Amelia Carolina Sparavigna
email: amelia.sparavigna@polito.it

Mauro Maria Baldi
email: mauro.baldi@polito.it

Overlapping Circles Grid Drawn with Compass and Straightedge on an Egyptian Artifact of 14th Century BC

Keywords: Geometry, History of Science, Geometric Patterns, Overlapping Circles Grids. Subject Areas: History of Science, History of Geometry

The study of the mathematics and geometry of ancient civilizations is a task which seems to be very difficult or even impossible to fulfil, if few written documents, or none at all, had survived from the past. However, besides the direct information that we can have from written documents, we can gain some indirect evidence on mathematics and geometry also from the analysis of the decorations we find on artifacts. Here, for instance, we will show that ancient Egyptians were able of making geometric constructions using compass and straightedge, quite before the Greek Oenopides of Chois, who lived around 450 BC, had declared some of their basic principles. In fact, a wood panel covered by an overlapping circles grid pattern, found in the tomb of Kha, an architect who served three kings of 18th Dynasty (1400-1350 BC), evidences that some simple constructions with compass and straightedge were used in ancient Egypt about nine centuries before Oenopides' time.

Introduction

Porphyry of Tyre, Neoplatonic philosopher of the mid third century AD, wrote in his "Life of Pythagoras" [START_REF] Porphyry | Life of Pythagoras[END_REF], that the great philosopher of Samos, who lived in the sixth century BC, learned the mathematical sciences from the Egyptians, Chaldeans and Phoenicians. According to Porphyry, old Egyptians excelled in geometry and Phoenicians in numbers and proportions. The Chaldeans were masters of "astronomical theorems, divine rites, and worship of the Gods". In the ancient world then, besides for its spectacular civilization, Egypt was also famous for its geometry. However, it is not easy to determine precisely the specific knowledge of Egyptians. In fact, today, we possess only a limited number of texts from ancient Egypt concerning mathematics and geometry. On geometry, we have the Moscow Mathematical Papyrus and the Rhind Mathematical Papyrus. From them, we find that the ancient Egyptians knew how to compute areas of several geometric shapes and the volumes of cylinders and pyramids. The Moscow Papyrus was most likely written during the 13th dynasty, based on older material probably dating to the 12th dynasty of Egypt, roughly 1850 BC [START_REF] Clagett | Ancient Egyptian Science: A Source Book[END_REF], whereas the Rhind Papyrus dates to around 1650 BC. It was copied by scribe Ahmes from a now-lost text from the reign of a king of the 12th dynasty. In this papyrus, we can find an approximation of number π, in a problem where a circle is approximated by an octagon [START_REF] Sparavigna | Number Pi from the Decorations of Ancient Artifacts[END_REF]. Besides from these documents, we can gain some indirect information on the geometry of ancient Egypt from the analysis of their of artifacts. For instance, in [START_REF] Sparavigna | Number Pi from the Decorations of Ancient Artifacts[END_REF][START_REF] Sparavigna | Hemaka's constant[END_REF], we have shown how it is possible to deduce from the geometric decorations of ancient objects some information on the value of π, approximated by a rational number. For instance, from some game disks (see Figure 1) found in the tomb of Hemaka, the chancellor of a king of the First Dynasty of Egypt, about 3000 BC, we deduced that probably Hemaka knew the value of π as 66/21=22/7=3.14. Here, we will show that ancient Egyptians, masters in drawing artistic figures, were also able of making constructions using compass and straightedge. It is clearly displayed by a wood panel, covered by an overlapping circles grid decoration, found in the tomb of architect Kha, who served three kings of 18th dynasty, and lived about 1400-1350 BC. Therefore, simple compass-straightedge constructions were already in use in ancient Egypt about nine centuries before the Greek Oenopides of Chois, who lived around 450 BC, had stated his principle that, for geometrical constructions, one should use no other means than compass and straightedge [START_REF] Christianidis | Classics in the History of Greek Mathematics[END_REF]. In fact, Oenopides' name is linked to two specific elementary constructions, one is concerning the drawing from a given point a straight line perpendicular to a given straight line, the other the drawing, on a given straight line and at a given point on it, an angle equal to a given angle [START_REF] Thomas | A Manual of Greek Mathematics[END_REF].

Architect Kha

Deir el Medina was a village of the ancient Egypt, inhabited by people working for the two great royal necropolises of the Valleys of the Queens and of the Kings. While excavating this village, the Italian Egyptologist Ernesto Schiaparelli discovered in February 1906 an untouched tomb [START_REF] Moiso | Ernesto Schiaparelli e la Tomba di Kha[END_REF]. It was the tomb of architect Kha, a high-ranking official of the 18th Dynasty, and his wife Merit. This tomb is rich of perfectly preserved contents and artifacts, which are showing what was the life of a wealthy family about 3400 years ago (in the Figure 2, a geometrically decorated coffin from the tomb). The contents of the tomb were transported to Turin, becoming one of the main attractions of its Museo Egizio. Among the several beautiful objects of Kha's tomb, we have discussed one in particular. It could be used as an inclinometer, a surveying instrument necessary to Kha for his work in building subterranean tombs. This device could also be the first protractor used in ancient time [START_REF] Sparavigna | The Architect Kha's Protractor[END_REF][START_REF] Sparavigna | A Possible Use of the Kha's Protractor[END_REF][START_REF] Marchant | Egyptian Tomb Mystery May Be World's First Protractor[END_REF]. Kha was an important foreman at Deir El-Medina, responsible for building projects during the reigns of Amenhotep II, Thutmose IV and Amenhotep III [START_REF] Hobson | Deir El Medina: The Painted Tombs, in Exploring the World of the Pharaohs: A complete guide to Ancient Egypt[END_REF]. These were kings of the eighteenth dynasty (c. 1543-1292 BC), the dynasty of the best known pharaohs, including Tutankhamun, Hatshepsut, the longest-reigning woman-pharaoh, and Akhenaten. The dynasty, also known as the Thutmosid Dynasty for the four kings named Thutmose, is the first of the three dynasties of the New Kingdom, the period in which ancient Egypt reached the peak of its power. During Kha's life, Egypt was a powerful and wealthy country. The court of Amenhotep III, "was luxurious beyond imagination, with the wealth of the Mediterranean world flowing into Egypt's coffers" [START_REF] Hawass | Bibliotheca Alexandria[END_REF]. Therefore, Kha the architect lived in a privileged environment where, as we can easily imagine, besides the luxury and exotic goods, also knowledge and science were flowing, from the Mediterranean and Eastern countries into the cultural elite of Egypt. So, besides the skills he used for his work as architect and engineer, Kha had also received education in mathematics and geometry for sure.

Overlapping circles grid drawn with compass and straightedge

Besides the coffins of Kha and Merit, the tomb contained all the objects necessary in their afterlife. There were the instruments that Kha used for his work and the make-up of Merit, such as ointments and kohl held in alabaster and faience vessels. Other objects in the tomb furniture include a bed, a chair, small tables and coffins, vessels and clothing garments. Geometric and polychrome figures are decorating the surface of coffins, such as that shown in Figure 2. The decoration consists of a very in-teresting and regular geometric pattern, clearly based on a regular lattice. However, it is not this artifact which is relevant to our discussion on the use in ancient Egypt of compass and straightedge. Among the objects found in the tomb, there is a wood panel which has, on its surface, a very remarkable drawing. The panel is shown in the Figure 3. Probably, it is one of the drawings which were usually made for painting on them the final polychrome decorations. It is made of parallel and perpendicular red lines and of black circles, which are giving a geometric pattern of repeating, overlapping circles of equal radii on the two-dimensional planar surface. Several examples of polychrome decorations having such patterns were shown in the 1856 book "The Grammar of Ornament", written by Owen Jones. As told in this book, the "formation of patterns by the equal division of similar lines, as by weaving, would give to a rising people the first notions of symmetry, arrangement, disposition, and the distribution of masses". Let us add that, besides the notions of symmetry and arrangement of masses, necessary for their architecture too, the ancient Egyptians had also some notions of geometric construction with compass and straightedge. Probably, these notions were well-known when Kha was living and used for the art of decoration during the New Kingdom. A further investigation is necessary to determine if they were developed in this period or before.

Conclusion

Assuming that artists were using geometric drawings -such as that shown in the Figure 3 -for polychrome decorations such as that in the Figure 2, from the analysis of these decorations we can deduce the geometric constructions the Egyptian were able of making with compass and straightedge. From the Figure 3, for instance, it is possible to argue that they were able of drawing from a point a straight line perpendicular to a given straight line. Then, thanks to a wood panel from the tomb of architect Kha, we can tell that ancient Egyptians used, about nine centuries before, a construction with compass and straightedge described by the Greek Oenopides of Chois in the fifth century BC.

Figure 1 .

 1 Figure 1. These sketches represent the gaming disks found in the tomb of chancellor Hemaka at Saqqara. Hemaka was an important official during the reign of the First Dynasty Egyptian king Den. The reign of this king lasted from 2975 BC to 2935 BC.

Figure 3 .

 3 Figure 3. A coffin with a geometric decoration from the Kha's Tomb (Museo Egizio, Torino).

Figure 4 .

 4 Figure 4. Wood panel found in the Kha's Tomb. Note the red lines creating the lattice for the centers of the black circles.


Bérenger Berthoul

Alain Batailly

Mathias Legrand

Laurent Stainier

Patrice Cartraud

B Érenger Berthoul

Abradable coating removal in turbomachines: a macroscopic approach accounting for various wear mechanisms

Keywords: Abradable coatings, turbomachines, wear

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Abradable coatings have long been used by aircraft engines manufacturers as a sacrificial material in order to reduce operating clearances in both compressor and turbine stages. These coatings advantageously mitigate potential parasitic leakage flows from a stage to the previous one and are easily worn off in case contacts occur with one of the blades thus preventing potentially hazardous structural damages. The nature of the material coating and its mechanical properties are adjusted depending on where it is applied in the engine: as an example, a coating sprayed on the casing of a low-pressure compressor does not require the same thermal resistance as one sprayed in the high-pressure turbine. Accordingly, there is a wide range of materials used as abradable coatings such as ceramics, aluminum/silicon alloys and nickel/graphite composites to name a few. These materials may be sprayed, sintered or even deposited with a honeycomb structure on the casing. Consequently, when blade-tip/abradable coating contacts occur, a large variety of mechanisms may be at play. It has been recently shown that if a blade/abradable contact does not threaten the blade integrity by itself, repeated contacts could lead to a resonance of the blade [START_REF] Millecamps | Interaction aube-carter : contribution de l'usure de l'abradable et de la thermomécanique sur la dynamique d'aube[END_REF].

The experimental setup described in [START_REF] Millecamps | Interaction aube-carter : contribution de l'usure de l'abradable et de la thermomécanique sur la dynamique d'aube[END_REF] involves the last stage of a full-scale aircraft engine low-pressure compressor operating under vacuum. Contact between the blade and the abradable coating is initiated through centrifugal effects. The interaction happens for only one blade, designed to be slightly longer. The blade tip shows high amplitude response, and the coating is worn out at six equally spaced locations along the casing circumference, called wear lobes.

Numerical simulations have already been carried out in [START_REF] Batailly | Numerical-experimental comparison in the simulation of rotor/stator interaction through blade-tip/abradable coating contact[END_REF] where the abradable is represented by a one-dimensional elastovisco-plastic material. This model allows for the prediction of the blade structural response of experiment [START_REF] Millecamps | Interaction aube-carter : contribution de l'usure de l'abradable et de la thermomécanique sur la dynamique d'aube[END_REF]. Nevertheless, it is mostly numerical, since the coating wear is taken into account by a permanent deformation of the abradable instead of a material loss.

A key element for the quantitative study of the movement of the blade is an accurate modeling of the interaction behavior. This paper presents a simplified macro-scale model of the interaction, based on the phenomenology and physics of abradable wear. The goal of this model is to depend only upon experimentally measurable material quantities. It has been implemented in the framework presented in [START_REF] Batailly | Numerical-experimental comparison in the simulation of rotor/stator interaction through blade-tip/abradable coating contact[END_REF].

The first section is a brief review of the phenomenology of abradable wear and its synthesis according to the underlying physical mechanisms. Then an attempt is made to model these mechanisms under simplifying assumptions. Finally, the numerical framework is briefly discussed before results are presented through two case studies. but where the normal to the coating is defined ( 100 µm).

Phenomenology of abradable wear

The large number of recent publications dealing with abradable coatings shows the need for a better understanding of the structural behavior of such materials when impacted by a rotating blade. From an experimental standpoint, several studies may be found regarding the static characterization of abradable coatings [START_REF] Peyraut | Finite element modeling of abradable materials -Identification of plastic parameters and issues on minimum hardness against coating's thickness[END_REF][START_REF] Johnston | The sensitivity of abradable coating residual stresses to varying material properties[END_REF]. Similarly, several abradability investigations carried out statically-or at very low speed-are available [START_REF] Yi | Friction and wear behaviour and abradability of abradable seal coating[END_REF] in addition to a few scratch tests [START_REF] Ma | Evaluation of abradable seal coating mechanical properties[END_REF]. In the context of turbomachinery, one should mention the pioneering work [START_REF] Borel | The wear mechanisms occurring in abradable seals of gas turbines[END_REF] on wear mechanisms in gas turbines as well as the work of Schmid [START_REF] Schmid | New high temperature abradables for gas turbines[END_REF] which will be our main source of information regarding the phenomenology of abradable materials. Recently, experimental investigations on the dynamic characterization of abradable coatings within a simulated turbomachine environment were reported with respect to: a high relative interaction speed [START_REF] Cuny | Experimental measurement of dynamic forces generated during short-duration contacts: application to blade-casing interactions in aircraft engines[END_REF] (from 100 to 500 m/s), elevated temperatures [START_REF] Dadouche | Experimental evaluation of abradable seal performance at high temperature[END_REF] (up to 600 ı C) as well as a combination of high interaction speed and elevated temperatures [START_REF] Mandard | Interacting force estimation during blade/seal rubs[END_REF][START_REF] Padova | Development of an experimental capability to produce controlled blade tip/shroud rubs at engine speed[END_REF]. While the influence of thermal conditions has long been investigated [START_REF] Novinski | Modified zirconia abradable seal coating for high temperature gas turbine applications[END_REF], experimental work carried out by industrials is rarely published.

To the authors' knowledge, there is no quantitative model of abradable wear in the literature. A few qualitative models of abradable wear based on experimental observations were proposed by Schmid [START_REF] Schmid | New high temperature abradables for gas turbines[END_REF]. A finite element micro model of the bulk material only is proposed in [START_REF] Faraoun | Modelling route for abradable coatings[END_REF]. Finally, two abradable coating models in the context of full 3D simulations have been described [LEGRAND2011, 15], but they have been developed focusing on numerical efficiency rather than physical rendering of wear phenomena.

Methodology

For a given abradable coating, wear and removal involve several complex mechanisms. In the following, the focus is made on three of these mechanisms, assumed dominant during the interaction with a rotating blade [START_REF] Schmid | New high temperature abradables for gas turbines[END_REF]: (1) cutting, (2) permanent deformation, and (3) micro-rupture. Further refinement of the model including other wear mechanisms such as transfer and fusion goes beyond the scope of this study. Based on the proportion of removal with respect to the coating thickness, see Fig. 2, the bulk behavior of the abradable material is assumed elastic.

For each phenomenon of interest, the modeling approach may be summed up as follows:

a brief description is provided based on existing publications, the underlying physics is detailed focusing on the forces the abradable coating is subjected to, a relation between the relevant quantities is given in order to quantify the phenomenon. Also, the implementation of the proposed wear model within an existing numerical strategy for the simulation of blade/casing interactions implies that: the macroscopic effect of wear is key. Thus computational power should not be dedicated to the solution of sophisticated PDEs to know the exact strain field within the abradable material, in this article, the proposed wear model is essentially macroscopic. This is justified by the fact that during a blade/casing interaction, the blade amplitudes of vibration do not seem to depend on micro scale quantities [START_REF] Millecamps | Interaction aube-carter : contribution de l'usure de l'abradable et de la thermomécanique sur la dynamique d'aube[END_REF] (such as the precise topology of the surface of the abradable at the scale of the rugosity), but mostly depend on their macroscopic averages (the topology at the scale of the blade edge). The quantities of interest are described in Tab. 1. From now on, as a first step to model the interaction, we simplify the problem by neglecting thermal effects and blade wear. They are not coupled within the time-stepping scheme. However thermal effects can be taken into account by using parameters measured in an experiment where the temperature in the contact region is representative of a real interaction.

as forces are hence given by unit length. The term normal refers to vectors in the direction of e r , and tangential refers to vectors in the direction of e  (cf. Fig. 2).

Cutting Kinematics Existing experimental observations [START_REF] Borel | The wear mechanisms occurring in abradable seals of gas turbines[END_REF] suggest that the interpretation of the cutting mechanism may be arduous: it is reported that after the interaction, the material still attached to the coating is left with no deformation nor damage. However, the material was not removed by brittle rupture because it has a shiny appearance [START_REF] Borel | The wear mechanisms occurring in abradable seals of gas turbines[END_REF]. The abradable coating is cleanly cut by the blade, yielding a very smooth surface with no observable permanent deformation.

Forces Cutting may be explained by an adiabatic shear band mechanism [START_REF] Wright | The physics and mathematics of adiabatic shear bands[END_REF]: due to the high velocity of the blade, deformation does not have time to propagate for elastic accommodation. Thus plastic deformations occur. The plastic work done is mainly dissipated as heat which does not have time to propagate (adiabatic). As a consequence, the material is soften on a narrow band where the material is easily sheared. So cutting is a kind of ductile rupture for high speed and/or high temperature. At the scale of the abradable material micro-structure, no deformation is observed, but it may be due to the fact that it should be localized at the extreme surface of the cut material. It should be noted that the development of shear bands in the abradable material can differ significantly according to the value of the blade incursion and two scenarii are considered:

when the incursion depth is of the order of magnitude of the material rugosity, cutting is a form of adhesive wear [START_REF] Rabinowicz | Friction and wear of materials[END_REF] at high velocity/high temperature. In this configuration, friction and adhesive wear should be considered parts of the same mechanism, namely plasticity at junctions [START_REF] Rabinowicz | Friction and wear of materials[END_REF], for larger incursion depths [START_REF] Mandard | Interacting force estimation during blade/seal rubs[END_REF], cutting should rather be assimilated to machining.

Formulations In the case of friction and adhesive wear, Coulomb friction and Archard's law will be used as they have proven to be a good approximation of the phenomena while using a reduced number of parameters:

F fr t D F n (1)
h ad D k ad e b n p (2)
For the modeling of machining, the "deck of card" model [START_REF] Groover | Fundamentals of Modern Manufacturing[END_REF] 1 is used, see Fig. 3. The force induced on the blade by machining is defined as:

F ma t D y cos sin h ma (3)
For the wear of a clean cut, the formula reduces to:

h ma D I I e ( 4
)
where I e , the elastic accommodation of the coating, is defined by

I e D y e a E a (5)
The shear band angle, can depend on the blade geometry, temperature and interaction velocity.

Permanent deformation

Kinematics material is transferred to the blade and gives it a saw tooth profile. The abradable material is then ploughed by the blade.

Forces material transfer is linked to adhesive wear and is left for the sake of simplicity. Ploughing can be seen as abrasive wear in the terminology of Rabinowicz [START_REF] Rabinowicz | Friction and wear of materials[END_REF].

Formulation wear due to abrasion can be modeled by another Archard's law:

h ab D k ab e b n p (6)
Regarding the force exerted on the blade, the following relation holds:

F ab t D F pp n pa (7)
where the number of particles along the blade edge n pa can be estimated by looking at the number of grooves on a post-mortem abradable specimen, and F pp , the ploughing force of one particle, can be estimated by scratch tests [START_REF] Ma | Evaluation of abradable seal coating mechanical properties[END_REF].

Micro-rupture

Kinematics packets of spray particles are pulled out of the surface by the blade, leaving a pock-marked abradable coating.

Forces the mechanism is associated to the rupture properties of the material. Abradable materials are quasi-brittle.

Formulation Assuming a simplified geometry, cf. Fig. 4, a statistical reasoning on energy dissipated when creating detached particles by brittle rupture gives:

F mr t D k mr (8)
where k mr is a factor depending on the form of the particle and is the surface energy of the abradable material. If the particle is a shell covering the coating before being detached, k mr D 2. If it is a cube, then k mr D 6. The worn thickness is then

h mr D l pa (9)

Summary

A model has been proposed for each mechanism. They shall now be gathered to obtain the complete model. First, it is assumed that friction/adhesive wear and abrasive wear are mutually independent and micro-rupture and machining happen for an elementary surface with respective proportions p mr and p ma , with 0 Ä p mr C p ma Ä 1, a zero proportion meaning that the mechanism is inactive. These proportions are dependent on interaction velocity, incursion and temperature. This allows for a representation of Schmid wear maps [START_REF] Schmid | New high temperature abradables for gas turbines[END_REF]. One should also note that the kinematics of the detached material is of importance: Schmid [START_REF] Schmid | New high temperature abradables for gas turbines[END_REF] notes that for velocity interactions greater than 100 m/s, most of the detached material is released as particles at the rear of the blade. But in another experiment at high temperature (300 ı C) and high incursions of the blade in the coating (50 µm) [START_REF] Baïz | Etude expérimentale du contact aube/abradable: contribution à la caractérisation mécanique des matériaux abradables et de leur interaction dynamique sur banc rotatif avec une aube[END_REF], there is a continuous chip at the front of the blade, and in yet another experiment, also at high incursions but at room temperature, particles are released at the front of the machining tool [START_REF] Cuny | Contribution à la caractérisation locale des couples de matériaux mis en jeu lors du contact rotor/stator dans une turbomachine[END_REF]. Also, some mechanisms happen at the front of the blade (micro-rupture and machining) with particles being released either at the front or at the rear, while other happen under the blade (friction/adhesive wear and abrasive wear). So the scenario of interaction should be: (1) compute the forces due to micro-rupture and machining together and ( 2) update the topology of the abradable once, (3) compute the forces due to friction/adhesive wear and abrasive wear and (4) update the topology again. With simplified notation such as h ad D h ad .h mr ; h ma /, the formula of abradable wear at the end of the interaction is thus:

h D h ad C h ab C p mr h mr C p ma h ma ( 10
)
Since all the mechanisms in this model are supposed to be caused by independent physical phenomena, the total energy given by the blade is the sum of the energies of each mechanisms (we neglect the kinetic energy of detached particles). This decomposition is also true for forces. Hence:

F t D F fr t C F ab t C p mr F mr t C p ma F ma t (11)

Numerical framework

Because our numerical framework has already been presented extensively in [START_REF] Legrand | Numerical investigation of abradable coating removal in aircraft engines through plastic constitutive law[END_REF] and [START_REF] Batailly | Numerical-experimental comparison in the simulation of rotor/stator interaction through blade-tip/abradable coating contact[END_REF], only its main characteristics are described here: the blade geometry is meshed in 3D and its finite element matrices are obtained. They are then reduced with a Craig-Bampton component mode synthesis [START_REF] Craig | Coupling of substructures for dynamics analyses[END_REF], in order to reduce the number of degrees of freedom while keeping unchanged the blade tip degrees of freedom for easy contact handling. Centrifugal hardening is taken into account. Then the blade/coating interaction problem is simulated with an explicit time integration scheme. The abradable coating is meshed with 1D abradable elements. At each time step, the contact is handled with a predictioncorrection procedure: a blade displacement is predicted and the facing abradable element is searched. When a penetration is detected, the force exerted by the abradable on the blade (computed based on the formulas of the preceding section) is used for the correction of the blade displacement. The wear of the abradable element is also computed for updating the coating topology. At t D 0, the blade is separated from the abradable coating by a positive clearance. The casing is considered to be rigid. It is perfectly circular at the beginning, and then progressively ovalized, such that at some point of the simulation, the blade comes into contact with the coating and begins to wear it.

Results

Note

In the following, amplitudes are normalized with respect to reference data for confidentiality purposes. The model presented in [START_REF] Batailly | Numerical-experimental comparison in the simulation of rotor/stator interaction through blade-tip/abradable coating contact[END_REF] revealed consistency with experimental data: the coating wear pattern and the modal response of the blade is similar to experiments [START_REF] Millecamps | Interaction aube-carter : contribution de l'usure de l'abradable et de la thermomécanique sur la dynamique d'aube[END_REF]. However, it needs to be calibrated numerically [START_REF] Legrand | Numerical investigation of abradable coating removal in aircraft engines through plastic constitutive law[END_REF] since its wear law is mainly numerical. Our goal here is to show that we can reproduce the same kind of results with the presented model, as a first step toward using phenomenologically based wear mechanisms. Parameter values are determined from our industrial partner data and literature [START_REF] Schmid | New high temperature abradables for gas turbines[END_REF][START_REF] Rabinowicz | Friction and wear of materials[END_REF]. Here we only use the adhesive and micro-rupture wear mechanisms. They are believed to be the predominant ones in the original experiment as incursions and temperature were low.

A wide angular velocity range ( 2 OE0:08 I 0:8) is considered, and one simulation is carried out for each angular velocity that remains constant over time. The results are analysed through two kinds of graphs: Wear maps (e.g. Fig. 5) the abradable coating wear profile facing the trailing edge-where most sophisticated patterns are observed-is plotted at the end of each simulation. For each angular velocity, meaning for one column of the map, a color code indicates the quantity of abradable removal along the casing circumference: from white where the coating is intact to black where maximum wear is observed [START_REF] Batailly | Unilateral contact induced blade/casing vibratory interactions in impellers: analysis for rigid casings[END_REF]. Interaction maps (e.g. Fig. 6) the Fourier transform (spectrum) of the blade edge displacement is plotted for each angular velocity. Low amplitudes are in blue, and frequency peaks in yellow and red. When looking at the wear maps in Fig. 5, the same kind of patterns can be noticed: strong wear around D 0:4 and D 0:7, with respectively 6 and 4 wear lobes. Both simulations are in accordance with the actual experiment [START_REF] Millecamps | Interaction aube-carter : contribution de l'usure de l'abradable et de la thermomécanique sur la dynamique d'aube[END_REF] where 6 lobes were found around D 0:4. Note however that the level of wear of the simulation with adhesive and micro-rupture wear laws is much less than with the elasto-plastic wear law. This is because these wear maps are taken after 50 blade turns, where there is no more wear in the case of the elasto-plastic model simulation, whereas it is still going on in the case of the adhesive and micro-rupture simulation.

The interaction maps (see Fig. 6) for the displacement at the edge of the blade also shows similar patterns: the engine order lines represent the contact due to the ovalization of the casing and are the same in both diagrams. Other peaks (inside the red ellipsis of Fig. 6) representing the excited eigenfrequencies of the blade are a match from an interaction map to the other, even if they are more blurry in the case of an adhesive and micro-rupture wear.

Preliminary results for a parameter study

The goal of this study is to compare wear maps for the same structural configuration-blade shape, clearance, casing ovalizationbut with different wear laws. In Fig. 7, two wear maps are displayed, each one with a different wear mechanism. Except for the level of wear, the patterns are very similar. This seems to indicate that the blade trajectory is mainly governed by the structural configuration of the blade. Note however that the interaction is not finished when the wear maps are extracted, and the wear patterns could diverge significantly from each other had the temporal integration been carried out longer.

Conclusion

The numerical simulation of blade/casing interaction in the presence of abradable wear is a complex subject. This article derived an approach for modeling wear of the coating, based on the phenomenology of abradable wear mechanisms. Each wear phenomenon is modeled by a simple analytical equation. Then the blade/casing interaction is solved numerically by a 3D explicit numerical scheme. The first results show that an implementation of wear based on these phenomena is capable of reproducing experimental results. A more systematic parameter study is ongoing. It aims at quantifying the influence of each mechanism on the blade dynamics. Also, additional investigations regarding experimental parameters are required in order to calibrate more precisely the proposed wear model parameters: sliding, scratching and toughness tests, analysis of machining chips... This model will enable the introduction of thermal effects through a state variable taking into account temperature and which the parameters presented on this study (such as the friction coefficient) may depend on. The model is also expected to reproduce more complex frequency response, as found in experiments [START_REF] Millecamps | Interaction aube-carter : contribution de l'usure de l'abradable et de la thermomécanique sur la dynamique d'aube[END_REF]. The final objective of this work is to compare the blade response with different kind of abradable materials in order to select the one with minimal induced vibratory response.

Figure 1 .

 1 Figure 1. Partial view of the fan stage of an aircraft engine. A zoom over the red square is depicted in Fig. 2

Figure 2 .

 2 Figure 2. Schematic representation of coating removal. Proportions are respected.

Figure 3 .

 3 Figure 3. Deck of card model for machining. Material is removed by successive shear bands

Figure 4 .

 4 Figure 4. Micro-rupture wear. A particle is formed when the blade impacts the coating

ΩFigure 5 .

 5 Figure 5. Wear maps for the comparative study

Figure 6 .

 6 Figure 6. Interaction maps for the comparative study

Figure 7 .

 7 Figure 7. Wear maps for the parameter study

Table 1 .

 1 Inputs and outputs of the model

	inputs	penetration of the blade into the abradable
		blade edge velocity
	outputs contact forces exerted onto the blade
		updated abradable surface geometry

Note on scalesIn the following, the macro scale refers to the scale of the blade ( 10 mm), the micro scale is the scale of the abradable rugosity, where the surface of the abradable can no longer be considered smooth with a well-defined normal ( 1 µm) and the meso scale is very small compared to the blade,

Note Formulas of the following paragraphs are given in the plane orthogonal to the axis of rotation, and some quantities such

The continuous chip in experiments is visually similar to a "deck of card" morphology.

Acknowledgments

Thanks go to Snecma for its technical and financial support.


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Background of advanced logistics systems

  Nathalie Fabbe-Costes To cite this version: Nathalie Fabbe-Costes. Background of advanced logistics systems. [Research Report] CRET (Université d'Aix-Marseille II) et OCDE. 1991. hal-01291005


Nathalie Fabbe-Costes

teaching and research institutions in France or abroad, or from public or private research centers.


Emmanuel Chasseigne
email: emmanuel.chasseigne@lmpt.univ-tours.fr.

Naoyuki Ichihara
email: ichihara@gem.aoyama.ac.jp.

Qualitative properties of generalized principal eigenvalues for superquadratic viscous Hamilton-Jacobi equations

This paper is concerned with the ergodic problem for superquadratic viscous Hamilton-Jacobi equations with exponent m > 2. We prove that the generalized principal eigenvalue of the equation converges to a constant as m → ∞, and that the limit coincides with the generalized principal eigenvalue of an ergodic problem with gradient constraint. We also investigate some qualitative properties of the generalized principal eigenvalue with respect to a perturbation of the potential function. It turns out that different situations take place according to m = 2, 2 < m < ∞, and the limiting case m = ∞.

Introduction

In this paper we study the ergodic problem for the following superquadratic viscous Hamilton-Jacobi equation with exponent m > 2:

λ -∆u + 1 m |Du| m -f = 0 in R N , (1.1)
where Du and ∆u denote the gradient and the Laplacian of u : R N → R, respectively, and f : R N → R is assumed to be continuous on R N and to vanish as |x| → ∞. The unknown of (1.1) is the pair of a real constant λ and a function u. We denote by λ m the generalized principal eigenvalue of (1.1) which is defined by

λ m := sup{λ ∈ R | (1.
1) has a continuous viscosity subsolution u }.

(1.2)

Here and in what follows, unless otherwise specified, every solution (subsolution, supersolution) u is understood in the viscosity sense. We refer, for instance, to [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF][START_REF] Koike | A beginner's guide to the theory of viscosity solutions[END_REF] for the definition and fundamental properties of viscosity solutions. The objective of this paper consists of two parts, which we present as A and B below.

A. Convergence as m → ∞. We study the convergence of λ m as m → ∞. More precisely, let us consider the following ergodic problem with gradient constraint: max λ -∆u -f, |Du| -1 = 0 in R N .

(1.3)

Let λ ∞ denote the generalized principal eigenvalue of (1.3) defined, similarly as (1.2), by the supremum of λ ∈ R such that (1.3) has a continuous viscosity subsolution u.

Then we prove that λ m converges to λ ∞ as m → ∞. In this sense, ergodic problem (1.3) can be regarded as the extreme case of (1.1) where m = ∞. Note that (1.3) has been studied by [START_REF] Hynd | The eigenvalue problem of singular ergodic control[END_REF][START_REF] Hynd | An eigenvalue problem for fully nonlinear elliptic equations with gradient constraints[END_REF] for functions f that are smooth, convex, and of superlinear growth as |x| → ∞. In these papers, λ ∞ is derived from the limit of δv δ (0) as δ → 0, where v δ is the solution to the following equation:

max δv δ -∆v δ -f, |Dv δ | -1 = 0 in R N .
The present paper provides another characterization of λ ∞ in terms of λ m under a different type of assumptions on f .

B. Qualitative properties. We introduce a real parameter β and consider (1.1) and (1.3) with βf in place of f . We are interested in qualitative properties of the generalized principal eigenvalue λ m = λ m,β with respect to β. In order to illustrate our main results briefly, we assume, for a moment, that f is nonnegative in R N with compact support (this can be relaxed, see Section 4). Then it turns out that there exists a critical value β c ≤ 0 such that λ m,β = 0 for all β ≥ β c , while λ m,β < 0 for all β < β c . Notice here that the value of β c , especially, its negativity depends sensitively on m and N. More specifically, the following three situations occur according to the choice of m:

(a) if m = 2, then β c = 0 for N = 1, 2 and β c < 0 for all N ≥ 3;

(b) if 2 < m < ∞, then β c = 0 for N = 1 and β c < 0 for all N ≥ 2;

(c) if m = ∞, then β c < 0 for all N ≥ 1.

The quadratic case (a) has been proved in [START_REF] Ichihara | Criticality of viscous Hamilton-Jacobi equations and stochastic ergodic control[END_REF]Theorem 2.5], and the second claim in (b) (i.e., the case where 2 < m < ∞ and N ≥ 2) is also suggested by [START_REF] Ichihara | The generalized principal eigenvalue for Hamilton-Jacobi-Bellman equations of ergodic type[END_REF]Theorem 2.4] in a slightly different context. The essential novelty of this paper, compared with [START_REF] Ichihara | Criticality of viscous Hamilton-Jacobi equations and stochastic ergodic control[END_REF][START_REF] Ichihara | The generalized principal eigenvalue for Hamilton-Jacobi-Bellman equations of ergodic type[END_REF], lies in the simultaneous derivation of (b) and (c) in combination with the convergence result obtained in part A. In particular, claim (c) for N ≥ 2 can be derived by passing to the limit of (b) as m → ∞. To the best of our knowledge, such a qualitative analysis of λ m,β , especially for m = ∞, seems to be new. We remark that we consider not only nonnegative functions f but also sign-changing ones, which lead to a more complex picture where two critical parameters β -≤ β + will play the role of the above β c . For instance, if N ≥ 2 and 2 < m ≤ ∞, then there exist β -< 0 < β + such that λ m,β = 0 for any β ∈ [β -, β + ], while λ m,β < 0 outside this interval. See Section 4 for details.

Our study of critical value β c is strongly motivated by the stochastic control interpretation of λ m,β . Loosely speaking, if 2 ≤ m < ∞, then the principal eigenvalue λ m,β coincides with the optimal value of the following ergodic stochastic control problem:

Minimize lim sup T →∞ 1 T E T 0 1 m * |ξ t | m * + βf (X ξ t ) dt , subject to X ξ t = √ 2W t + t 0 ξ s ds, t ≥ 0, (1.4)
where m * := m/(m -1), and W = (W t ) and ξ = (ξ t ) denote, respectively, an Ndimensional standard Brownian motion and an (F t )-adapted control process defined on some filtered probability space (Ω, F , P ;

(F t )). If f ≥ 0 in R N and β ≥ 0, then this is nothing but a minimization problem of the total cost (1/m * )|ξ t | m * + βf (X ξ t )
. The situation becomes delicate as far as β < 0. Intuitively, the controller of the optimization problem (1.4) falls into a trade-off situation between minimizing the cost (1/m * )|ξ t | m * and maximizing the reward |β|f (X ξ t ). The dominant term depends on the magnitude of |β|, and the critical value β c is determined as the threshold at which the controller changes his/her optimal choice: either "minimize cost" or "maximize reward". In particular, the negativity of β c implies the existence of such "phase transition", which we intend to characterize in the present paper.

As to the limiting case where m = ∞, the value λ ∞,β is related to the following singular ergodic stochastic control problem:

Minimize lim sup

T →∞ 1 T E |η| T + T 0 βf (X η t ) dt , subject to X η t = √ 2W t + η t , t ≥ 0,
where η = (η t ) stands for an (F t )-adapted control process of bounded variations, and |η| T denotes its bounded variation norm. We refer, for instance, to [START_REF] Menaldi | Singular ergodic control for multidimensional Gaussian processes[END_REF] and references therein for more information on singular ergodic stochastic control and associated PDEs with gradient constraint. See also [START_REF] Ichihara | Large time asymptotic problems for optimal stochastic control with superlinear cost[END_REF][START_REF] Ichihara | Criticality of viscous Hamilton-Jacobi equations and stochastic ergodic control[END_REF][START_REF] Ichihara | The generalized principal eigenvalue for Hamilton-Jacobi-Bellman equations of ergodic type[END_REF] for the stochastic control interpretation of λ m,β for 2 ≤ m < ∞. In this paper, we focus only on the PDE aspect and do not discuss its probabilistic counterpart.

The organization of the paper is as follows. In the next section, we discuss the solvability of (1.1). Specifically, we prove that, for any λ ≤ λ m , there exists a viscosity solution u of (1.1). In Section 3, we prove the convergence of λ m as m → ∞. Section 4 is devoted to qualitative properties of λ m,β with respect to β.

Solvability of (1.1)

We collect some notation used throughout the paper. For any R > 0, B R stands for the open ball of radius R, centered at the origin. For given k ∈ N ∪ {0}, γ ∈ (0, 1], and p ∈ [1, ∞], let C k,γ (R N ) and W k,p (R N ) denote local Hölder (or Lipschitz if k = 0, γ = 1) spaces and Sobolev spaces, respectively. Recall that the Hölder/Lipschitz norm over B R of functions in C k,γ (R N ) depends on R, in general. We also denote by C ∞ c (R N ) the set of smooth functions with compact support. Finally, let C 0 (R N ) stand for the totality of continuous functions f ∈ C(R N ) such that f (x) → 0 as |x| → ∞.

Let m > 2 and consider the ergodic problem

λ -∆u + 1 m |Du| m = f in R N , u(0) = 0, (2.1)
where the constraint u(0) = 0 is imposed to avoid the ambiguity of additive constant with respect to u. Throughout this paper, we assume without mentioning that f satisfies the following:

(A1) f ∈ C 0 (R N ).
To begin with, we recall some regularity estimates that will be needed repeatedly.

Theorem 2.1. Let α := (m -2)/(m -1). (i) For any R > 0, there exists a constant M R > 0 such that

|u(x) -u(y)| ≤ M R |x -y| α , x, y ∈ B R ,
for any locally bounded upper semicontinuous viscosity subsolution u of (2.1), where M R depends on max B R |f -λ|, but is independent of any large m > 2.

(ii) Suppose that f ∈ C 0,1 (R N ). Then, for any R > 0, there exists a constant

K R > 0 such that |u(x) -u(y)| ≤ K R |x -y|, x, y ∈ B R ,
for any continuous viscosity solution u of (2.1), where K R may depend on the sup-norm and the Lipschitz norm of f -λ over a larger ball, say B R+1 , but is independent of any large m > 2.

Proof. This theorem is a direct consequence of [2, Theorems 1.1 and 3.1]. Notice here that the gradient term of the equation in [START_REF] Capuzzo Dolcetta | Hölder estimates for degenerate elliptic equations with coercive Hamiltonians[END_REF] is not (1/m)|Du| m but |Du| p with p > 2. However, by a careful reading of their proof, one can see that M R and K R can be taken uniformly with respect to any large m > 2.

It is obvious from Theorem 2.1 that any locally bounded upper semicontinuous viscosity subsolution of (2.1) belongs to C 0,α (R N ) with α = (m -2)/(m -1). Taking this fact into account, one can redefine the generalized principal eigenvalue of (2.1) by

λ m := sup{λ ∈ R | (2.1) has a viscosity subsolution u ∈ C 0,α (R N )}. (2.2) Note here that λ m = -∞. Indeed, (λ, u) = (inf R N f, 0) is a viscosity subsolution of (2.1), so that λ m ≥ inf R N f > -∞.
We first observe a few properties of λ m that can be verified by its very definition. In what follows, we often use the notation λ m (f ) to emphasize the dependence of λ m on the function f . Proposition 2.2. Let f, g ∈ C 0 (R N ). We denote by λ m (f ), λ m (g) the associated generalized principal eigenvalues of (2.1), respectively. Then the following

(i)-(iii) hold. (i) f ≤ g in R N implies λ m (f ) ≤ λ m (g). (ii) (1 -δ)λ m (f ) + δλ m (g) ≤ λ m ((1 -δ)f + δg) for any δ ∈ (0, 1). (iii) λ m (f + c) = λ m (f ) + c for any c ∈ R.
Proof. We first show (i). Let u ∈ C 0,α (R N ) be a viscosity subsolution of (2.1) with f . Then it is also a viscosity subsolution of (2.1) with g in place of f . Hence, λ m (f ) ≤ λ m (g) by definition. One can also verify (ii) similarly. The validity of (iii) is obvious from the definition of λ m . Hence, we have completed the proof.

The following result implies that, if f ∈ C 0,1 (R N ), then "viscosity subsolution" in the definition of λ m can be replaced by "classical subsolution".

Proposition 2.3. Suppose that f ∈ C 0,1 (R N ). Then, for any λ < λ m , there exists a classical subsolution u ∈ C ∞ (R N ) of (2.

1).

Proof. Fix any λ < λ m and construct a smooth subsolution u of (2.1). To this end, we follow the ingenious idea due to [START_REF] Barles | On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations[END_REF][START_REF] Krylov | On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients[END_REF]

. Set f ε (x) := min |e|<ε f (x + e) for ε > 0. Then, f ε ∈ C 0,1 (R N ) ∩ C 0 (R N ), f ε ≤ f in R N , and {f ε } converges to f uniformly in R N as ε → 0. Let λ (ε)
m be the generalized principal eigenvalue of (2.1) with f ε in place of f . Then, in view of Proposition 2.2 and by choosing ε > 0 sufficiently small, we may assume that λ < λ (ε) m ≤ λ m . In particular, for the above λ, there exists a viscosity subsolution u

(ε) ∈ C 0,α (R N ) of (2.1) with f ε in place of f . Since f ε ( • -e) ≤ f in R N for any |e| < ε, one can also see that u (ε) ( • -e) is a viscosity subsolution of (2.1) for any |e| < ε. Now, let {ρ δ } δ>0 ⊂ C ∞ c (R N ) be a family of mollifier functions, i.e., ρ δ ≥ 0 in R N , R N ρ δ (x) dx = 1, and supp ρ δ ⊂ B δ for all δ > 0. Set u (ε) δ (x) := (u (ε) * ρ δ )(x)
for δ < ε, where * stands for the usual convolution. Then, by noting the convexity of p → (1/m)|p| m , one can see, similarly as in the proof of [START_REF] Barles | On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations[END_REF]Lemma 2.7], that u := u (ε) δ is a smooth viscosity subsolution of (2.1). Since a smooth viscosity subsolution is a classical subsolution, we have completed the proof.

We next verify that λ m is nonpositive.

Proposition 2.4. One has λ m ≤ 0. In particular, λ m is finite.

Proof. In view of Proposition 2.2 (i), it suffices to consider the case where f ∈ C 0,1 (R N ). Fix any λ < λ m , and let u ∈ C ∞ (R N ) be a classical subsolution of (2.1). Existence of such u is guaranteed by virtue of Proposition 2.3. Then, for any nonnegative test

function η ∈ C ∞ c (R N ) such that R N η(x) m * dx = 1, where m * := m/(m -1), we have λ R N η m * dx + R N Du • D(η m * ) dx + 1 m R N |Du| m η m * dx ≤ R N f η m * dx. Noting D(η m * ) = η m * /m Dη and Du • D(η m * ) ≤ 1 m |Du| m η m * + 1 m * |Dη| m * ,
we see that, for any ε > 0,

λ = λ R N η m * dx ≤ ε + R N (f -ε) + η m * dx + 1 m * R N |Dη| m * dx,
where (f -ε) + denotes the positive part of f -ε. Furthermore, if we define η δ (x) := δ N/m * η(δx) for δ > 0, which still satisfies R N η δ (x) m * dx = 1 for any δ > 0, then plugging this into the above η, we have

λ ≤ ε + R N (f (x) -ε) + η δ (x) m * dx + δ m * m * R N |Dη(x)| m * dx. (2.3)
Sending δ → 0, we obtain λ ≤ ε. Since ε > 0 and λ < λ m are arbitrary, we conclude that λ m ≤ 0. Hence, we have completed the proof.

The following proposition states a stability of λ m (f ) with respect to f .

Proposition 2.5. Let f, g ∈ C 0 (R N ). Then |λ m (f ) -λ m (g)| ≤ max R N |f -g|. In particular, if {f n } ⊂ C 0 (R N ) converges as n → ∞ to some f ∈ C 0 (R N ) uniformly in R N , then λ m (f n ) converges to λ m (f ) as n → ∞. Moreover, if {u n } is a family of viscosity solutions of (2.1) with f = f n and λ = λ m (f n ), then, along a suitable subsequence, {u n } converges as n → ∞ to a viscosity solution u of (2.1) with λ = λ m (f ) locally uniformly in R N . Proof. Since f ≤ g + max R N (f -g) + in R N , we see, in view of Proposition 2.2 (i) and (iii), that λ m (f ) -λ m (g) ≤ max R N (f -g) + .
Changing the role of f and g, we obtain the first claim. The second claim is obvious from the first one. In order to verify the last claim, we observe from Theorem 2.1 that {u n } pre-compact in C(R N ). Applying Ascoli-Arzela theorem, we see that {u n } converges, along a suitable subsequence, to a function u ∈ C 0,α (R N ) locally uniformly in R N . By the stability property of viscosity solutions, we conclude that u is a viscosity solution of (2.1) with λ = λ m (f ). Hence, we have completed the proof.

We now state the main result of this section.

Theorem 2.6. For any λ ≤ λ m , there exists a viscosity solution u ∈ C 0,α (R N ) of (2.1). Moreover, if f ∈ C 0,1 (R N ), then for any λ ≤ λ m , there exists a classical solution u ∈ C 2 (R N ) of (2.1).

Proof. We first prove the latter claim. Let f ∈ C 0,1 (R N ) and fix any λ < λ m . Then, by virtue of Proposition 2.3, there exists a classical subsolution u -∈ C ∞ (R N ) of (2.1). Fix any R > 0 and consider the Dirichlet problem

λ -∆u + 1 m |Du| m -f = 0 in B R , u = u -on ∂B R , (2.4)
where

∂B R := {x ∈ R N | |x| = R}.
Then it is known (e.g. [11, Théorème I.1]) that there exists a unique classical solution u R ∈ C 2,γ (B R ) of (2.4) for some γ ∈ (0, 1). By virtue of Theorem 2.1 together with the Schauder estimate, we see that {u R -u R (0)} R>0 is pre-compact in C 2 (R N ). In particular, letting R → ∞ along a suitable subsequence {R j } if necessary, we see that {u R j } and their first and second derivatives converge as j → ∞ to a function u ∈ C 2 (R N ) and its corresponding derivatives, respectively, locally uniformly in R N , and that u is a classical solution of (2.1). In order to see that (2.1) with λ = λ m has a classical solution, we choose any sequence {λ (n) } such that λ (n) → λ m as n → ∞, and let u (n) denote the associated classical solution to (2.1) with λ = λ (n) . Then one can see, similarly as above, that {u

(n) -u (n) (0)} is pre-compact in C 2 (R N ).
Passing to the limit as n → ∞ along a suitable subsequence if necessary, we conclude that (2.1) with λ = λ m has a classical solution.

We now prove the former claim. Fix any f ∈ C 0 (R N ) and choose a sequence

{f n } ⊂ C ∞ (R N ) ∩ C 0 (R N ) which converges as n → ∞ to f uniformly in R N . Let λ (n)
be the generalized principal eigenvalue of (2.1) with f n in place of f . Then, in view of Proposition 2.5, we observe that λ (n) → λ m as n → ∞. Now, fix any λ < λ m . We may assume without loss of generality that λ < λ (n) for any n ≥ 1. For each n ≥ 1, let u (n) ∈ C 2 (R N ) denote a classical solution of (2.1) with f n in place of f . Then, by Theorem 2.1 and the stability of viscosity solutions, we conclude that, along a suitable subsequence, {u (n) } converges as n → ∞ to a viscosity solution u ∈ C 0,α (R N ) of (2.1) locally uniformly in R N . We can also construct a viscosity solution of (2.1) with λ = λ m similarly as in the previous case. Hence, we have completed the proof. Theorem 2.6 implies that the following representation formula for λ m holds:

λ m = max{λ ∈ R | (2.1) has a viscosity solution u ∈ C 0,α (R N )}. Furthermore, if f ∈ C 0,1 (R N ), then λ m = max{λ ∈ R | (2.1) has a classical solution u ∈ C 2 (R N )}.

Convergence as m → ∞

This section is devoted to the convergence of λ m as m → ∞. To be precise, we rewrite the limiting equation

max λ -∆u -f, |Du| -1 = 0 in R N , u(0) = 0, (3.1)
and redefine the generalized principal eigenvalue of (3.1) by

λ ∞ := sup{λ ∈ R | (3.1) has a viscosity subsolution u ∈ C 0,1 (R N )}. (3.2)
The following result is crucial to our convergence result.

Proposition 3.1. Let {m k } ⊂ R be an increasing sequence such that m k → ∞ as k → ∞. Let (λ m k , u k ) be a solution of (2.1) with m = m k for each k. Suppose that λ k converges to some λ ∈ R as k → ∞. Then, up to a subsequence, {u k } converges as k → ∞ to a function u ∈ C 0,1 (R N ) locally uniformly in R N . Moreover, (λ, u) is a solution of (3.1).

Proof. In view of Theorem 2.1 (i), we see that there exist a subsequence of {u k }, which we denote by {u k } again, and a function u ∈ C(R N ) with u(0) = 0 such that u k → u as k → ∞ locally uniformly in R N . Note that u ∈ C 0,1 (R N ) since the constant M R in Theorem 2.1 (i) does not depend on any large m > 2.

We now verify that u is a viscosity solution of (3.1). We first prove the subsolution property. Fix any x 0 ∈ R N and let φ ∈ C 2 (R N ) be any function such that max R N (uφ) = (u -φ)(x 0 ). As is standard, one can assume that the maximum is strict, so that there exists a sequence {x k } ⊂ R N such that u k -φ attains its local maximum at x k and x k → x 0 as k → ∞. Then, by the subsolution property of u k , we see that

λ m k -∆φ(x k ) + 1 m k |Dφ(x k )| m k -f (x k ) ≤ 0. ( 3.3)
We now suppose that |Dφ(x 0 )| > 1. Then there exists an η > 0 such that |Dφ(x k )| ≥ 1 + η for all sufficiently large k. In particular, we have

1 m k (1 + η) m k ≤ -λ m k + ∆φ(x k ) + f (x k ).
Sending k → ∞, we get a contradiction since the right-hand side remains bounded, whereas the left-hand side goes to infinity as k → ∞. Hence, we have |Dφ(x 0 )| ≤ 1. Furthermore, letting k → ∞ in (3.3), we conclude that λ -∆φ(x 0 ) -f (x 0 ) ≤ 0, which implies that u is a viscosity subsolution of (3.1).

We next prove the supersolution property. Fix any

x 0 ∈ R N and let ψ ∈ C 2 (R N ) be such that min R N (u -ψ) = (u -ψ)(x 0 ). If |Dψ(x 0 )| ≥ 1,
then there is nothing to prove, so we assume that |Dψ(x 0 )| < 1. In particular, there exists some η > 0 such that |Dψ(x k )| ≤ 1 -η for all sufficiently large k. Furthermore, there exists a sequence {x k } ⊂ R N such that u k -ψ attains its local minimum at x k and x k → x 0 as k → ∞. Then, by the supersolution property of u k , we have

λ m k -∆ψ(x k ) + 1 m k |Dψ(x k )| m k -f (x k ) ≥ 0.
Letting k → ∞ in the above inequality, we obtain λ -∆ψ(x 0 ) -f (x 0 ) ≥ 0. Hence, we conclude that u is a viscosity supersolution of (3.1).

We are now in position to state the main result of this section. In particular, λ ≤ λ ∞ .

To prove the reverse inequality, we set λ := lim inf m→∞ λ m . Fix any ε > 0 and let u ∈ C 0,1 (R N ) be a viscosity subsolution of (3.1) with λ = λ ∞ -ε. Then, noting that |Du| ≤ 1 in R N in the viscosity sense, we see that, for any m > 2, u is a viscosity subsolution of

λ ∞ -ε - 1 m -∆u + 1 m |Du| m -f ≤ 0 in R N . This implies λ ∞ -ε -1/m ≤ λ m for any m > 2, so that λ ∞ -ε ≤ λ. Since ε > 0 is arbitrary, we obtain λ ∞ ≤ λ ≤ λ ≤ λ ∞ .
Hence, we have completed the proof.

The next result states that Proposition 2.3 remains valid for m = ∞.

Proposition 3.3. Suppose that f ∈ C 0,1 (R N ). Then, for any λ < λ ∞ , there exists a classical subsolution u ∈ C ∞ (R N ) of (3.1). In particular,

λ ∞ = sup{λ ∈ R | (3.1) has a classical subsolution u ∈ C ∞ (R N )}.
Proof. Fix any λ < λ ∞ , and let

{ρ δ } δ>0 ⊂ C ∞ c (R N ) be such that ρ δ ≥ 0 in R N , R N ρ δ (x)dx = 1
, and supp ρ δ ⊂ B δ for all δ > 0. Let {λ m k } be a sequence of generalized principal eigenvalues of (2.1) with m = m k such that λ m k → λ ∞ as k → ∞. Such a sequence exists by virtue of Theorem 3.2. In what follows, we assume that λ < λ m k for all k ≥ 1. For each k ≥ 1, let u (k) ∈ C 2 (R N ) be a classical solution of (2.1) with m = m k (and the common λ). Taking a subsequence if necessary, one may also assume that {u (k) } converges as k → ∞ to a viscosity solution u ∈ C 0,1 (R N ) of (3.1) locally uniformly in R N . Now we set u (k) δ := u (k) * ρ δ , u δ := u * ρ δ , and f δ := f * ρ δ , where * stands for the usual convolution. We choose δ > 0 so small that sup

R N |f δ -f | < λ m k -λ for all k ≥ 1. Then, since u (k) is a classical solution of (2.1) with m = m k , we see that u (k) δ enjoys the inequality λ -∆u (k) δ + 1 m k |Du (k) δ | m k -f ≤ 0 in R N
for all k ≥ 1 and any sufficiently small δ > 0. This implies that u

(k) δ is also a classical subsolution of λ -∆u (k) δ -f ≤ 0 in R N .
Letting k → ∞ and noting the stability of viscosity solutions, we conclude that u δ is a smooth viscosity subsolution, and therefore, a classical subsolution of the same equation. On the other hand, since |u| ≤ 1 a.e. in R N , which can be verified as in the proof of Proposition 3.1, we see that |u δ | ≤ 1 in R N . Hence, u δ enjoys (3.1) at every point x ∈ R N , and we have completed the proof.

Remark 3.4. The first claim of Theorem 2.6 remains true for m = ∞. Namely, for any λ ≤ λ ∞ , there exists a viscosity solution u ∈ C 0,1 (R N ) of (3.1). To see this, fix any λ < λ ∞ and choose an m 0 so large that λ m > λ for any m > m 0 . Let u m , for m > m 0 , be a viscosity solution of (2.1). Then, by Proposition 3.1, we conclude that, along a subsequence, {u m } converges to a viscosity solution u ∈ C 0,1 (R N ) of (3.1). The existence of a viscosity solution u to (3.1) with λ = λ ∞ has been proved in Theorem 3.2. Hence, the first claim of Theorem 2.6 is also valid for m = ∞. We do not know if the second claim remains true for m = ∞.

Qualitative properties

In this section, we introduce real parameter β and consider the ergodic problem for m > 2:

λ -∆u + 1 m |Du| m -βf = 0 in R N , u(0) = 0, (4.1)
and its limiting equation as m → ∞:

max{λ -∆u -βf, |Du| -1} = 0 in R N , u(0) = 0. (4.2)
In the rest of this paper, we impose the following assumption on f in addition to (A1):

(A2) f ≡ 0 and |f (x)| ≤ C 0 x -m * in R N for some C 0 > 0, where x := (1 + |x| 2 ) 1/2
and m * := m/(m -1) with the convention that m * := 1 for m = ∞.

Let λ m,β and λ ∞,β be the generalized principal eigenvalues of (4.1) and (4.2), respectively. In view of Proposition 2.4 and Theorem 3.2, we observe that λ m,β ≤ 0 for any β ∈ R and 2 < m ≤ ∞. It is also easy to see that λ m,0 = 0 for any 2 < m ≤ ∞. Furthermore, we have the following.

Proposition 4.1. Let 2 < m ≤ ∞. If f -≡ 0, then λ m,β → -∞ as β → ∞, and if f -≡ 0, then λ m,β = 0 for any β > 0. Symmetrically, if f + ≡ 0, then λ m,β → -∞ as β → -∞, and if f + ≡ 0, then λ m,β = 0 for any β < 0.
Proof. We first consider the case where 2 < m < ∞. In view of Proposition 2.5, we may assume that f ∈ C 0,1 (R N ). Suppose that f -≡ 0, and choose any η ∈ C ∞ c (R N ) such that η ≥ 0 in R N , R N η(x) m * dx = 1, and supp η ⊂ supp f -. Then, taking a classical solution u ∈ C 2 (R N ) of (4.1) with λ = λ m,β , multiplying both sides of (4.1) by η, and applying integration by parts, we see as in the proof of Proposition 2.4 that

λ m,β ≤ -β R N f -(x)η(x) m * dx + 1 m * R N |Dη(x)| m * dx. (4.3)
Since the integral of f -η m * over R N is strictly positive, we conclude that λ m,β → -∞ as β → ∞. We now take the limit as m → ∞ in (4.3). Then, since m * → 1 as m → ∞, we see from Theorem 3.2 that the claim is also valid for m = ∞.

We now suppose that f -≡ 0. Then, for any β > 0, the pair (λ, u) = (0, 0) is a subsolution of (4.1) and (4.2). This implies that λ m,β = 0 for any 2 < m ≤ ∞ and β > 0. By choosing -f and -β in place of f and β, respectively, we see that the latter claim of this proposition is also valid. Hence, we have completed the proof.

From Propositions 2.2 (ii), 2.4, and 4.1, for each 2 < m ≤ ∞, one can define β -, β + by

β + := max{β ∈ R | λ m,β = 0}, β -:= min{β ∈ R | λ m,β = 0}.
Obviously, -∞ ≤ β -≤ 0 ≤ β + ≤ ∞, and β + (resp. β -) is finite if and only if f -≡ 0 (resp. f + ≡ 0). Moreover, since f ≡ 0, either β + or β -is finite. As is mentioned in the introduction, we wish to know whether 0 < |β ± | (< ∞). The main result of this section can be stated as follows.

Theorem 4.2. Let β + be defined as above, and let f -≡ 0.

(i) Suppose that N ≥ 2 and 2 < m ≤ ∞. Then β + > 0.

(ii) Suppose that N = 1 and 2 < m < ∞. Then β + = 0.

(iii) Suppose that N = 1 and m = ∞. Then

β + > 0 provided f -∈ L 1 (R).
Changing (β, f ) into (-β, -f ), one has the following symmetrical result as a corollary of Theorem 4.2.

Corollary 4.3. Let β -be defined as above, and let f + ≡ 0. (i) Suppose that N ≥ 2 and 2 < m ≤ ∞. Then β -< 0. (ii) Suppose that N = 1 and 2 < m < ∞. Then β -= 0. (iii) Suppose that N = 1 and m = ∞. Then β -< 0 provided f + ∈ L 1 (R).

Remark 4.4. If N ≥ 2 and f is sign-changing, then β -< 0 < β + for any 2 < m ≤ ∞. From the ergodic stochastic control point of view, this implies that there exist two different critical points β + and β -at which the controller changes his/her optimal strategy. We remark that, if f is nonnegative or nonpositive in R N , then there is only one such critical point.

In the rest of this section, we prove (i)-(iii) of Theorem 4.2 one by one. The key to the proof of claim (i) is the following estimate.

β 0 := (N -m * ) m * m * C 0 > 0,
where m * := m/(m -1) and C 0 > 0 is the constant in (A2). Then, for any |β| ≤ β 0 , there exists a subsolution u ∈ C ∞ (R N ) of (4.1) with λ = 0.

Proof. We define u : R N → R by u(x) := (K/α) x α , where α = (m -2)/(m -1) and K > 0 is some constant that will be specified later. Then, by direct computations, we see that Du = K x -m * x and ∆u = KN x -m * -Km * x -m * -2 |x| 2 . Thus,

-∆u + 1 m |Du| m = x -m * -KN + Km * |x| 2 x -2 + K m m |x| m x -mm * = x -m * -KN + Km * |x| 2 x -2 + K m m |x| m x -m ≤ x -m * -(N -m * )K + K m m . Since the function K → f (K) := (K m /m)-(N-m * )K attains its minimum -(1/m * )(N- m * ) m * at K = (N -m * ) 1/(m-1) =: K m , we choose K = K m in the definition of u to obtain -∆u + 1 m |Du| m + βf ≤ x -m * |β|C 0 - 1 m * (N -m * ) m * in R N .
This implies that u is a subsolution of (2.1) with λ = 0 provided |β| ≤ β 0 . Hence, we have completed the proof.

As a corollary of this proposition, one can prove claim (i) of Theorem 4.2.

Proof of Theorem 4.2 (i). Let β 0 be the constant taken from Proposition 4.5. Then, it is obvious that β + ≥ β 0 > 0 for any 2 < m < ∞. Moreover, since m * → 1 as m → ∞, we see that β + ≥ β 0 ≥ (N -1)/(2C 0 ) > 0 for any large m. Hence, letting m → ∞ and noting that λ m,β converges to λ ∞,β as m → ∞ for any β ∈ R, we conclude that λ ∞,β = 0 for any β ≤ (N -1)/(2C 0 ). This yields that β + > 0 for N ≥ 2 and m = ∞. Hence, we have completed the proof.

Remark 4.6. In the case where N ≥ 2 and 2 < m < ∞, the positivity β + > 0 has been observed in [8, Proposition 2.4] when f ∈ C 0,1 (R N ). The new ingredient here is that we have an explicit lower bound of β + , uniform in m, which leads to the positivity of β + not only for 2 < m < ∞ but also for m = ∞. Recall that β + = 0 for N = m = 2 (see [START_REF] Ichihara | Criticality of viscous Hamilton-Jacobi equations and stochastic ergodic control[END_REF]). This exhibits a striking contrast between quadratic and superquadratic cases.

In what follows, we concentrate on the case where N = 1, in which case the ergodic problem (4.1) takes the form

λ -u ′′ + 1 m |u ′ | m -βf = 0 in R, u(0) = 0. ( 4.4)
We first prove claim (ii) of Theorem 4.2.

Proof of Theorem 4.2 (ii). We may assume without loss of generality that f ∈ C 0,1 (R N ). We prove that λ m,β < 0 for any β > 0. We argue by contradiction assuming that λ m,β = 0 for some β > 0. Let C > 0 be such that C m = max R N (βf ) -, and let u ∈ C 2 (R) be a classical solution of (4.4) with λ = 0. Then, we see that

-u ′′ + 1 m |u ′ | m = βf ≥ -C m in R.
By changing the variable such as s = u ′ (x)/C, we have

C m-1 x ≥ u ′ (x)/C u ′ (0)/C m |s| m + m ds ≥ - R m |s| m + m ds > -∞ for all x ∈ R.
Sending x → -∞, we get a contradiction. Hence, λ m,β < 0 for all β > 0.

We finally prove claim (iii) of Theorem 4.2. Let N = 1 and m = ∞. In this case, (4.2) can be written as max{λ

-u ′′ -βf, |u ′ | -1} = 0 in R, u (
L := R f -(u)du, K := sup y x -f (u)du x, y ∈ R, x < y .
Then 2/L ≤ β + ≤ 2/K, where 2/L := 0 for L = ∞ and 2/K := 0 for K = ∞.

Proof. We first show that 2/L ≤ β + . We may assume L < ∞, otherwise the inequality is obvious. Notice here that L > 0 by assumption. We set β 0 := 2/L and construct a classical subsolution u ∈ C 2 (R) of ( 4

is a classical solution to (4.6). We now choose

C := 1 L 0 -∞ f -(u)du - ∞ 0 f -(u)du .
Then, noting that u ′ (x) = β 0 F (x) + C for all x ∈ R, we have

u ′ (x) ≤ 2 L ∞ 0 f -(u)du + C = 1, u ′ (x) ≥ - 2 L 0 -∞ f -(u)du + C = -1
for all x ∈ R. Hence, u with the above C is a subsolution of (4.5) with λ = 0 and β = β 0 , which implies that β + ≥ 2/L. We next show that β + ≤ 2/K. Recall that K > 0 by assumption. We argue by contradiction assuming that β + > 2/K. Fix any β such that 2/K < β < β + . Then, λ ∞,β = 0 by the definition of β + . Fix an arbitrary δ > 0. Then, in view of Proposition 3.3, there exists a classical subsolution u ∈ C ∞ (R) of (4.5) with λ = -δ. In particular, we have -δ -u ′′ -βf ≤ 0, |u ′ | ≤ 1 in R.

This yields that, for any x, y ∈ R with x < y, Letting δ → 0 and taking the supremum over all x, y ∈ R such that x < y, we obtain βK ≤ 2, which is a contradiction. Hence, we have completed the proof.

Claim (iii) of Theorem 4.2 is a direct consequence of the above proposition.

Remark 4.8. Suppose that f + ≡ 0, that is, f ≤ 0 in R. Then L = K = R |f (u)|du, so that β + = 2/ R |f (u)|du. This implies that β + > 0 if and only if f ∈ L 1 (R). where C ∈ R is a constant. Then, similarly as in the proof of Proposition 4.7, we see that u is a classical solution of (1.3) for any C ∈ [-1/2, 1/2]. In particular, uniqueness for u does not hold without any growth condition as |x| → ∞. We remark here that, if N = 1 and f is convex and superlinear with respect to x, then, up to an additive constant, there exists only one viscosity solution u of (1.3) which satisfies u(x)/|x| → 1 as |x| → ∞ (see [START_REF] Hynd | An eigenvalue problem for fully nonlinear elliptic equations with gradient constraints[END_REF]Proposition 5.1]). At this stage, we do not know any uniqueness result for (1.3) under our assumptions (A1)-(A2).

Theorem 3 . 2 .

 32 Let λ m and λ ∞ be the generalized principal eigenvalues of (2.1) and (3.1), respectively. Then, λ m converges to λ ∞ as m → ∞. Moreover, equation (3.1) with λ = λ ∞ has a viscosity solution u ∈ C 0,1 (R N ).Proof. Set λ := lim sup m→∞ λ m . Note that λ ≤ 0 in view of Proposition 2.4. Let (λ m k , u m k ) be a sequence of solutions to (2.1) with m = m k such that λ m k → λ as k → ∞. Then, by taking a subsequence if necessary, we see from Proposition 3.1 that {u m k } converges to a viscosity solution u ∈ C 0,1 (R N ) of (3.1) locally uniformly in R N .

Proposition 4 . 5 .

 45 Let N ≥ 2 and 2 < m < ∞. Set

Proposition 4 . 7 .

 47 Let N = 1 and m = ∞. Suppose that f -≡ 0, and set

. 5 ) 0 F

 50 with λ = 0 and β = β 0 . Let us consider the linear equation-u ′′ + β 0 f -= 0 in R, u(0) = 0. (4.6) Then, for any C ∈ R, the function u ∈ C 2 (R) defined by u(x) = β 0 x (y)dy + Cx, F (y) := y 0 f -(u)du, (

  (s) + δ)ds = u ′ (y) -u ′ (x) + δ(y -x) ≤ 2 + δ(y -x).

Remark 4 . 9 . 0 F 0 ( 1

 49001 As far as the uniqueness for u, up to an additive constant, is concerned, equation (1.3) with λ = λ ∞ may have multiple solutions in general. Indeed, let N = 1 and f (x) := -(1 -|x|) + in (1.3). Then, in view of Remark 4.8, it is not difficult to observe that λ ∞ = 0. Furthermore, we define u : R → R by u(x) := x (y)dy + Cx, F (y) := y -|u|) + du,

Acknowledgment

The first author's research was partially supported by Spanish grant MTM2011-25287. The second author's research was partially supported by JSPS KAKENHI Grant Number 15K04935.