Agnes Bouchez
email: agnes.bouchez@thonon.inra.fr

Noémie Pascault

Cècile Chardon

Marc Bouvy

Philippe Cecchi

Luc Lambs

Mélanie Herteman

François Fromard

Patrice Got

Christophe Leboulanger

Agnès Bouchez

Mangrove microbial diversity and the impact of trophic contamination

Keywords:

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Recent assessments suggest that about one-third of mangrove, sea grass and salt marsh areas around the world have already been lost over recent decades as a result of reclamation, deforestation, engineering and urbanization [START_REF] Lewis | Fate and effects of anthropogenic chemicals in mangrove ecosystems: a review[END_REF][START_REF] Peixoto | Bacterial communities reflect the spatial variation in pollutant levels in Brazilian mangrove sediment[END_REF][START_REF] Penha-Lopes | Monitoring anthropogenic sewage pollution on mangrove creeks in southern Mozambique: a test of Palaemon concinnus Dana, 1852 (Palaemonidae) as a biological indicator[END_REF], as well as transformation to provide aquaculture ponds [START_REF] Alongi | Present state and future of the world's mangrove forests[END_REF]. Many coastal lagoons in the tropics support dense mangrove forests, which are productive areas harboring a wide diversity of organisms. Adapted to intertidal zones, they are subjected to highly variable physicochemical conditions of salinity, flooding, light, and temperature, which give rise to the high diversity that characterizes mangrove ecosystems [START_REF] Feller | Biocomplexity in mangrove ecosystems[END_REF].

Mangroves are often established on nutrient-rich sediments, and are able to absorb excess nutrients without suffering any major structural or functional disturbance [START_REF] Saenger | Mangrove ecology, Silviculture and Conservation[END_REF]. [START_REF] Nedwell | Inorganic Nitrogen-metabolism in a eutrophicated tropical mangrove estuary[END_REF] was the first to show that pretreated wastewater discharge into a mangrove swamp in Fiji reduced eutrophication in adjacent coastal waters, and therefore suggested that mangroves might serve as the final stage in sewage treatment. Since then, most attempts to evaluate the potential role of mangrove to remove nutrients from sewage have been done in the form of experimental studies [START_REF] Chu | Tide-tank system simulating mangrove wetland for removal of nutrients and heavy metals from wastewater[END_REF][START_REF] Wu | Constructed mangrove wetland as secondary treatment system for municipal wastewater[END_REF], or on constructed pilot sites [START_REF] Yang | Potential use of mangroves as constructed wetland for municipal, sewage treatment in Futian, Shenzhen, China[END_REF][START_REF] Tam | Novel technology in pollutant removal at source and bioremediation[END_REF]. [START_REF] Yang | Potential use of mangroves as constructed wetland for municipal, sewage treatment in Futian, Shenzhen, China[END_REF] highlighted the need for complementary treatment to eliminate pathogenic bacteria. However, only a few studies have investigated natural mangrove sites in intertidal zones [START_REF] Wang | Maintenance of estuarine water quality by mangroves occurs during flood periods: a case study of a subtropical mangrove wetland[END_REF][START_REF] Penha-Lopes | Monitoring anthropogenic sewage pollution on mangrove creeks in southern Mozambique: a test of Palaemon concinnus Dana, 1852 (Palaemonidae) as a biological indicator[END_REF]. [START_REF] Wang | Maintenance of estuarine water quality by mangroves occurs during flood periods: a case study of a subtropical mangrove wetland[END_REF] explored changes in water quality in a subtropical mangrove estuary (China), and highlighted the fact that large quantities of nutrients may be trapped by the mangrove during flood periods. In a comparison of nonimpacted and peri-urban subtropical mangroves (Mozambique), [START_REF] Penha-Lopes | Monitoring anthropogenic sewage pollution on mangrove creeks in southern Mozambique: a test of Palaemon concinnus Dana, 1852 (Palaemonidae) as a biological indicator[END_REF] showed that both the structure and the fitness of a natural shrimp population (Palaemon concinnus) were impacted by nutrient levels. Tam showed as early as (1998) that adding wastewater to mangrove soils seems to stimulate the growth of microbial populations, probably because of the nutrients and carbon supplied in wastewater, but this compartment has not been thoroughly explored within the framework of domesticsewage discharge in mangroves. Estuarine waters are dynamic environments in which sediments, and marine and fresh water mix, resulting in salinity and nutrient gradients. Shifts in physical, chemical, and microbiological properties between freshwater and adjacent coastal marine environments occur over short periods of time, driven by tides and freshwater flows, which create an intense abiotic pressure that influences the composition of bacterioplankton communities [START_REF] Crump | Phylogenetic analysis of particleattached and free-living bacterial communities in the Columbia River, its estuary, and the adjacent coastal ocean[END_REF]. Microorganisms have large population sizes and display long dispersal distances, high reproductive rates, and remarkable genetic diversity, suggesting that they can cross environmental boundaries, including salinity, more readily than multicellular organisms [START_REF] Logares | Infrequent marine-freshwater transitions in the microbial world[END_REF].

On solid surfaces, microorganisms organize themselves into microbial biofilms, mainly consisting of heterotrophic bacteria and autotrophic eukaryotes that are often known collectively as the ''microphytobenthos'', which is embedded in a microbial organic matrix [START_REF] Decho | Microbial biofilms in intertidal systems: an overview[END_REF]. These biofilm communities are present in the upper layers of sediment [START_REF] Holguin | The role of sediment microorganisms in the productivity, conservation, and rehabilitation of mangrove ecosystems: an overview[END_REF] as well as on tree roots [START_REF] Toledo | Cyanobacteria and black mangroves in Northwestern Mexico-colonization, and diurnal and seasonal nitrogen-fixation on aerial roots[END_REF][START_REF] Gomes | Taking root: enduring effect of rhizosphere bacterial colonization in mangroves[END_REF], where they carry out a number of different functions, including nutrient transformation, sediment stabilization, plant-growth promotion, and even providing protected suitable areas for pathogens entering marine systems. The impacts of pollutants such as polycyclic aromatic hydrocarbons (PAH, for a review see [START_REF] Fernandez-Luqueno | Microbial communities to mitigate contamination of PAHs in soil-possibilities and challenges: a review[END_REF]) and phthalates have been studied in mangrove ecosystems. Conversely, changes in trophic conditions due to nutrient loading have received much less investigation in these tropical zones [START_REF] Underwood | Adaptations of tropical marine microphytobenthic assemblages along a gradient of light and nutrient availability in Suva Lagoon, Fiji[END_REF][START_REF] Ramanathan | A study of microbial diversity and its interaction with nutrients in the sediments of Sundarban mangroves[END_REF], unless they are threatened by urban sewage or aquaculture. Yet such changes, which have been extensively studied in freshwater ecosystems, are known to modify the functionality of biofilm communities, as well as their diversity (e.g. [START_REF] Pesce | Longitudinal changes in microbial planktonic communities of a French river in relation to pesticide and nutrient inputs[END_REF][START_REF] Berthon | Using diatom life-forms and ecological guilds to assess organic pollution and trophic level in rivers: a case study of rivers in south-eastern France[END_REF].

This study focuses on biofilm microbial communities in a tropical mangrove with two different trophic statuses, either exposed or not exposed to effluents from a domestic wastewater pretreatment device. We tested an experimental approach using artificial substrates to collect natural biofilms, in order to avoid sampling biases due to environmental heterogeneity. Microbial communities were also sampled in situ to collect pelagic, benthic and root biofilms in order to compare their structures and diversities. This approach provides a very useful way of extending our knowledge about the microbial communities associated to mangrove ecosystems, and of evaluating the impact of urban sewage on these communities.

Materials and methods

Study sites

Two sites in a mangrove located in Chirongui bay, southwest of Mayotte Island in the Indian Ocean (12°55 0 S, 45°09 0 E, Fig. 1) were investigated in this study. This area was described in a previous study by [START_REF] Herteman | Bioremediation capacity assessment of a mangrove impacted by domestic sewage. Application-Site Pilot Malamani[END_REF]. Domestic wastewater collected from the contiguous village of Malamani (400-equivalent inhabitants) was subjected to primary treatment in a sedimentation tank which reduced the suspended matter concentration by 50%. Pre-treated wastewater was then discharged at low tide into a 45 m  15 m mangrove zone dominated by Rhizophora mucronata, at the rate of 10 m 3 per 24 h. This site, designated the ''impacted plot'', was studied simultaneously with a ''reference plot'' located in the same mangrove zone, but not subjected to wastewater disposal. Our study was conducted in March 2009 at the end of the rainy season, during the spring tides phase, after this wastewater pre-treatment and discharge had been taking place for 1 year.

Sub-surface and porewater sampling

Salinity and pH were measured directly on the field in crab water holes (0 cm) and in deep piezometers (30 and 100 cm). Mean values were obtained from two-days measurements under similar hydrological conditions. Nutrient contents were measured 0.45lm filtered water sampled at the same depth, and transported to the analytical laboratory (ARVAM, Reunion, France) in a cool-box at 4 °C. Nutrients concentrations were determined following French standard operating procedures (AFNOR). Interference with high NaCl concentrations was avoided using blank samples on a salinity gradient. In addition, the carbon source level was investigated through the chemical oxygen demand (COD) and biological oxygen demand (BOD) of the pre-treated wastewater that were measured three times during the first year of wastewater pretreatment.

Collection of microbial communities

Six large frosted glass slides (47 cm 2 per slide) fixed in a perforated plastic box were used as artificial substrata allowing periphytic communities to colonize them [START_REF] Morin | Dynamics of benthic diatom colonization in a cadmium/zinc-polluted river (Riou-Mort, France)[END_REF]. One box was placed in each plot (the impacted plot and the reference plot) at soil level, and maintained with ropes to arching stilt roots of R. mucronata for 8 days before sampling. This corresponded to a high tide period during which the sampling system remained under water for about 40% of the time. At the end of this period, the 6 slides from both of the sampling sites were collected, and transported to the field laboratory within 2 h in a cool-box. Colonized biofilms were then carefully removed from each replicate glass slide separately, using a razor blade, and suspended in 100 mL of 0.2 lm-filtered mangrove water. Functional measurements were performed immediately, and structural and diversity measurements were carried out subsequently on frozen material. Microbial communities were also sampled from sediments, water, and natural biofilms from both plots at two dates (the beginning and end of the 8-day biofilm colonization period, respectively). Superficial sediments were sampled on the first two centimeters using a truncated syringe. Two replicates of 2 mL of sediments were randomly collected at each date from each plot (8 samples). Surface water was sampled with a sterile bottle from small natural pools at each date and from each plot (4 samples, 20 mL each). Root biofilms were sampled from the bark of aerial R. mucronata stilt roots just above soil level. Three samples of approximately 8 cm 2 of root bark were collected at each date from each plot (12 samples), and transported to the field laboratory in a cool-box. Natural root biofilms were then carefully removed from each bark sample and suspended in 10 mL of 0.2-lm filtered mangrove water. All samples were prepared (by being frozen or kept in liquid nitrogen) for later structural and diversity measurements.

Characterization of microbial communities

Fluorescence measurement

The phototrophic community in colonized biofilms was characterized using various different photosynthesis parameters measured using a Phyto-PAM fluorometer (Heinz Walz GmbH, Effeltrich, Germany) equipped with the Optical Unit ED101-US (measuring chamber) featuring a standard quartz cuvette. Before measurements, triplicate colonized biofilm suspensions were dark-adapted for 30 min. A 2-mL sample was then introduced into the measuring chamber and allowed to stabilize under modulated (non-actinic) light for a couple of minutes. The basal fluorescence of dark-adapted samples (Fo) was measured as a valid proxy for the Chl a biomass estimation. A short saturating pulse was then applied (wavelength peak: 655 nm; intensity: 4000 lmol photons m À2 s À1 ; duration: 200 ms) to induce the maximal fluorescence yield (Fm) from dark-adapted cells. The variable fluorescence (Fv = Fm À Fo), and the maximum efficiency of photosystem II, (UPSII = Fv/Fm), were then calculated, the second one being is used as a proxy of the sample's fitness [START_REF] Parkhill | Fluorescence-based maximal quantum yield for PSII as a diagnostic of nutrient stress[END_REF]. The fluorescence signals obtained at four different excitation wavelengths (470, 520, 645 and 665 nm, channels F1, F2, F3 and F4, respectively) depend on pigment composition. The F1 signal is mainly linked to diatoms and green algae, whereas the F2 is related to diatoms only. The F3 signal is linked to the Cyanobacteria, while the F4 is related to the whole algal community [START_REF] Jakob | Estimation of chlorophyll content and daily primary production of the major algal groups by means of multiwavelength-excitation PAM chlorophyll fluorometry: performance and methodological limits[END_REF]. In order to assess the pigment dominance of the phototrophic community of colonized biofilm samples indirectly, we used the ratios of the fluorescence of dark-adapted samples associated with channels F1, F2 and F3 to the fluorescence associated with channel F4. Following [START_REF] Izagirre | Effects of sediment deposition on periphytic biomass, photosynthetic activity and algal community structure[END_REF] the F1/F3 ratio was also used as an indication of dominance of green algae + diatoms versus Cyanobacteria.

Abundance of bacterial cells

Colonized biofilm suspensions (1.8 mL) from three replicates in each plot were preserved using 100 lL of 0.22-lm filtered 37% formalin in 2-mL cryovials, and immediately frozen inliquid nitrogen (À196 °C). Bacterial counts were performed with a FACSCalibur flow cytometer (Becton Dickinson, San Jose, CA, USA) equipped with an air-cooled argon laser (488 nm, 15 mW). Stained bacterial cells, excited at 488 nm, were counted from their right-angle light scatter (RALS) and green fluorescence (FL1) collected at 530/30 nm. These cell parameters were recorded on a four-decade logarithmic scale mapped onto 1024 channels. Fluorescent beads (0.94 lm, Polysciences Inc., Warrington, PA, USA) were added to each sample as an internal standard. Standardized RALS and FL1 values (sample RALS and FL1 divided by 0.94 mm-beads RALS and FL1, respectively) were used as an estimation of the relative size and nucleic acid content of bacterial cells, respectively [START_REF] Troussellier | Flow cytometric discrimination of bacterial populations in seawater based on SYTO 13 staining of nucleic acids[END_REF]. Bacterial cells tend to cluster into two distinct fractions based on differences in individual cell fluorescence (related to their nucleic acid content) and in the side and forward light scatter signal (related to their cell size). These fractions are defined as HNA cells (high nucleic acid content) and LNA cells (low nucleic acid content), respectively (see details in [START_REF] Bouvier | A comparative study of the cytometric characteristics of high and low nucleic-acid bacterioplankton cells from different aquatic ecosystems[END_REF].

Abundance of phytobenthic cells

Eukaryotic cells below the 20-lM size class (nano-and picoeukaryotes) in colonized biofilms were also counted with the same flow cytometer. Cells excited at 488 nm were detected and counted on the basis of their RALS properties, and their orange (585/42 nm) and red fluorescence (>650 nm) emissions, which are related to phycoerythrin and chlorophyll pigments, respectively. Fluorescent beads (0.94 lm) were also added to each sample as an internal standard. Cell concentrations were calculated as for the bacterial community.

Community level molecular fingerprinting

Water samples and colonized and natural biofilm suspensions were filtered through a 0.2-lm Nuclepore filter (Whatman). Nucleic acid extraction for sediment and biofilm samples was performed using the ''Ultraclean for Soil'' kit and ''PowerBiofilm'' DNA Isolation Kit (MoBio) respectively, according to the Manufacturer's instructions. For the water samples, after lysing the cells, nucleic acids were co-precipitated using GenElute ™ -LPA (Sigma-Aldrich) according to the Manufacturer's instructions, and dissolved in TE buffer. The integrity of the total DNA was checked using a NanoDrop 1000 spectrophotometer (Thermofisher Scientific). Polymerase Chain Reactions (PCRs) of the eukaryotic 18S rRNA and of the bacterial 16S rRNA gene fragments were performed using 15 ng of template DNA. Primers Euk1Af [START_REF] Sogin | Structural diversity of eukaryotic small subunit ribosomal-RNAs -evolutionary implications[END_REF] and Euk516r-GC [START_REF] Amann | Combination of 16S ribosomal-RNA-targeted oligonucleotide probes with flow cytometry for analyzing mixed microbial-populations[END_REF] were used to amplify a 560-bp fragment of the 18S rRNA gene, while primers 358f-GC [START_REF] Muyzer | Profiling of complex microbialpopulations by denaturing gradient gel-electrophoresis analysis of polymerase chain reaction amplified-genes coding for 16S ribosomal-RNA[END_REF] and 907rM [START_REF] Schauer | Seasonal changes in the taxonomic composition of bacterioplankton in a coastal oligotrophic system[END_REF] were used to amplify a 570-bp fragment of the 16S rRNA gene. All amplifications, including a negative control, were performed as described in [START_REF] Dorigo | Lotic biofilm community structure and pesticide tolerance along a contamination gradient in a vineyard area[END_REF].

PCR samples were analyzed by DGGE with an INGENYphorU-2Ã2 system (Ingeny International), using a 1-mm thick gel of 6% (w/v) polyacrylamide with a denaturing gradient, where 100% denaturant contained 7 M urea and 40% formamide. The eukaryotic and prokaryotic communities were studied with 35-65% and 40-80% gradients respectively. Nucleic acids were detected by UV transillumination on stained DGGE gels (SYBR Gold, Molecular Probes). As we were focusing on the prokaryotic community, all 36 samples (colonized and natural biofilms, sediment, and water samples) were studied on the same DGGE gel, whereas the eukaryotic and prokaryotic communities were studied separately in the 12 samples from the colonized biofilms. DGGE profiles were analyzed using GelCompar II software (Applied Math NV) leading to a simple matrix of the presence/absence of bands (prokaryotic community for all samples). For colonized biofilms, a more detailed analysis was carried out, leading to a matrix of the quantification of band intensities (eukaryotic and prokaryotic communities of colonized biofilms).

Statistical data analysis

The richness (number of species per sample) and Shannon index were estimated from the DGGE bands.

The prokaryotic diversity of all samples was compared on the basis of the DGGE presence/absence matrix using the Jaccard similarity index. The similarity matrix was used to perform Multi-Dimensional Scaling (MDS) analysis.

The diversity of colonized biofilm communities (prokaryotic and eukaryotic) was compared on the basis of the DGGE matrix of relative band intensities submitted to Principal Component Analysis (PCA).

The abundances, fluorescence measurements, and diversity index of the colonized biofilms were tested for a plot effect using a one-way ANOVA with a 5% threshold. Parameters showing a significant plot effect were then tested using a Student's t test.

All these statistical tests were carried out using XLSTAT Software (version 2011.1).

Results

The impacted and reference plots shared common characteristics for temperature (27.4 ± 0.04), pH (6.62 ± 0.03), salinity (34.5 ± 1.5) and cohesive sediment structure, (mostly silt: 76%; fine sand: 21%, clay: 3% at surface). However, they differed with regard to nutrient concentrations (Table 1). In the surface layer of the sediment (0-30 cm), NO À 3 , NO À 2 and NH þ 4 concentrations in the interstitial water were higher in the impacted plot than in reference plot, whereas no difference was observed in the deeper layers. Conversely, the PO 4 concentration was higher in the deeper layers of the impacted plot, whereas no difference was observed in the surface layers. Concerning the carbon substrate, a mean COD value of 167 ± 57 mg L À1 was observed for the pre-treated effluent with a relatively high COD/BOD ratio (2.2 ± 0.4), indicating that organic matter was loaded in the impacted plot by the effluent, but this organic matter was poorly biodegradable.

Microbial community structure

In the impacted plot exposed to wastewater disposal, the total bacterial cell density was significantly higher than the one reported in the reference plot (p = 0.018), in particular for the fraction with high nucleic acid content (HNA bacteria, p = 0.016), which are considered to consist of more active bacteria (Table 2). Conversely, the proxy associated to the biomass of the phototrophic community (basal fluorescence measured on dark-adapted samples by the Phyto PAM) did not reveal any significant difference between the two zones (F = 0.0967; p = 0.762). Different fluorescence ratios were used to carry out an indirect assessment of pigment dominance in the phototrophic community. Among these ratios, F1/F3 and F1/F4 revealed significant differences between the communities originating from the two plots (p = 0.026 and p < 0.001, respectively), highlighting the differing contributions made by green algae and diatoms to the biomass of the periphytic communities (Fig. 2). The contribution of green algae and diatoms was higher in the impacted plot than in the reference one. We did not observe any significant difference between the photosynthetic performances of biofilms from our slides from one plot to another (F = 0.00207; p = 0.965). However, the maximum efficiency of photosystem II values (UPSII below 0.3) were unusually low in samples from both plots.

Community level molecular fingerprinting

For prokaryotic community fingerprinting, all 36 samples (colonized and natural biofilms, sediment, and water) were studied on the same DGGE gel. The diversity indices based on these data are summarized in Table 3. A total of 51 different DGGE-Operational Taxonomic Units (DGGE-OTUs) were detected in the 36 samples. On average, 26, 16 and 10 DGGE-OTUs were identified in the sediments, biofilms, and water, respectively. Sediment and colonized biofilms displayed the highest species richness and diversity (Shannon Index). The MDS values based on these DGGE data are presented in Fig. 3. The MDS values found for sediment and water communities were sharply distinguished from each other, while those found for root biofilm communities were more scattered. Impacted and reference natural samples remained closely clustered. Colonized biofilm communities appeared at an intermediate position between the natural samples. They exhibited a clear discrimination between impacted and reference samples.

With regard to the prokaryotic community of colonized biofilms (12 samples), a total of 33 different DGGE-OTUs were detected, with an average of 19 DGGE-OTUs per sample for both the reference and impacted plots. The diversity indices based on these data are summarized in Table 3. No significant difference was detected for the diversity index (ANOVA, data not shown) between the impacted and reference communities. Fig. 4a reports the PCA analysis of prokaryotic diversity showing the ordination of sampling points with respect to the two principal axes. The first two axes (PC1 and PC2) accounted for more than 61% of the variance. PC1 appeared to be related to the origin of the plots, with a clear separation between samples from the impacted plot and those from the reference plot. The reference samples appeared to be more widely scattered along PC2 than the impacted ones.

For the eukaryotic community of colonized biofilms (12 samples), a total of 60 DGGE-OTUs were detected, with an average of 23 and 25 DGGE-OTUs per sample for the reference and impacted plots, respectively. Diversity indices based on these data are summarized in Table 3. No significant difference in the diversity index (ANOVA, data not shown) was detected between the impacted and reference communities. Fig. 4b reports the PCA analysis of eukaryotic diversity showing the ordination of sampling points with respect to the two principal axes. The first two axes (PC1 and PC2) accounted for more than 32% of the variance. PC1 also appeared to be related to the origin of the samples per plot, as shown in Fig. 4a.

Discussion

This study reports for the first time the microbial diversity of a mangrove in the tropical island of Mayotte which suffers anthropogenic impacts due to a demographic explosion. The differences found in the community composition add to our knowledge about microbial distribution in mangrove ecosystems. The loading of nutrients in aquatic ecosystems has often been reported to be a cause of functional, structural and diversity changes in biofilm microbenthic communities in freshwater ecosystems (e.g. [START_REF] Underwood | Distribution of estuarine benthic diatom species along salinity and nutrient gradients[END_REF][START_REF] Pesce | Longitudinal changes in microbial planktonic communities of a French river in relation to pesticide and nutrient inputs[END_REF], but relatively rarely in intertidal marine environments [START_REF] Chu | Tide-tank system simulating mangrove wetland for removal of nutrients and heavy metals from wastewater[END_REF][START_REF] Underwood | Adaptations of tropical marine microphytobenthic assemblages along a gradient of light and nutrient availability in Suva Lagoon, Fiji[END_REF][START_REF] Ramanathan | A study of microbial diversity and its interaction with nutrients in the sediments of Sundarban mangroves[END_REF]. We first had to validate the use of artificial substrates to collect recent biofilms as an efficient tool for the sampling and observation of biofilm communities in the mangrove. We then used the results obtained to characterize the biofilm community and demonstrate its possible links with nutrient concentrations.

Sampling microbial communities in the mangrove

The samples of microbial communities (bacteria and microalgae) were taken both from natural substrata and from an artificial glass substrate. Natural communities were also collected from several different compartments: water, sediments and roots.

Due to the tidal rhythm in the mangrove, it was necessary to avoid as much as possible differences related to temperature and desiccation stresses, which are known to be prime factors controlling intertidal biofilms [START_REF] Kaehler | Do factors influencing recruitment ultimately determine the distribution and abundance of encrusting algae on seasonal tropical shores?[END_REF]. For this reason, the colonization time could not exceed 8 days, which corresponds to a spring tide period. By their functional ability, evaluated from a photosynthesis process with low UPSII values (below 0.3), the photoautotrophic communities presented the functional indexes of a recent stage. Values higher than 0.5 indicate a satisfactory physiological acclimation to the growth conditions [START_REF] Parkhill | Fluorescence-based maximal quantum yield for PSII as a diagnostic of nutrient stress[END_REF][START_REF] Claquin | Effects of simulated benthic fluxes on phytoplankton dynamic and photosynthetic parameters in a mesocosm experiment (Bay of Brest, France)[END_REF], and non-limited algae presented UPSII values of around 0.6-0.7 [START_REF] Kromkamp | Changes in phytoplankton biomass and primary production between 1991 and 2001 in the Westerschelde estuary (Belgium/The Netherlands)[END_REF]. After one week of colonization time, the phototrophic community did not yet seem to be functioning efficiently, unlike the mature communities harvested from root barks, which had higher UPSII values (0.4 ± 0.1). Functional evaluation of photosynthesis proved to be too closely linked to the stage of biofilm development to be used Table 2 Cell abundances in cells mL À1 measured on colonized biofilms and ANOVA p-value. Data are cell number mean values ± standard deviation. Significant differences, at 5% threshold, between impacted and reference plots are indicated in bold type.

Cell abundances (cells mL À1 ) ANOVA p-value

Impacted plot Reference plot HNA bacteria 1.0 Â 10 6 ± 1.8 Â 10 5 5.1 Â 10 5 ± 1.3 Â 10 5 0.016 LNA bacteria 3.3 Â 10 5 ± 1.0 Â 10 4 4.0 Â 10 5 ± 1.8 Â 10 4 0.333 Total bacteria 1.3 Â 10 6 ± 1.3 Â 10 5 9.1 Â 10 5 ± 1.5 Â 10 5 0.018 Cyanobacteria 7.8 Â 10 1 ± 3.8 Â 10 1 8.0 Â 10 1 ± 2.1 Â 10 1 0.925 Picoeukaryotes 2.5 Â 10 3 ± 1.1 Â 10 3 1.5 Â 10 3 ± 2.5 Â 10 2 0.231 Nanoeukaryotes 8.9 Â 10 2 ± 4.4 Â 10 2 6.5 Â 10 2 ± 9.1 Â 10 1 0.411 Fig. 2. Fluorescence ratios of dark-adapted biofilm samples from colonized biofilms from the impacted and reference plots. as an accurate functional end-point for the characterization of phototrophic communities after such a short colonization period. The composition of the prokaryotic community of colonized biofilms was homogeneous between replicates, while natural root biofilms displayed a more heterogeneous composition. Moreover the differences in composition between the two sampling plots appeared more clearly with colonized biofilms, than with natural samples. The values of species richness and diversity (Shannon index) of bacterial communities in colonized biofilms were intermediate between those found for the different natural communities (higher than water and root biofilm communities, lower than sediment communities). The diversity pattern of recent prokaryotic communities taken from the artificial substrates provided a reliable image of the pattern of mature natural communities, as it occupies a central intermediate position in the molecular diversity revealed using DGGE. As bacterial communities are known to be the earliest colonizers [START_REF] Chan | Patterns of biofilm succession on a sheltered rocky shore in Hong Kong[END_REF], these values probably indicate good development of the bacterial community. On the other hand, the eukaryotic community as a later colonizer, was assumed to be less developed. For this community developed on artificial substrates, mean diversity and richness values were high (24.2 and 2.7 respectively) and in the same range as those (mean values of 23 and 2.8 respectively) observed for fully-developed biofilms in freshwater ecosystems [START_REF] Dorigo | In situ assessment of periphyton recovery in a river contaminated by pesticides[END_REF][START_REF] Villeneuve | Influence of slight differences in environmental conditions (light, hydrodynamics) on the structure and function of periphyton[END_REF]. However, to our knowledge, no such values are yet available for eukaryotic communities from biofilms in tropical mangrove ecosystems.

Finally, the colonization device, involving glass plate surfaces, gave an accurate microbial score. Even if this device still needs to be tested on more shore areas before it can be generalized, this simple system looks promising as a way of revealing the richness and diversity of bacterial communities in tidal mangroves. Eukaryotic community diversity also seemed to be well assessed, but it too needs further evaluation. From a functional point of view, it looks as if a longer colonization time is required, but unfortunately the tidal rhythm does not allow for longer colonization periods without encountering daytime drought. Long exposure to dry conditions during tidal emersion may be hugely selective; masking any selection pressure linked to pollution, and can lead to slow community development. An alternative might be to use permanent pools located within the mangrove in order to permit longer incubation of the artificial substrates and to obtain mature biofilms well fitted for functional assessment of communities.

The microbial pattern along the trophic gradient

In the Chirongui bay mangrove, [START_REF] Lambs | Tracing sewage water by N-15 in a mangrove ecosystem to test its bioremediation ability[END_REF][START_REF] Herteman | Effects of pretreated domestic wastewater supplies on leaf pigment content, photosynthesis rate and growth of mangrove trees: a field study from Mayotte Island, SW Indian Ocean[END_REF] first reported results at the vegetation and crab community scale, showing that their growth was boosted as a result of changes in the nutrient load in the mangrove sediments. However, analyses and observations of mangrove ecosystems as a whole are necessary to evaluate their bioremediation capacities as vegetative buffers against anthropogenic N-loading in coastal waters. The major aim of the work reported here was to assess the changes in microbial communities that may be attributable to changes in the nutrient load in the mangrove sediments. The effluent-receiving plot is characterized by having a greater nutrient load than the reference plot, particularly in the upper sediment layers for N forms, and in the deeper layers for P. As P and C levels, especially bioavailable forms, estimated respectively from the sediment upper layer and from the pre-treated effluent, were not very different from one plot to the other, these nutrients may not play a major role in the differential shaping of microbial communities in the two plots. The microbial communities observed in this study corresponded mainly to the surface oxic zone, and can therefore be expected to be more influenced by variations in N forms, while the other physico-chemical parameters did not exhibit major variations, and are typical of a seawater-dominated environment [START_REF] Saenger | Mangrove ecology, Silviculture and Conservation[END_REF].

Variations of the phototrophic community

The phototrophic microbial community was assessed on colonized biofilms, and clear differences of composition were detected between the impacted and reference plots. A higher contribution of green algae and diatoms characterized the photosynthetic potential of the phototrophic assemblage in the impacted plot. In the sediments of tropical mangrove forests, [START_REF] Farooq | Distribution of chlorophyll in the mangrove sediments of Sonmiani bay, Pakistan[END_REF] reported a rapid decrease within the first few centimeters in the abundance of the microphytobenthic community, which may be directly related to changes in nutrient concentrations in addition to the light gradient.

No difference was found between the two plots with regard to diversity indices assessed by molecular fingerprinting (species richness and the Shannon index). Variations in nutrients did not seem to modify the number of eukaryotic DGGE-units in biofilms. However, a clear composition difference was observed between the eukaryotic assemblages of the two plots. As [START_REF] Villeneuve | Influence of slight differences in environmental conditions (light, hydrodynamics) on the structure and function of periphyton[END_REF] pointed out, minor environmental changes that do not affect diversity can still result in modifications in the composition of community assemblages.

Within this community, diatoms seem to play an important role in the impacted plot, as shown by phytoPAM results. Diatoms are known to play a central role in coastal food webs, and they are routinely used to indicate the trophic status of rivers (Berthon et al.,Fig. 4. Principal Component Analysis of the relative band intensity matrix deduced from the analysis of DGGE gels derived from the amplification of the gene coding for 16S rRNA (a) and 18S rRNA (b) for colonized biofilm samples (black circles: samples from the impacted plot I, white circles: samples from the Reference plot R). 2011). In many aquatic ecosystems, nutrient composition is closely linked to the composition of the diatom community: for instance, within the Colne estuary in temperate England, ammonium concentrations have been shown to be important in determining the diatom species composition [START_REF] Underwood | Distribution of estuarine benthic diatom species along salinity and nutrient gradients[END_REF], whereas in tropical Jamaica, phosphate and organic nitrogen were the main drivers [START_REF] Podzorski | The reaction of epiphytic diatoms (Bacillariophyta) to environmental-change in broad river, Western Jamaïca[END_REF]. [START_REF] Dham | Nitrogen uptake by size-fractionated phytoplankton in mangrove waters[END_REF] were able to link seasonal changes in the uptake of nitrogenous nutrients with the plankton community composition in mangrove waters. Pennate diatoms and flagellates were dominant during periods of high nitrate uptake, and centric diatoms and blue-green algae during those of high ammonium uptake. More recently, Chen et al. ( 2010) investigated the effect of environmental changes on mangrove diatoms, and revealed species associations that were closely linked to nitrogen and phosphate resources. [START_REF] Wachnicka | Distribution of diatoms and development of diatom-based models for inferring salinity and nutrient concentrations in Florida Bay and adjacent coastal wetlands of south Florida (USA)[END_REF] studied the distribution of diatoms in diverse subtropical habitat types in freshwater, mangrove, and marine locations, and assessed their compositions. In a given type of location, epiphytic, planktonic, and sediment assemblages were compositionally similar, implying a high degree of mixing along the shallow, tidal, and storm-prone coast. It was therefore hypothesized that a nutrientlinked assemblage was found in our impacted plot. These findings suggest that the phototrophic community, and diatoms in particular, could be used as sentinels of environmental changes in such mangrove ecosystems.

Variations of the prokaryotic community

We observed differences in the structure of the prokaryotic microbial community. The impacted plot, exposed to pre-treated wastewater discharges, contained a significantly higher density of bacterial cells, and this was mainly due to the abundance of the active bacteria fraction (HNA bacteria). They formed the dominant population in all samples (between 49% and 85% of the total bacterial community) with a higher average percentage in the impacted plot (75%). These results confirmed suggestions that nutrients could stimulate the growth of microbial populations in mangroves [START_REF] Tam | Effects of wastewater discharge on microbial populations and enzyme activities in mangrove soils[END_REF]. However, it is also necessary to find out whether this increase may also result in efficient nutrient consumption and recycling. In an experimental study, [START_REF] Ramanathan | A study of microbial diversity and its interaction with nutrients in the sediments of Sundarban mangroves[END_REF] showed that in addition to the increase in total bacterial load, higher nutrient levels in sediments were also linked to higher abundances of N-2 fixing and nitrifying bacteria. The increase in bacterial density, combined with changes in the community composition, may therefore play a significant role in the fate of nutrients in our mangrove ecosystem.

Focusing on the colonized biofilms, a marked differentiation was observed between the microbial assemblages of the two plots. As shown by [START_REF] Peixoto | Bacterial communities reflect the spatial variation in pollutant levels in Brazilian mangrove sediment[END_REF], the composition of bacterial communities might be severly impacted by major environmental stressors, such as nutrients which may play a key role in shaping the community. This had previously been observed in mangrove sediments by [START_REF] Amano | Contribution of anammox bacteria to benthic nitrogen cycling in a mangrove forest and shrimp ponds, Haiphong, Vietnam[END_REF], who highlighted the presence of anammox and denitrification activities, although the contribution of anammox was less significant. [START_REF] Cao | Diversity and abundance of ammoniaoxidizing archaea and bacteria in polluted mangrove sediment[END_REF] revealed the presence of the amoA gene in polluted estuarine mangrove sediments, where high levels of nutrients and metals were recorded as a result of wastewater discharge from nearby rivers.

The high level of HNA bacteria, together with a specific community pattern in the impacted plot, both suggest the potential presence of a specific community particularly involved in nutrient consumption and transformation. However, our fingerprinting approach does not provide information about the identity of the species represented by differential DGGE-bands. Barcoding combined with high-throughput sequencing might offer a clearer understanding of the community assemblages by revealing the identity of species present in the different plots [START_REF] Dos Santos | Mangrove bacterial diversity and the impact of oil contamination revealed by pyrosequencing: bacterial proxies for oil pollution[END_REF][START_REF] Cleary | Assessment of variation in bacterial composition among microhabitats in a mangrove environment using DGGE fingerprints and barcoded pyrosequencing[END_REF]. The specific bacterial populations involved and their role in the aquatic system should be determined by means of further studies of functional groups, such as N-Cycle bacteria for example [START_REF] Santoro | Microbial nitrogen cycling at the saltwater-freshwater interface[END_REF].

Conclusion

Our results, obtained using a simple and efficient colonization device for microbial communities, give a preliminary insight regarding the impacts of pre-treated wastewater effluents on microbial communities in a mangrove. In particular, a denser bacterial community exhibiting a specific assemblage that may be able to enhance nutrient recycling and consumption, have been observed in the impacted plot. However, further information about these systems is required. Data mining remains necessary, and should focus essentially on the diversity, structure and function of microbial communities in mangroves, which have so far received little investigation. Even less is known about their temporal and spatial dynamics. Finally, deeper insight into the functional diversity of these prokaryotic and eukaryotic communities is particularly necessary in order to obtain a clearer understanding of their ability to transform the effluents from the pre-treatment device, especially in the case of prokaryotic communities active in nitrogen cycling. More broadly, these results emphasize the importance of protecting mangrove ecosystems in order to enhance their buffer role against eutrophication of adjacent coastal marine waters.

Fig. 1 .

 1 Fig.1. Mangrove study area and land cover on Mayotte Island in the Indian Ocean.

Table 3

 3 Species

		Richness	Shannon index
	Prokaryotic community
	Sediment	25.8 ± 2.3	3.086 ± 0.116
	Water	10.3 ± 1.5	2.159 ± 0.115
	Root biofilm	12.8 ± 3.3	2.279 ± 0.285
	Colonized biofilm	19.3 ± 1.4	2.677 ± 0.144
	Eukaryotic community
	Colonized biofilm	24.2 ± 2.7	2.692 ± 0.203

richness and Shannon diversity index deduced from DGGE fingerprinting of prokaryotic and eukaryotic communities of mangrove samples from sediment (n = 8), water (n = 4), root biofilm (n = 12), and colonized biofilm (n = 12). Data are mean values ± standard deviation. Fig.

3

. Multidimensional Scaling of the presence/absence matrix deduced from the analysis of the DGGE gel derived from the amplification of the gene coding for 16S rRNA for all mangrove samples (black: samples from the Impacted plot, white: samples from the Reference plot, squares: sediment, triangles: root biofilm, circles: colonized biofilm, diamonds: water).

Acknowledgements

This paper is a contribution to the ECOMET project, funded by the French Ministry of Ecology, Energy, Sustainable Development, and Sea (MEEDDM), Grant No. CV070000783. The experimental site in Chirongui Bay, Mayotte Island, was set up as part of a joint research program between the Water Syndicate of Mayotte (SIE-AM) and the Laboratory of Functional Ecology and the Environment (ECOLAB CNRS University of Toulouse), and is partially funded by the Ecological Engineering Program (2007Program ( , 2009Program ( -2010) ) of the French National Center for Scientific Research (CNRS).


Here, |σ (x)| is a the norm of the value of the stress at x. To evaluate it we use some norm on the vector space of matrices. The value of K t,σ clearly depends on the norm used and various norms are discussed in Section 4. Using the essential supremum, we ignore high stresses on sets of zero volume.

Noting that without specifying particular constitutive relations there is a class Σ t of stress fields that are in equilibrium with t, we define the optimal stress concentration factor as

K t = inf σ ∈Σ t K t,σ .
(1.2)

Next, realizing that an engineer usually does not know a-priori the nature of the loads acting on a body exactly, the generalized stress concentration factor is defined as

K = sup t K t = sup t      inf σ ∈Σ t ess sup x∈Ω |σ (x)| ess sup y∈∂Ω |t (y)|      (1.3)
where the supremum is taken over all traction fields -essentially bounded vector fields on ∂Ω -i.e., over all t ∈ L ∞ (∂Ω , R 3 ). It is noted that K is a purely geometric property of the body Ω .

In [START_REF] Segev | Generalized stress concentration factors[END_REF][START_REF] Segev | Generalized stress concentration factors for equilibrated forces and stresses[END_REF] we related the generalized stress concentration factor to the norms of the trace mappings on Sobolev spaces and L D-spaces. It turns out that for the formulation of equilibrium and stress theory, particularly in the context of stress concentration, the Sobolev space W 1 1 (Ω , R 3 ) and the related L D(Ω ) space are especially useful. We recall that the space L D(Ω ) contains integrable vector fields w such that the components of their associated stretching, or (infinitesimal, linear) strain, ε(w) = 1 2 (∇w + (∇w) T ), ε(w) i j = 1 2 (w i, j + w j,i ), (1.4) are also integrable (see [START_REF] Temam | Functions of bounded deformations[END_REF][START_REF] Temam | On the continuity of the trace of vector functions with bounded deformation[END_REF][START_REF] Temam | Mathematical Problems in Plasticity (a translation of Problemes mathematiques en plasticite Bordas[END_REF][START_REF] Ebobisse | Fine properties of functions with bounded deformation and applications in variations problems[END_REF]).

Assuming that Ω is an open subset of R n having a C 2 -boundary, for both spaces one has a well defined, linear, bounded trace mapping γ , such that for every vector field w defined on Ω , γ (w) is a vector field defined on ∂Ω satisfying the following compatibility condition. For every continuous vector field u defined on the closure Ω , γ acts as the restriction to the boundary, i.e., γ (u| Ω ) = u| ∂Ω .

(1.5)

In [START_REF] Segev | Generalized stress concentration factors[END_REF] we have shown that if we ignore the requirement that the total force and total torque on every subbody of Ω vanish, then, K = γ , for γ : W 1 1 (Ω , R 3 ) -→ L 1 (∂Ω , R 3 ).

(1.6)

For the case where the total forces and torques on the various subbodies do vanish, a more detailed analysis is required (see [START_REF] Segev | Generalized stress concentration factors for equilibrated forces and stresses[END_REF]). Letting R be the finite dimensional vector space of rigid vector fields, one has to consider the quotient spaces L D(Ω )/R and L 1 (∂Ω , R 3 )/R. For these spaces, one can define an induced trace mapping

γ /R: L D(Ω )/R -→ L 1 (∂Ω , R 3 )/R, (1.7)
and it turns out that

K = γ /R . (1.8)
Thus, apart from the mathematical interest in estimates on γ and γ /R , such estimates are very significant in stress analysis. It is our objective here to estimate these constants. Our method of estimation has a mechanical flavor. In a way, it is dual to the analysis leading to the relation between generalized stress concentration factors and the norms of the trace mappings.

For the particular case of the Sobolev space W 1 1 (Ω ), Motron obtained recently [START_REF] Motron | Around the best constants for the Sobolev trace map from W 1,1 (Ω ) into L 1 (∂Ω )[END_REF] some estimates on the bounds. The method used here is different and is based on the maximum principle for the Dirichlet problem. The bounds we obtain give a concrete estimate for γ . Subsequently, we extend our method to the space of L D-fields. Specifically, recalling that the L D-norm is given by

w L D = i w i L 1 + i,m ε(w) im L 1 , (1.9)
we obtain bounds on the constants A and B such that for all L D-fields w (so that max{A, B} bounds γ ). Section 2 presents the results and methods of [START_REF] Segev | Generalized stress concentration factors for equilibrated forces and stresses[END_REF] relating stress concentration to the norm of the trace mapping. The last subsection discusses the simplification to the case of sourceless vector fields (rather then stress tensors) which may serve as motivation for studying the norm of the trace mapping for the Sobolev space W 1 1 (Ω ). In Section 3 we introduce the basic method for obtaining the bounds on the trace mapping for the Sobolev space using harmonic vector fields. The main result of this section is Theorem 3.4. As background material for the discussion of the bounds for the trace mapping on L D(Ω ), Section 4 presents some standard results on norms of matrices. These are significant in the mechanical context as they are used on the space of stress matrices. For example, a yield criterion is usually a seminorm on the space of stress matrices. Section 5 presents additional preparatory material-the optimal boundary values for the stresses for a given boundary traction field. Section 6 studies the bounds on the norm of the trace mapping for L D(Ω ) using the method of harmonic tensor fields and the central result is Theorem 6.1. Finally, the concluding remarks of Section 7 discuss the mechanical interpretation of the preceding analysis.

Generalized stress concentration and the norm of the trace mapping

Basic definitions and notation

We consider an open set Ω ⊂ R 3 , where in some sections, the presentation is in the setting of R n . We assume that Ω is bounded and that it has a C 2 -boundary. (The results hold for less restrictive assumptions.) We will use the summation convention for repeated indices and subscripted comma followed by an index will indicate partial differentiation with respect to the corresponding variable.

A vector field on Ω is interpreted physically as a virtual velocity field on the body or alternatively as a field of virtual infinitesimal displacements. A rigid field in R 3 is a vector field of the form

w(x) = a + b × x, a, b ∈ R 3 .
Clearly, rigid fields may be restricted to subsets of R 3 . We denote the space of rigid fields by R and it is a 6-dimensional vector space.

The following definitions and results concerning L D-fields are due to Temam and Strang [START_REF] Temam | Functions of bounded deformations[END_REF][START_REF] Temam | On the continuity of the trace of vector functions with bounded deformation[END_REF][START_REF] Temam | Mathematical Problems in Plasticity (a translation of Problemes mathematiques en plasticite Bordas[END_REF]. Given an integrable vector field w on Ω , we consider the corresponding stretching (linear strain) field ε(w) defined by

ε(w) im = 1 2 (w i,m + w m,i ), (2.1)
where the comma implies distributional derivative relative to the corresponding spatial coordinate. The integrable vector field w is of integrable stretching, or w ∈ L D(Ω ), if the components of the corresponding stretching are also integrable over Ω . On the vector space L D(Ω ) of integrable stretchings it is natural to use the norm

w = w L D = w 1 + ε(w) 1 , (2.2)
where • p indicates the L p -norm. With this norm, L D(Ω ) is a Banach space and we have a continuous and linear

ε: L D(Ω ) -→ L 1 (Ω , R 6 ). (2.3)
A basic theorem whose classical version is due to Liouville (see [6, pp. 18-19]) states:

Proposition 2.1. Kernel(ε) = R.
Let W be a Banach space of velocity fields. In the discussion below W will be either L D(Ω ) or L 1 (∂Ω , R 3 ). We refer to an element χ ∈ W/R as a distortion. We have the natural projection mapping onto the quotient space

π : W -→ W/R (2.4)
and the induced norm in W/R is given by

χ = inf w∈χ w , or [w] = inf r ∈R w + r . (2.5)
Proposition 2.2. For both W = L 1 (∂Ω , R 3 ) and W = L D(Ω ) there are continuous and linear projection mappings

π R : W -→ R. (2.6) For w ∈ W, π R (w) = a + b × x is given by a = 1 |U | U w, b = I -1 U x × w , (2.7
)

where U = ∂Ω for W = L 1 (∂Ω , R 3 ), U = Ω for W = L D(Ω ), |U |
is the Hausdorff measure of U , and

I im = U (x k x k δ im -x i x m ) is the moment of inertia of U .
The space L D(Ω ) has the following properties (see [START_REF] Temam | Mathematical Problems in Plasticity (a translation of Problemes mathematiques en plasticite Bordas[END_REF]).

Approximation: The restrictions of fields in C ∞ (Ω , R 3 ) to Ω are dense in L D(Ω ).

Extensions: There is a continuous linear extension operator E: L D(Ω ) → L D(R 3 ).

Regularity: If w is any distribution on Ω whose corresponding stretching is L 1 , then w ∈ L 1 (Ω , R 3 ).

Trace mapping: There is a unique linear, surjective, continuous trace mapping

γ : L D(Ω ) -→ L 1 (∂Ω , R 3 ) (2.8)
such that γ (w| Ω ) = w| ∂Ω for all continuous vector fields w defined on Ω . Distortions:

On L D(Ω )/R, χ ε = ε(χ ) 1 (2.9)
is a norm that is equivalent to the quotient norm described above. Thus, there is a constant C(Ω ) (depending on Ω only) such that for every w ∈ L D(Ω )

inf r ∈R w + r C(Ω ) ε(w) 1 .
(2.10)

The infimum is attainable, i.e., for each w ∈ L D(Ω ), there is a rigid motion r 0 satisfying

w + r 0 = inf r ∈R w + r C(Ω ) ε(w) 1 . (2.11)
Equivalent norm: If p is a seminorm on L D(Ω ) such that p(r ) = 0 implies that r = 0 for every r ∈ R (so p is a norm on R), then,

p(w) + ε(w) 1 (2.12)
is a norm on L D(Ω ) that is equivalent to the original norm. In particular, using the trace mapping one can take

p(w) = γ (w) 1,∂Ω , (2.13)
so the following is a norm that is equivalent to original (2.2)

w × = γ (w) 1,∂Ω + ε(w) 1 . (2.14)
Furthermore, one may use the projection π R : L 1 (∂Ω , R 3 ) → R as in Proposition 2.2 and a norm • R to obtain the equivalent norm

w ⊕ = π R • γ (w) R + ε(w) 1 .
(2.15)

Forces are regarded as elements of the dual spaces to the corresponding spaces of virtual velocities. So for a generic space of velocities W, a force F will be a member of W * . The evaluation F(w) is interpreted as virtual work, or virtual power, performed by the generalized force for the corresponding generalized velocity.

In the case the space W of velocities is an L p -space, 1 p ∞, (e.g., L 1 (∂Ω , R 3 )) a force may be represented by an element of the corresponding L q space with q = p/( p -1) through integration. We will use the same symbol for the force and its representing field. For example, for t ∈ L 1 (∂Ω , R 3 ) * = L ∞ (∂Ω , R 3 ) we have

t (w) = ∂Ω t • w.
(2.16)

A force F ∈ W * acting on a body is equilibrated if F(r ) = 0 for all r ∈ R. An equilibrated force F is of the form F = π * (F 0 ) for some F 0 ∈ (W/R) * . Furthermore, π * is norm preserving, i.e., π * (F 0 ) = F 0 , (2.17) so one can usually identify an equilibrated F with F 0 .

Generalized stress concentration factors and norms of trace mappings

The central mathematical object that we find suitable for formulating the continuum mechanics problem, particularly, those notions related to stress concentration is (L D(Ω )/R) * -the dual to the space of L D-distortions. Specifically, as described below and in further detail in [START_REF] Segev | Generalized stress concentration factors for equilibrated forces and stresses[END_REF], elements of this space may be represented by essentially bounded stress fields and on the other hand, the dual of the trace mapping associates an element of (L D(Ω )/R) * with any equilibrated boundary traction field.

Consider the composite mapping π

• γ : L D(Ω ) → L 1 (∂Ω , R 3 )/R. It is noted that for any r ∈ R, γ (r ) is a rigid motion on ∂Ω , hence, π • γ (w + r ) = π • γ (w) + π • γ (r ) = π • γ (w)
. Thus, we have a well defined mapping

γ /R: L D(Ω )/R -→ L 1 (∂Ω , R 3 )/R, (2.18)
given by γ /R(χ ) = π • γ (w), for some w ∈ χ . Clearly,

π • γ = (γ /R) • π. (2.19)
In addition,

γ (w) + r 1 = γ (w + r ) 1 γ w + r . (2.20) Hence, [γ (w)] = inf r ∈R γ (w) + r 1 γ inf r ∈R w + r = γ [w] (2.21)
and we conclude that γ /R is indeed bounded and γ /R γ .

(2.22)

The dual mapping γ * : L ∞ (∂Ω , R 3 ) → (L D(Ω )) * may now be applied to traction fields and

(γ /R) * : (L 1 (∂Ω , R 3 )/R) * → (L D(Ω )/R
) * may be applied to equilibrated boundary traction fields to give L Dforces and equilibrated L D-forces respectively. Clearly, Proof. The mapping γ * is injective because γ is continuous and surjective. As the quotient space projection π: L 1 (∂Ω , R 3 ) → L 1 (∂Ω , R 3 )/R is also continuous and surjective, the same argument applies to γ /R.

π * • (γ /R) * = γ * • π * . ( 2
Remark 2.4. Henceforth, we will use the equivalent norm • ε as in (2.9) on L D(Ω )/R. We will also use the equivalent norm • ⊕ on L D(Ω ) as in (2.15). This implies that the quotient norm on L D(Ω )/R is actually equal (not only equivalent) to the norm induced by the strain. That is,

[w] = [w] ε = ε(w) 1 . (2.24)
It is noted that the last equality, to be used below frequently, is independent of the choice of a particular projection π

R : L D(Ω ) → R. Proposition 2.5. Any T ∈ (L D(Ω )/R) * is represented by some symmetric stress field σ ∈ L ∞ (Ω , R 6 ) * , in the form T = ε * (σ ), (2.25
)

where ε * : L ∞ (Ω , R 6 ) → (L D(Ω )/R) * is the dual mapping to ε: L D(Ω )/R -→ L 1 (Ω , R 6 ).
(2.26)

In addition, for the dual norm

• ε on (L D(Ω )/R) * , we have T ε = inf σ, T =ε * (σ ) σ ∞ , (2.27
)

and the infimum is attained for some σ ∈ L ∞ (Ω , R 6 ), i.e., T ε = σ ∞ .
(2.28)

Proof. Using the duality L 1 (Ω , R 6 ) * = L ∞ (Ω , R 6
), the assertion follows from the fact that ε:

L D(Ω )/R → L 1 (Ω , R 6
) is a norm-preserving (by our choice of norm as in Remark 2.4), linear injection and using the Hahn-Banach theorem (see [START_REF] Segev | Generalized stress concentration factors for equilibrated forces and stresses[END_REF] for the details).

Corollary 2.6. Let t ∈ L ∞ (∂Ω , R 3 ) be any equilibrated traction field so there is a t

0 ∈ (L 1 (∂Ω , R 3 )/R) * such that t = π * (t 0 ). Then, there exists some stress field σ ∈ L ∞ (Ω , R 6 ) such that (γ /R) * (t 0 ) = ε * (σ ), and γ * (t) = π * • ε * (σ ). (2.29)
In addition,

(γ /R) * (t 0 ) = inf ε * (σ )=(γ /R) * (t 0 ) σ ∞ , (2.30)
and

γ * (t) = inf π * •ε * (σ )=γ * (t) σ ∞ . (2.31)
Proof. We make repetitive use of (2.23) and (2.17) and Proposition 2.5. For example,

γ * (t) = γ * • π * (t 0 ) = π * • (γ /R) * (t 0 ) = (γ /R) * (t 0 ) = inf π * •ε * (σ )=π * •(γ /R) * (t 0 ) σ ∞ = inf π * •ε * (σ )=γ * •π * (t 0 ) σ ∞ = inf π * •ε * (σ )=γ * (t) σ ∞ . (2.32)
Remark 2.7. The conditions (2.29) are equivalent to the principle of virtual work -a weak form of the equations of equilibrium -of continuum mechanics (as it is assumed throughout that the body forces vanish). For example,

γ * (t) = π * • ε * (σ ), implies γ * (t)(w) = (π * • ε * )(σ )(w), (2.33)
so that for any w ∈ L D(Ω ),

t (γ (w)) = σ (ε • π(w)) = σ (ε(w)). (2.34)
Hence, for a vector field w that is the restriction of a differentiable field on Ω ,

∂Ω t i w i = Ω σ i j ε(w) i j . (2.35) Theorem 2.8. Let, γ /R = sup χ∈L D(Ω )/R (γ /R)(χ ) χ , (2.36)
be the norm of the trace mapping for distortions. Then, the generalized stress concentration factor K for the boundary traction problem, satisfies

K = γ /R .
(2.37)

Specifically, K = sup w∈C ∞ (Ω ) inf r ∈R w| ∂Ω + r 1 ε(w) 1 .
(2.38)

Proof. We have the standard

γ /R = (γ /R) * .
(2.39) However,

(γ /R) * = sup t 0 (γ /R) * (t 0 ) ε t 0 (2.40) = sup t 0 ∈((∂Ω ,R 3 )/R) * 1 t 0 inf ε * (σ )=(γ /R) * (t 0 ) σ ∞ (2.41) = sup t∈Image π * 1 t ∞ inf π * •ε * (σ )=γ * (t) σ ∞ , (2.42)
where we used Proposition 2.5 and Corollary 2.6. The condition π * • ε * (σ ) = γ * (t) is equivalent to the condition that the stress field σ is in equilibrium with t, i.e., σ ∈ Σ t , by Remark 2.7. The expression (2.38) simply uses the definition of γ /R and the fact that C ∞ (Ω ) is dense in L D(Ω ).

The scalar case and the trace mapping on the

Sobolev space W 1 1 (Ω )
The previous discussion is simplified considerably if we consider scalar fields ϕ ∈ W 1 1 (Ω ) instead of the vector fields w ∈ L D(Ω ). In this case the boundary data is also a scalar field, the analog of a stress is a vector field, and the vector space R of rigid motions is replaced by the real numbers.

Consider a sourceless vector field σ on Ω that satisfies boundary conditions for its boundary flux, i.e.,

σ i,i = 0, in Ω , (2.43) σ i ν i = t, on ∂Ω , (2.44)
for some given essentially bounded t: ∂Ω → R. Physically, σ may be thought of as a material flow field (say for an incompressible flow) for a given flux density t on the boundary. Alternatively, σ may be thought of as a heat flow field where there are no heat sources in Ω so t is the given heat flux on the boundary; or σ may be the electric displacement field and t is the charge density on the boundary.

The weak formulation of the problem is

Ω σ i ϕ ,i = ∂Ω tϕ. (2.45)
(The test function ϕ may be thought of as a potential field in the electrostatic example or as the reciprocal of the temperature in the heat transfer example.)

The analog of the stress concentration factor is then

K = sup t∈L ∞ (∂Ω ) K t = sup t∈L ∞ (∂Ω ) inf σ ∈Σ t σ ∞,Ω t ∞,∂Ω = sup t∈L ∞ (∂Ω ) inf σ ∈Σ t ess sup x∈Ω |σ (x)| ess sup y∈∂Ω |t (y)| .
(2.46)

We will refer to K as the generalized field concentration factor. Thus for example, for the interpretation of σ as a flow field, for a given flux density t, K t will be the smallest ratio between the maximal magnitude of the velocity and the maximal value of the given boundary flux. Thus, ϕ may be regarded as an element of the Sobolev space

W 1 1 (Ω ) so ϕ W 1 1 = ϕ 1 + ∇ϕ 1 .
(2.47)

On the Sobolev space, the trace mapping

γ : W 1 1 (Ω ) → L 1 (∂Ω ) (2.48)
is well defined as expected (see [START_REF] Adams | Sobolev Spaces[END_REF]). Our assumption that there are no sources in Ω implies that ∂Ω t = 0 which is equivalent to considering W 1 1 (Ω )/R and γ /R. Thus, with the obvious adaptation of the norm on W 1 1 (Ω ) so it is identical to the one on R ⊕ (W 1 1 (Ω )/R), the analog of Theorem 2.8 will be for the scalar case

K = γ /R , γ : W 1 1 (Ω ) -→ L 1 (∂Ω ).
(2.49)

3. Bounds on the W 1 1 -trace operator

The bounds obtained using normal vector fields

In this section we consider bounds for the trace operator

γ : W 1 1 (Ω ) -→ L 1 (∂Ω ). (3.1)
In particular, we are looking for bounds A and B satisfying

∂Ω |ϕ| A Ω |∇ϕ| + B Ω |ϕ| (3.2)
for every ϕ ∈ W 1 1 (Ω ). Let n be any C 1 -vector field on Ω and ψ the restriction to Ω of a W 1 1 -function defined in an open neighborhood of Ω . Then,

Ω n i ψ ,i = Ω (n i ψ) ,i - Ω n i,i ψ (3.3)
implies using the Gauss-Green theorem that

∂Ω n i ν i ψ = Ω n i ψ ,i + Ω n i,i ψ, (3.4)
where ν is the outwards pointing unit normal to ∂Ω .

Definition 3.1. The vector field n on Ω will be referred to as a normal field if the following conditions hold.

(i) n(y) = ν(y) for all y ∈ ∂Ω .

(ii) |n|(x) 1 for all x ∈ Ω , where here and in the rest of this section we use the Euclidean norm for elements of

R n so |n| = √ n i n i .
The existence of normal vector fields is discussed in some detail below. We will use N(Ω ) for the collection of all normal vector fields. For a normal field, Eq. (3.4) assumes the form We now estimate each of the integrals on the right hand side.

∂Ω ψ = Ω n i ψ ,i + Ω n i,i ψ. ( 3
Ω n i |ϕ| ,i Ω |n|||ϕ| ,i | Ω |∇ϕ|, (3.7)
where we used Definition 3.1(ii) and

||ϕ| ,i | = |ϕ ,i | |∇ϕ|. Also Ω n i,i |ϕ| max Ω |n i,i | Ω |ϕ|. (3.8)
It follows that

B(Ω ) = inf n∈N(Ω ) max x∈Ω |n i,i (x)| = inf n∈N(Ω ) n i,i ∞,Ω . (3.10)
Then, the following exact inequality holds

ϕ 1,∂Ω = ∂Ω |ϕ| Ω |∇ϕ| + B(Ω ) Ω |ϕ| = ∇ϕ 1 + B(Ω ) ϕ 1 .
(3.11)

Estimation using harmonic normal fields

We now consider the existence of normal vector fields. By the assumption that ∂Ω is C 2 , ν is a C 1 -vector field on ∂Ω and we can extend it to a vector field n with |n(x)| 1 for all x ∈ Ω (see for example Theorem 3.6.2 in [START_REF] Ziemer | Weakly Differentiable Functions[END_REF]). Furthermore, we can require that the extension is harmonic in the following sense. For a vector field n, we use n for the vector field ( n) j = (n j ) ,ii . The field n is harmonic in Ω if n = 0, for all x ∈ Ω .

(3.12)

Theorem 3.3.
Let Ω be a bounded open subset of R n such that ∂Ω is C 2 . Then, there exists a unique normal vector field n 0 ∈ N(Ω ) which is harmonic. In addition, for the supremum of the divergence, ∇ • n 0 = n 0i,i , we have

∇ • n 0 ∞,Ω = ∇ • n 0 ∞,∂Ω . (3.13)
Proof. For any fixed j, we have a classical Dirichlet problem

n j = 0 in Ω , n j = ν j on ∂Ω . (3.14)
Given our smoothness assumption on ∂Ω , there is a unique solution n 0 j to each such boundary value problem and we obtain the harmonic vector field n 0 . For |n 0 | 2 = n 0 j n 0 j we have,

(n 0 j n 0 j ) = (n 0 j n 0 j ) ,ii
= (2n 0 j,i n 0 j ) ,i = 2n 0 j,ii n 0 j + 2n 0 j,i n 0 j,i .

(3.15)

In the last line, the first term vanishes because n 0 j is harmonic and hence (|n 0 | 2 ) 0. We conclude that |n 0 | 2 is subharmonic in Ω . By the maximum principle for subharmonic functions max

x∈Ω |n 0 (x)| 2 = max y∈∂Ω |n 0 (y)| 2 = max y∈∂Ω |ν(y)| 2 = 1. (3.16)
Thus, in addition to the boundary conditions, n 0 satisfies the condition 3.1(ii), and so n 0 ∈ N(Ω ).

Next, we note that for the harmonic n 0 ,

(∇ • n 0 ) = (n 0 j, j ) ,ii = (n 0 j,ii ) , j = 0. (3.17)
Thus, ∇ • n 0 is also harmonic in Ω so by the maximum principle Eq. (3.13) holds.

It turns out that the harmonic vector field of Theorem 3.3 plays an important role in the computation of B(Ω ), the second constant of the Sobolev W 1 1 (Ω ) trace inequality. Continuing to use n 0 for the unique harmonic normal vector field we have Theorem 3.4. The Sobolev constant B(Ω ) is given by

B(Ω ) = inf n∈N (Ω ) ∇ • n ∞,Ω = ∇ • n 0 ∞,∂Ω . (3.18)
Proof. Let (n m ), m ∈ N, be a sequence of normal vector fields such that

lim m→∞ ∇ • n m = B(Ω ). (3.19)
Using normal tubular neighborhoods (e.g., [11, p. 110]), there is a δ > 0 such that we can parameterize an open neighborhood V of ∂Ω in Ω by

(y, z) ∈ ∂Ω × [0, δ), (3.20)
where for each x ∈ V , y(x) is a unique point on the boundary such that x is on the line through y which is normal to the boundary, and z is the distance to the boundary along the normal line where x is situated. For each m, let V m be the open set

V m = x ∈ Ω : y(x) ∈ ∂Ω , z(x) < δ/m , (3.21)
and let

Ω m = Ω -V m so ∂Ω m = x ∈ Ω : y(x) ∈ ∂Ω , z(x) = δ/m . (3.22)
Now for each m we construct the harmonic lifting (cf. [12, p. 24]) n m of n m as follows. Let n 0m be the solution of the Dirichlet problem on Ω m with the boundary conditions n 0m (x) = n m (x) for x ∈ ∂Ω m . Set

n m (x) = n 0m (x), for x ∈ Ω m , n m (x), for x ∈ V m . (3.23)
By the maximum principle,

n mi ∞,Ω n mi ∞,Ω and ∇ • n m ∞,Ω ∇ • n m ∞,Ω , (3.24) so lim m→∞ ∇ • n m ∞,Ω = B(Ω ). (3.25)
In addition, as the various n mi ∞,Ω are bounded by 1, the same applies to n mi and the sequence n mi is uniformly bounded. Thus, (using a standard normal family argument) by Ascoli's theorem, it has a subsequence that converges uniformly to a limit continuous normal field n. On any compact subset of Ω this gives a uniformly convergent sequence of harmonic functions whose limit is then also harmonic. Thus, n is harmonic. Also, the limit n satisfies the conditions of Definition 3.1 and so it is a normal vector field. Finally, by the uniqueness of the solution to the Dirichlet problem n = n 0 .

Remark 3.5. Within the framework of the AB-program in geometric analysis, [START_REF] Druet | The AB-program in geometric analysis: Sharp Sobolev inequalities and related probelms[END_REF], Motron proves in [START_REF] Motron | Around the best constants for the Sobolev trace map from W 1,1 (Ω ) into L 1 (∂Ω )[END_REF], using different methods, the following two theorems for bounds on the trace mapping on W 1 1 (Ω ). (1) For any ε > 0, there exists a B ε such that for any

ϕ ∈ W 1 1 (Ω ), ∂Ω |ϕ| (1 + ε) Ω |∇ϕ| + B ε Ω |ϕ|. (3.26) (2) Assuming that Ω is a connected bounded open subset of R n whose boundary is piecewise C 1 , there exists A > 0 such that for any ϕ ∈ W 1 1 (Ω ), ∂Ω |ϕ| A Ω |∇ϕ| + |∂Ω | |Ω | Ω |ϕ|. (3.27)
We note that even if we had the solution to the AB-program, we would not have a concrete bound on γ . What we sought were simultaneous bounds on both A and B.

In addition, for a normal vector field

sup x∈Ω |∇ • n||Ω | Ω ∇ • n = ∂Ω n • ν = |∂Ω |, (3.28) so inf n∈N ∇ • n ∞,Ω |∂Ω | |Ω | . (3.29)
However, the inequality is not exact and equality is not attainable even for the harmonic normal vector field. (Think of two circles connected by a narrow neck of width t. Across the narrow neck, ∇ • n has to be of order 1/t in order to satisfy the boundary conditions. The values of |Ω | and |∂Ω | are not significantly different from those of the two circles.) Thus, Motron's bound B is smaller than the value obtained here. On the other hand, the bounds A and B we obtain are exact in the sense that you cannot lower B without increasing A.

Norms of symmetric tensors

As noted earlier, the value of the stress concentration factor depends on the norm we choose to use on the space of stress matrices. Thus, for the sake of completeness, we review below some elementary properties of norms of symmetric matrices on R n . We will denote the norm of a matrix T by |T | (reserving • for norms on function spaces).

Operator norms

In general, for a linear mapping T : V → U between normed spaces, the operator norm of T is defined by

|T | o = sup v |T (v)| |v| , v = 0. (4.1)
The operator p-norm, |T | op , 1 p ∞, on the space of matrices is defined as the operator norm for the case where the p-norm is used on both V = R m and U = R n . By the compactness of the unit ball in R m , the supremum is attainable and

|T | op = max |v|=1 |T (v)| p . (4.2)
In case T : W → W is a linear transformation defined on the inner product space R n equipped with the Euclidean 2-norm, |T | o2 may be calculated by

|T | o2 = sup v,v ∈W |T (v) • v | |v||v | , v, v = 0. (4.
3)

The following relations hold for symmetric matrices on R n .

|T | o1 = |T | o∞ = max i j |T i j |, (4.4
)

|T | o2 = max{|λ 1 |, . . . , |λ n |}, (4.5)
where λ 1 , . . . , λ n are the (real) eigenvalues of T . The norm |•| o2 is usually referred to as the spectral radius norm.

Vector norms

We will also regard symmetric matrices as vectors in R n(n+1)/2 and use the p-norm for them. Thus,

|T | p = i, j |T i j | p 1/ p . (4.6)
In particular,

|T | 1 = i, j |T i j |, (4.7
)

|T | ∞ = sup i, j |T i j |, (4.8
)

|T | 2 = T i j T ji = λ 2 i 1/2 . (4.9)
We recall that the Frobenius norm of a matrix is |T | F = (T i j T i j ) 1/2 and it is identical to |•| 2 for symmetric matrices.

Dual norms

Being a finite dimensional space we may identify the space of symmetric matrices R n(n+1)/2 with its dual space. Thus, we may regard any symmetric matrix T as a linear functional so that T (S) = T i j Dual norms may be used also for the operator norms. In particular, we note that

|T | o2 * = i |λ i |. (4.11)
In closing this short review, it is noted that the norms containing the index 2 are associated with the Euclidean norm for vectors and may be expressed in terms of the eigenvalues. These norms are invariant under orthogonal transformations of coordinates.

The equivalence constants

Since all the norms listed above are equivalent, for each pair of norms |•| a and |•| b , there is a finite positive number

K a b = sup T |T | a |T | b . (4.12)
In particular, the following exact relations hold:

1 √ n |T | o2 |T | o1 √ n, 1 |T | 2 |T | o2 √ n, 1 |T | o2 |T | ∞ n, (4.13) 1 |T | o2 * |T | o2 n, 1 |T | o2 * |T | 2 √ n. (4.14)
Thus, for example, K o2 2 = 1. In the sequel, we will use |σ (x)| to denote the norm of the value of a symmetric tensor σ at x ∈ Ω using a generic norm on the space of matrices.

Optimal boundary conditions for stresses

Consider the following problem: given a unit vector ν in a Euclidean 3-dimensional space and a unit vector t, find a symmetric matrix σ such that (i) σ (ν) = t-the compatibility condition; (ii) |σ | = inf{|T |, T (ν) = t, T = T T }, i.e., σ is the optimal symmetric matrix that satisfies condition (i). The problem has an obvious mechanical interpretation. If ν denotes the normal to the boundary at some given point, and t denotes the value of the surface traction field at that point, then, σ (ν) = t is the boundary condition for the stress field σ . Thus, a matrix σ satisfying the conditions above is the optimal stress matrix that will satisfy the boundary condition. Obviously, the normalization condition on t causes no loss of generality.

Let t n = (t • ν)ν = ν ⊗ ν(t) be the normal component of t and let t t = ν × (t × ν) be the tangent component of t. Thus, denoting the angle between ν and t by θ, |t n | = cos θ and |t t | = sin θ. We choose a basis { f j } where f 1 = ν, f 2 is a unit vector in the direction of t t and f 3 completes the other two to form a right-hand oriented orthonormal basis. In this basis, the matrix of σ satisfying the condition σ (ν) = t has to satisfy σ 11 = cos θ, σ 12 = σ 21 = sin θ, and σ 13 = σ 31 = 0. The rest of the components cannot be determined by the compatibility condition above and should be determined by the requirement for minimal norm of σ . (In the case where ν and t are parallel, one can take any orthonormal basis containing ν.)

Optimal boundary conditions relative to the |•| ∞ -norm

We wish to regard σ as an element of the dual space of symmetric matrices. Then, using the basis f i as above, the compatibility condition implies that we have a linear functional σ 0 defined on the subspace V of symmetric matrices containing elements of the form

[ε] =   ε 11 ε 12 0 ε 12 0 0 0 0 0   . (5.1)
The functional σ 0 acts on elements of V by

σ 0 ([ε]) = ε 11 cos θ + 2ε 12 sin θ. (5.2)
Thus, on this subspace

sup{|σ 0 (ε)|; ε ∈ V, |ε| 1 = 1} = max{| cos θ|, | sin θ|}. (5.3)
The extension

[σ ] =   cos θ sin θ 0 sin θ 0 0 0 0 0   (5.4)
of σ 0 to the space of all symmetric matrices, has the same norm and as such it provides the optimal boundary condition.

It is noted that while the development depends on the basis chosen, the optimal norm depends only on the angle θ, an invariant quantity.

Optimal boundary conditions relative to the |•| 2 -norm

If one uses |σ | = |σ | 2 induced by the inner product in the space of symmetric matrices as in the previous section, the optimal matrix can be obtained by orthogonality conditions of the optimal stress to basis vectors for the matrices that correspond to the undetermined components of the stress. This implies that all the undetermined components should vanish. Thus, the optimal stress is given in the { f i }-basis by Eq. (5.4) also, and

|σ | 2 = 1 + sin 2 θ . (5.5)
While the optimal σ for the |•| ∞ -norm depended on the basis { f i } chosen, the construction here is rotation-invariant.

The case where t = e k

We now consider the special case where t = e k , where e k , k = 1, 2, 3 is a base vector. In this case, cos θ = ν • e k = ν k and sin θ = 1 -ν 2 k . Using

σ = cos θ ν ⊗ ν + sin θ (ν ⊗ f 2 + f 2 ⊗ ν), (5.6)
we may write the matrix for σ relative to the {e i }-basis. We first note that

f 2 = e k -cos θ ν sin θ = e k -ν k ν 1 -ν 2 k , (5.7)
thus,

σ = ν k ν ⊗ ν + ν ⊗ (e k -ν k ν) + (e k -ν k ν) ⊗ ν. (5.8)
Rearranging the terms we conclude that for the case t = e k , the optimal boundary conditions for the stress are

σ = -ν k ν ⊗ ν + (ν ⊗ e k + e k ⊗ ν), (5.9)
and the optimal values are Using the same notation as above and using the coordinate system in the plane where the x and y axes are along f 1 and f 2 , respectively, we are looking for a 2 × 2 symmetric matrix that will satisfy the condition t = σ (ν) of least spectral radius, i.e., minimizes max{|λ 1 |, |λ 2 |}. The condition σ (ν) = t implies that σ x x = cos θ and σ x y = sin θ. Thus, to determine σ completely, one has to determine the single number σ yy .

|σ | 2 = 2 -ν 2 k , |σ | ∞ = max |ν k |, 1 -ν 2 k , for the natural basis { f i }.
In two dimensions we have the explicit expression for the eigenvalues as

λ 1,2 = σ x x + σ yy 2 ± σ x x -σ yy 2 2 + σ x y 2 . (5.10) Setting a = cos θ 2 = σ x x 2 , z = σ yy 2 , so σ 2 x y = sin 2 θ = 4 -a 2 , (5.11)
we have

λ 1,2 = a + z ± 1 -3a 2 -2az + z 2 = a + z ± (z -a) 2 + 1 -4a 2 .
(5.12)

Minimizing |λ 1,2 | is like minimizing λ 2 1,2 so we differentiate with respect to z. Thus,

dλ 2 1,2 dz = 2λ 1,2 dλ 1,2 dz = 2λ 1,2 1 ± z -a (z -a) 2 + 1 -4a 2
that vanishes identically only if 4a 2 = 1. Hence, for extremum, θ = 0. In general,

1 -4a 2 = 1 -cos 2 θ > 0, (5.13)
which implies that dλ 2 1,2 dz (5.14) has the same sign as the eigenvalue. It follows that z = 0 can only make |λ 1,2 | larger and the infimum is attained for z = σ yy = 0.

Remark 5.1. We note that the problem of optimal boundary condition for the stress is a generalization of the requirement in Definition 3.1(i) for the boundary value of a normal vector field with the difference that now we consider matrices rather than vectors. Indeed, n = ν gives the smallest value for |n| 2 , the Euclidean norm of the vector field n, such that n • ν = n(ν) = 1.

Worst case optimal boundary conditions

By worst case optimal boundary conditions we refer to the inclination of t relative to the normal ν for which the norm of the optimal stress attains a maximal value. That is, we are looking for

D = sup t inf σ |σ | : σ (ν) = t, |t| = 1 .
(5.15)

The number D (the subscript indicating the particular norm chosen) and the corresponding σ and t depend only on the choice of norms. For example, for the |•| 2 -norm, D 2 = √ 2 is attained for any unit traction t perpendicular to ν. For the |•| ∞ -norm, relative to the boundary natural basis, D ∞ = 1 is attained for either traction that is parallel to ν or traction that is perpendicular to it. 5.6. The spaces N k (Ω ) and e k -optimal tensor fields For k = 1, 2, 3 we now choose a symmetric tensor field T k = (T k i j ) on ∂Ω that satisfies the following conditions. (i) If e k denotes the k-th base vector in R 3 , then, T k (ν) = e k so T k i j ν j = δ k i . (ii) Clearly, at each point y on the boundary there is a collection of matrices that satisfy condition (i) above. We choose T k (y) to be a symmetric matrix that satisfies condition (i) and that has the least norm (of our choice on the space of matrices). Thus,

|T k (y)| = inf S k |S k | : S k (ν(y)) = e k , (5.16)
where the infimum is taken over all symmetric matrices. Thus, in the terminology of the preceding subsections, T k is the optimal stress for the e k as boundary traction and the discussion of Section 5.3 applies.

Clearly, the fields T k depend on Ω only. They depend continuously on ν so by the assumption that ∂Ω is C (5.17)

As ∂Ω is assumed to be smooth, any angle between ν and any fixed vector is attained on the boundary, hence, the worst case optimal boundary conditions are attained on the boundary always. Thus,

sup y∈∂Ω |T k (y)| = sup y∈∂Ω inf S k |S k |: S k (ν(y)) = e k = D (5.18)
and depends only on the choice of a norm. We use the notation T k for a tensor field on the boundary satisfying the two conditions above. Using Whitney's extension theorem (cf. [10, Theorem 3.6.2]), T k can be extended to symmetric differentiable tensor fields σ k on Ω that satisfy the following condition We denote the class of symmetric C 1 -tensor fields σ k on Ω that satisfy the boundary condition σ k (y) = T k (y), y ∈ ∂Ω , and condition (5.19) above by N k (Ω ). We will refer to a tensor field σ k ∈ N k (Ω ) as an e k -optimal tensor field.

It is quite clear that the foregoing discussion applies in the continuum mechanics context to the optimal stress field for the normalized boundary traction t = e k . If σ k ∈ N k (Ω ), we have for the stress concentration factor and optimal stress concentration factor,

K e k ,σ k = K e k = D .
(5.20)

Although the total force on the body is not equilibrated, D , may serve as a bound.

6. Bounds of the L D(Ω )-trace operator Theorem 6.1. Let the constants A(Ω ) and B(Ω ) be given by A(Ω ) = 3D , (6.1)

B(Ω ) = k ∇ • σ k 0 ∞,∂Ω , (6.2)
where σ k 0 is the solution of the Dirichlet problem σ k i j = 0, in Ω , σ k i j = T k i j , on ∂Ω . Then,

w 1,∂Ω A(Ω ) ε(w) 1 + B(Ω ) w 1 , (6.3)
for all w ∈ L D(Ω ), is an exact estimate.

The following subsections present the proof. (We will use A and B for A(Ω ) and B(Ω ), respectively, in order to simplify the notation.)

The principle of virtual work

Let Ω be an open region in R 3 having a smooth boundary, σ = (σ i j ) a symmetric smooth tensor field on Ω and w = (w i ) an L D-vector field on Ω . Then, σ i j w i, j = (σ i j w i ) , j -σ i j, j w i .

(6.4) Also, by the symmetry of σ , σ i j w i, j = σ i j ε i j . (6.5) Thus, we may write

Ω σ i j ε i j = Ω (σ i j w i ) , j + Ω σ i j, j w i , (6.6)
and using the Green-Gauss theorem on the first term on the right-hand side we obtain

Ω σ i j ε i j = Ω σ i j 1 2 (w i, j + w j,i ) = ∂Ω σ i j w i ν j + Ω σ i j, j w i , (6.7)
where ν is the unit outward pointing normal. We will refer to the identity above as the principle of virtual work.

The bounds

We now write the principle of virtual work (6.7) for the vector field w = (|w i |). (We reserve the notation |v| for the norm of the vector v in a finite dimensional space.) Thus,

Ω σ i j 1 2 (|w i | , j + |w j | ,i ) = ∂Ω σ i j |w i |ν j + Ω σ i j, j |w i |. (6.8)
Let σ k satisfy σ k (ν) = e k on ∂Ω so σ k i j ν j = δ k i . Then, the identity above assumes the form

∂Ω |w k | = Ω 1 2 (|w i | , j + |w j | ,i )σ k i j - Ω σ k i j, j |w i |. (6.9) Consider the integrand integrand = 1 2 (|w i | , j + |w j | ,i )σ k i j (6.10)
of the first integral on the right. As the expression is invariant under orthogonal transformations, it may be evaluated in the principle coordinate system of the matrix (|w i | , j + |w j | ,i )/2 where the off-diagonal elements vanish. Thus, without loss of generality, we may write (we do not use the summation convention here)

integrand = i |w i | ,i σ k ii max i {|σ k ii |} i |w i | ,i |σ k | ∞ i |w i,i | (using |w i | , j = sign(w i )w i, j ) |σ k | ∞ |ε(w)| 1 ,
where the equality is clearly attainable. Thus,

∂Ω |w k | Ω |σ k | ∞ |ε(w)| 1 + Ω |σ k i j, j ||w i | sup x∈Ω |σ k (x)| ∞ Ω |ε(w)| 1 + sup i,x∈Ω {|σ k i j, j (x)|} Ω i |w i |. As sup i,x∈Ω |σ k i j, j (x)| = ∇ • σ k ∞,Ω , (6.11)
we have

w k 1,∂Ω = ∂Ω |w k | σ k ∞,Ω ε(w) 1 + ∇ • σ k ∞,Ω w 1 , (6.12)
where we use

T ∞ = |T | ∞ L ∞ , and T 1 = |T | 1 L 1 ,
for the respective norms of a tensor field T . Adding this equation for k = 1, 2, 3, we obtain for the L 1 -norm of the restriction of w to the boundary the following bound

w 1,∂Ω k σ k ∞,Ω ε(w) 1 + k ∇ • σ k ∞,Ω w 1 . (6.13)
Clearly, for and equality holds for σ k ∈ N k (Ω ), we conclude that A is attained for fields σ k ∈ N k (Ω ) and A = 3D . This proves the first part (Eq. (6.1)) of Theorem 6.1.

A = inf σ k k σ k ∞,Ω : σ k (ν) = e k (6.14) and B = inf σ k k ∇ • σ k ∞,Ω : σ k (ν) = e k ( 6

Estimating B

The procedure we use is completely analogous to that of Section 3.2 and the proofs of Theorems 3.3 and 3.4. For a tensor field σ we use σ for the Laplacian σ i j = σ i j,ll . We say that σ is harmonic if σ = 0. Proposition 6.2. There is a unique harmonic tensor field σ k ∈ N k (Ω ), i.e., σ k is e k -optimal. For the harmonic e k -optimal σ k , we have

∇ • σ k ∞,Ω = ∇ • σ k ∞,∂Ω . (6
.17)

Proof. Consider the Dirichlet problem

σ k = 0, in Ω , σ k = T k , on ∂Ω . (6.18)
The existence and uniqueness are standard. Let σ k be harmonic, then, for each component σ k i j (no sum on repeated indices) Hence, the solution is also in N k (Ω ). Finally,

((σ k i j ) 2 ) = (σ k i j σ k i j ) ,ll , = 2(σ k i j,l σ k i j ) ,l = 2σ k i j,l σ k i j,l + 2σ k i j,ll σ k i j 0. ( 6
(∇ • σ k ) = (σ k i j, j ) ,ll
= (σ k i j,ll ) , j = 0, (6.22) so by the maximum principle for the components of ∇ • σ k , (6.17) holds.

Proof of the second part of Theorem 6.1 (Eq. (6.2)). Clearly, as the three σ k fields are independent, we should look for We apply the normal family argument and uniqueness of solutions as in the proof of Theorem 3.4, to obtain

B k = inf σ k ∇ • σ k ∞,Ω : σ k (ν) = e k . ( 6
B k = ∇ • σ k 0 ∞,∂Ω , (6.28)
where σ k 0 is the solution of the Dirichlet problem (6.18). Eq. (6.2) now follows from (6.15).

Discussion

We may describe the foregoing analysis in the mechanical context. For a given boundary traction field, we constructed the optimal boundary condition for the stress field. In Section 5, the case where the traction vector was a unit vector was considered, but as the relation between stress and traction is linear, this causes no loss of generality. Thus, one can assign the optimal boundary condition for the stress field for any given boundary traction field.

Next, one can solve the Laplace equation for each of the stress components. Unlike the usual case of continuum mechanics, we have a unique solution without imposing constitutive relations and the equilibrium equations are not satisfied. For the harmonic solution of the boundary value problem, the maximal stresses occur on the boundary and these stresses are the smallest that satisfy the traction boundary conditions. Eq. (6.2) and its proof indicate that the maximal value of ∇ • σ is the smallest possible. In light of the usual equilibrium equations ∇ • σ + b = 0 of continuum mechanics (b being the body force field), we can interpret the field -∇ • σ as additional body forces one has to supply for the equilibrium condition to hold. Thus, for equilibrium, the harmonic stress field is associated with an additional body force field whose maximum is the least (and is attained on the boundary). It is noted that the total of the traction field, ∂Ω t, was not required to vanish so it is not possible for equilibrium to hold. With the foregoing limitations in mind, the harmonic stress field solves the problem of optimal stress field for a given traction.

Next, we note that the generalized stress concentration factor may be described as the largest optimal stress concentration factor when we can vary the boundary traction fields while keeping their maximal value on the boundary to t ∞,∂Ω = 1. Thus, in the analysis all the components of the traction fields are set to be 1 everywhere. Again, this precludes equilibrium as the total force on the body in each direction is equal to the area of the boundary. The mathematical analog of this limitation is that we obtain bounds on γ and not γ /R. However, the bound on γ gives a bound γ /R because γ /R γ (Eq. (2.22)).

.23) Proposition 2 . 3 .

 23 The mappings γ * and (γ /R) * are injective.

. 5 )

 5 Given a C 1 mapping ϕ on Ω , the distributional derivatives |ϕ| ,i of its absolute value |ϕ|(x) = |ϕ(x)| are clearly integrable and hence, |ϕ| is W 1 1 . Rewriting Eq. (3.5) for ψ = |ϕ| we obtain ∂Ω |ϕ| = Ω n i |ϕ| ,i + Ω n i,i |ϕ|. (3.6)

9 )

 9 This inequality is clearly exact and recalling the definition of normal vector fields we have Theorem 3.2. Let Ω be a bounded open set in R n having a C 2 -boundary and set

  S ji and assign to it the dual norm |T | p * |T | p * = sup S |T (S)| |S| p , (4.10) where we have the usual |•| p * = |•| q , for q = p/( p -1).

5. 4 .

 4 Example: The 2-dimensional case for the |•| o2 -spectral radius norm

  2 they are continuous. Consider now the worst value of |T k (y)| on the boundary, i.e., sup y∈∂Ω |T k (y)| = sup y∈∂Ω inf S k |S k | : S k (ν(y)) = e k .

  sup x∈Ω |σ k (x)| = sup y∈∂Ω |T k (y)| = D .(5.19)

  .15) this bound is the tightest and we have w 1,∂Ω A ε(w) 1 + B w 1 , for all w ∈ L D(Ω ). (6.16) Since in general sup x∈Ω |σ k (x)| ∞ sup y∈∂Ω |σ k (y)| ∞ sup y∈∂Ω |T k (y)| ∞ , T k optimal as in (5.16) = D

  .19) Thus, σ k i j is subharmonic in Ω . By the maximum principle for subharmonic functions max x∈Ω analogous property holds for |σ k i j |. Thus, using the boundary conditionsσ k ∞,Ω = max i, j,x∈Ω |σ k i j (x)| = max i, j,y∈∂Ω |σ k i j (y)| = max i, j,y∈∂Ω|T k i j (y)| = D ∞ . (6.21)

  .23) Thus, let (σ k m ), m ∈ N, be a sequence of e k -optimal tensor fields such thatlim m→∞ ∇ • σ k m = B k . (6.24)The sets V m and Ω m may be constructed just as in the proof of Theorem 3.4. Let σ k 0m be the solution of the Dirichlet problem in Ω m with the boundary condition σ k 0m (x) = σ k m (x), for x ∈ ∂Ω m , and define the harmonic lifting accordingly asσ k m (x) = σ k 0m (x), for x ∈ Ω m , σ k m (x), for x ∈ V m .

Acknowledgements

The research leading to this paper was partially supported by the Paul Ivanier Center for Robotics Research and Production Management at Ben-Gurion University.