Diego Javier Zea

Hugues Richard
email: richardh@rki.de

Elodie Laine
email: elodie.laine@sorbonne-universite.fr

Sequence analysis ASES: Visualising evolutionary conservation of alternative splicing in proteins

ASES is a versatile tool for assessing the impact of alternative splicing, initiation and termination of transcription on protein diversity in evolution. It identifies exon and transcript orthogroups from a set of input genes/species for comparative transcriptomics analyses. It computes an evolutionary splicing graph, where the nodes are exon orthogroups, allowing for a direct evaluation of alternative splicing conservation. It also reconstructs a transcripts' phylogenetic forest to date the appearance of specific transcripts and explore the events that have shaped them. ASES web server features a highly interactive interface enabling the synchronous selection of events, exons or transcripts in the different outputs, and the visualisation and retrieval of the corresponding amino acid sequences, for subsequent 3D structure prediction.

Introduction

Alternative splicing, alternative promoter usage or alternative polyadenylation have the potential to greatly expand proteomes by generating multiple mRNA transcripts from a single gene. These transcripts may lead to protein isoforms with different shapes and functions. Alternative splicing (AS) has been linked to morphological diversity [START_REF] Tovar-Corona | Alternative splice in alternative lice[END_REF], organ development [START_REF] Mazin | Alternative splicing during mammalian organ development[END_REF], disease susceptibility [START_REF] Park | The expanding landscape of alternative splicing variation in human populations[END_REF], immune adaptation [START_REF] Rotival | Defining the genetic and evolutionary architecture of alternative splicing in response to infection[END_REF] and interactome rewiring [START_REF] Yang | Widespread expansion of protein interaction capabilities by alternative splicing[END_REF], among others. Hence, there is a growing interest in assessing its functional impact at the protein level, and in determining the evolutionary origin of alternative protein isoforms. To this aim, we have recently developed a couple of efficient scalable computational methods. The first one, ThorAxe (Zea et al., 2021), adds an evolutionary dimension to the analysis of transcript variability by extending the notion of splicing graph to an ensemble of genes/species. It establishes a mapping between orthologous regions in a set of transcripts coming from different genes/species, and uses this mapping to decompose the transcripts into building blocks. The second one, Phylosofs [START_REF] Ait-Hamlat | Transcripts' evolutionary history and structural dynamics give mechanistic insights into the functional diversity of the jnk family[END_REF], starts from ThorAxe's description of transcripts and takes a step further by inferring plausible scenarios explaining transcripts' evolution. Both tools adopt an end-product perspective where the transcripts are represented by their translated amino acid sequences. This choice allows reasoning directly over the impact of AS-induced variations on the protein sequences and structures. Here, we present Alternative Splicing Evolution Server (ASES), a versatile tool to explore the contributions of AS to protein diversity in evolution. Starting from a set of transcripts annotated in Ensembl (Zerbino et al., 2018), it relies on ThorAxe and PhyloSofS to compute an evolutionary splicing graph (ESG) and a transcripts' phylogenetic forest (TPF). ASES allows configurable queries enabling users to interactively inspect results at all scales (residue, exon, AS-event, transcript). It provides a rich environment for identifying the mechanisms underlying protein functional diversification through AS.

Definitions and implementation

The ESG is a compact exon-centred representation of the transcript variability observed in a set of genes/species (Figure 1A). The nodes represent s-exons, i.e. minimal transcript building blocks defined across species. The edges represent consecutive co-occurrences of s-exons in the input transcripts. Hence, a transcript is a path in the ESG. We estimate the evolutionary conservation of a s-exon (node) or a s-exon junction (edge) by quantifying the amount of species and transcripts including it, respectively. The heuristic for computing the ESG unfolds in five main steps (Zea et al., 2021): (1) cluster similar exons together, (2) define sub-exons within each species, (3) create a multiple sequence alignment (MSA) for each cluster, (4) identify s-exons as continuous blocks in the MSA, (5) refine s-exons. The TPF is a transcript-centred representation of the evolution of AS (Figure 1B). Each tree gives a plausible scenario for the emergence of a new transcript in evolution (the root) and how it led to a set of transcripts observed in present-day genes/species (the leaves). The whole forest is embedded in the gene tree retrieved from Ensembl. We rely on Sankoff's algorithm to determine the most parsimonious TPF (Ait-Hamlat et al., 2020). We implemented ASES web server using Genie, a Julia web framework [START_REF] Bezanson | Julia: A fresh approach to numerical computing[END_REF], with JavaScript and JQuery for front-end interactivity. We used Cytoscape.js [START_REF] Franz | Cytoscape. js: a graph theory library for visualisation and analysis[END_REF], neXtProt viewer and BioJS MSAViewer [START_REF] Yachdav | Msaviewer: interactive javascript visualization of multiple sequence alignments[END_REF] to visualise the ESGs, the protein sequences and the MSAs, respectively. D3 JavaScript library [START_REF] Bostock | D 3 data-driven documents[END_REF] is used for the forest and the gene structure.

Usage

ASES accepts as input the name or the Ensembl (Zerbino et al., 2018) stable identifier of a gene, the species where it comes from, and a list of species for comparative analysis -human (Hs), gorilla (Gg), macaque (Mam), mouse (Mum), rat (Rn), boar (Ss), cow (Bt), opossum (Md), platypus (Oa), frog (Xt), zebrafish (Dr) and nematode (Ce) by default. Once the job is submitted, the computation is done in three steps. First, ThorAxe dynamically retrieves the annotations associated to the query gene (genomic exons, protein coding transcripts, gene tree...etc) and its orthologs in the considered species from Ensembl. By default, we consider only the one-to-one orthologs defined in Ensembl, and we exclude the species containing multiple orthologs from the analysis. Optionally, users can extend the analysis to multiple orthology relationships. Then, ThorAxe computes the ESG and prepares the input for Phylosofs, which in turn infers the TPF. This last step typically takes more time and can be turned off.

ASES generates interactive representations of the ESG and the TPF, an interactive map of the gene structure with explicit genomic coordinates, and two tables (Figure 1). The colouring is intended to ease and guide users navigation. For instance, in the ESG, the nodes and edges get darker with their conservation degree (Figure 1A). Hence, although the ESG may seem complex at first sight, many AS events (bubbles in the graph) are typically not conserved (in yellow). See Supplementary Table S1 for statistics at the human protein-coding genome scale. In MAPK9, we readily identify a conserved mutually exclusive event involving the s-exons 10_0 and 10_1. ASES dynamically links the ESG and the TPF, thereby facilitating the inspection of specific s-exons, events or transcripts.

Here, the path highlighted in the ESG (Figure 1A) corresponds to the human green transcript selected in the TPF (Figure 1B). It includes the s-exon 10_0 and originates from an ancestral transcript (green root) present in the ancestor common to eutheria (Gg, Hs, Bt, Ss, Mum, Rn, Mam) and metatheria (Md). Among the five transcripts predicted for the eutheria ancestor, three were lost in human (purple, blue and red triangles). The users can visualise the s-exon losses and gains along the phylogeny upon clicking on the branches. ASES also displays the protein isoform sequences, and the user may simply copy-paste them in a domainannotation or 3D-structure prediction web server. Going back to the ESG (Figure 1A), selecting one s-exon, 10_0 here, makes the associated MSA appear (Figure 1C). We can see that 10_0 has a 25 AA consensus and is almost perfectly conserved across 9 species. Users can also select sexons in the genomic map (Figure 1D). Here, we see that the s-exon 10_0 (see arrow) is absent from zebrafish. This observation is reflected by the TPF, where the red transcripts include 10_1 instead of 10_0. Finally, the tables give detailed information about each of the s-exons and events, e.g. genomic coordinates, conservation levels, list of species, length and sequences. All data can be downloaded for a local analysis. ASES can be used in complement to other ressources, for instance to investigate the evolutionary origin and per-s-exon conservation of transcripts annotated in APPRIS (Rodriguez et al., 2013) (see Supplementary Figure S1).

Fig. 1 .

 1 Fig. 1. ASES example outputs for the human gene MAPK9 and its orthologs in 9 species. A. ESG, with colors indicating conservation levels (node: proportion of species, edges: average of transcripts). The grey rectangles highlight the path corresponding to the green transcript selected in panel B. B. TPF, where the nodes (transcripts) with the same colour belong to the same tree. The grey leaves are orphan transcripts (no phylogeny). The set of leaves observed in a present-day gene/species are grouped in a grey rectangular box. The boxes are transparent for ancestral genes/species.The triangles indicate transcript losses. C. Consensus sequence and MSA of the s-exon 10_0 selected in panel A. The human sequence is on the first row. D. Close-up view of the gene structure, where each line corresponds to a gene/species and each color indicates a s-exon. The blue box pointed by the arrow is the human exonic region belonging to the s-exon 10_0 selected in panel A.

Funding

A grant of the French national research agency (MASSIV project, ANR-17-CE12-0009) provided a salary to D.J.Z.


Hicham Jabraoui

Alain Esteve

Mirko Schoenitz

Edward L Dreizin

Carole Rossi
email: rossi@laas.fr

Atomic scale insights into the first reaction stages prior to Al/CuO nanothermite ignition: influence of porosity

Keywords: nanothermites, nanoenergetics, heterogeneous kinetic, reactive interface, nanocomposite, molecular dynamic, ReaxFF

Introduction

Nanothermites are mixtures of nanosized reactants which are stable under moderate conditions and capable of interacting with each other with the release of large amounts of chemical energy after activation by a triggering stimulus (thermal, mechanical, or electrical).

Much research in nanothermites have focused on Al based systems such as Al/Fe 2 O 3 , Al/MoO 3 , Al/CuO, and Al/Bi 2 O 3 due to their high energy density [START_REF] Umbrajkar | Exothermic Reactions in Al-CuO Nanocomposites[END_REF][START_REF] Fischer | A Survey of Combustible Metals, Thermites, and Intermetallics for Pyrotechnic Applications[END_REF] , and combustion performance tunability [START_REF] Rossi | Engineering of Al/CuO Reactive Multilayer Thin Films for Tunable Initiation and Actuation[END_REF][START_REF] Xu | Tuning the Reactivity and Energy Release Rate of I2O5 Based Ternary Thermite Systems[END_REF] . Al-based nanothermites have found applications for joining or welding 5,6, , tunable generators of biocidal-agents [START_REF] Liu | Preparation, Ignition, and Combustion of Magnesium-Calcium Iodate Reactive Nano-Composite Powders[END_REF][START_REF] Nakpan | Inactivation of Aerosolized Surrogates of Bacillus Anthracis Spores by Combustion Products of Aluminum-and Magnesium-Based Reactive Materials: Effect of Exposure Time[END_REF] , actuation in initiators/detonators [START_REF] Glavier | Nanothermite/RDX-Based Miniature Device for Impact Ignition of High Explosives[END_REF][START_REF] Fu | Metal-Interlayer-Metal Structured Initiator Containing Al/CuO Reactive Multilayer Films That Exhibits Improved Ignition Properties[END_REF][START_REF] Salvagnac | Layered Al/CuO Thin Films for Tunable Ignition and Actuations[END_REF] and in single-use miniaturized microelectromechanical systems [START_REF] Rodríguez | A Microactuator Based on the Decomposition of an Energetic Material for Disposable Lab-on-Chip Applications: Fabrication and Test[END_REF][START_REF] Nicollet | Fast Circuit Breaker Based on Integration of Al/CuO Nanothermites[END_REF][START_REF] Apperson | Characterization of Nanothermite Material for Solid-Fuel Microthruster Applications[END_REF][START_REF] Zhu | Preparation, and Performance of a Planar Ignitor Inserted With PyroMEMS Safe and Arm Device[END_REF] . A key parameter to high reactivity in nanothermites is intimate interfacial contact between fuel (Al) and oxidizer. That is why various preparation methods have been employed to combine aluminum fuel and oxidizer together in close proximity, including sputter deposition [START_REF] Rossi | Nano-Engineering of Al/CuO Multilayers: Bridging the Gap Between Research and Applications[END_REF][START_REF] Bahrami | Magnetron Sputtered Al-CuO Nanolaminates: Effect of Stoichiometry and Layers Thickness on Energy Release and Burning Rate[END_REF][START_REF] Wang | Probing the Reaction Zone of Nanolaminates at ∼μs Time and ∼μm Spatial Resolution[END_REF] , arrested milling methods [START_REF] Schoenitz | Fully Dense Nano-Composite Energetic Powders Prepared by Arrested Reactive Milling[END_REF] , electrospray [START_REF] Dai | From Nanoparticles to On-Chip 3D Nanothermite: Electrospray Deposition of Reactive Al/CuO@NC onto Semiconductor Bridge and Its Application for Rapid Ignition[END_REF] , sol-gel processing [START_REF] Wang | Fabrication and Characterization of Al-CuO Nanocomposites Prepared by Sol-Gel Method[END_REF] , and self-assembly [START_REF] Palussière | The Role of Alkylamine in the Stabilization of CuO Nanoparticles as a Determinant of the Al/CuO Redox Reaction[END_REF][START_REF] Calais | DNA Grafting and Arrangement on Oxide Surfaces for Self-Assembly of Al and CuO Nanoparticles[END_REF][START_REF] Shende | Nanoenergetic Composites of CuO Nanorods, Nanowires, and Al-Nanoparticles[END_REF][START_REF] Malchi | Electrostatically Self-Assembled Nanocomposite Reactive Microspheres[END_REF] . Interestingly, in Al/CuO fully-dense materials such as composites prepared by arrested reactive milling (ARM) [START_REF] Stamatis | A Multi-Step Reaction Model for Ignition of Fully-Dense Al-CuO Nanocomposite Powders[END_REF] and reactive multilayers [START_REF] Rossi | Engineering of Al/CuO Reactive Multilayer Thin Films for Tunable Initiation and Actuation[END_REF] , the exothermic redox reaction is restricted to condensed phase reactions at the CuO-Al interface, where an amorphous product Al 2 O 3 layer grows [START_REF] Lanthony | Formation of Al/CuO Bilayer Films: Basic Mechanisms through Density Functional Theory Calculations[END_REF][START_REF] Xiong | Interface Reaction Processes and Reactive Properties of Al/CuO Nanothermite: An Ab Initio Molecular Dynamics Simulation[END_REF][START_REF] Tang | Ab Initio Molecular Dynamics Study of Thermite Reaction at Al and CuO Nano-Interfaces at Different Temperatures[END_REF] and the ignition temperature is reduced compared to other types of thermites to ~850 K. [START_REF] Petrantoni | Multilayered Al/CuO Thermite Formation by Reactive Magnetron Sputtering: Nano versus Micro[END_REF][START_REF] Mily | The Role of Terminal Oxide Structure and Properties in Nanothermite Reactions[END_REF] Recent studies also showed that the ignition temperature can be further decreased by adding micro or nano voids within the interfacial layer, which was unexpected. Mursalat et al., [START_REF] Mursalat | Low-Temperature Exothermic Reactions in Al/CuO Nanothermites Producing Copper Nanodots and Accelerating Combustion[END_REF] observed that modifying the milling process parameters, i.e., adding a premilling step for Al powders using acetonitrile as a process control agent produces nano-voids in the thermite, resulting in lowering the ignition temperature down to ~650 K [START_REF] Mursalat | Low-Temperature Exothermic Reactions in Al/CuO Nanothermites Producing Copper Nanodots and Accelerating Combustion[END_REF][START_REF] Julien | Integration of Gold Nanoparticles to Modulate the Ignitability of Nanothermite Films[END_REF] and accelerating the reaction. Wu et al., [START_REF] Wu | Engineered Porosity-Induced Burn Rate Enhancement in Dense Al/CuO Nanothermites[END_REF] Addressing this issue theoretically using molecular dynamics (MD) is the most direct way to provide better understanding [START_REF] Jeon | Reactive Molecular Dynamics Study of Chloride Ion Interaction with Copper Oxide Surfaces in Aqueous Media[END_REF][START_REF]Interface Enabling Cold Sintering[END_REF][START_REF] Marsden | From Graphene to Graphene Oxide: The Importance of Extended Topological Defects[END_REF] . Only a few theoretical efforts [START_REF] Lanthony | Formation of Al/CuO Bilayer Films: Basic Mechanisms through Density Functional Theory Calculations[END_REF][START_REF] Xiong | Interface Reaction Processes and Reactive Properties of Al/CuO Nanothermite: An Ab Initio Molecular Dynamics Simulation[END_REF][START_REF] Tang | Ab Initio Molecular Dynamics Study of Thermite Reaction at Al and CuO Nano-Interfaces at Different Temperatures[END_REF][START_REF] Lanthony | Elementary Surface Chemistry during CuO/Al Nanolaminate-Thermite Synthesis: Copper and Oxygen Deposition on Aluminum (111) Surfaces[END_REF] attempted to describe the early stage of reactions of Al with CuO using density functional theory (DFT) or ab initio molecular dynamics. They describe the Al/CuO reactivity upon annealing but over very short duration (20 ps) and in small systems (100s atoms), which are not representative of experimental conditions. It was concluded that the increase in the rates of both Al oxidation and the migration of Cu atoms in aluminum oxide occur at high temperatures (2000 K and 3000 K). The theoretical studies reported so far have not been able to investigate reactions at low temperatures, i.e., reactions leading to ignition. Such low-temperature reactions may be of critical importance as altering the structure of the reactive interfaces, and further affect the No ternary phase forms when CuO is separated from the alumina layer by a pore. The diffusion of Al and O species through the growing alumina is activated at 800 K in the presence of the ternary aluminate phase as compared to 600 K, for the case with a pore at the interface. As a direct consequence of the early enrichment of the alumina layer with Al, molecules of oxygen trapped in the pore are adsorbed and chemically incorporated into the alumina layer at 600 K, explaining the small exotherm at 600 K found on differential scanning calorimetric (DSC) traces of Al/CuO dense composites containing nanopores.

Noteworthy the pore O 2 pressure drop may trigger the decomposition of the adjacent CuO surface [START_REF] Glorieux | Density of Superheated and Undercooled Liquid Alumina by a Contactless Method[END_REF] , thus supporting the presence of copper-rich nanodots (local reduction of CuO) observed in the partially reacted composites with voids at the interfaces [START_REF] Mursalat | Low-Temperature Exothermic Reactions in Al/CuO Nanothermites Producing Copper Nanodots and Accelerating Combustion[END_REF] . Not only does this study bring consistent understanding of the Al/CuO pre-ignition mechanisms in porous versus fully-dense composites, but it also explains the decrease in the ignition temperature and increased reactivity experimentally observed for layered or milled Al/CuO composites containing nanopores.

Computational details

Using LAMMPS packages [START_REF] Plimpton | Fast Parallel Algorithms for Short-Range Molecular Dynamics[END_REF] , MD simulations of atomic scale interactions at the CuO-Al interfaces were performed using periodic boundary conditions. The Nosé-Hoover thermostat and barostat served respectively to maintain the temperature in the canonical NVT ensemble, and the pressure in the isothermal-isobaric NPT ensemble. The ReaxFF force field parameters used in this study were developed by Psofogiannakis et al. [START_REF] Psofogiannakis | ReaxFF Reactive Molecular Dynamics Simulation of the Hydration of Cu-SSZ-13 Zeolite and the Formation of Cu Dimers[END_REF] The CuO material was built considering a cubic unit-cell with lattice parameters a = b = c = 4.23 Å and α = β = γ = 90° as described in ref [START_REF] Cao | DFT Study on Structural, Electronic, and Optical Properties of Cubic and Monoclinic CuO[END_REF] , and replicating it in the three spatial directions (x, y and z) forming a supercell containing 640 O atoms and 640 Cu atoms. First, the CuO atoms in the supercell were allowed to relax at 300 K in the NPT ensemble over 0.1 ns. Then, an additional 0.1 ns relaxation of both the atomic positions and volume was performed for the supercell using a ReaxFF MD simulation at 300 K in the NVT ensemble.

As a result, two 23.46 Å × 23.46 Å x,y surfaces cleaved normal to the [001] z direction formed, one O-rich surface that was chosen to be in contact with either the pore or the alumina layer, and one Cu-rich surface, which was frozen to mimic the CuO bulk material.

This slab underwent a final relaxation step at 300 K over 0.1 ns in the NVT ensemble. Note that the x,y supercell dimension that used for the CuO-Al interface with a pore is calibrated here on the basis of CuO rather than Al due to the CuO stability that necessitates finer tuning of cell dimensions.

To build the 2-3 nm thick native amorphous alumina coating any pure Al surface in contact with air 43 , namely amAl 2 O 3 , we used non-reactive force field MD simulations adapted to both Born-Meyer-Higgins's potential formulation and long-range Coulomb interactions. We used the Ewald summation with a cutoff of 12.0 Å for the real part and a relative error in forces < 10 -6 , while the short-range cutoff was set to 8.0 Å [START_REF] Bouhadja | Structural and Dynamic Properties of Calcium Aluminosilicate Melts: A Molecular Dynamics Study[END_REF]

Results and discussions

Effect of pore on evolution of the Al/CuO interface

The porous and fully-dense structures were heated up from 200 K to targeted temperatures, e.g., 400, 500, 600, 700, 800 and 900 K using a heating rate of 10 13 K•s -1 , followed by a plateau of 5 ns. This duration was chosen because in all cases, the system reached a stationary state after ~3 ns, as shown in supporting information file Overall, these first observations strongly suggest that the thin ternary aluminate layer, involving both Al and Cu slows down diffusion-driven reaction in the alumina layer for the fully-dense system. Lack of such ternary layer in porous composites may explain their experimentally observed enhanced reactivity compared to fully-dense composites. Next, the diffusion-driven reactions in the growing alumina upon annealing are quantified for both systems to better understand the impact of the thin ternary aluminate layer on the system reactivity.

Growth of an Al-rich Al x O y in place of the initial amAl 2 O 3 : porous vs dense.

One important finding of this theoretical study is the interfacial region enrichment with Al atoms from the Al region leading to the formation of an Al-rich layer in contact with either the pore or CuO surface. The number of Al atoms migrating from the Al region into the growing alumina layer is plotted in Figure 5a as a function of the annealing temperature for both dense and porous configurations. We note that the Al enrichment is systematically more pronounced in the porous system. After 5 ns at 400 K, the amount of additional Al represents 41% of the total Al species in Al x O y for the dense system while it reaches 45% for the porous system. At 900 K, the amount of additional Al in the Al x O y exceeds 50%: it represents 56 and 64% of the total Al species for the dense and porous system, respectively. Concomitantly to the Al enrichment, the interfacial region increases in thickness by growing into the Al region (Figure 5b). The atomic composition of the alumina region is quantified at different annealing temperatures and plotted in Figure 5c for the two systems: starting at 1.5 in the native (un-annealed) amAl 2 O 3 , the O/Al ratio drops down to ~0.9 after annealing at 400 K.

Then, the O/Al atomic ratio decreases nearly linearly with the annealing temperature. For 800 K, the ratios become 0.58 and 0.72 for the porous and dense system, respectively. The observed trend confirms that Al enrichment is more pronounced in the porous system within the temperature range considered. It has to be noted that the decrease in the O/Al ratio is also characterized by a decrease in the average coordination number of Al (Figure 5d). in order to discriminate, which of the interface between those grown in porous or in dense system is the most reactive.

Al coordination in the growing Al x O y region: porous vs dense.

Aluminum 5-fold coordination (Al V ) coexisting with 4-and 6-fold coordination (Al IV , Al VI ) structurally characterizes the amorphous alumina [START_REF] Landron | Liquid Alumina: Detailed Atomic Coordination Determined from Neutron Diffraction Data Using Empirical Potential Structure Refinement[END_REF][START_REF] Glorieux | Density of Superheated and Undercooled Liquid Alumina by a Contactless Method[END_REF] . At 300 K, our amAl 2 O 3 is characterized by an Al-O distance of 1.78 Å and contains 4% of Al III , 77 % of Al IV and 19%

of Al V (supporting information Figure S4). Figure 6 gives the spatial distribution of the Al ions (Al I , Al II , Al III , Al IV and Al V ) in the Al x O y grown at different annealing temperatures. In both systems, upon annealing, we observe a decrease in the number of Al V , which disappears completely after annealing at 800 K. This is consistent with the continuous enrichment of Al x O y with Al as discussed in the previous section. f correspond to dense system annealed at 400, 500, 600, 700, 800, and, 900 K, respectively. Images g, h, i, j, k, l correspond to porous system annealed at 400, 500, 600, 700, 800, and, 900 K respectively. The dotted lines shown in the image corresponding to the fully dense system at 800 K and 900 K indicate the initial growth of alumina crystals from the Al x O y region on CuO.

As a main observation, after annealing at 600 K, the spatial arrangement of the different Al CuO spontaneously decomposes into Cu 2 O in the temperature range of 600 -1000 K. [START_REF] Zhu | Oxidation Mechanism of Cu2O to CuO at 600-1050°C[END_REF][START_REF] Honjo | Electron Diffraction Studies on Oxide Films Formed on Metals and Alloys Part 1. Oxidation of Pure Copper[END_REF] We can, therefore, conclude that CuO surface around the pore decomposes into Cu 2 O and releases oxygen at temperatures greater than 600 K, which again corroborates the experimental observations of both gaseous oxygen at the interfaces [START_REF] Abdallah | Structure and Chemical Characterization at the Atomic Level of Reactions in Al/CuO Multilayers[END_REF] and Cu nanodots at the particle interfaces in porous systems recovered after heating to 650 K. These results suggest that for the porous system, kinetics is limited by the oxidation reaction (initially spontaneous and exothermic) whereas for the dense system, it is clearly limited by diffusion.

Finally, the diffusion coefficients of both Al and O atoms through the differently grown Al x O y layers are calculated and compared with those of amorphous alumina (native glassy state) serving as the reference material. For that purpose, the mean square displacements (MSD) for Al and O in different aluminas were calculated and averaged to establish statistical significance [START_REF] Jabraoui | Leaching and Reactivity at the Sodium Aluminosilicate Glass-Water Interface: Insights from a ReaxFF Molecular Dynamics Study[END_REF] . After obtaining the average MSD for Al and O, the diffusion coefficients ( ) are determined as the slopes of the respective MSD curves [START_REF] Jabraoui | Leaching and Reactivity at the Sodium Aluminosilicate Glass-Water Interface: Insights from a ReaxFF Molecular Dynamics Study[END_REF] . Figure 8 gives the results as logarithm of the diffusion coefficients. As expected, the Log diffusion coefficients of Al and O in the amorphous alumina (light blue data in Figure 8a-b respectively. [START_REF] Gall | Self-Diffusion in α-Al2O3 I. Aluminium Diffusion in Single Crystals[END_REF][START_REF] Prot | Self-Diffusion in Α-2 3[END_REF] The diffusion coefficients of Al and O in the Al x O y layers remain of the same order of magnitude as their diffusion coefficients in amorphous Al 2 O 3 (see Table 1). However, the temperature effect on diffusion coefficients in the Al x O y layers depends on the presence of pores:

-in the dense system, both and increase suddenly above 500

Conclusions

This theoretical work investigated the influence of pores at the CuO-Al interfaces on the preignition reaction processes using ReaxFF molecular dynamics simulations. The chemical and structural evolution of the reacting interface between the Al and CuO layers upon annealing up to 900 K were studied at the atomic scale, considering two interfacial structures: with and without a pore (1 nm spacing between Al and CuO surfaces, containing 13 O 2 molecules).

One major finding of this theoretical study is that in both, porous and dense systems there is a molecules from the pore onto the alumina surface at temperatures of 600 K and above. This leads to massive depletion of oxygen available near the alumina surface (recognizable as a predominance of Al II ions at the pore surface compared to predominance of Al IV ions at the Al-CuO interface in dense system) rendering the interfacial Al x O y layer highly reactive at 600 K and above. In addition, the O 2 pressure drop in the pore is expected to accelerate reduction of adjacent CuO surface. These findings can thus explain the presence of copper nanodots and decrease in the ignition temperature observed experimentally for layered or milled Al/CuO composites containing nanopores.

Associated content Supporting Information

The Supporting Information is available free of charge on the ACS Publications website at

Authors information

Corresponding Author

Carole Rossi -LAAS-CNRS, University of Toulouse, 31077 Toulouse, France; orcid.org/0000-0003-3864-7574; E-mail: rossi@laas.fr Authors

  also showed that incorporation of micro-sized pores into Al/CuO reactive multilayers decreases the ignition threshold around the pores and provokes the acceleration of their burn rate (×18% for 20% void loading). The authors concluded that voids or pores in the CuO-Al interface may modify the low-temperature chemical reaction sequence: instead of being driven by purely condensed phase diffusion over the CuO-Al interface, when pores are incorporated, gaseous O 2 is released by CuO in contact with the pores at very low temperatures. This oxygen can diffuse into solid Al and Al 2 O 3 , leading to low ignition temperature (650 -670 K). Two possible ignition pathways in Al/CuO nanolaminates have been also observed by time-of-flight mass spectrometer in 35 : in nanolaminates with 2 to 6 bilayers, gaseous O 2 is released prior to the ignition (spark) leading to a low ignition temperature (670 K) and enhanced overall reactivity compared to samples with 6 and more bilayers where initiation is controlled by condensed phase reactions (no gaseous O 2 signal is seen prior to the ignition). Those experimental findings point to the complexity of mechanisms governing the Al/CuO reactivity and raise an important question about the role the porosity at the CuO-Al interfaces might be playing to affect both the ignition and combustion in these composites. Creating controlled nanopores at the CuO-Al interfaces and probing the chemical evolution of the pores' surroundings at the nanoscale over a very short time to study the initial reaction stages experimentally is not yet possible.

  reactivity and combustion in fully-dense Al/CuO composites.This study aims to understand how the structural features of the interface may affect the ignition characteristics. To this end, the evolution of the characteristics of the interface between Al (and associated Al 2 O 3 surface layer) and CuO in terms of chemical composition, structure, and growth rate are investigated theoretically using reactive force field (ReaxFF) MD simulations and varying two parameters: temperature from 400 -900 K, and the interface structure: interfaces with or without pore (1 nm spacing between Al 2 O 3 and CuO surfaces containing free oxygen molecules). Upon heating, simulations show the growth of an Al-rich alumina region in both porous and dense systems in place of the initial amorphous alumina barrier layer. Major differences between the porous and dense systems emerge when the temperature exceeds 600 K. The direct contact between the Al 2 O 3 and CuO leads to the formation of a ternary aluminate phase made of a mixture of Al, O and Cu atoms, which impedes the diffusion-driven reaction between Al and O, thus reducing the material reactivity.

3 and

 3 . Fitting the x,y CuO slab dimension, a random structure of 1500 atoms (600 Al and 900 O) was first equilibrated at 4000 K for 1 ns to reach its liquid equilibrium state in the NVT ensemble. It was then quenched to 300 K at 10 K•ps -1 to reach the vitreous state in the NPT ensemble using an integration time step of 0.1 fs. The results are illustrated in Figure1a. More details on this methodology are documented in refs[START_REF] Jabraoui | Atomic Insights into the Events Governing the Borosilicate Glass-Water Interface[END_REF][START_REF] Jabraoui | Thermodynamic and Structural Properties of Binary Calcium Silicate Glasses: Insights from Molecular Dynamics[END_REF][START_REF] Jabraoui | Elastic and Structural Properties of Low Silica Calcium Aluminosilicate Glasses from Molecular Dynamics Simulations[END_REF][START_REF] Jabraoui | Molecular Dynamics Simulation of Thermodynamic and Structural Properties of Silicate Glass: Effect of the Alkali Oxide Modifiers[END_REF][START_REF] Jabraoui | Effect of Sodium Oxide Modifier on Structural and Elastic Properties of Silicate Glass[END_REF] . Two amAl 2 O 3 surfaces were generated by adding two vacuum slabs along z in either part of the amAl 2 O 3 slab (Figure1b). Finally, this amAl 2 O 3 surface was annealed again at 300 K for 1 ns in the NVT ensemble with the same ReaxFF potential parameters[START_REF] Psofogiannakis | ReaxFF Reactive Molecular Dynamics Simulation of the Hydration of Cu-SSZ-13 Zeolite and the Formation of Cu Dimers[END_REF] . Then, to build the Al-amAl 2 O 3 interface, a pure Al slab containing 1008 atoms (in a face-centered cubic crystal structure) and cleaved normal to the [001] direction was connected to the as prepared amAl 2 O 3 slab. Further chemical stabilization of this interface was realized through annealing in the NVT ensemble at 200 K over 1 ns. Then, from these two elementary structures, Al/amAl 2 O 3 and CuO, we derived two model systems: the dense and the porous system. The dense system was built by placing the Al/amAl 2 O 3 slab in direct contact to the CuO slab whereas, in the porous system, Al/amAl 2 O CuO slabs were separated by a vacuum space, the nanopore being 23.46 Å × 23.46 Å × 10 Å in dimension. We randomly introduced 13 O 2 molecules in the pore, equivalent to the saturated vapor pressure (5080 kPa). The final size of the supercell is of 23.46 Å × 23.46 Å × 110 Å and 23.46 Å × 23.46 Å × 120 Å, for dense and porous systems, respectively (Figure 1). As a final stage both systems underwent a relaxation step at 200 K over 1 ns to chemically stabilize all surfaces, including the amAl 2 O 3 /CuO interface. All annealing simulations were performed in the NVT ensemble and the time step is 1 fs. The top and bottom surface layers in the supercell were kept frozen during the entire simulation.

Figure 1 .

 1 Figure 1. Schematic 2D views representing (a) the different steps of the model-system

Figure S1 .

 S1 A 10 13 K.s -1 heating rate is imposed as an atomic motion damping procedure calibrated so that to avoid undesirable thermal shock at the very first stage of the simulations, leading to biased bond breaking. The structural and chemical composition evolution of the initial amAl 2 O 3 region upon annealing are characterized for both systems.

Figure 2

 2 Figure2gives side view snapshots of the annealed dense and porous systems at 500 K (b,f),

Figure 2 .

 2 Figure 2. Two-dimensional views of initial (un-annealed) and final (post-annealed)

Figure 3a -

 3a Figure 3a-b and Figure 3c-d respectively give the depth of penetration of aluminum atoms

Figure 3 .

 3 Figure 3. Penetration depth (ΔZ) into the native alumina region of Al atoms initially

Figure 4 .

 4 Figure 4. Aluminate interface characteristics: (a) Cu-Al radial distribution function (RDF)

Figure 5 .

 5 Figure 5. (a) Evolution of the amount of Al ions added in the Al x O y region in dense and

Figure 6 .

 6 Figure 6. Snapshot after annealing process over 5 ns accompanied by the distribution of

  ions (Al I , Al II , Al III and Al IV ) in the Al x O y regions evolves differently for the two systems. In the dense system, the Al x O y interfacial layer includes three distinct zones: i) near the Al-Al x O y interface, there is a region composed mainly of weaklycoordinated Al I and Al II , characterizing a highly reactive zone, ii) a central region dominated by Al III , iii) close to the Al x O y -CuO interface, there is a region showing an ordered arrangement of Al IV ions (polyhedral), the highest coordinated Al ions, characterizing a zone with reduced reactivity. By contrast, in porous system, the different Al ions are more uniformly dispersed in the entire Al x O y layer with a predominance of Al I and Al II at both Al-Al x O y and Al x O y -pore surfaces (Figure 6d-e). These two weak Al coordination regions are more responsive to oxygen compared to the central zone dominated by the Al III ions. The predominance of Al II ions at the Al x O y -pore surface compared to predominance of Al IV ions at the Al x O y -CuO interface might explain the low-temperature reactivity of Al/CuO thermite with nanopores. Indeed, the Al x O y -pore is almost totally depleted from its oxygen atoms with only Al III , Al II and Al I ions present (Figure 6a-c). Accordingly, at 600 K, reoxidation is observed (loss of Al II and Al I ), due to chemisorption of the oxygens initially present in the form of O 2 molecules in the pore. This can explain the small exotherm at 600 K found on DSC traces of Al/CuO dense composites containing nanopores. 32 Importantly, the adsorption and reaction of the 13 O 2 molecules into the alumina layer at 700 K (12 O at 600 K and the remaining 14 O at 700 K) provokes a drop in the oxygen partial pressure in the pore which further prompts the CuO reduction into Cu 2 O. The temperature-pressure region of formation of CuO, Cu 2 O (supporting information file Figure S6) shows that below 10 mbar,

3. 4 .Figure S7 .

 4S7 Figure S7. At the end, the amount of oxygen consumed by Al/Al x O y of either the dense or the

Figure 7 .

 7 Figure 7. (a) amount of oxygen from gaseous O 2 consumed by the Al x O y -Al system over 3

Figure 8 .

 8 Figure 8. Log diffusion coefficients as a function of reciprocal temperature for Al (a) and O

growing Al 2 O 3 -

 23 y interface layer continuously enriched with Al ions from the Al fuel reservoir, replacing the initial amorphous alumina barrier layer. However, this evolution towards a more reactive interfacial layer occurs differently for the dense and porous composites at temperatures above 600 K. First, the direct contact between amAl 2 O 3 and CuO leads to the formation of a thin ternary copper aluminate phase (comprising ~10 -20 % of the whole Al x O y layer at 800 K) made of a mixture of Al, O and Cu atoms. This ternary phase impedes further growth of the Al x O y layer and thus reduces the nanothermite reactivity, evaluated in this study in terms of the diffusion coefficients for both O and Al. Instead, the presence of a pore between the amAl 2 O 3 and CuO activates the dissociative adsorption of the gaseous O 2



  DOI: Further analysis of the ReaxFF dynamic simulations: The temporal evolution of the quantity characterizes the nanothermite Al/CuO interface for dense and porous systems such as the total number of Al in the Al x O y interfacial region, the average coordination number (ACN) of Al with respect to oxygen and the O/Al ratio. Different oxidation states of Al in the Al x O y interfacial region for dense and porous systems as a function of temperature.  Initial structure for the study of Al x O y -Al surface reactivity resulting from the annealing of dense and porous systems at 700 K. The temperature-pressure region of formation of CuO, Cu 2 O, Cu

Table 1 .

 1 This confirms that the presence of pores activates the diffusion of O through the Al x O y layer at a lower temperature (~600 K against 800 K w/o pore). It has to be noted that Al and O diffuse at comparable rates in the porous systems, whereas Al diffuses faster than O in the dense system. Diffusion of oxygen and aluminum in amorphous material of pristine alumina and interfacial alumina in porous and dense systems.

	K;	increases from -17.7 to -16.6 and	increases from -17.8
	to -16.9.
	-in the porous system, both	and	increase suddenly above
	500 K;	increases from -17.5 to -17.0 and	increases from -
	17.8 to -17.0.

Acknowledgements

The authors grateful acknowledge support from the European Research Council (H2020 Excellent Science) Researcher Award (grant 832889 -PyroSafe). This work was performed using high power computing resources from CALMIP.

Funding sources C.R. received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grand agreement No. 832889pyrosafe).

Author contributions

All authors contributed equally in this work.

Competing financial interests

The authors declare no competing financial interests.


Ph.D Abdessatar Khelifi
email: ab-dessatar.khelifi@fsb.rnu.tn

Christian Daveau
email: christian.daveau@u-cergy.fr

Imen Balloumi
email: balloumiimen@hotmail.fr

Asymptotic behaviors for eigenvalues and eigenfunctions associated to Stokes operator in the presence of small boundary perturbations

Keywords: Mathematical Physics, Analysis and Geometry Stokes eigenvalue problem, small perturbation, asymptotic expansion Stokes eigenvalue problem, small perturbation, asymptotic expansion 2010 AMS subject classifications. 47A75, 35P05, 35Q30

Asymptotic behaviors for eigenvalues and eigenfunctions associated to Stokes operator in the presence of small boundary perturbations

Abdessatar Khelifi, Imen Balloumi, Christian Daveau

Introduction

The field of eigenvalue problems under shape perturbation has been an active research area for several decades. Several related problems belong to Stokes systems, which are further subdivided by assumptions on the underlying media and on the Dirichlet boundary conditions. The main objective of this paper is to present a schematic way to derive high-order asymptotic expansions for both eigenvalues and eigenfunctions for the Stokes operator caused by small perturbations of the boundary. Also, we rigorously derive an asymptotic formula which is in some sense dual to the leading-order term in the asymptotic expansion of the perturbations in the Stokes eigenvalues due to interface changes of the inclusion. This is inspired from the fact that the Stokes system can be viewed as the incompressible limit [START_REF] Ammari | Effective viscosity properties of dilute suspensions of arbitrarily shaped particles[END_REF]. Thus we rigourously extended, in Section 3.3, some results elaborated in [START_REF] Ammari | Reconstruction of small interface changes of an inclusion from modal measurements II: The elastic case[END_REF] to the Stokes case. The properties of eigenvalue problems under shape deformations have been a subject of comprehensive studies [START_REF] Albert | Genericity of simple eigenvalues for elliptic pde's[END_REF][START_REF] Albert | Topology of the nodal and critical points sets for eigenfunctions of elliptic operators[END_REF][START_REF] Ammari | Splitting of resonant and scattering frequencies under shape deformation[END_REF][START_REF] Babuska | Eigenvalue Problems[END_REF][START_REF] Kato | Perturbation Theory for Linear Operators[END_REF][START_REF] Ortega | Generic simplicity of the eigenvalues of the stokes system in two soace dimensions[END_REF] and the area continues to carry great importance [START_REF] Ammari | Reconstruction of small interface changes of an inclusion from modal measurements II: The elastic case[END_REF][START_REF] Ammari | Layer Potential Techniques in Spectral Analysis[END_REF][START_REF] Kelliher | Eigenvalues of the Stokes operator versus the Dirichlet Laplacian in the plane[END_REF][START_REF] Kohr | The interior Neumann problem for the Stokes resolvent system in a bounded domain in R n[END_REF][START_REF] Medkova | Boundary value problems for the Stokes equations with jumps in open sets[END_REF][START_REF] Jia | Shaoqin Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods[END_REF][START_REF] Khelifi | Asymptotic property and convergence estimation for the eigenelements of the Laplace operator[END_REF]. A substantial portion of these investigations discusses the properties of smoothness and analyticity of eigenvalues and eigenfunctions with respect to perturbations.

Let Ω ⊂ R 3 be a bounded open domain with boundary of class C 2 . We consider the following eigenvalue problem for the Stokes system with homogeneous boundary conditions:

     -µ∆v + ∇p = λv in Ω ∇.v = 0 in Ω v = 0 in ∂Ω.
Here v = (v 1 , v 2 , v 3 ) denotes the velocity field, µ the viscosity while the scalar function p is the pressure. Now, if we suppose that µ is a positive constant, then the first equation in the above system is equivalent to:

-∆v + ∇(p/µ) = ( λ/µ)v in Ω.

Then, if µ is constant we have the following eigenvalue problem

     -∆v + ∇p = λv in Ω ∇.v = 0 in Ω v = 0 in ∂Ω (1.1)
with p := p/µ and λ := λ/µ. It is well known that this eigenvalue problem admits a sequence of a no decreasing positive eigenvalues 0 < λ 1 ≤ λ 2 ≤ • • • ≤ λ n ≤ • • • tending to infinity as n → +∞. The eigenfunctions {v n } n≥1 ⊂ (H 1 0 (Ω)) 3 and the eigenpressures {p n } n≥1 ⊂ L 2 (Ω) may be taken so that {v n } n≥1 constitutes an orthonormal basis of H(Ω) := {v ∈ (H 1 0 (Ω)) 3 : ∇ • v = 0 in Ω}. The pressure p is determined up to an additive constant. We assume that the boundary ∂Ω is subjected to a small, smooth deformation and that the boundary of the deformed domain Ω δ is given by:

∂Ω δ := {x = x + δh(x)ν(x), x ∈ ∂Ω} (1.2)
where ν(x) is the outward normal vector on ∂Ω and h(x) is a real function in C 2 (∂Ω) that satisfies h(x) C 2 (∂Ω) < 1.

(1.3)

Obviously, the domain Ω δ is of class C 2 and the Dirichlet eigenvalue problem for the Stokes system can be defined in Ω δ as well.

In this paper, we derive the asymptotic of eigenvalues, eigenfunctions and the eigenpressures solutions to the Stokes system:

     -∆v δ + ∇p δ = λ δ v δ in Ω δ ∇.v δ = 0 in Ω δ v δ = 0 in ∂Ω δ .
(1.4)

Here we suppose that the eigenvalue λ 0 is simple. Then the eigenvalue λ δ is simple and is near to λ 0 associated to the normalized eigenfunction v δ .

To the best of our knowledge, this is the first work to rigorously investigate Stokes eigenvalue problem in the presence of the perturbation (1.2) and derive (formally) high-order terms in the asymptotic expansion of λ δ -λ 0 and v δ -v 0 when δ → 0. However, by the same method, one can derive asymptotic formula for the Neumann problem as well. Zuazua and Ortega have proved in [START_REF] Ortega | Generic simplicity of the eigenvalues of the stokes system in two soace dimensions[END_REF] the regularity of the eigenvalues and eigenfunctions of the Stokes system with respect to the perturbation parameter, by using the Lyapunov-Schmidt method. Their proofs are essentially inspired in the work of J.

Albert [START_REF] Albert | Genericity of simple eigenvalues for elliptic pde's[END_REF][START_REF] Albert | Topology of the nodal and critical points sets for eigenfunctions of elliptic operators[END_REF] for the Laplace operator. Our analysis and uniform asymptotic formulas of eigenvalues and eigenfunctions, which are represented by the single-layer potential involving the Green function, are considerably different from those in [START_REF] Kelliher | Eigenvalues of the Stokes operator versus the Dirichlet Laplacian in the plane[END_REF][START_REF] Ortega | Generic simplicity of the eigenvalues of the stokes system in two soace dimensions[END_REF]. In [START_REF] Ammari | Reconstruction of small interface changes of an inclusion from modal measurements II: The elastic case[END_REF] Ammari et al. have rigorously derived an asymptotic formula for the perturbations in the eigenvalues due to interface changes of an elastic inclusion. Their original work was based on several gradient estimates in Hölder space, and the first correction term is given explicitly. As the Stokes system can be viewed as the incompressible limit (the compression modulus is infinite), the formulas given in [START_REF] Ammari | Reconstruction of small interface changes of an inclusion from modal measurements II: The elastic case[END_REF] for the elastic system may be obtained by taking the limit of those in the mentioned paper as the compression modulus goes to infinity. The same spirit was fixed in [START_REF] Ammari | Effective viscosity properties of dilute suspensions of arbitrarily shaped particles[END_REF]. Taking into consideration the previous claims, we focus our attention in Section 3.3 to determine the asymptotic expansion and so the leading-order term for the Stokes eigenvalues. To benefit from the well known results and to use easily gradient estimates, we may work in Hölder space and we may use general Stokes operator defined by the strain rate tensor. This paper is organized as follows. In Section 1, we describe the main problem in this paper. In Section 2, we develop a boundary integral formulation for solving the eigenvalue problem (1.4), and we present some preliminary results. Finally, Section 3 is dedicated to develop asymptotic expansion of the perturbations in the eigenvalues, in eigenfunctions and in eigenpressures. More precisely, in the Section 3.1, we present some basic results related to shape perturbation and we develop high-order terms in the expansion of the integral operator. In Section 3.2, we (formally) provide by layer potentials an asymptotic expansion for the perturbed eigenvalues if the unperturbed eigenvalue is simple. But in section 3.3, we rigourously derive the leading-order term for the perturbed Stokes eigenvalues in Hölder space and for both simple and multiple eigenvalues. Here the problems to be study are more general than the one introduced in Section 1. In Section 3.4, we end our paper by developing asymptotic expansion of the perturbations in the eigenfunctions and in eigenpressures using the same problem configurations as Section 1.

Integral equations method

We now develop a boundary integral formulation for solving the perturbed eigenvalue problem (1.4). The components of the fundamental Stokes tensor Γ = (Γ ij ) 3 i,j=1 and those of the associated pressure vector P = (P ij ) 3 i,j=1 , which determine the fundamental solution (Γ, P ) of the Stokes system in R 3 , are given by (see for instance [START_REF] Ammari | A method of biological tissues elasticity reconstruction using magnetic resonance elastography measurements[END_REF][START_REF] Ladyzhenskaya | Mathematical theory of the viscous incompressible[END_REF])

   Γ ij (λ, x) = -1 4π δ ij e i √ λ|x| |x| -1 4πλ ∂ x i ∂ x j ( e i √ λ|x| -1 |x| ) P i (x) = -1 4π x i |x| 3 , (2.1)
where ∂x i denotes ∂/∂x i for i = 1, 2, 3 and i 2 = -1. We recall that the i th row Γ i of Γ satisfies

-∆Γ i + ∇P i (x) -λΓ i = e i δ(x) in R 3 ∇.Γ i = 0 in R 3 ;
in the sense of distributions and where (e i ; i = 1, 2, 3) is the orthonormal basis of R 3 . Note that we used the Einstein convention for the summation notation omitting the summation sign for the indices appearing twice.

The potential theory for the Stokes system

Let us denote by ϕ = (ϕ 1 , ϕ 2 , ϕ 3 ) a complex vector-valued function with class C 0 (∂Ω).

The hydrodynamic single-layer potential with density ϕ ∈ C 0 (∂Ω) 3 is the vector function S(λ)ϕ(x) defined by

S(λ)ϕ(x) := ∂Ω Γ(λ, |x -y|)ϕ(y) dσ(y), x ∈ R 3 \∂Ω. (2.2)
The pressure term Q corresponding to the single layer potential is the function given by Qϕ(x) := ∂Ω P (x, y)ϕ(y) dσ(y), x ∈ R 3 \∂Ω.

For a careful study of these potentials, one can refer to [START_REF] Ladyzhenskaya | Mathematical theory of the viscous incompressible[END_REF], [START_REF] Varnhorn | The Stokes equations[END_REF], [START_REF] Kohr | The interior Neumann problem for the Stokes resolvent system in a bounded domain in R n[END_REF].

Taking into account the well known properties of Green function Γ, one obtains the result that the pair (Sϕ, Qϕ) are smooth functions in each of the domains Ω and R 3 \Ω respectively. Also these functions are classical solutions to the Stokes system (1.1).

The continuity and jump relations of the Stokes surface potentials on the boundary ∂Ω are described in the following proposition (see [START_REF] Ammari | A method of biological tissues elasticity reconstruction using magnetic resonance elastography measurements[END_REF], [START_REF] Depauw | Solutions des équations de Navier Stokes incompressibles dans un domaine exterieur[END_REF] pp. 41-42 or [START_REF] Varnhorn | The Stokes equations[END_REF] p. 66 ): 3 and let S denotes the surface potential defined in (2.2). Then on the boundary ∂Ω the following continuity and jump relations are satisfied:

Proposition 2.1 Let ϕ ∈ C 0 (∂Ω)
(S(λ)ϕ) + = (S(λ)ϕ)| -= S(λ)ϕ ∂S(λ)(ϕ) ∂ν (x) ± = ± ϕ(x) 2 + ∂Ω ∂Γ(λ, |x -y|) ∂ν(x) ϕ(y) dσ(y).

Boundary integral formulation

In this section, we give a boundary integral formulation in order to solve the eigenvalue problems (1.1) and (1.4).

Proposition 2.2 Suppose h satisfies (1.3). Then, there exists δ 0 > 0 such that the map Ψ δ (x) defined by

Ψ δ (x) = x + δh(x)ν(x) (2.3)
is a C 2 -diffeomorphism from ∂Ω to ∂Ω δ for δ < δ 0 . In addition, the following equality holds

det (∇Ψ δ ) = 1 + tr∇(hν)δ + 1 2 tr∇(hν) 2 -tr ∇(hν) 2 δ 2 (2.4) + det (∇(hν)) δ 3 ,
where tr means the trace of a matrix. Moreover, we have tr∇(hν) = div(hν).

Proof. Recall that the function h(x) is C 2 on ∂Ω, then the map Ψ δ (x) is also C 2 . A simple calculation yields the equality (2.4). Consequently, for δ small enough the map

Ψ δ (x) is a C 2 -diffeomorphism from ∂Ω to ∂Ω δ .
We further denote Ψ -1 δ the reciproque function of Ψ δ (x). Thanks to Ψ -1 δ , we can define the operator A δ (λ) as follows:

A δ (λ)ϕ(x) = S Ω δ (λ)ϕ(Ψ -1 δ ) (Ψ δ (x)), ϕ ∈ (L 2 (∂Ω)) 3 (2.5)
where S Ω δ (λ) is the hydrodynamic single-layer potential given by (2.2) when we have replaced the boundary ∂Ω by ∂Ω δ .

For i, j ∈ {1, 2, 3}, we can define the j th -component of the vector-valued function A δ (λ) as follows:

(A δ (λ)ϕ) j (x) = S Ω δ (λ)ϕ j (Ψ -1 δ ) (Ψ δ (x)), ϕ ∈ (L 2 (∂Ω)) 3 .
(2.6)

The j th -component of the single-layer potential S Ω δ (λ) is given by

S Ω δ (λ)ϕ j (x) := ∂Ω δ Γ ji (λ, |x -ỹ|)ϕ i (ỹ) dσ δ (ỹ), x ∈ R 3 \∂Ω δ , j = 1, 2, 3, (2.7)
where ϕ i is the i th -component of the vector-valued function ϕ. Using Proposition 2.2, relations (2.6)-(2.7) and the continuity relations given by Proposition 2.1, we obtain for x ∈ ∂Ω that

A δ (λ)ϕ j (x) = ∂Ω Γ ji (λ, |Ψ δ (x) -Ψ δ (y)|) det (∇Ψ δ (y)) ϕ i (y) dσ(y), j = 1, 2 , 3.
(2.8) Let A 0 the operator defined as in (2.5) by

A 0 φ = S(λ)φ,
where φ ∈ (L 2 (∂Ω)) 3 . Then, we have the following result, which is a slight variation of the Lemma 6.1 due to Ammari et al. [START_REF] Ammari | Layer Potential Techniques in Spectral Analysis[END_REF][START_REF] Ammari | Splitting of resonant and scattering frequencies under shape deformation[END_REF] for the scalar eigenvalue problem. 3 is Fredholm of index zero in C\iR -. In addition the Dirichlet eigenvalues of the Stokes system (1.1) are exactly its real zeros.

Proposition 2.3 The operator-valued function

A 0 (λ) : H -1/2 (∂Ω) 3 → H 1/2 (∂Ω)
From Proposition 2.3 we know that if λ 0 is an eigenvalue of (1.1) then λ 0 is a real zero of A 0 (λ). Moreover, for ǫ 0 small enough, the function A -1 0 (λ) is meromorphic in D ǫ 0 (λ 0 ), where D ǫ 0 (λ 0 ) means the disc of center λ 0 and radius ǫ 0 , and λ 0 is its unique pole in D ǫ 0 . Furthermore we have the following Laurent expansion:

A -1 0 (λ) = (λ -λ 0 ) -1 ℓ 0 + R 0 (λ), (2.9)
where ℓ 0 : KerA 0 (λ 0 ) → KerA 0 (λ 0 ), and R 0 (λ) is a holomorphic function.

Our main results in this section are summarized in the following theorem.

Theorem 2.1 Suppose that the eigenvalue λ 0 of (1.1) is with multiplicity 1. Then, there exist a positive constant δ 0 (ǫ 0 ) such that for |δ| < δ 0 , the operator-valued function λ → A δ (λ) has a real zero λ(δ) in D ǫ 0 (λ 0 ). This zero is exactly the eigenvalue of the perturbed eigenvalue problem (1.4), and is an analytic function with respect to δ in ] -δ 0 , δ 0 [. It satisfies λ(0) = λ 0 . Moreover, the following assertions hold:

A -1 δ (λ) = (λ -λ(δ)) -1 ℓ(δ) + R δ (λ), ℓ(δ) : Ker(A δ (λ(δ)) → Ker(A δ (λ(δ)), (2.10) where R δ (λ) is a holomorphic function with respect to (δ, λ) ∈] -δ 0 , δ 0 [×D ǫ 0 (λ 0 ).
3 Asymptotic formula

High-order terms in the expansion of A δ

We now present some basic results related to shape perturbation. The methods to be used here differ from those in [START_REF] Ammari | Reconstruction of small interface changes of an inclusion from modal measurements II: The elastic case[END_REF][START_REF] Ammari | Layer Potential Techniques in Spectral Analysis[END_REF], but the expressions deduced are the same. To begin, let (τ 1 (x), τ 2 (x)) be the orthornormal basis of the tangent plan to the surface ∂Ω at a regular point x. Their cross product is then orthogonal to ∂Ω at the point x.

By changing their order, we can assume that τ 1 × τ 2 is a vector pointing towards the exterior of the surface ∂Ω. Then dividing it by its length yields the unit normal vector ν(x), that is:

ν 0 (x) = τ 1 (x) × τ 2 (x) |τ 1 (x) × τ 2 (x)| , (3.1)
for x ∈ ∂Ω. Evidently ν 0 = ν, where ν was introduced in section 1.

Set

τ δ 1 = grad Ψ δ • τ 1 , and τ δ 2 = grad Ψ δ • τ 2 .
Using Proposition 2.2, we find that:

τ δ 1 = τ 1 + δM τ 1 , and τ δ 2 = τ 2 + δM τ 2 , (3.2)
where the (3 × 3)-matrix M is given by:

M =   ∂ 1 (hν 1 ) ∂ 2 (hν 1 ) ∂ 3 (hν 1 ) ∂ 1 (hν 2 ) ∂ 2 (hν 2 ) ∂ 3 (hν 2 ) ∂ 1 (hν 3 ) ∂ 2 (hν 3 ) ∂ 3 (hν 3 )   with ν i means the i-th (i = 1, 2, 3) component of the vector ν.
For δ sufficiently small, one can see that the outward unit normal vector to ∂Ω δ is given by

ν δ (x) = τ δ 1 (x) × τ δ 2 (x) |τ δ 1 (x) × τ δ 2 (x)| , (3.3)
for x ∈ ∂Ω. Then, the following asymptotic expansion holds.

Proposition 3.1 Let ν 0 be given by (3.1). Then, the outward unit normal ν δ (x) to ∂Ω δ at x, can be expanded uniformly as

ν δ (x) = ν (0) (x) + ∞ n=1 δ n ν (n) (x), x ∈ ∂Ω,
where the vector-valued functions ν (n) are uniformly bounded. In particular, for x ∈ ∂Ω:

ν (0) (x) = ν(x), ν (1) = 1 |τ 1 × τ 2 | τ 1 × M τ 2 + M τ 1 × τ 2 -ν 0 • (τ 1 × M τ 2 + M τ 1 × τ 2 ) ν 0 .
Proof.

Considering the expansions (3.2) for δ sufficiently small, the relation (3.3) becomes:

ν δ = a + δb + δ 2 c |a + δb + δ 2 c| , (3.4)
where a, b, and c are vector-valued functions given by:

a = τ 1 × τ 2 , b = M τ 1 × τ 2 + τ 1 × M τ 2 , and c = M τ 1 × M τ 2 .
So that, by expanding the quotient (3.4) as δ tends to zero, we get the desired results.

Next, one can use Proposition 2.2 to get the uniformly convergent expansion for the surface element as follows: Proposition 3.2 Let ỹ = Ψ δ (y) where Ψ δ (y) is given by (2.3) for y ∈ ∂Ω. Then, the following expansion for the surface element dσ δ (ỹ) holds uniformly for y ∈ ∂Ω:

dσ δ (ỹ) = det (∇Ψ δ ) dσ(y) = σ 0 (y) + σ 1 (y)δ + σ 2 (y)δ 2 + σ 3 (y)δ 3 dσ(y), (3.5)
where

σ 0 ≡ 1, σ 1 (y) = ∇ • (hν), σ 2 (y) = 1 2 tr∇(hν) 2 -tr ∇(hν) 2 , and σ 3 (y) = det (∇(hν)). Set x = Ψ δ (x), x ∈ ∂Ω (3.6) ỹ = Ψ δ (y), y ∈ ∂Ω, (3.7)
and define Θ(x, y)

:= 1 δ Ψ δ (x) -Ψ δ (y) -(x -y) for δ = 0. (3.8)
Recall that Ψ δ is a C 2 vector-valued function on ∂Ω, then Θ is also a C 2 function. Moreover, the following holds.

Proposition 3.3 The vector-valued function Θ(x, y) is C 2 on ∂Ω×∂Ω and there exists a constant C > 0 that only depends on Ω and h such that:

|Θ(x, y)| ≤ C|x -y|, |Θ(x, y) • (x -y)| ≤ C|x -y| 2 ,
for all x, y ∈ ∂Ω.

Proof. Expression (3.8) shows that:

Θ(x, y) = h(x)ν(x) -h(y)ν(y).
Since ∂Ω is a C 2 surface, there exists a constant C ′ > 0 such that:

|ν(x) -ν(y)| ≤ C ′ |x -y|, and |ν(x) • (x -y)| ≤ C ′ |x -y| 2 ,
for all x, y ∈ ∂Ω.

The last inequalities and the C 2 regularity of h yield the results of the proposition. Now, by using (3.8) we obtain:

x

-ỹ = x -y + δΘ(x, y), (x, y) ∈ ∂Ω × ∂Ω, (3.9)
and the following results hold.

Lemma 3.1 Let r > 0 be a fixed real and m ≥ 2 be a fixed integer. The following asymptotic expansions

e i √ λ|x-ỹ| |x -ỹ| m = e i √ λ|x-y| |x -y| m 1 + δT (m) 1 (x, y) + n≥2 δ n T (m)
n (λ; x, y) , (3.10)

∂ xi ∂ xj e i √ λ|x-ỹ| -1 |x -ỹ| = R (ij) 0 (λ; x, y) + δR (ij) 1 (λ; x, y) + n≥2 δ n R (ij) n (λ; x, y) (3.11)
hold uniformly for (λ, x, y) ∈ B r (0)×∂Ω×∂Ω, where B r (0) is a ball in the complex plane of center zero and radius r. In addition the functions

T (m) n (λ; x, y) and R (ij)
n (λ; x, y) are smooth and bounded uniformly on B r (0) × ∂Ω × ∂Ω. The first coefficients are given by:

R (ij) 0 (λ; x, y) := ∂ x i ∂ x j e i √ λ|x-y| -1 |x -y| , and
R (ij) 1 (λ; x, y) = (x-y)•Θ(x, y) -λ 2 δ ij -3i √ λ δ ij r -iλ 3/2 r i r j r +7λ 2 r i r j r 2 +15i √ λ r i r j r 3 e i √ λr r 5 +(x-y)•Θ(x, y) 4i √ λ δ ij r -15 δ ij r 2 -λ 2 [-5 r i r j r 2 + (r i θ j + r j θ i ) (x -y) • Θ(x, y) -7i √ λ[-6 r i r j r 3 + 1 r (r i θ j + r j θ i ) (x -y) • Θ(x, y) ] +15[-7 r i r j r 4 + 1 r 2 (r i θ j + r j θ i ) (x -y) • Θ(x, y) e i √ λr r 5 +15 (x -y) • Θ(x, y) r 7 δ ij -[-7 r i r j r 2 + (r i θ j + r j θ i ) (x -y) • Θ(x, y) ] ,
where r = |x -y|, r j = x j -y j , θ j means the j th component of Θ, and δ ij means the Kronecker index.

Proof. For m = 2 and x = y we have:

|x -ỹ| 2 = |x -y + δΘ(x, y)| 2 = |x -y| 2 1 + δc (2) 1 (x, y) + δ 2 c (2) 2 (x, y) ,
where

c (2) 1 (x, y) = 2Θ(x, y) • (x -y) |x -y| 2 , c (2)
2 (x, y) = Θ(x,y)•Θ(x,y) |x-y| 2 . Proposition 3.1 shows that c (2)
1 (x, y) and c

(2) 2 (x, y) are bounded uniformly on ∂Ω × ∂Ω.

For m > 2, we have:

|x -ỹ| m = |x -y + δΘ(x, y)| m = |x -y| m 1 + δ Θ(x, y) |x -y| m ,
where x = y. Using (3.8), Proposition 2.2 and Proposition 3.3, we see that the regular vector-valued function (x, y) → Θ(x,y) |x-y| is well defined on ∂Ω × ∂Ω, and it is independent of δ. Therefore, we can expand:

|x -ỹ| m = c (m) 0 (x, y) + ∞ n=1 δ n c (m) n (x, y) uniformly on ∂Ω × ∂Ω, (3.12)
where the first coefficients c

(m) 0 (x, y) = |x -y| m , c (m) 1 (x, y) = m < x -y, Θ(x, y) > |x -y| m-2 and c (m) 2 (x, y) = m 2 |x -y| m | Θ(x,y) |x-y| | 2 + (m -2) < x-y |x-y| 2 , Θ(x,y) |x-y| > 2 .
Moreover, if m is even, then c (m) n (x, y) = 0 for n ≥ m + 1. Now combining (3.12) for m = 1 with the well known asymptotic expansion of the exponential function, we immediately get

e i √ λ|x-ỹ| = ∞ n=0 δ n K n (λ; x, y) uniformly on B r (0) × ∂Ω × ∂Ω, (3.13)
where the first coefficients K 0 (λ; x, y) = e i √ λc

(1) 0

= e i √ λ|x-y| , and the coefficients K n (λ; x, y) can be deduced recursively from c

n . Thanks to relations (3.9), (3.12) and (3.13), we can obtain the desired result given by (3.10) where the coefficients T (m) n can be deduced easily from c (m) n and K n .

To prove relation (3.11), we firstly expand

∂ xi ∂ xj e i √ λ|x-ỹ| -1 |x -ỹ| = 1 |x -ỹ| ∂ xi ∂ xj e i √ λ|x-ỹ| -1 + ∂ xi 1 |x -ỹ| ∂ xj e i √ λ|x-ỹ| -1 (3.14) +∂ xj 1 |x -ỹ| ∂ xi e i √ λ|x-ỹ| -1 + ∂ xi ∂ xj 1 |x -ỹ| e i √ λ|x-ỹ| -1 .
To simplify, let us denote r = |x -y|, r i = x i -y i , r = |x -ỹ|, and rj defined by the relation

∂ xj r = rj r . (3.15)
Then,

∂ xj ( 1 r3 ) = -3 rj r5 , and ∂ xi ∂ xj ( 1 r3 ) = -3 δ ij r5 + 15 ri rj r7 . (3.16)
Now, by using the following result

∂ xj e i √ λ|x-ỹ| -1 = ∂ xj (i √ λr e i √ λr = i √ λ rj r e i √ λr , (3.17)
we get that

∂ xi ∂ xj e i √ λ|x-ỹ| -1 = i √ λ ∂ xi ( rj r )e i √ λr + rj r ∂ xi e i √ λr = i √ λ δ ij r - ri rj r3 +i √ λ ri rj r2 e i √ λr .
(3.18) To find the desired result in (3.11), we may use (3.16), (3.17) and (3.18) to see that the relation (3.14) verifies:

∂ xi ∂ xj e i √ λ|x-ỹ| -1 |x -ỹ| 3 = i √ λ δ ij r4 -3 δ ij r5 -λ 2 ri rj r5 -7i √ λ ri rj r6 +15 ri rj r7 e i √ λr +3 δ ij r5 -15 ri rj r7 . (3.19)
On the other hand, the components of the vectorial relation (3.9) can be given as

follows xi -ỹi = x i -y i + δθ i (x, y). i = 1, 2, 3 (3.20)
where θ i (x, y) means the i th component of the vector-valued function Θ(x, y).

Then, by relations (3.15) and (3.20) we deduce that

rj = xi -ỹi = x i -y i + δθ i (x, y) = r i + δθ i (x, y), i = 1, 2, 3. (3.21)
Using both relations (3.9) and (3.21), we get the following expansion ri rj = α

(ij) 0 + α (ij) 1 δ + α (ij) 2 δ 2 , (3.22)
where the first coefficients: α

(ij) 0 = r i r j , α (ij) 1
= r i θ j + r j θ i and α

(ij) 2 = θ i • θ j .
Now regarding (3.12) and using the fact that c (m) 0

= 0 for each integer m. Then one can expand

1 rm = κ (m) 0 (x, y) + ∞ n=1 δ n κ (m) n (x, y) uniformly , (3.23)
where the first coefficients: κ

(ij) 0,m + δβ (ij) 1,m + ∞ n=2 δ n β (ij) n,m , (3.24)
where the first coefficients:

β (ij) 0,m (x, y) = α (ij) 0 (c (m) 0 ) -1 (x, y), and
β (ij) 1,m (x, y) = -α (ij) 0 c (m) 1 (c (m) 0 ) -2 (x, y) + α (ij) 1 (c (m) 0 ) -1 (x, y).
To achieve the proof, we may insert all expansions (3.13), (3.23) (for m ∈ {4, 5}), and (3.24)(for m ∈ {5, 6, 7} into (3.19). We get that

∂ xi ∂ xj e i √ λ|x-ỹ| -1 |x -ỹ| 3 = R (ij) 0 (x, y) + δR (ij) 1 (x, y) + ∞ n=2 δ n R (ij) n (x, y)
where the first coefficient:

R (ij) 0 (λ; x, y) = i √ λδ ij κ (4) 0 -3δ ij κ (5) 0 -λ 2 β (ij) 0,5 -7i √ λβ (ij) 0,6 +15β (ij) 0,7 K 0 +3δ ij κ (5) 0 -15β (ij) 0,7 = i √ λδ ij (c (4) 0 ) -1 -3δ ij (c (5) 0 ) -1 -λ 2 r i r j (c (5) 0 ) -1 -7i √ λr i r j (c (6) 0 ) -1 +15r i r j (c (7) 0 ) -1 e i √ λc (1) 0 +3δ ij (c (5) 0 ) -1 -15r i r j (c (7) 0 ) -1 .
Using the fact that c

(m) 0 = |x -y| m , we find that R (ij) 0 (λ; x, y) := ∂ x i ∂ x j e i √ λ|x-y| -1 |x -y| 3 .
Based on (3.19), we find that

R (ij) 1 (λ; x, y) = i √ λδ ij κ (4) 0 -3δ ij κ (5) 0 -λ 2 β (ij) 0,5 -7i √ λβ (ij) 0,6 + 15β (ij) 0,7 K 1 + i √ λδ ij κ (4) 1 -3δ ij κ (5) 1 -λ 2 β (ij) 1,5 -7i √ λβ (ij) 1,6 + 15β (ij) 1,7 K 0 + 3δ ij κ (5) 1 -15β (ij) 1,7 .
So that, by using the fact that

K 1 = i √ λΘ • (x-y) r e i √ λc (1)
0 , we get

R (ij) 1 (λ; x, y) = i √ λΘ• (x -y) r i √ λδ ij (c (4) 0 ) -1 -3δ ij (c (5) 0 ) -1 -λ 2 r i r j (c (5) 0 ) -1 -7i √ λr i r j (c (6) 0 ) -1 +15r i r j (c (7) 0 ) -1 e i √ λc (1) 0 + i √ λδ ij (c (4) 0 ) -2 c (4) 1 -3δ ij (c (5) 0 ) -2 c (5) 1 -λ 2 [-r i r j (c (5) 0 ) -2 c (5) 1 + (r i θ j + r j θ i )(c (5) 0 ) -1 ] -7i √ λ[-r i r j (c (6) 0 ) -2 c (6) 1 +(r i θ j +r j θ i )(c (6) 0 ) -1 ]+15[-r i r j (c (7) 0 ) -2 c (7) 1 +(r i θ j +r j θ i )(c (7) 0 ) -1 ] e i √ λc (1) 0 +3δ ij (c (5) 0 ) -2 c (5) 1 -15[-r i r j (c (7) 0 ) -2 c (7) 1 + (r i θ j + r j θ i )(c (7) 0 ) -1 ].
Using the explicit forms of the coefficients c given above, we immediately get the desired expression of R (ij) 1 (λ; x, y). The other coefficients can be deduced easily by the same manner in terms of c (m) n . Now using (3.20), we obtain for i, j = 1, 2, 3 that:

(x i -ỹi )(x j -ỹj ) = ĝ0 (x, y) + δĝ 1 (x, y) + δ 2 ĝ2 (x, y), (3.25)
where ĝ0 (x, y) = (x i -y i )(x j -y j ), ĝ1 (x, y) = θ i (x, y)(x j -y j ) + θ j (x, y)(x i -y i ), and ĝ2 (x, y) = θ i (x, y) • θ j (x, y). Now, by using (3.5), (3.12) and (3.25), we immediately get

(x i -ỹi )(x j -ỹj ) |x -ỹ| m dσ δ (ỹ) = T0 (x, y) + δ T1 (x, y) + n≥2 δ n Tn (x, y) dσ(y), (3.26)
where the Taylor coefficients Tn can be given explicitly with the aid of ĝ0 , ĝ1 , and ĝ2 .

Next, the following result holds.

Lemma 3.2

The following uniform expansion holds on R × ∂Ω × ∂Ω:

(x i -ỹi )(x j -ỹj )|x -ỹ|e i √ λ|x-ỹ| dσ δ (ỹ) = E δ (λ; x, y) dσ(y) = ∞ n=0
δ n E n (λ; x, y) dσ(y)

(3.27) with E 0 (λ; x, y) = ĝ0 (x, y)|x -y|e i √ λ|x-y| ,
and the other coefficients E n are deduced from those σ n , c

n , K n and ĝn .

Proof. From the proof of Lemma 3.1 (for m = 1) and from relations (3.5) and (3.25), one can get that

(x i -ỹi )(x j -ỹj )|x -ỹ|e i √ λ|x-ỹ| dσ δ (ỹ) = ĝ0 (x, y) + δĝ 1 (x, y)+ δ 2 ĝ2 (x, y) ∞ n=0 δ n c (1) n ∞ n=0 K n (λ; x, y) σ 0 (y) + δσ 1 (y)+ δ 2 σ 2 (y) + δ 3 σ 3 (y) dσ(y).
By collecting terms of equal powers in the above relation, one can deduce easily the uniform expansion (3.27) with

E 0 (λ; x, y) = ĝ0 (x, y)|x -y|e i √ λ|x-y| .
Now, from (2.1) we have

Γ ij (λ, |x -y|) = e i √ λr 4π δ ij r - xi xj r 3 + e i √ λr 4πλ i δ ij r 4 -4i xi xj r 4 -3irx i xj + 1 4πλ -3 δ ij r 5 + 15 xi xj r 7 + e iλr 4πλ -3 δ ij r 5 + 15 xi xj r 3
, where r = |x -y| and xi = x i -y i . Then, by inserting relations (3.10), (3.25), (3.26) and that given by Lemma 3.2 into above identity, we immediately get the following main results Proposition 3.4 Let the perturbed boundary ∂Ω δ defined by (1.2). Let x and ỹ given by (3.6)-(3.7), and the surface element dσ δ (ỹ) given by (3.5). Then, the components Γ ij , 1 ≤ i, j ≤ 3 of the fundamental Stokes tensor can be expanded uniformly as:

Γ ij (λ, |x -ỹ|) dσ δ (ỹ) = Γ (0) ij (λ, |x -y|) + δΓ (1) ij (λ, |x -y|) (3.28) + n≥2 δ n Γ (n) ij (λ, |x -y|) dσ(y), y ∈ ∂Ω
where the first coefficients:

Γ (0) ij (λ, |x -y|) := δ ij 4π T (1) 0 + i λ T (4) 0 -3 r 0 λ 2 -3 T (5) 0 λ 2 + 1 4π ( 15 λ 2 -1)T (3) 0 ĝ0 - 4i 4πλ T (4) 0 ĝ0 - 3i 4πλ E 0 + 15 4πλ 2 T0 , and
Γ (1) ij (λ, |x -y|) := δ ij 4π T (1) 1 + i λ T (4) 1 -3 r 1 λ 2 -3 T (5) 1 λ 2 + 1 4π ( 15 λ 2 -1) T (3) 0 ĝ1 + T (3) 1 ĝ0 - 4i 4πλ T (4) 0 ĝ1 + T (4) 1 ĝ0 - 3i 4πλ E 1 + 15 4πλ 2
T1 . In Proposition 3.4, the coefficients r 0 and r 1 are deduced from (3.12) for m = 5.

Now, introduce a sequence of components of integral operators (A (n)

i ) n≥0 , defined for any ϕ ∈ L 2 (∂Ω) 3 by:

A (n) ϕ i (x) = ∂Ω Γ (n) ij (x, y)ϕ j (y) dσ(y), for i, j ∈ {1, 2, 3} and n ≥ 0.

Using previous results, il is clear that we know explicitly the first terms

A (0) i , A (1)
i . For any positive integer N , we can by recursive method get the term A (N ) i . Then, the following theorem holds. Theorem 3.1 Let A δ be the operator defined by (2.5). Let Ψ δ (x) be the diffeomorphism defined as in Lemma 2.2. Let N be a positive integer. There exists a positive constant C depending only on N , and h C 2 such that for any φ ∈ L 2 (∂Ω δ ) 3 and i, j ∈ {1, 2, 3}, the i th -component A δ i defined by (2.8) satisfies:

A δ φ i oΨ δ -A (0) ϕ) i - N n=1 δ n A (n) ϕ i L 2 (∂Ω) ≤ Cδ N +1 ϕ L 2 (∂Ω) 3 ,
where ϕ = φoΨ δ .

Asymptotic expansion of the eigenvalues

To develop asymptotic behaviors of eigenvalues and eigenfunctions with respect to the parameter of perturbation δ, we may use the results of Theorem 3.1. Then, the following asymptotic expansion related to the operator A δ appears clearly.

Proposition 3.5 Suppose that we have all hypothesis of Theorem 3.1. Then, the operator A δ (λ) defined by (2.5) can be expanded uniformly for x ∈ ∂Ω as follows:

A δ (λ)ϕ = A (0) (λ)ϕ + δA (1) (λ)ϕ + δ 2 A (2) (λ)ϕ + • • • ; as δ → 0, (3.29)
where ϕ ∈ L 2 (∂Ω) 3 , the i th -component of the first term is given by

A (0) i (λ)ϕ i (x) = ∂Ω Γ (0) ij (x, y)ϕ j (y) dσ(y), for i, j ∈ {1, 2, 3},
and more generally, the i th -component of the term with order n is given by

A (n) (λ)ϕ i (x) = ∂Ω Γ (n) ij (
x, y)ϕ j (y) dσ(y), for i, j ∈ {1, 2, 3} and n ≥ 1.

The coefficients Γ

(n) ij are given by (3.28).

Now, based on the well known works [START_REF] Ammari | Layer Potential Techniques in Spectral Analysis[END_REF][START_REF] Ammari | Splitting of resonant and scattering frequencies under shape deformation[END_REF], we tray to give (formally) an asymptotic expansion related to the eigenvalues λ δ if the eigenvalue λ 0 is simple. But, To get an explicit asymptotic formula for λ δ we may develop our analysis in some Hölder space. This will be given in Theorem 3.2. Before that, let a j (δ) denotes:

a j (δ) := 1 2iπ tr ∂Dǫ 0 (λ -λ 0 ) j (A δ ) -1 (λ)∂ λ A δ (λ)dλ. (3.30)
The functions a j (δ) is analytic in a complex neighborhood of 0 and satisfies: a j (δ) = a j (δ). The following main result holds.

Proposition 3.6 Let λ 0 be a simple eigenvalue of the problem (1.1). Let the operator A δ be defined by (2.5). Then, there exists a small positive number δ 0 such that the eigenvalue λ(δ) is analytic in ] -δ 0 , δ 0 [ and satisfies:

λ(δ) = λ 0 + δλ 1 + n≥2 λ n δ n , (3.31)
where the first coefficients are (formally) given by:

               λ 1 = 1 2iπ tr ∂Dǫ 0 (λ -λ 0 ) (A (0) ) -1 (λ)A (1) (λ)(A (0) ) -1 (λ)∂ λ A (0) (λ) dλ, λ 2 = 1 2iπ tr ∂Dǫ 0 (λ -λ 0 ) (A (0) ) -1 A (1) (A (0) ) -1 ∂ λ A (1) + (A (0) ) -1 A (2) (A (0) ) -1 ∂ λ A (0) + (A (0) ) -1 A (1) (A (0) ) -1 2 ∂ λ A (0) dλ. So, λ δ -λ 0 = 1 2iπ tr ∂Dǫ 0 (λ -λ 0 )(A (0) ) -1 (λ)∂ λ A δ (λ)ϕ dλ+ 1 2iπ tr ∂Dǫ 0 (λ -λ 0 ) ∞ k=1 (A (0) ) -1 (λ) A (0) (λ) -A δ (λ) (A (0) ) -1 (λ) k ∂ λ A δ (λ) dλ
By using (2.9), we find that 1 2iπ tr

∂Dǫ 0 (λ -λ 0 )(A (0) ) -1 (λ)∂ λ A δ (λ) dλ = 0.
This result is a direct consequence of the fact that R 0 (λ) and ∂ λ A δ (λ) are holomorphic in the variable λ.

Now we have:

λ δ -λ 0 = 1 2iπ tr ∂Dǫ 0 (λ -λ 0 )(A (0) ) -1 (λ) A (0) (λ) -A δ (λ) (A (0) ) -1 (λ) ∂ λ A δ (λ) dλ+ 1 2iπ tr ∂Dǫ 0 (λ -λ 0 )(A (0) ) -1 (λ) A (0) (λ) -A δ (λ) (A (0) ) -1 (λ) 2 ∂ λ A δ (λ) dλ+ 1 2iπ tr ∂Dǫ 0 (λ -λ 0 ) k≥3 (A (0) ) -1 (λ) A (0) (λ) -A δ (λ) (A (0) ) -1 (λ) k ∂ λ A δ (λ) dλ.
Inserting expression (3.29) into above relation, we may get:

λ δ -λ 0 = 1 2iπ tr ∂Dǫ 0 (λ-λ 0 )(A (0) ) -1 δA (1) (A (0) ) -1 +δ 2 A (2) (A (0) ) -1 +• • • ∂ λ A (0) +δ∂ λ A (1) +δ 2 ∂ λ A (2) + • • • ϕ dλ + 1 2iπ tr ∂Dǫ 0 (λ -λ 0 )(A (0) ) -1 δA (1) (A (0) ) -1 + δ 2 A (2) (A (0) ) -1 + • • • 2 ∂ λ A (0) + δ∂ λ A (1) + δ 2 ∂ λ A (2) + • • • dλ + • • •
If we collect the same powers of δ, then we get the desired results.

Asymptotic formula in Hölder space

In section we suppose that any shape deformation like (1.2) occurs inside a bounded domain Ω in a Hölder space. Based on the works [START_REF] Ammari | Effective viscosity properties of dilute suspensions of arbitrarily shaped particles[END_REF][START_REF] Ammari | Reconstruction of small interface changes of an inclusion from modal measurements II: The elastic case[END_REF][START_REF] Ammari | A method of biological tissues elasticity reconstruction using magnetic resonance elastography measurements[END_REF][START_REF] Ammari | Layer Potential Techniques in Spectral Analysis[END_REF][START_REF] Boyer | Fabrie Mathematical tools for the study of the incompressible Navier-Stokes equations and related models[END_REF], we may advance the asymptotic expansions for simple and/or multiple eigenvalues of the Stokes operator and we may give explicitly the fist term correction. Since the Stokes system can be viewed as the incompressible limit and as done in [START_REF] Ammari | Effective viscosity properties of dilute suspensions of arbitrarily shaped particles[END_REF], our method in this section may be deeply based on the ones developed in [START_REF] Ammari | Reconstruction of small interface changes of an inclusion from modal measurements II: The elastic case[END_REF] for the elastic case.

Suppose that in this section that Ω ⊂ R 2 is a bounded domain with C 1,1 boundary.Let D be an open subset of Ω such that dist(∂Ω, D) ≥ d 0 > 0 representing an inclusion made of a different Newtonian fluid material. Assume that the boundary ∂D is of class C 2,1 . We denote by σ 0 and σ 1 the stress tensor fields in Ω\D and D, respectively. We assume that both Ω\D and D are occupied by isotropic and homogeneous Newtonian fluids. Then, the tensors σ 0 and σ 1 may be given by (σ s ) ijlk = µ s (δ ki δ lj + δ kj δ li ) for i, j, k, l = 1, 2 and s = 0, 1, (

where µ 0 and µ 1 are the viscosity constants of the flow in Ω\D and D, respectively. Given two (2 × 2) matrices A and B we denote by A : B the contraction, i.e., A : B = ij a ij b ij . Now, it is useful to introduce the strain rate tensor D for the flow as follows:

D(v) := 1 2 (∇v + (∇v) T ) = 1 2 ( ∂v i ∂x j + ∂v j ∂x i ) 1≤i,j≤3
.

Let σ D := σ 0 χ Ω\D + σ 1 χ D . Then according to [START_REF] Boyer | Fabrie Mathematical tools for the study of the incompressible Navier-Stokes equations and related models[END_REF], one can rewrite the eigenvalue problem (1.1) as the generalized one:

           -div(σ D D(v 0 ) -p 0 Id) = λ 0 v 0 in Ω ∇.v 0 = 0 in Ω v 0 = 0 in ∂Ω v 0 = 1 (3.34) where (v 0 , p 0 , λ 0 ) ∈ (H 2 (Ω)) 2 × H 1 (Ω) × R *
+ and Id means the identity. Here we refer the reader to ([10], Theorem IV.5.8) for more details about the regularity properties. On the other hand, if S mens the Stokes operator, it is well know that (see for example, [START_REF] Ammari | Effective viscosity properties of dilute suspensions of arbitrarily shaped particles[END_REF][START_REF] Boyer | Fabrie Mathematical tools for the study of the incompressible Navier-Stokes equations and related models[END_REF]) there exist an orthogonal projection (the Leray projection) P such that:

S(v) = P (-∆v), ∀v ∈ (H 2 (Ω)) 2 ∩ H(Ω).
In consequence, the Stokes operator enters the general framework about spectral properties of -∆.

The δ-perturbation, denoted by D δ , of the domain D is given by

∂D δ := {x = x + δh(x)ν(x), x ∈ ∂D} (3.35)
where here h(x) is assumed to be real function in C 1,1 (∂D) that satisfies h C 1,1 (∂D) < c 0 for some positive constant c 0 .

(3.36)

Let σ D δ := σ 0 χ Ω\D δ + σ 1 χ D δ . Therefore, (1.4) can be generalized to the following perturbed eigenvalue problem

           -div(σ D δ D(v δ ) -p δ Id) = λ δ v δ in Ω ∇.v δ = 0 in Ω v δ = 0 in ∂Ω v δ = 1 (3.37) where (v δ , p δ , λ δ ) ∈ (H 2 (Ω)) 2 × H 1 (Ω) × R * + .
Our purpose in this section is to develop a rigourously asymptotic behavior of the eigenvalue of (3.37) assumed to be more precisely. The main result is the following. Theorem 3.2 Suppose that Ω ⊂ R 2 with class C 1,1 . Let λ 0 be a simple eigenvalue of the problem (3.34). Assume that we have (3.36), and the pressure still the same if we have (3.35). Then, the first term correction in the asymptotic expansion of the eigenvalue λ δ is given through the following formula:

λ δ -λ 0 = δ ∂D h(x)V[D(v e 0 )](x) : D(v e 0 (x))ds(x) + O(δ 1+β ),
where β some positif constant and V means the viscous moment tensor (VMT) given by V

[D(v e 0 )] = [ 1 σ 1 (V 1 ∇v e 0 (x)τ ) ⊗ τ + ( σ 0 σ 1 ∇v e 0 (x)ν) ⊗ ν].
Here, we denote by ν, τ respectively the outward normal vector and the tangent vector to ∂D and V 1 a given operator expresses the transmission conditions for v 0 . Proof. To derive the corresponding formulae for the eigenvalues we will use an idea close to Osborn'theorem [START_REF] Osborn | Spectral approximation for compact operators[END_REF] which gives estimates for the convergence of the eigenvalues of a sequence of compact operators. For any f ∈ L 2 (Ω), define the linear operator T ǫ f = u ǫ , where u ǫ is the solution to the problem

-div(σ D δ D(u δ ) -p δ Id) = f in Ω, u δ = 0 on ∂Ω. (3.38)
For the same given function f , we define the linear operator T f = u 0 , where u 0 is the solution to the problem

-div(σ D D(u 0 ) -p 0 Id) = f in Ω, u 0 = 0 on ∂Ω. (3.39)
Clearly T ǫ and T are are compact self-adjoint operators. One may use standard energy estimates based on Korn and Poincar'e inequalities to get these facts. Moreover, it is easy to see that {T δ ; δ ≥ 0} with T 0 = T are collectively compact and that T δ → T pointwise as δ → 0. To get the last result, one may prove that T δ f converges to T f in L 2 (Ω) for every f ∈ L 2 (Ω). By assumption, in the presence of (3.35), there exist a function p such that p δ = p 0 = p. Then, let u δ = T δ f and u 0 = T δ f . For any w ∈ H 1 (Ω), we have:

Ω ∇ • (σ D δ D(u δ ) -pId).w = Ω f.w, so, Ω (σ D δ D(u δ ) -pId) : D(w) = - Ω f.w. Similarly, Ω (σ D D(u 0 ) -pId) : D(w) = - Ω f.w.
Consequently, choosing w = u δ -u 0 and subtracting these two equations we get

Ω (σ D δ D(u δ ) -σ D D(u 0 )) : D(u δ -u 0 ) = 0 which gives Ω σ D δ D(u δ -u 0 ) : D(u δ -u 0 ) = - Ω (σ D δ -σ D )D(u 0 ) : D(u δ -u 0 ).
Hence, by using successively Korn's inequality and Hölder's one, one can get

∇(u δ -u 0 ) L 2 (Ω) ≤ C ∇u 0 L 2 (D δ △D) , and
I 2 = δ µ 2 0 ∂D∩{h<0} h(x)V[D(v e 0 )](x) : D(v e 0 )(x)ds(x) + O(δ 1+ α 2(α+2) ),
where α > 0 and V[D(v e 0 )] deduced from (3.48). Therefore, with β = inf(η, α 2(α+2) ). To achieve the proof, we recall that

≺ (T δ -T )v 0 , v 0 ≻= I 1 + I 2 = δ µ 2 0 ∂D h(x)V[D(v e 0 )] (
λ δ -λ 0 = λ 0 λ δ (1/λ 0 -1/λ δ ) = λ 0 λ δ (μ 0 -μδ )
and we take into consideration that λ

0 λ δ = λ 2 0 + δ(λ 0 λ 1 ) + δ 2 (λ 0 λ 2 ) + • • • .
Now we consider the case of a multiple eigenvalue. Let the operators T and T δ be defined as in the proof of Theorem 3.2. We suppose that μ0 be a nonzero eigenvalue of T with multiplicity m ≥ 1. Then, for δ small, T δ has a set of m eigenvalues μj

µ 0 -μδ - 1 m m j=1 ≺ (T δ -T )u 0,j , u 0,j ≻ ≤ C (T -T δ )u 0,j 2 L 2 (Ω) , (3.49)
Moreover, for each j = 1, • • • , m, there is an eigenfunction u δ,j corresponding to μj δ such that u δ,j L 2 (Ω) = 1, and

u δ,j -u 0,j L 2 (Ω) ≤ C (T -T δ )| Ker(T -μ 0 I) . (3.50)
For more details, we can refer to [START_REF] Ammari | Layer Potential Techniques in Spectral Analysis[END_REF][START_REF] Khelifi | Asymptotic property and convergence estimation for the eigenelements of the Laplace operator[END_REF]. Suppose now that the eigenvalue λ 0 of the problem (3.34) is with geometric multiplicity m and let v 0,j ; j = 1, • • • , m be L 2 -orthonormal eigenfunctions corresponding to λ 0 . Let λ j δ be the eigenvalues of problem (3.37) generated by splitting from λ 0 and let v j δ be the associated eigenfunction (normalized with respect to L 2 ) such that v j δ → v 0,j as δ → 0. Based on (3.49) and (3.50), one can use similar approaches as in the proof of Theorem 3.2 to get the following result.

Corollary 3.1 Suppose that Ω ⊂ R 2 with class C 1,1 . Let λ 0 be an eigenvalue, of the problem (3.34), with (geometric) multiplicity m ≥ 1. Assume that we have (3.36), and the pressure still the same if we have (3.35). Then, there exist m eigenvalues λ j δ , j = 1, • • • , m (repeated according to their multiplicities) such that λ j δ → λ 0 as δ → 0 and the following asymptotic expansion holds:

1 λ 0 - 1 m m j=1 1 λ j δ = δ m m j=1 ∂D h(x)V[D(v e 0,j )](x) : D(v e 0,j )(x)ds(x) + O(δ 1+β ),
where β some positif constant and V given by Theorem 3.2.

Asymptotic formula for eigenfunctions and eigenpressures

This section is dedicated to develop asymptotic expansions for both eigenfunctions and eigenpressures associated to (1.4). Here we suppose that we have all hypothesis and conditions of the Section 1 and Section 2. Define the operator:

B δ (λ)ϕ(x) = (W(λ)ϕ)(Ψ -1 δ )(Ψ δ (x)),
where ϕ ∈ L 2 (∂Ω), Ψ δ given by Section 2.2, and W(λ) is the operator associated to hydrodynamic double layer potential [START_REF] Kohr | The interior Neumann problem for the Stokes resolvent system in a bounded domain in R n[END_REF][START_REF] Ladyzhenskaya | Mathematical theory of the viscous incompressible[END_REF][START_REF] Varnhorn | The Stokes equations[END_REF]. Then, the following main result holds.

Theorem 3.3 Suppose that we have (1.2) and let λ 0 be a simple eigenvalue of (1.1). Let A δ be the operator defined by (2.5), and M δ = A δ + B δ . Let O 0 be a bounded neighborhood of Ω in R 3 . Then there exists a constant δ 1 > 0 smaller than δ 0 such that the eigenfunction v δ corresponding to the eigenvalue, λ δ , in (H 1 (Ω δ )) 3 ∩ H(Ω δ ) can be chosen to depend holomorphically in (x, δ) ∈ O 0 ×] -δ 1 , δ 1 [. Moreover this eigenfunction satisfies the following asymptotic formulae

v δ (x) = v 0 (x) + n≥1 v n (x)δ n , (3.51)
where the function v 0 is the eigenfunction solution of (1.1) associated to λ 0 . The terms v n are computed from the Taylor coefficients of the operator valued function M δ and of those of the function a(δ) = (a ij (δ)) 1≤i,j≤3 .

Proof. From [START_REF] Kohr | The interior Neumann problem for the Stokes resolvent system in a bounded domain in R n[END_REF][START_REF] Medkova | Boundary value problems for the Stokes equations with jumps in open sets[END_REF] we deduce that there exist a continuous function ϕ(t, δ), which is analytic in R 2 ×] -δ 0 , δ 0 [ and such that

  2 (x, y). By using (3.22)-(3.23), we get that ri rj rm = β

  x) : D(v e 0 )(x)ds(x) + O(δ 1+ α 2(α+2) ), and by considering the above relation, (3.40) and (3.46) we obtain μ0 -μδ = δ μ2 0 ∂D h(x)V[D(v e 0 )](x) : D(v e 0 )(x)ds(x) + O(δ 1+β )

δ

  (counted according to their multiplicity) such that μj δ → μ0 for eachj = 1, • • • , m as δ → 0. Let μδ = 1 m m j=1 μjδ , and let {u 0,1 , u 0,2 , • • • , u 0,m } is an orthonormal basis for Ker(T -μ0 I), then as written in (3.40) there exists a constant C (independent of δ) such that

v

  δ (x) = S(λ δ )ϕ + W(λ δ )ϕ, x ∈ Ω (3.52) solves the eigenvalue problem(1.4). Moreover, the function given by U (δ)(x) = M(λ δ )ϕ(Ψ -1 , δ) satisfies the eigenvalue problem(1.4) in Ω δ with the boundary conditions:U (δ)| ∂Ω δ = 0. Here, M(λ δ )ϕ(Ψ -1 , δ) = M δ (λ)ϕ. Now, by (3.52), we deduce that v δ (x) = U (δ)(x) = M(λ δ )ϕ(Ψ -1 , δ) is jointly analytic with respect to (x, δ) in { x-Ψ δ=0 (y) ≤ z 0 }×]-δ 0 , δ 0 [,where z 0 is a positive constant. The function v δ (x) is jointly analytic in the variables (x, δ) ∈ O 0 ×] -δ 0 , δ 0 [. We shall now give the asymptotic expansion of the function v δ (x) when δ tends to 0. Integral equation (2.2) gives usv δ (x) = ∂Ω M (λ δ , |x -Ψ δ (y)|)ϕ(y, δ)|∇Ψ δ (y)|dσ(y),where M is the kernel of the operator M δ . The perturbed eigenvalue λ δ is in a small neighborhood of λ 0 for small values of δ. Then we have the following Taylor expansionM (λ δ ), |x -Ψ δ (y)|)|∇Ψ δ (y)| = M (λ 0 , |x -Ψ(y)|)|∇Ψ(y)| + k≥1 δ k M k (x, y),which holds uniformly in x ∈ O 0 and y ∈ ∂Ω. The analyticity of the function ϕ(y, δ) with respect to δ immediately gives ϕ(t, δ) = ϕ 0 (y) + k≥1 δ k ϕ k (y), uniformly in y ∈ ∂Ω. Substituting the last two asymptotic into (3.53) we findv δ (x) = v δ=0 (x) + k≥1 δ k [ k n=1 ∂Ω ϕ k-n (y)M n (x, y)dσ(y)].The next result provide us with the asymptotic expansion of the eigenpressures. Corollary 3.2 Suppose that we have all hypothesis of Theorem (3.3). Then the eigenpressures p δ solution of (1.4) have the following uniform asymptotic expansion:p δ (x) = p 0 (x) + n≥1 p n (x)δ n ,(3.53)where the function p 0 is the eigenpressure solution of (1.1) associated to λ 0 . The terms p n are computed from the Taylor coefficients λ n and v n = (v 1 n , v 2 n , v 3 n ) as follows:p n (.; x i ; .) = ∆v i n dx i + n k=0 λ k v i n-k dx i , where i = 1, 2, 3.Proof. From system (1.4) we have∇p δ = ∆v δ + λ δ v δ .Hence, we can expand the function p δ in powers of δ as we have done for λ δ and v δ . Moreover, we have:∂ i p δ = ∆v i δ + λ δ v i δ , for i = 1, 2, 3. (3.54) To get the Taylor coefficients p n , introduced in the formula (3.53), one can insert both asymptotic expansions (3.31) and (3.51) into relation (3.54), and integrate with the convenable variable.

Proof. Let λ δ be the eigenvalue of the problem (1.4). It is well known that λ δ is also simple such that λ δ → λ 0 as δ → 0. On the other hand, if we take the curl of the first equation in (1.4), we see that there exists a function w = w(v δ ) called vorticity associated to v δ such that:

That is, w is an eigenfunction of the negative Laplacian, but with homogeneous boundary conditions on the velocity v δ . Moreover, let u δ be the stream function for v δ given as in Lemma 2.10 of [START_REF] Kelliher | Eigenvalues of the Stokes operator versus the Dirichlet Laplacian in the plane[END_REF]. Then, w = ∆u δ and ∇u δ = 0 on ∂Ω. Since u δ is determined only up to a constant, we can then assume that u δ = 0 on ∂Ω. Thus, u δ satisfies the following eigenvalue problem for the Dirichlet biharmonic operator:

Note that Temam [START_REF] Temam | Theory and numerical analysis[END_REF] exploits the similar correspondence between the Stokes problem and the biharmonic problem in the proof of the regularity of solutions to the Stokes system and to justify several results. Moreover, as pointed out by Ashbaugh in [START_REF] Ashbaugh | On universal inequalities for the low eigenvalues of the buckling problem[END_REF], there is a similar correspondence between the eigenvalue problems for the Dirichlet Laplacian and system (3.32) with the boundary condition replaced by ∆u δ = 0. Then, one can exploit this correspondence to use the approach used in [START_REF] Ammari | Splitting of resonant and scattering frequencies under shape deformation[END_REF] in order to develop an asymptotic expansion for the eigenvalue.

On the other hand, it is well known [START_REF] Ammari | Layer Potential Techniques in Spectral Analysis[END_REF][START_REF] Ammari | Splitting of resonant and scattering frequencies under shape deformation[END_REF] that there exits a polynomial-valued function δ → Q δ (λ) of degree 1, analytic in ] -δ 0 , δ 0 [ and of the form:

such that the perturbation λ δ -λ 0 is precisely its zero. For the existence of Q δ one can follow the general approach used, for example, in [START_REF] Ammari | Splitting of resonant and scattering frequencies under shape deformation[END_REF] for the case of Laplace operator. Writing:

Then we have

Therefore, by (3.30) we have

On the other hand, for δ in a small neighborhood of 0, the following Neumann series converges uniformly with respect to λ in ∂D ǫ 0 :

where △ denotes the symmetric difference. It then follows by the Poincaré inequality that

Consequently, using the last inequality and the fact that |D δ △D| → 0 as δ → 0 and that ∇u 0 ∈ L 2 (Ω) we obtain that T δ → T pointwise as δ → 0 in L 2 (Ω).

Hence all hypotheses hold for the theorem of Osborn. Now if we set,

then according to the problem (3.38)(resp. (3.39)) we can see that ( µ δ , v δ ) (resp.( µ 0 , v 0 )) is eigenpair of T δ (resp. of T 0 ). So, a theorem of Osborn [START_REF] Osborn | Spectral approximation for compact operators[END_REF] yields

where C is independent of δ and v 0 is the solution of (3.34). Furthermore, if v δ is the solution to (3.37), then

According to [START_REF] Ammari | Reconstruction of small interface changes of an inclusion from modal measurements II: The elastic case[END_REF][START_REF] Boyer | Fabrie Mathematical tools for the study of the incompressible Navier-Stokes equations and related models[END_REF] we can extend the regularity results obtained by De Giorgi and Nash in the scalar case [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF] to get

where C is a positif constant and for d 0 > 0 we have

To compute the term ≺ (T δ -T )v 0 , v 0 ≻ appearing in (3.40), we may firstly recall that T v 0 = µ 0 v 0 and T δ v 0 = ũδ where ũδ is the solution to

From (3.42) and (3.43), we have

Now, as done in [START_REF] Ammari | Reconstruction of small interface changes of an inclusion from modal measurements II: The elastic case[END_REF] we put x t := x+th(x)ν(x) for x ∈ ∂D and t ∈ [0, δ]. It is clearly that the Jacobian determinant of the change of variables (x, t)

for δ small enough, and hence we get

). On the other hand, based on the result of Li and Nirenberg proved in [START_REF] Li | Estimates for elliptic systems from composite material[END_REF], the following gradient estimate holds:

where ũδ is the solution of (3.42), and the positive constant C is independent of δ.

Now using Lemma 2.2 in [START_REF] Ammari | Reconstruction of small interface changes of an inclusion from modal measurements II: The elastic case[END_REF], and the relations (3.43)-(3.46), one may obtain:

On the other hand, we use the gradient estimates (3.42) and (3.43) for ũδ and v 0 , to approximate

for δ sufficiently small. To proceed with our proof we may investigate the transmission conditions of v 0 along the interface ∂D. One can easily see from the equation in (3.34) that v 0 satisfies: v e 0 = v i 0 and (σ 0 D(v e 0 ))ν = (σ 1 D(v i 0 ))ν where v e 0 = v 0 | Ω\D and v i 0 = v 0 | D . Let τ be the unit tangential vector field to ∂D, and following the approach used in [START_REF] Ammari | Reconstruction of small interface changes of an inclusion from modal measurements II: The elastic case[END_REF], one can find that

where

Here I 2 is the (2 × 2) identity matrix, I 4 is the identity 4-tensor and the viscosity constants µ 0 , µ 1 are given in (3.33).

Applying the transmission conditions (3.47) for the function ũδ , one may get

. Now, regarding Lemma 3.2 in [START_REF] Ammari | Reconstruction of small interface changes of an inclusion from modal measurements II: The elastic case[END_REF], which still applied to Stokes case, on may approximate