Cosimo Prete

Laurent Galmiche

Fifonsi-Gwladys Quenum-Possy-Berry
email: fifonsi-gwladys.quenum-possy-berry@gendarmerie.interieur.gouv.fr

Clémence Allain
email: callain@ppsm.ens-cachan.fr

Nicolas Thiburce

Thomas Colard

Lumicyano™: a new fluorescent cyanoacrylate for a one-step luminescent latent fingermark development

Keywords: Fluorescent cyanoacrylate, latent fingermark, non-porous surfaces, semi-porous substrates, dye staining, Lumicyano. 1

Latent fingermarks developed by cyanoacrylate fuming often lack contrast; therefore further enhancement is required, such as dye staining. This second step is part of the conventional detection sequences performed by forensics practitioners. Dye-staining or powder dusting aims at improving contrast and at increasing the legibility of details, yet their use may at times be limited. Indeed powder dusting may not be effective due to unexpected adherence to the background, and poor affinity to the cyanoacrylate. In the same way staining processes can dye a whole semi-porous surface or may wash the marks.

To avoid that second step, a new luminescent cyanoacrylate (Lumicyano™) which allows one-step development without changing the fuming chamber settings (80% humidity rate, 120°C fuming temperature) was developed and assessed. This study aimed at comparing Lumicyano™ to a conventional two-step process. A detailed sensitivity study was conducted on glass slides, as well as the processing of various non-porous and semi-porous substrates, usually considered as problematic for a dye staining step.

The results indicate that Lumicyano™ detects fingermarks with equal or better sensitivity and ridge details than currently used cyanoacrylate. Secondly in luminescent mode, good ridges clarity and excellent contrast are observed, even if Lumicyano™ is sometimes less bright than the two-step process. Furthermore, conventional enhancement can still be carried out if needed. As a conclusion, Lumicyano™ makes it possible to avoid an enhancement step which can be detrimental to further examinations, particularly on rough or semi-porous surfaces.

ACKNOWLEDGEMENTS:

We warmly thank Pr Pierre Audebert, head of the PPSM research team, who initiated this research subject and supported us with helpful discussions throughout.

We are also greatful for the time the IRCGN fingerprint unit staff spent on the operational evaluation and particularly to the practitioners who assessed the processed fingermarks. INTRODUCTION Among the lot of finger mark development techniques reported in the literature (for a review see [START_REF] Bleay | Fingerprint Source Book[END_REF] and references therein), cyanoacrylate fuming is an extremely simple and efficient method described for the first time in the late 1970s, in Japan and slightly later in UK and Canada [START_REF] Yamashita | Latent print development, in: The Fingerprint sourcebook[END_REF][START_REF] German | Cyanoacrylate (Superglue) Fuming Tips[END_REF].

Items with potential latent marks are placed in a tightly closed fumigation chamber under 80% humidity atmosphere and the cyanoacrylate evaporates upon heating at 120 °C [START_REF] Bleay | Finger mark development techniques within scope of ISO 17025[END_REF]. Although the chemical process is still under debate and not well understood [START_REF] Czekanski | A mechanistic model for the superglue fuming of latent fingerprints[END_REF][START_REF] Wargacki | Understanding the chemistry of the development of latent fingerprints by superglue fuming[END_REF][START_REF] Wargacki | Enhancing the quality of aged latent fingerprints developed by superglue fuming: loss and replenishment of initiator[END_REF][START_REF] Thiburce | Design of a Control Slide for Cyanoacrylate Polymerization: Application to the CA-Bluestar Sequence[END_REF], the anionic polymerization of cyanoacrylate takes place, in this condition, probably initiated by a variety of compounds contained in the residue constituting the mark, such as amino acids, fatty acids and proteins.. Thus, latent marks are efficiently detected on non-porous and semi-porous substrates as a sticky white material that forms along the papillary ridges.

However, the main limitation of this technique arises from the white colour taken by the detected fingermarks, which often lacks contrast with the light coloured substrates. Several post-treatments have been proposed to overcome this issue. They consist in dusting the fingermark with coloured or fluorescent dye powders [START_REF] Liu | The effectiveness of strong afterglow phosphor powder in the detection of fingermarks[END_REF], or staining the CA-developed marks with a fluorescent dye solution such as Ardrox [START_REF] Mc Carthy | Evaluation of ARDROX as a luminescent stain for cyanoacrylate processed latent impression[END_REF], Basic Yellow 40, or Rhodamine 6G [START_REF] Mazella | Additional study of cyanoacrylate stains[END_REF]. However, these post-treatments are time-consuming and sometimes contain carcinogenous compounds. For example, Rhodamine 6G itself is suspected to be carcigenous [START_REF]Rhodamine 6G[END_REF] and Ardrox and Basic Yellow 40 solutions can contain methanol: the use of these fluorescent solutions requires appropriate protection for laboratory workers, such as fume-cupboards, and adequate waste elimination, which are expensive and require space. Moreover, staining treatment with fluorescent dyes is not possible on various substrates, particularly on semi-porous ones. Indeed the staining solution may also stain the background.

Finally, on non-porous surfaces, to avoid too a high background reaction, the stain has to be rinsed and fingerprints may be washed away; this rinsing step is also detrimental to electrical equipment which cannot be immersed.

Few studies [START_REF] Morimoto | A new method to enhance visualization of latent fingermarks by sublimating dyes, and its practical use with a combination of cyanoacrylate fuming[END_REF][START_REF] Takatsu | Vapor-phase Staining of Cyanoacrylate-Fumed Latent Fingerprints Using p-Dimethylaminobenzaldehyde[END_REF] propose staining methods based on the sublimation of a fluorescent dye. These methods are usually difficult to use because of the high temperature needed to obtain the sublimation of the dye or because of the high exposition time of the cyanoacrylate-fumed latent fingerprint to the dye vapour-phase. In both cases, these post-treatments are also time-consuming.

Considering all these limitations, the development of a one-step fluorescent cyanoacrylate method has been a topic of outstanding interest for several years [START_REF] Weaver | One-step fluorescent cyanoacrylate fingerprint development technology[END_REF][START_REF] Spring | Visualisation of latent fingerprints with cyanoacrylates, Secondary Visualisation of latent fingerprints with cyanoacrylates[END_REF] . Hahn and Ramotowski [START_REF] Hahn | Evaluation of a Novel One-Step Fluorescent Cyanoacrylate Fuming Process for Latent Print Visualization[END_REF] have recently described such a fluorescent cyanoacrylate which requires a very high temperature (230 °C) for fumigation. This severely limits the application of this new product, since it requires a modification of the fuming chambers and since there is a risk, at such a high temperature, of cyanoacrylate decomposition to product lethal cyanide gas [START_REF] Fung | Investigation of hydrogen cyanide generation from the cyanoacrylate fuming process used for latent fingermark detection[END_REF].

We report here the use of a new fluorescent cyanoacrylate, Lumicyano™, developed by Crime Scene Technology (CST) in association with the Ecole Normale Supérieure (ENS) de Cachan, containing a easily sublimable low-molecular weight fluorophore that allows the one-step fluorescent revelation of latent fingerprints under standard humidity and temperature fumigation conditions (80% humidity, 120 °C). The fluorescence can be observed either under UV (315-340 nm) or visible (450-550 nm) intense light irradiation, ensuring a good compatibility with the lightning material available within most police forces.

The quality of this new fingermark development process has been assessed by the French Gendarmerie Forensic Science Institute (IRCGN). First, a detailed comparison with a usual twostep process (cyanoacrylate fuming followed by staining with Basic Yellow 40) has been conducted on glass slides, on fresh as well as on aged fingermarks (up to 4 weeks). Then, the techniques have been applied to a variety of non-porous and some semi-porous substrates.

MATERIALS AND METHODS

Spectroscopy

Spectroscopic measurements were done in order to show that absorption and emission spectra were scarcely changed by the different states of the fluorophore, and above all, to provide practitioners with useful data to choose accurate light source and filters (luminescence mode).

UV-Vis absorption spectra were recorded using an Agilent Cary 5000 spectrophotometer.

Fluorescence emission spectra were recorded using a Horiba Jobin-Yvon Fluorolog-3 spectrofluorimeter. For measurements in solution, a right angle configuration was used and optical density was adjusted below 0.1 to avoid reabsorption artifacts. For solid state measurements (thin films and fingermarks), a front-face configuration was used. Fluorescence quantum yield was measured using Rhodamine 6G in ethanol as a reference.

Physical examination

To investigate in details the quality of the polymerization process, physical examination has been undergone with two devices: a QUANTA 400 Environmental Scanning Electron Microscope (ESEM) made by FEI and a digital microscope (VHX 2000 manufactured by Keyence), equipped with lens Z100 or Z20. Samples were illuminated by a Vilber-Lourmat UV lamp (P = 6 W,  max = 312 nm).

Samples preparation

Surfaces

Different substrates were used during this study. To compare the Lumicyano™ efficiency and sensitivity with the conventional sequence (cyanoacrylate followed by dye staining), "cut edge frosted" microscope slides provided by VWR were used 1 . The slides were used directly after being picked out of their pack.

In a second stage, to check the Lumicyano™ versatility, miscellaneous types of items were used. This last step aims at illustrating the ability of Lumicyano™ to be easily used on "operational" cases under normal practical environmental conditions.

Fingermarks

For the first part of the study, as the comparison should be the most controlled and reproducible as possible in our lab, participants were asked to wash and dry their hands, and then waited for 20 minutes without touching anything. Before depositing marks on microscope slides, they were asked to pass each hand on another in order to evenly distribute secretions.

During the second part of the study (processing of miscellaneous surfaces), eccrine, sebaceous, and uncontrolled marks were deposited on the various surfaces. As for sebaceous marks, participants were asked not to put any cream or product on their face before coming to work. Then they performed the same process as for eccrine marks but after having dried their hands, they wipe the wings of their nose with their thumbs.

Depletion series and donors

To compare the two processes, nine donors (7 male and 2 female of different ages) were chosen for the intervariability of their fingermark quality. A ranking of donors, based on conventional cyanoacrylate fuming developments on eccrine marks, has been made. Donor A has been judged as being the best one and donor I as being the worst one.

Several depleted marks were planted successively. By doing so, the quantity of secretion decreases from the first to the last marks. Prior to this study, tests have been carried out to choose an accurate number of marks for the depletion series. Four donors (included into the study) deposited 10 times a mark on a slide pausing three seconds. For two of them (considered as "bad donors") there were no significant prints revealed after the third one (included). As the quality of fingermarks is dependent on numerous factors (donors' health, contact force and duration…), we chose not to go beyond the third depletion, thus acknowledging the problem of depletion discussed in other studies [START_REF] Fieldhouse | Consistency and reproducibility in fingermark deposition[END_REF]. With this methodology, each series consists of 27 marks (9 donors giving 3 fingermarks each).

During the second part of the study, no depletion was used but only two fingermarks were planted because the first one might be too charged with secretions.

Deposition protocol

Two microscope slides were put side by side and donors asked to leave their fingermarks at the junction of the two slides. Therefore the two halves of the mark can be processed separately without other handlings. Donors let their finger two seconds at the junction.

For the study on miscellaneous surfaces, each sample was also cut into two before the fingermarks were left.

Ageing

Time and ageing conditions have an effect on cyanoacrylate fuming efficiency [START_REF] Wargacki | Enhancing the quality of aged latent fingerprints developed by superglue fuming: loss and replenishment of initiator[END_REF]. To check the Lumicyano™ ability to detect fresh and old marks, three ageing times were chosen (i.e., a day, a week and 4 weeks), this period choice being inspired of Kent's recommendations [START_REF] Kent | Standardzing Protocols for Fingerprint Reagent Testing[END_REF]. All the depletion series were stored in a ventilated closet at room temperature in a dark space. Rubbing is avoided by putting the slides on trays and placing the trays on the upper shelves of the closet, humidity is not accurately controlled (the variation is under the environmental condition of our laboratory), but outdoor major variations are contained thanks to the ventilation system; thus marks would undergo a constant dry. This methodology provided us 81 marks (3 series of 27 marks).

Secondly, on miscellaneous surfaces deposited marks were treated within two days after deposition.

Processing

Cyanoacrylate fuming

Two kinds of processes were compared:  Lumicyano™ is produced by Crime Scene Technology (France). This red liquid is composed of 99% cyanoacrylate and 1% fluorophore.

 Cyanobloom (low viscosity) purchased from Foster + Freeman, followed by enhancement.

The cyanoacrylate fuming cabinet is the MVC 5000 (2003 model version 6.2b), made by Mason

Vactron and supplied by Foster + Freeman. In this study, the fuming time was of 20 minutes.

Parameters are those commonly advocated and in accordance with our accredited protocol under ISO 17025: a maximum of 80% humidity rate and 120°C fuming temperature [START_REF] Paine | The effect of relative humidity on the effectiveness of the cyanoacrylate fuming process for fingermark development and on the microstructure of the developed marks[END_REF][START_REF] Champod | Fingerprints and other ridge skin impression[END_REF].

For a given ageing period, all depletion series left halves are processed with 3.5 g of Cyanobloom.

Afterwards, the chamber is let wide open to cool down the system at room temperature. The second halves are processed the same way with 3.5 g of Lumicyano™. Slides are hanged, always in the same order, on metal bars at the chamber upper level.

Image recording

After having being processed, the halves of each fingermark were gathered and photographed under white light (both in the first and the second part of the study).

Afterwards halves treated with Lumicyano™ are lit under UV light (312 nm), photos were taken within 4 hours after the fumigation.

Enhancement

After at least 24 hours drying in a ventilated chamber, slides processed with cyanoacrylate (Cyanobloom or Lumicyano™) were dyed with Basic Yellow 40 dissolved in methanol (1.5 g for 1 L methanol [START_REF] Champod | Fingerprints and other ridge skin impression[END_REF]). Slides are stained by drop-casting a BY40 solution on the mark, then washed with distilled water, and finally let to dry at room temperature in a ventilation box during a few minutes (until the rinsing water evaporates).

Concerning tests on miscellaneous surfaces, enhancement has also been processed with YELLOWescent™ latent print powder manufactured by SIRCHIE, or MAGNETA FLAKE™ fingerprint powder manufactured by CSI Ltd, depending on surfaces type and colour.

Imaging

UV light is provided by:  a gas-discharged lamps (provided by Consort) 30 W at 312 nm wavelength, at 5 cm from the slides in the first part of the study.

 a SUPERLITE 400 made by Lumatec®, at 320 nm to 400 nm (filterwheel on position Nr3, 1.8 W/cm 2 ) to light the miscellaneous surfaces in the second part of the study. In the first part of the study, pictures are taken with a NIKON D300 camera and an AF-S VR Micro-Nikkor 105mm f/2.8G IF-ED lens, always at the same distance from the slides (42cm).

In the second part of the study, the former camera and also a NIKON D800 were used with the same lens.

Camera white balance set up was chosen in order to get a picture colour (through a UV or the Tiffen n°15, deep yellow filters) similar to what is seen by eyes only protected by (respectively) UV or yellow goggles provided with the Crimescope. Other camera setups were chosen to get the best contrast. Contrast is not further improved by image processing, but only by taking in photo fluorescent marks through a UV or the yellow filter. For each series, technicians were provided with an instruction sheet and were asked to assess fingerprints about 3 evaluation criteria: contrast (meaning the ability of the fingerprint ridges to part with the background), ridges continuity (meaning the possibility of watching unbroken ridges and the ridges flow), level of details (meaning the possibility of watching second (minutiae) and even third level details (pores and ridge edges)).

Still to avoid bias, assessors gave a letter to each mark (for each mark a particular half is pointed to):  A, if the half pointed to is judged to give the best result;  B, if the half pointed is judged to give the lower result;

 or C if they cannot decide in between; without knowing which half is being processed by Lumicyano.

Those letters are collected from the assessors and are related to a score depending on how is the mark positioned (A=1 B = -1 if dealing with Lumicyano™, A=-1 or B = 1 otherwise, C = 0).

Finally, all marks are added and averaged by the number of practitioners and fingerprints. Two thresholds were set up: the first one for results slightly more or less optimal (the same on both parts of equality) and the second for results more or less optimal (the same on both parts of equality).

Results are summarized in tables 2, 3 and 4.

RESULTS AND DISCUSSION

Spectroscopy

The UV-visible absorption spectrum of the fluorophore contained in Lumicyano™ was recorded in acetonitrile solution (Figure 1). It shows two absorption bands: one, intense, in the UV range at 326 nm and a weaker transition in the visible range (511 nm). The fluorescence emission spectrum of the fluorophore in acetonitrile solution shows a broad and intense (quantum yield = 0.27) emission band centred at 562 nm (yellow). The same absorption and fluorescence spectra were recorded in the solid state on a polymerized Lumicyano™ film obtained by sticking 2 microscope slides with 2 drops of Lumicyano™. The spectra in solution and in the solid state are very similar, which indicates that the structure of the fluorophore is not modified during the polymerization process.

A microscope slide coated with grease and a latent fingermark left on a microscope slide have been fumigated with Lumicyano™ in the standard conditions. In both cases, the polycyanoacrylate deposit is opaque and it is not possible to record absorption spectra. We only recorded fluorescence spectra (Figure 1). No spectral difference appears between the homogeneous films obtained from the grease coated microscope slide and the processed fingermark. In both cases, the fluorescence spectrum is similar to those obtained for the fluorophore in acetonitrile solution and for the fluorophore in a polymerized Lumicyano™ film obtained without sublimation. This confirms again that the structure of the fluorophore is not modified during the sublimation process. Table 1 summarizes the absorption maxima and fluorescence emission maxima for the studied samples. The small variation in the maximum values can easily be explained by a change of the local environment of the fluorophore and especially by a modification of the local polarity [START_REF] Valeur | Effect of polarity on fluorescence emission[END_REF].

Morphology of the polymer

SEM Pictures

To investigate in details the quality of the polymerization process, fingermarks detected using the Lumicyano™ were imaged using scanning electron microscopy. SEM pictures show a very high and homogeneous concentration of noodle-type polymer on the ridges (Figure 2a) and a very low polymer development between the ridges (Figure 2b). Several studies [START_REF] Thiburce | Design of a Control Slide for Cyanoacrylate Polymerization: Application to the CA-Bluestar Sequence[END_REF][START_REF] Paine | The effect of relative humidity on the effectiveness of the cyanoacrylate fuming process for fingermark development and on the microstructure of the developed marks[END_REF][START_REF] Mankidy | Facile catalytic growth of cyanoacrylate nanofibers[END_REF][START_REF] Mankidy | Influence of initiators on the growth of poly(ethyl 2-cyanoacrylate) nanofibers[END_REF][START_REF] Lewis | Processes Involved in the Development of Latent Fingerprints Using the Cyanoacrylate Fuming Method[END_REF] have shown that the noodle-type polymerization is associated with a good macroscopic development of the latent fingermark. This kind of polymer corresponds to a maximized contrast between the ridges and the background, and it has been demonstrated that it is obtained for a humidity rate close to 80 % [START_REF] Paine | The effect of relative humidity on the effectiveness of the cyanoacrylate fuming process for fingermark development and on the microstructure of the developed marks[END_REF]. Doiphode et al. [START_REF] Doiphode | Nanofibers and spheres by polymerization of cyanoacrylate monomer[END_REF] have shown that heating the monomer at low temperature (upper than 100°C but lower than 180°C) results in a low monomer concentration in the vapour phase, small monomer droplets being formed in the vapour. They suggest that polycyanoacrylate fibres grow from tiny spots on the substrate where water molecules and polymerization initiators are present according to a vapour-liquid-solid (VLS) process. In such a process, the droplet having initiated the

Magnified fluorescent pictures

Fingermarks developed using the Lumicyano™ were examined under UV illumination using a digital optical microscope in order to obtain magnified fluorescence pictures of the ridges. An excellent contrast can be observed (Figure 3a), with a perfect co-deposition of the fluorescent dye and the polymer of cyanoacrylate, ensuring an adequate signal-to-noise ratio (contrary to what is generally observed with post-staining treatments using fluorescent dyes in solutions). Each polymer fiber is fluorescent, indicating that the polymer and the fluorophore are intimately linked (Figure 3b). This strong affinity between the polymer and the fluorophore leads to a strong selectivity of the deposit of fluorophore which can occur everywhere the polymer is present and only on the polymer.

Sensitivity of the Lumicyano™

In preliminary tests, different amounts of cyanoacrylate (2.5 g, 3 g, 3.5 g, 4 g and 4.5 g) were tested.

3.5 g was chosen as the most adapted quantity, but good results were also obtained with the 4 g recommended by Foster + Freeman.

It has been observed that if the Lumicyano™ fuming process is stopped before the whole glue was fumed, a strongly coloured cyanoacrylate is left in the fuming cup and the detected fingermark appears far less luminescent than it should be. As a consequence, it is recommended to evaporate completely the amount of Lumicyano™ used in a development process. Preliminary results indicate that overexposure (that is to say, ridges fusion due to an excess of cyanoacrylate deposition) seldom occurs with the Lumicyano™. We suppose this could be related to the high quality (low viscosity)

of the cyanoacrylate monomers used in the Lumicyano™. A detailed study on this topic is under completion in our laboratories. Furthermore, if marks are under-developed, a subsequent treatment with Lumicyano™ is still possible.

From the results collected from examination under white light (see Figure 4), it can be concluded that the Lumicyano™ develops fingermarks better than, Cyanobloom for 1 day and 1 week old fingermarks. Besides, development with Lumicyano and Cyanobloom are slightly the same for 4 weeks old fingermarks.

Observation of the marks shows that the "1 day" and "1 week" Cyanobloom-processed marks were slightly overdeveloped compared to the Lumicyano™ ones, whereas the "4 weeks" Lumicyanoprocessed marks were very slightly underdeveloped (remember the amount of cyanoacrylate recommended by Foster&Freeman is up to 4 g, so quantity can be increased to solve that point).

This can explain the discrepancies between results. Moreover, the gap between developments quality is narrow (see Tables 2 to 4) and it has been difficult for technicians to score one "better" than the other.

Anyway, the same amount of cyanoacrylate has been used for all the processes. So, the score of the Lumicyano™ on "4 weeks" marks may be improved by slightly increasing the amount of Lumicyano.

Overall, Lumicyano™-processed fingermarks are slightly better defined than Cyanobloomprocessed ones. The pictures of the half-fingermarks processed using the Lumicyano™, and taken under UV light, show a good contrast. Contrary to Cyanobloom, Lumicyano™ allows to take fluorescent pictures directly, without any additional staining step.

Examination of the scores given to the comparison file II shows that the contrast obtained with Lumicyano™ in fluorescence mode is better than the contrast obtained with the half-fingermarks processed using Cyanobloom followed by BY40 staining (see scoring Tables 2 to 4, and see Figure 5 for an example). As for file I, it has been difficult for technicians to score one by comparison with the other regarding the contrast. Nevertheless, when considering the ridges continuity and the ridges detail level, comparison file II shows that BY40 staining enhances the detection process. Indeed, this happens when there are just few polymers (lack of secretions) or when optimized polymerization conditions are not fulfilled. An under-development leads to a weaker fluorescence signal from Lumicyano™. Consequently, as BY40 is bright and has a strong affinity for cyanoacrylate, the second-step with BY40 staining makes ridges more continuous and brings more ridge details. However, there is still a risk of marks being washed or unwanted background staining. Above all, no digital enhancement has been made on any picture of this study.

Indeed, in case of reaction of the background with the dye, the Lumicyano™ process provides a better signal to noise ratio and digital processing may have improved the Lumicyano™ results in comparison with the two-step process.

Finally, comparison file III (see tables 2 to 4) scoring shows that, after an enhancement with BY40, the fingermarks initially processed with the Lumicyano™ are able to catch up those processed with Cyanobloom. The resulting contrast and ridge definition are increased and may overpass the characteristics of the others. This is consistent with the good quality of the polymerisation observed by SEM. (++ means Lumicyano™ is more optimal than the two-step process. + means Lumicyano™ is slightly more optimal than the two-step process. = means the two treatments are almost equivalent (less than a ninth of the points is won by one of the treatment).

-means the two-step process is slightly more optimal than Lumicyano™.

--means the two-step process is more optimal than Lumicyano™) Treatments were undergone on the same on day old fingermark.

Fading

The fluorescence of the Lumicyano TM -processed marks decreases after some time, which depends on the environmental conditions (humidity, temperature) and also strongly on the substrate. For example, fading is fast on glass substrates (see figure 6) but ongoing experiments show that some

Results on miscellaneous surfaces

On non-porous surfaces, such as white plastic bags, clingwraps, etc., an enhancement step is often required when dealing with conventional cyanoacrylate fuming. The point is that detected fingermarks are likely to be washed away on that kind of surfaces when the dye is rinsed or are likely to be wiped away by the brush when powder is applied.

No real scoring has been possible given the insufficient number of processed fingermarks in this part of the experiments. Nevertheless, one main observation can be made: when the Lumicyano TM fuming has sufficiently developed the fingermarks, the fluorescence is high enough to avoid staining and give sufficient contrast. If the fluorescence is not satisfactory, a subsequent enhancement process is still possible, in a similar way to any non-fluorescent cyanoacrylate.

Concerning semi-porous surfaces such as postcards, glossy photo paper or cigarette packs, the problem is dye-staining generally results in the staining of the whole item. Thus powdering is required and still may wipe away the fingerprint.

We came to the same conclusions as for difficult non-porous surfaces: when cyanoacrylate fuming has sufficiently developed the fingermarks, fluorescence is bright enough and gives a very good contrast (contrary to Cyanobloom alone) without any risk of detail loss (contrary to Cyanobloom + powder).

It must also be noticed, that Lumicyano™ has a second absorption peak between 450 nm and 550 nm (515 nm centred, see figure 1). A green light combined with orange google and filter could thus be used, but the absorption peak is five times lower than the UV one. However, the use of a green light (for example a green LASER "Tracer" from Coherent and its associated orange bandpass filter) is an alternative for surfaces with a high UV response. Hence with white glossy photopaper, laser illumination makes it possible to photograph prints with a very good contrast without any further enhancement, and the results are far better than after powder dusting (see Figure 7).



  Non-porous surfaces: o black trash bags in polyethylene, o clingwraps for fridge and microwave use, supplied by Systeme U. o Coca Cola (Olympic Games edition) and Canada dry cans.  Semi-porous surfaces: o photo paper for printer DS80 manufactured for the media set n°DM81280 by Dai Nippon Printing Co. Ltd, o Camel cigarette packs, o postcards n°5227 printed by Hüber edition.

  White and blue (445 nm) lights are provided by a CrimeScope® CS-16-500 (manufactured by SPEX forensics, HORIBA/Jobin&Yvon, 2006 model) at an angle of 45°, 25 cm away from the slides.

2. 6 .

 6 ScoringFour fingerprint examiners from IRCGN -but not included in the study processes -were asked to compare the two cyanoacrylate processes. Three comparison files were created, inspired by the work of Wilkinson and Misner [23]:  In file I, half-fingermarks developed with Lumicyano™ are compared with halffingermarks developed with Cyanobloom, both photographed under white light.  In file II, half-fingermarks developed with Lumicyano™ and photographed under UV light are compared with half-fingermarks developed with Cyanobloom + BY40 and photographed under blue light.  In file III, half-fingermarks developed with Lumicyano™ + BY40 are compared with halffingermarks developed with Cyanobloom + BY40, both photographed under blue light.

Figure 1 :

 1 Figure 1: Normalized absorption spectrum (red curves, left scale) and normalized emission spectrum (yellow curves, right scale) of the fluorophore in acetonitrile solution. Normalized emission spectrum of a fingermark fumigated using Lumicyano™ (dashed black line, right scale).

  polymerization is gradually carried to the tip of the polymer fibre during his growth. A monomer molecule coming in contact with the fibre tip is quickly polymerized in the presence of water, while the monomer molecule coming on the side of the fibre polymerizes more slowly. The use of the Lumicyano™ in the standard conditions of fumigation (80% humidity, 120°C) leads to the formation of the expected noodle-type polymer fibre. This shows that the fluorophore does not disturb the VLS process.

Figure 2 :

 2 Figure 2: SEM pictures of a latent fingerprint revealed with Lumicyano™. Magnitude 1200x (a), 270x (b)

Figure 3 :

 3 Figure 3: Magnified fluorescent pictures of a latent fingermark detected using the Lumicyano™, observed under UV light (312 nm). (a) x200, (b) x400.

Figure 4 :

 4 Figure 4: Optical comparison under white light between Cyanobloom (left side) and Lumicyano™ (right side), for three fingermark ageing periods, i.e., one day (a), one week (b), and four weeks (c).

4 :

 4 Results of the comparison made by practitioners through three ageing times of the latent fingerprints (1 day, 1 week, 4 weeks). Results are means of marks given by 4 practioners on 27 fingermarks and are expressed on a verbal scale: I = Lumicyano™ vs. Cyanobloom, in visible light II = Lumicyano™ in UV vs. (Cyanobloom + BY40 at 445 nm), and III = (Lumicyano™ + BY40 at 445 nm) vs. (Cyanobloom + BY40 at 445 nm);

Figure 5 :

 5 Figure 5: Optical comparison of Cyanobloom+BY40 under blue light: 445nm and deep yellow filter (left side), with the one-step Lumicyano™ fumigation under UV light and deep yellow filter (right side) .Treatments were undergone on the same on day old fingermark.

Figure 6 :

 6 Figure 6: Illustration of the luminescence fading on glass slides, for one day old fingermarks revealed by Lumicyano TM . Left side: picture taken just after treatment. Right side pictures taken after 0 (a), 2 (b), 6 (c), 24 (d) and 48 (e) hours.

Figure 7 :

 7 Figure 7: Comparison of development on a piece of white glossy photo paper: (Cyanobloom + MAGNETA FLAKE™ powder) vs Lumicyano™ under green light (Laser 532nm) (a), (Cyanobloom+ BY40 under blue light: 445nm and deep yellow filter) vs. Lumicyano™ under green light (Laser 532nm) (b)

Table 1 :

 1

		Absorption wavelength (nm)	Emission wavelength (nm)
	Fluorophore in acetonitrile solution	326/ 511	562
	Lumicyano™ film	328/ 515	566
	Lumicyano™ fumigated film	-	557
	Lumicyano™ fumigated fingerprint	-	557

Absorption and emission wavelengths of the Lumicyano™ fluorophore in different conditions and states.

Table 2

 2

	: after 1 day
	File	I	II	III
	Contrast	++	+	=
	Ridges continuity	+	-	=
	Ridges details level	+	=	=
	Table 3: after 1 week
	File	I	II	III
	Contrast	++	+	++
	Ridges continuity	++	-	++
	Ridges details level	+	-	+

Table 4 :

 4 after 4 weeks

During the test period, we run out of slides, and as the new slides sent by VWR were different and lightly UVluminescent, we change to "ground color frosted white" by VWR which made no significant difference with the first ones-.


A Chambolle
email: antonin.chambolle@cmap.polytechnique.fr

M Goldman
email: goldman@mis.mpg.de

M Novaga

Fine properties of the subdifferential for a class of one-homogeneous functionals

We collect some known results on the subdifferentials of a class of one-homogeneous functionals, which consist in anisotropic and nonhomogeneous variants of the total variation. It is known that the subdifferential at a point is the divergence of some "calibrating field". We establish new relationships between Lebesgue points of a calibrating field and regular points of the level surfaces of the corresponding calibrated function.

Introduction

In this note we recall some classical results on the structure of the subdifferential of first order one-homogeneous functionals, and we give new regularity results which extend and precise previous work by G. Anzellotti [START_REF] Anzellotti | Pairings between measures and bounded functions and compensated compactness[END_REF][START_REF] Anzellotti | On the minima of functionals with linear growth[END_REF][START_REF] Anzellotti | Traces of bounded vector fields and the divergence theorem[END_REF].

Given an open set Ω ⊂ R d with Lipschitz boundary, and a function u ∈ C 1 (Ω) ∩ BV (Ω), we consider the functional J(u) := Ω F (x, Du)

where F : Ω × R d → [0, +∞) is continuous, and F (x, •) is a smooth and uniformly convex positively one-homogeneous functional on R d for all x ∈ Ω. The functional J can be canonically relaxed to the whole of BV (Ω) (see [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]Section 5.5]) and we still write, in analogy with the notation commonly used for the total variation, J(u) = Ω F (x, Du) for u ∈ BV (Ω),

where F (x, Du) is in general a Radon measure in Ω defined by F (x, Du) := F x, Du |Du| |Du| with Du |Du| the Radon-Nikodym derivative of Du with respect to |Du|. Since BV (Ω) ⊂ L d/(d-1) (Ω), it is natural to consider J as a convex, l.s.c. function on the whole of L d/(d-1) (Ω), with value +∞ when u ∈ BV (Ω). In this framework, for any u ∈ L d/(d-1) (Ω) such that J(u) < +∞, that is u ∈ BV (Ω), we can define the subdifferential of J at u, in the duality (L d/(d-1) , L d ), as

∂J(u) := g ∈ L d (Ω) : J(v) ≥ J(u) + Ω g(v -u) dx ∀v ∈ L d/(d-1) (Ω) .
Notice that a function g ∈ L d (Ω) belongs to ∂J(u) if and only if u is a minimizer of the functional J(v) -Ω gvdx among all v ∈ L d/(d-1) (Ω).

The goal of this paper is to investigate the particular structure of the functions u and g, when g ∈ ∂J(u). Let F * (x, z) := sup

w∈R d z • w -F (x, w)
be the Legendre-Fenchel or convex conjugate of F . Notice that F * takes values in {0, +∞}, and F * (x, z) = 0 if and only if F • (x, z) ≤ 1, where F • denotes the convex polar of F defined in (4) below.

Given u ∈ L d/(d-1) , the functional J(u) can also be expressed by duality as

J(u) = sup - Ω udiv z dx : z ∈ C ∞ c (Ω; R d ) , F * (x, z(x)) = 0 ∀x ∈ Ω .
It follows that a function g ∈ ∂J(u) has necessarily the form g = -div z, for some vector field z ∈ L ∞ (Ω; R d ) with F * (x, z(x)) = 0 a.e. in Ω. Since by a formal integration by parts one gets z • Du = F (x, Du), two natural questions arise:

• in what sense is this relation true?

• can one assign a precise value to z on the support of the measure Du?

The first question has been answered by Anzelotti in the series of papers [START_REF] Anzellotti | Pairings between measures and bounded functions and compensated compactness[END_REF][START_REF] Anzellotti | On the minima of functionals with linear growth[END_REF][START_REF] Anzellotti | Traces of bounded vector fields and the divergence theorem[END_REF]. However, for the particular vector fields we are interested in, we can be more precise and obtain pointwise properties of z on the level sets of the function u. Indeed, we shall show that z has a pointwise meaning on all level sets of u, up to H d-1 -negligible sets (which can be much more than |Du|-a.e., as illustrated by the function u = +∞ n=1 2 -n χ (0,xn) , defined in the interval (0, 1), with (x n ) a dense sequence in that interval).

We will therefore focus on the properties of the vector fields z ∈ L ∞ (Ω, R d ) such that F * (x, z(x)) = 0 a.e. in Ω and g = -div z ∈ L d (Ω), and such that there exists a function u such that for any φ ∈ C ∞ c (Ω),

- Ω div z uφ dx = Ω u z • ∇φ dx + Ω φF (x, Du) .
In particular, one checks easily that u minimizes the functional

Ω F (x, Du) - Ω gu dx (1)
among perturbations with compact support in Ω. Conversely, given g ∈ L d (Ω) with g L d sufficiently small, there exist functions u which minimize (1) under various types of boundary conditions, and corresponding fields z. This kind of functionals appears in many contexts including image processing and plasticity [START_REF] Andreu-Vaillo | Parabolic Quasilinear Equations Minimizing Linear Growth Functionals, collection[END_REF][START_REF] Temam | Problèmes mathématiques en plasticité (French)[END_REF]. Notice also that, by the Coarea Formula [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF], it holds

Ω F (x, Du) - Ω gu dx = R ∂ * {u>s} F (x, ν) dH d-1 (x) - {u>s} g dx ds ,
where ν is the unit normal to {u > s}, and one can show (see for instance [START_REF] Chambolle | An algorithm for mean curvature motion[END_REF]) that any level set of the form {u > s} or {u ≥ s} is a minimizer of the geometric functional

E → ∂ * E F (x, ν) dH d-1 (x) - E g dx . (2)
defined for sets E of finite perimeter. The canonical example of such functionals is given by the total variation, corresponding to F (x, Du) = |Du|. In this case, (2) boils down to

P (E) - E g dx. (3)
In [START_REF] Barozzi | The mean curvature of a set of finite perimeter[END_REF], it is shown that every set with finite perimeter in Ω is a minimizer of (3) for some g ∈ L 1 (Ω). If F is even in ν and when g ∈ L d (Ω), the boundary ∂E is only of class C 0,α out of a singular set (see [START_REF] Ambrosio | Partial regularity for quasi minimizers of perimeter[END_REF]). However, if g ∈ L p (Ω) with p > d, and E is a minimizer of (2), then ∂E is locally C 1,α for some α > 0, out of a closed singular set of zero H d-3 -measure [START_REF] Almgren | Regularity and singularity estimates on hypersurfaces minimizing elliptic variational integrals[END_REF][START_REF] Schoen | A new proof of the regularity theorem for currents which minimize parametric elliptic functionals[END_REF] (some regularity assumption on F is required, see also Remark 3.11 below). Since the Euler-Lagrange equation of (2) relates z to the normal of E, understanding the regularity of z is closely related to understanding the regularity of ∂E.

Our main result is that the Lebesgue points of z correspond to regular points of ∂{u > s} or ∂{u ≥ s} (Theorem 3.7), and that the converse is true in dimension d ≤ 3 (Theorem 3.10).

Preliminaries 2.1 BV functions

We briefly recall the definition of function of bounded variation and set of finite perimeter. For a complete presentation we refer to [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF].

E (t) := x ∈ R d : lim r↓0 |E ∩ B r (x)| |B r (x)| = t .
We denote by ∂E := E (0) ∪ E (1) c the measure theoretical boundary of E. Let spt (|Dχ E |) be the support of the measure |Dχ E |: we define the reduced boundary of E by:

∂ * E := x ∈ spt (|Dχ E |) : ν E (x) := lim r↓0 Dχ E (B r (x)) |Dχ E |(B r (x)) exists and |ν E (x)| = 1 ⊂ E ( 1 2 ) .
The vector ν E (x) is the measure theoretical inward normal to the set E.

Proposition 2.3. If E is a set of finite perimeter then Dχ E = ν E H d-1 ∂ * E, P (E) = H d-1 (∂ * E) and H d-1 (∂E \ ∂ * E) = 0.
Definition 2.4. We say that x is an approximate jump point of u ∈ BV (Ω) if there exist ξ ∈ S d-1 and distinct a, b ∈ R such that

lim ρ→0 1 |B + ρ (x, ξ)| B + ρ (x,ξ)
|u(y) -a| dy = 0 and lim

ρ→0 1 |B - ρ (x, ξ)| B - ρ (x,ξ)
|u(y) -b| dy = 0, where B ± ρ (x, ξ) := {y ∈ B ρ (x) : ±(yx) • ξ > 0}. Up to a permutation of a and b and a change of sign of ξ, this characterize the triplet (a, b, ξ) which is then denoted by (u + , u -, ν u ). The set of approximate jump points is denoted by J u .

The following proposition can be found in [2, Proposition 3.92].

Proposition 2.5. Let u ∈ BV (Ω). Then, defining

Θ u := {x ∈ Ω : lim inf ρ→0 ρ 1-d |Du|(B ρ (x)) > 0}, there holds J u ⊂ Θ u and H d-1 (Θ u \ J u ) = 0.

Anisotropies

Let F (x, p) : Ω × R d → [0, +∞) be a continuous functions, which is convex and positively one-homogeneous in the second variable: F (x, λp) = λF (x, p) for all λ > 0, x, p; and such that there exists c 0 > 0 with

c 0 |p| ≤ F (x, p) ≤ 1 c 0 |p| ∀(x, p) ∈ R d × R d .
We say that F is uniformly elliptic if for some δ > 0, the function p → F (p) -δ|p| is still a convex function. We define the polar function of F by

F • (x, z) := sup {F (x,p)≤1} z • p (4) so that (F • ) • = F . It is easy to check that [F (x, •) 2 /2] * = F • (x, •) 2 /2
, where as before the * denotes the convex conjugate with respect to the second variable. In particular, if differentiable,

F (x, •)∇ p F (x, •) and F • (x, •)∇ z F • (x, •) are inverse monotone operators. Also, one has that F * (x, z) = 0 if and only if F • (x, z) ≤ 1, and F * (x, z) = +∞ else. If F (x, •) is differentiable then, for every p ∈ R d , F (x, p) = p • ∇ p F (x, p) (Euler ′ s identity) and z ∈ {F • (x, •) ≤ 1} with p • z = F (x, p) ⇐⇒ z = ∇ p F (x, p). If F is elliptic and of class C 2 (R d × R d \ {0}), then F • is also elliptic and C 2 (R d × R d \ {0}).
We will then say that F is a smooth elliptic anisotropy. Observe that, in this case, the function F 2 /2 is also uniformly δ 2 -convex (this follows from the inequalities D 2 F (x, p) ≥ δ/|p|(Ip ⊗ p/|p| 2 ) and F (x, p) ≥ δ|p|). In particular, for every x, y, z ∈ R d , there holds

F 2 (x, y) -F 2 (x, z) ≥ 2 (F (x, z)∇ p F (x, z)) • (y -z) + δ 2 |y -z| 2 , (5)
and a similar inequality holds for F • . We refer to [START_REF] Schneider | Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications[END_REF] for general results on convex norms and convex bodies.

Pairings between measures and bounded functions

Following [START_REF] Anzellotti | Pairings between measures and bounded functions and compensated compactness[END_REF] we define a generalized trace [z, Du] for functions u with bounded variation and bounded vector fields z with divergence in L d .

Definition 2.6 (Anzelotti [5]). Let Ω be an open set with Lipschitz boundary, u ∈ BV (Ω) and z ∈ L ∞ (Ω, R d ) with div z ∈ L d (Ω). We define the distribution [z, Du] by [z, Du], ψ = - Ω u ψ div z dx - Ω u z • ∇ψ dx ∀ψ ∈ C ∞ c (Ω).
Proposition 2.7 (Anzelotti [START_REF] Anzellotti | Pairings between measures and bounded functions and compensated compactness[END_REF]). The distribution [z, Du] is a bounded Radon measure on Ω and if ν is the inward unit normal to Ω, there exists a function [z, ν] ∈ L ∞ (∂Ω) such that the generalized Green's formula holds,

Ω [z, Du] = - Ω udiv z dx - ∂Ω [z, ν]u dH d-1 .
The function [z, ν] is the generalized (inward) normal trace of z on ∂Ω.

Given z ∈ L ∞ (Ω; R d ), with div z ∈ L d (Ω)
, we can also define the generalized trace of z on ∂E, where E is a set of locally finite perimeter. Indeed, for every bounded open set Ω with Lipschitz boundary, we can define as above the measure [z, Dχ E ] on Ω. Since this measure is absolutely continuous with respect to

|Dχ E | = H d-1 ∂ * E we have [z, Dχ E ] = ψ z (x)H d-1 ∂ * E with ψ z ∈ L ∞ (∂ * E) independent of Ω.
We denote by [z, ν E ] := ψ z the generalized (inward) normal trace of z on ∂E. If E is a bounded set of finite perimeter, by taking Ω strictly containing E, we have the generalized Gauss-Green Formula

E div z dx = - ∂ * E [z, ν E ]dH d-1 . Anzellotti proved the following alternative definition of [z, ν E ] [6, 7] Proposition 2.8 (Anzelotti [6, 7]). Let (x, α) ∈ R d × R d \ {0}. For any r > 0, ρ > 0 we let C r,ρ (x, α) := {ξ ∈ R d : |(ξ -x) • α| < r, |(ξ -x) -[(ξ -x) • α]α| < ρ}.
There holds

[z, α](x) = lim ρ→0 lim r→0 1 2rω d-1 ρ d-1 Cr,ρ(x,α) z • α dy
where ω d-1 is the volume of the unit ball in R d-1 .

3 The subdifferential of anisotropic total variations

Characterization of the subdifferential

The following characterization of the subdifferential of J is classical and readily follows for example from the representation formula [10, (4.19)]. where the equality holds in the sense of measures. Moreover, for every t ∈ R, for the set

E = {u > t} there holds [z, ν E ] = F (x, ν E ) H d-1
-a.e. on ∂E. We will say that such a vector field is a calibration of the set E for the minimum problem (2).

Remark 3.2. In [START_REF] Anzellotti | Pairings between measures and bounded functions and compensated compactness[END_REF], it is proven that if

z ρ (x) := 1 |Bρ(x)| Bρ(x) z(y) dy, then z ρ • ν E weakly* converges to [z, ν E ] in L ∞ loc (H d-1 ∂ * E). Using (5) it is then possible to prove that if z calibrates E then z ρ converges to ∇ p F (x, ν E ) in L 2 (H d-1 ∂ * E) yielding that up to a subsequence, z φ(ρ) converges H d-1 -a.e. to ∇ p F (x, ν E ).
Unfortunately this is still a very weak statement since it is a priori impossible to recover from this the convergence of the full sequence z ρ .

The main question we want to investigate now is whether we can give a classical meaning to

[z, ν E ] (that is understand if [z, ν E ] = z • ν E ).
We observe that a priori the value of z is not well defined on ∂E which has zero Lebesgue measure.

We let S := spt (Du) ⊂ Ω be the support of the measure Du, that is, the smallest closed set in Ω such that |Du|(Ω \ S) = 0. We will show that essentially in S, z is well-defined, as soon as g ∈ L d (Ω).

The next result is classical, for a proof we refer to [START_REF] Massari | Frontiere orientate di curvatura media assegnata in L p (Italian)[END_REF][START_REF] Gonzalez | Massari Variational mean curvatures[END_REF].

Lemma 3.3 (Density estimate).

There exists ρ 0 > 0 (depending on g ∈ L d (Ω)) and a constant γ > 0 (which depends only on d), such that for any B ρ (x) ⊂ Ω with ρ ≤ ρ 0 , and any level set

E of u (that is, E ∈ {{u > s}, {u ≥ s}, {u < s}, {u ≤ s}, s ∈ R}), if |B ρ (x) ∩ E| < γ|B ρ (x)| then |B ρ/2 (x)∩E| = 0.
As a consequence, E 0 and E 1 are open, ∂E is the topological boundary of E 1 , and (possibly changing slightly γ)

if x ∈ ∂E, then H d-1 (∂E ∩ B ρ (x)) ≥ γρ d-1 .
This result is not true anymore if g ∈ L d (Ω) [START_REF] Gonzalez | Massari Variational mean curvatures[END_REF]. If ∂Ω is Lipschitz, it is true up to the boundary.

Corollary 3.4. It follows that u ∈ L ∞ loc (Ω) and u ∈ C(Ω \ Θ u ).
Proof. For any ball B ρ (x) ⊂ Ω and inf

B ρ/2 (x) u < a < b < sup B ρ/2 (x) u, one has +∞ > |Du|(B ρ (x)) ≥ b a P ({u > s}, B ρ (x)) ds ≥ (b -a)γ ρ 2 d-1
, so that osc B ρ/2 (x) (u) must be bounded and thus u ∈ L ∞ loc (Ω). Moreover, if x ∈ Ω \ Θ u we find that lim ρ→0 osc Bρ(x) (u) = 0 so that u is continuous at the point x.

We remark that if sets (E n ) n satisfy the density estimate of Lemma 3.3 and converge in L 1 to some limit set, then one easily deduces that the convergence also holds in the Hausdorff (or Kuratowski, is the sets are unbounded) sense. Applying this principle to the level sets of u, we find that all points in the support of Du must be on the boundary of a level set of u: Proposition 3.5. For any x ∈ S, there exists s ∈ R such that either x ∈ ∂{u > s} or x ∈ ∂{u ≥ s}.

Proof. First, if x ∈ S then |Du|(B ρ (x)) = 0 for some ρ > 0 and clearly x cannot be on the boundary of a level set of u. On the other hand, if x ∈ S, then for any ball B 1/n (x) (n large) there is a level s n (uniformly bounded) with ∂{u > s n } ∩ B 1/n (x) = ∅ and by Hausdorff convergence, we deduce that either x ∈ ∂{u > s} or x ∈ ∂{u ≥ s} where s is the limit of the sequence (s n ) n (which must actually converge).

The following stability property is classical (see e.g. [START_REF] Chambolle | Plane-like minimizers and differentiability of the stable norm[END_REF]). Proposition 3.6. Let E n be local minimizers of (2), with a function g = g n ∈ L d (Ω), and converging in the L 1 -topology to a set E. Assume that the sets E n are calibrated by z n (see

Prop. 3.1), that z n * ⇀ z weakly- * in L ∞ and g n → g = -div z ∈ L d (Ω), in L 1 (Ω) as n → ∞.
Then z calibrates E, which is thus also a minimizer of (2).

Let us observe that, if z n * ⇀ z and F • (x, z n (x)) ≤ 1 a.e. in Ω, then in the limit one still gets F • (x, z(x)) ≤ 1 a.e. in Ω, thanks to the continuity of F and the convexity in the second variable.

The Lebesgue points of the calibration.

The next result shows that the regularity of the calibration z implies some regularity of the calibrated set.

Proof. We reason as in [START_REF] Chambolle | Plane-like minimizers and differentiability of the stable norm[END_REF]Th. 4.5] and let z ρ (y) := z(x + ρy). Since x is a Lebesgue point of z, we have that z ρ → z in L 1 (B R ), hence also weakly- * in L ∞ (B R ) for any R > 0, where z ∈ R d is a constant vector.

We let E ρ := (Ex)/ρ and g ρ (y) := g(x + ρy) (so that div z ρ = ρg ρ ). Observe that E ρ minimizes

∂ * Eρ∩BR F (x + ρy, ν Eρ (y)) dH d-1 (y) + ρ Eρ∩BR g ρ (y) dy ,
with respect to compactly supported perturbations of the set (in the fixed ball B R ). Also,

ρg ρ L d (BR) = g L d (BρR) ρ→0 -→ 0 .
By Lemma 3.3, the sets E ρ (and the boundaries ∂E ρ ) satisfy uniform density bounds, and hence are compact with respect to both local L 1 and Hausdorff convergence.

Hence, up to extracting a subsequence, we can assume that E ρ → Ē, with 0 ∈ ∂ Ē. Proposition 3.6 shows that z is a calibration for the energy ∂ Ē∩BR F (x, ν Ē (y)) dH d-1 (y), and that Ē is a minimizer calibrated by z.

It follows that [z, ν Ē ] = F (x, ν Ē (y)) for H d-1 -a.e. y in ∂ Ē, but since z is a constant, we deduce that Ē = {y • ν ≥ 0} with ν/F (x, ν) = ∇ p F • (x, z) 1 . In particular the limit Ē is unique, hence we obtain the global convergence of E ρ → Ē, without passing to a subsequence.

We want to deduce that x

∈ ∂ * E, with ν E (x) = F (x, ν E (x))∇ p F • (x, z
), which is equivalent to [START_REF] Anzellotti | On the minima of functionals with linear growth[END_REF]. The last identity is obvious from the arguments above, so that we only need to show that

lim ρ→0 Dχ Eρ (B 1 ) |Dχ Eρ |(B 1 ) = ν . ( 7
)
Assume we can show that

lim ρ→0 |Dχ Eρ |(B R ) = |Dχ Ē |(B R ) = ω d-1 R d-1 (8)
for any R > 0, then for any

ψ ∈ C ∞ c (B R ; R d ) we would get 1 |Dχ Eρ |(B R ) BR ψ • Dχ Eρ = - 1 |Dχ Eρ |(B R ) BR∩Eρ div ψ(x) dx -→ - 1 |Dχ Ē |(B R ) BR∩ Ē div ψ(x) dx = 1 |Dχ Ē |(B R ) BR ψ • Dχ Ē and deduce that the measure Dχ Eρ /(|Dχ Eρ |(B R )) weakly- * converges to Dχ Ē /(|Dχ Ē |(B R )).
Using again ( 8)), we then obtain that

lim ρ→0 Dχ Eρ (B R ) |Dχ Eρ |(B R ) = ν (9)
for almost every R > 0. Since Dχ

Eρ (B µR )/(|Dχ Eρ |(B µR )) = Dχ E ρ/µ (B R )/(|Dχ E ρ/µ |(B R ))
for any µ > 0, (9) holds in fact for any R > 0 and (7) follows, so that x ∈ ∂ * E. It remains to show [START_REF] Auer | Differentiability of the stable norm in codimension one[END_REF]. First, we observe that, by minimality of E ρ and Ē plus the Hausdorff convergence of ∂E ρ in balls, we can easily show the convergence of the energies lim ρ→0 ∂Eρ∩BR

F (x + ρy, ν Eρ (y)) dH d-1 (y) + ρ Eρ∩BR g ρ (y) dy = ∂ Ē∩BR F (x, ν Ē (y)) dH d-1 (y)
and, by the continuity of F , lim ρ→0 ∂Eρ∩BR

F (x, ν Eρ (y)) dH d-1 (y) = ∂ Ē∩BR F (x, ν Ē (y)) dH d-1 (y) . ( 10
)
Then, [START_REF] Anzellotti | Traces of bounded vector fields and the divergence theorem[END_REF] follows from Reshetnyak's continuity theorem where, instead of using the Euclidean norm as reference norm, we use the uniformly convex function F (x, •) and the convergence of the measures F (x, Dχ Eρ ) to F (x, Dχ Ē ) (see [START_REF] Reshetnyak | Weak Convergence of Completely Additive Vector Functions on a Set[END_REF][START_REF] Chambolle | Plane-like minimizers and differentiability of the stable norm[END_REF]).

Corollary 3.8. For any x ∈ S let E x ∈ {{u > u(x)}, {u ≥ u(x)}} be the upper level set of u such that x ∈ ∂E x . Then, the equality

z(x) = ∇ p F x, Dχ E x |Dχ E x | (x) (11)
holds Lebesgue a.e. in S = spt (Du).

Remark 3.9. In the inhomogeneous isotropic case F (x, p) = a(x)|p|, with a(•) periodic, a similar result has been proved by Auer and Bangert in [START_REF] Auer | Differentiability of the stable norm in codimension one[END_REF]Th. 4.2]. As a consequence they obtain differentiability properties of the so-called stable norm associated to the functional J (see also [START_REF] Chambolle | Plane-like minimizers and differentiability of the stable norm[END_REF] for the anisotropic version of their result).

In dimension 2 and 3 we can also show the reverse implication, proving that regular points of the boundary correspond to Lebesgue points of the calibration. The idea is to show that the parameters r, ρ in Proposition 2.8 can be taken of the same order. Theorem 3.10. Assume the dimension is d = 2 or d = 3. Let x, s be as in Proposition 3.5, E be a minimizer of (2) and assume x ∈ ∂ * E. Then x is a Lebesgue point of z and (6) holds at x.

Proof. We divide the proof into two steps.

Step 1. We first consider anisotropies F which are not depending on the x variable. Without loss of generality we assume x = 0. By assumption, there exists the limit

ν := lim ρ→0 Dχ E (B ρ (0)) |Dχ E |(B ρ (0))| (12)
and, without loss of generality, we assume that it coincides with the vector e d corresponding to the last coordinate of y ∈ R d . Also, if we let E ρ := E/ρ, the sets E ρ , E c ρ , ∂E ρ converge in B 1 (0), in the Hausdorff sense (thanks to the uniform density estimates), respectively to {y d ≥ 0}, {y d = 0}, {y d ≤ 0}. We also let z ρ (y) := z(ρy) and g ρ (y) := g(ρy), in particular -div z ρ = ρg ρ . We let

ω(ρ) = sup x∈Ω g L d (Bρ(x)∩Ω) (13)
which is continuously increasing and goes to 0 as ρ → 0, since |g| d is equi-integrable. We introduce the following notation: a point in R d is denoted by y = (y ′ , y d ), with y ′ ∈ R d-1 . We let D s := {|y ′ | ≤ s}, z := ∇F (ν) and D t s = {D s + λz : |λ| ≤ t} and denote with ∂D s the relative boundary of D s in {y d = 0}.

We choose s ≤ 1, 0 < t ≤ s, (t is chosen small enough so that

D t s ⊂ B 1 (0), that is t < (1/|z|) √ 1 -s 2 ). We integrate in D t s the divergence ρg ρ = -div z ρ = div (z -z ρ ) against the function (2χ E -1)t -ν•y F (ν)
, which vanishes for y d = ±tF (ν) if ρ is small enough (given t > 0), so that ∂E ρ ∩ B 1 (0) ⊂ {|y d | ≤ tF (ν)}. For y on the lateral boundary of the cylinder D t s , let ξ(y) be the internal normal to ∂D s + (-t, t)z at the point y. Using the fact that z ρ is a calibration for E ρ , we easily get that for almost all s,

D t s ρg ρ (2χ E -1)t - ν • y F (ν) dy = ∂Ds+(-t,t)z (2χ E -1)t - ν • y F (ν) [(z -z ρ ), ξ] dH d-1 -2t ∂Eρ∩D t s z • ν Eρ -F (ν Eρ ) dH d-1 + D t s 1 - z ρ • ν F (ν) dy . ( 14
)
Now since F • (∇F (ν)) = 1, there holds z • ν Eρ -F (ν Eρ ) ≤ 0 and using that z • ξ(y) = 0 on ∂D s + (-t, t)z, we get

D t s 1 - z ρ • ν F (ν) dy ≤ D t s ρg ρ (2χ E -1)t - ν • y F (ν) dy ∂Ds+(-t,t)z (2χ E -1)t - ν • y F (ν) z ρ • ξ dH d-1 . (15)
We claim that for |ξ| ≤ 1 with ξ • z = 0, there holds

(ξ • z ρ ) 2 ≤ C(F (ν) -ν • z ρ ) (16) Since (ξ • z ρ ) 2 ≤ |z ρ | 2 -[z ρ • (z/|z|)] 2 it is enough to prove |z ρ | 2 -[z ρ • (z/|z|)] 2 ≤ C(F (ν) -ν • z ρ ).
Using that ν/F (ν) = ∇F • (z), from (5) applied to

F • together with F • (z) = 1 ≥ F • (z ρ ), we find (F (ν) -ν • z ρ ) = F (ν)(1 -z ρ • ∇F • (z)) ≥ C|z ρ -z| 2 .
which readily implies [START_REF] Paolini | Regularity for minimal boundaries in R n with mean curvature in L n[END_REF]. We thus have ∂Ds+(-t,t)z

(2χ Eρ -1)t - ν • y F (ν) (z ρ • ξ) dH d-1 ≤ 2C F (ν)t ∂Ds+(-t,t)z 1 - z ρ • ν F (ν) dH d-1 ≤ 2CF (ν)t √ t ∂Ds+(-t,t)z 1 - z ρ • ν F (ν) dH d-1 1 2 H d-2 (∂D s ) . (17)
Now, we also have

ρ D t s (2χ Eρ -1)t - ν • y F (ν) g ρ dy ≤ 2tρ 1-d D ρt ρs g dy ≤ 2tρ 1-d g L d (Bρs(0)) |D ρt ρs | 1-1/d ≤ ct 2-1/d s d-2+1/d ω(ρs) (18)
where here, c = 2H d-1 (D 1 ) 1-1/d , and ω is defined in [START_REF] Gonzalez | Massari Variational mean curvatures[END_REF]. We choose a < 1, close to 1, and t ∈ (0, (

1/|z|) √ 1 -a 2 ). If ρ > 0 is small enough (so that ∂E ρ ∩ B 1 is in {|y d | ≤ tF (ν)}), letting f (s) := D t s 1 - zρ•ν F (ν)
dy, we deduce from ( 15), ( 17) and ( 18) that for a.e. s with t ≤ s ≤ a, one has (possibly increasing the constant c)

f (s) 2 ≤ c s d-2 t 3 f ′ (s) + t 4-2/d s 2d-4+2/d ω(ρs) 2 . ( 19
)
Unfortunately, this estimate does not seem to give much information for d >

f (s) 2 ≤ c st 3 f ′ (s) + t 10/3 s 8/3 ω(ρs) 2 . ( 20
)
Given M > 0, we fix a value t > 0 such that log(a/t) ≥ cM . If ρ is chosen small enough, then ∂E ρ ∩ B 1 (0) ⊂ {|y d | < tF (ν)}, and (20) holds. It yields (assuming f (t) > 0, but if not, then the proposition is proved)

- f ′ (s) f (s) 2 + 1 ct 3 1 s ≤ ct 1/3 s 5/3 ω(ρs) 2 f (s) 2 ≤ ct 1/3 s 5/3 ω(aρ) 2 f (t) 2 (21)
where we have used the fact that t ≤ s ≤ a and f, ω are nondecreasing. Integrating (21) from t to a, after multiplication by t 3 we obtain

t 3 f (a) - t 3 f (t) + log(a/t) c ≤ 3c 8 t 10/3 (a 8/3 -t 8/3 ) ω(aρ) 2 f (t) 2 .
Hence we get

t 3 f (t) + ca 8/3 t -8/3 ω(aρ) 2 t 6 f (t) 2 ≥ M. (22)
Eventually, we observe that

f (t) = D t t 1 - z(ρy) • ν F (ν) dy = 1 ρ d D ρt ρt 1 - z(x) • ν F (ν) dx , so that (22) can be rewritten      D ρt ρt 1 - z(x) • ν F (ν) dx (ρt) 3      -1 ≥ -1 + 1 + 4M ca 8/3 t -8/3 ω(aρ) 2 2ca 8/3 t -8/3 ω(aρ) 2 (23)
The value of t being fixed, we can choose the value of ρ small enough in order to have 4M ca 8/3 t -8/3 ω(aρ) 2 < 1, and (using

√ 1 + X ≥ 1 + X/2 -X 2 /8 if X ∈ (0, 1)), (23) yields      D ρt ρt 1 - z • ν F (ν) dy (ρt) 3      -1 ≥ M -M 2 ca 8/3 t -8/3 ω(aρ) 2 ≥ 3 4 M . (24)
It follows that

lim sup ε→0 D ε ε 1 - z • ν F (ν) dy ε 3 ≤ 4 3 M -1 (25)
and since M is arbitrary, 0 is indeed a Lebesgue point of z, with value z = ∇F (ν) (recall that 1

-z(x)•ν F (ν) ≥ (C/F (ν))|z(x) -z| 2 ).
Step 2. When F depends also on the x variable, the proof follows along the same lines as in Step 1, taking into account the errors terms in ( 15) and [START_REF] Reshetnyak | Weak Convergence of Completely Additive Vector Functions on a Set[END_REF]. Keeping the same notations as in Step 1 and setting z := ∇ p F (0, ν) we find that since F • (0, z) ≤ 1, there holds z • ν Eρ ≤ F (0, ν Eρ ) and thus

∂Eρ∩D t s z • ν Eρ -F (ρx, ν Eρ )dH d-1 ≤ ∂Eρ∩D t s |F (0, ν Eρ ) -F (ρx, ν Eρ )|dH d-1 ≤ Cρs d-1
where the last inequality follows from t ≤ s and the minimality of E ρ inside D t s . Now since

(F • ) 2 (0, z ρ ) -(F • ) 2 (ρx, z ρ ) ≥ (F • ) 2 (0, z ρ ) -1 ≥ 2 ν F (0, ν) • (z ρ -z) + δ 2 |z ρ -z| 2
we find that (16) transforms into,

(ξ • z ρ ) 2 ≤ C (F (0, ν) -ν • z ρ ) + ((F • ) 2 (0, z ρ ) -(F • ) 2 (ρx, z ρ ))
for every |ξ| ≤ 1 and ξ • z = 0, from which we get ∂Ds+(-t,t)z

(2χ Eρ -1)t - ν • y F (ν) (z ρ • ξ) dH d-1 ≤ 2CF (0, ν)t √ t ∂Ds+(-t,t)z 1 - z ρ • ν F (0, ν) dH d-1 1 2 H d-2 (∂D s ) + 2Ct ∂Ds+(-t,t)z (F • ) 2 (0, z ρ ) -(F • ) 2 (ρx, z ρ ) 1/2 dH d-1 ≤ CF (0, ν)t √ t ∂Ds+(-t,t)z 1 - z ρ • ν F (0, ν) dH d-1 1 2 H d-2 (∂D s ) + Ctρ 1/2 s d-1 t .
Using these estimates, we finally get that, setting as before f (s) := D t s 1 -zρ•ν F (0,ν) dy, there holds f (s)2 ≤ c s d-2 t3 f ′ (s) + t 4-2/d s 2d-4+2/d ω(ρs) 2 + ρts d-1 + ρ 1/2 t 2 s d-1 .

From this inequality, the proof can be concluded exactly as in Step 1.

Remark 3.11. Assuming F has some regularity (Lipschitz in the first variable, and C 2,β and even in the second, see [START_REF] Schoen | A new proof of the regularity theorem for currents which minimize parametric elliptic functionals[END_REF]), then for d = 2 or d = 3 and g ∈ L p (Ω) with p > d, ∂E is of class C 1,α for some α > 0. In this case, [START_REF] Chambolle | An algorithm for mean curvature motion[END_REF] holds everywhere in spt (Du).

Eventually, we can also give a locally uniform convergence result (valid in dimension d = 2, 3, with the assumption 2 that F is even in dimension 3). Proposition 3.12. For all x ∈ Ω we let z ρ (x) := 1 |B ρ (0)| Bρ(x)∩Ω z dy .

Then, F • (x, z ρ (x)) → 1 locally uniformly on S.

Proof. Given K ⊂ Ω a compact set, we can check that for any t > 0, there exists ρ 0 > 0 such that for any x ∈ K ∩ S, if E x is the level set of u through x, then for any ρ ≤ ρ 0 , the boundary of (E xx)/ρ ∩ B 1 (0) lies in a strip of width 2t, that is, there is ν x ρ ∈ S d-1 with ∂((E xx)/ρ) ∩ B 1 (0) ⊂ {|y • ν x ρ | ≤ t}). Indeed, if this is not the case, one can find t > 0, ρ k → 0, x k ∈ K ∩ S, such that ∂((E x kx k )/ρ k ) ∩ B 1 (0) is not contained in any strip of width 2t. Up to a subsequence we may assume that x k → x ∈ K ∩ S, and from the bound on the perimeter, that (E x kx k )/ρ k converges (in the Kuratowski sense) to a local minimizer of ∂E F (0, ν E )dH d-1 and is thus a halfspace. 3 It yields that ∂((E x kx k )/ρ k ) ∩ B 1 (0) converges in the Hausdorff sense (thanks to the density estimates) to a hyperplane. We easily obtain a contradiction.

The thesis follows when we observe that the proof of Proposition 3.10 can be reproduced by replacing the direction ν E x (x) (which exists only if x lies in the reduced boundary of E x ) with the direction ν x ρ given above.

Remark 3.13. In the Euclidean case (F = | . |) it has already been observed in [START_REF] Ambrosio | Partial regularity for quasi minimizers of perimeter[END_REF] that the blow-ups are flat at each point of the boundary of a set with curvature in L d (but for a closed set of maximal dimension d -8), however the spiral example in [START_REF] Gonzalez | Boundaries of prescribed mean curvature[END_REF] shows that even if d = 2, the orientation of the limit line may not be unique.

Proposition 3 . 1 .

 31 Let F be a smooth elliptic anisotropy and g ∈ L d (Ω) then u is a local minimizer of (1) if and only if there exists z ∈ L ∞ (Ω) with div z = g, F * (x, z(x)) = 0 a.e. and [z, Du] = F (x, Du),

Theorem 3 . 7 .

 37 Let E = {u > t} or E = {u ≥ t}, and let x ∈ ∂E be a Lebesgue point of z. Then, x ∈ ∂ * E and z(x) = ∇ p F (x, ν E (x)).

  Definition 2.1. Let Ω be an open set of R d , we say that a function u ∈ L 1 (Ω) is a function of bounded variation if We denote by BV (Ω) the set of functions of bounded variation in Ω (when Ω = R d we simply write BV instead of BV (R d )). We say that a set E ⊂ R d is of finite perimeter if its characteristic function χ E is of bounded variation and denote its perimeter in an open set Ω by P (E, Ω) := Ω |Dχ E |, and write simply P (E) when Ω = R d .

	Ω	z∈C 1 |z|∞≤1 c (Ω) |Du| := sup	Ω	u div z dx < +∞.
	Definition 2.2. Let E be a set of finite perimeter and let t ∈ [0; 1]. We define

  3. It seems it allows to conclude only whenever d ∈ {2, 3}. Since the case d = 2 is simpler, we focus on d = 3. Estimate (19) becomes

We use here thatF (x, •)∇F (x, •) = [F • (x, •)∇F • (x, •)] -1 , so that z = ∇F (x, ν Ē (y)) implies both F • (x, z) = 1 and ν Ē (y)/F (x, ν Ē )(y) = ∇F • (x, z)

Probably just technical.

If d = 2, this Bernstein result readily follows from the strict convexity of F , see [12, Prop 3.6] whereas for d = 3, see [21, Thm.

4.1], where it is assumed that F is even. In the case of the area i.e when F (x, Du) = |Du| and d ≤ 7, see also[START_REF] Gonzalez | Massari Variational mean curvatures[END_REF] Rem 3.2].

A counterexample.

We provide an example where g ∈ L d-ε (Ω), with ε > 0 arbitrarily small, and Theorem 3.10 does not hold.

Let Ω = B 1 (0) be the unit ball of R d and let E = Ω ∩ {x d ≤ 0}. We shall construct a vector field z : Ω → R d such that z = ν E on ∂E ∩Ω, |z| ≤ 1 everywhere in Ω, divz ∈ L d-ε (Ω), but 0 is not a Lebesgue point of z. Notice that E minimizes the functional (3) with g = divz.

Letting r n → 0 be a decreasing sequence to be determined later, and let B n = B rn (x n ) with x n = 2r n e d . Without loss of generality, we may assume r n+1 < r n /4 so that the balls B n are all disjoint. We define the vector field z as follows:

if we choose r n converging to zero sufficiently fast, so that g = -div z ∈ L d-ε (Ω). However, since z • e d ≤ 1/2 in B rn/2 (x n ), we also have

On the other hand, for δ ∈ (0, 1/6 d ) we have

if we take the sequence r n converging to 0 sufficiently fast. It follows that 0 is not a Lebesgue point of z.