A high surface quality is necessary for automotive industry especially for exterior parts. Most outdoors polymers are painted for aesthetic reasons as metallic parts. The quality aspect must be indistinctive of the nature of the substrate, metal or polymer, to produce a global high quality aspect of the vehicle. Whatever the material, metal or polymer, sometimes defects appear in small locations which lead to the rejection of the automotive part. Here is presented a specific defect, aligned dots on the painted polymer part which are not present before painting process. The analytical method is almost universal but the defect is specific. It results from the conjugation of polymer nature and process and from the nature of painting.

I. INTRODUCTION.

High gloss surfaces with a well controlled color are the main characteristics of body-car painting. For design reasons the color should be homogeneous on all the exterior surfaces of the car, even if a metal or interference paintings are applied. The design imposes a global appearance with various material exhibiting very different properties. For example stiff door and roof-car are necessary to preserve safety of passengers while soft carbumpers are necessary for pedestrians and to recover the initial aspect of the car after a little hurt. The way is to use very different materials (metals, polymers) which are juxtaposed and painted [START_REF] Andrews | Black and beyondcolour and the mass-produced motor car[END_REF]. Polymers are widely used to form bumpers, but also door profiles, spoilers and supports for registration plate. The choice of polymer results from weight, mechanical properties and from design considerations. Complex shapes with low radius curvature are easy to produce with polymer materials, by injection molding and much more difficult by metal forming. Some polymers, like polypropylene and polyethylene, require a surface treatment (flaming, plasma …) to bond the paint-layer and to insure a good adhesion. Many other polymers (polyamide, ABS-PC, …) include chemical functions to form bonds with painting. As an effect of a good compromise shock-resistance, weight and price, elastomer-modified polypropylene is widely used to form E.Raynaud is (on leave) with MINES-Paristech, CEMEF, Centre de Mise en Forme des Matériaux, CNRS UMR 7635, BP207 1 rue Claude Daunesse 06904 Sophia-Antipolis cedex France E. Darque-Ceretti is with the same laboratory (e-mail: evelyne.darqueceretti @ensmp.fr).

B. Monasse is with the same laboratory (corresponding author phone: 330493957434; fax: 330492389752; e-mail: Bernard.monasse@ensmp.fr).

P. Duquesne is with Mecaplast, 4 av Prince Albert II-98014 Monaco (pduquesne@mecaplast.com) F. Georgi is with the same enterprise(fgeorgi@mecaplast.com) bumpers. Other parts (spoilers, support for registrationplate) need a higher rigidity. So ABS-PC (acrylonitrilebutadiene-styrene 65% polycarbonate 35% blend) is preferred for this application [START_REF] Khan | Rheological and mechanical properties of ABS/PC blends[END_REF]. After injection-molding, ABS-PC is easily painted as a result of its chemical functions.

Whatever the material (iron, aluminum, polypropylene, ABS-PC, …) the painting is usually applied without any trouble for wide diffusion products. A wide range of defects are observed such as peeling-off the paint layer, local change of brightness or of color, pit or pluck of the paint. These defects are specific of the material, of the paint and processing conditions. In automotive industry, their appearance is redhibitory for the aspect quality of the vehicle and implies a rejection of the part [START_REF] Wolff | Analysis of paint defects by mass spectroscopy (LAMMA®/ToF-SIMS)[END_REF].

The visual defect here analyzed consists in align dots on painted parts. This defect is not present on ABS-PC parts after injection-molding step. It appears in specific zones only at the end of painting process. Only some parts present this defect after painting. The location of the defect is systematic but its appearance seems to be random.

II. MATERIALS AND METHODS

A. Polymer aspect and defect location

The same ABS-PC, white color, was injected in two different moulds to fix the shape: a support of registration plate and a spoiler. After the injection-molding step no defect is visible on the surface even if a microscopy observation is applied. Small polymer barbs are polished on the edge of the part. Following the industrial process each geometry is painted with a different paint: a blue cap of registration plate (fig. 1) and a white spoiler (fig. 2). More than 99% of the painted surface is glossy with a homogeneous color. The defects appear only very small zones located always at the same location for each geometry (fig. 1,2): near the mold-gate (fig. 1a), at the contact of weld-lines (fig. 1b) and at the end of filling zone (parting-line) (fig. 1c,2). The weld-line results from a bypass of an obstacle in the mould by the molten polymer. All these locations are tightly connected to process geometry. They appear on flat surface with a complex flow (fig. 1b) and on regions with a very small radius curvature (fig. 1a,1c,2). For optical reasons they are more visible on a flat surface than on a small radius.

B. Painting

A part of the answer comes from the painting process which must be detailed. The first step is the deposit of a primer layer (7 to 8 µm thick). Then it was desolvated during 8 min. This polyurethane primer presents a high energy of adhesion with the substrate and allows fixing layers superior to the polymer. Then a basecoat including colored pigments is then applied. Its thickness depends on color: 30 μm for white and 17 μm for blue paint. The deposit is done in two steps with 9 min total desolvation time. A 25 to 35 μm thick topcoat is deposited and desolvated during 10 min. Lastly the whole part is heated at 85°C during 40 min in a drying oven.

The defect does not appear during these intermediate periods for painting. They are observed after cooling from the oven.

III. RESULTS

It is necessary to observe with a closer view the defects to define their nature and origin. They appear as dots distributed along a line (fig. 3) and along a weld-line. This parting-line is usual after injection-molding process. The painting is insufficient to mask this geometrical defect and this line is usually observed after painting. The dots are strongly connected to this processing detail. An observation by optical microscopy under reflection (Leica DMRX) shows out that each of these dots seems to be a bubble (fig. 4). The bubble is centered on the parting-line and its size varies from 50 μm, for the smallest, up to 400 μm. Their size is sufficiently large to be observed, 60-70 μm is the norm for a detection of an isolated detail (1" arc) with naked eye. Hypothetic individual bubbles with a size smaller than 10 μm could be undetected following the industrial test of quality. The size of the defect and their correlation (alignment) are crucial for their detection. The dark ring around a bright circle including some details results from a protuberance of the surface which produces this optical effect (dark-field reflection) (fig. 4). The blue pigments inside the basecoat can be also observed as clear dots on figure 4. The bubbles can be observed in more detail with SEM (FEI/Philips XL30ESEM) (fig. 5). A small bubble (diameter ~200μm) is presented in figure 5 which is significant of all the bubbles observed on the 10 mm 200 µm a b painted parts. They are formed by a cavity partly covered by a very thin foil which mainly produces the protuberance observed by optical microscopy. The foil is not complete and has been broken (fig. 5a). A hole is shown out in the bottom of the bubble (fig. 5b). From these observations it is suspected that the bubble is located in the paint layers with no clear evidence of its presence in a specific layer.

It is necessary to observe the inner part of the material with the three paint layer to locate the bubble. Thin layers (5 μm thick) were cut-out in planes perpendicular to the surface and perpendicular to the line defined in figures 3 and 4. These cut were done all through the bubble (fig. 6). The bubble is confined in the topcoat and results from deformation of the material only in this superficial layer. These bubbles and volcanoes have very similar shapes. An observation of superficial part of ABS-PC shows the presence of a crack perpendicular to the polymer surface. This crack is stopped at the boundary with the primer layer (fig. 6a to 6c) aside in the centre of the bubble where the primer and the basecoat are broken (fig. 6d). This rupture is observed as a hole in the bottom of the bubble (fig. 5b). In that point, there is a direct connection between the crack inside the polymer and the bubble.

IV. INTERPRETATION

The strong correlation between these different experimental facts strongly suggests an explanation. A simple one is to suppose a crack is provoked by some painting solvents which initiate the crack in the polymer. In combination with this primary effect, the solvent diffuses inside the crack. At the end of the industrial process the painted part is heated. The solvent contained inside the crack is vaporized and strongly expanse. Gas escape from the crack at the weak points of the primer and basecoat to form holes (fig. 5b). Then the gas inflates the bubble in the topcoat up to the rupture of the thin foil (fig. 5a).

This model implies various hypotheses and critical points for the formation of these defects. A direct connection of the cracks with the parting-line is required. The model implies the role of polymer cracking by some solvent. It also implies that the solvent diffuses inside the crack very early, during the primer or basecoat deposit. A random weakness of the primer and basecoat along the surface explains why the holes, and therefore the bubbles, are randomly distributed along the line. Finally, this interpretation implies that the topcoat is still a liquid-like material during the bubble inflation. These various points should be checked to strengthen the model.

The effect of solvents can be studied by immersion of injected parts inside some of these solvents during a time of painting (10 min at room temperature) (fig. 7). In the location of the bubbles, the primer and the basecoat layers have an almost constant thickness when the topcoat is highly deformed up to form a very thin foil (fig. 5). This observation implies that during bubble expansion the primer and basecoat were almost solid materials and topcoat almost a liquid. It must be noticed that the diffusion coefficient of solvent in material in solid-state is considerably lower than in a liquid-state. This general behavior implies that the solvent widens polymer crack was present either in primer or in basecoat solutions.

Thus these solvents open a crack only along the line described in figures 3 and 4 and along the weld-line. The rest of the surface, almost the complete surface, is unaffected by the contact with the solvents. The crack formation is not the result of an intrinsic sensitivity of polymer to the solvents. A local detail of the injectedpolymer should explain this sensitivity. This detail has been searched along the line after injection-molding. You should remember that polymer is polished in that zone before painting. Microscopic observations have been done on injected-parts before and after polishing (fig. 8). Observation under light polarization shows-out the molecular orientation parallel to the surface (fig. 8a) which is partly erased by polishing action. The orientation is complex in this initial protuberance [START_REF] Fritch | Injection-molding of ABS, mold quality in quality with the machine[END_REF]. It leads to a molecular orientation almost perpendicular to the surface after polishing and an initial crack can be detected in this particular zone (fig. 8b). The diffusion of solvents inside the polymer part is easier as an effect of this initial crack and of molecular orientation which favors the solvent diffusion. The solvent only widens the crack. This effect of a complex orientation on polymer cracks formation is observed on this zone without polishing and on weld-line where a complex orientation is also present. The cracks result from a local molecular orientation in conjunction with contact with solvents. The heating bursts the solvent and highly deforms the topcoat to form the bubbles. This is the critical step for the formation of the visual defect.

V. CONCLUSION

The quality of a painted part is largely conditioned by the presence of detectable defects. Their size and location is crucial. All the surface need to present the same high quality. In this paper, a specific defect is analyzed with a method with a wide experimental range. The present defect results from the conjunction of the nature of polymer (ABS-PC) of paint layers (solvents) and of processing condition (molecular orientation). To prevent this defect formation, various ways are possible: modification of processing conditions to prevent crack initiation, modification of the mold-geometry to avoid complex flow (welding-line, parting-line), curing between basecoat and topcoat deposit to evaporate the solvent inside the crack, change in solvent composition of the paint. The devil is in the detail.

Figure 1 .Figure 2 .

 12 Figure 1. A cap of registration plate (five defective zones are bordered)

Figure 3 :

 3 Figure 3 : Macrography of dots aligned on the top the spoiler

Figure 4 :

 4 Figure 4 : A bubble (dot) observed by optical microscopy on a line defect

Figure 5 :

 5 Figure 5: SEM observation of a bubble with a broken foil a) and a hole b)

Figure 6 :

 6 Figure 6: optical microscopy under transmission of thin cuts through half a bubble: from a border (a) to the diameter (d)

Figure 7 :

 7 Figure 7: Macrograph of a crack on the surface after immersion inside solvents

Figure 8 :

 8 Figure 8: optical microscopy observation under polarization with a lambda plate of thin cuts (e = 5 μm) from injected parts before (a) and after polishing (b)


Rosane Ushirobira

Denis Efimov

Géry Casiez

Nicolas Roussel

Wilfrid Perruquetti

A Forecasting Algorithm for Latency Compensation in Indirect Human-Computer Interactions

Human-computer interactions are greatly affected by the latency between the human input and the system visual response. The compensation of this latency is an important problem for the HCI (human-computer interaction) community. In this work, a simple forecasting algorithm is developed for latency compensation in indirect interaction using a mouse, based on numerical differentiation. Several differentiators are compared, including a novel algebraic version. An optimized procedure is developed for tuning the parameters of the algorithm. The efficiency is demonstrated on real data, measured with a 1 ms sampling time.

I. INTRODUCTION

Human interactions with computing systems are largely affected by the latency between human gestures and system visual responses [START_REF] Mackenzie | Lag as a determinant of human performance in interactive systems[END_REF]. This problem remains relatively unexplored and it is only quite recently that some works studied the latency in this context [START_REF] Ng | Designing for low-latency direct-touch input[END_REF], [START_REF] Pavlovych | The tradeoff between spatial jitter and latency in pointing tasks[END_REF]. For example, it has been recognized that a latency as low as 2 ms can be perceived by a human [START_REF] Ng | Designing for low-latency direct-touch input[END_REF] (by itself, average end-toend latency estimation for human-computer interactions is a complex problem [START_REF] Bérard | Two touch system latency estimators: high accuracy and low overhead[END_REF]). It is on direct touch systems, such as smartphones, tablets and touch-screens, that latency is the most perceivable. Furthermore, the compensation of latency is of great importance for some kind of interactions, e.g. drag and drop tasks for tablet computers or drawing using any type of input devices, from computer mouse till touch-based devices (touchpads, optical finger navigation sensors, touch screens, etc.). The solutions for latency compensation can be realized on the hardware level (by using more reactive and performing components), and that may significantly increase the price of the computer system, or on the software level (by code optimization or design of special prediction algorithms), and that may be a less expensive solution for end-users.

An end-to-end latency around 50 ms is known to affect performance in mouse-based pointing tasks. MacKenzie and Ware have shown the degradation of performance is particularly important from 75 ms latency [START_REF] Mackenzie | Lag as a determinant of human performance in interactive systems[END_REF]. Pavlovych and Stuerzlinger did not notice any drop of performance up to 58 ms of latency when acquiring targets as small as 14 pixels R. Ushirobira This work was supported by ANR (TurboTouch project, ANR-14-CE24-0009). It was also partially supported by the Government of Russian Federation (Grant 074-U01) and the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031).

wide [START_REF] Pavlovych | The tradeoff between spatial jitter and latency in pointing tasks[END_REF] while Teather et al. measured a 15% decrease of performance between 35 and 75 ms end-to-end latency [START_REF] Teather | Effects of tracking technology, latency, and spatial jitter on object movement[END_REF]. More recently Deber et al. found a perception threshold for latency in indirect touch pointing tasks around 55 ms [START_REF] Deber | How much faster is fast enough?: User perception of latency & latency improvements in direct and indirect touch[END_REF]. A recent work on the measure of end-to-end latency on modern operating systems, with no application-specific code running, found an average latency ranging from 46 to 83 ms with jitter ranging from 5 to 25 ms, depending on the operating system and toolkit [START_REF] Casiez | Looking through the eye of the mouse: A simple method for measuring endto-end latency using an optical mouse[END_REF]. This means that latency, close or above the 50 ms threshold, is still present in high-end machines with up-to-date operating systems and toolkit, even in the best case scenario.

Prediction or forecasting deals with the value estimation of some variable of interest at a specified future instant of time [START_REF] Hyndman | Forecasting: principles and practice[END_REF]. Many methods devoted to forecasting exist, whose selection depends on constraints and on the context of the addressed problem (i.e. whether process statistics or models are available, or its periodicity, or the available computational power for a prediction algorithm realization, etc.). If it is the prediction of human movements that must be accomplished, then statistical methods cannot be applied due to the diversity of possible gestures. Models based on famous Fitts's or Accot-Zhai steering laws [START_REF] Fitts | The information capacity of the human motor system in controlling the amplitude of movement[END_REF], [START_REF] Accot | More than dotting the i's-foundations for crossing-based interfaces[END_REF] allow pointing time prediction. However, even though target prediction models have been studied in some works (e.g. [START_REF] Asano | Predictive interaction using the delphian desktop[END_REF], [START_REF] Lank | Endpoint prediction using motion kinematics[END_REF], [START_REF] Pasqual | Mouse pointing endpoint prediction using kinematic template matching[END_REF]), the design of a bio-mechanical prediction model with an additional requirement of being sufficiently quick for an online implementation remains a difficult problem.

This paper presents the development of a latency compensation algorithm for a particular case of mouse-based interaction. It is supposed that the signal from a mouse-based sensor is received with a lag, and its future implementation and processing will also introduce an additional delay before a system reaction will be sent back to human. It is then necessary to predict the output of the sensor in a future instant of time to compensate the overall latency in the system. Since the model of the human gesture itself is assumed unavailable and there is no statistics on a human practice, then some basic signal-based prediction methods have to be applied. In this work a time-series approach based on differentiation from [START_REF] Esfandiari | Output feedback stabilization of fully linearizable systems[END_REF], [START_REF] Levant | Robust exact differentiation via sliding mode technique[END_REF], [START_REF] Levant | High-order sliding modes: differentiation and output feedback control[END_REF], [START_REF] Fliess | Non-linear estimation is easy[END_REF], [START_REF] Perruquetti | Finite-time observers: Application to secure communication[END_REF], [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF], [START_REF] Efimov | A hybrid robust non-homogeneous finitetime differentiator[END_REF], [START_REF] Ushirobira | Algebraic parameter estimation of a multi-sinusoidal waveform signal from noisy data[END_REF] is employed.

The outline of this work is as follows. The computer system equipped with mouse sensors and the problem statement are presented in Section II. Applied differentiation algorithms are introduced in Section III, including a new algebraic differentiator. Our latency compensation algorithm is proposed in Section IV. Discussion is given in Section V. In this case, the human input comes from a touch screen, it is propagated and processed by the computer hardware and software on different levels (operating system, executed application and graphic software) and the visual response appears on the same screen. All these levels (including the touch screen itself twice) are sources of delay in the response and in this particular case, the interaction is very sensitive to a latency since one may easily compare one's own gesture and the corresponding response. A scheme of indirect interaction is presented in Fig. 2, where the human input comes from an independent device, such as a touchpad or a computer mouse, and after a similar process, the response appears on the screen. Latency in this case is less evident in general, since it is harder to compare exactly one's own input and the corresponding output, that depends on transfer functions implemented in the system [START_REF] Casiez | No more bricolage! Methods and tools to characterize, replicate and compare pointing transfer functions[END_REF].

There are several sources of latency in both types of interaction. For instance, whatever the touch sensing technology is, the computation of contact points can require a significant amount of time. But in both cases, direct and indirect interactions, affecting layers include the device driver, the input management subsystem, the graphics software subsystem (window server), one or more GUI toolkits, the executed application and the display hardware, and each of them can contribute in the delay propagation. In all these processes, there is no guarantee about the average end-to-end delay (from human gesture till visual feedback). In addition, nowadays the screen frequency is 60 Hz and there is a low expectancy of a higher frequency in a next future.

Hence, taking in mind future development of mouse-based devices, an assumption that the measurements of a human input after the mouse sensor are available with 1 ms sampled time seems to be reasonable. For example, Logitech G500 mouse 1 is available on the market providing measurements of its position with 1 ms USB rate. Using the indirect interaction 1 http://support.logitech.com/product/gaming-mouse-g500.

scheme given in Fig. 2 with this mouse and Libpointing2 software, we were able to simulate a part of operating system responsible for interactions with a mouse-based sensor, then integrate and test a forecasting algorithm on this level.

For a forecasting algorithm development, we have the following problem statement: the raw measurements

y(t i ) = x(t i ) + v(t i )
at time instants t i are available for i = 0, 1, . . . , K (K > 0), where x(t i ) corresponds to a coordinate x(t) ∈ R provided by the sensor at t i , sampled irregularly with an average close to 1 ms and v(t i ) is a bounded noise of unknown nature. For a given lag L > 0, it is necessary to forecast the value x(t i + L) using previously measured data:

x(t i + L) = F(ȳ(t i ), ti ),
where ȳ(t i ) = [y(t 0 ), . . . , y(t i )], t = [t 0 , . . . , t i ] and the map F(•) represents the forecasting algorithm to be developed.

III. NUMERICAL DIFFERENTIATION ALGORITHMS

Due to a rather general problem statement and the absence of a model for this human-computer interaction process, in this work we will use a simple Taylor series expansion for forecasting. If we would presume the human gesture as a smooth signal of time then

x(t i + L) = +∞ j=0 L j j! x (j) (t i ).
In practice, x(t) is not infinitely differentiable and it is difficult to calculate derivatives up to an arbitrary order, hence a constant M > 0 is selected such that

x(t i + L) = M j=0 L j j! x (j) (t i ), (1)
where x (j) (t i ) are the estimates of x (j) (t i ) calculated by a numerical differentiation algorithm. The error of Taylor series truncation has order L M +1 and the error of the most differentiators (the difference |x (j) (t i ) -x (j) (t i )|) is proportional to the amplitude of the noise v ([14], [START_REF] Levant | Robust exact differentiation via sliding mode technique[END_REF], [START_REF] Levant | High-order sliding modes: differentiation and output feedback control[END_REF], [START_REF] Perruquetti | Finite-time observers: Application to secure communication[END_REF], [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF], [START_REF] Ushirobira | Algebraic parameter estimation of a multi-sinusoidal waveform signal from noisy data[END_REF]). Therefore, there exists M such that the forecasting algorithm (1) has a reasonable accuracy. In this work, we will apply and experimentally compare several numerical differentiators, their selection is motivated by possibilities of on-line implementation, required computational power, robustness with respect to noise and delay in the estimates.

A. Algebraic Differentiator

The algebraic time derivative estimation is based on concepts of differential algebra and operational calculus. A more detailed description of the approach can be found in [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF], [START_REF] Ushirobira | Algebraic parameter estimation of a multi-sinusoidal waveform signal from noisy data[END_REF], [START_REF] Fliess | Non-linear estimation is easy[END_REF]. A moving horizon version of this technique is summarized below adapted from the mentioned references. For a real-valued signal y(t), analytic on some time interval, consider its approximating N th degree polynomial function, originated from its truncated Taylor expansion:

y(t) = N i=0 a i i! t i , (2)
where the terms a i are the unknown constant coefficients representing the derivatives of the signal. The aim is to estimate these time derivatives of y(t) up to order N . Notice that in our case, we assume that y(t) is the measured signal from a signal x(t) with some negligible noise, so we may consider only y(t). The errors from this assumption can be found for instance in [START_REF] Dayan | Analyse d'erreurs d'estimateurs des dérivées de signaux bruités et applications[END_REF]. This estimation can be done on a moving time horizon using integrals of y(t). For any T > 0, the jth order time derivative estimate y (j) (t), j = 0, 1, 2, • • • , N, of the signal y(t) defined in (2) satisfies the following convolution [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF]:

y (j) (t) = ˆT 0 Hj(T, τ )y(t -τ )dτ, j = 0, 1, • • • , N, (3)
for all t ≥ T , where the convolution kernel,

Hj(T, τ ) = (N + j + 1)!(N + 1)! T N +j+1 × N -j κ 1=0 j κ 2 =0 (T -τ ) κ 1 +κ 2 (-τ ) N -κ 1 -κ 2 κ1!κ2!(N -κ1 -κ2)!(κ1 + κ2)!(N -κ1 + 1)
.

The kernel H j (T, τ ) depends on the order j of the time derivative to be estimated and on an arbitrary constant length of time window T > 0. For example, if we are interested in calculating the first-order time derivative, a degree-one polynomial can be selected y(t) = a 0 + a 1 t. By applying the above result to this signal, we obtain the following firstorder time derivative estimate

ẏ(t) = 6 T 3 ˆT 0 (T -2τ ) y(t -τ )dτ. (4)
The effect of the time integral, presented in equation ( 4), is obviously to dampen the impact of the measurement noise on the estimate. This noise dampening effect can also be used to filter out the noise from the original signal y(t). In that case, instead of calculating the first order time derivative, we have to estimate the zero order derivative (see [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] for details).

The differentiation error, caused by truncation in (2) and the integration over T , can be evaluated as a function of the second derivative amplitude.

Theorem 1. [23] If ess sup t≥0 |ÿ(t)| = ÿ < +∞, then ess sup t≥T | ẏ(t) -ẏ(t)| ≤ ÿ T 72 .
Since in our case only discrete observations are available, we need to discretize equation [START_REF] Bérard | Two touch system latency estimators: high accuracy and low overhead[END_REF]. Assume that the sampling is almost regular, i.e. t i = iT s with T s is the sampling time, then for y k = y(kT s ) and ẏk = ẏ(kT s ), the discretized approximation of equation ( 4) can be written as following:

ẏk = 6 m 2 T s m l=0 1 -2 l m y k-l (5)
where m > 0 denotes the number of summation steps, i.e. T = mT s . Remark 1. One point to be noted here is the effect of delay on differentiation. This delay appears from the integration on the interval [0, T ] in equation ( 4). Again this integration helps to reduce the noise effect. Moreover, it was shown in [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] that delayed version of the differentiator gives better performance than the non-delayed version. So, in our case, we trade-off delay with robustness to noise and the quality of the differentiators. Finally, the algebraic differentiator (5) has only one tuning parameter m (or T in (4)).

Higher order time derivatives can also be computed in the same way using equation (3). However, in that case, higher order polynomials have to be considered. The new solution proposed in this paper is a consequent connection of the first order differentiators [START_REF] Teather | Effects of tracking technology, latency, and spatial jitter on object movement[END_REF]. Since (5) is a discrete-time filter of order m, then to realize the second order differentiator, after self-product of the polynomial from (5), we obtain another discrete-time filter of order 2m + 1 (in Matlab, the function conv can be used for this purpose). Repeating these procedure we can obtain filters for estimation of i th derivative of order im + 1. For T s = 0.001, the amplitude Bode plots of these filters together with the corresponding plots of "ideal" differentiating filters

s i | s= 2 Ts z-1 z+1
are shown in Fig. 3. As we can conclude from these plots, below the frequency 10 Hz all algebraic differentiators up to order four are rather close to the "ideal" ones.

B. Homogeneous finite-time differentiator

Consider a sufficiently smooth signal y(t). The aim of this differentiator is to estimate ẏ(t), . . . , y (n-1) (t). Assume that y (n) (t) = θ(t) and set z = y ẏ • • • y (n-1) T . Then,

ż = Az + Θ(t), y = Cz, with A =       0 1 0 0 0 0 0 1 0 0 . . . . . . . . . . . . . . . 0 0 0 0 1 0 0 0 0 0       , C = 1 0 • • • 0 and Θ(t) = (0 . . . 0 θ(t))
T ∈ R n . The following homogeneous finite-time differentiator can be proposed [START_REF] Perruquetti | Finite-time observers: Application to secure communication[END_REF]:

˙ z 1 = z 2 -k 1 z 1 -y α , ˙ z i = z i+1 -k i z 1 -y iα-(i-1) , i = 2, . . . , n -1, (6) ˙ z n = -k n z 1 -y nα-(n-1)
where x γ = |x| γ sign(x) for all x ∈ R and γ > 0, z ∈ R n is the estimate of z, k 1 , ..., k n form a Hurwitz polynomial and a suitable choice of (α 1 , ..., α n ) ∈ R n >0 ensuring the system homogeneity. Then y (j-1) (t) = z j (t), j = 1, . . . , n after some finite time of transients. The differentiation error dynamics takes the form: 1) .

ė1 = e 2 -k 1 e 1 α , ėi = e i+1 -k i e 1 iα-(i-1) , i = 2, . . . , n -1, ( 7
)
ėn = θ(t) -k n e 1 nα-(n-
Due to the term θ(t), it is impossible to get the convergence of the error to zero without having any additional knowledge about the signal θ(t). This problem can be overcome by assuming that y(t) is locally polynomial and that on a small time interval, θ(t) = 0. In this case, it is possible to recover the time derivatives.

C. Higher order sliding mode and linear high-gain differentiators

A way to compensate exactly θ(t) in the last equation of ( 7) is to select α = 1 -1 n . Then a homogeneous observer can be obtained, and in the last equation e 1

nα-(n-1) = e 1 0 = sign(e 1 ).

Therefore, if |θ(t)| < k n then the sign function can compensate exactly the influence of θ. Such a differentiation algorithm is called Higher Order Sliding Mode (HOSM) differentiator [START_REF] Levant | High-order sliding modes: differentiation and output feedback control[END_REF]. For n = 2, its very popular version, supertwisting differentiator, has been proposed in [START_REF] Levant | Robust exact differentiation via sliding mode technique[END_REF]:

ż1 = z 2 -k 1 z 1 -y 0.5 , ż2 = -k 2 sign(z 1 -y).
Its non-homogeneous modification has been given in [START_REF] Efimov | A hybrid robust non-homogeneous finitetime differentiator[END_REF]. However, the sign function is discontinuous, then a chattering phenomenon may appear in the numerical realization due to measurement noise [START_REF] Levant | Robust exact differentiation via sliding mode technique[END_REF], [START_REF] Levant | High-order sliding modes: differentiation and output feedback control[END_REF]. According to [START_REF] Levant | High-order sliding modes: differentiation and output feedback control[END_REF], the system (7) is globally finite-time stable for this selection of α and for k 1 , . . . , k n forming a Hurwitz polynomial. Another limiting case is by selecting α = 1, then the system ( 6) is reduced to a linear one (α i = r i = 1 for all i = 1, . . . , n) and for sufficiently high values of gains k 1 , ..., k n (forming a Hurwitz polynomial) a differentiator from [START_REF] Esfandiari | Output feedback stabilization of fully linearizable systems[END_REF] can be obtained. In this case, the error system is globally exponentially stable. As for the case α ∈ 1 -1 n-1 , 1 , for α = 1 the influence of θ is not rejected but just attenuated by high gains k 1 , ..., k n . Remark 2. To switch between a HOSM differentiator, a highgain linear differentiator or homogeneous one, it is necessary to fix the gains k 1 , ..., k n forming a Hurwitz polynomial and take α = 1 -1 n , α = 1 or α ∈ 1 -1 n-1 , 1 , respectively. Thus, by tuning only α, it is possible to switch between

(s + λ) n+1 = s n + n j=1 k j s n-j
parametrized by λ > 0. Consequently, two parameters, α and λ, can be used for tuning these three differentiators.

IV. LATENCY COMPENSATION

Let us consider [START_REF] Mackenzie | Lag as a determinant of human performance in interactive systems[END_REF]. By selecting some M > 0 and by taking estimates of derivatives x (j) (t i ) calculated for j = 1, . . . , M by one of numerical differentiators given in Section III, we obtain a forecasting algorithm.

A. Noise and overshoot attenuation

One problem is that the estimates x (j) (t i ), j = 1, . . . , M (independently on used differentiation algorithm from Section III) are sensitive to the measurement noise v (imported by the mouse) and sampling/quantization errors (imported by software), especially for higher values of j. Thus, the predicted values x(t i + L) contain these errors/noises, and it is desirable to decrease the dependence of x(t i +L) on them. The influence of the noise is most important for small values of derivatives (when only noise is present in x (j) (t i )) and for high amplitudes of the signal x (j) (t i ) (these big deviations of x (j) (t i ) may also be originated by the noise). Of course, some preliminary filtering of y(t) can be used to decrease the noise influence in the differentiated signal. However, any additional filtering will induce an additional delay in y(t) to be compensated. Thus, in this work we applied a static nonlinear transformation χ j : R → R, j = 1, . . . , M of x (j) (t i ), that combines saturation and dead-zone, in order to reduce noise influence on x(t i + L) for big and small values of x (j) (t i ) respectively. The form of the transformation χ j is given in Fig. 4, where the parameter σ j > 0 determines the dead-zone width and σ j > 0 is responsible for saturation amplitude. The forecasting algorithm (1) takes the form:

x(t i + L) = M j=0 L j j! χ j [ x (j) (t i )]. (8)
Another way to improve accuracy of prediction and decrease sensitivity to noises is to adapt the value of L in the right hand-side of (8) depending on derivative value:

L(γ) = max    0, L - M j=1 γ j |χ j [ x (j) (t i )]|    ,
where γ = [γ 1 , . . . , γ M ] T ∈ R M are tuning parameters. The idea is also that if the values of some derivatives are high, then to avoid overshooting, it is desirable to decrease the interval of prediction L. Then (8) can be rewritten as:

x(t i + L) = M j=0 L j j! χ j [ x (j) (t i )]. (9)
To realize [START_REF] Fitts | The information capacity of the human motor system in controlling the amplitude of movement[END_REF], we must select (or tune) the parameters σ j , σ j , γ j for j = 1, . . . , M , and the type of differentiator, e.g. algebraic (just one tuning parameter m), HOSM, linear highgain or homogeneous one (two tuning parameters α and λ).

B. Parameter tuning

To tune the parameters and select the type of differentiator, the following criteria have been chosen for e(t i ) = x(t i + L) -x(t i + L):

J 2 = K i=k e 2 (t i ), J ∞ = max k≤i≤K |e(t i )|,
where 0 < k < K is selected to avoid the influence of transients at initial instants of time. Notice that J ∞ helps to minimize the overshooting in forecasting, and J 2 is responsible for overall closeness of x(t i + L) and x(t i + L).

It is difficult to tune all these parameters simultaneously due to a high nonlinearity and interrelations among them, then an iterative tuning procedure can be applied.

First, for parameters σ j , σ j and α, λ, it is not possible to calculate a gradient with respect to J 2 or J ∞ , therefore, a heuristic minimization algorithm has to be applied [START_REF] Russell | Artificial Intelligence: A Modern Approach[END_REF]. The simplest one consists in generating a grid of admissible values for these parameters and calculating next the corresponding values of J 2 and J ∞ . For instance, if θ ∈ Θ ⊂ R 2M +2 where θ = [σ 1 , σ 1 , . . . , σ M , σ M , α, λ] ∈ R 2M +2 and Θ defines the set of admissible values, then

θ * = arg min θ∈Θ {κ 2 J 2 (θ) + κ ∞ J ∞ (θ)}
with κ 2 > 0 and κ ∞ > 0 are weighting coefficients. To evaluate the shape of Θ, some preliminary statistics on measurements of x(t i ) and derivatives x (j) (t i ) can be used, e.g. histograms of these signals with taking σ j , σ j in order to cover a desired percentage of values for x (j) .

It is possible to calculate the gradient of J 2 with respect to

γ = [γ 1 , . . . , γ M ] T ∇ γ J 2 (γ) =      K i=k e(t i ) M j=1 L j-1 (γ) (j-1)! if L > γ T d(t i ), ×χ j [ x (j) (t i )]d(t i ) 0 otherwise, where d(t i ) = [|χ 1 [ x (1) (t i )]|, . . . , |χ M [ x (M ) (t i )]|] T .
Then a least square algorithm can be applied to optimize the values of these parameters. The same for the parameter m in (5).

C. Experimental results

To test our proposed forecasting algorithms, two datasets have been collected using Logitech G500 mouse with sampling period 1 ms and Libpointing software, with lags 100 ms and 50 ms. More precisely, to produce a dataset, we record human movements of a mouse with Libpointing and add a predefined artificial lag. Optimization has been performed for each dataset separately either for homogeneous or for algebraic differentiators.

The results of application of the developed forecasting algorithm are shown in Fig. 5 for the lag 100 ms and in Fig. 6 for the lag 50 ms. These results are obtained by algebraic and homogeneous differentiators with M = 4. The values of criteria for these scenarios are given in Tab. I. As we can conclude from these results, homogeneous differentiators provide a worse quality of forecasting (both criteria J 2 and J ∞ take higher values) and the predicted signal has more chattering due to noise and quantization, while the new algebraic differentiators give smoother curves. For the case of latency 50 ms the results of forecasting have less overshoots and errors than for 100 ms.

Another set of simulation results is devoted to analysis As we can conclude, the second derivative improves significantly the quality of forecasting, the third and fourth derivatives also influence positively on the quality of prediction (the predicted signal becomes more oscillatory with these derivatives), but with less impact.

V. CONCLUSION

The problem of latency compensation for human-computer interaction through a mouse device has been introduced. A simple forecasting algorithm has been developed, based on truncated Taylor series and numerical differentiators, including a novel algebraic technique. To decrease the dependence of predicted values on measurement and numerical noises, a nonlinear transformation has been applied to derivatives (since filtering would introduce a delay). Nonlinear optimization tools have been used for parameter tuning. The results of the developped forecasting algorithm applied on real data have been included to demonstrate efficiency of the approach. The implementation of this algorithm in the software of touch-based devices is planned. In future works, a comparison with other existing methods will be studied.

  , D. Efimov and W. Perruquetti are with Inria, Non-A team, 40 av. Halley, 59650 Villeneuve d'Ascq, France. G. Casiez and N. Roussel are with Inria, Mjolnir team, 40 av. Halley, 59650 Villeneuve d'Ascq, France. D. Efimov, W. Perruquetti, G. Casiez and N. Roussel are also with CRIStAL (UMR CNRS 9189), 59650 Villeneuve d'Ascq, France. D. Efimov is also with Department of Control Systems and Informatics, University ITMO, 49 av. Kronverkskiy, 197101 Saint Petersburg, Russia.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Direct interaction via touch device

Fig. 3 .

 3 Fig. 3. Amplitude Bode plots for algebraic differentiators

Fig. 4 .

 4 Fig. 4. Static transformation

Fig. 5 .

 5 Fig. 5. Forecasting for the lag 100 ms

4 tFig. 7 .

 47 Fig. 7. Forecasting using different derivatives

Libpointing[START_REF] Casiez | No more bricolage! Methods and tools to characterize, replicate and compare pointing transfer functions[END_REF] is a software toolkit that allows to bypass the system's transfer functions to receive raw asynchronous events from an HID pointing device. See http://libpointing.org/.


Maher Belgacem

Abderrahman Boukricha

Cloaking via change of variables for second order quasi-linear elliptic differential equations

Keywords:

) obtained in the context of electric impedance tomography formulated by a linear differential operator in divergence form. Contents 1. Introduction 2. Preliminaries 3. Quasilinear elliptic equations 4. Change of variables 5. Cloaking via change of variables 6.

Introduction

Since ages, the electromagnetic invisibility has long been fascinating men and has had a big interest in the fantasy literature and researchers. From Plato who evoked the invisibility in the myth of Gyges (the myth of the magic ring making any wearer invisible) to the contemporary character of Harry Potter, she was associated with the magic (wizardry, witchery), the supernatural. Its irruption into the scientific and technical sphere could only arouse curiosity or surprise a wide audience.

However just in 2006, Simultaneously Ulf Leonhardt( [START_REF] Leonhardt | Optical conformal mapping[END_REF]) and John Pendry ( [START_REF] Pendry | Controlling electromagnetic fields[END_REF]) proposed similar concepts to achieve perfect invisibility. The initial idea can be however simple: If one follows the Snell-Descartes Laws, one can also distorts the light path and arranged that in order to avoid a particular area, that area and all objects that can contain become de facto invisible! Of course, this physical idea could germinate in a context where various ingredients seemed to make it realistic.

The key point discovered by Leonhardt and Pendry in 2006 is that the form of Maxwell equations that describe the propagation of electromagnetic waves are invariant under (conformal)coordinates changes. Therefore the path of the light may be decomposed: In one hand, in a set of intrinsic equations valid whatever the environment and which do not depend on the geometry (the new medium dielectric permittivity ε(x) and the new medium magnetic permeability µ(x) are depending only on the transformation and not on the geometry). In the other hand in a set of equations that characterize the environment and depend on geometry. It is thus possible to associate the change of geometry, that is to say the transformations of the space or coordinates in optical properties and to translate deformation of space in effects equivalent materials: This is the main idea of the transformational Optic, a new way of designing optical devices (the conformal transformations, applicable to two-dimensional case, have been long used by opticians and are special cases for which the properties of material are isotropic). So just, imagine a clever but simple distortion of the space, and apply the principle of transformational optic in order to obtain the optical characteristics of an invisibility Cape/cloak.

Of course, the question that immediately arises is whether we have materials at our disposal, which exhibit suitable characteristics. Let us say immediately that these materials do not exist in nature. One of the most active areas of modern optics is precisely the conception of metamaterials, that is to say structures in which the basic cells, more or less similar, are of sizes of the order of magnitude equal or below the wavelength and contain constituted substructures of conventional materials. The basic cells and their orderly arrangement give the total structure of new optical properties very different from those of basic materials and impossible to find in the range of natural homogeneous materials. For an exhaustive knowledge/history of the electromagnetic invisibility, see(A. Nicolet [START_REF] Nicolet | Les capes d'invisibilité et l'optique transformationnelle, Images de la Physique[END_REF])

The mecanism, using the change of variables, for obtaining electromagnetic invisibility/cloaking was similar to that of Greenleaf, Lassas and Uhlmann (cf [START_REF] Greenleaf | Anisotropic conductivities that cannot be detected by EIT[END_REF]) obtained earlier in 2003 for constructing conductivity ensuring a cloaking in the case of electric impedance tomography. Later Kohn, Shen, Vogelius and Weinstein (cf [START_REF] Kohn | Cloaking via change of variables in Electric Impedance Tomography[END_REF]) in 2008 and that of Kohn, Onofrei, Vogelius and Weinstein (cf [START_REF] Kohn | Cloaking via change of variables for the Helmoltz equation[END_REF])in 2010 use the same mecanism to generalize and improve the results of Greenleaf et al not only for electric impedance tomography but also for the Helmholtz equation. Another adoptations use the change of variables based cloaking to elastic sensing in 2006 ( [START_REF] Milton | On cloaking for elasticity and physical equations with a transformation invariant form[END_REF]) or to acoustic in 2007 ( [START_REF] Cummer | One path to acoustic cloaking[END_REF])

Let us remark hier that the Dirichlet-to-Neumann-Map and the solution of the Calderón problem ( [START_REF] Brander | Calderon's problem for the p-Laplacian:First order derivative of conductivity on the boundary[END_REF] and [START_REF] Calderón | On an inverse boundary value problem[END_REF]) play a crucial role for previous research on the cloaking.

the linear partial differential equations in divergence form formalising the context of Electric Impedance Tomography are generalized in nonlinear partial differential equations in the framework of the weighted p-Laplace equations (cf [START_REF] Brander | Calderon's problem for the p-Laplacian:First order derivative of conductivity on the boundary[END_REF] and [START_REF] Heinonen | Nonlinear potential theory of degenerate elliptic equations[END_REF]). In relation with this nonlinear generalization and for the investigation of the Calderón problem, interesting results are obtained (cf [START_REF] Brander | Calderon's problem for the p-Laplacian:First order derivative of conductivity on the boundary[END_REF] and [START_REF] Hauer | The p-Dirichlet-to-Neumann operator with applications to elliptic and parabolic problems[END_REF]).

The p-Laplace equation has applications in e.g. image processing, fluid mechanics, plastic mouling and modelling of sand-pile. In contrast to previous investigations on the cloaking, the weighted p-Laplace equations are however, for every p = 2, not invariant under admissible (conformal)change of variables (i.e. coordinates change). Since the p-Laplace operators belong to the class of Leray-Lions (cf. [START_REF] Leray | Quelques resulatats de Visik sur les problemes elliptiques non lineaires par les methodes de Minty-Browder[END_REF]), we will prove in this paper the following useful results : An admissible (conformal) coordinates change, transforms a quasi-linear elliptic differential equation in a quasi-linear elliptic differential equation given by operators in the class of ). We can hence obtain the method of coordinates change for the investigation of cloaking in this nonlinear context formalized by Leray-Lions.

The paper is organized as follows: We begin, in section 2, by introducing frequently used notions needed for the determination of the nonlinear variational Dirichlet problem, introduced in section 3, for second order quasi-linear elliptic differential equations in the class of [START_REF] Hauer | The p-Dirichlet-to-Neumann operator with applications to elliptic and parabolic problems[END_REF]) and genralizing the weighted p-Laplace equations ( [START_REF] Heinonen | Nonlinear potential theory of degenerate elliptic equations[END_REF]). In section 3, we recall, on Lipschitz domain Ω in R d with d ≥ 2, functions from Ω × R d to R d satisfying structural assumptions defining second order quasi-linear elliptic differential operators/equation belonging to the class of Leray-Lions. We then use ( [START_REF] Hauer | The p-Dirichlet-to-Neumann operator with applications to elliptic and parabolic problems[END_REF]) to recall the solution of the Dirichlet problem given a boundary function in the Sobolev-Slobodečki space W 1-1 p ,p (∂Ω) and to recall the Dirichlet-to-Neumann Map. In section 4, we define admissible change of variables and we prove that it transforms a second order quasi-linear elliptic differential equation in the class of Leray-Lions to a equation of the same art. Moreover this admissible change of varianbles leaves invariant the Dirichlet-to-Neumann-Map.

In section 5, we adopt the framework of second order quasi-linear elliptic differential equations to define the notion of cloaking. Further, we consider, for Ω the ball B 2 of center x = 0 and radius 2, the admissible (nonsingular) change of variables F ρ for 0 < ρ < 1 used in ( [START_REF] Greenleaf | Anisotropic conductivities that cannot be detected by EIT[END_REF] and [START_REF] Kohn | Cloaking via change of variables in Electric Impedance Tomography[END_REF]), and define an associated regular near cloak. We then prove that the unit ball B 1 is almost invisible (in the sence of item2 of theorem 2) as ρ is sufficiently small. Further we prove in theorem 3 that the regular near cloak does not focus on the radial case For the singular change of variales F , used in previous works(see among others ( [START_REF] Greenleaf | Anisotropic conductivities that cannot be detected by EIT[END_REF], [START_REF] Kohn | Cloaking via change of variables in Electric Impedance Tomography[END_REF], [START_REF] Kohn | Cloaking via change of variables for the Helmoltz equation[END_REF])), we define an associeted singular cloak (given on the shell B 2 \B 1 by a transformation by F to a given second order quasi-linear elliptic differential operator and on B 1 by an arbitrary second order quasi-linear elliptic differential operator)and we show that it gives a perfect invisibilty for the ball B 1 . We prove, in theorm 4, that the potentiel outside the cloaked region with Dirichlet data ϕ ∈ W 1-1 p ,p (∂Ω) is the same obtained for ϕ ∈ H 1 2 (∂Ω) in (cf. [START_REF] Kohn | Cloaking via change of variables in Electric Impedance Tomography[END_REF] pages 17 and 19 ) in the linear case of second order differential operator in divergence form modelling the electric impedance tomographie. The last section is devoted to the application of our results to the linear case treated in ( [START_REF] Greenleaf | Anisotropic conductivities that cannot be detected by EIT[END_REF], [START_REF] Kohn | Cloaking via change of variables in Electric Impedance Tomography[END_REF])and we prove that our theorem 5 is a generalisation, at the framework of equations in the class of Leray-Lions, of theorem 3 in (cf. [START_REF] Kohn | Cloaking via change of variables in Electric Impedance Tomography[END_REF]) A direct application as for the Laplacian (2-Laplacian) to the p-laplacian for p = 2 will be treated in a next work... Because of the nonlinear character of our work, we have developed other methods not related to previous results such as from [START_REF] Friedman | Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence[END_REF] or from [START_REF] Kohn | Cloaking via change of variables in Electric Impedance Tomography[END_REF]in the linear case.

Preliminaries

Recall on Sobolev spaces and related fields We assume hier that Ω is a bounded Lipshitz domain in R d with d ≥ 2. For p ∈ [1, +∞], denote by L p (Ω) and W 1,p (Ω)the usual Lebesgue and first sobolev spaces. We denote by C 0,1 (Ω) the space of all Lipshitz continuous functions on the closure Ω of Ω. The boundary ∂Ω is equiped with the (d -1) dimensional Hausdorf measure dH when we work with the lebesgue space L p (∂Ω). Morever, C(∂Ω) denotes the set of all real valued continous functions on ∂Ω.

Trace on the boundary

Since Ω is a Lipshitz domain, by J. Nečas ([18] Theorem 4.2, 4.6 and 3.8) the mapping u → u | ∂Ω from C 0,1 (Ω) to C 0,1 (∂Ω) has a unique continuous extension mapping

T r : W 1,p (Ω) -→ L p * (∂Ω) called trace operator with p * = p(d-1) d-p if 1 ≤ p < d , p * ≥ 1 if p = d and p * = ∞ .
For conveniance, it is written u | ∂Ω instead of T r (u) for u ∈ W 1,p (Ω) even if u does not belong to C(Ω) and we call u | ∂Ω or T r (u) the trace of u.

Sobolev-Slobodečki space on the boundary

It is well Known (cf. J. Nečas [START_REF] Nečas | les methodes directes en theorie des equations elliptiques[END_REF] section 3.8) that

• ker(T r ) = {u ∈ W 1,p (Ω)|T r u = 0} the kernel of T r concide with the Sobolev space W 1,p 0 (Ω).

• Rg(T r ) = {T r u|u ∈ W 1,p (Ω)} the range of T r coincide with the Sobolev-Slobodečki space W 1-1 p ,p (∂Ω)(cf. [START_REF] Hajlasz | Traces of Sobolev Functions on Fractal Type Sets and Characterization of Extension Domains[END_REF]) defined as the linear subspace of all ϕ ∈ L p (∂Ω) with finite semi-norm

[ϕ] p p = ∂Ω ∂Ω |ϕ(x) -ϕ(y)| p |x -y| d-2+p dxdy
and equiped with the norm

ϕ W 1-1 p ,p (∂Ω) = ϕ L p (∂Ω) +[ϕ] p for every ϕ ∈ W 1-1 p ,p (∂Ω)
Remark 2.1 We have the following important results:

• The trace operator T r has a linear bounded right inverse

Z : W 1-1 p ,p (∂Ω) → W 1,p (Ω)
(cf. [START_REF] Nečas | les methodes directes en theorie des equations elliptiques[END_REF] theorem 5.7)

• Another crucial propriety of a Lipshitz domain (cf.D. Hauer [START_REF] Hauer | The p-Dirichlet-to-Neumann operator with applications to elliptic and parabolic problems[END_REF] Lemma 2.1) Ω is that: for 1 < p < ∞ and for 1 ≤ q < ∞ the space C ∞ (Ω) lies dense in W 1,p (Ω) and the set

{v → v | ∂Ω , v ∈ C ∞ (Ω)} is dense in W 1-1 p ,p (
∂Ω) and in L q (∂Ω).

3 Second order quasi-linear elliptic differential equations/operators thoughout this paper, we assume that Ω is a bounded domain in R d with a lipschitz boundary ∂Ω, d ≥ 2 and 1 < p < ∞ Definition 3.1 A function a : Ω × R d -→ R d satisfies a structural assumptions, needed for the defintion of a second order quasi-linear elliptic operator if :

1. a is a Caratheodory function satisfying the following assumptions:

For some constants 0 < α ≤ β < ∞ 2. < a(x, ξ), ξ >≥ α|ξ| p for a.e x ∈ Ω and every ξ ∈ R d 3. |a(x, ξ)| ≤ β|ξ| p-1 for a.e x ∈ Ω and every ξ ∈ R d 4. < (a(x, ξ 1 ) -a(x, ξ 2 ), (ξ 1 -ξ 2 ) >> 0 with ξ 1 = ξ 2 and for a.e x ∈ Ω 5. a(x, λξ) = λ|λ| p-2 a(x, ξ) ∀λ ∈ R, λ = 0 and for a.e x ∈ Ω Where <, > is the scalar product in R d and |ξ| =< ξ, ξ >

Remark 3.2

1. Under the previous assymptions, the Caratheodory function a defines a second order quasi-linear elliptic operator A, belonging to the class of Leray-Lions operators (cf. [START_REF] Leray | Quelques resulatats de Visik sur les problemes elliptiques non lineaires par les methodes de Minty-Browder[END_REF]) as follows:

Au := -div(a(x, ∇u)) in D (Ω) for every u ∈ W 1,p
Loc (Ω). Hier is ∇u = ∇ x u is the gradient of u relatively to the variable x.

2. To the quasi-linear elliptic operator A is developed by (cf. [START_REF] Heinonen | Nonlinear potential theory of degenerate elliptic equations[END_REF]) a nonlinear potential theory.

3. In a very recent work by Daniel Hauer in (cf. [START_REF] Hauer | The p-Dirichlet-to-Neumann operator with applications to elliptic and parabolic problems[END_REF]) a Dirichlet-to-Neumann map associated with the second order quasi-linear elliptic operator A, was defined and Follow-up of applications to elliptic and parabolic problems.

4. By taking a(x, ξ) = σ(x)|ξ| P -2 ξ for ξ ∈ R d and σ is a continous and strongly elliptic matrix on Ω in the sence that for some constants 0 < m < M < ∞:

m|ξ| 2 ≤< σ(x)ξ, ξ >≤ M |ξ| 2 for all x ∈ Ω and ξ ∈ R d .
It is well known that the function a is a function satisfying the previous strutural assumptions and the associated operator A is called the weighted p-Laplace operator (cf. [START_REF] Heinonen | Nonlinear potential theory of degenerate elliptic equations[END_REF])(resp.the classical p-Laplace ∆ p obtained by σ(x) = Id R d ) and is a prototype of quasi-linear elliptic differential operators belonging to the class of Leray-Lions operators.

5. If p = 2 then the weighted 2-Laplace operator is well known as a differential operator in divergence form known in physic as the PDE in electrostatics which is later known as a mathematical formalism for the Electric Impedance Tomography as follows: In what follows in this section, we will review (cf.D.Hauer [START_REF] Hauer | The p-Dirichlet-to-Neumann operator with applications to elliptic and parabolic problems[END_REF]) some basic facts about quasi-linear elliptic equations given by -div(a(x, ∇u)) = 0 Where a satisfies a structural assumptions. Let ϕ ∈ W 1-1 p ,p (∂Ω)(the Sobolev-Slobodečki space on the boundary∂Ω). We consider the following nonlinear variational Dirichlet problem:

∇.(σ∇u) = i,j ∂ ∂x i (σ ij (x)) ∂u ∂x j in Ω The electrical conductivity σ(x) = σ ij (x) is,
-div (a(x, ∇u)) = 0 in Ω, u = ϕ on ∂Ω (1)
Definition 3.3 1. We call u ∈ W 1,p Loc (Ω) a weak solution of -div (a(x, ∇u)) = 0 in Ω, (2)
if u satisfies the following integral equation:

Ω (a(x, ∇u))∇vdx = 0 for all v ∈ W 1,p 0 (Ω), (3) 2.
A weak solution will be called a-harmonic function on Ω.

For later use, we need the following compactness results concerning weak solutions of (2), which is an immediate consequence of (D. Hauer [START_REF] Hauer | The p-Dirichlet-to-Neumann operator with applications to elliptic and parabolic problems[END_REF] Lemma 2.3 ) and (L.Boccardo and F. Murat [START_REF] Boccardo | Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations[END_REF] Theorem 2.1 and Remark 2.1). Lemma 3.4 If (u n ) is a bounded sequence in W 1,p (Ω) of weak solution of (2), then there is subsequence u kn of u n and a weak solution u ∈ W 1,p (Ω) of ( 2) such that (u kn ) converges to u weakly in W 1,p (Ω), strongly in L P , ∇u k converges to ∇u strongly in (L q (Ω)) d for every q < p(thus a.e in Ω) and a(x, ∇u kn ) converges to a(x, ∇u) a.e in Ω and weakly in L P (Ω).

Proposition 3.5 Let ϕ ∈ W 1-1 p ,p (
∂Ω), then there exists a unique weak solution of (2) which is solution of the problem [START_REF] Boccardo | Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations[END_REF].

Proof Let Φ ∈ W 1,p (Ω) be such that Φ | ∂Ω = ϕ , it is well known that the operator v → A(v) = -div(a(x, ∇v + ∇Φ))
satisfies the assumptions of the Minty-Browder theorem [J.L.Lions [START_REF] Lions | Quelques methodes de resolution des problemes aux limites non lineaires[END_REF], thm 2.1]. Hence the equation ( 2) admits a weak solution u ∈ W 1,p (Ω) satisfying u -Φ ∈ W 1,p 0 (Ω). We have then

Ω a(x, ∇u)∇vdx = 0 ∀v ∈ W 1,p 0 (Ω)
Let u 1 be another weak solution with (u

1 -Φ) ∈ W 1,p 0 (Ω) then u -u 1 ∈ W 1,p 0 (Ω) and Ω a(x, ∇u 1 )(∇u -∇u 1 )dx = Ω a(x, ∇u)(∇u -∇u 1 )dx = 0 and thus Ω (a(x, ∇u) -a(x, ∇u 1 )(∇u -∇u 1 )dx = 0.
By the strict monotonicity of the function a (assumption 4) we obtain ∇u = ∇u 1 and by the Poincaré Inequality we get u = u 1 .

We arrive to the following definition Definition 3.6 For a given boundary value ϕ ∈ W 1-1 p ,p (∂Ω), we call a function u ∈ W 1,p a W 1,p -solution of the Dirichlet problem [START_REF] Boccardo | Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations[END_REF] on Ω if (u -Zϕ) ∈ W 1,p 0 (Ω)and u is a weak solution of (2). Z is the linear bounded right inverse of the trace function operator on the boundary ∂Ω (Remark 2.

1).

In what follows, we will denote for every ϕ ∈ W 1-1 p ,p (∂Ω) by P ϕ = P a ϕ the W 1,psolution of the Dirichlet problem (1) associated with the function a on Ω.

Remark 3.7

• for every λ ∈ R and ϕ ∈ W 1-1 p ,p (∂Ω), we have

P λϕ = λP ϕ and |P ϕ| ≤ P |ϕ|
• By (cf. [START_REF] Heinonen | Nonlinear potential theory of degenerate elliptic equations[END_REF] Theorem 3.70) we will choose P ϕ a continous function on Ω.

• For useful Properties of the application P:

W 1-1 p ,p (∂Ω) -→ W 1,p (Ω) (cf D. Hauer [9] lemma 2.5).
Then c is a function from Ω × R d -→ R d satisfying a structural assumptions.

Dirichlet-to-Neumann Map (DNP) associated with a quasilinear elliptic operator

If for a given boundary value ϕ ∈ W 1-1 p ,p (∂Ω), P ϕ and a(x, ∇P ϕ) are smooth enough up to the boundary ∂Ω and that ν denote the outward pointing unit normal vector on ∂Ω, the co-normal derivative of P ϕ (associated with the quasi-linear elliptic operator A defined by the function a as at the beginning of this section), on ∂Ω is formally defined by the dot product a(x, ∇P ϕ).ν on ∂Ω.

It is well known, (e.g. D. Hauer [START_REF] Hauer | The p-Dirichlet-to-Neumann operator with applications to elliptic and parabolic problems[END_REF]), that the Dirichlet-to-Neumann operator Λ associated with A assign to each Dirichlet boundary data ϕ the correspending co-normal derivative of P ϕ. We formally set: Λϕ = a(x, ∇P ϕ).ν

Multiplying the equaliy by some function ψ ∈ C ∞ (Ω) with respect to the inner product on L 2 (∂Ω) and applying Green's formula yields

∂Ω (Λϕ)ψ | ∂Ω dH = Ω a(x, ∇P ϕ)∇ψdx = Ω a(x, ∇P ϕ)∇Zψdx H is the d -1 dimensional
W -(1-1 p ),p (∂Ω) of W 1-1 p ,p (
∂Ω) . This justifies why the following definition makes sense and is consistent to the case of smooth functions. Definition 3.9 We call the mapping Λ :

W 1-1 p ,p (∂Ω) -→ W -(1-1 p ),p (∂Ω), defined by < Λϕ, ψ >= Ω a(x, ∇P ϕ)∇Zψdx
for every ϕ, ψ ∈ W 1-1 p ,p (∂Ω), the Dirichlet-to-Neumann map associated with the second order quasi-linear differential operator A defined by the Function a(see Remark 3.1 1.).

We recall hier the following interesting results (cf. D.Hauer [START_REF] Hauer | The p-Dirichlet-to-Neumann operator with applications to elliptic and parabolic problems[END_REF] Proposition 3.3). Proposition 3.10 The mapping Λ :

W 1-1 p ,p (∂Ω) -→ W -(1-1 p ),p (
∂Ω) has the following properties:

1. < Λϕ, ψ >= Ω a(x, ∇P ϕ)∇P ψdx for every ψ, ϕ 2. Λ is continuous and monotone, that is for every

ϕ 1 ,ϕ 2 < Λϕ 1 -Λϕ 2 , ϕ 1 -ϕ 2 >≥ 0 3. There exists C 1 > 0 such that ||Λϕ|| W -(1-1 p ),p (∂Ω) ≤ C 1 ||ϕ|| p-1 W 1-1 p ,p (∂Ω)
for every ϕ ∈ W 1-1 p ,p (∂Ω).

4. There exists a constant C 2 > 0 such that

< Λϕ, ϕ >≥ C 2 ||ϕ|| P W 1-1 p ,p (∂Ω)
for every ϕ ∈ W

1-1 p ,p m (∂Ω) Hier W 1-1 p ,p m (∂Ω) is the subspace of all ϕ ∈ W 1-1
p ,p (∂Ω) satisfying the so-called compatibility condition ∂Ω ϕdx = 0 equipped with the induced norm and W

-(1-1 p ),p m (∂Ω) is the dual space of W 1-1 p ,p m (∂Ω)

Change of variables

Along this entire section we consider a Lipschitz domain Ω and a function a satisfying a structural assumptions (cf. section 3). Let U be an open subset in Ω and G a change of variables given by y = G(x) (where G is an invertible, differentiable and oriented preserving ) from U to G(U ) and DG(resp.DG T ) the (resp. transposed of the) Jacobian matrix of G with elements a ij = ∂yi ∂xj . We set for u and v in W 1,p (U ) I := U a(x, ∇ x u)∇ x vdx. By the change of variables y = G(x) we get:

I = G(U ) 1 det DG(G -1 (y))
DG(G -1 (y))a(G -1 (y), (DG(G -1 (y)) T ∇ y ũ(y))∇ y ṽ(y)dy

Where ũ(y) = u(G -1 (y)) and ṽ(y

) = v(G -1 (y)) If we set a G (y, ξ) := 1 det DG(G -1 (y)) DG(G -1 (y))a(G -1 (y), (DG(G -1 (y))) T ξ) for y ∈ G(U ) then: U a(x, ∇ x u)∇ x vdx = G(U )
a G (y, ∇ y ũ)∇ y ṽdy Proposition 4.1 Let U be an open subset in Ω and G a differentiable function from U to G(U ). We have for every x ∈ U and i ∈ {1, ..., d}, the eigenvalues λ i (x), β i (x) of the symmetric matrices (DG(x)) T DG(x) and DG(x)(DG(x)) T are nonneegative and hence for every θ ∈ R d and x ∈ U :

(min(λ i (x), i ∈ {1, ..., d}))

1 2 |θ| ≤ |DG(x)θ| ≤ (max(λ i (x), i ∈ {1, ..., d})) 1 2 |θ|

and

(min(β i (x), i ∈ {1, ..., d}))

1 2 |θ| ≤ |(DG(x)) T θ| ≤ (max(β i (x), i ∈ {1, ..., d})) 1 2 |θ|

Proof

Let for i ∈ {1, ..., d}, λ i (x), β i (x) the eigenvalues and u i , v i the orthonormal bases of eigenvectors respectively for the symmetric matrices (DG(x)) T DG(x) and DG(x)(DG(x)) T . Then for every θ ∈ R d , there exist real numbers θ i , µ i such that θ = n i=1 θ i u i = n i=1 µ i v i . We thus have:

|DG(x)θ| 2 =< (DG(x)) T DG(x)θ, θ >=< n i=1 θ i λ i (x)u i , n i=1 θ i u i >= n i=1 θ 2 i λ i (x)
and

|DG(x)) T θ| 2 =< DG(x)(DG(x)) T θ, θ >=< n i=1 µ i β i (x)v i , n i=1 µ i v i >= n i=1 µ 2 i β i (x)
the sought inequalities are then easy to obtain. given by a

G(U ) G (y, ξ) = a G (y, ξ)I G(U ) (y) satisfies on G(U ) × R d a

structural assumptions as by a

Proof

We set x = G -1 (y). Let for i ∈ {1, ..., d}, λ i (y), β i (y) the eigenvalues and u i , v i the orthonormal bases of eigenvectors respectively for the symmetric matrices (DG(x)) T DG(x) and DG(x)(DG(x)) T . From the hypothesis on the determinent of the matrix (DG(x)) and since the real numbers λ i (x), β i (x) are nonnegative, we easily obtain that the positiv real numbers α 1 = min(λ i (x), i ∈ {1, ..., d}), x ∈ U , α 2 = M ax(λ i (x), i ∈ {1, ..., d}, x ∈ U ), δ 1 = min(β i (x), i ∈ {1, ..., d}, x ∈ U ) and δ 2 = M ax(β i (x), i ∈ {1, ..., d}, x ∈ U ) are positive. By the previous corollary we have;

α 1 |θ| ≤ |DG(x)θ| ≤ α 2 |θ| and δ 1 |θ| ≤ |(DG(x)) T θ| ≤ δ 2 |θ|
for every θ ∈ R d and every x ∈ U . For α and β the constants given by the assumptions 2 and 3 on the function a, the previous inequalities and the hypothesis, we have

< a G (y, ξ), ξ >≥ α det DG(x) |(DG(x)) T ξ| p ≥ α(α 1 ) p det DG(x) |ξ| p ≥ α(α 1 ) p c 2 |ξ| p
for every y ∈ G(U ). It follows that the function a

G(U ) G
satisfies on G(U ) the assumption 2 with the constant α(α1) p c2 . Moreover, since

|a G (y, ξ)| = 1 det DG(x) |(DG(x))a(x, (DG(x)) T ξ)| we have |a G (y, ξ)| =≤ α 2 c 1 |a(x, (DG(x)) T ξ)| ≤ βα 2 c 1 |(DG(x)) T ξ| p-1 ≤ βα 2 (δ 2 ) p-1 c 1 |ξ| p-1
we hence obtain that the function a

G(U ) G
satisfies on G(U ) the assumption 3 with the constant β = (α2) p c1 . The other assumptions 4 and 5 and that the function a

G(U ) G is a Caratheodory function from G(U ) × R d to R d are easy verified.
a G (y, ξ) := 1 det DG(x) DG(x)a(x, (DG(x)) T ξ) if y / ∈ G(E), := 0 if y ∈ G(E) (5)
Where y = G(x) and E is the negligible closed subset in the definition of the admissible change of variables.

Theorem 1

Let G be an admissible change of variables on Ω and a G the transformed of the Caratheodory function a by G then 1. a G is a function on Ω × R d satisfying a structural assumptions as a in section 3.

a function u is an a-harmonic function on Ω if and only

if ũ = u • G -1 is an a G -
harmonic function on Ω. The notion of a-harmonic function is referred hier to the definition 3.2 2.

Proof

We have, by the change of variables y = G(x), the following relations between the Caratheodory Functions a and a G :

Ω\E a(x, ∇ x u)∇ x vdx = Ω\G(E) a G (y, ∇ y ũ)∇ y ṽdy Since E and G(E) are negligeable we have Ω a(x, ∇ x u)∇ x vdx = Ω a G (y, ∇ y ũ)∇ y ṽdy
The function a G satisfies bei Proposition 4.4 a same structural assumptions on Ω as the function a and because G is an admissible change of variables(G(x) = x on the boundary), we have v ∈ W 1,p 0 (Ω) if and only if ṽ ∈ W 1,p 0 (Ω). It is then easy to see that u is an a-harmonic function on Ω if and only if ũ is an a G -harmonic on Ω Remark 4.7

• the function a G is called the push-forward of a by the change of variables G (cf. [START_REF] Heinonen | Nonlinear potential theory of degenerate elliptic equations[END_REF] or [START_REF] Kohn | Cloaking via change of variables in Electric Impedance Tomography[END_REF]).

• For the weighted p-Laplace given by a(x, ξ) = σ(x)|ξ| P -2 ξ we have:

a G (y, ξ) = 1 det DG(G -1 (y)) DG(G -1 (y))σ(G -1 (y))(DG(G -1 (y))) T |(DG(G -1 (y))) T ξ| p-2 ξ
This shows that the transformation by G of the weighted p-Laplace operator has not the same form of a p-Laplace

We denote in what follows by P b ϕ, for every ϕ ∈ W 1-1 p ,p (∂Ω), the solution of the Dirichlet problem (1) with boundary value ϕ defined in section 3 for every function b satisfying a structural assumptions in section 3, Corollary 4.8 We have:

P a ϕ = P a G ϕ for all ϕ ∈ W 1-1 p ,p (∂Ω) where P a ϕ = P a ϕ • G -1 .
Proof Since P ϕ = P a ϕ, for every ϕ ∈ W 1-1 p ,p (∂Ω), then it is an a-harmonic solution, it follows that P ϕ is, for every ϕ ∈ W 1-1 p ,p (∂Ω) a solution of the Dirichlet problem (1) defined in section 3 for the Caratheodory Function a G . Since a G is by Proposition 4.4 a function from Ω × R d -→ R d satisfying a structural assumptions (section 3), we hence obtain the corollary.

< Λϕ, ψ > = Ω a(x, ∇ x P a ϕ)∇ x Zψdx = Ω a G (y, ∇ y P a ϕ)∇ y Zψdy = Ω a G (y, ∇ y P a G ϕ)∇ y Zψdy =< Λ G (ϕ), ψ > (6)
Thus for every ϕ, Λ(ϕ) and Λ G (ϕ) determine identical linear form on W 1-1 p ,p (∂Ω). It follows that

Λ(ϕ) = Λ G (ϕ) for every ϕ ∈ W 1-1 p ,p (∂Ω) and hence Λ = Λ G on W 1-1 p ,p (∂Ω)
5 Cloaking via change of variables Definition 5.1 (cf. [START_REF] Kohn | Cloaking via change of variables in Electric Impedance Tomography[END_REF][START_REF] Kohn | Cloaking via change of variables for the Helmoltz equation[END_REF]) We say a region of space is cloaked for a particular class of measurements if its contents -and even the existence of the cloak are invisible using such measurements.

Remark 5.2 Let Ω be a relativ compact open set in R d such that the Dirichlet problem associated with a given partial differential operator is solvable for a class R of resolutive functions defined at the boundary(in a local setting) or at the complement (in a nonlocal setting) of Ω. We can then consider, for the definition of the cloaking, the class of solutions S(f ) f ∈R of the partial differential operator as the particular class of measurements.

We introduce in what follows a definition for the cloaking in the framework of second order quasi-linear elliptic differential equations. We now explain how change-of-variables-based cloaking for quasi-linear elliptic operators work. Our demarche is much more general but for simplicity we focus on the radial case: Ω = B 2 where B 2 is the ball of center x = 0 and radius 2 and the region D to be cloaked is B 1 , the concentric ball of radius 1.

We start by explaining how B 1 can be nearly-cloaked using the following admissible change of variables: Fixing a small parameter ρ > 0, consider the piecewise-smooth change of variables from B 2 to B 2 :

F ρ (x) = x ρ if |x| ≤ ρ, 2-2ρ 2-ρ + 1 2-ρ |x| x |x| if ρ ≤ |x| ≤ 2, ( 8
)
F ρ expends the ball B ρ = {x ∈ R d / |x| ≤ ρ} to B 1 , while mapping the full domain B 2 to itself. Lemma 5.4 We have for every x ∈ B 2 \ B ρ DF ρ (x) = ( 1 2 -ρ + 1 |x| 2 -2ρ 2 -ρ )I - 2 -2ρ 2 -ρ 1 |x| xx T detDF ρ (x) = ( 1 2 -ρ )( 1 2 -ρ + 1 |x| 2 -2ρ 2 -ρ ) d-1 = (|x| + 2(1 -ρ)) d-1 (2 -ρ) d |x| d-1
where I is the identity matrix and x = x |x| Proof An easy calculation gives the Jacobian matrix DF ρ (x) which implies that it is symmetric, x is an eigenvector with eigenvalues

The regular near cloak associated with F ρ

Let b be an arbitrary Caratheodory function from B 2 × R d to R d satisfying structural assymptions of section 3. Given a function a satisfying the same assumptions as b, we define the "regular near cloak" associated with F ρ as follows:

a ρ b (y, ξ) = b(y, ξ) for y ∈ B 1 and ξ ∈ R d , a Fρ (y, ξ) for y ∈ B 2 \B 1 and ξ ∈ R d (9)
Lemma 5.5 For every 0 < ρ < 1, a ρ b is, by lemma 3.7 and theorem 1, a function from B 2 ×R d to R d satisfying structural assumptions of section 3 and u ∈ W 1,p is an a ρ b -harmonic solution if and only if uoF ρ is an (a ρ b ) F -1 ρ -harmonic solution for the following Caratheodory function:

(a ρ b ) F -1 ρ (x, ξ) = b F -1 ρ (x, ξ) for x ∈ B ρ and ξ ∈ R d , a(x, ξ) for x ∈ B 2 \B ρ and ξ ∈ R d (10)
Which gives that an a ρ b -harmonic function is an a-harmonic function outside the closed ball Bρ for every 0 < ρ < 1.

In what follows, let ϕ ∈ W 1-1 p ,p (∂B 2 ) and P ρ ϕ the W 1,p solution of the following quasilinear elliptic equation

-div(a ρ b (x, ∇ x u)) = 0 in B 2 , u = ϕ on ∂B 2 . ( 11
)
We recall hier the following Lemma 5.6 Let K be a non-empty compact subset of R d and U an open set containing K, then there exists a function θ ∈ C ∞ c (R d ) such that 0 ≤ θ ≤ 1 , θ = 1 on K and θ = 0 outside U Proposition 5.7 There exist constants C 1 and C 2 such that for every ϕ ∈ W 1-1 p ,p (∂B 2 ) and 0 < ρ < 1, we have

• B2\Bρ |∇ x (P ρ ϕ • F ρ (x))| p dx 1 p ≤ C 1 ϕ W 1-1 p ,p (∂B2) • B1 |∇ x P ρ ϕ(x))| p dx 1 p ≤ C 2 ϕ W 1-1 p ,p (∂B2)
Proof Let 0 < ρ 0 < 1 and θ the function associated by the previous lemma with K = B ρ0 and U = B 1 . Let v = (1 -θ)P ϕ and w := P ρ ϕ • F ρ -(1 -θ)P ϕ on B 2 , Since F ρ is an admissible change of variables, then w ∈ W 1,p (B 2 ) and T r(w) = 0, where T r(u) is the Trace on the boundary ∂B 2 of any function u ∈ W 1,p (B 2 ), hence w ∈ W 1,p 0 (B 2 ). Let

I = B2 (a ρ b ) F -1 ρ (x, ∇ x P ρ ϕ • F ρ (x))(∇ x P ρ ϕ • F ρ (x))dx
By the Lemma 5.5 and since v = 0 on B ρ0 we get that for every 0 ≤ ρ ≤ ρ 0

I = Bρ b F -1 ρ (x, ∇ x P ρ ϕ•F ρ (x)))(∇ x P ρ ϕ•F ρ (x))dx+ B2\Bρ a(x, ∇ x P ρ ϕ•F ρ (x))∇ x P ρ ϕ•F ρ (x)dx and hence I ≤ B2\Bρ a(x, ∇ x P ρ ϕ • F ρ (x))∇ x vdx
for α and β the contants given by the structural assumptions 2 and 3 on the function a and by the Hoelder inequality we have for p such that

1 p + 1 p = 1 α B2\Bρ |∇ x P ρ ϕ•F ρ (x)| p dx ≤ β B2\Bρ (|∇ x P ρ ϕ•F ρ0 (x)| p-1 ) p dx 1 p B2\Bρ |∇ x v(x)| p dx 1 p
Which gives:

B2\Bρ |∇P ρ ϕ • F ρ | p dx 1 p ≤ β α B2 |∇ x v| p dx 1 p = β α ∇ x v p It follows, since θ ∈ C ∞ c (R d ) that ∇ x v p ≤ ( ∇ x θ ∞ +1) P ϕ W 1,p (B2)
An easy calculation yields

P ϕ W 1,p (B2) ≤ (1 + 2K β α ) Zϕ W 1,p (B2)
where K is the Poincaré constant on B 2 . For Z the norm of the linear bounded right inverse of the trace, we get

Zϕ W 1,p (B2) ≤ Z ϕ W 1-1 p ,p (∂B2)
putting

C 1 = (1 + 2K( β α )) β α ( ∇ x θ ∞ +1)
Z we obtain the result sought. Further bei the change of variable we have

Bρ b F -1 ρ (x, ∇ x P ρ ϕ • F ρ (x)))(∇ x P ρ ϕ • F ρ (x))dx = B1 b(y, ∇ y P ρ ϕ(x))(∇ y P ρ ϕ(y))dy hence B1 b(y, ∇ y P ρ ϕ(y))(∇ y P ρ ϕ(y))dy ≤ B2\Bρ a(x, ∇ x P ρ ϕ • F ρ (x))∇ x vdx
thus for α b the constant corresponding to the assumption 2 on the function b we have

α b B1 |∇ x P ρ ϕ(x))| p dx ≤ β B2\Bρ (|∇ x P ρ ϕ • F ρ (x)| p-1 ) p dx 1 p B2\Bρ |∇ x v(x)| p dx 1 p dx ≤ β B2\Bρ (|∇ x P ρ ϕ • F ρ (x)| p )dx 1 p ∇v p ≤ β( β α ∇v p ) p p ∇v p (12) it follows that B1 |∇ x P ρ ϕ(x))| p dx 1 p ≤ ( β α b ) 1 p ( β α ) 1 p ∇v p = ( α α b ) 1 p ( β α ) ∇v p
Again putting,

C 2 = ( α α b ) 1 p (1 + 2K β α ) β α ( ∇θ ∞ +1) Z
we obtain the sought result.

Corollary 5.8 We have

B2 |∇ x (P ρ ϕ • F ρ (x))| p dx 1 p ≤ (C 1 + C 2 ) ϕ W 1-1 p ,p (∂B2)
for every ϕ ∈ W 1-1 p ,p (∂B 2 ), every 0 < ρ < 1 and every p ≤ d.

Proof

We have Since F ρ (x) = x ρ on B ρ , by a change of variables we easy obtain

Bρ |∇ x (P ρ ϕ • F ρ (x))| p dx ≤ ρ d-p B1 |∇ y (P ρ ϕ))| p dy hence Bρ |∇ x (P ρ ϕ • F ρ (x))| p dx 1 p ≤ C 2 ϕ W 1-1 p ,p (∂B2)
.

Proposition 5.9 There exists a constant C such that

P ρ ϕ • F ρ W 1,p (B2) ≤ C ϕ W 1-1 p ,p (∂B2)
for every ϕ ∈ W 1-1 p ,p (∂B 2 ), every 0 < ρ < 1 and every p ≤ d.

Proof Since F ρ (x) = x for x ∈ ∂B 2 , then (P ρ ϕ • F ρ -P ϕ) ∈ W 1,p 0 (B 2 ), by the Poincaré inequality we have P ρ ϕ • F ρ -P ϕ p ≤ K ∇(P ρ ϕ • F ρ -P ϕ) p therefore

P ρ ϕ•F ρ p ≤ K ∇P ρ ϕ•F ρ p + P ϕ p +K ∇P ϕ p ≤ K ∇P ρ ϕ•F ρ p +(1+K) P ϕ W 1,p (B2) thus P ρ ϕ • F ρ W 1,p (B2) ≤ (C 1 + C 2 )(K + 1) + (K + 1)(1 + 2K β α Z ϕ W 1-1 p ,p (∂B2)
and it follows that

C = (2 + α α b ) 1 p )(K + 1)(1 + 2K β α ) β α ( ∇θ ∞ +1) Z
satisfies the sought inequality.

Proposition 5.10 Let p ≤ d and u ρ (x) := P ρ ϕ • F ρ (x) for every x ∈ B 2 . Then u ρ is, for every 0 < ρ < 1, an a-harmonic function outside the closed ball Bρ , which converge as ρ → 0 to P ϕ, the a-harmonic function solution of the problem [START_REF] Boccardo | Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations[END_REF] with Dirichlet data ϕ. Moreover u ρ converge to P ϕ as ρ tends to zero, weakly in W 1,p (B 2 ) and strongly in L p , ∇P ρ ϕ converge to ∇P ϕ a.e on Ω and a(x, ∇P ρ ϕ) converge to a(x, ∇P ϕ) a.e in B 2 and weakly in L p (Ω) Proof By the Lemma (5.5) and the transformed of a ρ b by the admissible change of variables F -1 ρ , u ρ is for every 0 < ρ < 1 an a-harmonic function outside the closed ball B ρ . By the proposition 5.9, the theorem of Banach-Alaoglu, the lemma 3.4, a diagonal procedure and the trace theorem, we obtain that u ρ converge to P ϕ as ρ tends to zero, weakly in W 1,p (B 2 ) and strongly in L p and ∇u ρ converge to ∇P ϕ a.e on B 2 and a(x, ∇u ρ ) converge to a(x, ∇P ϕ) a.e in B 2 and weakly in L p (B 2 )

lim ρ →0 B2\Bρ |(a(x, ∇ x P ρ ϕ • F ρ ) -a(x, ∇ x P ϕ))f (x)|dx = 0
The convergence does not depend on the Caratheodory function b Proof We have for every measurable subset E of B 2

| E I B2\Bρ a(x, ∇ x P ρ ϕ • F ρ )f (x)dx| ≤ E I B2\Bρ |a(x, ∇ x P ρ ϕ • F ρ )f (x)|dx ≤ E I B2\Bρ β|∇ x P ρ ϕ • F ρ )| p-1 f (x)dx ≤ β B2\Bρ |∇ x P ρ ϕ • F ρ )| p dy 1 p E |f (x)| p dx 1 p ≤ β(C 1 ϕ W 1-1 p ,p (∂B2) ) p-1 E |f (x)| p dx 1 p .
(13) Where C 1 is the constant given in Proposition 5.6. We easy see that the set of functions {a(x, ∇ x P ρ ϕ • F ρ )I B2\Bρ , 0 < ρ < 1} is uniformly integrable relatively to the parameter ρ and the caratheodory function b. By the Proposition 5.10 and the Vitali Convergence Theorem we get the result sought.

We finish this subsection with the two following nearly cloking theorems: for every ϕ, ψ ∈ W 1-1 p ,p (∂B 2 ) and 0 < ρ < ρ 0 we have

Theorem 2 For every ψ ∈ W 1-1 p ,p (∂B 2 ), let f = (1 -θ)P ψ where θ is a C ∞ c (R d
< Λ ρ b ϕ, ψ > -< Λϕ, ψ >= B2\Br a(x, ∇ x P ρ ϕ • F ρ ) -a(x, ∇ x P ϕ) ∇ x f dx
for every r with ρ < r < ρ 0 2. for every ϕ, ψ

∈ W 1-1 p ,p (∂B 2 ) lim ρ →0 | < Λ ρ b ϕ, ψ > -< Λϕ, ψ > | = 0 Proof Let ψ, ϕ ∈ W 1-1 p ,p (∂B 2
). Since the functions f and P ρ ϕ • F ρ have the function ψ as trace on the Boundary of B 2 , then by propositions 3.9, 4.7 and the lemma 5.5 we have

< Λ ρ b ϕ, ψ >=< (Λ ρ b (F -1 ρ ))ϕ, ψ > . Let I :=< Λ ρ b ϕ, ψ > -< Λϕ, ψ > .
By the choose of the function f , we have ∇f = 0 on B ρ and then

I = B2 (a ρ b ) F -1 ρ (x, ∇ x P ρ ϕ • F ρ )∇ x P ρ ψ • F ρ dx - B2 a(, ∇P ϕ)∇P ψdx = B2 (a ρ b ) F -1 ρ (x, ∇ x P ρ ϕ • F ρ )∇f dx - B2 a(, ∇P ϕ)∇f dx = B2\Bρ (a(x, ∇ x P ρ ϕ • F ρ ) -a(x, ∇P ϕ))∇ x f dx (14)
Hence

< Λ ρ b ϕ, ψ > -< Λϕ, ψ >= B2\Bρ (a(x, ∇ x P ρ ϕ • F ρ ) -a(x, ∇P ϕ))∇ x f dx
Since ∇f = 0 on B ρ0 we get the sought result.

By proposition 5.13, we then obtain the validity of item 2.

Remark 5.14 The ball B 1 is almost invisible as ρ is sufficiently small.

We have focused on the radial symmetric setting. However our arguments did not use this symmetry in any essential way. Indeed, the Rademacher theorem, an aproximation result for L 1 (Ω) functions by smooth functions and the same previous arguments proves: Theorem 3 Let G : B 2 → Ω be a Lipschitz continuous map with a Lipschitz continuous inverse, and let

D = G(B 1 ) then H ρ = G • F ρ • G -1 : Ω → Ω is peacewise Lipschitz; moreower • H ρ expands G(B ρ ) to D and • H ρ (x) = x

Singular Cloak

In what follows, let

F (x) = lim ρ →0 F ρ (x) = (1 + 1 2 |x|) x |x| for x = 0.
If we set y = F (x), then F -1 (y) = 2(|y| -1) y |y| for |y| > 1. By the lemma 5.2 we have:

• DF = 1 2 + 1 |x| I - 1 |x| xx T where x = x |x| For x = 0 • For x = 0 det(DF )(x) = 1 2 1 2 + 1 |x| d-1 = (|x| + 2) d-1 2 d |x| d-1
We hence obtain from the previous calculus Remark 5.16 (DF ) -1 is uniformly bounded on B 2 but F is not an admissible change of variables and as by previous papers (cf. [START_REF] Kohn | Cloaking via change of variables in Electric Impedance Tomography[END_REF], [START_REF] Kohn | Cloaking via change of variables for the Helmoltz equation[END_REF]) it will be called a singular change of variables.

Let b be an arbitrary function from B 2 × R d -→ R d satisfying a structural assumption as the function a Definition 5.17 (Singular cloak associated with F and a)

We define the singular cloak associated with the singular change of variables F , the Function, denoted by a F b , from

B 2 × R d -→ R d given as follows a F b (y, ξ) = b(y, ξ) for y ∈ B 1 and ξ ∈ R d , a F (y, ξ) for y ∈ B 2 \B 1 and ξ ∈ R d (16)
where a F (y, ξ) := 1 det DF(x) DF (x)a(x, (DF (x))ξ) for every x > 0 and y = F (x)

The following theorems give, for the singular change of variables F , that the associate singular cloak makes the ball B 1 perfectly invisible in the sence of the definitions 5.1 and 5.3.

Theorem 4

Let ϕ ∈ W 1-1 p ,p (∂B 2 ) and u a locally bounded function from B 2 to R which is a W 1,p loc ( B2 \ ∂B 1 ) solution of the following "Dirichlet problem"

   Ω (a F b (y, ∇ y u))∇ y vdx = 0 for all v ∈ W 1,p 0 (B 2 ), u = ϕ on ∂B 2 (17)
then there exist a constant C such that u has the following form u(y) = P ϕ(F -1 (y)) for almost every y such that 1 < |y| ≤ 2 u(y) = C for almost every y ∈ B 1

Proof Let ε > 0 and B ε the ball with center x = 0 and radius ε. We have for

x ∈ B 2 , |x| > ε if and only if y = F (x) > 1 + ε 2 .
From the previous properties of the jacobian matrix DF it follows that if

v ∈ W 1,p 0 (B 2 \ B ε ) then ṽ = v • F -1 ∈ W 1,p 0 (B 2 \ B 1+ ε 2 )
. By the hypothesis on u and a variable changes we have

B2\B 1+ ε 2 a F b (y, ∇u)∇ṽdy = B2\Bε a(x, ∇ x (u • F ))∇ x vdx = 0
for every ε > 0, Thus the locally bounded function u • F on B 2 is a-harmonic on B 2 \ {0}, since the point {0} is a-polar (i.e polar for quasilinear elliptic structure given by a, (cf.

[10] theorem 10.1), then by the removable singularuty theorem (cf. By the theorem 2, we have < Λϕ, ψ >= lim ρ→0 < Λ ρ b ϕ, ψ > and we thus obtain the sought results.

we end this section by the following very interesting result.

  as in remark 4, a strongly elliptic matrixvalued fonction on Ω. It relates voltage u and the associated electric fields ∇u to the resulting current σ∇u.

Lemma 3 . 8

 38 Let D be a measurable subset in Ω, a and b are two Caratheodory functions respectively from (Ω\D) × R d -→ R d and from D × R d -→ R d satisfying a structural assumptions given at the beginning of this section. let c be the function from Ω × R d -→ R d given by c(x, ξ) := b(x, ξ) for x ∈ D and ξ ∈ R d , a(x, ξ) for x ∈ Ω\D and ξ ∈ R d

  Hausdorf measure on the boundary ∂Ω. If in addition Λϕ ∈ L P (∂Ω), then by an approximation argument using Remark 2.1 item 2, we can conclude that ∂Ω ΛϕψdH = Ω a(x, ∇P ϕ)∇Zψdx for every ψ ∈ W 1-1 p ,p (∂Ω) If ϕ and ψ belong to W 1-1 p ,p (∂Ω), the integral on the right-hand side of this equation exists, we easly see that the functional ψ → Ω a(x, ∇P ϕ)∇Zψdx belong to the dual space

Corollary 4 . 2

 42 If the Jacobian matrix of the change of variables G is symmetric then for every θ ∈ R d and x ∈ U , we have(min(|σ i (x)|, i ∈ {1, ..., d}))|θ| ≤ |DG(x)θ| ≤ (M ax(|σ i (x)|, i ∈ {1, ..., d}))|θ|where σ i (x), i ∈ {1, ..., d}) are the eigenvalues of the symmetric matrix DG(x).

Proposition 4 . 3

 43 Let U be an open subset in Ω and G a change of variables from U to G(U ) . Assume that there exist c 1 , c 2 positive constants such that c 1 < detDG(G -1 (y)) < c 2 for every y ∈ G(U ), then the function a G(U ) G

Definition 4 . 4 (

 44 admissible change of variables) We will say that a piece-wise smooth function G(y = G(x)) from Ω to Ω is an admissible change of variables, if G is invertible, G(x) = x at the boundary ∂Ω, there exists a subset E such that E as well G(E) are closed and of Lebesgue measure zero and G and G -1 are continuously differentiable respectively on Ω \ E and on Ω \ G(E). Moreover there exist two positive constants c 1 and c 2 such that c 1 < detG(x) < c 2 for every x ∈ Ω \ E Lemma 4.5 A function G from Ω to Ω is an admissible change of variables if and only if G -1 is an admissible change of variables. Definition 4.6 Let a be a function from Ω×R d to R d satisfying a structural assumptions and G an admissible change of variables. Then the transformed a G of a by G is defined as follows:

Proposition 4 . 9

 49 Let G be an admissible change of variables on Ω, and Λ G is the Dirichletto-Neumann map associated with a G then Λ = Λ G Proof We have for every ϕ, ψ in W 1-1 p ,p (∂Ω) and by the definition 3.8 of the Dirichlet-to-Neumann Map associated with the Caratheodory function a < Λϕ, ψ >= Ω a(x, ∇ x P a ϕ)∇ x Zψdx Since P a ϕ is a solution of the Dirichlet Problem (1) and (Zψ -Zψ) ∈ W 1,p 0 (Ω), from the previous corollary 4.4 and from the definition of the Dirichlet-to-Neumann Map associated with the Caratheodory function a G , we have:

Definition 5 . 3 7 )

 537 Let a be a fixed function satisfying, as in the beginning of setion3, a structural assumptions and a measurable subset D with D ⊂ Ω ⊂ Ω ⊂ Ω , Ω and Ω are open subset. Let a c be function from (Ω \D) × R d -→ R d satisfying locally a structural assumptions. We say that a c cloaks the region D if its every extention a D across D in the following form a D (x, ξ) := b(x, ξ) for x ∈ D and ξ ∈ R d , a c (x, ξ) for x ∈ Ω\D and ξ ∈ R d (produce on Ω the same Dirichlet-to-Neumann-map as a, regardless of the choice of the Caratheodory function b satisfying on Ω × R d a structural assumptions.

  1 2-ρ , and x⊥ = {y/y⊥x} is a d -1 dimensional eigenspace with eigenvalue ( 1 2-ρ + 1 |x| 2-2ρ 2-ρ ). Moreover for every 0 < ρ < 1 the key properties of F ρ are • F ρ is continuous and piecewise-smooth • F ρ and F -1 ρ are admissible change of variables on B 2 (in the sense of definition 4.5).

Corollary 5 . 11 Corollary 5 . 12

 511512 We have lim ρ →0 P ρ ϕ(y) = P ϕ • F -1 (y) for every 1 < |y| ≤ 2 where F (xFor every 0 < ρ < 1 the functionP ρ ϕ • F ρ is, locally uniformly bounded on B 2 \ B ρ .Proof By Remark 3.7 item 1 and the Harnack inequality (cf.[START_REF] Heinonen | Nonlinear potential theory of degenerate elliptic equations[END_REF]) Proposition 5.13 For every ϕ ∈ W 1-1 p ,p (∂B 2 ) and function f ∈ L p (B 2 ) we have

1 ρ

 1 ) function from B 2 to R associated, according to the Lemma 5.6, with the compact set K = B ρ0 for a fixed 0 < ρ 0 < 1 and the open set U = B 2ρ0 . For every 0 < ρ < ρ 0 , let Λ ρ b , Λ ρ b (F -1 ρ ) and Λ are the Dirichlet-to-Neumann Maps associated respectively with a ρ b , (a ρ b ) F -and a. Then 1.

  at the boundary ∂Ω in the same way as in the beginning of this subsection, for given function a und b from Ω × R d -→ R d we define on Ω × R d the regular near cloak associated with H ρ as follows : a ρ b (y, ξ) = b(y, ξ) for y ∈ D and ξ ∈ R d , a Hρ (y, ξ) for y ∈ Ω\D and ξ ∈ R d (15) Then for every ϕ, ψ ∈ W 1-1 p ,p (∂B 2 ) lim ρ →0 | < Λ ρ b ϕ, ψ > -< Λϕ, ψ > | = 0 where Λ ρ b and Λ are the Dirichlet to Neumann maps associated respectively with a ρ b and a. Remark 5.15 The subset D ⊂ Ω is almost invisible as ρ is sufficiently small.

Then 1 . 1 2

 11 [START_REF] Heinonen | Nonlinear potential theory of degenerate elliptic equations[END_REF] theorem 7.36) the function u•F is extendable to an a-harmonic Function. Since u•F = u = ϕ at the boundary of B 2 , we hence obtain u • F = P ϕ and thereforeu = P ϕ • F -1 on B 2 \ B1 Let v ∈ C ∞ c (B 2 ) and v 1 (y) = v(0) for |y| ≤ 1 and v 1 (y) = v • F -1 (y) for |y| > 1.It is easy to verify that v 1 ∈ W 1,p 0 (B 2 ) and by the hypothesis we haveB1 b(y, ∇ y u)∇ y v 1 dy + B2\ B1 a F (y, ∇ y u)∇ y v 1 dy = 0 for every v ∈ C ∞ c (B 2 ).By the change of variables y = F (x) we getB2\ B1 a F (y, ∇ y u)∇ y v 1 dy = B2\{o} a(x, ∇ x (u • F ))∇ x vdx.By the previous calculus we have u • F = P ϕ and then B2\{o} a(x, ∇ x (u • F ))∇ x vdx = 0, hence B1 b(y, ∇ y u)∇ y vdy = 0 for every v ∈ C ∞ c (B 2 ), it follows that b(y, ∇ y u) = 0 almost every where in B1 . Since the Function b satisfies a structural assumptions of section 3, by the assumption 2 we then obtain ∇ y u = 0 a.e and u is constant a.e in B 1 . Corollary 5.18 Let ϕ ∈ W 1-1 p ,p (∂B 2 ) and v the function from B 2 to R such thatv(y) = P ϕ(F -1 (y)) if 1 < |y| ≤ 2 v(y) = P ϕ(0) for y ∈ B 1([START_REF] Pendry | Controlling electromagnetic fields[END_REF] v is the unique continuous function satisfying the conditions of the previous theorem 2. v is the potentiel outside the cloaked region with Dirichlet data ϕ ∈ H (∂Ω) obtained in (cf.[START_REF] Kohn | Cloaking via change of variables in Electric Impedance Tomography[END_REF] pages 17 and 19 ) in the linear case modelling the electric impedance tomographie.By theorem 2, the existence of the limit of Λ ρ b , as ρ tends to zero, leads to the following definition of the Dirichlet-to-Neumann map: Definition 5.19 The mapping from W 1-1 p ,p (∂B 2 ) to its dual W -(1-1 p ),p (∂B 2 ) defined, for every ϕ, ψ ∈ W 1-1 p ,p (∂B 2 ), by < Λ F b ϕ, ψ >:= lim ρ→0 < Λ ρ b ϕ, ψ > will be called the Dirichlet-to-Neumann map associated with the singular cloak a F b (Definition 5.17).

Theorem 5 2 .

 52 For every function b from B 2 × R d -→ R d satisfying a structural assumptions of section 3, we have 1. Λ F b = Λ ; where Λ is the Dirichlet-to-Neumann-map associated with a. For every ϕ, ψ ∈ W 1-1 p ,p (∂B 2 ) we have< Λ F b ϕ, ψ >= B2\ B1 a F (y, ∇ y P ϕ • F -1 )∇ y P ψ • F -1 dy Proof Let ϕ, ψ in W 1-1 p ,p (∂B 2 ), by the Proposition 3.9 for the Dirichlet-to-Neumann-map associated with the function a and by the change of variablesF from B 2 \ {o} to B 2 \ B 1 , we have < Λϕ, ϕ > = B2\{o} a(x, ∇ x P ϕ)∇ x P ϕdx = B2\ B1 a F (y, ∇ y P ϕ • F -1 )∇ y P ϕ • F -1 dy = B2\ B1 1 detDF (F -1 ) a(x, DF (F -1 )∇ y P ϕ • F -1 )DF (F -1 )∇ y P ϕ • F -1 dy ≥ α B2\ B1 1 detDF (F -1 (y))|DF (F -1 (y))∇ y P ϕ • F -1 | p dy.

( 20 )

 20 This yields by the Hoelder Inequality that the function1 detDF (x) a(F -1 (y), DF (F -1 (y))∇ y P ϕ • F -1 (y))DF (F -1 (y))∇ y P ψ • F -1 (y)is integrable on B 2 \ B1 and hence by a change of variables y = F (x) we getB2\ B1 a F y, ∇ y P ϕ • F -1 ∇ y P ψ • F -1 dy = B2a(x, ∇ x P ϕ)∇ x P ψdx =< Λϕ, ψ > .

Corollary 5 . 20 3 6 6 . 1

 520361 For every function a satisfying a structural assumptions of section 3, every singular cloak a F b (Definition 5.16), associated with the singular change of variables F and a, Cloaks (makes perfectly invisible) the unit ball B 1 in the sense of definition 5.Applications Electric Impedance TomographyAs given in Remark 3.1 item 5, differential operators in divergence form known in physic as the PDE in electrostatics is a mathematical modelisation for the Electric Impedance Tomography as follows:∇.(σ∇u) = i,j ∂ ∂x i (σ ij (x)) ∂u ∂x j in Ω

Acknowledgements

The authors are indebted to the Faculty of Mathematics of the University of Bielefeld, Germany, for pleasant working conditions. Especially they expresses their warmest thanks to Professor Michael Röckner for his support.

Where σ is a continous and strongly elliptic summetric matrix on Ω in the sence that for some constants 0 < m < M < ∞: m|ξ| 2 ≤< σ(x)ξ, ξ >≤ M |ξ| 2 for all x ∈ Ω and ξ ∈ R d . It is then easy to see that we have the case p = 2 and that the differential operator ∇.(σ∇) is the second order quasi-linear elliptic operator corresponding to the following function satisfying a structural assumptions with α = m and β = M a(x, ξ) = σ(x)ξ for every x ∈ Ω and ξ ∈ R d .

We have then easy obtained for the singular cloak (see definition 5.16) in this case

Our theorem 5 is then a generalisation, in the framework of the class of Leray-Lions operators, of theorem 3 of R.V.Kohn, H Shen, M.S.Vogelius, M.I.Weinstein (cf. [START_REF] Kohn | Cloaking via change of variables in Electric Impedance Tomography[END_REF]) in the linear setting of operators in divergence form.