Marie Brandt

Joseph Cowell

Margaret L Aulsebrook

Dr Gilles Gasser
email: gilles.gasser@chimieparistech.psl.eu

Thomas L Mindt
email: thomas.mindt@lbiad.lbg.ac.at.

Radiolabelling of the Octadentate Chelators DFO* and oxoDFO* with Zirconium-89 and Gallium-68

Keywords: chelators, zirconium-89, gallium-68, positron emission tomography (PET)

Radiolabelling of the octadentate chelators DFO* and

oxoDFO* with zirconium-89 and gallium-68

Introduction

Radiolabelled antibodies (Ab) have become an indispensable tool in nuclear medicine for the diagnosis (imaging) of tumours due to their excellent specificity towards various antigens important in oncology. [START_REF] Barbet | Radiolabeled antibodies for cancer imaging and therapy[END_REF] In particular, the number of applications of Abs labelled with positron emitting radionuclides (immunoPET) is steadily increasing. [START_REF] Brandt | An Overview of PET Radiochemistry, Part 2: Radiometals[END_REF][START_REF] Mcinnes | Copper, gallium and zirconium positron emission tomography imaging agents: The importance of metal ion speciation[END_REF] Several immunoPET agents are currently being evaluated in clinical trials. [START_REF] Bensch | 89Zr-atezolizumab imaging as a non-invasive approach to assess clinical response to PD-L1 blockade in cancer[END_REF][START_REF] Dijkers | Development and Characterization of Clinical-Grade 89Zr-Trastuzumab for HER2/neu ImmunoPET Imaging[END_REF][START_REF] Yoon | Evaluation of Next-Generation Anti-CD20 Antibodies Labeled with 89Zr in Human Lymphoma Xenografts[END_REF][START_REF] Bahce | Pilot study of 89Zr-bevacizumab positron emission tomography in patients with advanced non-small cell lung cancer[END_REF] In general, such agents consist of a tumour-targeting monoclonal Ab functionalised with a chelator carrying a positron emitting radiometal.

Abs exhibit slow pharmacokinetics and the time until a sufficient accumulation in the target is achieved, can take days to weeks. [START_REF] Ryman | Pharmacokinetics of Monoclonal Antibodies[END_REF] In the context of nuclear imaging, this results in the necessity of radionuclides with a matching physical half-life, e.g., zirconium-89 ( 89 Zr, t1/2 = 78.41 h). [START_REF] Carmon | Application of Immuno-PET in Antibody-Drug Conjugate Development[END_REF][START_REF] Van Dongen | Immuno-PET: a navigator in monoclonal antibody development and applications[END_REF] The only chelator currently used in clinical trials utilizing 89 Zr-labelled Abs is the siderophore desferrioxamine (DFO 1, Figure 1). However, the hexadentate chelator DFO 1 is not ideal for the complexation of zirconium since the metal ion mostly has coordination numbers >6 [START_REF] Racow | General Approach to Direct Measurement of the Hydration State of Coordination Complexes in the Gas Phase: Variable Temperature Mass Spectrometry[END_REF][START_REF] Adams | Multifunctional Desferrichrome Analogues as Versatile 89Zr(IV) Chelators for ImmunoPET Probe Development[END_REF]. Therefore, a chelator with a higher denticity, e.g., hepta-or octadentate, would be the more optimal choice. It has been suggested that the unsaturated coordination sphere of zirconium in complexes with DFO results in slow release of zirconium in vivo. This in turn leads to nonspecific uptake of the osteophilic radiometal in bones, an increased background in the PET image, and an additional unnecessary radiation dose to the patient. [START_REF] Dijkers | Development and Characterization of Clinical-Grade 89Zr-Trastuzumab for HER2/neu ImmunoPET Imaging[END_REF][START_REF] Holland | 89Zr-DFO-J591 for immunoPET of prostate-specific membrane antigen expression in vivo[END_REF] As a consequence, several octadentate chelating systems for the stable complexation of 89 Zr have been reported in the recent past. [START_REF] Heskamp | 89Zr-Immuno-Positron Emission Tomography in Oncology: State-of-the-Art 89Zr Radiochemistry[END_REF][START_REF] Adams | Multifunctional Desferrichrome Analogues as Versatile 89Zr(IV) Chelators for ImmunoPET Probe Development[END_REF][START_REF] Guérard | Investigation of the complexation of natZr(iv) and 89Zr(iv) by hydroxypyridinones for the development of chelators for PET imaging applications[END_REF][START_REF] Tieu | Octadentate Zirconium(IV)-Loaded Macrocycles with Varied Stoichiometry Assembled From Hydroxamic Acid Monomers using Metal-Templated Synthesis[END_REF] Several of these new chelating systems utilize 4 hydroxamate moieties for octadentate chelation of the radiometal in order to minimize the transchelation rate in vivo. A promising chelator of this series is an octadentate version of DFO, termed DFO* 2 (Figure 1). [START_REF] Patra | An octadentate bifunctional chelating agent for the development of stable zirconium-89 based molecular imaging probes[END_REF] It has been shown that DFO* 2 forms more stable complexes with 89 Zr than DFO 1 in vitro and in vivo. In a direct comparison of [ 89 Zr]Zr-DFO-trastuzumab with analogous [ 89 Zr]Zr-DFO*-trastuzumab in a N87-tumour mouse model, it was shown that the application of DFO* 2 led to significantly less nonspecific uptake of radioactivity in bone structures as well as in the skin, liver, spleen, and ileum. [START_REF] Vugts | Comparison of the octadentate bifunctional chelator DFO*-pPhe-NCS and the clinically used hexadentate bifunctional chelator DFO-pPhe-NCS for 89Zr-immuno-PET[END_REF] Also, the superiority of DFO* 2 in comparison to DFO 1 and DFO-squaramide [START_REF] Rudd | A desferrioxamine B squaramide ester for the incorporation of zirconium-89 into antibodies[END_REF] for late time point PET imaging was demonstrated in primates utilizing a humanized IgG Ab against the herpes simplex viral protein gD. [START_REF] Berg | Total-body PET and highly stable chelators together enable meaningful (89)Zr-antibody-PET studies up to 30 days post-injection[END_REF] Based on published preclinical data, it has been concluded that DFO* 2 is a significant improvement to the commonly applied DFO 1.

A potential limitation of DFO* 2 is represented by its low solubility in water. This in turn could interfere with the conjugation of the chelator to proteins that are affected in their binding and folding behaviour by the presence of higher concentrations of organic co-solvents, such as for example human serum albumin [START_REF] Batista | Selective DMSO-induced conformational changes in proteins from Raman optical activity[END_REF] or G10 antibodies [START_REF] Melnikova | Antigen-binding activity of monoclonal antibodies after incubation with organic solvents[END_REF]. Consequently, three water-soluble versions of DFO* 2 with oxygen-containing backbones were published independently by two groups in 2017 and 2020. Codd and co-workers reported a biosynthetic approach using Streptomyces pilosus that provides small amounts of a water-soluble version of DFO* 2 with four oxygen atoms in the ligand scaffold. [START_REF] Richardson-Sanchez | Exploiting the biosynthetic machinery of Streptomyces pilosus to engineer a water-soluble zirconium(iv) chelator[END_REF] Three years later, the octadentate ligand DFOB-PPH N O C O with two oxygen atoms in the ligand backbone, synthesized from readily available desferrioxamine, was published by the same group. [START_REF] Brown | Improved Access to Linear Tetrameric Hydroxamic Acids with Potential as Radiochemical Ligands for Zirconium-89 PET Imaging[END_REF] Alternatively, we employed a solidphase synthesis for the potentially large scale preparation of oxoDFO* 3 (Figure 1). Using the shake-flask-method, logD7.4 values of -1.5 for oxoDFO* 3 and -2.1 for the corresponding zirconium complex nat Zr-oxoDFO* 3 were determined. [START_REF] Briand | A solid phase-assisted approach for the facile synthesis of a highly water-soluble zirconium-89 chelator for radiopharmaceutical development[END_REF] In comparison, the published logD value for DFO* 2 is logD7.4 = -0.4397. [START_REF] Briand | A solid phase-assisted approach for the facile synthesis of a highly water-soluble zirconium-89 chelator for radiopharmaceutical development[END_REF] Very recently, a theoretical study by Holland was published predicting that oxoDFO* 3 is an excellent candidate for the stable complexation of zirconium with an estimated thermodynamic stability constant of log = 54.16. [START_REF] Holland | Predicting the Thermodynamic Stability of Zirconium Radiotracers[END_REF] Here, we present the first 89 Zr-labelling of the water-soluble, octadentate chelator oxoDFO* 3. In addition, we were interested in exploring the utility of our DFO* and oxoDFO* chelators for the chelation of other radiometals. In that regard, we also report the results of our investigation into the ability of DFO* 2 and oxoDFO* 3 to complex the radiometal gallium-68 ( 68 Ga, t1/2 = 67.7 min).

Experimental Section

Materials and Methods

All chemicals and solvents were obtained from commercial sources and used as received.

Millipore water (Milli Q® 8/16 Direct System, Merck) and ethanol (gradient grade ≥99%, Sigma-Aldrich) were used for HPLC. DFO 1 (mesylate salt) was purchased from Sigma Aldrich. DFO* 2 and oxoDFO* 3 as well as the corresponding 89 Zr-and non-radioactive zirconium ( nat Zr) complexes were synthesized according to previously published methods. [START_REF] Briand | A solid phase-assisted approach for the facile synthesis of a highly water-soluble zirconium-89 chelator for radiopharmaceutical development[END_REF][START_REF] Patra | An octadentate bifunctional chelating agent for the development of stable zirconium-89 based molecular imaging probes[END_REF] 89 Zr was obtained in 0.1 M oxalic acid as [ 89 Zr]Zr-oxalate ( 89 Zr-oxalate) from Perkin Elmer in medical quality. [ 68 Ga]GaCl3 ( 68 GaCl3, ca. 1-1.5 GBq) was eluted from an IRE Elit GalleEo TM 68 Ge/ 68 Ga generator (Fleurus) with 0.1 M HCl according to the procedure recommended by the manufacturer. The eluate (450-550 MBq in 1.1 mL) was collected in an evacuated vial and used without any further processing.

ESI-MS spectra were recorded with a ThermoFisher Scientific LTQ-Orbitrap Velos. RadioTLC was performed using either iTLC plates (glass fibre, impregnated with silica gel, Varian) or RP-TLC plates (silica gel 60 RP-18 on aluminium, Merck) with the solvent system indicated in the description of individual experiments. [START_REF] Vanasschen | Radiolabelling with isotopic mixtures of 52g/55Mn(II) as a straight route to stable manganese complexes for bimodal PET/MR imaging[END_REF] RadioTLC images for quantification were obtained using a Packard Instant Imager (measurement time: 1 min).

High performance liquid chromatography (HPLC) was conducted with a Chromaster preassembled system equipped with a 5160 pump and a 5140 UV detector (Hitachi / VWR). Radio-HPLC was performed on a Merck Hitachi L-6200A intelligent pump with a Merck Hitachi UV detector L-7400 and a Packard Radiomatic Flo-One Beta detector equipped with a BGO cell for the detection of radioactivity. For all HPLC analyses, a Synergi Fusion column (RP, 4 µ, 80 Å, 30 x 2 mm, Phenomenex) was used under isocratic conditions (5% Ethanol / 20 mM phosphate buffer, pH 7) with a flow rate of 0.4 mL/min and a UV wavelength of 220 nm.

Radiosyntheses 89 Zr-labelling reactions were carried out by adapted published methods. [START_REF] Patra | An octadentate bifunctional chelating agent for the development of stable zirconium-89 based molecular imaging probes[END_REF][START_REF] Vugts | Comparison of the octadentate bifunctional chelator DFO*-pPhe-NCS and the clinically used hexadentate bifunctional chelator DFO-pPhe-NCS for 89Zr-immuno-PET[END_REF] In brief, stock solutions of the chelators (1 mM) were prepared in water (DFO 1, oxoDFO* 3) or DMSO (DFO* 2). 89 Zr-oxalate (20-30 MBq) was transferred to a LoBind Eppendorf® flask and diluted with 0.1 M oxalic acid (pH 1) to a total volume of 200 µL. 90 µL Na2CO3 (2 M) solution was added and the mixture was left standing for 3 min at room temperature (RT, pH 6-7). 2-50 µL of chelator stock solutions (2-50 nmol) was diluted with water to a volume of 200 µL and added to the neutralized solution of 89 Zr. HEPES buffer (0.5 M, pH 7.4) was added to a final volume of 1 mL (resulting in final precursor concentrations of 2-50 M) and the mixture was incubated at RT for up-to 2 h. The progress of the reaction was monitored by iTLC (Figure S2) and RP-TLC (Figure S3) with citrate buffer (0.1 M, pH 4) as mobile phase.

Transchelation Experiments

The 89 Zr-and 68 Ga-complexes of DFO 1, DFO* 2 and oxoDFO* 3 were prepared with a chelator concentration of 10 µM in HEPES buffer as described above (2 h at RT for 89 Zr and 10 min at 95 °C for 68 Ga). 10 µL of the solutions were transferred to LoBind Eppendorf® tubes containing 200 µL of solutions of DTPA (5 or 50 mM, corresponding to 10 4 and 10 5 fold excess DTPA, respectively) and gently shaken at RT (final pH 6). RadioTLC (iTLC) of the reaction mixtures were performed at different time points to determine the transchelation of the radiometals to the challenging chelator DTPA (mobile phase for 89 Zr: 0.05 M DTPA, pH 5; for 68 Ga: 0.05 M EDTA, pH 4.5). [START_REF] Deri | Alternative chelator for (8)(9)Zr radiopharmaceuticals: radiolabeling and evaluation of 3,4,3-(LI-1,2-HOPO)[END_REF]

Results and Discussion

Radiolabelling Reactions

First 89 Zr-labelling reactions were carried out at different concentrations of oxoDFO* 3 using reported reaction conditions. [START_REF] Zeglis | A practical guide to the construction of radiometallated bioconjugates for positron emission tomography[END_REF][START_REF] Patra | An octadentate bifunctional chelating agent for the development of stable zirconium-89 based molecular imaging probes[END_REF] We found that an oxoDFO* 3 concentration of approximately 10 M resulted in quantitative complexation of 89 Zr (≈20-30 MBq) at RT within 2 h (Figure 2B). Similar results have been reported for DFO 1 and DFO* 2 [START_REF] Patra | An octadentate bifunctional chelating agent for the development of stable zirconium-89 based molecular imaging probes[END_REF] and have been reproduced in this work as well. Importantly, the applied conditions (RT, pH 7.4) are suitable and also used by default for the 89 Zr-labeling of proteins, e.g., Abs. [START_REF] Vugts | Comparison of the octadentate bifunctional chelator DFO*-pPhe-NCS and the clinically used hexadentate bifunctional chelator DFO-pPhe-NCS for 89Zr-immuno-PET[END_REF][START_REF] Zeglis | A practical guide to the construction of radiometallated bioconjugates for positron emission tomography[END_REF] The chemical identity of the formed complex 89 Zr-3 was confirmed by HPLC comparison with the nonradioactive reference compound nat Zr-3 (Figures 3 andS1). [START_REF] Briand | A solid phase-assisted approach for the facile synthesis of a highly water-soluble zirconium-89 chelator for radiopharmaceutical development[END_REF] It should be noted that the broad peaks seen in the HPLC trace are due to an isocratic mobile phase. Application of standard reversed phase (RP) HPLC columns failed to separate the compounds, wherefore a special reversed phase/borderline ion exchange (Synergi Fusion, also see Supporting Information) phase had to be used. The broad peaks do not indicate the formation of multiple isomers. The identity of nat Zr-3 was confirmed by LRMS (m/zcalculated= 912. [START_REF] Zeglis | A practical guide to the construction of radiometallated bioconjugates for positron emission tomography[END_REF] [M+H] + , m/zobserved = 912.37; [M+H] + , Figure S4).

Next, we looked into the radiolabelling kinetics in more detail. Shortly after the start of the reaction, a transient intermediate, which was not further characterized, formed in minor amounts as detected by iTLC (Rf 89 Zr-intermediate = 0.2; not observed by RP-TLC). This intermediate converted slowly to the desired radiometal complex (Rf 89 Zr-3 = 0) within 60 min and without release of 89 Zr (Rf free 89 Zr 4+ as its citrate = 0.8-1, Figure S3). Deri et al. have reported similar observations during 89 Zr-labeling studies of the chelator HOPO. [START_REF] Deri | Alternative chelator for (8)(9)Zr radiopharmaceuticals: radiolabeling and evaluation of 3,4,3-(LI-1,2-HOPO)[END_REF] They suggested that the intermediate might correspond to an initially formed kinetic product, which reacts further to the thermodynamically stable final complex. [START_REF] Deri | Alternative chelator for (8)(9)Zr radiopharmaceuticals: radiolabeling and evaluation of 3,4,3-(LI-1,2-HOPO)[END_REF] The transient intermediate formed is rapidly and fully converted to the desired radiolabelled product within 2 h, wherefore its appearance was not considered as a limitation for applications. The clean formation of desired product 89 Zr-3 with a radiochemical yield (RCY) of ≥97% after approximately 90-120 min could be confirmed. Figure 2A shows the formation of 89 Zr-3 (red curve) as well as the formation of the intermediate (dots) and its conversion to 89 Zr-3. The radiolabelling efficiency and achievable molar activity (up to 3 GBq/µmol) was comparable to those of the reported analogous chelator DFO* 2. [START_REF] Patra | A novel octadentate BFCA for zirconium-89[END_REF] Since oxoDFO* 3 primarily designed for a combination with temperature sensitive proteins, kinetic studies were only performed at room temperature.

Assessment of the Stability of 89 Zr-Complexes

To evaluate the stability of the 89 Zr-complexes of DFO 1, DFO* 2, and oxoDFO* 3, they were incubated at RT and pH 6 with an excess of DTPA (5 mM or 50 mM). We chose a slightly acidic pH because 89 Zr-DFO has been reported to be more stable under these conditions. [START_REF] Deri | Alternative chelator for (8)(9)Zr radiopharmaceuticals: radiolabeling and evaluation of 3,4,3-(LI-1,2-HOPO)[END_REF] The extent of transchelation of 89 Zr was determined by TLC at different time points up to 48 h (Rf 89 Zr-DTPA = 0.9-1, Rf 89 Zr-1,2,3 = 0). The experiments performed with 5 mM of DTPA indicated superior stability of 89 Zr-2 or 3 in comparison to 89 Zr-1. While [ 89 Zr]Zr-DFO* ( 89 Zr-2) and 89 Zr-3 were stable in the presence of 5 mM DTPA over a time period of 48 h, 50% of [ 89 Zr]Zr-DFO ( 89 Zr-1) converted to 89 Zr-DTPA within 24 h (Figure 4). Even though transchelation was observed for all 89 Zr-complexes at the higher concentration of DTPA of 50 mM, octadentate DFO* 2 and oxoDFO* 3 clearly formed the more stable complexes with 89 Zr than hexadentate DFO 1. These results are in agreement with previously published data [START_REF] Patra | An octadentate bifunctional chelating agent for the development of stable zirconium-89 based molecular imaging probes[END_REF] and reflect the increased stability of Zr-complexes as a result of octadentate coordination of the metal. Furthermore, the challenging experiments at 50 mM allowed for the determination of the stabilities of 89 Zr-2 or 3, relative to one another. Interestingly, 89 Zr-3 exhibited slightly improved stability at the higher concentration of DTPA than 89 Zr-2 in these experiments. Since the benefit of the improved stability of 89 Zr-2 has already been demonstrated in mice and primates, [START_REF] Vugts | Comparison of the octadentate bifunctional chelator DFO*-pPhe-NCS and the clinically used hexadentate bifunctional chelator DFO-pPhe-NCS for 89Zr-immuno-PET[END_REF][START_REF] Berg | Total-body PET and highly stable chelators together enable meaningful (89)Zr-antibody-PET studies up to 30 days post-injection[END_REF] the analogous, water-soluble chelator oxoDFO* 3 is likely to be suitable for in vivo as well. 68 Ga-Labeling Experiments Hydroxamate-containing ligands have also found applications in combination with other (PET) radiometals. [31-33] particular, complexes of DFO 1 with 68 Ga have been used for the labelling of affibodies [34] and peptides [35,36] as well as for infection imaging. [37,38] This is despite the fact that the stability of 68 has been the subject of contradictory discussions. [START_REF] Kaeppeli | Comparison of desferrioxamine and NODAGA for the gallium-68 labeling of exendin-4[END_REF] To further study the utility of our DFO* 2 and oxoDFO* 3 chelators, we decided to evaluate their suitability as 68 Ga chelators. 68 Ga-labelling experiments were performed at different pH and temperatures and the stability of the Ga-complexes formed were studied. HEPES was chosen as a buffer at two different pH values (pH 4.5 and 7.4) due to its favourable properties to reduce the formation of 68 Ga-colloids [START_REF] Bauwens | Optimal buffer choice of the radiosynthesis of (68)Ga-Dotatoc for clinical application[END_REF] and in accordance with the 68 Ga-radiolabelling conditions used in our clinical routine facilities (pH 4.5) This also allowed us to compare the 68 Ga results directly to the radiolabelling results for 89 Zr, which were also performed at pH 7.4.

In brief, better radiolabelling results were achieved at pH 4.5 than at pH 7.4. For the latter pH, radiolabelling yields were 22-40% for chelators 1, 2 & 3. However, complete complexation of 68 Ga 3+ could not be achieved at RT and pH 4.5 within a reasonable reaction time for the shortlived radiometal (Table 1). Unlike in the case of 89 Zr, an elevated temperature (95 °C) was necessary to accomplish quantitative 68 Ga-radiolabelling. Interestingly, these findings are in contrast with results by Tsionou et al., who reported quantitative 68 Ga-radiolabelling of DFO at room temperature within 10 minutes at slightly different pH in ammonium acetate buffer. [START_REF] Tsionou | Comparison of macrocyclic and acyclic chelators for gallium-68 radiolabelling[END_REF] Table 1. Radiolabelling of DFO 1, DFO*2 and oxoDFO* 3 with 68 GaCl3 in HEPES at pH 4.5 and stability of the corresponding complexes 68 Ga-1,2,3 (n = 2).

Ligand

T a) Values after incubation for 1 h / 2 h in 5 mM DTPA at pH 6.

The stability of the 68 Ga-complexes was assessed by the same transchelation experiments with an of DTPA (5 mM) as described above. In general, we found that 68 Ga-1,2,3 were significantly less stable under the applied conditions than the corresponding 89 Zr-complexes with a decreasing stability of 68 Ga-1 ≈ 68 Ga-2 > 68 Ga-3 (Table 1). Surprisingly, approximately one third of the radiometal was found already transchelated to the challenging chelator DTPA for 68 Ga-1 after 2 h, although 68 Ga-complexes of DFO 1 conjugated to peptides and proteins have been reported to be stable in vitro and in vivo. [START_REF] Bauman | Development of 68Ga-and 89Zr-Labeled Exendin-4 as Potential Radiotracers for the Imaging of Insulinomas by PET[END_REF][START_REF] Kaeppeli | Comparison of desferrioxamine and NODAGA for the gallium-68 labeling of exendin-4[END_REF][START_REF] Oroujeni | Preclinical Evaluation of [(68)Ga]Ga-DFO-ZEGFR:2377: A Promising Affibody-Based Probe for Noninvasive PET Imaging of EGFR Expression in Tumors[END_REF][START_REF] Tsionou | Comparison of macrocyclic and acyclic chelators for gallium-68 radiolabelling[END_REF] Transchelation experiments in vitro are appropriate for a relative comparison of the stability of different (radio)metal complexes but only of limited value to predict their stability in vivo.

Conclusion

In this proof of concept study, we report the first 89 Zr-labelling of the water-soluble, octadentate chelator oxoDFO* 3. Quantitative complexation of 89 Zr-oxalate was achieved at RT and pH 7.4 within 2 h, reaction conditions well-suited for the radiometallation of Abs. The stability of the formed complex 89 Zr-3 as determined in vitro by transchelation experiments was found to be similar to that of 89 Zr-2. [START_REF] Patra | A novel octadentate BFCA for zirconium-89[END_REF] Since the applicability of DFO* 2 for 89 Zr-immunoPET has already been demonstrated preclinically, [START_REF] Berg | Total-body PET and highly stable chelators together enable meaningful (89)Zr-antibody-PET studies up to 30 days post-injection[END_REF][START_REF] Vugts | Comparison of the octadentate bifunctional chelator DFO*-pPhe-NCS and the clinically used hexadentate bifunctional chelator DFO-pPhe-NCS for 89Zr-immuno-PET[END_REF] oxoDFO* also represents a promising candidate in particular in cases where the application of co-solvents for conjugation chemistry is an issue due to the sensitivity of the protein of interest.

DFO 1, DFO* 2, and oxoDFO* 3 can also be applied for the complexation of 68 Ga, however, elevated temperatures are required for quantitative radiolabelling and the formed products 68 Ga-1,2,3 exhibit a lower stability in vitro than the corresponding 89 Zr-complexes. The employment of alternatives to commonly used chelators for 68 Ga-labeling of biological molecules [START_REF] Tsionou | Comparison of macrocyclic and acyclic chelators for gallium-68 radiolabelling[END_REF] has been reported to be beneficial in certain cases [START_REF] Kaeppeli | Comparison of desferrioxamine and NODAGA for the gallium-68 labeling of exendin-4[END_REF] and thus, further studies are warranted.

In depth investigations towards the application of bifunctional versions of oxoDFO* 3 [START_REF] Briand | A solid phase-assisted approach for the facile synthesis of a highly water-soluble zirconium-89 chelator for radiopharmaceutical development[END_REF] for the 89 Zr-labelling of proteins and antibodies including in vivo testing have now become justified by our proof-of-concept study. Efforts to raise funds for the preclinical evaluation of 89 in vivo are initiated. In addition, the assessment of the suitability of the novel chelators for complexation of other (radio)metals is currently ongoing and will be reported in due time.

Figure 1 .

 1 Figure 1. Hexadentate DFO 1 and its octadentate analogues DFO* 2 and oxoDFO* 3.

  For68 Ga-labelling reactions, 3.3 µL of stock solutions of DFO 1, DFO* 2 and oxoDFO* 3 (3.3 nmol),) were diluted with either 300 µL HEPES buffer (0.5 M, pH 7.4) or HEPES buffer (1.5 M, pH 4.5) in a LoBind Eppendorf® flask, resulting in a final precursor concentration of 10 M. 20 µL 68 GaCl3 (50-200 MBq) were added to each vial. Radiolabelling reactions were performed at RT for 2 h or in a heating block at 95 °C for 10 min. The progress of the reactions was monitored with iTLC and 0.05 M EDTA pH 4.5 as mobile phase. Rf ( 68 Ga-EDTA = 1; 68 Ga-1,2,3 = 0.

Figure 2 .

 2 Figure 2. A) Formation of 89 Zr-3 using 10 µM precursor at RT in HEPES (pH 7.4, RT, n = 3); B) RCY for the radiolabelling of oxoDFO* 3 with 89 at different concentrations of oxoDFO* 3 after a reaction time of 2 h at RT (n=2-3). The blue curve indicates the sum of the intermediate and 89 Zr-3.

Figure 3 .

 3 Figure 3. HPLC-chromatograms of oxoDFO* 3, 89 Zr 4+ , nat Zr-3 and 89 Zr-3. The identity of nat Zr-3 (peak at Rt = 5.9 min) was also confirmed by mass spectrometry. Slightly different Rt-values is the result of the serial arrangement of the γ-and UV-detectors. UV traces were measured at λ = 220 nm.

Figure 4 .

 4 Figure 4. Transchelation curves for 89 Zr-1, 2, 3 measured in 5 mM and 50 mM DTPA at pH 6 and RT. Data points of 89 Zr-2 (orange line) and 89 Zr-3 (grey line) are overlapping; n = 2-3. These results are additionally summarized in a table, which can be found in the supporting material.

This work was funded by the Swiss National Science Foundation (grant Nr. SNSF 205321_157216 to G.G. and T.L.M). The authors thank Katarina Benčurová (Ludwig Boltzmann Institute Applied Diagnostics) and Jonas Aronow (Medical University of Vienna

Conflict of interest

Thomas Mindt and Gilles Gasser are co-inventors on pending patent applications describing DFO* and its derivatives. These patent applications are owned by the University of Zurich and the University of Basel.The patent owners have granted an exclusive license to the company ABX Advanced Biochemicals Compounds. The inventors are unrelated to this company.

Supporting information

Additional experimental material (HPLC chromatograms, iTLC pictures, MS spectra, stability data and general notes) is available at http//:doi.xxxxxxxxxxxxxxxxxxxxxx

Graphical abstract

The water-soluble, octadentate chelator oxoDFO* provides stable complexes with the positron emitter Zirconium-89. The radiolabelling can be performed at room temperature and neutral pH and thus, oxoDFO* represents a promising chelator for applications in immunoPET.


M Bonnivard

J Olivier

From adherence to slip in nanofluidics: a mathematical justification based on a drop of viscosity. The scalar case

From adherence to slip in nanofluidics: a mathematical justification based on a drop of viscosity. The scalar case

Introduction

Context and goals

Nanofluidics is the study of the flow of materials at the nanoscale [START_REF] Bocquet | Nanofluidics, from bulk to interfaces[END_REF]. This is the scale of flow in Carbon Nanotubes which are physical systems of great interest nowadays [START_REF] Holt | Fast mass transport through sub-2-nanometer carbon nanotubes[END_REF][START_REF] Whitby | Fluid flow in carbon nanotubes and nanopipes[END_REF]. At this scale, flows present some striking features, such as the capacity for the material to slip much more easily that one would expect [START_REF] Bocquet | Flow boundary conditions from nano-to microscales[END_REF][START_REF] Secchi | Massive radius-dependent flow slippage in carbon nanotubes[END_REF][START_REF] Ramya | Enhanced water permeability and tunable ion selectivity in subnanometer carbon nanotube porins[END_REF]. The origin of this slip is the subject of current debate in the physics community [START_REF] Faucher | Critical knowledge gaps in mass transport through single-digit nanopores: A review and perspective[END_REF][START_REF] Lauga | Microfluidics: The No-Slip Boundary Condition[END_REF].

In classical fluid mechanics, the interaction between the fluid and the walls of a bounded domain is usually modeled via appropriate boundary conditions. Probably the three most used boundary conditions are (let us note u the velocity field of the flow)

No-slip (or adherence): imposing that u vanishes at the wall.

Slip: imposing that the normal component of u vanishes and that the tangential part of the normal stress is proportional to the tangential part of the velocity field. The inverse of the proportionality factor has the dimension of a length called the slip length.

Perfect slip: imposing that the normal component of u vanishes and that the tangential part of the normal stress also vanishes.

Let us point that, although one could expect the averaging procedure usually applied to obtain macroscopic flow equations to fail at the nanoscale, Stokes equations remain surprisingly efficient in nanofluidics [START_REF] Gravelle | Large permeabilities of hourglass nanopores: From hydrodynamics to single file transport[END_REF][START_REF] Karniadakis | Simple Fluids in Nanochannels[END_REF]. From a mathematical perspective, one successful strategy initiated in the early 2000's [START_REF] Casado-Díaz | Why viscous fluids adhere to rugose walls: A mathematical explanation[END_REF][START_REF] Jäger | On the roughness-induced effective boundary conditions for an incompressible viscous flow[END_REF] to explain the occurrence of adherence or partial slip on solid walls, consists in modeling microasperities on the surface and analyzing their effect on the flow by an homogenisation process, imposing only a mild non penetration boundary condition on the rugous wall, i.e. that the normal component of the fluid velocity vanishes. This so-called "rugosity effect" has been studied quite extensively in the last decades, which has led to a rather complete description of the asymptotic effect of rough patterns on viscous flow [START_REF] Bucur | Boundary behavior of viscous fluids: influence of wall roughness and friction-driven boundary conditions[END_REF][START_REF] Bucur | On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries[END_REF][START_REF] Jäger | On the roughness-induced effective boundary conditions for an incompressible viscous flow[END_REF].

In this paper, we consider a completely different interpretation of the apparent slip length measured in nanoscopic devices, proposed in [START_REF] Myers | Why are why are slip lengths so large in carbon nanotubes? Microfluid[END_REF], where the author postulates that the source of this slip arises from a "depletion layer with reduced viscosity near the wall". This hypothesis is supported by experimental evidence [START_REF] Poynor | How water meets a hydrophobic surface[END_REF] and Molecular Dynamics simulations [START_REF] Joseph | Why are carbon nanotubes fast transporters of water?[END_REF] bringing out that the viscosity drops near the wall of the nanotube. In [START_REF] Myers | Why are why are slip lengths so large in carbon nanotubes? Microfluid[END_REF], the flow is modeled as a Stokes flow in an infinite cylindrical pipe, with no-slip boundary condition, but two viscosities: a "bulk" one at the center of the pipe and a "wall" one near the walls which is smaller than the bulk one. In this model, the fluid is supposed to adhere at the wall. Yet, by solving the equation in this simple geometry, the author was able to describe the resulting flow as if it had an effective slip length and a constant viscosity equal to the one in the bulk, computing this length in terms of the viscosity drop and the sizes of the depletion layer and tube radius. From this result we started wondering: is there a general mathematical framework to study the passing from a model with varying viscosity and no-slip to a model with constant viscosity and slip?

The question is to link two models with different equations in the same domain and different boundary conditions. Many different problems of this type are (or have been) studied with the help of asymptotic analysis in PDE: one expresses the parameters of one of the model as function of small parameter ε and by studying the behaviour of the solution as ε goes to 0, one proves that the solution converges to the second model. The now classical problem that might be the most closely related to ours, and that we already mentioned, is the rugosity effect. In that case, one attempts to link a model posed on a family of domains depending on a parameter ε and "converging" in some sense with the perfect slip boundary condition on each domain, and to explain how one obtains a positive slip length, or no-slip on the limit domain. In a certain sense, the problem that we propose to address is the opposite: whereas rugosity aims at explaining how a fluid can "slow down" because of the wall, our problem is to explain and justify mathematically the "speeding up" of the fluid caused by a drop of viscosity near the wall.

The model

The model we will study is as follows. Our spatial domain will be the square

Ω =]0, 1[×]0, 1[
and is loosely thought of as representing half of a longitudinal section of nanotube. In this framework one can think of x 1 to be the longitudinal coordinate along the tube while x 2 is the radial coordinate away from the axis. The axis of the tube would be at x 2 = 0 and the carbon layer is at x 2 = 1. In this setting, what we call the lateral boundaries x 1 = 0 and x 1 = 1 do not have much of an influence on the problem as the sequel will show. The depletion layer will thus be located around x 2 = 1 and we will model it to have a typical size of ε. More precisely, let us introduce a function d ∈ C 1 ([0, 1]) and the function γ ε defined on [0, 1] by γ ε (x 1 ) = 1 -εd(x 1 ) .

(

) 1
We denote by Γ ε the graph of γ ε , defined by

Γ ε = {(x 1 , γ ε (x 1 )), x 1 ∈ [0, 1]} . (2)
In out setting, the depletion layer in our setting is defined as

B ε = {(x 1 , x 2 ) ∈ Ω, γ ε (x 1 ) < x 2 < 1} ⊂ Ω .
To simplify notation we will note

Ω ε = Ω \ B ε .
To the macroscopic variable x = (x 1 , x 2 ) ∈ B ε , we associate the microscopic variable y = (y 1 , y 2 ) defined by

y 1 = x 1 , y 2 = 1 -x 2 ε .
The depletion layer is then described in microscopic variable by

B ε = {(y 1 , 1 -εy 2 ), (y 1 , y 2 ) ∈ ω} ,
where ω is defined by ω = (y 1 , y 2 ) ∈ (0, 1) 2 , 0 < y 2 < d(y 1 ) .

In this paper we will not study the fluid problem directly, but present a toy model that we believe contains the main features of the full fluid problem. Basically, we will replace the vectorial Stokes operator with a 2d scalar elliptic operator to present the basic ideas and tools. The extension to the full vectorial fluid problem is left for subsequent work.

Our equation is thus a stationary diffusion equation with a space varying coefficient to account for the depletion layer. To stay close to [START_REF] Myers | Why are why are slip lengths so large in carbon nanotubes? Microfluid[END_REF], the diffusion will take two constant values, 1 in Ω ε and a smaller value in the depletion layer B ε . To parametrize the drop of viscosity inside the depletion later we introduce the parameter 0 < α < 2, such that inside B ε the viscosity is ε α . In other words, our viscosity µ ε is defined by

µ ε (x 1 , x 2 ) = 1 Ωε (x 1 , x 2 ) + ε α 1 Bε (x 1 , x 2 ) (3)
where 1 Ωε and 1 Bε stand for the indicator functions of Ω ε and B ε respectively. We will work with the PDE in Ω where the unknown is u ε and f is a right-hand side defined on Ω:

-div(µ ε ∇u ε ) = f . (4)
As usual this system is supplemented with boundary conditions. For the sake of simplicity we have chosen to put an homogeneous Dirichlet boundary condition at the bottom x 2 = 0 and on the lateral boundary condition x 1 = 0 and x 1 = 1. As mentioned before, our analysis does not rely much on the fact that we know that u ε vanishes at the lateral boundaries and we think that other boundary conditions such as periodic ones for instance could be analysed exactly as in our method. On the other hand, the Dirichlet boundary condition on x 2 = 0 is really important in our analysis and changing it to something else would need a careful look at our study to see what stays and what must be changed. So in the sequel we will always have

u ε = 0 at x 2 = 0 , u ε = 0 at x 1 = 0 and x 1 = 1 . (5)
Finally our main concern is with the boundary condition at x 2 = 1. We will study the system with an homogeneous Neumann boundary condition and an homogeneous Dirichlet boundary condition in the sequel.

To conclude this part, let us stress that we make the following analogy with the fluid problem: the no-slip property will be modeled by a Dirichlet boundary condition, the slip condition property will be modeled by a Robin (also called Fourier) boundary condition while the perfect slip boundary condition will be modeled by a Neumann boundary condition.

A 1d example

In this section we would like to give a foretaste of what is to come, by simplifying even more the model to a 1d case, and taking the external force f to be constant (equal to 1). We will not give full details of the computation but try and present the main features of our model. In this 1d model the geometry simplifies and B ε just becomes ]1 -ε, 1[ and the viscosity is defined on the segment [0, 1] by

µ ε = 1 ]0,1-ε[ + ε α 1 ]1-ε,1[ . (6)
The problem reduces here to finding u ε ∈ H 1 (0, 1) satisfying

-(µ ε u ε ) = 1 in ]0, 1[ u ε (0) = 0
and either u ε (1) = 0 for the Neumann case or u ε (1) = 0 in the Dirichlet case.

Neumann case

As in the paper, let us start with the Neumann case. It is easy to compute by hand the solution of this problem:

u ε (x) = 1 2 (1 -(1 -x) 2 ) for 0 ≤ x ≤ 1 -ε 1 2 (1 -ε 2 ) + 1 2ε α (ε 2 -(1 -x) 2 ) for 1 -ε ≤ x ≤ 1
What is to be noticed in this case are the following facts:

• Whatever the value of α, the sequence of functions u ε converges to the function ū

(x) = 1 2 (1 -(1 -x) 2 ). Moreover ū is "determined" by what happens in the intervals ]0, 1 -ε[.
For instance ū is also the limit of the sequence of functions

ūε (x) = u ε (x) for 0 ≤ x ≤ 1 -ε u ε (1 -ε) for 1 -ε ≤ x ≤ 1
This will be important in our analysis as the sequence (ū ε ) is obviously bounded in H 1 (0, 1) whereas we have

1 0 (u ε (x)) 2 dx = 1-ε 0 (u ε (x)) 2 dx + 1 1-ε (u ε (x)) 2 dx = 1 -ε 3 3 + ε 3 ε 2α
which means that (u ε ) is not necessarily bounded in H 1 (0, 1).

• The limit function ū is the solution to the system

     -u = 1 in ]0, 1[ u(0) = 0 u (1) = 0
This means that the limit problem seems to keep the same boundary condition at x = 1. This is coherent with our view: if the fluid is allowed to slip perfectly right from the start, then it cannot be "sped up" by getting slip as it is already perfectly slipping.

• The energy of the system is defined by

1 0 µ ε (u ε ) 2 and thus is equal to 1 0 µ ε (x)(u ε (x)) 2 dx = 1 -ε 3 3 + ε 3 ε α
We conclude from this that the energy of the system is not necessarily bounded if α is too large. This explains why we will put some restrictions on the values of α. In this particular example, the right-hand side f is a very simple and regular function, and even in this case we could not guarantee from qualitative properties alone on the data that the energy would be bounded. But since our goal is to study a physically sensible system, we will have to ensure that the energy remains bounded, and hence include this as an hypothesis in our framework. In the one-dimensional Neumann case, this corresponds to assuming that α < 3.

• Finally one can compute the difference u ε -ū to be

u ε (x) -ū(x) = 0 if 0 ≤ x ≤ 1 -ε 1 ε α -1 1 2 (ε 2 -(1 -x) 2 ) if 1 -ε < x < 1
From this expression, it is easy to compute

1 0 (u ε -ū ) 2 = 1 ε α -1 2 ε 3
3 which means that u ε does not converge strongly to ū in H 1 (0, 1) if α is larger than 3/2. It is not surprising that the sequence does not converge in the H 1 norm to its limit as it is customary in rugosity problems for instance [START_REF] Bonnivard | Computation of the effective slip of rough hydrophobic surfaces via homogenization[END_REF][START_REF] Bucur | On the asymptotic limit of flows past a ribbed boundary[END_REF][START_REF] Casado-Díaz | Asymptotic behavior of a viscous fluid with slip boundary conditions on a slightly rough wall[END_REF]. So let us look at the limit in L p norm instead. In this case we have

1 0 (u ε -ū) p = 1 ε α -1 p 1 2 p 1 1-ε (ε 2 -(1 -x) 2 ) p dx .
We change variable and introduce the microscopic variable y = (1 -x)/ε, a variable that we will also use later on. We the obtain:

1 0 (u ε -ū) p = 1 ε α -1 p ε 2p+1 2 p 1 0
(1 -y 2 ) p dy .

From this we obtain the convergence of u ε to ū in L p as long as α < 2p+1 p . On the one hand, setting p = 1 in this inequality yields α < 3. This means that in the limit case α = 3 where the energy is still bounded, there is not even a convergence in L 1 even if both u ε and the limit ū are uniformly bounded in L 2 . On the other hand, the inequality is satisfied for all p if α ≤ 2 and in particular for p = 2. Finally, if α > 2 we can write p < 1 α-2 which means the strength of the convergence decreases as α increases. This discussion on strong convergence in L p shows that taking large values of α can make the analysis more difficult, and gives another insight on the necessity to impose an upper bound on the value of α in order to get useful information on the limit when ε goes to zero.

Dirichlet case

To end this section we would like to give some elements of the same 1d model with a Dirichlet boundary condition. This time we consider an arbitrary (continuous or integrable for instance) f and try to solve:

     -(µ ε u ε ) = f in ]0, 1[ u ε (0) = 0 u ε (1) = 0 (7)
We will not go into detail but there is a kernel K ε defined on [0, 1] 2 such that the solution u ε can be written as

u ε (x) = 1 0 K ε (x, s)f (s)ds , (8)
where for s > x,

K ε (x, s) = x 0 µ -1 ε 1 s µ -1 ε 1 0 µ -1 ε , (9)
and for x > s, K ε (x, s) = K ε (s, x). Using definition (3), a straightforward computation gives for s > x

K ε (x, s) =      x(1-ε-s+ε 1-α ) 1-ε+ε 1-α on 0 ≤ x ≤ s ≤ 1 -ε x 1-s ε α 1-ε+ε 1-α for s ∈ [1 -ε, 1], x ∈ [0, 1 -ε] 1 - 1 1-ε+ε 1-α 1-x ε α 1-s ε α on 1 -ε ≤ x ≤ s ≤ 1 .
Hence, we can study the limiting process of u ε as ε goes to 0 by studying the limiting process of K ε . Interestingly we have three possible regimes:

subcritical α < 1: in this regime K ε converges (at least pointwise) to K 0 (x, s) = x(1 -s) for 0 ≤ x ≤ s ≤ 1 critical α = 1: in this regime K ε converges (at least pointwise) to K 1 (x, s) = x 1 -s 2 for 0 ≤ x ≤ s ≤ 1 overcritical α > 1: in this regime K ε converges (at least pointwise) to K 2 (x, s) = x for 0 ≤ x ≤ s ≤ 1 and again K i (s, x) = K i (x, s). Now define u i (x) = 1 0 K i (x, s)f (s)ds.
Interestingly enough, these functions are solutions to diffusion problems with different boundary conditions namely:

α < 1: u 0 solves a Dirichlet problem at x = 1:      -u 0 = f u 0 (0) = 0 u 0 (1) = 0 (10) α = 1: u 1 solves a Robin problem at x = 1:      -u 1 = f u 1 (0) = 0 u 1 (1) + u 1 (1) = 0 (11) α > 1: u 2 solves a Neumann problem at x = 1:      -u 2 = f u 2 (0) = 0 u 2 (1) = 0 (12)
Again these computations can be done by hand and the various convergence can be more or less strengthened depending on the value of α just like in the Neumann case. What is promising is that this 1d version of our toy model seems to lead exactly to the type of discussion we want to obtain. The sequel of the article is dedicated to understanding these questions in 2d in a framework general enough to avoid computing solutions exactly, which is rarely possible.

Outline of the paper

The rest of the paper is organized as follows. In Section 2, we state our main results on the asymptotic behaviour of u ε , solution to (4) and satisfying (5) as well as Neumann or Dirichlet boundary conditions on x 2 = 1, assuming a certain energy bound (Theorems 1 and 2). Relying on explicit computations, we also establish in Theorem 3 that this bound is automatically satisfied by u ε in the case of a flat interface. Sections 3 and 4 are dedicated to the proof of Theorems 1 and 2; Section 3 gathers compactness results on u ε and Section 4 characterizes the limit ū as the solution to the Poisson problem -∆ū = f in Ω, with zero boundary condition on x 2 = 0, x 1 = 0, x 1 = 1, completed with the appropriate boundary condition on x 2 = 1, that expresses the effect of the drop of viscosity. In Section 5, we prove Theorem 3, in the case of Dirichlet boundary condition on x 2 = 1; to lighten up the presentation, most of the explicit computations are postponed to Appendix A. Finally, Section 6 discusses possible generalizations of our results.

Main results

Let α ∈]0, 2[ and for any ε > 0, let µ ε be defined by (3).

The Neumann case

First, from standard elliptic theory (see [START_REF] Evans | Partial differential equations[END_REF]) we have the following result for fixed ε > 0: Proposition 1. Let V N be defined by

V N = {v ∈ H 1 (Ω), v |x 1 =0 = v |x 1 =1 = 0, v |x 2 =0 = 0}, (13)
which is a Hilbert space when endowed with the scalar product

(v, w) = Ω ∇v • ∇w for any v, w ∈ V N .
Let f be in L 2 (Ω). Then there exists a unique function

u ε in V N , solution of the following problem      -div(µ ε ∇u ε ) = f in Ω u ε = 0 on x 1 = 0, x 1 = 1 and x 2 = 0 ∂ 2 u ε = 0 on x 2 = 1 (14)
which means for all φ ∈ V N Ω\Bε

∇u ε • ∇φ + ε α Bε ∇u ε • ∇φ = Ω f φ . (15)
Before we can state our results let us give a definition.

Definition 1. We say that a family (u ε ) ε>0 of functions in H 1 (Ω) satisfy the energy bound if there exists a real constant C > 0 independent of ε such that

Ω\Bε |∇u ε | 2 + ε α Bε |∇u ε | 2 ≤ C . (16)
Now let us state our main results for the Neumann case. First the general result:

Theorem 1. Let (u ε ) ε be the family of solutions obtained from Proposition 1 and suppose that this family satisfy the energy bound [START_REF] Holt | Fast mass transport through sub-2-nanometer carbon nanotubes[END_REF]. Then u ε converges weakly in L 2 (Ω) toward ū the variational solution to the problem:

     -∆ū = f in Ω ū = 0 on x 1 = 0, x 1 = 1 and x 2 = 0 ∂ 2 ū = 0 on x 2 = 1
To us it is not standard elliptic theory that the family of solutions given by Proposition 1 satisfy automatically an energy bound ( 16) and we take it as an hypothesis in Theorem 1. We suspect that the validity of this hypothesis might not be true in general but depends on f . Nevertheless, this hypothesis is physically meaningful: it means that there is finite energy in the system independently of the scale at which we look at it which is a reasonable assumption for this procedure to make sense.

The Dirichlet case

Again from standard elliptic theory we can rely on the following theorem: Proposition 2. Let V D be defined as

V D = H 1 0 (Ω) (17)
Let f be in L 2 (Ω). Then there exists a unique function

u ε in V D solution of the following problem      -div(µ ε ∇u ε ) = f in Ω u ε = 0 on x 1 = 0, x 1 = 1 and on x 2 = 0 u ε = 0 on x 2 = 1 (18)
which means for all φ ∈ V D

Ω\Bε ∇u ε • ∇φ + ε α Bε ∇u ε • ∇φ = Ω f φ .
Our main results are thus:

Theorem 2. Let (u ε ) ε be the family of solutions obtained from Proposition 2 and suppose that this family satisfy the energy bound [START_REF] Holt | Fast mass transport through sub-2-nanometer carbon nanotubes[END_REF]. Then u ε converges weakly in L 2 (Ω) toward a function ū such that if 0 < α < 1, ū is the variational solution to the problem:

     -∆ū = f in Ω ū = 0 on x 1 = 0, x 1 = 1 and on x 2 = 0 ū = 0 on x 2 = 1 if α = 1
, ū is the variational solution to the problem:

     -∆ū = f in Ω ū = 0 on x 1 = 0, x 1 = 1 and on x 2 = 0 ∂ 2 ū + 1 d(x 1 ) ū = 0 on x 2 = 1 (19)
if 1 < α < 2, ū is the variational solution to the problem:

     -∆ū = f in Ω ū = 0 on x 1 = 0, x 1 = 1 and on x 2 = 0 ∂ 2 ū = 0 on x 2 = 1 (20)

The flat interface case

The previous results relied on the hypothesis that the family of solutions of ( 14) or ( 18) satisfy the energy bound ( 16) and we have already explained the relevance of this hypothesis in our physical context.

In the case of a flat interface (d(x 1 ) = 1 and γ ε (x 1 ) = 1 -ε), however, we can use the simplified geometry to compute more finely the solutions of ( 14) and ( 18) using Fourier analysis and the tensorial nature of the geometry to check that the energy bound ( 16) is satisfied. This analysis is based on the following lemma: Lemma 1. From basic Fourier theory (see for instance Th. II.10 p.52 and Problem 12 p. 64 in [START_REF] Reed | Methods of Modern Mathematical Physics, Volume I : Functional Analysis[END_REF]) we can state that

• For any g in L 2 (]0, 1[×]a, b[) there exists a unique sequence (g k ) k≥1 of functions in L 2 (a, b) such that k≥1 g k 2 L 2 (a,b) < +∞ and such that the series k≥1 g k (x 2 ) sin(kπx 1 ) converges to g in L 2 (]0, 1[×]a, b[). Moreover g 2 L 2 (]0,1[×]a,b[) = 1 2 k≥1 g k 2 L 2 (a,b) • Moreover, g is in H 1 (]0, 1[×]a, b[) with zero lateral traces if, and only if g k ∈ H 1 (a, b) for all k ≥ 1 and k≥1 1 0 (k 2 π 2 |g k (x 2 )| 2 + |g k (x 2 )| 2 )dx 2 < +∞ and one has ∂ 1 g 2 L 2 (]0,1[×]a,b[) = 1 2 k≥1 k 2 π 2 g k 2 L 2 (a,b) and ∂ 2 g 2 L 2 (]0,1[×]a,b[) = 1 2 k≥1 g k 2 L 2 (a,b)
Then we can state:

Theorem 3. Assume γ ε = 1 -ε and f is in L 2 (Ω).
Then by Lemma 1 there exists a family

(f k ) k≥1 of functions in L 2 (0, 1) such that f (x 1 , x 2 ) = k≥1 f k (x 2 ) sin(kπx 1 )
with equality in the space L 2 (Ω).

1. For all k ≥ 1 and for all ε > 0,

there exist c - k in H 2 (0, γ ε ) and c + k in H 2 (γ ε , 1) solutions in the strong sense of                    -c - k + k 2 π 2 c - k = f k on ]0, γ ε [ ε α (-c + k + k 2 π 2 c + k ) = f k on ]γ ε , 1] c - k (0) = 0 c - k (γ ε ) = c + k (γ ε ) c - k (γ ε ) = ε α c + k (γ ε ) c + k (1) = 0 (21)
2. For all ε > 0,

k≥1 1 0 (k 2 π 2 |c ± k (x 2 )| 2 + |c ± k (x 2 )| 2 )dx 2 < +∞ (22) so one can define u - ε in H 1 (Ω ε ) and u + ε in H 1 (B ε ) by u ± ε (x 1 , x 2 ) = k≥1 c ± k (x 2 ) sin(kπx 1 ). 3. u - ε does not belong to H 2 (Ω ε ) but it is in {v ∈ H 1 (Ω ε ), ∆v ∈ L 2 (Ω ε )} and similarly u + ε belongs in {v ∈ H 1 (B ε ), ∆v ∈ L 2 (B ε )} and we have                          -∆u - ε = f in Ω ε -ε α ∆u + ε = f in B ε u - ε |x 2 =0 = 0 u ± ε |x 1 =0 = u ± ε |x 1 =1 = 0 u - ε |x 2 =γε = u + ε |x 2 =γε ∂ x 2 u - ε |x 2 =γε = ε α ∂ x 2 u + ε |x 2 =γε u + ε |x 2 =1 = 0
4. The function u ε defined by

u ε (x) = u - ε (x) if x ∈ Ω ε u + ε (x) if x ∈ B ε is in H 1 (Ω)
and is the variational solution to (18).

5. The family (u ε ) ε satisfies (16) as long as α ≤ 2.

3 Asymptotic analysis of a sequence (u ε ) satisfying the energy bound [START_REF] Holt | Fast mass transport through sub-2-nanometer carbon nanotubes[END_REF] The aim of this section is to analyze the behaviour as ε goes to 0 of a family of function (u ε ) ε in V which can be V N (recall that it is defined by ( 13)) or V D (defined by ( 17)) satisfying the energy bound [START_REF] Holt | Fast mass transport through sub-2-nanometer carbon nanotubes[END_REF]. Thus in the rest of this section, (u ε ) ε will be a sequence of functions in V satisfying ( 16) but not necessarily the solution of a problem for the moment.

Modifying u ε in B ε

Since u ε vanishes on x 2 = 0, by Poincaré inequality, there exists a constant C > 0 such that

Ωε |u ε | 2 ≤ C Ωε |∇u ε | 2 . ( 23
)
In this inequality, the constant C can be chosen independent of ε since the domains Ω \ B ε are uniformly bounded in the x 2 direction. Combining [START_REF] Reed | Methods of Modern Mathematical Physics, Volume I : Functional Analysis[END_REF] with the energy bound [START_REF] Holt | Fast mass transport through sub-2-nanometer carbon nanotubes[END_REF], we obtain the existence of a constant C > 0 such that

∀ε > 0 Ω\Bε |u ε | 2 + |∇u ε | 2 ≤ C . ( 24
)
Since the family of functions γ ε defined by ( 1) is uniformly Lipschitz, there exists a family of linear extension operators

E ε : H 1 (Ω ε ) → H 1 (Ω) such that for any ε > 0, ∀w ∈ H 1 (Ω ε ) E ε (w) |Ωε = w and a constant C > 0 such that ∀ε > 0 E ε L(H 1 (Ωε),H 1 (Ω)) ≤ C
(see for instance [START_REF] Chenais | On the existence of a solution in a domain identification problem[END_REF]). Throughout the paper, we will note ūε the function defined in H 1 (Ω) by

ūε = E ε (u ε ) . ( 25
)
Combining the previous bound with [START_REF] Secchi | Massive radius-dependent flow slippage in carbon nanotubes[END_REF], we get that the sequence

(ū ε ) ε is bounded in H 1 (Ω). Since H 1 (Ω) is compactly embedded in L 2 (Ω)
, there exists a function ū in H 1 (Ω) such that, up to extracting a subsequence, ūε converges to ū weakly in H 1 (Ω) and strongly in L 2 (Ω). Moreover, by continuity of the trace operator H 1 (Ω) → H 1/2 (∂Ω), the weak limit ū also belongs to V.

Weak convergence of

u ε to ū in L 2 (Ω)
To prove that u ε also converges to ū albeit only weakly in L 2 (Ω), we rely on a density argument. First, we establish the existence of a constant C > 0 such that

∀ε > 0 Ω u 2 ε ≤ C . (26)
Let ε > 0 be fixed. By density of regular functions in H 1 (Ω), it is not restrictive to assume that u ε is regular. Since u ε vanishes on {x 2 = 0}, integrating on vertical lines, there holds for every

(x 1 , x 2 ) ∈ B ε 1 2 u ε (x 1 , x 2 ) 2 = γε(x 1 ) 0 ∂ 2 u ε (x 1 , s) u ε (x 1 , s) ds + x 2 γε(x 1 ) ∂ 2 u ε (x 1 , s) u ε (x 1 , s) ds .
Integrating the previous relation on B ε gives

1 2 Bε u 2 ε = 1 0 1 γε(x 1 ) γε(x 1 ) 0 ∂ 2 u ε (x 1 , s) u ε (x 1 , s) ds dx 2 dx 1 + 1 0 1 γε(x 1 ) x 2 γε(x 1 ) ∂ 2 u ε (x 1 , s) u ε (x 1 , s) ds dx 2 dx 1 =: I 1 + I 2 .
The first integral can be simplified as

I 1 = 1 0 (1 -γ ε (x 1 )) γε(x 1 ) 0 ∂ 2 u ε (x 1 , s) u ε (x 1 , s) ds dx 1
so that by definition of γ ε , Cauchy-Schwarz inequality and the energy bound ( 16),

|I 1 | ≤ ε d ∞ Ωε |∂ 2 u ε | |u ε | ≤ ε C Ωε u 2 ε 1/2
.

By Poincaré Inequality, since γ ε (x 1 ) ≤ 1 for any x 1 ∈ (0, 1),

Ωε u 2 ε ≤ Ωε |∇u ε | 2 ≤ C . (27)
Hence,

|I 1 | ≤ ε C .
Interverting the integration order between x 2 and s variables in I 2 yields

I 2 = 1 0 1 γε(x 1 ) 1 s ∂ 2 u ε (x 1 , s) u ε (x 1 , s) dx 2 ds dx 1 = 1 0 1 γε(x 1 ) (1 -s) ∂ 2 u ε (x 1 , s) u ε (x 1 , s) ds dx 1 .
As a result, we get the following bound:

|I 2 | ≤ ε d ∞ Bε |∂ 2 u ε | |u ε | ≤ ε d ∞ ε α Bε |∇u ε | 2 1/2 1 ε α Bε u 2 ε 1/2 ≤ ε C 1 ε α Bε u 2 ε 1/2
, where we have once again used the energy bound on ∇u ε . Summing up the estimates on I 1 and I 2 , we deduce that

Bε u 2 ε ≤ C ε 1 + 1 ε α/2 Bε u 2 ε 1/2 . Setting I ε = Bε u 2 ε 1/2
, we see that I ε satisfies

I 2 ε -Cε 1-α 2 I ε -Cε ≤ 0 .
Hence, the nonnegative quantity I ε is bounded by the only positive root of the polynomial

X 2 -Cε 1-α 2 X -Cε, that is I ε ≤ 1 2 Cε 1-α 2 + √ C 2 ε 2-α + 4Cε .
This proves that Bε u 2 ε goes to zero as ε goes to zero, which combined with (27), yields [START_REF] Whitby | Fluid flow in carbon nanotubes and nanopipes[END_REF]. Let ψ ∈ L 2 (Ω) and fix δ > 0. Let C ∞ c (Ω) be the space of infinitely differentiable functions with compact support in Ω. By density, there exists

ψ ∈ C ∞ c (Ω) such that Ω (ψ -ψ) 2 ≤ δ 2 . Since ψ has a compact support in Ω, there exists ε 0 > 0 such that for any 0 < ε < ε 0 , ψ ≡ 0 in B ε , so that Ω u ε ψ = Ωε u ε ψ = Ωε ūε ψ = Ω ūε ψ .
By strong convergence of ūε to ū in L 2 (Ω), we can pass to the limit in the last integral and deduce that lim

ε→0 Ω u ε ψ = Ω ū ψ . ( 28
)
Now, consider the following decomposition:

Ω (u ε -ū) ψ = Ω u ε (ψ -ψ) + Ω (u ε -ū) ψ + Ω ū ( ψ -ψ) .
By Cauchy-Schwarz inequality and ( 26), there exists a constant C > 0 such that for any ε > 0,

Ω u ε (ψ -ψ) + Ω ū ( ψ -ψ) ≤ C δ.
Using (28), we can pass to the lim sup in the previous equality to obtain

lim sup ε→0 Ω (u ε -ū) ψ ≤ C δ ,
which concludes the proof since δ is arbitrary.

4 Identification of the limit ū when u ε is the solution of (14) or [START_REF] Joseph | Why are carbon nanotubes fast transporters of water?[END_REF] In this section we will prove Theorem 1 and Theorem 2. Based on the hypothesis of these theorems, and the results of Section 3 we already know that up to extraction, the family of solutions to ( 14) and ( 18) converge in the weak L 2 (Ω) topology to some limit ū. This section aims at characterizing this limit. We will prove that ū can be seen as the weak solution of an identified elliptic problem. This problem having a unique solution, a standard argument shows that the whole sequence (u ε ) ε also converges to ū. For this reason, we work in the sequel of this section as if the family (u ε ) ε converges and discard any further reference to any extraction of this sequence.

The Neumann case

In this paragraph, we will characterize the weak limit ū ∈ V N as the unique solution to the elliptic problem [START_REF] Lauga | Microfluidics: The No-Slip Boundary Condition[END_REF]. Let φ ∈ V N be a given test function. Using the function ūε defined by [START_REF] Ramya | Enhanced water permeability and tunable ion selectivity in subnanometer carbon nanotube porins[END_REF], the weak formulation (15) can be rewritten as

Ωε ∇ū ε • ∇φ + ε α Bε ∇u ε • ∇φ = Ω f φ
or equivalently, as

Ω ∇ū ε • ∇φ - Bε ∇ū ε • ∇φ + ε α Bε ∇u ε • ∇φ = Ω f φ .
Let us pass to the limit in every term of the left-hand side of the previous equality. By weak convergence of (ū ε ) ε toward ū in H 1 (Ω), there holds immediately

lim ε→0 Ω ∇ū ε • ∇φ = Ω ∇ū • ∇φ .
By Cauchy-Schwarz inequality,

Bε ∇ū ε • ∇φ ≤ Bε |∇ū ε | 2 1/2 Bε |∇φ| 2 1/2 ≤ C Bε |∇φ| 2
∇ū ε • ∇φ = 0 .
Lastly, using the energy bound [START_REF] Holt | Fast mass transport through sub-2-nanometer carbon nanotubes[END_REF] and Young inequality, we obtain the estimate

ε α Bε ∇u ε • ∇φ ≤ ε α 2 ε α/2 Bε |∇u ε | 2 + ε -α/2 Bε |∇φ| 2 ≤ ε α/2 2 ε α Bε |∇u ε | 2 + Ω |∇φ| 2 ≤ C ε α/2 , whence lim ε→0 ε α Bε ∇u ε • ∇φ = 0 .
Gathering the previous computations, we obtain that

Ω ∇ū • ∇φ = Ω f φ ,
which proves that ū is the variational solution to problem [START_REF] Lauga | Microfluidics: The No-Slip Boundary Condition[END_REF], since we have already established that ū ∈ V N .

The Dirichlet case

One major difference between the Neumann and the Dirichlet cases is that, in the Dirichlet case, the boundary condition u ε = 0 on x 2 = 1 may or may not pass to the limit as ε goes to zero, depending on the exponent α.

Case α < 1. We claim that if 0 < α < 1, the limit ū actually satisfies the same boundary condition ū = 0 on x 2 = 1 , in other words, ū ∈ V D . The distinction between the cases α < 1 and α ≥ 1 comes from the following estimate of the L 2 -norm of the trace of u ε on Γ ε (defined by ( 2)):

Γε u 2 ε ≤ C ε 1-α . ( 29
)
The previous inequality can be obtained using the energy bound, the boundary condition u ε (x 1 , 1) = 0 for a.e. x 1 ∈ (0, 1) and integrating on vertical lines {x 1 } × (1 -εd(x 1 ), 1), as follows:

Γε u 2 ε = 1 0 u ε (x 1 , 1 -εd(x 1 )) 2 1 + ε 2 d (x 1 ) 2 dx 1 ≤ 1 + ε 2 d ∞ 1 0 u ε (x 1 , 1 -εd(x 1 )) 2 dx 1 ≤ 1 + ε 2 d ∞ 1 0 1 1-εd(x 1 ) |∂ 2 u ε (x 1 , s)| ds 2 dx 1 ≤ 1 + ε 2 d ∞ ε d ∞ 1 0 1 1-εd(x 1 ) |∂ 2 u ε (x 1 , s)| 2 ds dx 1 ≤ C ε Bε |∇u ε | 2 ≤ C ε 1-α .
Applying a similar argument to the function ūε , which is not vanishing on x 2 = 1, we obtain the following trace inequality:

{x 2 =1} ū2 ε ≤ C ε Bε |∇ū ε | 2 + Γε ū2 ε .
Since ūε is bounded in H 1 (Ω), and ūε and u ε have the same trace on Γ ε , (29) implies that

{x 2 =1} ū2 ε ≤ C ε + ε 1-α . Hence, if 0 < α < 1, lim ε→0 {x 2 =1} ū2 ε = 0. Denote by T : H 1 (Ω) → H 1/2 ({x 2 = 1}
) the trace operator. Since T is linear and continuous and ūε converges weakly to ū in H 1 (Ω),

T ūε converges weakly to T ū in H 1/2 ({x 2 = 1}) (30)
(see, for instance, [5, Theorem III.9]). By compact embedding of

H 1/2 ({x 2 = 1}) into L 2 ({x 2 = 1}), this convergence is strong in L 2 ({x 2 = 1}). But lim ε→0 {x 2 =1} ū2 ε = 0 so T ūε converges strongly to zero in L 2 ({x 2 = 1}), whence T ū = 0. This proves that ū ∈ V D if 0 < α < 1.
It remains to prove that for any φ ∈ V D ,

Ω ∇ū • ∇φ = Ω f φ.
This can be achieved following the same steps as in Section 4.1, starting from the equality

Ωε ∇ū ε • ∇φ + ε α Bε ∇u ε • ∇φ = Ω f φ .
Case α ≥ 1. In this case we need to do a finer analysis of the behaviour of u ε in B ε . To this aim, we introduce a rescaled function v ε , depending on the micro-variable y = (y 1 , y 2 ) ∈ ω and defined by

v ε (y) = ε α+1 2 u ε (y 1 , 1 -εy 2 ) for a.e. y = (y 1 , y 2 ) ∈ ω . (31)
Notice that v ε vanishes on {y 2 = 0}. Setting x = (x 1 , x 2 ) and using the change of variables

y 1 = x 1 , y 2 = 1 -x 2 ε , (32)
there holds

ω |∂ 1 v ε (y)| 2 dy = ε α Bε |∂ 1 u ε (x)| 2 dx , ω |∂ 2 v ε (y)| 2 dy = ε α+2 Bε |∂ 2 u ε (x)| 2 dx .
Hence, the energy bound ( 16) yields the existence of a constant C > 0 such that for any ε > 0,

ω |∇v ε | 2 ≤ C, ω |∂ 2 v ε | 2 ≤ C ε 2 .
What is mainly of interest to us is that up to extraction there exists w in L 2 (ω) such that

1 ε ∂ 2 v ε w weakly in L 2 (ω) . (33)
Now we can end our proof of Theorem 2 by identifying the limit ū. Let φ ∈ V N and define φ ∈ H 1 (ω) by

φ(y 1 , y 2 ) = -1 - y 2 d(y 1 ) φ(y 1 , 1) for a.e. (y 1 , y 2 ) ∈ ω . (34)
Then for every ε, the function φ ε defined by

φ ε (x) = φ(x) + 1 Bε (x) φ x 1 , 1 -x 2 ε for a.e. x = (x 1 , x 2 ) ∈ Ω is in V D ,
so it is an admissible test function for problem [START_REF] Joseph | Why are carbon nanotubes fast transporters of water?[END_REF]. Testing against this function, we obtain

Ω f φ + Bε f (x) φ x 1 , 1 -x 2 ε dx = Ωε ∇u ε • ∇φ + ε α Bε ∇u ε • ∇φ + ε α Bε ∂ 1 u ε (x) ∂ 1 φ x 1 , 1 -x 2 ε dx - Bε 1 ε ∂ 2 u ε (x) ∂ 2 φ x 1 , 1 -x 2 ε dx .
Using the functions ūε introduced in Section 3 by ( 25) and v ε defined by (31), and the change of variables (32), this can be rewritten as

Ω f φ + Bε f (x) φ x 1 , 1 -x 2 ε dx = Ω ∇ū ε • ∇φ - Bε ∇ū ε • ∇φ + ε α Bε ∇u ε • ∇φ + ε α-1 2 Bε ∂ 1 v ε x 1 , 1 -x 2 ε ∂ 1 φ x 1 , 1 -x 2 ε dx + Bε 1 ε 2 ∂ 2 v ε x 1 , 1 -x 2 ε ∂ 2 φ x 1 , 1 -x 2 ε dx = Ω ∇ū ε • ∇φ - Bε ∇ū ε • ∇φ + ε α Bε ∇u ε • ∇φ + ε α-1 2 ε ω ∂ 1 v ε ∂ 1 φ + ω 1 ε ∂ 2 v ε ∂ 2 φ . ( 35
)
Let us examine the limit of each term of the previous equality as ε goes to zero.

• By Cauchy-Schwarz inequality and using once again the change of variables (32),

Bε f (x) φ x 1 , 1 -x 2 ε dx ≤ Bε f (x) 2 dx 1/2 Bε φ x 1 , 1 -x 2 ε 2 dx 1/2 ≤ Ω f (x) 2 dx 1/2 ε ω φ(y) 2 dy 1/2 ≤ ε 1/2 f L 2 (Ω) φ L 2 (ω) .
Hence,

1 ε α-1 2 Bε f (x) φ x 1 , 1 -x 2 ε dx ≤ ε 2-α 2 f L 2 (Ω) φ L 2 (ω) ,
and since 0 < α < 2, we deduce that lim ε→0

1 ε α-1 2 Bε f (x) φ(x 1 , 1-x 2 ε ) dx = 0.
• By the same arguments as in Subsection 4.1, we get

lim ε→0 Ω ∇ū ε • ∇φ = Ω ∇ū • ∇φ , lim ε→0 Bε ∇ū ε • ∇φ = 0 , lim ε→0 ε α Bε ∇u ε • ∇φ = 0. • Since ∂ 1 v ε is uniformly bounded in L 2 (ω)
, applying Cauchy-Schwarz inequality we immediately obtain that

lim ε→0 ε ω ∂ 1 v ε ∂ 1 φ = 0.
• Let w ∈ L 2 (ω) be the function satisfying (33), then by definition of the weak convergence,

lim ε→0 ω 1 ε ∂ 2 v ε ∂ 2 φ = ω w ∂ 2 φ .
In the case α > 1, it is enough to pass to the limit in (35) and obtain the relation

Ω ∇ū • ∇φ = Ω f φ .
This proves that ū is the solution to [START_REF] Lauga | Microfluidics: The No-Slip Boundary Condition[END_REF].

In the critical case α = 1, in order to establish that ū is the solution to [START_REF] Karniadakis | Simple Fluids in Nanochannels[END_REF], we need to rewrite the limit ω w ∂ 2 φ as a surface integral on {x 2 = 1}. To this aim, we define for a.e. y = (y 1 , y 2 ) ∈ ω, the quantity W (y 1 , y 2 ) by We claim that W (y 1 , d(y 1 )) = ū(y 1 , 1) for a.e. y 1 ∈ (0, 1) ,

W (y 1 , y 2 ) =
so that the previous inequality can be rephrased as

ω w ∂ 2 φ = 1 0 1 d(x 1 ) ū(x 1 , 1) φ(x 1 , 1) dx 1 .
Hence, passing to the limit in (35) in the case α = 1 yields

Ω ∇ū • ∇φ + 1 0 1 d(x 1 ) ū(x 1 , 1) φ(x 1 , 1) dx 1 = Ω f φ ,
which proves that ū is the weak solution to [START_REF] Karniadakis | Simple Fluids in Nanochannels[END_REF]. It remains to prove relation (37). Taking α = 1 and applying the definition of v ε (31) at y 2 = d(y 1 ), there holds

v ε (y 1 , d(y 1 )) ε = u ε (y 1 , 1 -ε d(y 1
)) for a.e. y 1 ∈ (0, 1) .

We will prove that, as sequences of functions of y 1 :

(i) v ε (y 1 , d(y 1 )) ε W (y 1 , d(y 1 )) weakly in L 2 (0, 1) ; (ii) u ε (y 1 , 1 -ε d(y 1 )) → ū(y 1 , 1) strongly in L 2 (0, 1).
Then by uniqueness of the limit, the identity (37) follows.

To prove (i), we notice that since v ε vanishes on y 2 = 0,

v ε (y 1 , d(y 1 )) ε = d(y 1 ) 0 ∂ 2 v ε (y 1 , y 2 )
ε dy 2 for a.e. y 1 ∈ (0, 1) .

For any ζ ∈ L 2 (0, 1), there holds

1 0 v ε (y 1 , d(y 1 )) ε ζ(y 1 ) dy 1 = 1 0 d(y 1 ) 0 ∂ 2 v ε (y 1 , y 2 ) ε dy 2 ζ(y 1 ) dy 1 = ω ∂ 2 v ε (y) ε ζ(y 1 ) dy .
Using property (33), we obtain w(y 1 , y 2 ) dy 2 for a.e. y 1 ∈ (0, 1), so the previous limit can be rewritten as

This proves (i).

To prove (ii), we take advantage of the fact that by construction, u ε and ūε have the same trace on y 2 = 1 -εd(y 1 ), and introduce for a.e. y 1 ∈ (0, 1) the decomposition:

u ε (y 1 , 1 -εd(y 1 )) -ū(y 1 , 1) = (ū ε (y 1 , 1 -εd(y 1 )) -ūε (y 1 , 1)) + (ū ε (y 1 , 1) -ū(y 1 , 1)) .
We estimate

1 0 |u ε (y 1 , 1 -εd(y 1 )) -ū(y 1 , 1)| 2 dy 1 by 1 0 |u ε (y 1 , 1 -εd(y 1 )) -ū(y 1 , 1)| 2 dy 1 ≤ 2 1 0 |ū ε (y 1 , 1 -εd(y 1 )) -ūε (y 1 , 1)| 2 dy 1 + 1 0 |ū(y 1 , 1) -ūε (y 1 , 1)| 2 dy 1 .
Integrating on vertical lines and using that ∇ū ε is uniformly bounded in L 2 (Ω) and Cauchy-Schwarz inequality, we obtain

1 0 |ū ε (y 1 , 1 -ε d(y 1 )) -ūε (y 1 , 1)| 2 dy 1 = 1 0 1 1-ε d(y 1 ) ∂ 2 ūε (y 1 , s) ds 2 dy 1 ≤ ε d ∞ 1 0 1 1-ε d(y 1 ) |∂ 2 ūε (y 1 , s)| 2 ds dy 1 ≤ ε d ∞ Ω |∇ū ε | 2 ≤ C ε .
As a consequence of (30), up to a subsequence, ūε (y 1 , 1) converges strongly in L 2 y 1 (0, 1) to ū(y 1 , 1), hence lim ε→0 1 0 |ū(y 1 , 1) -ūε (y 1 , 1)| 2 dy 1 = 0. Combining this property with the previous upper bound, we deduce that

lim ε→0 1 0 |u ε (y 1 , 1 -ε d(y 1 )) -ū(y 1 , 1)| 2 dy 1 = 0 , which proves (ii).

The flat interface

This section is devoted to the proof of Theorem 3. This approach being fairly standard we will concentrate on the points that matter in our specific case.

Proof of Theorem 3.1

For each k the function f k is in L 2 (0, 1). Then we can use Duhamel's Formula to solve the differential equations of (21) under the form:

c - k (x) = A - k ch(kπx) + B - k sh(kπx) - e kπx 2kπ
x 0 e -kπs f k (s)ds + e -kπx 2kπ

x 0 e kπs f k (s)ds

c + k (x) = A + k ch(kπx) + B + k sh(kπx) - e kπx 2kπ
x γε e -kπs f k (s) ε α ds + e -kπx 2kπ

x γε e kπs f k (s) ε α ds where A ± k , and B ± k are 4 degrees of freedom which can be set uniquely to enforce the four remaining equations of [START_REF] Myers | Why are why are slip lengths so large in carbon nanotubes? Microfluid[END_REF].

We will see subsequently more precise formulas for c ± k but on this version we can clearly see that c ± k are H 1 functions on their respective domains since the primitives are primitives of L 2 functions (see [START_REF] Brézis | Analyse fonctionnelle : théorie et applications[END_REF] for instance). But using the differential equations it is also clear that their second derivative (in the distribution sense) is also L 2 and thus c ± k are in fact H 2 in their respective domain and solution in the strong sense of the differential equations which concludes the proof of the first point of Theorem 3.

Proof of Theorem 3.3 and Theorem 3.4

We delay the proof of the second point of Theorem 3 as it is actually the main difficulty in the proof of the theorem.

From Lemma 1 we already know that

u - ε is in H 1 (Ω ε ) and u + ε is in H 1 (B ε ).
Taking a test function φ in H 1 (Ω ε ) and using again Lemma 1 to define the φ k associated, we obtain by Parseval formula that:

Ωε ∇u - ε • ∇φ = 1 2 k≥1 γε 0 c - k (x 2 )φ k (x 2 ) + k 2 π 2 c - k (x 2 )φ k (x 2 ) dx 2 = 1 2 k≥1 γε 0 f k (x 2 )φ k (x 2 )dx 2
by construction of c - k and thus reversing the Parseval formula yields:

Ωε ∇u - ε • ∇φ = Ωε f φ
This means that we have -∆u - ε = f at least in the distributional sense in Ω ε , and justifies that

u - ε is in {v ∈ H 1 (Ω ε ), ∆v ∈ L 2 (Ω ε )}. A similar argument shows that u + ε is in {v ∈ H 1 (B ε ), ∆v ∈ L 2 (B ε )}.
To prove the trace equalities, we start by using Lemma 1 and the estimation of Theorem 3.2 to confirm that u ± ε have zero lateral trace. Now for the property of traces on x 2 = 0, x 2 = γ ε and x 2 = 1, one can see that since for all k, c ± k are in H 2 of their respective domains (from the previous paragraph), then c ± k are in C 1,1/2 and thus, u

+ ε actually belongs to C 1 x 2 ([0, γ ε ], H 1 0 (0, 1)) and u + ε in C 1 x 2 ([γ ε , 1], H 1 0 (0, 1)).
Thus the remaining trace equalities can be checked directly on the Fourier coefficients and come from the construction of the c ± k . Now we can end the proof of Theorem 3.4. First, considering that u ± ε are in H 1 and their traces coincide on x 2 = γ ε the reasoning behind Theorem 1.7.1 p75 in [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF] applies and shows that u ε is at least H 1 (Ω) and

∇u ε = ∇u - ε in Ω ε ∇u + ε in B ε . Now let us take v in C ∞ c (Ω).
Since Ω ε and B ε are rectangles, they are rectilinear polygons. Then from [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF] Lemma 1.5.3.8 p. 61 and the remark following it, we can state that

Ωε ∇u - ε • ∇v = - Ωε ∆u - ε v + ∂Ωε ∂ ν u - ε v Bε ∇u + ε • ∇v = - Bε ∆u + ε v + ∂Bε ∂ ν u + ε v
where ∂ ν stand for the normal derivative in the sense of the trace. The boundary integral are in general in the sense of the duality H -1/2 , H 1/2 but here we have enough regularity and ∂ ν u ± ε are actually in L 2 of the boundary.

Given that v is 0 on ∂Ω the previous formulas can be recast as

Ωε ∇u - ε • ∇v = - Ωε ∆u - ε v + x 2 =γε ∂ x 2 u - ε v Bε ∇u + ε • ∇v = - Bε ∆u + ε v - x 2 =γε ∂ x 2 u + ε v
Consequently adding the first line to ε α times the second and using the properties of u ± ε yields:

Ωε ∇u - ε • ∇v + ε α Bε ∇u + ε • ∇v = Ωε (-∆u - ε )v + Bε (-ε α ∆u + ε )v + x 2 =γε (∂ x 2 u - ε -ε α ∂ x 2 u + ε )v = Ω f v
and thus we have

Ωε ∇u ε • ∇v + ε α Bε ∇u ε • ∇v = Ω f v .
In the Dirichlet case, by density of C ∞ c (Ω) in H 1 0 (Ω), we conclude that u ε is the weak solution of (18). In the Neumann case, the previous argument can be easily adapted to take into account the boundary x 2 = 1 and obtain that u ε is also the solution to [START_REF] Gravelle | Large permeabilities of hourglass nanopores: From hydrodynamics to single file transport[END_REF].

Proof of Theorem 3.2 and Theorem 3.5

Since our goal is to check [START_REF] Holt | Fast mass transport through sub-2-nanometer carbon nanotubes[END_REF] for u ε , we use its construction as a Fourier series to see that because of Parseval formula :

Ωε |∇u ε | 2 + ε α Bε |∇u ε | 2 = 1 2 k≥1 γε 0 (|(c - k ) (x 2 )| 2 + k 2 π 2 |c - k (x 2 )| 2 )dx 2 + ε α 2 k≥1 1 γε (|(c + k ) (x 2 )| 2 + k 2 π 2 |c + k (x 2 )| 2 )dx 2
Consequently, we will prove Theorem 3.2 with enough detail to show simultaneously ( 22) and ( 16) in this setting which will yield Theorem 3.

The main tool is to use Duhamel's formula to represent the solutions of (21) which will be slightly different between the Neumann case and the Dirichlet case. We will present in the sequel the rationale in the case of a Dirichlet boundary condition and most technical details will be found in the Appendix.

Firstly let us introduce K defined on

[0, 1] × [0, 1] × R * + by K(y, s, λ) = sh(λ(1-s))sh(λy) λsh(λ) if y ≤ s sh(λ(1-y))sh(λs) λsh(λ) if y ≥ s ( 38
)
This function is useful because of the following Lemma:

Lemma 2. Let λ > 0. If β is in L 2 (0, 1) the function v(y) = 1 0 K(y, s, λ)β(s)ds is in H 2 (0, 1) and a strong solution to -v + λ 2 v = β v(0) = v(1) = 0
Moreover, we have v (y) = 1 0 ∂ y K(y, s, λ)β(s)ds where ∂ y K is the almost everywhere derivative of K with respect to y. This lemma is proved using Duhamel's formula in the Appendix. Thus we can rescale c ± k to be defined on [0, 1] and using the previous lemma, solve the equations up to solutions of the homogeneous equation. Then setting the degrees of freedom with the various conditions at x = 0, x = γ ε and x = 1 yields the unique solution of ( 21) via the formulas:

c - k (x) = γε 0 γ ε K x γ ε , t γ ε , kπγ ε + sh(kπε)sh(kπt)sh(kπx) kπDε α sh(kπγ ε ) f k (t)dt + 1 γε sh(kπ(1 -t))sh(kπx) kπDε α f k (t)dt (39) c + k (x) = 1 γε ε 1-α K 1 -x ε , 1 -t ε , kπε + sh(kπγ ε )sh(kπ(1 -t))sh(kπ(1 -x)) kπDε α sh(kπε) f k (t)dt + γε 0 sh(kπt)sh(kπ(1 -x)) kπDε α f k (t)dt ( 40
)
where D is a shorthand for

D = ch(kπγ ε ) sh(kπε) ε α + sh(kπγ ε )ch(kπε) (41)
With these formulas we obtain via Cauchy-Schwarz and Young inequalities:

γε 0 |c - k (x)| 2 dx ≤ 2 γε 0 γε 0 γ ε K x γ ε , t γ ε , kπγ ε + sh(kπε)sh(kπt)sh(kπx) kπDε α sh(kπγ ε ) 2 dtdx γε 0 |f k (t)| 2 dt + 2 γε 0 1 γε sh(kπ(1 -t))sh(kπx) kπDε α 2 dtdx 1 γε |f k (t)| 2 dt (42) 1 γε |c + k (x)| 2 dx ≤ 2 1 γε 1 γε ε 1-α K 1 -x ε , 1 -t ε , kπε + sh(kπγ ε )sh(kπ(1 -t))sh(kπ(1 -x)) kπDε α sh(kπε) 2 dtdx 1 γε |f k (t)| 2 dt + 2 1 γε γε 0 sh(kπt)sh(kπ(1 -x)) kπDε α 2 dtdx γε 0 |f k (t)| 2 dt (43)
It is then a matter of estimating the double integrals in the previous inequality and sort out their dependence on ε and k. The details of the computations can again be found in appendix but we find that (using a generic numerical constant C independent of ε and k that can change from line to line):

1 γε γε 0 sh(kπt)sh(kπ(1 -x)) kπDε α 2 dtdx ≤ C εγ ε k 2 π 2 1 γε 1 γε ε 1-α K 1 -x ε , 1 -t ε , kπε + sh(kπγ ε )sh(kπ(1 -t))sh(kπ(1 -x)) kπDε α sh(kπε) 2 dtdx ≤ C ε 2-2α k 2 π 2 γε 0 γε 0 γ ε K x γ ε , t γ ε , kπγ ε + sh(kπε)sh(kπt)sh(kπx) kπDε α sh(kπγ ε ) 2 dtdx ≤ C γ ε 2 k 2 π 2
From these estimations and the fact that γ ε = O(1) we infer:

k 2 π 2 γε 0 |c - k (x)| 2 dx ≤ C(1 + ε) f k 2 L 2 (44)
ε α k 2 π 2 1 γε |c + k (x)| 2 dx ≤ C(ε 2-α + ε α+1 ) f k 2 L 2 (45)
Now we have to conduct a similar analysis for the first derivatives of c ± k . It follows the same pattern, we first differentiate equations ( 39) and ( 40):

(c - k ) (x) = γε 0 ∂ y K x γ ε , t γ ε , kπγ ε + sh(kπε)sh(kπt)ch(kπx) Dε α sh(kπγ ε ) f k (t)dt + 1 γε sh(kπ(1 -t))ch(kπx) Dε α f k (t)dt (46)
(c + k ) (x) = 1 γε -ε -α ∂ y K 1 -x ε , 1 -t ε , kπε - sh(kπγ ε )sh(kπ(1 -t))ch(kπ(1 -x)) Dε α sh(kπε) f k (t)dt - γε 0 sh(kπt)ch(kπ(1 -x)) Dε α f k (t)dt (47)
Then again using Cauchy-Schwarz and Young inequality we obtain:

γε 0 |(c - k ) (x)| 2 dx ≤ 2 γε 0 γε 0 ∂ y K x γ ε , t γ ε , kπγ ε + sh(kπε)sh(kπt)ch(kπx) Dε α sh(kπγ ε ) 2 dtdx γε 0 |f k (t)| 2 dt + 2 γε 0 1 γε sh(kπ(1 -t))ch(kπx) Dε α 2 dtdx 1 γε |f k (t)| 2 dt (48) 1 γε |(c + k ) (x)| 2 dx ≤ 2 1 γε 1 γε ε -α ∂ y K 1 -x ε , 1 -t ε , kπε + sh(kπγ ε )sh(kπ(1 -t))ch(kπ(1 -x)) Dε α sh(kπε) 2 dtdx 1 γε |f k (t)| 2 dt + 2 1 γε γε 0 sh(kπt)ch(kπ(1 -x)) Dε α 2 dtdx γε 0 |f k (t)| 2 dt (49)
and finally using the estimation from the appendix we obtain

γε 0 1 γε sh(kπ(1 -t))ch(kπx) Dε α 2 dtdx ≤ Cεγ ε γε 0 γε 0 ∂ y K x γ ε , t γ ε , kπγ ε + sh(kπε)sh(kπt)ch(kπx) Dε α sh(kπγ ε ) 2 dtdx ≤ Cγ ε 2 1 γε γε 0 sh(kπt)ch(kπ(1 -x)) Dε α 2 dtdx ≤ C γ ε ε α kπ 1 γε 1 γε ε -α ∂ y K 1 -x ε , 1 -t ε , kπε + sh(kπγ ε )sh(kπ(1 -t))ch(kπ(1 -x)) Dε α sh(kπε) 2 dtdx ≤ Cε 2-2α
Injecting the previous estimations in ( 48) and ( 49) and adding respectively with (44) and ( 45) yields

k 2 π 2 γε 0 |c - k (x)| 2 dx + γε 0 |(c - k ) (x)| 2 dx ≤ C(1 + ε) f k 2 L 2 (50)
ε α k 2 π 2 1 γε |c + k (x)| 2 dx + ε α 1 γε |(c + k ) (x)| 2 dx ≤ C(1 + ε 2-α + 1 kπ + ε 1+α ) f k 2 L 2 (51)
For fixed ε > 0, this yields Theorem 3.2 because the right-hand sides are summable since f is in L 2 (Ω).

Now we can end the proof of Theorem 3.5. From Theorem 3.4 we obtain that

Ωε |∇u ε | 2 + ε α Bε |∇u ε | 2 = Ωε |∇u + | 2 + ε α Bε |∇u -| 2
Then from Lemma 1 we can state that:

Ωε |∇u ε | 2 + ε α Bε |∇u ε | 2 = 1 2 k≥1 γε 0 k 2 π 2 |c - k | 2 + γε 0 |(c - k ) | 2 + ε α 2 k≥1 1 γε k 2 π 2 |c + k | 2 + 1 γε |(c + k ) | 2 ≤ C(1 + ε) f 2 L 2 (Ω) + C(1 + ε 2-α + ε 1+α ) f 2 L 2
Thus, as long as α ≤ 2 the solution to ( 18) satisfy ( 16) and Theorem 3 is proved.

Further comments

In this section, we examine some possible extensions of our results, in two directions:

1. considering more general boundary conditions on lateral boundaries and at the bottom;

2. imposing a smooth transition between the bulk viscosity and the reduced viscosity near the wall, rather than a jump of viscosity.

Adaptation to more general boundary conditions

For the sake of clarity, we have only considered homogeneous Dirichlet boundary conditions on lateral boundaries x 1 = 0, x 1 = 1 and at the bottom x 2 = 0, but our results apply to more general boundary conditions.

As regards lateral boundary conditions, our analysis can be directly adapted to more general conditions, such as nonhomogeneous Dirichlet conditions, Neumann conditions or periodic ones. As a matter of fact, the condition imposed on the lateral boundaries did not play a significant role in the asymptotic analysis of the model.

Concerning the condition on the lower boundary, the requirement that u ε vanishes on x 2 = 0 is useful from a technical point of view, since it allows us to deduce from the energy bound a uniform bound of u ε in H 1 (see Section 3.1), and gain compactness. More fundamentally, to ensure that limit problems of Robin [START_REF] Karniadakis | Simple Fluids in Nanochannels[END_REF] or Neumann types [START_REF] Lauga | Microfluidics: The No-Slip Boundary Condition[END_REF] are well-posed, one should typically impose a (not necessarily homogeneous) Dirichlet boundary condition u ε = g on x 2 = 0, or at least on a given subset of positive measure. Introducing a suitable lift of this boundary value g, one could reduce the problem to an homogeneous one and apply the same analysis.

Smooth drop of viscosity near the boundary

In the model described in Section 1.2, we have considered the case of a jump of viscosity along the interface Γ ε located at x 2 = 1 -ε d(x 1 ). Starting from Dirichlet boundary condition on x 2 = 1, in the critical case α = 1, we have obtained in the limit a friction coefficient 1/d(x 1 ) on x 2 = 1 (see system [START_REF] Karniadakis | Simple Fluids in Nanochannels[END_REF]). In the case where d(x 1 ) ≡ d is constant, this corresponds to a slip length equal to d (the bulk viscosity being normalised to 1 in Ω ε ).

In this paragraph, we investigate the possible effect of a smooth drop of viscosity near the boundary x 2 = 1, on the value of the slip length obtained in the limit system. To illustrate this effect, we consider the simplified one-dimensional setting of Subsection 1.3.2, introducing

• a sequence (δ ε ) ε>0 such that 0 < δ ε < ε for any ε > 0,

• a continuous, nonnegative function τ : [0, 1] → [0, +∞[ such that τ (0) = 0 and τ (1) = 1, and replacing definition (6) of µ ε by

µ ε (x) =        1 if 0 ≤ x ≤ 1 -ε (ε -α -1) τ x-(1-ε) δε + 1 -1 if 1 -ε ≤ x ≤ 1 -ε + δ ε ε α if 1 -ε + δ ε ≤ x ≤ 1 . ( 52
)
With this new definition of µ ε , the viscosity transitions smoothly between the bulk value 1 and the boundary value ε α ; the transition region is of length δ ε and the function τ gives the viscosity profile in the transition area. The case of a jump of viscosity considered in the rest of the paper corresponds to the limit case δ ε = 0.

The solution u ε of problem [START_REF] Bucur | Boundary behavior of viscous fluids: influence of wall roughness and friction-driven boundary conditions[END_REF] where µ ε is given by (52), can be represented by formula [START_REF] Bucur | On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries[END_REF], where the kernel K ε is defined by [START_REF] Casado-Díaz | Why viscous fluids adhere to rugose walls: A mathematical explanation[END_REF]. In order to pass to the limit in u ε , it is enough to derive the expression of K ε (x, s) for 0 ≤ x ≤ s ≤ 1 -ε, since this expression converges to the kernel of the limit u as ε goes to zero. We perform the following computations for 0 ≤ x ≤ s ≤ 1 -ε :

x 0 µ -1 ε = x 0 1 = x 1 s µ -1 ε = 1-ε s 1 + 1-ε+δε 1-ε (ε -α -1)ν y -(1 -ε) δ ε + 1 dy + 1 1-ε+δε ε -α = 1 -ε -s + δ ε (ε -α -1) 1 0 ν(y)dy + δ ε + ε -α (ε -δ ε ) = 1 -ε -s + ε 1-α + δ ε (ε -α -1) 1 0 ν(y)dy -1 = 1 -ε -s + ε 1-α + δ ε ε (ε 1-α -ε) 1 0 ν(y)dy -1
and in particular,

1 0 µ -1 ε = 1 -ε + ε 1-α + δ ε ε (ε 1-α -ε) 1 0 ν(y)dy -1 .
If 0 < α < 1, we see that K ε (x, s) → x(1 -s) so that u ε converges to the solution u 0 of problem [START_REF] Casado-Díaz | Asymptotic behavior of a viscous fluid with slip boundary conditions on a slightly rough wall[END_REF], as it was the case when the viscosity was given by [START_REF] Bucur | On the asymptotic limit of flows past a ribbed boundary[END_REF].

If α = 1, up to a extracting a subsequence, we can assume the existence of λ ∈ [0, 1] such that lim ε→0 δ ε /ε = λ, and define the parameter θ ∈]0, 1] by

θ = 1 2 + λ 1 0 τ (y)dy -1 .
In that case, K ε (x, s) → K θ (x, s) := x(1 -θs), and the function u θ (x) = The factor θ/(1 -θ) in the previous boundary condition can be interpreted as a friction coefficient, which means that the slip length is multiplied by a factor (1 -θ)/θ, taking into account the transition between the bulk viscosity and the boundary viscosity through the limit λ and the profile τ . Notice that in the case λ = 0, we retrieve the limit problem [START_REF] Chenais | On the existence of a solution in a domain identification problem[END_REF] since θ = 1/2. Finally, if α > 1, K ε (x, s) → x so that u ε converges again to the solution of problem [START_REF] Evans | Partial differential equations[END_REF], with Neumann boundary conditions on x = 1.

This study of a simplified 1d model suggests that working with a smooth drop of viscosity near the boundary may result in a large panel of slip lengths. This could be an interesting lead to explore in order to interpret certain large values of slip length that are reported in the experimental fluid mechanics literature.

Note that we thus have v (y) = 1 0 ∂ y K(y, s, λ)β(s)ds where ∂ y K is the almost everywhere derivative of K with respect to y meaning that we are allowed to derive once under the integral sign the formula for v. Moreover we have v (1) = -1 0 sh(λs) sh(λ) β(s)ds. From this formula we check that v is in H 1 (]0, 1[) and thus v is in H A.2 Proof of (39) and (40)

Let us start from expressions (39) and (40) which we recall here for convenience:

c - k (x) = γε 0 γ ε K x γ ε , t γ ε , kπγ ε + sh(kπε)sh(kπt)sh(kπx) kπDε α sh(kπγ ε ) f k (t)dt + 1 γε sh(kπ(1 -t))sh(kπx) kπDε α f k (t)dt c + k (x) = 1 γε ε 1-α K 1 -x ε , 1 -t ε , kπε + sh(kπγ ε )sh(kπ(1 -t))sh(kπ(1 -x)) kπDε α sh(kπε) f k (t)dt + γε 0 sh(kπt)sh(kπ(1 -x)) kπDε α f k (t)dt • Since K(0, •, •) = 0 we easily check that c - k (0) = c + k (1) = 0. • Since K(1, •, •) = 0 and 1 -γ ε = ε we see that c - k (γ ε ) = c + k (γ ε ). • We can rewrite c - k = p - k + h - k and c + k = p + k + h + k where p - k (x) = γε 0 γ ε K x γ ε , t γ ε , kπγ ε f k (t)dt h - k (x) = γε 0 sh(kπε)sh(kπt) kπDε α sh(kπγ ε ) f k (t)dt + 1 γε sh(kπ(1 -t)) kπDε α f k (t)dt sh(kπx) p + k (x) = 1 γε ε 1-α K 1 -x ε , 1 -t ε , kπε f k (t)dt h + k (x) = 1 γε sh(kπγ ε )sh(kπ(1 -t)) kπDε α sh(kπε) f k (t)dt + γε 0 sh(kπt) kπDε α f k (t)dt sh(kπ(1 -x))
• It is obvious from the expressions that -(

h ± k ) + k 2 π 2 h ± k = 0. Now we write p - k (γ ε y) = γ ε γε 0 K y, t γ ε , kπγ ε f k (t)dt = 1 0 K(y, s, kπγ ε )γ ε 2 f k (γ ε s)ds p + k (1 -εy) = ε 1-α 1 γε K y, 1 -t ε , kπε f k (t)dt = 1 0 K(y, s, kπε)ε 2-α f k (1 -εs)ds
Thus from Lemma 2 we obtain that

- d 2 dy 2 (p - k (γ ε y)) + (kπγ ε ) 2 p - k (γ ε y) = γ ε 2 f k (γ ε y) - d 2 dy 2 (p + k (1 -εy)) + (kπε) 2 p - k (1 -εy) = ε 2-α f k (1 -εy) which is equivalent to -γ ε 2 (p - k ) (γ ε y) + γ ε 2 (kπ) 2 p - k (γ ε y) = γ ε 2 f k (γ ε y) -ε 2 (p + k ) (1 -εy) + ε 2 (kπ) 2 p - k (1 -εy) = ε 2-α f k (1 -
εy) Simplifying and setting x = γ ε y in the first equation and x = 1 -εy in the second yields

-(p - k ) + (kπ) 2 p - k = f k on (0, γ ε ) , -(p + k ) + (kπ) 2 p + k = 1 ε α f k on (γ ε , 1) . Combining with the equation satisfied by h ± k , we obtain -(c - k ) + k 2 π 2 c - k = f k on (0, γ ε ) , -(c + k ) + k 2 π 2 c + k = f k ε α on (γ ε , 1) .
• Finally derivating (recall that the proof of Lemma 2 showed that we are allowed to differentiate p ± k under the integral sign) and setting x = γ ε yields

(p - k ) (γ ε ) = γε 0 ∂ y K 1, t γ ε , kπγ ε f k (t)dt = - γε 0 sh kπγ ε t γε sh(kπγ ε ) f k (t)dt (h - k ) (γ ε ) = γε 0 sh(kπε)sh(kπt) Dε α sh(kπγ ε ) f k (t)dt + 1 γε sh(kπ(1 -t)) Dε α f k (t)dt ch(kπγ ε ) (p + k ) (γ ε ) = - 1 ε α 1 γε ∂ y K 1, 1 -t ε , kπε f k (t)dt = 1 ε α 1 γε sh kπε 1-t ε sh(kπε) f k (t)dt (h + k ) (γ ε ) = - 1 γε sh(kπγ ε )sh(kπ(1 -t)) Dε α sh(kπε) f k (t)dt + γε 0 sh(kπt) Dε α f k (t)dt ch(kπε)
We have using ( 41)

sh(kπε)ch(kπγ ε ) Dε α -1 = sh(kπε)ch(kπγ ε ) -ch(kπγ ε )sh(kπε) -ε α sh(kπγ ε )ch(kπε) Dε α = - sh(kπγ ε )ch(kπε) D 1 - sh(kπγ ε )ch(kπε) D = ch(kπγ ε )sh(kπε) ε α D .
This yields

(c - k ) (γ ε ) = - γε 0 ch(kπε)sh(kπt) D f k (t)dt + 1 γε sh(kπ(1 -t))ch(kπγ ε ) Dε α f k (t)dt (c + k ) (γ ε ) = - γε 0 sh(kπt)ch(kπε) Dε α f k (t)dt + 1 γε ch(kπγ ε )sh(kπ(1 -t)) D(ε α ) 2 f k (t)dt Thus we do have (c - k ) (γ ε ) = ε α (c + k ) (γ ε
). This ends the proof that the formulas (39) and (40) define the solutions to [START_REF] Myers | Why are why are slip lengths so large in carbon nanotubes? Microfluid[END_REF] with the Dirichlet boundary condition.

(kπ(1 -x))sh(kπt) kπDε α 2 dxdt = 1 (kπ) 2 D 2 ε 2α 1 γε sh(kπ(1 -x)) 2 dx γε 0 sh(kπt) 2 dt = 1 (kπ) 2 D 2 ε 2α sh(2kπε) 4kπ - ε 2 sh(2kπγ ε ) 4kπ - γ ε 2
Now we use that sh(2θ) = 2sh(θ)ch(θ) so we can write:

γε 0 1 γε sh(kπ(1 -t))sh(kπx) kπD 2 dxdt = εγ ε 4(kπ) 2 sh(kπε) 2 ch(kπγ ε ) 2 D 2 ε 2α × coth(kπε) kπε - 1 sh(kπε) 2 th(kπγ ε ) kπγ ε - 1 ch(kπγ ε ) 2
It is routine to check that the functions θ → coth(θ)/θ -1/sh(θ) 2 and θ → th(θ)/θ -1/ch(θ) 2 are bounded functions over R. Moreover from (41) we see that

sh(kπε) 2 ch(kπγ ε ) 2 D 2 ε 2α ≤ 1
Thus as announced, there is a numerical constant C > 0 (independent of k and ε) such that:

γε 0 1 γε sh(kπ(1 -t))sh(kπx) kπDε α 2 dtdx ≤ C εγ ε (kπ) 2 .

Now we turn to

γε 0 γε 0 γ ε K x γ ε , t γ ε , kπγ ε + sh(kπε)sh(kπt)sh(kπx) kπDε α sh(kπγ ε ) 2 dtdx 1 γε 1 γε ε 1-α K 1 -x ε , 1 -t ε , kπε + sh(kπγ ε )sh(kπ(1 -t))sh(kπ(1 -x)) kπDε α sh(kπε) 2 dtdx
We set x = γ ε y and t = γ ε s in the first integral and x = 1 -εy and t = 1 -εs in the second one which yields: in the case (λ, M ) = (kπγ ε , sh(kπε)/ε α ) and (λ, M ) = (kπε, sh(kπγ ε )) As we can check, λK(y, s, λ) + M sh(λs)sh(λy)

γε 0 γε 0 γ ε K x γ ε , t γ ε , kπγ ε + sh(kπε)sh(kπt)sh(kπx) kπDε α sh(kπγ ε ) 2 dtdx = γ ε 2 (kπ) 2 1 0 1 0 kπγ ε K(y, s, kπγ ε ) + sh(kπε)sh(kπγ ε s)sh(kπγ ε y) Dε α sh(kπγ ε ) 2 dsdy 1 γε 1 γε ε 1-α K 1 -x ε , 1 -t ε , kπε + sh(kπγ ε )sh(kπ(1 -t))sh(kπ(1 -x)) kπDε α sh(kπε) 2 dtdx = ε 2 (kπ) 2 ε 2α
Dsh(λ)
is invariant through exchange of y and s thus we have: Expanding the square and performing the integrations yield now:

1 0 1 0 λK(y, s, λ) + M sh(λs)sh(λy) Dsh(λ) 2 dsdy = M ch(λ) D 2 1 2λ - 1 sh(2λ) 2 + 1 4sh(λ) 2 2 - th(λ) λ - sh(λ) 2 λ 2 1 - M D ch(λ) + 1 2 1 th(λ) - 1 λ 1 2λ - 1 sh(2λ)
First we remark that since sh(λ) ≥ λ for non negative λ by convexity, we have

2 - th(λ) λ - sh(λ) 2 λ 2 ≤ 1 - th(λ) λ .
Then we claim that it is again routine to check that

λ → 1 2λ - 1 sh(2λ) 2 λ → 1 sh(λ) 2 1 - th(λ) λ λ → 1 th(λ) - 1 λ 1 2λ - 1 sh(2λ)
are bounded functions over R.

Finally, when (λ, M ) = (kπγ ε , sh(kπε)/ε α ), we note that One can check that θ → th(θ) θ + 1 ch(θ) 2 and θ → coth(θ) θ -1 sh(θ) 2 are bounded functions over R. Moreover, as sh(kπε)ch(kπγ ε ) ≤ ε α D we obtain a numerical constant such that Now we have (Dε α ) 2 = (sh(kπε)ch(kπγ ε )) 2 + 2ε α sh(kπε)ch(kπε)sh(kπγ ε )ch(kπγ ε ) + (ε α sh(kπγ ε )ch(kπε)) 2 = (sh(kπε)ch(kπγ ε )) 2 + 1 2 ε α sh(2kπε)sh(2kπγ ε ) + (ε α sh(kπγ ε )ch(kπε)) 2 ≥ 1 2 ε α sh(2kπε)sh(2kπγ ε )

M D ch(λ) = sh(kπε) ε α ch(kπγ ε ) ch(kπγ ε ) sh(kπε)
Moreover, the functions θ → 1 θ -1 sh(θ) and θ → 1 + θ sh(θ) are bounded functions over R so that we have a numerical constant C such that

y 2 0 1 , 1 )

 211 w(y 1 , s) ds .(36)For a.e. y 1 ∈ (0, 1), the function W (y 1 , •) is in H 1 (0, 1) as a function of y 2 and vanishes at y 2 = 0. Coming back to the definition of φ (34), we get ω W (y 1 , d(y 1 )) dy 1 .

lim ε→0 1 0

 1 v ε (y 1 , d(y 1 )) ε ζ(y 1 ) dy 1 = ω w(y) ζ(y 1 ) dy . By definition (36), W (y 1 , d(y 1 )) = d(y 1 ) 0

1 0W

 1 1 , y 2 ) dy 2 ζ(y 1 ) dy 1 = (y 1 , d(y 1 )) ζ(y 1 ) dy 1 .

1 0 0 s( 1 - 1 xx( 1 - 1 x( 1 - 1 0 1 0

 101111111 K θ (x, s)f (s)ds satisfies u θ (x) = x θx)f (s)ds + θs)f (s)ds , u θ (x) = -θ x 0 sf (s)ds + θs)f (s)ds , u θ (x) = -f (x) ,hence u θ (1) = (1 -θ) sf (s)ds and u θ (1) = -θ sf (s)ds. Thus, u θ satisfies Robin boundary condition u θ (1) + θ 1 -θ u θ (1) = 0 .

1 y

 1 2 (]0, 1[) as announced. Derivating a second time yieldsv (y) = -λ sh(λy) λsh(λ) β(y)ch(λ(1 -y)) + λ 2 y 0 sh(λs) λsh(λ) β(s)ds sh(λ(1 -y)) -λ sh(λ(1 -y)) λsh(λ) β(y)ch(λy) + λ 2 sh(λ(1 -s)) λsh(λ) β(s)ds sh(λy) = -sh(λ(1 -y))ch(λy) + sh(λy)ch(λ(1 -y)) sh(λ) β(y) + λ 2 v(y) = -β(y) + λ 2 v(y)using hyberbolic trigonometry.

A. 3

 3 Proof of the estimates needed for the L 2 norm of c ± k As stated in Section 5.3 we now have to compute various estimates to obtain controls over the L 2 norms of c ± k and (c ± k )

1 0 1 0 2 dsdy

 112 kπεK(y, s, kπε) + sh(kπγ ε )sh(kπεs)sh(kπεy) Dsh(kπε)We thus see that evaluating these integrals amounts to computing

1 ] 2 dtdx ≤ C γ ε 2 ( 2 dtdx 4 ( 1 = 4 (

 1222414 ε α + ch(kπε)sh(kπγ ε ) ∈ [0, 1]and when (λ, M ) = (kπε, sh(kπγ ε )) we haveM D ch(λ) = sh(kπγ ε )ch(kπε) ch(kπγ ε ) sh(kπε) ε α + ch(kπε)sh(kπγ ε ) ∈ [0,Thus we obtain, as claimed in Section 5.3, a numerical constant C such that: kπγ ε + sh(kπε)sh(kπt)sh(kπx) kπDε α sh(kπγ ε ) sh(kπγ ε )sh(kπ(1 -t))sh(kπ(1 -x)) kπDε α sh(kπε)2 dtdx ≤ C ε 2 (kπ) 2 ε 2αA.4 Proof of the estimates needed for the L 2 norm of (c ± k )Again we start with the estimation of Changing variables t = 1 -t and x = 1 -x in the first integral shows that Dε α ) 2 sh(2kπ(1 -a)) 2kπ(1 -a) εγ ε sh(kπε) 2 ch(kπγ ε ) 2 Dε α ) 2 × th(kπγ ε )

2 dtdx ≤ C εγ ε 4 .

 24 (1 -t))ch(kπx) Dε αOn the other hand, setting a = γ ε yields

∂∂∂ 2 dsdy∂

 2 Finally we turn to the last to integrals to estimate: ε + sh(kπε)sh(kπt)ch(kπx)Dε α sh(kπγ ε ) ε )sh(kπ(1 -t))ch(kπ(1 -x)) Dε α sh(kπε)2Again, setting x = γ ε y and t = γ ε s in the first integral and x = 1 -εy and t = 1 -εs in the second one yields ε + sh(kπε)sh(kπt)ch(kπx)Dε α sh(kπγ ε ) y K(y, s, kπγ ε ) + sh(kπε) ε α sh(kπγ ε s)ch(kπγ ε y)) Dsh(kπγ ε ) ε )sh(kπ(1 -t))ch(kπ(1 -x))Dε α sh(kπε) y K(y, s, kπε) + sh(kπγ ε ) sh(kπεs)ch(kπεy) Dsh(kπε) y K(y, s, λ) + M sh(λs)ch(λy) Dsh(λ)with again (λ, M ) = (kπγ ε , sh(kπε)/ε α ) and (λ, M ) = (kπε, sh(kπγ ε )). Here a difficulty comes from the fact that we don't have symmetry with respect to the exchange of y and s.Using (38) we can state∂ y K(y, s, λ) = sh(λ(1-s))ch(λy) sh(λ) if y ≤ s ch(λ(1-y))sh(λs)Let us call u the integration variable in both integrals. Expanding the squares, regrouping similar terms and using hyperbolic trigonometry yield: y K (y, s, λ) + M sh(λs)ch(λy) Dsh(λ)

2 dtdx ≤ Cγ ε 2 1 γε 1 γεε 2 ≤

 2112 As in the previous section, for (λ, M ) = (kπγ ε , sh(kπε)/ε α ) and (λ, M ) = (kπε, sh(kπγ ε )) we have M ch(λ)/D ∈]0, 1[ and moreover we know that λ →1 (2λ) 2 -1 sh(2λ) 2 , λ → coth(λ) λ -1 sh(λ) 2 and λ → 1 sh(λ) 2 2 -sh(λ) 2 λ 2 -th(λ)λ are bounded from above functions. This allows us to conclude as claimed that there is a numerical constant such thatγε 0 γε 0 ∂ y K x γ ε , t γ ε , kπγ ε + sh(kπε)sh(kπt)ch(kπx) Dε α sh(kπγ ε ) -α ∂ y K 1 -x ε ,1-t ε , kπε + sh(kπγ ε )sh(kπ(1 -t))ch(kπ(1 -x)) Dε α sh(kπε) Cε 2-2α

A Details of the computations in the Dirichlet case

A.1 Proof of Lemma 2

We can start with the proof of Lemma 2. Using the definition of K, the function v(y) = 1 0 K(y, s, λ)β(s)ds can be written as

From [START_REF] Brézis | Analyse fonctionnelle : théorie et applications[END_REF] we know that a function is in

With this knowledge, it is clear from the previous formula that thusly defined, v is in H 1 (]0, 1[). Moreover, setting in the formula y = 0 and 1 yields v(0) = v(1) = 0 so it is actually in H 1 0 (]0, 1[). We now derive v to obtain: