Francesca Nanni
email: fnanni@ing.uniroma2.it

Giovanni Ruscito

Debora Puglia

Andrea Terenzi

J M Kenny

Gualtiero Gusmano

Effect of carbon black nanoparticle intrinsic properties on the self-monitoring performance of glass fibre reinforced composite rods

Keywords: Self-monitoring (B), polymer-matrix composites (A), pultrusion (E)

Self-monitoring composite rods, made of an internal conductive core surrounded by an external structural skin, were manufactured and tested. Both parts were made of glass fibre-epoxy Electrical conductivity was achieved in the inner core by incorporating as an alternative high surface area or low surface area carbon black in the resin. Self-monitoring performance was assessed by simultaneous mechanical and electrical resistance measurements. The aim was to correlate the electrical resistance variation to stress. Only one type of material showed appropriate self-monitoring properties, since increase of electrical resistance was recorded at increasing loading (both monothonic and cyclic tensile loading), while electrical resistance recovery at high loads was found in the other case. Calorimetric analysis, rheological measurements and SEM observations were carried out to explain this result. Filler dispersion seems to

Introduction

In past years, self-monitoring polymer composite materials were proposed as innovative monitoring systems, that can provide contemporarily structural and sensing properties [START_REF] Schulte | Load and failure analyses of CFRP laminates by means of electrical resistivity measurements[END_REF][START_REF] Chung | Structural Health monitoring by electrical resistance measurement Smart Mater[END_REF][START_REF] Muto | Hybrid composites with selfdiagnosis function preventing fatal fracture[END_REF]. Such materials gained particular interest in the field of civil engineering applications, where the use of polymer composite reinforcing rods has become more and more attractive, in order to overcome corrosion problems, typical of steel reinforcements. Usual polymers employed in these applications are epoxy, vinylesterepoxy, polyester, etc. [START_REF] Micelli | Durability of FRP rods for concrete structures Construction and[END_REF]. Polymer composite rods generally present linear elastic failures, i.e. a lack of ductility, which can lead to sudden fractures [START_REF] Wu | Self-Diagnosis of structures strengthened with hybrid carobfibre-reinforced polymer sheets Smart Mater[END_REF]. Therefore these reinforcements are usually applied together with health-monitoring systems, to prevent catastrophic failures. Both traditional (such as strain gauges, piezoelectrics [START_REF] Tracy | Identifying Impact Load in Composite Plates Based on Distributed Piezoelectric Sensors[END_REF]) and innovative monitoring systems (fibre optics, etc. [START_REF] Zhou | Smart FBG-OFBG bars and their applications in reinforced concrete beams in Structural Health Monitoring and Intelligent Infrastructures[END_REF][START_REF] Kabashima | Damage detection of satellite structures by optical fiber with small diameter[END_REF]), usually make use of sensors that, placed either inside or outside the concrete structure, are invasive, complicated, expensive and, in the case of external sensor, do not allow continuous monitoring throughout structure service life. To overcome such drawbacks, an alternative monitoring method has been proposed more recently, which makes use of selfmonitoring materials, providing both the structural and sensing functions [START_REF] Indada | Health monitoring of concrete structure using self-diagnosis materials[END_REF][START_REF] Nanni | Self-sensing CF-GFRP rods as mechanical reinforcement and sensors of concrete beams[END_REF][START_REF] Li | Sensors and actuators based on carbon nanotubes and their composites: a review[END_REF].

Such materials are usually realized with polymer composites, since they offer intrinsic versatility (i.e. possibility to include different phases within the matrix), ease of fabrication and low costs, which are attractive features for mass production, as in the case of civil engineering. The working principle of self-monitoring materials is based on the correlation between the change of electrical resistance of a conductive phase and the stress/strain occurring to the material [START_REF] Okuhara | Memorizing maximum strain in carbon-fiber-reinforced plastic composites by measuring electrical resistance under pre-tensile load[END_REF][START_REF] Nanni | Self-sensing polymer composite materials for civil applications[END_REF]. In the case of polymeric composites, usually the conductive phase is carbon either in the form of long fibres (CFRP) [START_REF] Wang | Interlaminar interface in carbon fibre polymer-matrix composites, studied by contact electrical resistivity measurement[END_REF][START_REF] Todoroki | Electrical resistance change of unidirectional CFRP due to applied load[END_REF][START_REF] Ho Kang | Self-diagnosis of damage in fibrous composites using electrical resistance measurements[END_REF] or particles (CPRP) [START_REF] Okuhara | Carbon-matrix composites with continous glass fiber and carbon black for mazimum strain sensing[END_REF][START_REF] Nanni | Assessment of self-diagnosis properties of CP_GFRP nanocomposites for civil applications[END_REF][START_REF] Okuhara | Fiber reinforced composites as self-sensing materials for memorizing damage histories[END_REF] or, more recently nanotubes [START_REF] Thostenson | Carbon nanotubi networks: sensing of distributed strain and damage for life prediction and self-healing[END_REF][START_REF] Gao | Coupled carbon nanotube network and acoustic emission monitoring for sensing of damage development in composites[END_REF]. In the first case, when dealing with unidirectional CFRP rods, carbon fibres can operate both as reinforcement and sensitive part and, therefore, are particularly applied in civil engineering applications, where they are usually used in combination with glass fibres to form hybrid composites (CF-GFRP), that can allow to reach some pseudo-ductility [START_REF] Bakis | Self monitoring, pseudo-ductile, hybrid FRP reinforcement rods for concrete applications[END_REF][START_REF] Krestis | A review of the tensile, compressive, flexural and shear properties of hybrid fibrereinforced plastics[END_REF]. From the monitoring point of view, this solution present low sensitivity at low strains (unless prestressing is provided [START_REF] Okuhara | Memorizing maximum strain in carbon-fiber-reinforced plastic composites by measuring electrical resistance under pre-tensile load[END_REF]) and generally low maximum electrical resistance variation (about 10%) until fracture. Therefore the authors in [START_REF] Nanni | Design, manufacturing and testing of sel-sensing carbon fibre-glass fibre reinforced polymer rods[END_REF] suggested to employ such system as non-continuous health monitoring, and to tailor the composite composition (i.e. glass/carbon fibre ratio) to use this material as a "guard sensor", generating an alarm warning at specific stress. Good self-monitoring results were, instead, obtained in CFRP laminates to monitor cracking and delamination at interlaminar interface [START_REF] Wang | Chung DDL Self-sensing attained in carbon-fiber-polymer-matrix structural composites by using the interlaminar interface as sensor Smart[END_REF]. The use of carbon particle as conductive phase, instead, has been more recently proposed and showed good self-monitoring potentialities. Okuhara et al in [START_REF] Okuhara | Self-diagnosis function of FRP containing electrically conductive phase[END_REF] reported that, differently from CFRP, the introduction of micrometer carbon flakes in the resin allows to achieve more sensitivity, in particular at low strain, and to perform continuous monitoring. This system presents an interesting "memory" function, in the sense that a residual electrical resistance can be found after cyclic loading, that can be somehow correlated to the maximum stress applied to the material. Moreover, Okuhara et al. in [START_REF] Okuhara | Development of conductive FRP containing carbon phase for self-diagnosis structures[END_REF] and Inada et al. al [START_REF] Indada | Health monitoring of concrete structure using self-diagnosis materials[END_REF] observed that carbon particle type and geometrical shape play a significant role in the self-monitoring behaviour. They showed that the use of graphite carbon flakes is particularly suitable for high sensitivity samples, while the use of spherical carbon black particles increase the material capability to memorize maximum applied load. The self-monitoring efficiency of carbon nanoparticles within polymer matrix encouraged and stimulated the research in the more innovative and promising field of carbon nanotube (CNT) loaded polymers [START_REF] Thostenson | Real-time in situ sensing of damage evolution in advanced fiber composites using carbon nanotube networks[END_REF][START_REF] Chou | Real-time in situ sensing of damage evolution in advanced fiber composites using carbon nanotube networks[END_REF][START_REF] Park | Strain-dependent electrical resistance of multi-walled carbon nanotube/polymer composite films[END_REF], whose penetration in industrial manufacturing is not shortly expected due to difficulty in CNT manipulation and processing. CNT loaded polymers in the form of film have shown [START_REF] Chou | Real-time in situ sensing of damage evolution in advanced fiber composites using carbon nanotube networks[END_REF][START_REF] Park | Strain-dependent electrical resistance of multi-walled carbon nanotube/polymer composite films[END_REF] to be very promising as strain sensing, but have to be used as external sensors, while good self-monitoring has been achieved in [START_REF] Thostenson | Real-time in situ sensing of damage evolution in advanced fiber composites using carbon nanotube networks[END_REF] in monitoring delamination of cross-ply laminates with CNT dispersed in the resin. More recently a very interesting work [START_REF] Wichmann | Piezoresistive response of epoxy composites with carbon nanoparticles under tensile load[END_REF] compared the self-monitoring performance of CB-epoxy and MWCNT-epoxy composites and evidenced that self-monitoring in CNT samples shows a distinct dependency of the electrical resistivity on mechanical load, that was attributed to the peculiar nanofiller fibre-like structure.

The different self-monitoring behaviour of CFRP and CPRP systems is related to their different conditions of electrical conductivity, which, in turn, are linked to the material microstructure. In the case of aligned long carbon fibres, the system can be assimilated to a bunch of conductive wires. The electrical current flows either in fibres direction and in transverse direction by means of bridge contacts between neighbouring fibres. This occurrence explains the low sensitivity of such system since, even when some carbon fibres are broken and the longitudinal current flow is obstructed, electrical conductivity is still insured by means of bridge contacts among broken fibres, that allow current flow in transverse direction [START_REF] Schulte | Load and failure analyses of CFRP laminates by means of electrical resistivity measurements[END_REF]. In the case of carbon particles, instead, the electrical model is that of a number of conductors randomly dispersed in an insulating matrix. The theory for such systems reports that conductivity is achieved when a percolation pattern is formed [START_REF] Kirkpatrik | Percolation and Conduction[END_REF][START_REF] Brosseau | Dielectric and microstructure properties of polymer carbon black composites[END_REF], i.e. a sufficient numbers of contacts between conductive particles is present to insure current flow. In this conditions, the system quickly changes from being insulator to being conductive. When such system undergoes increasing strain, separation between conductive particles occurs, resulting in higher electrical resistance.

Sensitivity is enhanced since the conductive particles gradually separate under strain, with a consequent continuous enhancement of resistivity. Moreover, beyond matrix yielding and after loading, a permanent separation between particles occurs, which generates the "memory effect". In this latter system, the conductivity and, hence the self-monitoring performance, depends on a variety of factors usually related to the peculiar matrix/filler intrinsic properties, such as filler content, intrinsic conductivity, surface area, geometrical shape, etc. [START_REF] Mallette | Carbon black-filled PET/HDPE blends: effect of CB structure on rheological and properties[END_REF][START_REF] Traina | Time temperature dependence of the electrical resistivity of high density polyethylene/carbon black composites[END_REF]. Some of these features affect conductivity since they interact with particle dispersion and formation of aggregates, particularly important when dealing with nanofillers. In fact, when carbon nanoparticles (CnP) are mixed to an insulating polymer matrix different mesostructures can appear, depending on the filler intrinsic properties. In particular, it is well known [START_REF] Balberg | A comprehensive picture of the electrical phenomena in carbon-black polymer composites[END_REF] that carbon nanoparticles with high surface area and Oil Absorption Number (OAN) present the socalled "high structure", i.e. highly branched aggregates, which leads to the formation of the conductive network at lower carbon content (low percolation threshold).

In this research two types of hybrid self-monitoring composite rods, made of an internal conductive core surrounded by an external structural part, were manufactured and fully characterized. Both the internal core and the external part were made of glass fibre-epoxy, nevertheless, electrical conductivity was achieved in the inner core by incorporating carbon nanoparticles within the resin. In particular, the manufactured selfmonitoring composite materials contain, as an alternative, two types of carbon black nanoparticles with different surface areas, OAN and particle size. The aim was to correlate the composite self-monitoring performance to the conductive filler properties, by characterizing the filler interaction with the epoxy matrix. Tensile tests were carried out, on both kind of samples, together with electrical resistance measurements, to assess the self-monitoring performance, while their microstructure was observed by field emission scanning electron microscope (FE-SEM). DSC and rheological measurements were performed in order to correlate the measured self-monitoring performance to the material microstructure of both systems. Prior to self-monitoring testing, percolation behaviour of both systems was assessed to find conductivity threshold.

Materials and methods

The manufactured composite materials consisted of an internal electrically conductive core surrounded by an insulating sheath (Figure 1).

Figure 1 , Table 1

Both parts are made of unidirectional glass fibres (Cofitech, Caselle, Italy, 475W, 2400 tex) in epoxy resin (pure epoxy resin with isofordiammine cycloalifatic hardener, SP system, Gurit, Newport, UK). The manufacturing process is hand-pultrusion. A twostep process was employed: first a glass fibre bundle was pulled through the epoxy resin previously loaded with carbon particles to gain the necessary electrical conductivity, further three additional glass fibres were pultruted around the central conductive core to obtain the insulating sheath and to insure good mechanical properties. Cure reaction was carried out at 60°C for 3h. Table 1 reports the main characteristics of the realized samples. In final specimens the internal conductive core was made longer than the external sheath to provide electrical contacts, that were applied at specimen ends using a highly conductive silver paint (Electrolube), to allow electric resistance variation measurements (digital multimiter Keithley DMM 2700). The electrical resistance of both wires and contacts between wires and specimens surface is included in the twopoints measurement scheme adopted. Nevertheless, preliminary electrical resistance measurements carried out with both two-points and four-points methods showed a difference in specimen resistance evaluation of around 1 that, being far lower than the specimens electrical resistance (table 4), can be considered negligible [START_REF] Park | Strain-dependent electrical resistance of multi-walled carbon nanotube/polymer composite films[END_REF]37].

Specimens ends were provided of external metallic cylindrical tabs for tensile testing (Figure 1a). In Figure 1d a particular showing sample gripping in tensile test jig, with the electrical contact above the metallic cylinder not affected by any gripping pressure.

The The CnP/resin mixture was prepared by adding the particles to the resin and mechanically stirring at 600 rpm for 1h. Successively the hardener was added to the mixture (38 phr) that was kept under mechanically stirring for further 5 min and sent to the pultrusion process. Two types of carbon nanoparticles (Printex XE 2b Evonik-Degussa, Essen, Germany and Super P, Timcal Ltd, Bodio, Switzerland) were used as an alternative to prepare the electrically conductive element: both are spherical carbon black powders with particle average diameters of respectively 30 and 40 nm, but different surface areas. Table 2 reports the main characteristics of the chosen CnP.

Table 2

Percolation was assessed by measuring electrical resistivity of samples with different carbon contents (Keithley DMM 2700) (Table 3). To this aim 10x10x3mm 3 CnP-epoxy samples [START_REF] Flandin | Anomalous percolation transition in carbon black -epoxy composite materials[END_REF] were cast in moulds, cured following the above reported procedure, and provided by electrical contacts made of silver paint on opposite sides. Different carbon loadings were chosen in the two types of samples due to the diverse CnP density and efficiency in reaching percolation. The goal was to obtain for the self-monitoring testing samples with similar initial electrical resistance (samples HSA_3 and LSA_4 in Table 3).

Table 3 Table 3

Microstructure of both conductive elements was observed by field emission scanning electron microscope (FE-SEM, Leo Supra 35) to find out the percolation pattern and evaluate filler dispersions. Samples for SEM analysis were microtomed and polished. Monotonic tensile and cyclic tests were carried out at 2 mm/min (Instron 5569J) on hybrid CnP-GFRP composite samples with filler content close to percolation (HSA_3 and LSA_4 samples). In the case of cyclic testing the load was progressively increased following the law: F i = F (i-1) + 250 (N), with F maximum applied load (N) and i, number of cycle. Together with the mechanical tests, electrical resistance variation was acquired (DMM 2700 Keithley Instruments) to assess the self-monitoring performance.

Calorimetric measurements were performed to evaluate the interaction between the filler and the pre-polymer, in order to verify whether it could affect the curing behaviour of the epoxy system, with particular attention to the final degree of cure of the material and on the glass transition of fully cured samples. Tests have been carried out using a differential scanning calorimeter (Mettler Toledo 822e, temperature scanning in the range -50°C/250°C, heating rate 10°C/min). Isothermal cure of the three systems (neat epoxy, HSA_3 and LSA_3) was also considered, in order to estimate the final degree of cure of samples, cured for different times at 30, 60 and 90°C.

Rheological tests were carried out with a rotational rheometer ARES with parallel plate geometry (φ = 25 mm, frequency sweep test between 0,01 and 100 rad/s at temperature of 30°C and strain of 0,03%), to evaluate and compare the viscosity of the two CNP/epoxy filled systems. In order to minimize the wall effects, the gap within the plates was maintained higher than 100 times the mean particle diameter.

Results and discussion

Percolation

Percolation curves of both systems are reported in Figure 2, while Table 4 reports more in detail samples electrical characteristics. It was found that in the two systems, percolation occurs at different CnP content, respectively at 2.2 %vol (HSA sample) and at 3.8 %vol (LSA sample). Such results are in good agreement (Table 5) with those obtained by applying Janzen equation [START_REF] Janzen | On the critical conductive filler loading in antistatic composites[END_REF] (eq.1):

OAN Vol C * * 4 1 1 %) ( ρ + = Φ (1)
which correlates the conductive filler content at percolation with carbon particles OAN and density ( ).

Table 5

The slight differences found between experimental and theoretical values are in the usual range, as reported in [40], when working with highly structured carbon fillers.

Percolation is affected by particles surface area: small surface areas, in fact, correspond to the formation of isolated and compact carbon particle mesostructures, the so-called "low structure" carbon black [START_REF] Kirkpatrik | Percolation and Conduction[END_REF], that difficultly interact to form a conductive network.

Particles with higher surface area, instead, form highly branched agglomerates that more easily interconnect each other [START_REF] Janzen | On the critical conductive filler loading in antistatic composites[END_REF]40], reaching percolation at lower filler content.

Surface area is in turn correlated to the particles diameter: the higher the former, the smaller the latter. The experimental results are consistent with all these considerations.

In fact, we found percolation to occur at lower filler content in the case of HSA nanoparticles due to their high surface area, low diameter and high OAN. Moreover, even above percolation, HSA specimens present higher conductivity at lower filler content.

Self-monitoring assessment

Figure 3, Figure 5a and Table 6 report the self-monitoring performance of a HSA_3 sample. The reported results show a very interesting self-monitoring behaviour, since a strict correlation between mechanical and electrical measurements was found, which make such material suitable for continuous monitoring. Nevertheless, it has to be remarked that this system offers limited sensitivity at low strain (Figure 3), that is however typical of carbon black filled systems [START_REF] Nanni | Assessment of self-diagnosis properties of CP_GFRP nanocomposites for civil applications[END_REF]. Sensitivity was evaluated at low strain (Figure 5c) by means of the gauge factor (GF), i.e. the ratio between electrical resistance variation and strain (( R/R 0 )/ ). In the case of HSA_3 it is around 6, which is higher than that of usual metal alloys used for foil strain gauges (that lies between 0.7 and 5) [START_REF] Park | Strain-dependent electrical resistance of multi-walled carbon nanotube/polymer composite films[END_REF]. In this system, the self-monitoring performance can be referred to the presence of a network of carbon particles that, prior to loading, allows goods dc electrical conductivity. Successively, the progressive increase of stress and, hence of strain, and/or the inset of damage causes particles separation, resulting in an increase of electrical resistance. Under increasing loading, in fact, always less conductive pathways are available for current flow, with a consequent decrease of conductivity. In previous work [START_REF] Okuhara | Memorizing maximum strain in carbon-fiber-reinforced plastic composites by measuring electrical resistance under pre-tensile load[END_REF] it was evidenced that the abovementioned mechanism of selfmonitoring behaviour, implying particle separation at increasing loads, plays an important role in determining system sensitivity (i.e. electrical resistance variation with stress or strain) too. In particular sensitivity at high loads showed to decrease at increasing carbon content, where the high amount of conductive particles always insure enough electrical current flow. Therefore, in this research, specimens with carbon content close to that of percolation were chosen. The good self-monitoring performance is kept even under progressive cyclic loading, where, moreover, the presence of a clear residual electrical resistance after few cycles testifies the ability of such system to memorize maximum applied load, in agreement with what reported in the cited work [START_REF] Indada | Health monitoring of concrete structure using self-diagnosis materials[END_REF][START_REF] Okuhara | Carbon-matrix composites with continous glass fiber and carbon black for mazimum strain sensing[END_REF].

Very poor self-monitoring results were found, instead, in LSA_4 samples (Figure 4, Figure 5b and Table 6). In monotonic tensile tests, in fact, the electrical resistance measurements show a self-monitoring behaviour up to about 5000N, above which an evident decrease of electrical resistance was recorded, even under increasing loading.

Even system sensitivity at low strain (GF reported in Figure 5 c) is unsatisfying, being around 3, almost the half of that found in the case of HSA_4 sample and in the same range of that of metal alloys. Besides, some loading drops were recorded in the monotonic tensile test (Figure 4), that most probably have to be referred to some debonding occurring between the CnP sensitive part and the external glass skin. The phenomenon of electrical resistance recovery at high loads was recorded even under cyclic loading, where such occurrence is even more evident: the electrical resistance variation progressively decreases its amplitude, starting from 4500N, up to a dramatic change at the final cycles, where it is not anymore able to follow the cyclic loading. The recorded electrical resistance recovery can be explained only as an enhancement of electrical conductivity at high loads, which, considering the peculiar mechanism of electrical conductivity in such samples, means that particles, formerly separated, come near and a uniform distribution of carbon nanoparticles is again achieved. To explain the surprising behaviour found in the case of LSA samples two possible, even concomitant, mechanisms were hypothesized. The first one is related to the degree of filler dispersion within the matrix. An inadequate dispersion implies the presence of carbon nanoparticles agglomerates that can collapse under high loads, releasing carbon nanoparticles (i.e. a larger number of conductive elements dispersed in the matrix), thus increasing system conductivity. Nevertheless, since this behaviour was not observed in the case of HSA_samples, it should be presumed that, in this system, a more efficient filler dispersion was achieved, resulting in a more uniform microstructure. If this hypothesis is correct, this is probably due to the nanoparticles intrinsic properties. A second possible explanation could be found in a different interaction between HSA and LSA nanoparticles and the epoxy resin. It can be supposed that LSA particles influence the curing process, not allowing to reach perfect cure. Under these conditions a certain grade of viscoelasticity could remain in the material, that can be responsible of nanoparticles flow at high loads, with consequent internal rearrangement and enhancement of conductivity due to the increased number of contacts among nanoparticles. In order to verify both theories, DSC, rheological measurements and specimens microstructure investigations were carried out, whose results are reported in the following subsections.

DSC Measurement

Calorimetric measurement were performed, in order to evaluate the curing behaviour of the two HSA_3 and LSA_4 self-monitoring systems. DSC measurements were carried out by dynamic heating, under the above described conditions, respectively on neat resin and resin loaded with the two nanopowders (Table 7). Tg of cured systems was evaluated too (Table 8) .

Table 7

To complete the DSC analysis, isothermal heating measurements were carried out at 30°C, 60°C and 90°C. Table 8 reports the results, in which the final degree of cure α (defined as the ratio between isothermal heat and total dynamic heat of reaction) is calculated for every system. The measurements of heat of reaction for the filled systems (epoxy + HSA and epoxy + LSA), both in dynamic and isothermal tests, were done normalizing the experimental value to the specific nanoparticles content.

Table 8

The thermal analysis demonstrated that the curing behaviour of all systems is very similar, even if in the case of filled systems the heat produced is reduced of an amount at least equivalent to the percentage of filler content. Moreover, in agreement with literature [START_REF] Valentini | Dynamics of amine functionalized nanotubes/epoxy composites by dielectric relaxation spectroscopy[END_REF][START_REF] Tao | Effects of carbon nanotube fillers on the cuuring processes of epoxy resin-based composites[END_REF][START_REF] Chung | Calorimetric study of the effect of carbon fillers on the curing epoxy[END_REF], it was found that the presence of thermally conductive filler decreases peak temperature of few degrees [START_REF] Xie | Cure Kinetic Study of Carbon Nanofibers/Epoxy Composites by Isothermal DSC[END_REF]. The degree of cure is higher at 60°C than 30°C, as expected, while the decrease of conversion at 90°C can be explained considering the plausible evaporation of the hardener at the high temperatures. In any case, no outstanding differences were found in the final degree of cure of samples containing the two nanoparticles, at least not enough to justify the different selfmonitoring behaviour due to the presence of uncured areas in LSA specimens.

Rheological Measurement

Rheological measurements were carried out in order evaluate the efficiency of carbon nanoparticles dispersion of HSA_3 and LSA_4 samples [START_REF] Khare | Rheology of Confined Polymer Melts[END_REF][START_REF] Lakdawala | Rheology of Polymers Containing Carbon Black[END_REF]. In Figure 6 complex viscosity ( * = ' -i '' ) as a function of angular frequency ( ) is reported. As commonly known [START_REF] Gandhi | Dynamic and Mechanical Behaviour of Polymer Containing Carbon Black[END_REF] ,

' and '' are respectively given by the ω η

' ' ' G = and ω η ' ' ' G = (3)
with G ' elastic or storage modulus and G '' viscous or loss modulus. G' and G'' for samples with different nanofillers are plotted in Figures 7.

Figure 7

The results show that the two specimens present pretty much the same viscosity, nevertheless, considering that HSA_3 sample contains a much lower filler content, it is possible to conclude that it presents a more uniform filler dispersion. The literature [START_REF] Wagener | A rheological method to compare the degree of exfoliation of nanocomposites[END_REF] suggested a method to quantify the dispersion level of nanoparticles in a polymeric matrix with the evaluation of the shear thinning effect in polymer-nanoparticles viscosity curves. Clear correlation between the shear thinning exponent n and the tensile modulus can be done: the higher the exponent n, the more efficient and well dispersed is the reinforcement of the respective composite. Higher values of viscosity and G'' for the HSA system confirm that, for the CnPs dispersions, higher surface area of HSA particles gives better dispersion of the nanoparticles in the epoxy system.

Microstructure

SEM micrographs of HSA_3 and LSA_4 bulk specimens (Figure 8), taken following the suggestions reported in [START_REF] Kovacs | Analyzing the quality of carbon nanotube dispersions in polymers using scanning electron microscopy[END_REF], confirm that the two systems present different microstructures. In particular, a more uniform filler distribution is achieved in the case of HSA_3 specimens, where the dark areas, related to the presence of carbon nanoparticles, are well dispersed throughout the specimen. LSA microstructure, instead, is composed of more coarse and compacted nanoparticles agglomerates surrounded by neat resin (white areas) that, anyhow, form a continuous patter that allow electrical conductivity. These observations strengthen the theory that considers filler dispersion the major feature affecting self-monitoring properties, more that its interaction with the resin. In the case of LSA_4 samples, in fact, carbon nanoparticles form "low structure" aggregates, with few connections among each others. In our opinion such aggregates under high loads collapse releasing nanoparticles that start to contribute to electrical conductivity and hence, leading to the electrical resistance recovery.

Conclusions

In this paper, two self-monitoring hybrid composite materials, containing different carbon nanoparticles as conductive phase, were prepared and tested. The results showed that only HSA samples containing high surface area nanoparticles show true selfmonitoring behaviour, while low surface area nanoparticles are not suitable for such applications, since electrical resistance recovery was found at high loads. DSC measurements evidenced that both types of nanoparticles do not interact with resin cure.

Rheological measurements demonstrated that HSA samples show a more uniform filler dispersion, while large aggregates are present in the case of LSA ones. This occurrence could be responsible for the electrical resistance recovery, due to aggregates breakage at high loads, with consequent release of a number of carbon nanoparticles in the matrix, that increase electrical conductivity. SEM observations confirm the different microstructures of the two types of specimens, validating this theory.

Finally, it is possible to conclude that the main result of the performed research is that conductive filler dispersion and its intrinsic properties are very important features when aimed to prepared self-monitoring materials. In particular, the use of nanofillers with high surface areas, high OAN and low particle dimensions are recommended to achieve a reliable self-monitoring system.

LIST OF CAPTIONS

Figure 2 ,

 2 Figure 2 , Table 4

Figure 3

 3 Figure 3

Figure 4

 4 Figure 4

Figure 5 ,

 5 Figure 5 , Table 6

Figure 6

 6 Figure 6

Figure 8

 8 Figure 8

Figure 1 Figure 2 Figure 3 Figure 4

 1234 Figure 1 Error! Main Document Only. (a) Sketch of hybrid CnP-GFRP, (b) sample cut and open to show inner part and (c) SEM micrograph of the internal conductive core

Figure 5

 5 Figure 5 Stress/strain curve (dotted light gray line) and R/R 0 % /strain curve (dark gray line)

Figure 6

 6 Figure 6Viscosity of HSA_3 and LSA_4 samples as a function of angular frequency.

Figure 7 GFigure 8

 78 Figure 7 G' and G'' of HSA_3 and LSA_4 samples as a function of angular frequency

Figure 1 Figure 2 Figure 3 Figure 4

 1234 Figure 1 (a) Sketch of hybrid CnP-GFRP, (b) sample cut and open to show inner part, (c) SEM

Figure 5 Figure 6 Figure 7 GFigure 8

 5678 Figure 5 Stress/strain curve (dotted light gray line) and ΔR/R 0 % /strain curve (dark gray line) for a) HSA_3 sample, b) LSA_4 sample. Electrical resistance variation against strain up to 1%strain are reported in figure

Table 1

 1 TABLES Specimens main characteristics

	Internal	Total	Specimens	Fibre volume fraction	Fibre volume
	conductive core	diameter	length (mm)	in conductive core	fraction in external
	diameter (mm)	(mm)			sheaths
	1.3±0.2	3±0.4	200±8	0.53±0.04	0.62±0.05

Table 2

 2 Carbon nanoparticles main characteristics

	Carbon nanoparticles Printex XE 2B	Supplier Degussa	Surface area (BET) [m 2 /g] 1000	OAN (ASTMD2414-04) [cm 3 /g] 3.8	Density ( ) 130 [g/dm 3 ](Pour ASTM D1513)	Particle size [nm] 30
	Super P	Timcal	62	2.9	160 [kg/m 3 ](in the bag, Int. method26)	40

Table 3

 3 Carbon nanoparticles content in manufactured samples

	CnP type	Acronym	Sample #	CnP (% Vol)
			HSA_1	1.2±0.15
	Xe-2B	HSA_# (High Surface Area)	HSA_2 HSA_3 HSA_4 HSA_5	1.7±0.17 2.2±0.12 2.7±0.16 3.7±0.22
			HSA_6	4.7±0.18
			LSA_1	2±0.13
	Super P	LSA_# (Low Surface Area)	LSA_2 LSA_3 LSA_4 LSA_5	3±0.18 3.5±0.2 3.8±0.23 4.5±0.28
			LSA_6	6±0.38

Table 4

 4 Electrical resistivity measurements for HSA and LSA samples (measurements repeated on three samples of each type)

		HSA		LSA
	Sample # Resistivity [ *cm]	Sample #	Resistivity [ *cm]
	HSA_1	Insulator	LSA_1	Insulator
	HSA_2	Insulator	LSA_2	Insulator
	HSA_3	495±50	LSA_3	9850±1580
	HSA_4	304±15	LSA_4	470±59
	HSA_5	275±17	LSA_5	115±7.8
	HSA_6	89±3.5	LSA_6	70±6

Table 5

 5 Percolation Threshold obtained by Janzen Equation and experimental data for both

		systems
	Sample	Φ Φ Φ Φ C (Vol %) Janzen Value	Φ Φ Φ Φ C (Vol %) Experimental Value
	HSA_3	3.3	2.2
	LSA_4	4.3	3.8

Table 6

 6 Main mechanical and self-monitoring results of HSA_3 and LSA_4 type samples (tests repeated on 3 different specimens of each type)

	Sample	Initial Electrical Resistance (kΩ Ω Ω Ω)	Maximum Load (N)	Specimen cross section (mm 2 )	Tensile strength (MPa)	Maximum Strain (%)	Electrical Resistance Variation at fracture(%)	Strain @ 1000N (%)	Electrical Resistance Variation @1000N(%)
	HSA_3	290±16	7365±203	21±0.7	347±21	3±0.36	32 ±4.5	0.24±0.05	0.8±0.3
	LSA_4	330±20	6882±110	22±0.8	313±16	4.1±0.55	13 * ±5.5	0.33±0.06	0.9±0.4

*

Maximum electrical resistance variation during test =16.44%±1.6 @ 5000N

Table 7

 7 Results of dynamic heating of HSA and LSA systems.

		∆ ∆ ∆ ∆H (J/g)	T peak (°C) T g (°C)
	Epoxy	512.90±11. 3 96.7±1.1 96.7±1.0
	Epoxy + HSA_3 415.04±9.8	95.2±0.8 96.4±0.9
	Epoxy + LSA_4	401.01±0.3	94.2±1.2 95.8±1.1

Table 8

 8 Results of isothermal heating of HSA and LSA systems.

	T iso = 30°C	T iso = 60°C	T iso = 90°C


Habib Ammari

Hajer Bahouri

David Dos

Santos Ferreira

Isabelle Gallagher
email: isabelle.gallagher@math.jussieu.fr

Isabelle Gallagher Stability

Stability estimates for an inverse scattering problem at high frequencies

Keywords: 2000 Mathematics Subject Classification. 35R30, 35B30. Key words and phrases. Inverse scattering problem, Lipschitz stability, resolution

Introduction

The first aim of this paper is to establish Lipschitz stability results for the inverse scattering problem of determining the low-frequency component (lower than the operating frequency) of the compactly supported potential from scattering or near-field measurements. It is known that, in general, the problem is exponentially unstable [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF][START_REF] Alessandrini | Examples of instability in inverse boundary-value problems[END_REF][START_REF] Alessandrini | Open issues of stability for the inverse conductivity problem[END_REF][START_REF] John | Continuous dependence on data for solutions of partial differential equations with a prescribed bound[END_REF][START_REF] Mandache | Exponential instability in an inverse problem for the Schrödinger equation[END_REF]. However, taking advantage of a priori information may improve stability and give accurate reconstruction algorithms [START_REF] Ammari | Direct reconstruction methods in ultrasound imaging of small anomalies[END_REF][START_REF] Ammari | Reconstruction of small inhomogeneities from boundary measurements[END_REF][START_REF] Ammari | The generalized polarization tensors for resolved imaging. Part II: Shape and electromagnetic parameters reconstruction of an electromagnetic inclusion from multistatic measurements[END_REF]. The Lipschitz stability results proved in this paper together with the recent analysis of the local convergence of the nonlinear Landweber iteration in [START_REF] De Hoop | Local analysis of inverse problems: Hölder stability and iterative reconstruction[END_REF] show that the low-frequency component of the potential can be determined from the data in a linearly stable way. Moreover, they precisely quantify the resolution limit, which is defined as the characteristic size of the smallest oscillations in the potential that can be stably recovered from the data. Since Rayleigh's work, it has been admitted that the resolution limit in inverse scattering is of order π over the operating frequency [START_REF] Born | Principles of Optics[END_REF]. This is nothing else than a direct application of the uncertainty principle in inverse scattering [START_REF] Bertero | Resolution and super-resolution in inverse diffraction[END_REF][START_REF] Chen | Inverse scattering via Heisenberg's uncertaintly principle[END_REF][START_REF] Donoho | Uncertaintly principles and signal recovery[END_REF][START_REF] Slepian | Some comments on Fourier analysis, uncertainty and modeling[END_REF]. It is well-known that if the support of the potential is a point support, then the reconstructed location of the point potential from the scattering data has finite size of order of the Rayleigh resolution limit [START_REF] Ammari | An Introduction to Mathematics of Emerging Biomedical Imaging[END_REF][START_REF] Bertero | Resolution and super-resolution in inverse diffraction[END_REF]. Having this in mind, the results of this paper prove that the Fourier transform of the potential can be reconstructed in a linearly stable way for all frequencies (dual variable to the space one) smaller than the operating frequency, and therefore, justify the notion of resolution limit. More intriguingly, again in view of [START_REF] De Hoop | Local analysis of inverse problems: Hölder stability and iterative reconstruction[END_REF], they prove that the stability of the reconstruction of the potential increases at high operating frequencies.

The second aim of the paper is to show that infinite resolution can be achieved from near-field measurements. Here, the near-field operator approximates Sommerfeld's radiation condition and is equivalent to the measurements of the Cauchy data at a finite distance. Moreover, if the potential is supported near the boundary, then infinite resolution can be achieved in the monotone case. In fact, a Lipschitz stability result holds for both the low and high frequency components of the potential. It should be noted that the scattering amplitude can be recovered from the near-field operator. However, approximating the near-field operator from the scattering amplitude is a severely ill-posed problem [START_REF] Karp | A convergent 'farfield' expansion for two-dimensional radiation functions[END_REF][START_REF] Novikov | A multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0[END_REF][START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF][START_REF] Taylor | Estimates for approximate solutions to acoustic inverse scattering problems[END_REF]] and therefore cannot be of any practical and realistic use. It was shown in [START_REF] Karp | A convergent 'farfield' expansion for two-dimensional radiation functions[END_REF] that in order to compute the near-field operator from the scattering amplitude one needs to differentiate the scattering amplitude an infinite number of times.

The results of this paper extend to medium scattering the recent results in [START_REF] Ammari | Multistatic imaging of extended targets[END_REF][START_REF] Ammari | Resolution and stability analysis in full aperture, linearized conductivity and wave imaging[END_REF][START_REF] Zhao | Analysis of the response matrix for an extended target[END_REF], where a stability and resolution analysis was performed for linearized conductivity and wave imaging problems. They can be also used to justify the hopping (or continuation in the frequency) reconstruction algorithms proposed in [START_REF] Bao | Regularity and stability for the scattering map of a linearized inverse medium problem[END_REF][START_REF] Chen | Inverse scattering via Heisenberg's uncertaintly principle[END_REF][START_REF] Chen | On the inverse scattering problem for the Helmholtz equation in one dimension[END_REF][START_REF] Coifman | An improved operator expansion algorithm for direct and inverse scattering computations[END_REF].

In connection with our results in this paper, we also refer to the works by Isakov [START_REF] Isakov | Increased stability in the Cauchy problem for some elliptic equations[END_REF] and Isakov and Kindermann [START_REF] Isakov | Subspaces of stability in the Cauchy problem for the Helmholtz equation[END_REF], Bao, Lin, and Triki [START_REF] Bao | A multi-frequency inverse source problem[END_REF][START_REF] Bao | Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data[END_REF], Nagayasu, Uhlmann and Wang [START_REF] Nagayasu | Depth dependent stability estimates in electrical impedance tomography[END_REF][START_REF] Nagayasu | Increasing stability in an inverse problem for the acoustic equation[END_REF], as well as Derveaux, Papanicolaou, and Tsogka [START_REF] Derveaux | Resolution and denoising in near-field imaging[END_REF]. In [START_REF] Isakov | Increased stability in the Cauchy problem for some elliptic equations[END_REF][START_REF] Isakov | Subspaces of stability in the Cauchy problem for the Helmholtz equation[END_REF], an evidence of increasing stability in wave imaging when frequency is growing was given. In [START_REF] Bao | A multi-frequency inverse source problem[END_REF], stability estimates for the inverse source problem were established and the conversion of the logarithmic type stability to a Lispchitz one first proved. Numerical results to illustrate the stability of the source reconstruction problem were presented in [START_REF] Bao | Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data[END_REF]. In [START_REF] Bao | On the stability of an inverse problem for the wave equation[END_REF], Lipschitz stability estimates for the time-dependent wave equation were obtained. In [START_REF] Nagayasu | Depth dependent stability estimates in electrical impedance tomography[END_REF], a stability estimate for a linearized conductivity problem was derived and its dependence on the depth of the inclusion highlighted. In [START_REF] Nagayasu | Increasing stability in an inverse problem for the acoustic equation[END_REF], it is shown that the ill-posedness of the inverse acoustic problem decreases when the frequency increases and the stability estimate changes from logarithmic type for low frequencies to a Lipschitz type for large frequencies. In [START_REF] Derveaux | Resolution and denoising in near-field imaging[END_REF], the enhancement of resolution in the near-field was studied and numerically illustrated. Our results in this paper confirm these important observations in a quite general situation and precisely quantify them.

Our paper is organized as follows. Section 1 is devoted to the stability of the reconstruction of the potential from the scattering amplitude (called also far-field pattern) in the high frequency regime. Theorem 2 proves that the low-frequency component of the potential can be determined in a stable way from the scattering amplitude. The threshold frequency determines the resolution limit. Section 2 extends the results of Section 1 to the nearfield measurements. Theorem 3 shows that the same results as those in Section 1 hold for reconstructing the potential from measurements of the near-field operator. In Section 3 we show that we gain infinite resolution for potentials supported near the boundary. If the potential is supported near the boundary, then infinite resolution can be achieved in the monotone case. Theorem 4 provides a Lipschitz stability result for both the low and high frequency components of the potential. Finally, in Appendix A, we provide useful results on Bessel's functions.

Finally, we mention that the letter C will be used to denote a universal constant which may vary from line to line. We also use A B to denote an estimate of the form A ≤ CB for some constant C. We also use the classical notation x = 1 + |x| 2 .

1. Far field pattern 1.1. Definitions and notations. Let Ω be a bounded domain in the Euclidean space R d of dimension d ≥ 2, let q ∈ C ∞ 0 (R d ) be a real-valued potential supported in Ω. We use the classical notation D = -i∂ for derivatives and consider the Helmholtz equation with potential

D 2 u -λ 2 u + qu = 0 (1.1) at frequency λ ∈ R * + := R + \ {0}.
Plane waves e iλx•ω propagating along the direction ω in S d-1 are solutions of the free Helmholtz equation

D 2 u -λ 2 u = 0. (1.2)
Here S d-1 denotes the unit sphere in R d . More generally, plane waves generate the set of solutions in the space of tempered distributions, S ′ (R d ), of the free Helmholtz equation: all solutions of (1.2) with polynomial growth are superpositions of elementary plane waves e iλx•ω when ω varies on the sphere S d-1 . It will be useful later on to adopt Melrose's notation in [START_REF] Melrose | Geometric scattering theory[END_REF] to designate these solutions: for g ∈ C ∞ (S d-1 ), we shall write

Φ 0 (λ)g(x) = S d-1 e iλx•ω g(ω) dω.
Obviously Φ 0 (λ)g is a solution of (1.2) which belongs to S ′ (R d ).

To guarantee the uniqueness of solutions to the Helmholtz equation (1.1), one can impose conditions on the behavior of solutions at infinity. More precisely, we are interested in solutions which can be decomposed

u = e iλx•ω + u scat = u in + u scat (1.3)
as the sum of an incoming planar wave and a scattered wave satisfying Sommerfeld's radiation condition

∂ ∂|x| -iλ u scat = o 1 |x| d-1 2
as |x| → +∞, (1.4) uniformly with respect to the direction θ = x |x| at fixed frequency λ ∈ R * + . The following result from [START_REF] Melrose | Geometric scattering theory[END_REF]Lemma 2.4] holds.

Proposition 1.1. There exists a unique solution to the Helmholtz equation (1.1) with potential q ∈ C ∞ 0 (Ω) of the form (1.5) ϕ q (x, ω, λ) = e iλx•ω + ϕ scat q (x, ω, λ), where the scattered wave ϕ scat q satisfies Sommerfeld's radiation condition (1.4) and which is given by

(1.6) ϕ scat q = -R q (λ)(e iλx•ω q).
Here R q denotes the meromorphic continuation of the perturbed resolvent. Furthermore ϕ q depends smoothly on (x, ω, λ) ∈ R d × S d-1 × R * + and is bounded.

We choose to denote Φ q (λ)g = S d-1 ϕ q (x, ω, λ)g(ω) dω the operator with kernel ϕ q (x, ω, λ) given by (1.5).

Theorem 1. The scattered wave in the solution (1.3) to the Helmholtz equation (1.1) given by Proposition 1.1 assumes the form

ϕ scat q (x) = e iλ|x| |x| d-1 2 a q x |x| , ω, λ + O 1 |x| d+1 2 as |x| → +∞, (1.7)
where a q is a smooth function on

S d-1 × S d-1 × R * + and a q x |x| , ω, λ = - 1 2iλ λ 2πi d-1 2
q(y)ϕ q (y, ω, λ)e -iλ x |x| •y dy.

Proof. We consider the Green function G λ corresponding to the free Helmholtz equation

(D 2 y -λ 2 )G λ (x, y) = δ(x -y), (1.8)
subject to Sommerfeld's radiation condition (1.4), with δ being the Dirac delta function. Let R > 0 be large enough so that the ball of radius R contains the support of q. By definition of the Green function, for all |x| ≤ R we have

u(x) = |y|≤R (D 2 y -λ 2 )G λ (x, y)u(y) dy,
and if u is a solution to the Helmholtz equation (1.1) we deduce by Green's formula that for any

|x| ≤ R u(x) = - |y|≤R G λ (x, y)q(y)u(y) dy - |y|=R ∂G λ ∂r (x, y) u(y) -G λ (x, y) ∂u ∂r (y) dσ(y).
Along the same lines, it is possible to derive a similar identity for the plane wave

u in (x) = e iλω•x u in (x) = - |y|=R ∂G λ ∂r (x, y) u in (y) -G λ (x, y) ∂u in (y) ∂r (y) dσ(y),
taking into account the fact that (D 2λ 2 )u in = 0. Subtracting the two identities gives the following representation formula for the scattered wave

u scat = u -u in u scat (x) = - |y|≤R G λ (x, y) q(y) u(y) dy - |y|=R ∂G λ ∂r (x, y) u scat (y) -G λ (x, y) ∂u scat ∂r (y) dσ(y).
The Green function of the free Helmholtz equation is explicitly given by

G λ (x, y) = 1 4i λ 2π d-2 2 |x -y| -d-2 2 H (1) d/2-1 λ|x -y| , (1.9)
where H

G λ (x, y) = 1 2iλ λ 2πi d-1 2 e iλ|x-y| |x -y| -d-1 2 1 + O 1 λ|x -y| . (1.10) Since we have |x -y| = |y| - x • y |y| + O |x| 2 |y|
we find for |y| large enough and fixed λ:

e iλ|x-y| = e iλ |y| e -iλ y |y| •x 1 + O |x| 2 |y| .
We therefore obtain for fixed x, λ and large R = |y|

G λ (x, y) = 1 2iλ λ 2πi d-1 2 |y| -d-1 2 e iλ |y| e -iλ y |y| •x 1 + O 1 R . (1.11)
Analogously, we have

∂G λ ∂r (x, y) = iλc d (λ)|y| -d-1 2 e iλ|y| e -iλ y |y| •x 1 + O 1 R , with c d (λ) = 1 2iλ (λ/2πi) d-1
2 . This leads to

u scat (x) = - |y|≤R G λ (x, y)q(y)u(y) dy + c d (λ) R d-1 2 |y|=R ∂u scat ∂r -iλu scat e iλR-iλ y |y| •x 1 + O 1 R dσ(y).
Sommerfeld's radiation condition implies that the second right-hand side term tends to zero when R tends to infinity, so we get

u scat (x) = -G λ (x, y)q(y)u(y) dy .
Using once again the asymptotic formula (1.11) together with the fact that the Green function is symmetric we get

u scat (x) = - 1 2iλ λ 2πi d-1 2 e iλ|x| |x| d-1 2 q(y)u(y)e -iλ x |x| •y 1 + O |y| 2 |x| dy.
To summarize, we have that

u scat (x) = e iλ|x| |x| d-1 2 a q x |x| , ω, λ + O 1 |x| d+1 2 , with a q x |x| , ω, λ = - 1 2iλ λ 2πi d-1 2 q(y)u(y)e -iλ x |x| •y dy.
This proves that the scattered part of any solution of Helmholtz' equation subject to Sommerfeld's radiation condition takes the form announced in Theorem 1.

Definition 1.2. We define the scattering amplitude associated with the po-

tential q ∈ C ∞ 0 (R d ) by the smooth function a q : S d-1 × S d-1 × R * + → C given by a q (θ, ω, λ) = - 1 2iλ λ 2πi d-1 2
q(y)ϕ q (y, ω, λ)e -iλθ•y dy. (1.12)

We denote

A q (λ)g(θ) = S d-1 a q (θ, ω, λ)g(ω) dω, θ ∈ S d-1 ,
the corresponding operator with kernel a q (θ, ω, λ).

It is easy to get an asymptotic expansion of

Φ q (λ)g(x) = S d-1 e iλx•ω + ϕ scat q (x, ω, λ) g(ω) dω
as |x| → +∞ using the stationary phase and Theorem 1:

(1.13) Φ q (λ)g = 2π λ|x| d-1 2
e -iλ|x| e i(d-1) π 4 g( -θ)

+ e iλ|x| e -i(d-1) π 4 g(θ) + λ 2π d-1 2 A q (λ)g(θ) + O 1 |x| with θ = x/|x| ∈ S d-1
. The operator which maps the coefficient of e -iλ|x| into the coefficient of e iλ|x| is given by

g(-θ) → i -d+1 g(θ) + λi 2π d-1 2 A q (λ)g(θ) .
This is, after renormalization and composition with the antipodal map, the scattering matrix [START_REF] Melrose | Geometric scattering theory[END_REF].

Definition 1.3. The scattering matrix is the operator given by

S q (λ) = Id + λi 2π d-1 2 A q (λ).
Integration by parts allows to relate the values of the potential inside the domain with the scattering matrix S q (λ).

Lemma 1.4. We have the following identities

(q 1 -q 2 )u 1 u 2 dx = -2iλ 2π λ d-1 S d-1 g 1 g 2 -S q 1 (λ)g 1 S q 2 (λ)g 2 dω, (q 1 -q 2 )u 1 u 2 dx = -2iλ 2π λi d-1 S d-1 ǧ2 S q 1 (λ)g 1 -ǧ1 S q 2 (λ)g 2 dω,
for any pair of solutions 1

u 1 = Φ q 1 (λ)g 1 , u 2 = Φ q 2 (λ)g 2 ,
to the Helmholtz equations (1.1) related to the potentials q 1 , q 2 .

Proof. Let R be large enough so that the ball of radius R contains the support of both potentials q 1 , q 2 . By Green's formula, we have

|x|≤R (q 1 -q 2 )u 1 u 2 dx = |x|=R ∂ r u 1 u 2 -u 1 ∂ r u 2 dσ(x)
and using the asymptotic formula (1.13) on the functions u 1 = Φ q 1 (λ)g 1 and u 2 = Φ q 2 (λ)g 2 , we deduce

∂ r u 1 u 2 -u 1 ∂ r u 2 = -2iλ 2π λR d-1 g 1 (-θ) g 2 (-θ) -S q 1 (λ)g 1 (θ) S q 2 (λ)g 2 (θ) + O 1 R d , which implies |x|≤R (q 1 -q 2 )u 1 u 2 dx = -2iλ 2π λ d-1 S d-1 g 1 g 2 -S q 1 g 1 S q 2 g 2 dθ + O 1 R .
Letting R tend to infinity provides the first identity.

The proof of the second identity is similar, since

∂ r u 1 u 2 -u 1 ∂ r u 2 = -2iλ 2π λR d-1 g 1 (-θ) S q 2 g 2 (θ) -S q 1 (λ)g 1 (θ) g 2 (-θ) + O 1 R d .
This completes the proof of the lemma.

Choosing q 1 = q 2 = q in Lemma 1.4, we obtain the following properties of the scattering matrix:

t S q (λ) = P • S q (λ) • P, S q (λ) * S q (λ) = Id recalling that P g = ǧ is the antipodal map. Here, t denotes the transpose and * the transpose conjugate. Besides, using the first relation together with Lemma 1.4, we finally get

(q 1 -q 2 )u 1 u 2 dx = 2iλ 2π λi d-1 2 S d-1 ǧ2 A q 1 (λ) -A q 2 (λ) g 1 dω, (1.14)
for any pair of solutions

u 1 = Φ q 1 (λ)g 1 , u 2 = Φ q 2 (λ)g 2 ,
to the Helmholtz equations (1.1) with Sommerfeld's radiation condition (1.4) related to the potentials q 1 , q 2 . 1.2. Stability estimates at high frequencies. The first stability result we shall prove in this paper states that low frequencies (i.e. smaller than 2λ) may be recovered in a stable way from the scattering amplitude.

Theorem 2. For all ε, M, R > 0 and all α > d there exist C ε , λ 0 > 0 such that the following stability estimate holds true. Let q 1 , q 2 ∈ C ∞ 0 (R d ) be two potentials supported in the ball centered at 0 and of radius R such that q 1 L ∞ , q 2 L ∞ ≤ M . Then for all λ ≥ λ 0 ,

(1.15) |ξ|≤(2-ε)λ ξ -α ( q 1 -q 2 )(ξ) 2 dξ ≤ C ε λ 3 a q 1 -a q 2 2 L 2 + C ε λ -2 q 1 -q 2 2 L ∞ .
The proof of Theorem 2 relies on estimates on solutions to the Helmoltz equations stated below, as well as on the following lemma, stating that one can relate the Fourier transform of the potential and the scattering amplitude.

Lemma 1.5. For all M, R > 0 there exist constants C, λ 0 > 0 such that for all potentials q ∈ C ∞ 0 (R d ) supported in the ball B(0, R) and satisfying q L ∞ ≤ M, we have the following approximation:

a q (θ, ω, λ) + 1 2iλ λ 2πi d-1 2 q λ(θ -ω) ≤ C λ d-5 2 q 2 L 2 (1.16)
for all λ ≥ λ 0 .

Lemma 1.5 is based on the following classical estimate on the free resolvent R 0 (λ). We refer to [START_REF] Burq | Semi-classical estimates for the resolvent in nontrapping geometries[END_REF] for an exposition by Burq of an elementary proof due to Zworski. This type of a priori estimates have a long history and go back to the work of Agmon [START_REF] Agmon | Spectral properties of Schrödinger operators and scattering theory[END_REF] on weighted estimates on the resolvent and the limiting absorption principle.

Proposition 1.6. Let χ ∈ C ∞ 0 (R d )
, there exists a constant C > 0 such that for all λ > 1 we have

χR 0 (λ)χ L(L 2 (R d )) ≤ Cλ -1 .
Proof of Lemma 1.5. We start with the expression defining the scattering amplitude (see Definition 1.2):

a q (θ, ω, λ) = - 1 2iλ λ 2πi d-1 2
q(y)ϕ q (y, ω, λ)e -iλθ•y dy so by (1.5) we deduce that

a q (θ, ω, λ) + 1 2iλ λ 2πi d-1 2 q λ(θ -ω) = - 1 2iλ λ 2πi d-1 2 q(x)ϕ scat q (x, ω, λ) e -iλx•θ dx.
The resolvent identity reads (see for instance Formula (2.3) in [START_REF] Melrose | Geometric scattering theory[END_REF])

(1.17) R q (λ) = R 0 (λ) -R 0 (λ) q R q (λ) which implies that if χ ∈ C ∞ 0 (R d
) is a cutoff function which equals 1 on the ball B(0, R), then χR q (λ)χ = χR 0 (λ)χ -χR 0 (λ)χ q χR q (λ)χ .

If we apply this identity to -qe iλx•ω , we get in view of (1.6)

χϕ scat q = -χR 0 (λ)χ qe iλx•ω ) + χR 0 (λ)χ qχϕ scat q
and using the estimate of Proposition 1.6, we obtain

χϕ scat q L 2 ≤ Cλ -1 q L 2 + Cλ -1 q L ∞ χϕ scat q L 2 .
Taking λ ≥ 2CM we deduce

χϕ scat q L 2 ≤ 2Cλ -1 q L 2 .
Using Cauchy-Schwarz's inequality, we get

q ϕ scat q e -iλx•ω dx ≤ 2Cλ -1 q 2 L 2
and this completes the proof of the lemma.

Note that the main ingredient in the proof of Lemma 1.5 consists in combining the estimate for the non-perturbed resolvent (Proposition 1.6) with the resolvent identity (1.17), along with the fact that q is compactly supported.

This lemma is the basis of the reconstruction of the potential from the scattering amplitude at high frequencies, since one can choose θ, ω ∈ S d-1 such that λ(θω) = ξ for any fixed frequency ξ with |ξ| ≤ 2λ and let λ tend to infinity. We use a similar approach to prove a stability estimate at high frequencies.

Proof of Theorem 2. Again, we start with the expression defining the scattering amplitude and then express the potential in terms of the scattering amplitude: for j ∈ {1, 2} we write

q j λ(θ -ω) = -2i(2πi) d-1 2 λ -d-3
2 a q j (θ, ω, λ)q j (x)ϕ scat q j (x, ω, λ) e -iλx•θ dx.

Taking the difference of the two expressions yields

(1.18) ( q 1 -q 2 ) λ(θ -ω) ≤ 2(2π) d-1 2 λ -d-3 2 (a q 1 -a q 2 )(θ, ω, λ) + q 1 -q 2 L 2 ϕ scat q 1 L 2 (|x|≤R) + q 2 L 2 ϕ scat q 1 -ϕ scat q 2 L 2 (|x|≤R)
where we recall that the supports of q 1 , q 2 are contained in the ball of center 0 and radius R > 0. As in the proof of Lemma 1.5 we have

ϕ scat q 1 L 2 (|x|≤R) ≤ Cλ -1 q 1 L 2 . (1.19)
In light of (1.6), the difference of the two scattered waves satisfies

ϕ scat q 1 -ϕ scat q 2 = -R q 1 (λ) -R q 2 (λ) (q 1 e iλx•ω -R q 2 (λ) (q 1 -q 2 )e iλx•ω
and by the resolvent identity (1.17), the first term on the right-hand side reads

R q 2 (λ)(q 1 -q 2 )R q 1 (λ)(q 1 e iλx•ω = -R q 2 (λ)(q 1 -q 2 ) ϕ scat q 1 . Let χ ∈ C ∞ 0 (R d
) be a cutoff function which equals 1 on the ball B(0, R), then χ(ϕ scat q 1ϕ scat q 2 ) = -χR q 2 (λ)χ (q 1q 2 )ϕ scat q 1

-χR q 2 (λ)χ (q 1q 2 )e iλx•ω .

Therefore along the same lines as in the proof of Lemma 1.5, the resolvent identity (1.17) and Proposition 1.6 give rise to

(1.20) ϕ scat q 1 -ϕ scat q 2 L 2 (|x|≤R) λ -1 q 1 -q 2 L ∞ ϕ scat q 1 L 2 (|x|≤R) + λ -1 q 1 -q 2 L 2 .
Indeed invoking the resolvent identity (1.17), we infer that χR q 2 (λ)χ = χR 0 (λ)χ -χR 0 (λ)χ q 2 χR q 2 (λ)χ .

Thus applying this identity to (q 1q 2 )ϕ scat q 1 , we obtain

χR q 2 (λ)χ (q 1 -q 2 )ϕ scat q 1 L 2 ≤ Cλ -1 q 1 -q 2 L ∞ ϕ scat q 1 L 2 (|x|≤R) + Cλ -1 q 2 L ∞ χR q 2 (λ)χ (q 1 -q 2 )ϕ scat q 1 L 2 , which implies for λ > 2CM χR q 2 (λ)χ (q 1 -q 2 )ϕ scat q 1 L 2 ≤ 2Cλ -1 q 1 -q 2 L ∞ ϕ scat q 1 L 2 (|x|≤R) .
We have also

χR q 2 (λ)χ (q 1 -q 2 )e iλx•ω L 2 λ -1 q 1 -q 2 L 2 ,
which achieves the proof of Estimate (1.20).

Taking into account those bounds, we obtain in view of (1.18) the estimate

(1.21) ( q 1 -q 2 ) λ(θ -ω) λ -d-3 2 (a q 1 -a q 2 )(θ, ω, λ) + λ -1 q 1 -q 2 L ∞ . We denote r(ξ) = 1 -|ξ| 2 when |ξ| ≤ 1. For ξ ∈ R d with |ξ| ≤ 2λ, choose η ∈ ξ ⊥ with norm |η| = r(ξ/2λ). The vectors θ = η + ξ 2λ , ω = η - ξ 2λ ,
have length one, taking the square of (1.21) and integrating the resulting estimate with respect to η yields

( q 1 -q 2 )(ξ) 2 λ -2 q 1 -q 2 2 L ∞ + λ -d+3 r(ξ/2λ) -d+2 |η|=r(ξ/2λ) η⊥ξ a q 1 -a q 2 η + ξ 2λ , η - ξ 2λ , λ 2 dη.
When |ξ| ≤ (2ε)λ we have r(ξ/2λ) ≥ ε(4ε)/2 therefore multiplying the previous inequality by ξ -α ≤ 1 and integrating with respect to ξ we get

|ξ|≤(2-ε)λ ξ -α ( q 1 -q 2 )(ξ) 2 dξ ≤ C ε λ -2 q 1 -q 2 2 L ∞ + C ε λ 3 |ξ|≤1 |η|=r(ξ) η⊥ξ (a q 1 -a q 2 ) η + ξ, η -ξ, λ 2 dη dξ.
We consider the following 2d -2 dimensional submanifold of S 2d-1 Σ = (ξ, η) ∈ S 2d-1 : ξ, η = 0 and the following diffeomorphism

ϕ : Σ → S d-1 × S d-1 (ξ, η) → (ξ + η, η -ξ), with inverse ϕ -1 : S d-1 × S d-1 → Σ (θ, ω) → θ -ω 2 , θ + ω 2
for which we have

Σ F (η + ξ, η -ξ) dη ∧ dξ = S d-1 ×S d-1 F (θ, ω) ϕ -1 * (dη ∧ dξ) = 2 -d S d-1 ×S d-1 F (θ, ω) dω ∧ dθ.
Then we finally get

|ξ|≤(2-ε)λ ξ -α ( q 1 -q 2 )(ξ) 2 dξ ≤ C ε λ 3 S d-1 ×S d-1 a q 1 θ, ω, λ -a q 2 θ, ω, λ 2 dθ dω + C ε λ -2 q 1 -q 2 2 L ∞
and this completes the proof of our estimate.

In particular, we recover the uniqueness of the potential from the scattering amplitude at high frequencies.

Corollary 1.7. If for all (θ, ω, λ) belonging to S d-1 × S d-1 × R * + , we have a q 1 (θ, ω, λ) = a q 2 (θ, ω, λ) then q 1 = q 2 .

Near field pattern

(D 2 -λ 2 + q)u = 0 in Ω, (∂ ν -iλ)u| ∂Ω = f ∈ L 2 (∂Ω). (2.1)
This problem has a unique solution u ∈ H 1 (Ω) for all f ∈ L 2 (∂Ω). Writing a variational formulation of (2.1) and using a unique continuation argument shows the uniqueness of a solution to (2.1). The existence follows from Fredholm's alternative [START_REF] Melenk | On generalized finite element methods[END_REF].

Remark 2.1. Other classical boundary conditions are either Dirichlet or Neumann boundary conditions. However, one has to make the additional assumption that λ 2 is not a Dirichlet (or Neumann) eigenvalue of D 2 + q, for the Dirichlet problem corresponding to (2.1) to have a unique solution for all f ∈ H 1 2 (∂Ω). Unfortunately, this does not make sense if one wants to take the high frequency limit λ → ∞. To bypass this difficulty, we study the boundary problem (2.1). Indeed, this condition is natural. It approximates Sommerfeld's radiation condition at high frequencies [START_REF] Engquist | Absorbing boundary conditions for the numerical simulation of waves[END_REF][START_REF] Keller | Exact nonreflecting boundary conditions[END_REF].

Remark 2.2. It follows from [START_REF] Melenk | On generalized finite element methods[END_REF] that the unique solution u ∈ H 1 (Ω) of (2.1) satisfies

Du L 2 (Ω) + λ u L 2 (Ω) ≤ C qu L 2 (Ω) + f L 2 (∂Ω)
for some constant C independent of λ. Therefore, as λ → ∞, we have

Du L 2 (Ω) + λ 2 u L 2 (Ω) ≤ C f L 2 (∂Ω) .
Definition 2.3. The near field pattern is the map

Nq(λ) : L 2 (∂Ω) → L 2 (∂Ω) f → u| ∂Ω .
The typical inverse problem on the near field pattern is whether it uniquely determines the potential q. This was solved (for smooth potentials) by Sylvester and Uhlmann [START_REF] Sylvester | A global uniqueness theorem for an inverse boundary value problem[END_REF] in dimension d ≥ 3 in the case of the Dirichletto-Neumann map. Reconstruction methods were proposed by Nachman in [START_REF] Nachman | Reconstruction from boundary measurements[END_REF] and stability issues were studied by Alessandrini [START_REF] Alessandrini | Open issues of stability for the inverse conductivity problem[END_REF]. It was shown by Mandache in [START_REF] Mandache | Exponential instability in an inverse problem for the Schrödinger equation[END_REF] that the logarithmic stability result of Alessandrini in [START_REF] Alessandrini | Open issues of stability for the inverse conductivity problem[END_REF] is optimal.

Stability estimates at high frequencies.

The following result is the counterpart of Theorem 2 in the near-field context. Theorem 3. For all M, R > 0 and all α > d there exist C, λ 0 > 0 such that the following stability estimate holds true. Let q 1 , q 2 ∈ C ∞ 0 (R d ) be two potentials supported in the ball centered at 0 and of radius R such that q 1 L ∞ , q 2 L ∞ ≤ M . Then for all λ ≥ λ 0 ,

(2.2) |ξ|≤2λ ξ -α q 1 -q 2 (ξ) 2 dξ ≤ C Nq 1 -Nq 2 2 + Cλ -2 q 1 -q 2 2 L ∞ .
Proof. We start with two solutions u 1 , u 2 of the equations (D 2λ 2 + q j )u j = 0 and computing

Ω (q 1 -q 2 )u 1 u 2 dx = Ω ∆u 1 u 2 dx - Ω u 1 ∆u 2 dx = ∂Ω (∂ ν -iλ)u 1 u 2 dσ - ∂Ω u 1 (∂ ν -iλ)u 2 dσ yields the formula Ω (q 1 -q 2 )u 1 u 2 dx = ∂Ω u 2 Nq 1 (λ)u 1 -u 1 Nq 2 (λ)u 2 dσ.
Choosing q 1 = q 2 shows that the application Nq j is symmetric, and using this additional information, we get the formula

Ω (q 1 -q 2 )u 1 u 2 dx = ∂Ω u 2 Nq 1 (λ) -Nq 2 (λ) u 1 dσ.
Let us write q = q 1q 2 and choose

u 1 = ϕ q 1 = e iλx•θ 1 + ϕ scat q 1
and u 2 = ϕ q 2 = e iλx•θ 2 + ϕ scat q 2 , with θ 1 , θ 2 ∈ S d-1 . Recall from the proof of Lemma 1.5 that the scattered waves satisfy the estimate

ϕ scat q 1 L 2 (B(0,R)) + ϕ scat q 2 L 2 (B(0,R)) ≤ Cλ -1 . It follows that Ω e iλx•(θ 1 +θ 2 ) q(x) dx = ∂Ω Nq 1 (λ) -Nq 2 (λ) u 1 u 2 dσ - Ω q(x)(ϕ scat q 1 + ϕ scat q 2 + ϕ scat q 1 ϕ scat q 2 ) dx
and therefore

q λ(θ 1 + θ 2 ) ≤ Nq 1 (λ) -Nq 2 (λ) u 1 L 2 (∂Ω) u 2 L 2 (∂Ω) + C λ q L ∞ ≤ C Nq 1 (λ) -Nq 2 (λ) + C λ q L ∞ for all (θ 1 , θ 2 ) ∈ S d-1 × S d-1 .
We notice that

S d-1 × S d-1 → B(0, 2) (θ 1 , θ 2 ) → θ 1 + θ 2
is a submersion when θ 1 , θ 2 are not colinear. This implies that

q(ξ) ≤ C Nq 1 (λ) -Nq 2 (λ) + C λ q L ∞ , ξ ∈ B(0, 2λ). (2.3)
Multiplying this estimate by ξ -α /2, taking the square and integrating on the ball B(0, 2λ) completes the proof of the theorem.

3. The case of a potential located near the boundary 3.1. Definitions and notations. In this section we show, in the model case of the unit disk, that if the potential is supported close to the boundary of the disk, then a larger range of frequencies may be recovered by the near field Nq(λ) than in the general case treated in the previous two sections. More precisely, introducing radial coordinates (x 1 , x 2 ) = (r cos θ, r sin θ), with (r, θ) ∈ R + × [0, 2π[, we consider the following model problem in two space dimensions:

(3.1) (D 2 -λ 2 + q λ )u n = 0 in B = x ∈ R 2 , |x| ≤ 1 , ∂ r -iλ u n|∂B = e inθ .
We suppose that q λ is a smooth, radial function, with support included in D κ λ for some fixed constant κ > 0, where (3.2)

D κ λ = r ∈ [0, 1], 1 -κλ -1 < r < 1 .
In the following for simplicity we shall drop the index λ in the notation of the potential.

The following result shows that in the monotone case, one can improve on the frequency band recovered in the general case (see Theorems 2 and 3). We recall that Nq(λ) denotes the operator norm of Nq(λ) in L(L 2 ).

Theorem 4. Let q 1 and q 2 be two smooth radial potentials supported on D κ λ as defined in (3.2) and such that q 1 ≥ q 2 . There are positive constants λ 0 and C such that the following holds. Let λ → K(λ) be any function such that λ ≤ K(λ). Then the following stability estimate is valid for all λ ≥ λ 0 :

|ξ|≤K(λ) ( q 1 -q 2 )(ξ) 2 dξ ≤ CK 2 (λ) λ 4 Nq 1 (λ) -Nq 2 (λ) 2 + C 2 q 1 ,q 2 λ 4 q 1 -q 2 2 L ∞ , where C q 1 ,q 2 = max q 1 L ∞ , q 2 L ∞ , q 1 L ∞ q 2 L ∞ .
Remark 3.1. One has trivially that |ξ|≤K(λ)

( q 1 -q 2 )(ξ) 2 dξ ≤ CK 2 (λ) q 1 -q 2 2 L ∞ ≤ CK 2 (λ) λ 2 q 1 -q 2 2
L ∞ , so the estimate provided in Theorem 4 is of a different nature.

Remark 3.2. The proof of Theorem 4 is presented in Section 3.2, as an immediate consequence of Lemma 3.3 proved in Section 3.3. That lemma relates the Laplace transform of a function to the near field operator. It holds in much more generality than Theorem 4, without the additional assumption that q 1q 2 ≥ 0. However we are unable to relate the Fourier transform of a function to its Laplace transform in general (this fact is well-known to be difficult and in general very unstable); for nonnegative functions however the relation is very easy and enables us to conclude.

3.2.

Proof of Theorem 4. We start by stating a lemma, proved in Section 3.3, which relates the Laplace transform of a function to the near field operator. It is stated in the framework of general, non radial functions. We define the operator T : g → T g by

T g(r) = g(1 -r)
as well as the Laplace transform L: L(g)(t) = R g(s) e -s t ds.

For any function θ → f (θ) we call c k (f ) its Fourier transform, for k ∈ Z:

c k (f )(r) = 2π 0 f (r, θ) e -i k θ dθ.
Lemma 3.3. Let q 1 and q 2 be two functions supported on D κ λ . There are three positive constants λ 0 , K and C such that if t ≥ 2Kλ > 2Kλ 0 , then

L c k T (r(q 1 -q 2 ) (t) ≤ C λ 2 Nq 1 (λ) -Nq 2 (λ) + C q 1 ,q 2 λ 2 q 1 -q 2 L ∞ where C q 1 ,q 2 = max q 1 L ∞ , q 2 L ∞ , q 1 L ∞ q 2 L ∞ .
Proof of Theorem 4. Let q = q 1 -q 2 . Since q is radial, we have q(x) = Q(|x|) and q(ξ) = Q(|ξ|) with for all ρ > 0,

Q(ρ) = 2π 0 ∞ 0 e -irρ cos θ Q(r) rdrdθ.
In particular recalling that Q is nonnegative, we find that for all ξ ∈ R 2 ,

| q(ξ)| ≤ 2π ∞ 0 Q(r) rdr ≤ 2π ∞ 0 e ζ 0 (1-r) e -ζ 0 (1-r) Q(r) rdr
for any ζ 0 ∈ R. Then we can apply Lemma 3.3 to the particular case of a radial function, so choosing k = 0 and ζ 0 = 3Kλ 0 we infer that

| q(ξ)| ≤ Ce ζ 0 κ/λ L T (rQ) (ζ 0 ) (3.3) ≤ Ce 3Kκ λ 2 Nq 1 (λ) -Nq 2 (λ) + C q 1 ,q 2 λ 2 q 1 -q 2 L ∞ .
The end of the proof of Theorem 4 follows easily by taking the L 2 norm in ξ.

3.3.

Proof of Lemma 3.3. The method of proof follows the ideas developed in the proofs of Theorems 2 and 3, adapting the estimates to our special situation where the potentials are located near the boundary. The heart of the matter consists to approximate the solutions to Helmholtz equation (3.1) by separable solutions in radial coordinates involving Bessel functions, which allows in light of Debye's formula to relate the Laplace transform of the potential to its near field operator.

More precisely, we shall look for solutions to (3.1) under the following form, for n ∈ Z:

u n (r, θ) = J |n| (λr) λ(J ′ |n| (λ) -iJ |n| (λ)) e inθ + v n = z n (r, λ) e inθ + v n ,
where J n is a Bessel function of the first kind (see Appendix A), solution to

∂ 2 r + 1 r ∂ r - n 2 r 2 + 1 J n (r) = 0.
Suppose that for ℓ ∈ {1, 2}, u ℓ n solves (3.1) with potential q ℓ . Then writing

u ℓ n (r, θ) = z n (r, λ) e inθ + v ℓ n (r, θ), we find that (D 2 -λ 2 + q ℓ )v ℓ n = -q ℓ z n e inθ in B, (∂ r -iλ)v ℓ n|∂B = 0.
Due to Property (A.6) we have if 1λ -1 ≤ r ≤ 1 and |n| ≥ Kλ for K large, that λ z n (r, λ) is bounded, for λ ≥ λ 0 , by a constant depending only on K and λ 0 . For now on we shall denote by C(λ 0 , K) such a constant, which may change from line to line.

Therefore, arguing as in the proof of Lemma 1.5 and taking advantage of the fact that on the support of q ℓ , r varies in an interval of size λ -1 , we find as soon as λ is large enough compared to q ℓ L ∞

(3.4) v ℓ n L 2 ≤ C(λ 0 , K) λ q ℓ (r, θ) z n (r, λ) L 2 ≤ C(λ 0 , K) λ 5 2 q ℓ L ∞ .
Now going back to the computations of Section 2 we consider two positive integers n and m such that n, m ≥ Kλ, and we write

B (q 1 -q 2 )u 1 n u 2 -m dx = ∂B (N q 1 (λ) -Nq 2 (λ)
e inθ e -imθ dθ.

Therefore decomposing

u 1 n (r, θ) = z n (r, λ) e inθ + v 1 n (r, θ) and u 2 -m (r, θ) = z m (r, λ) e -imθ + v 2 -m (r, θ), we get 2π 0 Nq 1 (λ) -Nq 2 (λ) e inθ e -imθ dθ = B (q 1 -q 2 )e i(n-m)θ z n (r, λ) z m (r, λ) rdr dθ + B (q 1 -q 2 )e inθ z n (r, λ) v 2 -m rdr dθ (3.5) + B (q 1 -q 2 )e -imθ z m (r, λ) v 1 n rdr dθ + B (q 1 -q 2 )v 1 n v 2 -m rdr dθ.
As λ z n (r, λ) is bounded by a constant C(λ 0 , K), we have (recalling that q 1 and q 2 are compactly supported in an interval in r of size λ -1 )

B (q 1 -q 2 )e inθ z n (r, λ) v 2 -m r dr dθ ≤ C(λ 0 , K) λ 3 2 q 1 -q 2 L ∞ v 2 -m L 2
and similarly

B (q 1 -q 2 )e -imθ z m (r, λ) v 1 n r dr dθ ≤ C(λ 0 , K) λ 3 2 q 1 -q 2 L ∞ v 1 n L 2 . Finally B (q 1 -q 2 )v 1 n v 2 -m rdr dθ ≤ q 1 -q 2 L ∞ v 1 n L 2 v 2 -m L 2 .
Thus by (3.4), we get from (3.5), for n, m ≥ Kλ > Kλ 0 ,

B (q 1q 2 )e i(n-m)θ z n (r, λ) z m (r, λ) rdr dθ

≤ 2π 0 Nq 1 (λ) -Nq 2 (λ) e inθ e -imθ dθ + C q 1 ,q 2 λ 4 q L ∞ ,
where q = q 1q 2 and

C q 1 ,q 2 = C(λ 0 , K) max q 1 L ∞ , q 2 L ∞ , q 1 L ∞ q 2 L ∞ .
This gives rise to

(3.7) B (q 1 -q 2 )e i(n-m)θ z n (r, λ) z m (r, λ) rdr dθ ≤ Nq 1 (λ) -Nq 2 (λ) + C q 1 ,q 2 λ 4 q L ∞ , which leads in light of (A.6) to B (q 1 -q 2 )e i(n-m)θ e -(n+m)(1-r) rdr dθ ≤ C(λ 0 , K) λ 2 Nq 1 (λ) -Nq 2 (λ) + C q 1 ,q 2 λ 2 q L ∞
, for all n, m ≥ Kλ. In conclusion we have for any k ∈ Z, any ℓ ∈ N and any j ≥ 2Kλ > 2Kλ 0 , (3.8) B (q 1q 2 )e -i kθ e -j(1-r) rdr dθ ≤ C(λ 0 , K) λ 2 Nq 1 (λ) -Nq 2 (λ) + C q 1 ,q 2 λ 2 q L ∞ . The conclusion follows from (3.3). Lemma 3.3 is proved.

Concluding remarks

In this paper we have shown that the low-frequency component of the potential can be determined in a stable way from the scattering measurements and justified the resolution limit. We have also proved that in the near-field we have in the monotone case infinite resolution in reconstructing the potential near the boundary. We think that the result holds in the general case. However, its proof seems to be out of reach. In fact, even though a sampling (or interpolation) formula for the Laplace transform does exist [START_REF] Boumenir | Sampling eigenvalues in Hardy spaces[END_REF][START_REF] Rybkin | A new interpolation formula for the Titchmarsh-Weyl mfunction[END_REF], making norm-estimates similar to those in Theorem 4 is very challenging. Our results can be extended in many directions. It would be very interesting to study the limited-view case and show, as in [START_REF] Ammari | Limited view resolving power of conductivity imaging from boundary measurements[END_REF], that we recover infinite resolution from near-field measurements on the overlap of the source and receiver apertures. Another challenging problem is to understand how probe interaction can improve local resolution by converting where z is any point of the complex plane cut along the segment ] -∞, 0]. The motivation for introducing the Hankel functions is that the linear combination of J n (z) and Y n (z) have very simple asymptotic expansions for large |z|: it is thus well-known that H (1) n (z) = 2 πz (2πn tanh α) (2πn tanh α)

1 2 1 + O 1 n , J ′ n (n sech α) = (tanh α)
1 2 e -n(α-tanh α) (4πn) Without loss of generality we may assume that n ∈ N, then defining cosh α 1 = n λr and cosh α 2 = n λ , it is easy to see that under the above assumptions (λ ≪ n and r ∼ 1), we have necessarily cosh α i ≫ 1 for i ∈ {1, 2}, hence α i ≫ 1. This implies cosh α i ∼ e α i , sinh α i ∼ e α i , and tanh α i ∼ 1, which gives rise to Re z n (λ, r) = J n (n sech α 1 )J ′ n (n sech α 2 ) λ(J ′2 n (n sech α 2 ) + J 2 n (n sech α 2 )) and Im z n (λ, r) = J n (n sech α 1 )J n (n sech α 2 ) λ(J ′2 n (n sech α 2 ) + J 2 n (n sech α 2 )) • Finally, taking advantage of (A.5) we get Re z n (λ, r) ∼ e -n(α 1 -α 2 ) λ ∼ e -n log( 1

r ) λ •
The computation is identical for Im z n (λ, r), so using the fact that r is near to 1, we obtain the desired conclusion.

1

 1 is the Hankel function of first kind and order d/2 -1 (cf. Appendix A). The asymptotic behavior of Hankel functions (A.3) implies that for |y| large enough

2. 1 .

 1 Definitions and notations. Instead of considering the Helmholtz equation on the whole Euclidean space (with Sommerfeld's radiation condition) we focus on the Cauchy problem with Robin boundary condition on a bounded open set Ω ⊂ R d with smooth boundary

  |z| → ∞. Furthermore, we have the following Debye formulas whose proof can be found for instance in[START_REF] Olver | Asymptotics and special functions[END_REF] Chapter 9.4] and [1, Chapter 9]:J n (n sech α) ∼ e -n(α-tanh α)

5 )

 5 J n (n sech α) = e -n(α-tanh α)

1 n

 1 as n → ∞, where sech z denotes the hyperbolic secant of z defined by sech z = 1 cosh z with cosh z the hyperbolic cosine.Debye's formula gives rise to the following asymptotic behavior for the function introduced in Section 3:z n (λ, r) = J |n| (λr) λ(J ′ |n| (λ) -iJ |n| (λ))defined for n ∈ Z. Let us prove that for |n| ≫ λ and r ∼ 1,

We use the notation ǧ(ω) = g(-ω) = P g(ω) for the antipodal map.

Acknowledgements. We are very grateful to J. Sjöstrand for taking the time to discuss some properties of the Laplace transform with us.

This work was supported by the ERC Advanced Grant Project MULTIMOD-267184 and the Institut Universitaire de France.

evanescent modes of the potential to propagating ones [START_REF] Carney | Near-field tomography, in Inside out: inverse problems and applications[END_REF]. These problems will be the subject of forthcoming works.

Appendix A. Bessel functions

Bessel's equation arises when finding separable solutions to the Helmholtz equation in spherical coordinates, and writes as follows:

It is well known (see for instance [START_REF] Lebedev | Special functions and their applications[END_REF][START_REF] Olver | Asymptotics and special functions[END_REF] and the references therein) that one of the solutions of Bessel's equation is the entire function J n (z) known as the Bessel function of the first kind of order n, and defined for arbitrary z ∈ C by the convergent series

in the case where n ∈ N and by

To find a general solution of Bessel's equation (A.1), we need a second solution of (A.1) which is linearly independent of J n (z). For such a solution, we usually choose Y n (z) the Bessel function of the second kind which is entire in the complex plane cut along the segment ] -∞, 0] and defined for n ∈ N by

where ψ(m + 1) = -γ + 1 + 1 2 + ... + 1 m , γ being the Euler constant. We also define Y -n (z) = (-1) n Y n (z).

Since J n and Y n are linearly independent, the general expression for solutions of (A.1) is a linear combination of Bessel functions of the first and second kinds, i.e, u(z) = A J n (z) + B Y n (z), where A and B are constants.

Another basis of solutions to the differential equation (A.1) is given by the Bessel functions of the third kind or Hankel functions, denoted by H n . These functions are defined by the formulas (A.2)

H (1) n (z) = J n (z) + iY n (z) and H (2) n (z) = J n (z) -iY n (z),