0 in Q, y ∈ Ω and 0 < r < r y ( ) = min (dist(y, Γ)/3, √ ), we have r 3 u L 2 ((t0+ ,t1-),H 1 (B(y,2r))) (2.1) ≤ C u ϑ L 2 ((t0,t1),H 1 (B(y,r))) u 1-ϑ L 2 ((t0,t1),H 1 (B(y,3r))
) . The proof of Theorem 2.1 is based on a Carleman inequality for a family of parabolic operators. To this end, let Z be an arbitrary set and consider the family of operators

L z = div(A z ∇ • ) -∂ t , z ∈ Z, where, for each z ∈ Z, A z = (a ij z
) is a symmetric matrix with W 1,∞ (Ω) entries and there exists 0 < κ ≤ 1 so that (2.2)

A z (x)ξ • ξ ≥ κ|ξ| 2 , x ∈ Ω, ξ ∈ R n and z ∈ Z,

and

(2.3)

a ij z W 1,∞ (Ω) ≤ κ -1 , 1 ≤ i, j ≤ n, z ∈ Z. Pick ψ ∈ C 2 (Ω) without critical points in Ω and set Σ = Γ × (t 0 , t 1 ). Let g(t) = 1 (t -t 0 )(t 1 -t) and ϕ(x, t) = g(t) e 4λ ψ ∞ -e λ(2 ψ ∞+ψ(x)) , χ(x, t) = g(t)e λ(2 ψ ∞+ψ(x)) .

Theorem 2.2. (Carleman inequality)

There exist three positive constants C, λ 0 and τ 0 , only depending only on ψ, Ω, κ and T 0 , so that

C Q λ 4 τ 3 χ 3 u 2 + λ 2 τ χ|∇u| 2 e -2τ ϕ dxdt (2.4) ≤ Q (L z u) 2 e -2τ ϕ dxdt + Σ λ 3 τ 3 χ 3 u 2 + λτ χ|∇u| 2 + (λτ χ) -1 (∂ t u) 2 ) e -2τ ϕ dσdt,
for all u ∈ H 1 ((t 0 , t 1 ), H 2 (Ω)), z ∈ Z, λ ≥ λ 0 and τ ≥ τ 0 .

Proof. Since the dependance of the constants will be uniform with respect to z ∈ Z, we drop for simplicity the subscript z in L z and its coefficients. On the other hand, as

C ∞ (Q) is dense in H 1 ((t 0 , t 1 ), H 2 (Ω)), it is enough to prove (2.4) when u ∈ C ∞ (Q).
Let Φ = e τ ϕ , u ∈ C ∞ (Q) and set w = Φ -1 u that we extend by continuity at t = 0 and t = T by setting w(•, 0) = w(•, T ) = 0. Then straightforward computations give P w = [Φ -1 LΦ]w = P 1 w + P 2 w + cw, where

P 1 w = aw + div (A∇w) -τ ∂ t ϕw, P 2 w = B • ∇w + bw -∂ t w, with a = a(x, t, λ, τ ) = λ 2 τ 2 χ 2 |∇ψ| 2 A , B = B(x, t, λ, τ ) = -2λτ χA∇ψ, b = b(x, t, λ, τ ) = -2λ 2 τ χ|∇ψ| 2 A , c = c(x, t, λ, τ ) = -λτ χdiv (A∇ψ) + λ 2 τ χ|∇ψ| 2 A . Here |∇ψ| A = A∇ψ • ∇ψ = A 1/2 ∇u .
We obtain by making integrations by parts

Q aw(B • ∇w)dxdt = 1 2 Q a(B • ∇w 2 )dxdt (2.5) = - 1 2 Q div(aB)w 2 dxdt + 1 2 Σ a(B • ν)w 2 dσdt and Q div (A∇w)B • ∇wdxdt (2.6) = - Q A∇w • ∇(B • ∇w)dxdt + Σ (B • ∇w)(A∇w • ν)dσdt = - Q B ∇w • A∇wdxdt - Q ∇ 2 wB • A∇wdxdt + Σ (B • ∇w)(A∇w • ν)dσdt.
Here B = (∂ j B i ) is the Jacobian matrix of B and

∇ 2 w = (∂ 2 ij w) is the Hessian matrix of w. But Q B j ∂ 2 ij wa ik ∂ k wdxdt = - Q B j a ik ∂ 2 jk w∂ i wdxdt - Q ∂ j B j a ik ∂ k w∂ i wdxdt + Σ B j ν j a ik ∂ k w∂ i wdσdt. Therefore Q ∇ 2 wB • A∇wdxdt = - 1 2 Q div(B)A + à ∇w • ∇wdxdt (2.7) + 1 2 Σ |∇w| 2 A (B • ν)dσdt, where à = (ã ij ) with ãij = B • ∇a ij .
It follows from (2.6) and (2.7)

Q div (A∇w)B • ∇wdxdt = 1 2 Q -2AB + div(B)A + Ã ∇w • ∇wdxdt (2.8) + Σ (B • ∇w) (A∇w • ν) dσdt - 1 2 Σ |∇w| 2 A (B • ν)dσdt.
A new integration by parts yields

Q div (A∇w)bwdxdt = - Q b|∇w| 2 A dxdt - Q w∇b • A∇wdxdt + Σ bwA∇w • νdσdt.
This and

- Q w∇b • A∇wdxdt ≥ - Q (λ 2 χ) -1 |∇b| 2 A w 2 dxdt - Q λ 2 χ|∇w| 2 A dxdt
(where we used the inequality

|AX • Y | ≤ |X| 2 A + |Y | 2 A , for all X, Y ∈ R n ) imply Q div (A∇w)bwdxdt ≥ - Q (b + λ 2 χ)|∇w| 2 A dxdt (2.9) - Q (λ 2 χ) -1 |∇b| 2 A w 2 dxdt + Γ bw(A∇w • ν)dσdt.
One more time, integrations by parts entail (2.10)

Q aw∂ t wdxdt = 1 2 Q a∂ t w 2 dxdt = - 1 2 Q ∂ t aw 2 dxdt, (2.11) Q ∂ t ϕw∂ t wdxdt = 1 2 Q ∂ t ϕ∂ t w 2 dxdt = - 1 2 Q ∂ 2 t ϕw 2 dxdt, Q ∂ t ϕwB • ∇wdxdt = 1 2 Q ∂ t ϕB • ∇w 2 dxdt (2.12) = - 1 2 Q div(∂ t ϕB)w 2 dxdt + 1 2 Σ ∂ t ϕ(B • ν)w 2 dσdt. Also, (2.13)
Q div(A∇w)∂ t wdxdt = - Q A∇w • ∇∂ t wdxdt + Σ (A∇w • ν)∂ t wdσdt.
But an integration by parts with respect to t gives

Q A∇w • ∇∂ t wdxdt = - Q A∇∂ t w • ∇wdxdt = - Q ∇∂ t w • A∇wdxdt,
where we used w(•, 0) = w(•, T ) = 0. Whence

Q A∇w • ∇∂ t wdxdt = 0.
This identity in (2.13) entails

(2.14) Q div(A∇w)∂ t w = Σ (A∇w • ν)∂ t wdσdt.
Now a combination of (2.5), (2.8) to (2.12) and (2.14) gives

Q P 1 wP 2 wdxdt - Q c 2 w 2 dxdt (2.15) ≥ Q f w 2 dxdt + Q F ∇w • ∇wdxdt + Σ g(w)dσdt, where f = - 1 2 div(aB) + ab -(λ 2 χ) -1 |∇b| 2 A -c 2 + 1 2 ∂ t a - τ 2 ∂ 2 t ϕ + τ 2 div(∂ t ϕB) -τ b∂ t ϕ, F = -AB + 1 2 div(B)A + Ã -(b + λ 2 χ)A, g(w) = 1 2 aw 2 (B • ν) - 1 2 |∇w| 2 A (B • ν) + (B • ∇w)(A∇w • ν) + bw(A∇w • ν) - τ 2 ∂ t ϕ(B • ν)w 2 -(A∇w • ν)∂ t w.
We obtain, by using the elementary inequality (α -β

) 2 ≥ α 2 /2 -β 2 , α > 0, β > 0, P w 2 2 ≥ ( P 1 w + P 2 w 2 -cw 2 ) 2 ≥ 1 2 P 1 w + P 2 w 2 2 -cw 2 2 ≥ Ω P 1 wP 2 wdx - Ω c 2 w 2 dx. Whence (2.15) implies (2.16) P w 2 2 ≥ Q f w 2 dxdt + Q F ∇w • ∇wdxdt + Σ g(w)dσdt.
In light of the following inequalities, where C is a constant depending only on T 0 and ψ,

|∂ t ϕ| ≤ Cχ 2 , |∂ 2 t ϕ|, |∇∂ t ϕ| ≤ Cχ 3 |(A∇w • ν)∂ t w| ≤ λτ χ|A∇w • ν| 2 + (λτ χ) -1 (∂ t w) 2 ,
straightforward computations show that there exist four positive constants C 0 , C 1 , λ 0 and τ 0 , only depending only on ψ, Ω, T 0 and κ, such that, for all λ ≥ λ 0 and τ ≥ τ 0 , so that

f ≥ C 0 λ 4 τ 3 χ 3 , F ξ • ξ ≥ C 0 λ 2 τ χ|ξ| 2 , for any ξ ∈ R n , |g(w)| ≤ C 1 λ 3 τ 3 χ 3 w 2 + λτ χ|∇w| 2 + (λτ χ) -1 (∂ t w) 2 . Hence C Q (λ 4 τ 3 χ 3 w 2 + λ 2 τ χ|∇w| 2 )dxdt ≤ Q (P w) 2 dxdt (2.17) + Σ (λ 3 τ 3 χ 3 w 2 + λτ χ|∇w| 2 + (λτ χ) -1 (∂ t w) 2 )dσdt.
As ∇w = Φ -1 (∇u + λτ χu∇ψ), we obtain

|∇w| 2 = Φ -2 |∇u| 2 + λ 2 τ 2 χ 2 |∇ψ| 2 u 2 + 2λτ χu∇u • ∇ψ .
Therefore we find, by using an elementary inequality,

|∇w| 2 ≥ Φ -2 |∇u| 2 + λ 2 τ 2 |∇ψ| 2 u 2 -4λ 2 τ 2 u 2 |∇ψ| 2 - 1 2 |∇u| 2
and then

|∇w| 2 ≥ Φ -2 1 2 |∇u| 2 -3λ 2 τ 2 χ 2 u 2 ∇ψ 2 ∞ .
Consequently, modifying λ 0 if needed, we get

(2.18) λ 2 τ χ|∇w| 2 + λ 4 τ 3 χ 3 w 2 ≥ CΦ -2 λ 2 τ χ|∇u| 2 + λ 4 τ 3 χ 3 τ 2 u 2 .
On the other hand, it is not hard to establish the inequality

(2.19) (∂ t w) 2 ≤ Φ -2 (∂ t u) 2 + Cτ 2 χ 2 u 2 .
The expected inequality follows then by combining (2.17), (2.18) and (2.19).

From the preceding proof it is obvious that Theorem 2.2 holds whenever L z is replaced by L. That is we have Theorem 2.3. (Carleman inequality) There exist three positive constants C, λ 0 and τ 0 , only depending only on ψ, Ω, κ and T 0 , so that

C Q λ 4 τ 3 χ 3 u 2 + λ 2 τ χ|∇u| 2 e -2τ ϕ dxdt (2.20) ≤ Q (Lu) 2 e -2τ ϕ dxdt + Σ λ 3 τ 3 χ 3 u 2 + λτ χ|∇u| 2 + (λτ χ) -1 (∂ t u) 2 ) e -2τ ϕ dσdt, for all u ∈ H 1 ((t 0 , t 1 ), H 2 (Ω)), λ ≥ λ 0 and τ ≥ τ 0 . Proof of Theorem 2.1. Let u ∈ H 1 ((t 0 , t 1 ), H 2 (Ω)) satisfying Lu = 0 and set Q(µ) = B(0, µ) × (-1, 1), µ > 0. Fix (y, s) ∈ Ω × (t 0 , t 1 ) and 0 < r < r (y,s) = min dist(y, Γ)/3, √ s -t 0 , √ t 1 -s ≤ r 0 = r 0 (diam(Ω), T 0 ). Let w(x, t) = u(rx + y, r 2 t + s), (x, t) ∈ Q(3), Then L r w = div(A r ∇w) -∂ t w = 0 in Q(3), where A r (x) = (a ij (rx + y)).
Clearly, the family (A r ) satisfies (2.2) and (2.3) uniformly with respect to r ∈ (0, r (y,s) ).

Let

χ ∈ C ∞ 0 (U ) satisfying 0 ≤ χ ≤ 1 and χ = 1 in K, with U = {x ∈ R n ; 1/2 < |x| < 3} and K = {x ∈ R n ; 1 ≤ |x| ≤ 5/2} . Theorem 2.2 applied to χw when Ω is substituted by U and g(t) = 1/(1 -t 2 ) gives, for λ ≥ λ 0 and τ ≥ τ 0 , C Q(2)\Q(1) λ 4 τ 3 ϕ 3 w 2 + λ 2 τ ϕ|∇w| 2 e -2τ ϕ dxdt (2.21) ≤ Q(3) (L r (χw)) 2 e -2τ ϕ dxdt, the constant C only depends on κ. But supp(L r (χw)) ⊂ [{1/2 ≤ |x| ≤ 1} ∪ {5/2 ≤ |x| ≤ 3}] × (-1, 1) and (L r (χw)) 2 ≤ Λ(w 2 + |∇w| 2 )
, where Λ = Λ(r 0 ) is independent on r. Therefore, fixing λ and changing τ 0 if necessary, (2.21) implies, for τ ≥ τ 0 ,

C Q(2) w 2 + |∇w| 2 e -2τ ϕ dxdt ≤ Q(1)
w 2 + |∇w| 2 e -2τ ϕ dxdt (2.22) + Q(3)\Q(5/2) w 2 + |∇w| 2 e -2τ ϕ dxdt.
Let 0 < ρ < 1 to be specified later and choose ψ(x) = -|x| 2 in (2.22) (which is without critical points in U ). In that case ϕ(x, t) = g(t) e 36λ -e λ(18-|x| 2 ) .

We have

ϕ(x, t) ≤ g(-1 + ρ) e 36λ -e 14λ ≤ 1 ρ e 36λ -e 14λ = α ρ , (x, t) ∈ B(2) × (-1 + ρ, 1 -ρ), ϕ(x, t) ≥ g(0) e 36λ -e 18λ = e 36λ -e 18λ = β, (x, t) ∈ Q(1), ϕ(x, t) ≥ g(0) e 36λ -e 47 4 λ = e 36λ -e 47 4 λ = γ, (x, t) ∈ Q(3) \ Q (5/2) . As β α < 1 < γ α , we can fix θ ∈ (0, 1) so that 1 ρ := θ β α + (1 -θ) γ α > 1. Set a = 2(1 -θ)(γ -β) and b = 2θ(γ -β) and Q(2) = B(0, 2) × (-1 + ρ, 1 -ρ). Then (2.22) yields C Q(2)
w 2 + |∇w| 2 dxdt ≤ e aτ Q(1)
w 2 + |∇w| 2 dxdt + e -bτ Q(3) w 2 + |∇w| 2 dxdt.
Similarly to the elliptic case [3, Theorem 2.17 and its proof, pages 19 to 21] (see also the proof of Proposition 4.1), we obtain from this inequality the following one

C w L 2 ((-1+ρ,1-ρ),H 1 (B(2))) ≤ w ϑ L 2 ((-1,1),H 1 (B(1))) w 1-ϑ L 2 ((-1,1),H 1 (B(3)))
, with ϑ = a a+b . We get by making a change of variable, where τ = 1 -ρ,

r u L 2 ((s-τ r 2 ,s+τ r 2 ),H 1 (B(y,2r))) (2.23) ≤ C u ϑ L 2 ((s-r 2 ,s+r 2 ),H 1 (B(y,r))) u 1-ϑ L 2 ((s-r 2 ,s+r 2 ),H 1 (B(y,3r))) .
Here and until the end of this proof, the generic constant C only depends on Ω, κ and

T 0 . Fix 0 < < (t 1 -t 0 )/2. Let s 0 = t 0 + and s k = s k-1 + 2τ r 2 , k ≥ 1, in such a way that (s k-1 , s k ) = ((s k-1 + τ r 2 ) -τ r 2 , (s k-1 + τ r 2 ) + τ r 2 ).
We consider q the smallest integer so that (t 1 -

) -s q-1 ≤ 2τ r 2 or equivalently (t 1 -) -s q-2 > 2τ r 2 . Whence (2.24) q < t 1 -t 0 -2 2τ r 2 + 3 < δ 2τ r 2 + 3diam(Ω) 2 r 2 = T 0 2τ + 3diam(Ω) 2 1 r 2 . Let r < r y ( ) = min (dist(y, Γ)/3, √ ). It follows from (2.23) that r u L 2 ((s k-1 ,s k ),H 1 (B(y,2r))) ≤ C u ϑ L 2 ((t0,t1),H 1 (B(y,r))) u 1-ϑ L 2 ((t0,t1),H 1 (B(y,3r))) , with s q = t 1 -. Thus r q k=1 u L 2 ((s k-1 ,s k ),H 1 (B(y,2r))) ≤ Cq u ϑ L 2 ((t0,t1),H 1 (B(y,r))) u 1-ϑ L 2 ((t0,t1),H 1 (B(y,3r))) . In consequence r u L 2 ((t0+ ,t1-),H 1 (B(y,2r))) (2.25) ≤ Cq u ϑ L 2 ((t0,t1),H 1 (B(y,r))) u 1-ϑ L 2 ((t0,t1),H 1 (B(y,3r))) . Estimate (2.24) in (2.25) yields Cr 3 u L 2 ((t0+ ,t1-),H 1 (B(y,2r))) ≤ u ϑ L 2 ((t0,t1),H 1 (B(y,r))) u 1-ϑ L 2 ((t0,t1
),H 1 (B(y,3r))) , r < r y ( ). The proof is then complete.

Quantifying the uniqueness of continuation from an interior data

We start with a Hardy inequality for vector valued functions. Lemma 3.1. Let X be a Banach space with norm • and s ∈ (0, 1/2). There exists a constant c > 0 so that, for any u ∈ H s ((t 0 , t 1 ), X), we have

u δ s L 2 ((t0,t1),X)
≤ c u H s ((t0,t1),X) .

Here

δ = δ(t) = min{|t -t 0 |, |t -t 1 |}.
Proof. Let u ∈ H s ((t 0 , t 1 ), X). From the usual Hardy's inequality in dimension one (see for instance [START_REF] Grisvard | Elliptic Problems in Nonsmooth Domains[END_REF]) we have

(3.1) t1 t0 u(t) 2 δ 2s (t) dt ≤ c u(•) 2 H s ((t0,t1)) . But u(•) 2 H s ((t0,t1)) = u 2 L 2 ((t0,t1),X) + t1 t0 t1 t0 | u(τ ) -u(t) | 2 |τ -t| 1+2s dtdτ ≤ u 2 L 2 ((t0,t1),X) + t1 t0 t1 t0 u(τ ) -u(t) 2 |τ -t| 1+2s dtdτ = u 2 H s ((t0,t1),X) .
Whence the result follows.

In the rest of this paper we shall often apply Hardy's inequality in Lemma 3.1 to functions from H k ((t 0 , t 1 ), H), where k ≥ 1 is an integer and H is a Hilbert space. This is made possible by [11, Remark 9.5, page 46] saying that H s ((t 0 , t 1 ), H), 0 < s < 1, can be seen as an interpolated space between L 2 ((t 0 , t 1 ), H) and H 1 ((t 0 , t 1 ), H). Precisely, we have

H s ((t 0 , t 1 ), H) = [L 2 ((t 0 , t 1 ), H), H 1 ((t 0 , t 1 ), H)] 1-s , 0 < s < 1.
We readily obtain from Lemma 3.1 the following corollary. Corollary 3.1. Let H be a Hilbert space and s ∈ (0, 1/2). There exists a constant c > 0 so that, for any u ∈ H 1 ((t 0 , t 1 ), H), we have

u δ s L 2 ((t0,t1),H) ≤ c u H 1 ((t0,t1),H) ,
where δ is as in Lemma 3.1.

Next, we prove Proposition 3.1. Let s ∈ (0, 1/2). There exist ω Ω, only depending on Ω, and three constants c > 0, C > 0 and σ 0 > 0, only depending on Ω, κ, T 0 , s and α, so that, for any u ∈ X (Q) satisfying Lu = 0 in Q and 0 < σ < σ 0 , we have

u L 2 ((t0,t1),L 2 (Γ)) + ∇u L 2 ((t0,t1),L 2 (Γ)) (3.2) ≤ C σ min(α,s)/2 u X (Q) + e e c/ √ σ u L 2 ((t0,t1),H 1 (ω)) .
Proof. Since Ω is Lipschitz, it has the uniform interior cone property (see for instance [START_REF] Henrot | Variation et optimisation de formes[END_REF]). That is there exist R > 0 and θ ∈ 0, π 2 so that, for any x ∈ Γ, we may find ξ = ξ(x) ∈ S n-1 for which

C(x) = {x ∈ R n ; |x -x| < R, (x -x) • ξ > |x -x| cos θ} ⊂ Ω. Fix x ∈ Γ and let ξ = ξ(x) be as in the definition above. Let 0 < ≤ 0 < min ((3R/(2 sin θ)) 2 , (t 1 -t 0 )/4 , set y 0 = y 0 (x) = x + (R/2)ξ and ρ = √ sin θ/3. Let N = [R/(2ρ)], the integer part of R/(2ρ), if R/(2ρ) ∈ N and N = R/(2ρ) -1 if R/(2ρ) ∈ N. Define then x 0 = x + (R/2 -N ρ)ξ.
Furthermore, consider the sequence

y j = x + (R/2 -jρ)ξ, 0 ≤ j ≤ N.
By construction, B(y j , 3ρ) ⊂ C(x), 0 ≤ j ≤ N and, as |y j+1 -y j | = ρ, we have

B(y j+1 , ρ) ⊂ B(y j , 2ρ), 0 ≤ j ≤ N -1.
Let u ∈ X (Q). We use in the sequel the temporary notation

M = M (u) = u X (Q) .
Set I j = (t j 0 , t j 1 ), where t j i = t i + (-1) i j , with i = 0, 1 and 0

≤ j ≤ N . Note that N ≤ (R/(2ρ)) = (3R/(2 sin θ)) √ . Then I N = ∅ if (3R/ sin θ) √ < t 1 -t 0 .
This condition always holds provided that we substitute 0 by min( 0 , (t 1 -t 0 ) 2 sin 2 θ/(9R 2 )).

In the rest of this proof C is a generic constant only depending on Ω, κ, α, s and T 0 .

Using that I j+1 = (t j 0 + , t j 1 -) and noting that ρ < √ , we get from (2.1)

ρ 3 u L 2 (Ij+1,H 1 (B(yj ,ρ))) ≤ CM 1-ϑ u ϑ L 2 (Ij ,H 1 (B(xj ,ρ))) , 0 ≤ j ≤ N -1.
the constant ϑ, only depending on Ω, κ and T 0 , satisfies 0 < ϑ < 1.

Whence

u L 2 (I N ,H 1 (B(y N ,ρ))) ≤ (Cρ -3 ) (1-β)/(1-ϑ) M 1-β u β L 2 (I0,H 1 (B(y0,ρ))) , with β = ϑ N +1 .
In this inequality, modifying C if necessary, we may assume that Cρ

-3 ≥ 1. Thus u L 2 (I N ,H 1 (B(y N ,ρ))) ≤ Cρ -3/(1-ϑ) M 1-β u β L 2 (I0,H 1 (B(y0,ρ))
) , Let J = I N . Since B(y 0 , ρ) ⊂ B(y 0 , R sin θ/6) ⊂ C(x) and y N = x 0 , the last inequality entails

(3.3) u L 2 (J,H 1 (B(x0,ρ))) ≤ Cρ -3/(1-ϑ) M 1-β u β L 2 (I0,H 1 (B(y0,R sin θ/6))) . Define ω = ỹ∈Γ B(y 0 (ỹ), R sin θ/6).
It is worth mentioning that ω only depends on Ω. We get from (3.27)

(3.4) u L 2 (J,H 1 (B(x0,ρ))) ≤ Cρ -3/(1-ϑ) M 1-β u β L 2 (I0,H 1 (ω))) . Now, since u is Hölder continuous, we have |u(x, t)| ≤ [u] α |x -x| α + |u(x, t)|, x ∈ B(x 0 , ρ), t ∈ J.
Here and henceforth

[w] α = sup (x1,t1),(x2,t2)∈Q (x 1 ,t 1 ) =(x 2 ,t 2 ) |w(x 1 , t 1 ) -w(x 2 , t 2 )| |x 1 -x 2 | α + |t 1 -t 2 | α/2 , w ∈ C α,α/2 (Q). Whence |S n-1 |ρ n J |u(x, t)| 2 dt ≤ 2n[u] 2 α B(x0,ρ)×J |x -x| 2α dxdt (3.5) + 2n B(x0,ρ)×J |u(x, t)| 2 dxdt, Similarly, where 1 ≤ i ≤ n, |S n-1 |ρ n J |∂ i u(x, t)| 2 dt ≤ 2n[∂ i u] 2 α B(x0,ρ)×J |x -x| 2α dxdt (3.6) + 2n B(x0,ρ)×J |∂ i u(x, t)| 2 dxdt.
We have

(3.7) |x -x| ≤ |x -x 0 | + |x 0 -x| ≤ ρ + (R/2 -N ρ) ≤ 2ρ.
Therefore, we have as a consequence of a combination of (3.5), (3.6) and (3.7)

J |u(x, t)| 2 dt + J |∇u(x, t)| 2 dt ≤ C M 2 ρ 2α + ρ -n u L 2 (J,H 1 (B(x0,ρ)))
which, in light of (3.28), yields

J |u(x, t)| 2 dt + J |∇u(x, t)| 2 dt ≤ C M 2 ρ 2α + ρ -n-6/(1-ϑ) M 2(1-β) u 2β L 2 (I0,H 1 (ω)) .
Integrating over Γ both sides of this inequality with respect to x, we find

u L 2 (Γ×J) + ∇u L 2 (Γ×J) ≤ (3.8) C M ρ α + ρ -n/2-3/(1-ϑ) 0 M 1-β u β L 2 (I0,H 1 (ω)) .
Bearing in mind that J = (t 0 + N , t 1 + N ), we get by applying Corollary 3.1, for some fixed s ∈ (0, 1/2),

u L 2 ((t0,t0+N ),L 2 (Γ)) , u L 2 ((t1-N ,t1),L 2 (Γ)) ≤ c(N ) s u H 1 ((t0,t1),L 2 (Γ)) , ∇u L 2 ((t0,t0+N ),L 2 (Γ)) , ∇u L 2 ((t1-N ,t1),L 2 (Γ)) ≤ c(N ) s ∇u H 1 ((t0,t1),L 2 (Γ)) .
Therefore, as the trace operator

u ∈ H 1 ((t 0 , t 1 ), H 2 (Ω)) → (u, ∇u) |Σ ∈ H 1 ((t 0 , t 1 ), L 2 (Γ)) n+1 , is bounded, we obtain u L 2 ((t0,t0+N ),L 2 (Γ)) , u L 2 ((t1-N ,t1),L 2 (Γ)) ≤ c(N ) s u H 1 ((t0,t1),H 2 (Ω)) , ∇u L 2 ((t0,t0+N ),L 2 (Γ)) , ∇u L 2 ((t1-N ,t1),L 2 (Γ)) ≤ c(N ) s u H 1 ((t0,t1),H 2 (Ω)) .
These inequalities together with (3.8) give

C u L 2 (Σ) + ∇u L 2 (Σ) ≤ (ρ α + (N ) s )M + ρ -n/2-3/(1-ϑ) M 1-β u β L 2 (I0,H 1 (ω))
. We obtain by applying Young's inequality to the last term

C u L 2 (Σ) + ∇u L 2 (Σ) ≤ (ρ α + (N ) s + α/2 )M (3.9) + ρ -(n/2+3/(1-ϑ))/β -(1-β)α/(2β) u L 2 (I0,H 1 (ω)) .
Next, we have (3.10)

ρ α + (N ) s + α/2 ≤ C min(α,s)/2
and, as β = θ N +1 , we have β = O(e -c/ √

), from which we deduce in a straightforward manner that

(3.11) ρ -(n/2+3/(1-ϑ))/β -(1-β)α/(2β) ≤ Ce e c √ .
We end up by observing that (3.10) and (3.11) in (3.9) give the expected inequality.

The a priori estimate in the following lemma is well adapted to our purpose. It does not involve neither the initial time data nor the final time data.

Lemma 3.2.

There exists a constant C > 0, only depending on Ω, κ and T 0 , so that, for any u ∈ H 1 ((t 0 , t 1 ), H 2 (Ω)) satisfying Lu = 0 in Q, we have

(3.12) C u L 2 ((t0,t1),H 1 (Ω)) ≤ u H 1 ((t0,t1),L 2 (Γ)) + ∇u L 2 ((t0,t1),L 2 (Γ)) .
Proof. In this proof C is a generic constant that can only depend on Ω, κ and T 0 . Let u ∈ H 1 ((t 0 , t 1 ), H 2 (Ω)) satisfying Lu = 0 in Q and set v = e -t u. Then v solves the following equation

(3.13) div(A∇v) -v -∂ t v = 0 in Q.
Let 0 < < (t 1 -t 0 )/2 and choose χ ∈ C ∞ 0 ((t 0 , t 1 )) satisfying 0 ≤ χ ≤ 1, χ = 1 in (t 0 + , t 1 -) and, for some universal constant c, |χ | ≤ c/ .

We multiply (3.13) by χv and integrate over Q. We then get by making an integration by parts

- Q χA∇v • ∇vdxdt - Q χv 2 dxdt + Σ χvA∇v • νdσdt + 1 2 Q v 2 χ dxdt = 0,
from which we deduce in a straightforward manner

Q χA∇u • ∇udxdt + Q χu 2 dxdt ≤ (3.14) e 2(t1-t0) C Σ (u 2 + |∇u| 2 )dσdx + 1 2 Q u 2 |χ |dxdt .
On the other hand, as supp(χ ) ⊂ (t 0 , t 1 ) \ (t 0 + , t 1 -), we have

J 2 = Q u 2 |χ |dxdt ≤ c t0- t0 Ω u 2 dxdt + c t1 t1- Ω u 2 dxdt. Therefore (3.15) lim sup →0 J 2 ≤ Ω u 2 (x, t 0 )dx + Ω u 2 (x, t 1 )dx.
We rewrite (3.14) in the form

(3.16) C u L 2 ((t0+ ,t1-),H 1 (Ω)) ≤ u L 2 ((t0,t1),L 2 (Γ)) + ∇u L 2 ((t0,t1),L 2 (Γ)) + J .
We apply Hardy's inequality in Corollary 3.1. We obtain

u L 2 ((t0,t0+ ),H 1 (Ω)) , u L 2 ((t1-,t1),H 1 (Ω)) ≤ C s u H 1 ((t0,t1),H 1 (Ω)) .
This and (3.16) produce

C u L 2 ((t0,t1),H 1 (Ω)) ≤ u L 2 ((t0,t1),L 2 (Γ)) + ∇u L 2 ((t0,t1),L 2 (Γ)) + s u H 1 ((t0,t1),H 1 (Ω)) + J .
Making → 0, we get by using (3.15)

C u L 2 ((t0,t1),H 1 (Ω)) ≤ u L 2 ((t0,t1),L 2 (Γ)) + ∇u L 2 ((t0,t1),L 2 (Γ)) + u(•, t 0 ) L 2 (Ω) + u(•, t 1 ) L 2 (Ω) .
We complete the proof by using the following inequality

C u(•, t 0 ) L 2 (Ω) + u(•, t 1 ) L 2 (Ω) ≤ u H 1 ((t0,t1),L 2 (Γ)) + ∇u L 2 ((t0,t1),L 2 (Γ)) .
To prove this inequality we proceed similarly to the proof of observability inequalities for parabolic equation. First, if s 0 = (3t 0 + t 1 )/4 and s 1 = (t 0 + 3t 1 )/4, we get as a straightforward consequence of the Carleman inequality in Theorem 2.3,

(3.17) C u L 2 ((s0,s1)×Ω) ≤ u H 1 ((t0,t1),L 2 (Γ)) + ∇u L 2 ((t0,t1),L 2 (Γ)) . Next pick ψ ∈ C ∞ ([t 0 , t 1 ]) so that 0 ≤ ψ ≤ 1, ψ = 0 in [t 0 , s 0 ] and ψ = 1 in [s 1 , t 1 ]. Then v = ψu is the solution of the IBVP    -div(A∇v) + ∂ t v = ψ u in Q, v = u on Σ, v(•, t 0 ) = 0. Hence Q A∇v • ∇vdxdt + Σ v(A∇v • ν) + 1 2 Q ∂ t v 2 dxdt = Q ψ uvdxdt. But Q ∂ t v 2 dxdt = Ω v 2 (x, t 1 )dx.
That is we have

Q A∇v • ∇vdxdt + Σ vA∇v • ν + 1 2 Ω v 2 (x, t 1 )dx = Q ψ uvdxdt.
We deduce from this identity

C v(•, t 1 ) 2 L 2 (Ω) + ∇v 2 L 2 (Q) (3.18) ≤ u 2 L 2 ((t0,t1),L 2 (Γ)) + ∇u 2 L 2 ((t0,t1),L 2 (Γ)) + ψ u L 2 (Q) v L 2 (Q) . Noting that w → Ω |∇w| 2 dx + Γ w 2 (x)dσ(x) 1 2
defines an equivalent norm on H 1 (Ω), we get

v 2 L 2 (Q) ≤ c Ω ∇v 2 L 2 (Q) + u 2 L 2 ((t0,t1),L 2 (Γ)) .
We obtain then from Young's inequality

ψ u L 2 (Q) v L 2 (Q) ≤ 1 2 ψ u 2 L 2 (Q) + c Ω 2 ∇v 2 L 2 (Q) + c Ω 2 u 2 L 2 ((t0,t1),L 2 (Γ)) .
This inequality in (3.18), with sufficiently small, yields

C u(•, t 1 ) L 2 (Ω) = C v(•, t 1 ) L 2 (Ω) (3.19) ≤ u L 2 ((t0,t1),L 2 (Γ)) + ∇u L 2 ((t0,t1),L 2 (Γ)) + ψ u L 2 (Q) .
Bearing in mind that supp(ψ ) ⊂ [s 0 , s 1 ], we deduce from (3.17)

ψ u L 2 (Q) ≤ C u L 2 (Ω×(s0,s1)) ≤ u H 1 ((t0,t1),L 2 (Γ)) + ∇u L 2 ((t0,t1),L 2 (Γ)) . This in (3.19) yields C u(•, t 1 ) L 2 (Ω) ≤ u H 1 ((t0,t1),L 2 (Γ)) + ∇u L 2 ((t0,t1),L 2 (Γ)) .
As the Carleman estimate in Theorem 2.3 still holds for the backward parabolic equation div(A∇u) + ∂ t u = 0, we have similarly

C u(•, t 0 ) L 2 (Ω) ≤ u H 1 ((t0,t1),L 2 (Γ)) + ∇u L 2 ((t0,t1),L 2 (Γ)) .
The proof is then complete.

If u ∈ Y (Q) satisfies Lu = 0 in Q then ∂ t u ∈ X (Q) and L∂ t u = 0 in Q.
Proposition 3.1 applied to both u and ∂ t u together with Lemma 3.2 produce the following result. Corollary 3.2. Let s ∈ (0, 1/2). There exist ω Ω, only depending on Ω, and three constants c > 0, C > 0 and σ 0 > 0, only depending on Ω, κ, T 0 , s and α, so that, for any u ∈ Y (Q) satisfying Lu = 0 in Q and 0 < σ < σ 0 , we have

(3.20) u L 2 ((t0,t1),H 1 (Ω)) ≤ C σ min(α,s)/2 u Y (Q) + e e c/ √ σ u H 1 ((t0,t1),H 1 (ω)) .
We now quantify the uniqueness of continuation from an interior subdomain to an another interior subdomain. Prior to that, we define the geometric distance d D g on a bounded domain D of R n by

d D g (x, y) = inf { (ψ); ψ : [0, 1] → D Lipschitz path joining x to y} , where (ψ) = 1 0 ψ(t) dt is the length of ψ.
Observe that, according to Rademacher's theorem, any Lipschitz continuous function ψ :

[0, 1] → D is almost everywhere differentiable with ψ(t) ≤ k a.e. t ∈ [0, 1],
where k is the Lipschitz constant of ψ. In particular, (ψ) is well defined.

The following lemma will be used to prove the next proposition. We provide its proof in Appendix A. Let ω Ω, ω Ω and s ∈ (0, 1/2). There exist three constants γ > 0, C > 0 and 0 > 0, only depending on Ω, κ, T 0 , s, ω and ω, so that, for any u ∈ H 2 ((t 0 , t 1 ), H 1 (Ω)) satisfying Lu = 0 in Q and 0 < < 0 , we have

(3.21) C u H 1 ((t0,t1),H 1 (ω)) ≤ s u H 2 ((t0,t1),H 1 (Ω)) + e γ/ u H 1 ((t0,t1),H 1 (ω)) .
Proof. Pick Ω 0 a Lipschitz domain so that Ω 0 Ω, ω Ω 0 and ω Ω 0 . Set then d 0 = dist(Ω 0 , Γ). Fix 0 < < 0 := min d 2 0 /9, 1 and let 0 < δ < √ . Let x 0 ∈ ω, x ∈ ω and let ψ : [0, 1] → Ω 0 be a Lipschitz path joining x 0 to x so that (ψ) ≤ d Ω0 g (x 0 , x) + 1. For simplicity's sake, we use in this proof the notation

d = d Ω0 g L ∞ (Ω0×Ω0) . Let τ 0 = 0 and τ k+1 = inf{τ ∈ [τ k , 1]; ψ(τ ) ∈ B(ψ(τ k ), δ)}, k ≥ 0.
We claim that there exists an integer N ≥ 1 so that ψ(1) ∈ B (ψ(τ N ), δ/2). If not, we would have ψ(1) ∈ B (ψ(τ k ), δ/2), for any k ≥ 0. As the sequence (τ k ) is non decreasing and bounded from above by 1, it converges to τ ≤ 1. In particular, there exists an integer k 0 ≥ 1 so that ψ(t k ) ∈ B (ψ(τ ), δ/2), k ≥ k 0 . But this contradicts the fact that |ψ(τ k+1 ) -ψ(τ k )| = δ, for any k ≥ 0.

Let us check that N ≤ N 0 , where N 0 depends only on d and δ. Pick 1 ≤ j ≤ n so that max

1≤i≤n |ψ i (τ k+1 ) -ψ i (τ k )| = |ψ j (τ k+1 ) -ψ j (τ k )| . Then δ ≤ n |ψ j (τ k+1 ) -ψ j (τ k )| = n τ k+1 τ k ψj (t)dt ≤ n τ k+1 τ k ψ(t) dt.
In consequence, where τ N +1 = 1,

(N + 1)δ ≤ n N k=0 τ k+1 τ k ψ(τ ) dτ = n (ψ) ≤ n(d + 1).
Therefore

N ≤ N 0 = n(d + 1) δ .
Here

[n(d + 1)/δ] is the integer part of n(d + 1)/δ. Let x k = ψ(t k ), 0 ≤ k ≤ N . If |z -x k+1 | < δ then |z -x k | ≤ |z -x k+1 | + |x k+1 -x k | < 2δ.
In other words, B(x k+1 , δ) ⊂ B(x k , 2δ).

Remark that we have also

B(x k , 3δ) ⊂ Ω, for each k. Define I k = (t 0 + k , t 1 -k ) k ≥ 0. If t k 0 = t 0 + k and t k 1 = t 1 -k then I k = (t k 0 , t k 1 ) and I k+1 = (t k 0 + , t k 1 -). Let u ∈ H 2 ((t 0 , t 1 ), H 1 (Ω) satisfying Lu = 0 in Q.
In this proof we use the following temporary notation

M = u H 1 ((t0,t1),H 1 (Ω)) .
Taking into account that δ < √ , we have from the three-cylinder inequality (2.1)

u L 2 (I k+1 ,H 1 (B(x k+1 ,δ))) ≤ C 0 δ -3 M 1-ϑ u ϑ L 2 (I k ,H 1 (B(x k ,δ)))
, the constants C 0 and ϑ, 0 < ϑ < 1, only depend on Ω, κ and T 0 .

Set

Λ k = u L 2 (I k ,H 1 (B(x k ,δ))) , 0 ≤ k ≤ N , and Λ N +1 = u L 2 (I N +1 ,H 1 (B(x,δ/2)))
. We can then rewrite this inequality in the form

(3.22) Λ k+1 ≤ C 0 δ -3 M 1-ϑ Λ ϑ k . Let β = ϑ N +1 . We get in a straightforward manner from (3.22) Λ N +1 ≤ (C 0 δ -3 ) 1-ϑ N +2 1-ϑ M 1-β Λ β 0 .
Substituting if necessary C 0 by max(C 0 , 1), we may assume that C 0 ≥ 1. Then the last inequality gives

Λ N +1 ≤ Cδ -3 1-ϑ M 1-β Λ β 0 .
From here and until the end of the proof, C is a generic constant, depending only on Ω, κ, ω, ω, s and T 0 .

Young's inequality then leads, for σ > 0,

Λ N +1 ≤ Cδ -3 1-ϑ ((1 -β)σ β 1-β M + βσ -1 Λ 0 ) ≤ Cδ -3 1-ϑ (σ β 1-β M + σ -1 Λ 0 ).
If δ is sufficiently small B(x 0 , δ) ⊂ ω. On the other hand, ω can be recovered by O(δ -n ) balls of radius δ 2 . Whence, bearing in mind that

I N +1 ⊃ J = (t 0 + (N 0 + 1) , t 1 -(N 0 + 1) ), we have (3.23) u L 2 (J,H 1 (ω)) ≤ δ -3 1-ϑ -n σ β 1-β M + σ -1 u L 2 ((t0,t1),H 1 (ω)) .
Then we take σ in (3.23) in order to satisfy

δ -3 1-ϑ -n σ β 1-β = δ.
In that case

σ -1 ≤ δ -m β , with m = (4 -ϑ)/(1 -ϑ) + n. But β = ϑ N +1 = e -(N +1)| ln ϑ| ≥ ϑe -N0| ln ϑ| ≥ ϑe -n| ln ϑ|(d+1) δ . Hence δ -3 1-ϑ -n σ -1 ≤ e γ/δ . This inequality in (3.23) yields (3.24) C u L 2 (J,H 1 (ω)) ≤ δM + e γ/δ u L 2 ((t0,t1),H 1 (ω)) .
Let ℵ = 2n(d + 1) + √ 0 . Substituting 0 by min( 0 , (t 1 -t 0 )/(3ℵ)), we may assume that J 0 = (t 0 -ℵ , t 1 +ℵ ) = ∅. Taking δ = √ /2, we get in straightforward manner that J ⊃ J 0 . Hence (3.24) yields

(3.25) C u L 2 (J0,H 1 (ω)) ≤ √ M + e 2γ/ √ u L 2 ((t0,t1),H 1 (ω)) .
We get by applying again Hardy's inequality in Lemma 3.1, for some fixed s ∈ (

) u L 2 ((t0,t0+ℵ ),H 1 (ω)) , u L 2 ((t1-ℵ ,t1),H 1 (ω)) ≤ cℵ s s M. 0, 1/2), (3.26
Then (3.25) and (3.26) entail

C u L 2 ((t0,t1),H 1 (ω)) ≤ s/2 M + e 2γ/ √ u L 2 ((t0,t1),H 1 (ω)) .
Substituting by 2 and 2γ by γ, we obtain

(3.27) C u L 2 ((t0,t1),H 1 (ω)) ≤ s u H 1 ((t0,t1),H 1 (Ω)) + e γ/ u L 2 ((t0,t1),H 1 (ω)) . As ∂ t u ∈ H 1 ((t 0 , t 1 ), H 1 (Ω) satisfies L∂ t u = 0 in Q, (3.27) is applicable with u substituted by ∂ t u. That is we have (3.28) C ∂ t u L 2 ((t0,t1),H 1 (ω)) ≤ s ∂ t u H 1 ((t0,t1),H 1 (Ω)) +e γ/ ∂ t u L 2 ((t0,t1),H 1 (ω)) .
Putting together (3.27) and (3.28) to obtain the expected inequality.

We are now ready to prove the result quantifying the uniqueness of continuation from an interior data. Prior to do that, we need to introduce a definition. Set * = e -e and, for µ > 0 and 0 ≤ * ,

Φ 0 ,µ ( ) =    0 if = 0, (ln ln | ln |) -µ if 0 < ≤ 0 , if ≥ 0 . Theorem 3.1.
Let ω Ω and s ∈ (0, 1/2). There exist two constants C > 0 and 0 < 0 ≤ * , only depending on Ω, κ, ω, α, s and T 0 , so that, for any u ∈ Y (Q) satisfying Lu = 0 in Q, we have

C u L 2 ((t0,t1),H 1 (Ω)) ≤ u Y (Q) Φ 0,µ I(u, ω) u Y (Q) .
Here µ = min(s, α)/4 and I(u, ω) = u H 1 ((t0,t1),H 1 (ω)) .

Proof. From Corollary 3.2, there exist ω Ω and three constants c > 0, C > 0 and σ 0 > 0 so that, for any u ∈ Y (Q) satisfying Lu = 0 in Q and 0 < σ < σ 0 , we have

C u L 2 ((t0,t1),H 1 (Ω)) ≤ σ µ u Y (Q) + e e c/ √ σ u H 1 ((t0,t1),H 1 (ω)) .
Here µ = min(s, α)/2. But according to Proposition 3.2, there exist three constants γ > 0, C > 0 and 0 > 0 so that, for any u ∈ X 1 (Q) satisfying Lu = 0 in Q and 0 < < 0 , we have

C u H 1 ((t0,t1),H 1 (ω)) ≤ s u Y (Q) + e γ/ u H 1 ((t0,t1),H 1 (ω)) .
The last two inequalities yield

C u L 2 ((t0,t1),H 1 (Ω)) ≤ (σ µ + s e e c/ √ σ ) u Y (Q) (3.29) + e γ/ e e c/ √ σ u H 1 ((0,T ),H 1 (ω)) ,
for any u ∈ Y (Q) satisfying Lu = 0 in Q, 0 < σ < σ 0 and 0 < < 0 .

We assume, by reducing σ 0 if needed, that σ µ 0 e -e c/ √ σ 0 < s 0 . We get, by taking so that s = σ µ e -e c/ √ σ ,

(3.30) C u L 2 ((t0,t1),H 1 (Ω)) ≤ σ µ u Y (Q) + e e e c/ √ σ u H 1 ((t0,t1),H 1 (ω)) , for any u ∈ Y (Q) satisfying Lu = 0 in Q, and 0 < σ < σ 0 . Fix u ∈ Y (Q), non identically equal to zero, satisfying Lu = 0 in Q. Let M = u L 2 ((t0,t1),H 1 (Ω)) u Y (Q) and N = u H 1 ((t0,t1),H 1 (ω)) u Y (Q) .
Then (3.30) can be rewritten as If N ≥ 0 then obviously we have

(3.33) M ≤ 1 ≤ N 0 .
The expected inequality follows then from (3.32) and (3.33).

Stability of parabolic Cauchy problems

An additional step is necessary to prove our stability estimate for the Cauchy problem. It consists in quantifying the uniqueness of continuation from the Cauchy data to an interior subdomain. Proposition 4.1. Let ν ∈ (0, 1/2). There exist ω Ω, only depending on Ω and Γ 0 , and two constants C > 0 and c > 0, only depending on Ω, κ, Γ 0 , ν and T 0 , so that, for any u ∈ H 2 ((t 0 , t 1 ), H 2 (Ω)) satisfying Lu = 0 in Q and 0 < < (t 1 -t 0 )/2, we have

C u H 1 ((t0,t1),L 2 (ω)) ≤ ν u H 2 ((t0,t1),H 1 (Ω)) (4.1) + e c/ 2 u H 2 ((t0,t1),L 2 (Γ0)) + ∇u H 1 ((t0,t1),L 2 (Γ0)) .
Proof. Pick 0 < < (t 1 -t 0 )/2, 0 < η < and let s ∈ [t 0 + , t 1 -]. Let x ∈ Γ 0 be arbitrarily fixed and let R > 0 so that B(x, R) ∩ Γ ⊂ Γ 0 . Take x 0 in the interior of R n \ Ω sufficiently close to x is such a way that ρ = dist(x 0 , K) < R, where K = B(x, R) ∩ Γ 0 (think to the fact that Ω is on one side of its boundary). Fix then r > 0 in order to satisfy B(x 0 , ρ + r) ∩ Γ ⊂ Γ 0 and B(x 0 , ρ + θr) ∩ Ω = ∅, for some 0 < θ < 1.

Let φ ∈ C ∞ 0 (B(x, ρ + r)) satisfying φ = 1 on B(x, ρ + (θ + 1)r/2). Set, where 0 < δ < 1 is a constant to be specified in the sequel,

Q 0 = [B(x 0 , ρ + r) ∩ Ω] × (s -η, s + η), Q 1 = [B(x 0 , ρ + θr) ∩ Ω] × (s -δη, s + δη), Q 2 = {[B(x 0 , ρ + r) \ B(x 0 , ρ + (θ + 1)r/2)] ∩ Ω} × (s -η, s + η), Σ 0 = [B(x 0 , ρ + r) ∩ Γ] × (s -η, s + η).
We apply Theorem 2.3, with Q substituted by Q 0 , ψ = (ρ + r) 2 -|x -x 0 | 2 , g(t) = 1/[(t -s + η)(s + η -t)] and λ fixed, to φu so that u ∈ H 1 ((t 0 , t 1 ), H 2 (Ω)) satisfies Lu = 0 in Q in order to obtain

C Q1 u 2 e -2τ ϕ dxdt ≤ Q0 (L(φu)) 2 e -2τ ϕ dxdt (4.2) + Σ0 (u 2 + |∇u| 2 + (∂ t u) 2 )e -2τ ϕ dxdt.
Here and henceforth, C is a generic constant that can only depend on Ω, κ, ν, Γ 0 and T 0 .

But

L(φu) = Lφu + 2A∇φ • ∇u. Whence supp(L(φu)) ∩ Q 0 ⊂ Q 2 together with (4.2) yield C Q1 u 2 e -2τ ϕ dxdt ≤ Q2 (u 2 + |∇u| 2 )e -2τ ϕ dxdt (4.3) + Σ0 (u 2 + |∇u| 2 + (∂ t u) 2 )e -2τ ϕ dxdt. Let α = η -2 e 4λ(ρ+r) 2 -e λ(2(ρ+r) 2 -(ρ+θr) 2 ) := η -2 α, β = η -2 e 4λ(ρ+r) 2 -e 2λ(ρ+r) 2 := η -2 β, γ = η -2 e 4λ(ρ+r) 2 -e λ(2(ρ+r) 2 -(ρ+(θ+1)r/2) 2 ) := η -2 γ.
Then it is straightforward to check that

ϕ(x, t) ≤ α 1 -δ in Q 1 , ϕ(x, t) ≥ β in Σ 0 , ϕ(x, t) ≥ γ in Q 2 .
Noting that

β α = β α < 1 < γ α = γ α , we can choose 0 < κ < 1 so that 1 1 -δ := κ β α + (1 -κ) γ α > 1. With this choice of δ, (4.3) yields C Q1 u 2 dxdt ≤ e -4bη -2 τ Q2 (u 2 + |∇u| 2 )dxdt (4.4) + e 4aη -2 τ Σ0 (u 2 + |∇u| 2 + (∂ t u) 2 )dxdt. Here a = (1 -κ)(γ -β)/2 and b = κ(γ -β)/2. Let η = /2, s 0 = t 0 + -δ /2, s 1 = s 0 + δ /2 . . . s k = s 0 + kδ /2. Let K = K( ) so that K k=0 (s k -δ /2, s k + δ /2) ⊃ [t 0 + , t 1 -]. If Q k j (resp. Σ k 0 ) denotes Q j (resp. Σ 0 ), j = 1, 2
, when s is substituted by s k , then it follows from (4.4)

C K k=0 Q k 1 u 2 dxdt ≤ e a -2 τ K k=0 Q k 2 (u 2 + |∇u| 2 )dxdt + e -b -2 τ K k=0 Σ k 0 (u 2 + |∇u| 2 + (∂ t u) 2 )dxdt.
Note that the intervals s k -2 , s k + 2 overlap, but their union can cover at most two times a subdomain of (t 0 , t 1 ). Whence CI ≤ e a -2 τ N + e -b -2 τ M, τ ≥ τ 0 , where we used the temporary notations

I = u L 2 ([B(x0,ρ+θr)∩Ω)]×(t0+ ,t1-)) , M = u L 2 ((t0,t1),H 1 (Ω)) , N = u H 1 ((t0,t1),L 2 (Γ0)) + ∇u L 2 ((t0,t1),L 2 (Γ0)) .
In (4.5), we get by substituting τ by 2 τ (4.6) CI ≤ e aτ M + e -bτ N, τ ≥ τ 0 / 2 .

Set

τ 1 = ln(N/M ) a + b . If τ 1 ≥ τ 0 / 2 then τ = τ 1 in (4.6) yields (4.7) CI ≤ M ϑ N 1-ϑ , with ϑ = b a+b . When τ 1 < τ 0 / 2 , we have M < e (a+b)/ 2 τ0 N.
This inequality entails (4.8)

I ≤ M = M ϑ M 1-ϑ ≤ M ϑ e (1-ϑ)(a+b)τ0/ 2 N 1-ϑ .
So, in any case, one of estimates (4.7) and (4.8) holds. In other words, we proved

e -c/ 2 u L 2 ([B(x0,ρ+θr)∩Ω)]×(t0+ ,t1-)) ≤ u ϑ L 2 ((t0,t1),H 1 (Ω)) u H 1 ((t0,t1),L 2 (Γ0)) + ∇u L 2 ((t0,t1),L 2 (Γ0)) 1-ϑ .
Fix ω B(x 0 , ρ + θr) ∩ Ω. Then the last inequality implies

e -c/ 2 u L 2 (ω×(t0+ ,t1-)) ≤ u ϑ L 2 ((t0,t1),H 1 (Ω)) u H 1 ((t0,t1),L 2 (Γ0)) + ∇u L 2 ((t0,t1),L 2 (Γ0)) 1-ϑ . Hence C u L 2 (ω×(t0+ ,t1-)) (4.9) ≤ σ γ e c/ 2 M + σ -1 e c/ 2 u H 1 ((t0,t1),L 2 (Γ0)) + ∇u L 2 ((t0,t1),L 2 (Γ0)) ,
for σ > 0, where γ = 1-ϑ ϑ . We get, once again from Hardy's inequality in Lemma 3.1,

u L 2 (ω×(t0,t0+ )) , u L 2 (ω×(t1-,t1)) ≤ C ν M 1 (u), where M 1 (u) = u H 1 ((t0,t1),H 1 (Ω)) .
Combined with (4.9) this inequality yields

C u L 2 (ω×(t0,t1)) ≤ σ γ e c/ 2 + ν M 1 + σ -1 e c/ 2 u H 1 ((t0,t1),L 2 (Γ0)) + ∇u L 2 ((t0,t1),L 2 (Γ0)) .
In this inequality, we take σ so that σ γ = ν e -c/ 2 . Noting that σ -1 ≤ -ν/γ , we find

C u L 2 (ω×(t0,t1)) ≤ ν M 1 (u) (4.10) + e c/ 2 u H 1 ((t0,t1),L 2 (Γ0)) + ∇u L 2 ((t0,t1),L 2 (Γ0)) .
As we have seen in the preceding proof, inequality (4.10) still holds when u is substituted by ∂ t u. That is we have

C ∂ t u L 2 (ω×(t0,t1)) ≤ ν M 1 (∂ t u) (4.11) + e c/ 2 u H 2 ((t0,t1),L 2 (Γ0)) + ∇u H 1 ((t0,t1),L 2 (Γ0)) .
We add side by side (4.10) and (4.11) in order to obtain the expected inequality.

Remark 4.1. In the preceding proof we assumed that u ∈ H 2 ((t 0 , t 1 ), H 2 (Ω)). But a quick inspection of the proof of this proposition shows that if fact the result can be extended to functions from H 2 ((t 0 , t 1 ), H 1 (Ω)) ∩ H 1 ((t 0 , t 1 ), H 2 (Ω)).

Proposition 4.1 gives only an estimate in H 1 ((t 0 , t 1 ), L 2 (ω)). But we can obtain from it an estimate in H 1 ((t 0 , t 1 ), H 1 (ω)) by using the following Caccioppoli type inequality for the parabolic equation Lu = 0. Lemma 4.1. Let ω 0 ω 1 Ω. There exists a constant C > 0, only depending on Ω, κ, T 0 , ω 0 and ω 1 , so that, for any u ∈ H 2 ((t 0 , t 1 ), H 2 (Ω)) satisfying Lu = 0 in Q, we have (4.12)

C u H 1 ((t0,t1),H 1 (ω0)) ≤ u H 2 ((t0,t1),L 2 (ω1)) .

Proof. Let u ∈ H 2 ((t 0 , t 1 ), H 2 (Ω)) satisfying Lu = 0 in Q. As we have done in the preceding proofs, it is sufficient to prove C u L 2 ((t0,t1),H 1 (ω0)) ≤ u H 1 ((t0,t1),L 2 (ω1)) , because this inequality holds for both u and ∂ t u. By Green's formula, for any v ∈ L 2 ((t 0 , t 1 ), H 1 0 (Ω)), we have As D is Lipschitz, if x ∈ Γ then there exist a neighborhood U of x in R n and a bijective map φ : Q → U so that φ : Q → U and φ -1 : U → Q are Lipschitz continuous, and Here k = k + k -.

φ(Q -) = Ω ∩ U φ(Q 0 ) = U ∩ Γ.
A compactness argument shows that D can be recovered by finite number of open subsets U j with U j is either a ball or an open subset of the form U . As D is a domain then necessarily any U j intersect at least U for some = j. In consequence, any arbitrary two points x, y ∈ D can be joined by a Lipschitz path consisting on finite number (independent on x and y) of line segments and paths of the form ψ. Whence, in light of (A.1), we may find a constant C > 0 depending only on D so that d D g (x, y) ≤ C for any x, y ∈ D.

It is worth mentioning the following consequence of Lemma 3.3.

Corollary A.1. Let D be a Lipschitz bounded domain of R n . Then there exists a constant κ > 0 so that d D g (x, y) ≤ κ|x -y| for any x, y ∈ D. Proof. We proceed by contradiction. Assume then that there is no κ > 0 so that d D g (x, y) ≤ κ|x -y|, for any x, y ∈ D. In particular, for any positive integer i, we may find two sequences (x i ) and (y i ) in D so that x i = y i and d D g (x i , y i ) > i|x i -y i |, for each i. Thus (A.2)

|x i -y i | ≤ 1 i d D g L ∞ (D×D) , for each i.
Subtracting if necessary a subsequence, we may assume that x i converges to some x ∈ D. Using (A.2) we see that y i converges also to x. Fix j so that x ∈ U j , where U j is as in the preceding proof. According to (A.1), we have d D g (y, z) ≤ K|y -z|, for any y, z ∈ U j , for some constant K > 0.

On the other hand, there exists a positive integer i 0 so that x i , y i ∈ U j , for i ≥ i 0 . Hence, for any i ≥ max(i 0 , K), we get i|x i -y i | ≥ K|x i -y i | ≥ d D g (x i , y i ) > i|x i -y i |. This leads to the expected contradiction.

Lemma 3 . 3 .

 33 Let D be a bounded Lipschitz domain of R n . Then d D g ∈ L ∞ (D × D). Proposition 3.2.

( 3 .

 3 31) CM ≤ σ µ + e e e c/ √ σ N, 0 < σ < σ 0 . Define the function by (σ) = σ µ e -e e c/ √ σ . If N < min( (σ 0 ), * ) = 0 then there exists σ so that (σ) = N . Changing c if necessary, we have 1 N ≤ e e e c/ √ σ . Or equivalently σ ≤ [ln ln | ln N |] -1/2 It follows readily by taking σ = σ in (3.31) that (3.32) CM ≤ [ln ln | ln N |] -µ/2 .

∂φ 2 (

 2 t uvdxdt = 0. Let φ ∈ C ∞ 0 (ω 1 ) satisfying 0 ≤ φ ≤ 1 and φ = 1 in ω 0 . Taking v = φ 2 u in (4.13) we get in straightforward manner ∂ t u) 2 dxdt.Combining (4.14), (4.15) and (4.16), we end up getting C ∇u L 2 ((t0,t1),L 2 (ω0)) ≤ u L 2 (ω1×(t0,t1)) + ∂ t u L 2 (ω1×(t0,t1)) .

For

  x, y ∈ D ∩ U defineψ(t) = φ(φ -1 (x) + t[φ -1 (y) -φ -1 (x)]), t ∈ [0, 1].Clearly, noting that Q -is convex, ψ is a Lipschitz path in D joining x to y. We have in addition, where k + and k -are the respective Lipschitz constants of ψ andψ -1 , |ψ(t) -ψ(s)| ≤ k + |t -s||φ -1 (y) -φ -1 (x)| ≤ k + k -|t -s||x -y|, t, s ∈ [0) dt ≤ k|x -y| ≤ kdiam(D).

MC is supported by the grant ANR-17-CE40-0029 of the French National Research Agency ANR (project MultiOnde). MY is partially supported by Grant-in-Aid for Scientific Research (S) 15H05740 of Japan Society for the Promotion of Science.

Or equivalently

C u L 2 ((t0,t1),H 1 (ω0)) ≤ u H 1 ((t0,t1),L 2 (ω1)) as expected.

An immediate consequence of Caccioppoli's inequality (4.12) and Proposition 4.1 (applied both to u and ∂ t u), we have Corollary 4.1. Let ν ∈ (0, 1/2). There exist ω Ω, only depending on Ω and Γ 0 , and two constants C > 0 and c > 0, only depending on Ω, κ, T 0 , ν and Γ 0 , so that, for any u ∈ H 3 ((t 0 , t 1 ), H 2 (Ω)) satisfying Lu = 0 and 0 < < (t 1 -t 0 )/2, we have

We are now in position to complete the proof of Theorem 1.1. We recall that

If M = u Z (Q) then, in light of inequality (3.30) in the end of the proof of Theorem 3.1 and inequality (4.17), we get, for 0 < < (t 1 -t 0 )/2 and 0 < σ < σ 0 ,

the constants C > 0, c > 0 and σ 0 > 0 only depend on Ω, κ, T 0 , ν and Γ 0 .

The rest of the proof in quite similar to that of Theorem 3.1.

As we have noted above, we have L∂ j t u = 0 in Q as soon as Lu = 0 in Q, for any integer j ≥ 0. This observation enables us to state the following variante of Theorem 1.1, where

Theorem 4.1. Let Γ 0 be a nonempty open subset of Γ and s ∈ (0, 1/2). Then there exist two constants C > 0 and 0 < 0 ≤ * , depending on Ω, κ, T 0 , α, s and Γ 0 , so that, for any integer j ≥ 0 and u ∈ Z j (Q) satisfying Lu = 0 in Q, we have

with µ = min(α, s)/4 and

Appendix A

We are grateful to Tom ter Elst [START_REF] Elst | [END_REF] for having communicated to us the proofs of Lemma 3.3 and Corollary A.1 bellow. We reproduce in this appendix these proofs.

Proof of Lemma 3.3. Let D be a Lipschitz domain of R n and introduce the notations