A A Rekik

F Azaïs

N Dumas

F Mailly

P Nouet

Modeling the influence of etching defects on the sensitivity of MEMS convective accelerometers

Keywords: MEMS, convective accelerometer, etching defects, fault modeling I

In this paper, a behavioral model that includes the influence of etching defects on the sensitivity of MEMS convective accelerometers is presented. Starting from an existing behavioral model, new physically-based expressions have been derived to introduce etching defects in the simulation of thermal conduction in the sensor. In addition, a semi-empirical model has been introduced for thermal convection. Finally, a very good agreement is obtained between the behavioral model and FEM simulations.

INTRODUCTION

In the field of Micro-Electro Mechanical Systems (MEMS) manufacturing, alternate test methods based on electrical test stimuli raise interest to facilitate testing and to reduce its cost [START_REF] Charlot | Generation of electrically induced stimuli for MEMS self-test[END_REF]. Fault injection model can be used for evaluating test coverage [START_REF] Dumas | Evaluation of a fully electrical test and calibration method for MEMS capacitive accelerometers[END_REF] or for optimizing test stimuli [START_REF] Natarajan | Alternate Electrical Tests for Extracting Mechanical parameters of MEMS Accelerometer Sensors[END_REF]. In the case of mechanically actuated structures, high-level models allowing the injection of manufacturing defects are increasingly accurate for representative studies [START_REF] Da Silva | Designing MEMS for Reliability (DFR)[END_REF]. However, such models do not exist for convective sensors. Presently, evaluation of the influence of geometrical and material properties in convective accelerometers is only possible through Finite Element Modeling (FEM) [START_REF] Luo | Study on linearity of a micromachined convective accelerometer[END_REF][START_REF] Chaehoi | Experimental and finiteelement study of convective accelerometer on CMOS[END_REF], which is very time consuming and deals with a limited number of physical domains. The development of more compact models would ease the study of alternate test methods and more generally manufacturability issues.

In a previous study on MEMS convective accelerometers, we have shown that most of the parametric faults that affect the device specifications can be detected with electrical test measurements [START_REF] Luo | Study on linearity of a micromachined convective accelerometer[END_REF]. In particular, easily detected faults concern variations of both electrical and thermal resistances, and lateral geometrical parameters. Faults that escape to electrical tests are related to the convective behavior of the device. However, the correlations that may exist between the conductive and convective parameters were not taken into account in the model used for fault injection. In addition, the influence of the cavity depth was not included in this model, although it has been shown that this parameter significantly affects the device sensitivity [START_REF] Rekik | A study of package effects on the behavior of MEMS convective accelerometers[END_REF]. It is therefore the objective of this paper to develop a high-level behavioral model to inject faults related to the cavity depth and to represent the potential correlations between conductive and convective phenomenon.

In this paper, we first briefly describe the accelerometer together with its previously published high-level behavioral model. In the second part, we detail the model development considering the main blocks of the model (heater source, fluid conduction, fluid convection and transduction). Finally, in the last part, we validate this new semi-empirical model with FEM simulations for different set of parameters and we illustrate benefits of such a model for alternate testing of MEMS convective sensors.

II. DEVICE UNDER TEST

A. Device Description

The device under test is a convective accelerometer obtained by Front-Side Bulk Micromachining (FSBM) of a CMOS die fabricated in a 0.8 µm technology from Austria Microsystems® (Fig. 1). Main lateral dimensions are the halfwidth of both the heater beam (r 1 ) and the cavity (r 2 ), and the distance between the heater and one detector (d). The three thin bridges are composed of the CMOS process back-end layers (oxide, polysilicon, aluminum, and nitride). In particular, polysilicon is used to implement resistors, for both heating (R H ) and temperature sensing (R D1 , R D2 ). The heater R H is powered with an electrical voltage (U H ) to create a temperature gradient in the bottom (i.e. etched silicon) and top (i.e. package) cavities: the temperature is then maximum at the heater location and minimum at the cavities boundaries. In absence of acceleration, the temperature detectors (R D1 , R D2 ) are located on identical temperature gradients for symmetry reasons. Under acceleration along the AA'-axis, the cavity temperature distribution deforms due to free convection and detectors may measure the differential temperature. Indeed, polysilicon resistivity exhibit high temperature dependence (Temperature Coefficient of Resistance, TCR=9×10 -4 /°C) and the thermal signal is easily converted into a voltage by means of a Wheatstone bridge. This voltage is then amplified by an instrumentation amplifier. For more details on sensor 978-1-4244-7791-3/10/$26.00 ©2010 IEEE manufacturing and characterization, please refer to previous works from some of the authors [START_REF] Chaehoi | Experimental and finiteelement study of convective accelerometer on CMOS[END_REF][START_REF] Leman | Modeling and system-level simulation of a CMOS convective accelerometer[END_REF].

B. Device model

For system-level and electronic interface circuit design, a behavioral model of the sensor was developed and implemented in both Matlab/Simulink® and Cadence® [START_REF] Leman | HDL Modeling of convective accelerometers for system design and optimization[END_REF]. This model (Fig. 2) is composed of four main blocks. First, the heater temperature (T H ) is calculated from the external power supply (U H ). Then, regarding fluid conduction, an analytical expression based on solving the heat conduction in a cylindrical geometry approximation is used to compute the common mode temperature (T CM ) of the detectors. This expression involves lateral sensor dimensions (r 1 , r 2 and d).

Regarding fluid convection, the differential temperature (T D ) resulting from an acceleration is calculated with an empirical expression that involves fitting parameters. Finally, regarding transduction, both detector resistances (R Di ) are deduced from their TCR. This model was calibrated with silicon data and FEM simulation.

C. Manufacturing defects and fault modeling

Process scattering will affect some model parameters. For instance, the nominal value of the electrical resistances depends on polysilicon doping, thickness, width and length (a typical 3 dispersion of about 20% is usually considered). In the same way, the resistance temperature coefficient TCR will be affected by doping concentrations. All these variations are global variations that affect every block of the model in the same way. In addition, process scattering will also introduces local variations that should be modeled as mismatch errors.

Regarding lateral dimensions, these are parameters that are rather well-controlled in the manufacturing process. They are nevertheless subject to variations, but with a low dispersion. Note that the distance d between the heater and the detector bridge is set by the mask and therefore should not change from one fabricated device to another.

Note that contrary to lateral dimensions, the cavity depth is a parameter that is very difficult to control. Indeed, it is very sensitive to the etching process (etching solution composition, etching time and etching solution movements). As the optical control is not an easy task during etching, the cavity depth is likely to be reduced if ideal conditions are not met during the etching step.

III. BEHAVIORAL MODELING

This section details the development of the behavioral model that includes the influence of the cavity depth, considering the 4 main blocks of the initial model, i.e. the heater source, fluid conduction, fluid convection and transduction.

A. Heater source

During normal operation, the heating bridge is powered with a dc voltage (U H ) in order to set an initial temperature distribution in the cavity. The average heater temperature (T H ) is directly linked to the electrical power (P H ) with a linear relationship:

 H 2 H H A H H A H R U Rth T P Rth T T           
where T A is the ambient temperature, R H is the electrical resistance of the beam and Rth H is the thermal resistance of the heater beam.

The temperature dependence of the electrical resistance is taken into account in the power dissipation with:

   H 0 H H T TCR 1 R R         
where TCR is the temperature coefficient given by the foundry and R H0 is the nominal value of the heater resistance at a reference temperature T 0 . This electrical resistance is obviously independent of the cavity depth but just depends on the size of the heating resistance and on the material properties.

In contrast, the thermal resistance of the beam depends not only on the beam dimension but also on its geometrical environment. In order to evaluate the influence of the cavity depth on this parameter, we have performed a number of FEM simulations for different values of the cavity depth between 35µm and 490µm and different values of the heater temperature from 350K to 600K, and we have recorded the heat transfer coefficient h H (in W.m -2 .K -1 ). This coefficient writes:

 S Rth 1 h H H       
where S=54.7×10 3 µm² is the exchange surface between the beam and the fluid given by 2(2r 1 +e)L, with L the beam length and e the beam thickness.

Figure 3 reports the heat transfer coefficient (h H ) with respect to the cavity depth h 1 for different heater temperatures T H . The higher the heater temperature is, the higher the heat transfer coefficient. This phenomenon is only due to the variation of air thermal conductivity which increases with temperature. This dependence must be included in the model in order to enable the simulation of the device under different biasing conditions. Two behaviors can be distinguished:

 for high values of the cavity depth, the heat transfer coefficient is constant,  for low values of the cavity depth, the heat transfer coefficient increases when the cavity depth decreases.

This can be explained by the shape of the hot bubble created by the heater, as illustrated in figure 4. When the cavity depth is high enough, the lateral silicon walls of the cavity limit the size of the hot bubble. The size of this hot bubble is therefore not affected by the value of the cavity depth and the heat transfer coefficient just depends on the heater temperature and lateral dimensions. In contrast, for low values of the cavity depth, this parameter becomes the limiting factor and the size of the hot bubble tends to reduce. In this case, the heat transfer coefficient depends on the heater temperature, the heater width and the distance between the heater and the bottom of the cavity. The transition between these two behaviors occurs around h 1 =175µm, which corresponds to half the distance between the heater and the lateral cavity border (r 2 ). Note that this threshold does not depend on the heater temperature. From this analysis, we have derived an analytical model for the heat transfer coefficient. This model relies on the assumption that we have a radial heat flow in a cylindrical geometry. Considering the expression of air conductivity from 100K to 600K:

  3 3 2 2 1 0 1 ) ( T T T T          
with  0 the air conductivity extrapolated at 0K, and  1 ,  2 and  3 the coefficients of thermal variation of conductivity for air.

Integrating the Fourier law for heat conduction

) ( grad T air    
in a cylindrical geometry, we can express:

                                   A H 4 A 4 H 3 A H 3 A 3 H 2 A H 2 A 2 H 1 1 2 1 0 H T T T T 4 T T T T 3 T T T T 2 1 R R ln R h  
where R 1 and R 2 correspond to the radius of the cylinders representing the heater and the hot bubble respectively.

Note that this equation includes the dependence with respect to the heater temperature T H . To include the dependence with respect to the cavity depth, we have then established the radius of the equivalent cylinder R 2 =f(h 1 ), which would give the heat transfer coefficient obtained by FEM simulations (Figure 5). First, we have determined the different values of h H for different values of the etching cavity (h 1 ). Second, we have calculated the cylinder equivalent radius R 2 to minimize the error using equation ( 4) to represent the values of h H previously determined for each value of h 1 . In that step, we consider a cylinder equivalent radius R 1 =16µm for the heater. This value is a trade-off between a cylinder with a perimeter equivalent to the heater bridge perimeter and a cylinder that contains the heater beam. Finally, we express the equivalent radius of the hot bubble R 2 to reflect a linear relationship for low values of h 1 and a saturation at r 2 =350µm for high values of h 1 with a transition at r 2 /2:

 4 4 2 4 1 1 2 2 2 r h h r R              
The order of the denominator polynomial has been chosen for adequate transition between both regions (Figure 5).

B. Fluid conduction

The initial temperature of both detectors, i.e. the common mode temperature T CM , is also controlled by heat conduction in the air. The same assumption of a radial heat flow in a cylindrical geometry can therefore be used. The relationship between the common mode temperature T CM and the heater temperature T H can then be obtained by solving heat conduction equation at a distance d from the heater:

                                                     4 4 3 3 3 2 2 2 1 2 1 2 4 4 3 3 3 2 2 2 1 4 3 2 ln ln 4 3 2 A H A H A H A H A CM A CM A CM A CM T T T T T T T T R R R d T T T T T T T T      



In the same way than in the previous section, we have determined the size of the equivalent cylinder R 2 ' that gives the best fit between the common mode temperatures T CM calculated with equation ( 6) and measured in FEM simulations. Results are summarized in figure 6 for various cavity depth h 1 . Similarly, we observe a linear relationship for low values of h 1 and a saturation at r 2 =350µm for high values of h 1 , with a transition at r 2 /2. However, in the linear region, the reduction of the size of the equivalent cylinder is less important than in the previous case. This is due to the fact that the common mode temperature is measured on a specific axis (i.e. sensitive axis) whereas the total heat flow is an average effect along all the directions and is dominated by the smaller distance.

As a result we have to consider, for the determination of the common mode temperature T CM , an equivalent cylinder with a radius R 2 ' different than the one used for the determination of the heat transfer coefficient. The relationship between this radius and the cavity depth can be analytically expressed by:

d 2 r h h ) d r ( ' R 4 4 2 4 1 1 2 2             
As illustrated in fig. 6, there is a good agreement between this expression and values established using FEM simulations.

C. Fluid convection

The sensor sensitivity is proportional to the differential temperature between both detectors in presence of acceleration along the sensitive axis. This temperature variation is due to free convection and can be analytically solved only for very simple geometries. For complex geometries FEM simulations are mandatory to calculate the differential temperature. Figure 7 shows the differential temperature T D -T CM =ΔT D /2 according to the cavity depth h 1 and to the heater temperature T H . Results are given for an acceleration of 1g along the sensitive axis and for the nominal distance d=175m. As for the heat transfer coefficient and for the same reasons, the differential temperature (ΔT D /2) saturates for high values of h 1 . For small depths, the differential temperature is proportional to h 1 and the transition between the two regions occurs around h 1 =150µm. For a given depth, the differential temperature is proportional to (T H -T A ) 1.8 . Equation 8gives the semi-empirical expression established to represent this behavior, which involves some fitting parameters and sensor dimensions:

  a ) T T ( ) r r )( r r ( ) r d )( r d ( 4 10 150 h h 10 84 . 10 T T 8 . 1 A H 1 2 2 1 2 1 4 4 6 4 1 1 6 CM D                  
where 10.84×10 -6 K -1.8 /g is a constant linked to the thermophysical properties of air and to the height of the top cavity (10mm for these simulations).

Figure 7 shows a good agreement between this expression and FEM simulations, with a relative error that remains below 6%.

D. Transduction

The role of the detectors is to convert thermal signal variations into electrical resistance variations thanks to the temperature sensitivity of polysilicon. Using a Wheatstone bridge, these resistance variations can then be translated into voltage variations for further conditioning. The electrical resistance of each detector R Di is simply given by:

                         0 0 2 0 2 1 1 Di i D D CM Di Di R Vbias Rth T T T TCR R R  
where R Di0 is the nominal value of each detector resistance at reference temperature T 0 . , and V biasi is the biasing voltage of each detector.

Note that this expression includes self-heating of the detectors, and therefore involves the thermal resistance of the detectors. This thermal resistance Rth D depends on the cavity depth as it is influenced by the geometrical environment. However because the detectors have a high resistance value of 50k, the self-heating impact is almost negligible. Consequently, we have not modeled the dependence of the cavity depth on the detector thermal resistance.

IV. VALIDATION & EXPLOITATION

A. Model validation

Model validation with respect to FEM simulations is based on numerous experiments for different values of the etching cavity depth. First, we have compared the calculated heater thermal resistance with the value determined from FEM simulations. Results are summarized in figure 8.a. It can be seen that we have a very good agreement whatever the heater temperature, with a relative error that remains below 2%. Then, we have calculated the detectors' common mode temperature and we have reported these values together with results from FEM simulations (Fig. 8.b). Here again a good agreement is observed whatever the heater temperature and h 1 are, with a relative error that remains below 4%.

B. Model exploitation

Interestingly, the common mode temperature (Fig. 8.b) and the differential temperature (Fig. 7) present very similar behaviors. As an illustration, figure 9 plots the differential temperature T D /2 resulting from a 1g acceleration vs. the common mode temperature T CM , measured in FEM simulations on devices with different cavity depths ranging from 35µm up to 490µm. It clearly appears that, for a given heater temperature, both temperatures are strongly correlated. This is very interesting because the differential temperature T D results from the application of acceleration and therefore directly relates to the device sensitivity, while the common mode temperature T CM just depends on the biasing conditions of the structure. In other words, it means that etching defects that affect the device sensitivity also affect the detectors' common mode temperature in a similar way. Hence, verifying the common mode temperature in absence of acceleration may be a way to verify the device sensitivity to a given acceleration. However, it should be underlined that this result is obtained for a fixed heater temperature, and for a device with all parameters at nominal value except the cavity depth that varies from 35µm to 490µm. Practically, the heater temperature T H cannot be imposed but only the biasing voltage of the heater beam U H , and the device parameters obviously exhibit some dispersions. To further explore this possibility, the behavioral model that includes the influence of the cavity depth can be very helpful. Indeed it is not realistic to perform Monte-Carlo FEM simulations, but on the contrary the behavioral model easily supports Monte-Carlo simulations at system-level. In this objective, we have performed experiments considering the following dispersions on the parameters' model:  A Gaussian variation with a standard deviation of 3=20% on the value of the electrical resistances. This variation corresponds to the typical uncertainty given by the foundry and is a global variation, i.e. it affects all resistances in the same amount.  A Gaussian variation with a standard deviation of 3=2µm on the value of the geometrical parameters related to horizontal dimensions, i.e. the heater dimension r 1 and the distance r 2 between the heater and the border of the cavity. Indeed lateral dimensions are parameters that are rather well-controlled in the manufacturing process and therefore exhibit a low dispersion. Note that the distance d between the heater and the detectors is not subject to process variations due to mask self-alignment.  A Gaussian variation with a standard deviation of 3=10% on TCR has been set as it is well-known that this parameter is quite sensitive to doping fluctuations.  A uniform variation on the value of the cavity depth h 1 between 35µm and 490µm, since it is the objective to investigate the influence of this parameter. Potential asymmetries have also been introduced in the model. In particular, a mismatch error with a standard deviation of 3=2% has been considered for the nominal value of the electrical resistance of the detectors R Di0 and the value of the reference resistors R REFi . a) b)

Simulations have been performed on 1,000 runs and we have first observed the dispersion on the heater temperature (Fig. 10.a) for a fixed biasing voltage U H =2V of the heater resistance. It clearly appears that the influence of the cavity depth on the heater thermal resistance has a significant impact on the achieved heater temperature. By taking into account this dispersion, the proposed model hence permits to achieve realistic fast system-level simulations, which would be extremely time-consuming in FEM since it requires 3D simulations involving both solid and fluidic elements. Figure 10.b gives the differential temperature T D /2 resulting from a 1g acceleration vs. the common mode temperature T CM , for a biasing of the heater beam at 2V. These results confirm that, even in presence of process variations, a rather good correlation exists between both temperatures. It is worth noting that the nominal value of the cavity depth is h 1 =390µm, which corresponds to an average heater temperature of about 540K when the heater beam is biased at 2V, which in turns translates into a common mode temperature in the range of 360K to 380K. In this region, we can observe a linear relationship between the common mode temperature T CM and the differential temperature T D /2. More generally, the correlation that exists between the common mode temperature T CM and the differential temperature T D offers an attractive perspective towards a lowcost test solution to verify the device sensitivity without the need for applying a calibrated acceleration. Indeed, the detectors' common mode temperature does not depend on the acceleration and is therefore a parameter that may be evaluated using only an electrical test setup.

V. CONCLUSION

In this paper, we have developed a behavioral model that includes the influence of etching defects on the sensitivity of MEMS convective accelerometers. Indeed, if lateral dimensions are rather well-controlled during the manufacturing process, this is not the case for the cavity depth which is very sensitive to the etching process conditions. Moreover, this parameter influences both the conductive and convective behavior of the structure, resulting in a significant impact on the sensitivity of manufactured devices.

Based on an existing behavioral model, we have established new physically-based expressions for the heater thermal resistance (which drives the actual heater temperature for a given biasing voltage) and for the common mode temperature of detectors. A semi-empirical expression has also been established for the differential temperature of detectors. All these expressions involve not only the lateral dimensions of the sensors but also the cavity depth h 1 , the latter being the more likely parametric fault in an industrial process. A very good agreement has been found between this behavioral model and FEM simulations.

Finally, the proposed model allows the realistic simulation of a convective accelerometer with a single electrical input, i.e. the biasing voltage of the heater beam. It allows evaluating process scatterings using behavioral models as Monte-Carlo simulations are achievable in a reasonable CPU time compared to FEM simulations. It is therefore very useful to explore potential correlations between different parameters. In particular, we have established that a correlation between the common mode temperature of the detectors and the differential temperature of the detectors exists.

Future work involves further exploration of such correlations in order to define alternate test procedures able to verify and/or estimate sensor sensitivity. Another perspective concerns the refinement of the model in order to include the influence of the package on the behavior of the final product.

Figure 1 .

 1 Figure 1. SEM picture of the prototype and geometrical parameters: r1=20µm, r2=350µm, d=175µm, h1=390µm

Figure 2 .

 2 Figure 2. Block diagram of the sensor model

Figure 3 .

 3 Figure 3. Heat transfer coefficient vs. cavity depth for different heater temperatures

Figure 4 .

 4 Figure 4. Temperature profile within cavity for two different values of the cavity depth -a) h1=490µm and b) h1=75µm

Figure 5 .

 5 Figure 5. Size of the equivalent cylinder representing the hot bubble vs. cavity depth

Figure 6 .

 6 Figure 6. Size of the equivalent cylinder that governs the common mode temperature vs. cavity depth

Figure 7 .

 7 Figure 7. Differential temperature (ΔTD/2) vs. cavity depth for different heater temperatures (solid line: model, dots: FEM)

Figure 8 .

 8 Figure 8. Comparison between behavioral model and FEM simulation for the heater thermal resistance RthH (a) and for the common mode temperature TCM (b) (solid line: model, dots: FEM)

Figure 9 .

 9 Figure 9. FEM simulation results: differential temperature (ΔT D /2) due to a 1g acceleration vs. common mode temperature

Figure 10 .

 10 Figure 10. Behavioral model Monte-Carlo simulation results: dispersion on the heater temperature with UH=2V (a) and differential temperature (ΔTD/2) due to a 1g acceleration vs. common mode temperature (b)


V Kerzérho
email: v.a.kerzerho@ewi.utwente.nlazais

F Azais

M Comte

P Cauvet

S Bernard

M Renovell
email: renovell@lirmm.fr

ANC-based method for testing converters with random-phase harmonics

Keywords: converter testing, harmonic characterization, random-phase harmonics

In this paper, we present an extension of the ANC ("Analogue Network of Converters")-based method to characterize the harmonic components of a set of converters with random-phase harmonics using only digital test resources. The ANC-based method was primarily developed under the assumption that the harmonics' phase is proportional to the input phase. This assumption is not valid for all converter architectures, where filtering effects may affect the harmonics' phase. The improved ANC-based method is able to calculate the magnitude of the harmonic components regardless of their phase. The simulation results and the experiments show an excellent agreement between the values measured using the method and the values measured with a usual test setup, for the THD and SFDR parameters.

I. INTRODUCTION

The democratization of portable data and communication applications like mobile phones, laptops, or audio MP3 players is currently the mainspring of the lookout for integration of many different functions into a single package. Systems-in-Package (SiPs) or Systems-On-Chip (SOCs) that integrate very different analogue or mixed-signal blocks have been developed toward this aim.

Although they offer clear benefits such as device extreme miniaturization or connection length reduction, they imply in compensation very significant test challenges. Indeed in many mixed-signal circuits, the test of the analogue blocks may represent up to 90% of the whole test effort while these blocks represent only 10% of the whole chip area. The reason of such a challenge is twofold. Firstly, analogue testing is made of a long sequence of parameter characterization that is performed using very expensive instruments able to accurately measure analogue signals. Secondly, controllability and observability of deeply embedded analogue blocks are much reduced and the possibility of external testing may be limited, as far as the number of pads is greatly reduced. Consequently the internal accessibility of the whole system decreases drastically. Also, as signals become faster and systems are operated at higher speeds, external testing become more susceptible to noise, crosstalk and probing problems.

To overcome these problems, several authors have proposed different Built-In-Self-Test (BIST) techniques where signals are internally generated and/or analyzed [START_REF] Toner | A BIST scheme for an SNR test of a sigmadelta ADC[END_REF][START_REF] Toner | A BIST Technique for a Frequency Response and Intermodulation Distortion Test of a Sigma-Delta ADC[END_REF][START_REF] Ohletz | Hybrid Built In Self Test (HBIST) for Mixed Analog/Digital Integrated Circuits[END_REF][START_REF] Sunter | A simplified polynomial-fitting algorithm for DAC and ADC BIST[END_REF][START_REF] Azais | Towards an ADC BIST scheme using the histogram test technique[END_REF][START_REF] Azais | Implementation of a linear histogram BIST for ADCs[END_REF][START_REF] Arabi | A New Built-In Self Test Approach For Digital-to-Analog and Analog-to-Digital Converters[END_REF]. Another possible and less expensive solution consists in introducing Design-For-Test (DFT) features to internally transform the analogue signals into digital signals that are made controllable and observable from the chip I/Os [START_REF] Nagi | A Signature Analyzer for Analog and Mixed-Signal Circuits[END_REF][START_REF] Mahoney | DSP-Based Testing of Analog and Mixed-Signal Circuits[END_REF][START_REF] Burns | An Introduction to mixed-signal IC test and measurement[END_REF][START_REF] Huertas | Test and design-for-testability in mixed-signal integrated circuits[END_REF]. As a result, only digital signals are externally handled by non-expensive digital test equipment (Low Cost Tester).

In an example prototype for Set-Top-Box applications, like in most current mixed-signal systems, the converters are among the main components: 2 Analogue-to-Digital Converters (ADC) and 6 Digital-to-Analogue Converters (DAC) are embedded in the same SiP. The today specifications for these converters require a 10-bit resolution, but the next generations will make use of 12-bit converters. Testing this whole set of converters is a very complex task requiring a long test time because of the above mentioned problems of accessibility, signal integrity, accuracy of converter parameter measurements.

In this context, an original DFT technique called "Analogue Network of Converters" (ANC) was developed that permits to test the whole set of embedded ADCs and DACs [START_REF] Kerzérho | Analogue Network of Converters": a DFT Technique to Test a Complete Set of ADCs and DACs Embedded in a Complex SiP or SOC[END_REF]. An extremely small circuitry is added to the original chip allowing the application of a fully digital lowcost test to the System-in-Package/System-On-Chip. The basic principle consists in setting a test path made of DACs and ADCs. The fault masking is avoided by using a postprocessing algorithm that discriminates harmonics from the different converters inserted in the test path. The ANC-based method can also be exploited to test standalone ADCs using low resolution arbitrary waveform generators as it breaks the rule stating that the test equipment resolution should be at least 2 bits higher than the ADC under test [START_REF] Kerzérho | ADC Production Test Technique Using Low-Resolution Arbitrary Waveform Generator[END_REF].

The ANC-based method was developed with the assumption that the phase of the harmonic components is strictly proportional to the input signal nominal phase.

978-1-4244-7791-3/10/$26.00 ©2010 IEEE However our experiments did not confirm this theoretical assumption. Indeed, we could observe that some harmonics may have a phase that is different from zero, despite the input signal phase is set to zero. In this paper, we present an extension of the primarily developed method to the estimation of the harmonics with a phase which is not correlated to the phase of the input signal.

The paper is organized as follows. Section II gives the fundamental knowledge on data converter testing based on spectral analysis and highlights the issues related to the phase of harmonics. Section III depicts the summarized principle of the ANC-based method. The mathematical developments of the improved ANC-based method are then presented in section IV, followed by validations in section V. Finally section VI concludes the paper.

II. CONVERTERS, HARMONICS AND PHASES

A very popular test for data converters relies on spectral analysis. It is usually implemented in a DSP-based test architecture [START_REF] Mahoney | DSP-Based Testing of Analog and Mixed-Signal Circuits[END_REF][START_REF]IEEE standard for terminology and test methods for analog-to-digital converters[END_REF]. Basically, a 1-tone sine-wave stimulus is sourced to the converter input, then an FFT analysis is performed on the output sample set. The resulting spectrum is subsequently processed to compute the dynamic parameters such as the Total Harmonic Distortion (THD) or the Spurious Free Dynamic Range (SFDR). The spectrum also permits to estimate an essential static parameter, i.e. the Integral Non Linearity (INL), using the method proposed in [START_REF] Janik | A Spectral approach to estimate the INL of A/D converter[END_REF].

A typical spectrum considering an ADC driven by an ideal sine-wave at frequency f 0 is illustrated in figure 1. This spectrum exhibits a fundamental bin at f 0 , the harmonic components being located at multiple values of the fundamental frequency, and noise at other frequencies. The fundamental component corresponds to the input sine-wave while other components are induced by the converter nonidealities. From this spectrum, the dynamic parameters are evaluated. In particular, the typical definitions for the THD and SFDR parameters are given by:

              2 2 2 10 log 10 ) ( F H dB THD k k (1)           ) max( log 20 ) ( 10 k H F dB SFDR ( 2
)
where F is the amplitude of the fundamental and H k the amplitude of the k th harmonic.

To introduce the theoretical fundamentals, let us consider an ideal sine-wave applied to an ADC. Using a Fourier series expansion, the output signal can be expressed by equation [START_REF] Ohletz | Hybrid Built In Self Test (HBIST) for Mixed Analog/Digital Integrated Circuits[END_REF] given hereafter. This equation distinguishes the sampled sinewave x(n) that would be delivered by an ideal ADC from the sum of all harmonic contributions introduced by the converter non-idealities. Hence H 0 corresponds to the offset contribution of the converter and H 1 corresponds to the contribution of the gain error and non linearity of the converter on the fundamental. The total amplitude of the fundamental component, previously defined as F, is the sum of H 1 and the contribution of the ideal ADC, which is included in x(n) in equation ( 3):

                 0 cos k k i n k n k H n x n s     (3)
where n is the sample index,  i the input phase,  k the phase of the k th harmonic, ( ) the noise that affect the converted signal, and  n the nominal sampling phase given by:

n N M n   2  ( 4
)
where N is the number of samples and M the number of cycles (i.e. signal periods) in the sample set.

The theoretical developments that mathematically link the harmonic components to the converter INL are given in [START_REF] Janik | A Spectral approach to estimate the INL of A/D converter[END_REF].

The expression of the output signal can be easily linked to the spectrum presented in figure 1 by developing equation ( 3):

                    0 ) sin( sin ) cos( cos k n k i k n k i k n k k H k k H n x n s        (5)
The sum term corresponds to the Fourier series expansion made of the complex harmonic values resulting from the FFT. Indeed, ) cos(

k i k k H   
is equal to the real part of the harmonic computation and ) sin(

k i k k H   
is equal to the imaginary part. Now if we consider the case of a converter with harmonics' phase proportional to the input phase:

) (   i k k  (6)
and by setting the input phase  i to zero, equation ( 5) becomes:

             0 cos k n k n k H n x n s   (7)
In this case, the imaginary part of the spectrum is cancelled and only the real part of the spectrum directly corresponds to the amplitude of the spectrum.

However if we consider the case of a converter with harmonics' phase not correlated to the input phase, setting the input phase  i to zero leads to:

                 0 sin ) sin( cos ) cos( k n k k n k k n k H k H n x n s      (8)
In this case, it clearly appears that both the real and the imaginary parts of the spectrum have to be considered and used in the computation of the dynamic parameters.

III. ANC FUNDAMENTAL PRINCIPLE

The Analogue Network of Converters, presented in [START_REF] Kerzérho | Analogue Network of Converters": a DFT Technique to Test a Complete Set of ADCs and DACs Embedded in a Complex SiP or SOC[END_REF], is a DfT concept developed to enable the fully digital test of a set of DACs and ADCs embedded in a complex system. It can also be used to relax the requirements on the test equipment in case of stand-alone converter testing [START_REF] Kerzérho | ADC Production Test Technique Using Low-Resolution Arbitrary Waveform Generator[END_REF].

In the ANC-based method, the analogue outputs of the DACs are basically connected to the analogue inputs of the ADCs, and only digital inputs and outputs are required for test stimulus application and test response analysis. As illustrated in figure 2, the method requires at least two DACs and one ADC. The idea is then to exploit different configurations and different test conditions in order to separate the harmonic contribution of each converter. The mathematical developments were established considering only static non-idealities of the converters, meaning that the harmonics' phase of each converter was not taken into account. In this case, the model presented in equation ( 3) simplifies and becomes:

                0 cos k i n k n k H n x n s    (9)
We are therefore in the situation where setting the initial phase  i to zero results in the cancellation of the imaginary part of the spectrum, and the real part directly leads to the amplitude of the spectrum.

To illustrate the method, let us first consider a test configuration in which the DAC1 output is directly connected to the ADC1 input. The spectrum of the resulting output signal is computed and the values of the harmonic components H k m are extracted. Obviously, the output signal is affected by errors of both converters. In other words, the resulting spectrum includes the harmonic contribution of DAC1 as well as the harmonic contribution of ADC1. According to equation ( 9), setting a zero initial phase shift, we can write the following equation:

FS k FS k m k Hadc1 Hdac1 H   (10)
where Hdac1 k FS and Hadc1 k FS correspond to the amplitude of the k th harmonic of DAC1 and ADC1 respectively, for a fullscale input signal (FS).

Equation [START_REF] Burns | An Introduction to mixed-signal IC test and measurement[END_REF] establishes a relation between the harmonic contributions of the two converters involved in the test configuration. In this equation, the left member is known and corresponds to the amplitude of the k th spectral bin measured on the spectrum, while the right member corresponds to the unknowns under calculation.

This short example demonstrates the relationship between one configuration and its resulting equation, which leads to the fundamental idea of the ANC-based method. By using different configurations, we can establish different equations. The objective is then to determine the adequate set of configurations that results in a system of independent equations, which can be solved in order to discriminate the harmonic contribution of each converter. We have identified 3 different configurations (DAC1/ADC1, DAC2/ADC1 and DAC1+DAC2/ADC1) associated with 4 different test conditions (related to amplitude and phase of test stimuli) that permit to obtain a system of five independent equations:

                                          
       k Hadc1 k Hdac2 Hdac1 H Hadc1 k Hdac2 Hdac1 H Hadc1 Hdac2 H Hadc1 Hdac2 H Hadc1 Hdac1 H FS k FS k FS k m k FS k FS k FS k m k FS k FS k m k FS k FS k m k FS k FS k m k
Solving this system permits to compute the amplitude of the harmonic components, for each individual converter.

IV. EXTENSION OF THE ANC-BASED METHOD

A limitation of the ANC-based method described in the previous section is that the phase of the harmonics is not taken into account in the mathematical developments. However, the method was proven efficient in practical experimentations with real converters [START_REF] Kerzérho | A novel DFT technique to test a complete set of ADC's and DAC's embedded in a complex SiP[END_REF]. This can be explained by the fact that in all experiments, the converters under study presented a phase of harmonics linearly proportional to the input signal nominal phase. In such a situation, when setting the input signal nominal phase to zero, the imaginary part of the spectrum is cancelled and the real part directly gives the amplitude of the spectrum. The system of equations established from the model given by equation ( 9) is therefore valid and leads to a correct estimation of the harmonic contribution of each individual converter. However the assumption that the harmonics' phase is linearly proportional to the input signal nominal phase is not always verified and highly depends on the converter architecture. We have seen that this assumption is often verified for types of architecture that use a sample and hold stage, but it cannot be assessed for other architectures where filtering and noise may affect the harmonics' phase. Our objective is consequently to improve the ANC-based method so that it can cover all the real-life cases.

The fundamental principle of this improved version of the ANC-based method is the same, but the mathematical developments have now to be established using the complete model valid in the general case, i.e. the model described by equation [START_REF] Ohletz | Hybrid Built In Self Test (HBIST) for Mixed Analog/Digital Integrated Circuits[END_REF]. So let us consider again the configuration in which the DAC1 output is directly connected to the ADC1 input. Obviously, the output signal is still affected by errors of both converters. Now, considering the general case described in equation ( 3) and setting the input signal nominal phase  i to zero, we can write the following equations:

                  FS k adc1 FS k FS k dac1 FS k m k FS k adc1 FS k FS k dac1 FS k m k Hadc1 Hdac1 H Hadc1 Hdac1 H , , , , sin sin ) Im( cos cos ) Re(     (10)
Similarly, we can derive equations for all configurations and test conditions involved in the ANC-based method. We finally obtain a system of 10 independent equations given hereafter.

This system can be solved to determine the real and imaginary parts of harmonic contribution related to the different converters. From this, the dynamic parameters such as THD and SFDR can be computed for each converter. Solving this system of equations therefore makes the independent characterization of each converter possible.

                                                                                                                                                                                                                                             
                                          FS k adc1 FS k adc1 FS k FS k dac2 FS k dac2 FS k FS k dac1 FS k m k FS k adc1 FS k adc1 FS k FS k dac2 FS k dac2 FS k FS k dac1 FS k m k FS k adc1 FS k FS k dac2 FS k FS k dac1 FS k m k FS k adc1 FS k FS k dac2 FS k FS k dac1 FS k m k FS k adc1 FS k FS k dac2 FS k m k FS k adc1 FS k FS k dac2 FS k m k FS k adc1 FS k FS k dac2 FS k m k FS k adc1 FS k FS k dac2 FS k m k FS k adc1 FS k FS k dac1 FS k m k FS k adc1 FS k FS k dac1 FS k m k k k Hadc1 k k Hdac2 2 , 2 2 / , 1 2 / , 1 2 / , 5 , , 2 , 2 2 / , 1 2 / , 1 2 / , 5 , 2 / , 2 / 2 / , 2 / , 4 , 2 / , 2 / 2 / , 2 / , 4 , 2 / , 2 / 2 / , 2 / 3 , 2 / , 2 / 2 / , 2 / 3 , , , 2 , , , 2 , , , 1 , , , 1 ,

V. VALIDATION

The theoretical extension of the ANC-based test method has been validated through simulation, using the converter model described in [START_REF] Kerzérho | Analogue Network of Converters": a DFT Technique to Test a Complete Set of ADCs and DACs Embedded in a Complex SiP or SOC[END_REF]. Converter non-idealities are included in the model based on real-life INL values extracted from measurements on several data converters. Sampling jitter and thermal noise are also considered. In addition, we also include a possible injection of an arbitrary phase shift on the converter harmonics.

In order to validate the improved ANC-based method, we have considered a set of three converters, i.e. two DACs and one ADC. First, we have simulated each data converter in a stand-alone configuration and we have applied a usual test to get reference values for the THD and SFDR parameters. Then, we have simulated the set of converters in the different configurations and test conditions defined in the ANC-based method. From these five acquisitions, we have established the system of independent equations that links the measured harmonic contribution to the individual harmonic contribution of each converter (considering only the real part of the spectrum for the original ANC-based method and both the real and imaginary parts of the spectrum for the improved ANCbased method) and we have solved the system. Finally, we have computed the THD and SFDR parameters using the estimated amplitude of the harmonic components.

Results are summarized in Table I and II, which compare the values of the THD and SFDR parameters computed using the stand-alone configuration to the values of the THD and SFDR parameters computed using either the original or the improved ANC-based method, with and without phase shift injected on the converters harmonics. Analyzing these results, it can be seen that when the converters harmonics are not affected by a random phase shift, both the original and the improved versions of the ANC-based method produce an accurate measurement of the THD and SFDR parameters, with an estimation error that remains below 0.7dB. However when the converters' harmonics are affected by a random phase shift, the performances of the original ANC-based method degrades with an estimation error that increases up to 4.3dB for the THD parameter of DAC1. In contrast, the improved ANC-based method gives very accurate measurements, with an estimation error lower than 0.2dB. These results clearly validate the efficiency of the improved ANC-based method.

VI. CONCLUSION

In this paper, we have presented an extension of the ANCbased method in order to handle converters with random-phase harmonics. Indeed, the ANC-based method was originally developed under the assumption that the harmonics' phase is proportional to the input phase. Although this assumption is often verified, it is not valid for all converter architectures. New mathematical developments based on a general model that includes the harmonics' phase of converters were therefore established. The efficiency of the improved ANCbased test was evaluated through simulations, demonstrating an excellent agreement between the THD and SFDR values measured with a classical test setup and the THD and SFDR values computed with the method.

Compared to the initial ANC-based test method, no additional resources are required, and the number of test configurations and test acquisitions remains the same. Only the size of the independent equation system is different, growing from 5 to 10. However, it is worth noting that this computation is performed off-chip using the tester digital signal processing resources and has a negligible impact on the cost of the test procedure. These developments therefore offer an efficient generic method to perform the harmonic characterization of a set of data converters using only digital external test equipment, without any assumption on the converter architecture and potential phase shift of the converter harmonics.

Figure 1 .

 1 Figure 1. Output spectrum of an ADC under an ideal sine-wave stimulus

Figure 2 .

 2 Figure 2. Analog Network of Converters (ANC) concept

TABLE I

 I

	.	MEASUREMENT OF THD AND SFDR VALUES WITHOUT
		PHASE SHIFT ON HARMONICS' PHASE
			THD (dB)		SFDR (dB)
		DAC1	DAC2	ADC	DAC1	DAC2	ADC
	Ref	-77.9	-74.3	-66.1	78.9	76.1	70.5
	Original ANC	-77.2	-74.1	-65.8	79.3	75.6	69.8
	Inproved ANC	-77.3	-74.2	-65.7	79.3	75.6	69.5
	TABLE II.	MEASUREMENT OF THD AND SFDR VALUES WITH
		PHASE SHIFT INJECTED ON HARMONICS' PHASE
			THD (dB)		SFDR (dB)
		DAC1	DAC2	ADC	DAC1	DAC2	ADC
	Ref	-77.9	-74.4	-66.1	78.9	76.1	70.2
	Original ANC	-73.6	-73.2	-67.0	76.8	76.6	68.5
	Inproved ANC	-77.7	-74.4	-66.2	78.8	76.1	70.4


Guy Barles
email: barles@lmpt.univ-tours.fr

Annalisa Cesaroni

Matteo Novaga
email: novaga@math.unipd.it

Homogenization of fronts in highly heterogeneous media

Keywords: Homogenization, propagation of fronts, heterogeneous media, evolution by mean curvature, viscosity solutions AMS subject classifications : 35B27, 35K55, 35J20, 53C44, 49L25

We consider the evolution by mean curvature in a highly heterogeneous medium, modeled by a periodic forcing term, with large L ∞ -norm but with zero average. We prove existence of a homogenization limit, when the dimension of the periodicity cell tends to zero, and show some properties of the effective velocity.

Introduction

This paper deals with mean curvature flow in a heterogeneous medium, represented by a Z nperiodic function g ∈ Lip(R n ) acting on the flow as an additive forcing term. This problem has been already considered in the literature, starting from the papers [START_REF] Craciun | Effective motion of a curvature-sensitive interface through a heterogeneous medium[END_REF][START_REF] Lions | Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications[END_REF]. In particular, in [START_REF] Dirr | Pulsating wave for mean curvature flow in inhomogeneous medium[END_REF][START_REF] Cardaliaguet | A discussion about the homogenization of moving interfaces[END_REF] the authors consider the following scaling

V ε (x) = ε H(x) + g x ε for x ∈ Γ ε (t), (1)
where Γ ε (t) is the evolving hypersurface, V ε (x) is its normal velocity at x ∈ Γ ε (t), and H(x) its mean curvature. Under appropriate assumptions on the function g, one can prove that the evolution law (1) converges, as ε → 0, to the first-order anisotropic law

V (x) = c(ν(x)) for x ∈ Γ(t),
where Γ(t) denotes the limit of Γ ε (t), V (x) the limiting normal velocity at x ∈ Γ(t) and c is a continuous function of the normal vector ν(x) to Γ(t) at x, and can be determined by solving a suitable cell problem. Since c = 0 when g has zero average, a natural question is what happens if we consider a different time-scaling of (1), namely

V ε (x) = H(x) + 1 ε g x ε for x ∈ Γ ε (t), (2)
under the additional assumption that g has zero average. The object of the present paper is the study of the limit of the evolution law (2) as ε → 0. We assume that the evolving hypersurface Γ ε (t) is a graph of a function u ε (•, t) with respect to a fixed hyperplane, independent of ε, and that the forcing term g does not depend on the variable orthogonal to such hyperplane. In this case, the function u ε satisfies the equation

u ε t (t, x) = tr I - Du ε ⊗ Du ε 1 + |Du ε | 2 D 2 u ε + 1 ε g x ε 1 + |Du ε | 2 in (0, +∞) × R n , (3)
where we require that the initial datum u 0 : R n → R is a uniformly continuous function such that u 0 (x)qx is bounded for some q ∈ R n . Another motivation for considering (3) comes from the following homogenization problem:

u ε t = tr I - Du ε ⊗ Du ε 1 + |Du ε | 2 D 2 u ε + g x ε 1 + |Du ε | 2 in (0, +∞) × R n . (4)
Since the function g is bounded, one can show that, as ε → 0, u ε → u locally uniformly in R n × [0, +∞). Moreover, u solves the limit equation

u t = tr I - Du ⊗ Du 1 + |Du| 2 D 2 u + ˆ[0,1] n g dx 1 + |Du| 2 in (0, +∞) × R n . ( 5
)
This result, derived in [START_REF] Cesaroni | Curvature flow in heterogeneous media[END_REF] when n = 1, can be obtained in general dimensions by the so-called perturbed test function method [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF], which is by now a standard tool in viscosity solutions theory applied to homogenization problems. More precisely, one considers the formal asymptotic expansion u ε (x, t) = u(x, t)

+ ε 2 χ x ε , x, t (6)
where the corrector χ(ξ, x, t) = ψ(ξ, Du(x, t)) is periodic in ξ and solves, for every fixed p = Du(x, t), the cell problem

tr I - p ⊗ p 1 + |p| 2 D 2 ξξ χ = ˆ[0,1] n g(x)dx -g(ξ) 1 + |p| 2 in R n .
Plugging the expansion (6) in (4) and using the comparison principle for viscosity solutions, one obtains that u solves [START_REF] Barles | Uniqueness results for quasilinear parabolic equations through viscosity solutions methods[END_REF].

In this paper, we apply the same method to Equation (3), and we prove that, under a suitable assumption on the function g (see Section 2), the limit function u solves the anisotropic parabolic equation

u t = tr A(Du) D 2 u in (0, +∞) × R n ,
where A(p) is a smooth function depending on g, with values in the set of positive definite symmetric matrices. Obviously, when g ≡ 0 we have A(p) = I -p⊗p 1+|p| 2 . When n = 1, thanks to an explicit representation formula for A(p), we can further show that 0 < A(p) ≤ 1 1 + p 2 for all p ∈ R and that lim |p|→∞ A(p)(1 + p 2 ) = 0, when g ≡ 0. In particular, this implies that the presence of g has the effect of decreasing the speed of the front in the limit, without stopping the motion.

A further consequence of our result is the nonexistence of compact embedded solutions of the prescribed curvature problem

H + g = 0, (7)
for all g such that A(p) = 0 for some p ∈ R n . For similar results in the nonperiodic setting, we refer to [START_REF] Kirsch | An obstruction to the existence of immersed curves of prescribed curvature[END_REF] and references therein. The paper is organized as follows. In Section 2 we describe the standing assumptions on the forcing term g. Section 3 and 4 are devoted to the analysis of two ergodic problems in R n , which permit to define the limit parabolic operator F (p, X) = tr (A(p) X). In Section 4 we consider the case of planar curves (i.e. n = 1), and obtain a more explicit description of the function A(p) (see Proposition 4.1). Finally, Section 5 contains the main homogenization result (see Theorem 5.3) and a discussion of its consequences for the related prescribed curvature problem [START_REF] Bensoussan | Perturbation methods in optimal control[END_REF].

Standing assumptions

In this section we state some conditions on the forcing term g, which will hold throughout the paper. Setting Q := (0, 1) n , our first condition is: (G1) g : R n → R is Lipschitz continuous, Z n -periodic and ´Q g(y)dy = 0.

The requirement that g has zero average on Q is necessary due to the homogenization result for Equations ( 1) and ( 4) discussed in the Introduction.

We also need a condition ensuring that the oscillation of g on Q is not too large. Let us first define the space BV per (Q) of functions which have periodic bounded variation in Q. We refer to [START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF] for a general introduction to functions of bounded variation and sets of finite perimeter.

It is a classical result that any u ∈ BV (Q) admits a trace u Q on ∂Q (see e.g. [3, Thm 3.87]). Let ∂ 0 Q := ∂Q ∩ {y : n i=1 y i = 0} and let σ : ∂ 0 Q → ∂Q be the function σ(y) := y + n i=1 λ i (y)e i , where λ i (y) = 1 if y i = 0 and λ i (y) = 0 otherwise. The periodic total variation of u ∈ BV (Q) is defined as

|Du| per (Q) := |Du|(Q) + ˆ∂0Q |u Q (y) -u Q (σ(y))| dH n-1 (y). ( 8
)
The space BV per (Q) is the space BV (Q) endowed with the norm

u BVper(Q) := u L 1 (Q) + |Du| per (Q).
For every E ⊆ Q we define the periodic perimeter of E as

Per per (E, Q) := |Dχ E | per (Q) (9)
where χ E is the characteristic function of E. We observe that BV per (Q) coincides with BV (T n ), where T n := R n /Z n is the n-dimensional torus. In particular, the following Coarea Formula holds [3, Thm 3.40]

|Du| per (Q) = ˆR Per per ({u > t}, Q) dt for all u ∈ BV per (Q). ( 10
)
The second condition we assume on g is:

(G2) there exists δ < 1 such that for every E ⊆ Q of finite perimeter

ˆE g(y) dy ≤ δ Per per (E, Q). (11)
Note that, when n > 1, condition (G2) is satisfied whenever g L n (Q) < C(Q), where C(Q) is the isoperimetric constant of T n . Indeed, since g has zero average, possibly exchanging E with Q \ E in [START_REF] Cesaroni | Curvature flow in heterogeneous media[END_REF] we can assume that |E| ≤ 1/2. By the isoperimetric inequality on T n [START_REF] Ros | The isoperimetric problem. In Global theory of minimal surfaces[END_REF], we then obtain ˆE g(y) dy

≤ g L n (Q) |E| n-1 n ≤ g L n (Q) C(Q) Per per (E, Q) = δ Per per (E, Q) where δ = g L n (Q) /C(Q) < 1. When n = 1, condition (G2) is equivalent to max y∈[0,1] ˆy 0 g(s) ds -min y∈[0,1] ˆy 0 g(s) ds < 2 (12)
which is the same condition assumed in [START_REF] Cardaliaguet | A discussion about the homogenization of moving interfaces[END_REF].

3 The cell problems and the effective operator

In this section, we consider two ergodic problems in Q, see [START_REF] Crandall | Users guide to viscosity solutions of second order partial differential equations[END_REF], [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF], whose solutions are useful to define the limit problem as ε → 0 of the singularly perturbed Equations (3).

Lemma 3.1 Under the standing assumptions, for every p ∈ R n the equation

-div Dχ + p 1 + (Dχ + p) 2 = g(y) in R n (13)
admits a Z n -periodic solution χ(y; p) ∈ C 2+α (R n ), for all α < 1, which is unique up to an additive constant. Moreover χ depends smoothly on p.

Proof We observe that ( 13) is the Euler-Lagrange equation of the functional

J p (u) := ˆQ 1 + (Du + p) 2 -gu dy + ˆ∂0Q |u Q (y) -u Q (σ(y))| dH n-1 (y)
which is a convex lower semicontinuous functional on BV per (Q) (see [START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF]). We claim that J p is coercive on the subspace

BV 0 per (Q) := u ∈ BV per (Q) : ˆQ u = 0 .
Notice that |Du| per (Q) is an equivalent norm on BV 0 per (Q) [START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF]Thm 3.44]. By condition (G2) and the Coarea Formula (10), recalling that ´Q g dy = 0, we have

ˆQ gu dy = ˆR ˆ{u>t} g dy dt ≤ δ ˆR Per per ({u > t}, Q) dt = δ|Du| per (Q), (14)
for all u ∈ BV per (Q). Therefore, we obtain

J p (u) ≥ ˆQ |Du + p| -ˆQ gu dy + ˆ∂0Q |u Q (y) -u Q (σ(y))| dH n-1 (y) ≥ -|p| -ˆQ gu dy + |Du| per (Q) ≥ -|p| + (1 -δ)|Du| per (Q) ,
which proves that J p is coercive on BV 0 per (Q). Since J p is coercive and lower semicontinuous on BV 0 per (Q), it admits a minimizer χ(•; p). Moreover, by convexity, χ(•; p) is the unique minimizer of J p on BV per (Q), up to an additive constant.

Finally, by [START_REF] Giusti | On the equation of surfaces of prescribed mean curvature[END_REF], any solution χ(y; p) of ( 13) is Lipschitz continuous, and hence of class C 2+α (R n ) by standard elliptic regularity [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]. Also, by differentiation of (13) in the p-variables, the same regularity holds for any derivative of χ(y; p) with respect to p.

2

Remark Notice that the periodic function ψ p (y) = χ(y; p) + p • y solves the prescribed curvature problem

-div Dψ p 1 + (Dψ p ) 2 = g(y), y ∈ R n .
In particular, the graph of ψ p is a plane-like solution of the geometric equation H = g, lying at a bounded distance from the hyperplane {(y, p • y) : y ∈ R n }. We refer to [START_REF] Caffarelli | Planelike minimizers in periodic media[END_REF] for a general analysis of such solutions.

Lemma 3.2 For any p ∈ R n and M ∈ S n there exists a unique constant F (p, M ) such that there exists a Z n -periodic solution ψ(y; p, M ) ∈ C 2+α (R n ), for all α < 1, to the cell problem

F (p, M ) = tr I - (p + D y χ) ⊗ (p + D y χ) 1 + |p + D y χ| 2 D 2 ψ + M + 2D 2 py χM -2 (p + D y χ) T D 2 yy χ Dψ + (D p χ) T M 1 + |p + D y χ| 2 - (p + D y χ) • (Dψ + (D p χ) T M ) (1 + |p + D y χ| 2 ) 2 (p + D y χ) + g(y) (p + D y χ) • (Dψ + (D p χ) T M ) 1 + |p + D y χ| 2 , ( 15
)
where χ(y; p) is the solution to (13) with χ(0; p) = 0. Moreover ψ(y; p, M ) is in C 2 (R n ) and is unique up to an additive constant.

Finally, there exists a n × n symmetric matrix A(p), depending smoothly on p, such that

F (p, M ) = tr(A(p)M ). ( 16
)
Proof Observe that, using the fact that χ solves [START_REF] Crandall | Users guide to viscosity solutions of second order partial differential equations[END_REF], Equation ( 15) can be rewritten as

tr B(y, p)D 2 ψ + b(y, p) • Dψ = F (p, M ) -tr B(y, p)(I + 2D 2 py χ)M -(b(y, p)) T M D p χ , (17)
where B(y, p) := I -

(p+Dyχ)⊗(p+Dy χ) 1+|p+Dy χ| 2
is a symmetric, positive definite matrix and

b(y, p) := - 2(p + D y χ) T D 2 yy χ 1 + |p + D y χ| 2 + 3g(y) 1 + |p + D y χ| 2 + 2∆ yy χ 1 + |p + D y χ| 2 (p + D y χ).
Moreover, recalling that χ is Z n -periodic and smooth, we get that ( 17) is a uniformly elliptic equation and that both B(y, p) and b(y, p) are Z n -periodic in y. The existence of a unique constant F (p, M ) such that (17) admits a continuous periodic solution ψ is then a well-known fact (see [4, Thm II.2], [1, Thm 7.1]). Finally, ψ is unique up to an additive constant and is of class C 2+α with respect to y, for all α < 1, by elliptic regularity [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF].

We also have an explicit characterization of F (p, M ). Indeed, consider the differential operator L p (φ) := tr B(y, p)D 2 φ + b(y, p) • Dφ and let L ⋆ p be its formal adjoint. Then, the equation L ⋆ p (m) = 0 admits a Z n -periodic solution m(y; p) > 0, which is unique up to a multiplicative constant (see [START_REF] Bensoussan | Perturbation methods in optimal control[END_REF]Thm II.4.2]), so that we may fix ´Q m(y; p)dy = 1. Notice that m(y; p), as well as χ(y; p), depends smoothy on p by elliptic regularity.

In [7, Thm II.6.1] (see also [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF]Thm 2.1], [1, Corollary 6.2]) it is proved that

F (p, M ) = ˆQ tr B(y, p)(I + 2D 2 py χ(y; p))M + (b(y, p)) T M D p χ(y; p) m(y; p) dy.
This formula implies, in particular, the regularity of F with respect to p, since the functions b(y, •) and B(y, •) are smooth, due to the regularity properties of χ and m. Finally, the above representation formula for F also implies [START_REF] Giusti | On the equation of surfaces of prescribed mean curvature[END_REF], since F (p, •) is a linear function of M for any fixed p, and hence which can be written as tr[A(p)M ] for some symmetric matrix A(p).

2

Remark Notice that, always by elliptic regularity, from [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF] it follows that the map g → A(p) (when defined) is continuous with respect to the Lipschitz norm of g. In particular, since A(p) = I -p⊗p 1+|p| 2 when g = 0, we get that there exists δ > 0 such that, if g Lip < δ, then A(0) = 0. Notice that, possibly reducing δ, this condition implies [START_REF] Cesaroni | Curvature flow in heterogeneous media[END_REF].

The effective operator in dimension 1

When n = 1, we obtain a much more explicit description of the effective operator F (p, M ) (see Proposition 4.1). This is done by solving explicitly the two cell problems ( 13) and [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF]. Lemma 4.1 Under the standing assumptions, for every p ∈ R the periodic solution to (13) such that χ(0; p) = 0 is given by

χ(y; p) = -py + ˆy 0 c(p) -G(s) 1 -(c(p) -G(s)) 2 ds, (18)
where G ′ (y) = g(y), ´1 0 G(y)dy = 0, and

c = f -1 with f (c) = ´1 0 c-G(y) √ 1-(c-G(y)) 2 dy. Proof We denote by m := max [0,1] G and m = min [0,1] G.
Integrating once the equation, easy computations show that every solution to (13) satisfies 2 for some constant c. Therefore solving the problem necessarily requires max [0,1] (Gc) < 1 and min [0,1] (Gc) > -1. It is possible to find c satisfying this condition if and only if m-m < 2. This is ensured by [START_REF] Cesaroni | Curvature flow in heterogeneous media[END_REF]. In this case it is sufficient to choose c ∈ (m-1, m+1). For such constants c, we get that χ y (y

1 1+(p+χy ) 2 = 1-(c-G(y))
) = -p + c-G(y) √ 1-(c-G(y)) 2 . We define the function f : (m -1, m+ 1) → R by f (c) := ˆ1 0 c -G(y) 1 -(c -G(y)) 2
dy.

Straightforward computations show that f is strictly increasing and we claim that f (m-1, m+1) = (-∞, +∞). Indeed we remark that, by assumption (G1), G ∈ C 1,1 (R) and, if y 0 is a maximum point of G, then for y close to y 0 we have

|G(y) -G(y 0 )| ≤ k|y -y 0 | 2 . As a consequence, if c = m -1, in a small neighborhood of y 0 we have 1 -(m -1 -G(y)) 2 = 1 -(G(y 0 ) -1 -G(y)) 2 = 2(G(y) -G(y 0 )) + (G(y) -G(y 0 )) 2 ≤ k|y -y 0 | 2 ,
for some constant k > 0. Possibly changing the constant k, we then get c -G(y)

1 -(c -G(y)) 2 ≤ - k |y -y 0 | .
This inequality shows that the function is not integrable for c = m -1 and therefore f (c) → -∞ as c → m -1. An analogous argument holds when c = m + 1. Since we are looking for periodic solutions to Equation ( 13), we impose the condition ´1 0 χ y (y)dy = 0. This gives f (c) = p, that is, c = f -1 .

Notice that the function c is smooth and

c ′ (p) = 1 ´1 0 [1 -(c(p) -G(y)) 2 ] -
0 < A(p) = c ′ (p) ´1 0 1 -(c(p) -G(y)) 2 dy ≤ 1 1 + p 2 ∀p ∈ R, (19)
where G(y) and c(p) are as in Lemma 4.1. Moreover if g ≡ 0 there exists a constant K g such that

0 < A(p)(1 + p 2 ) ≤ K g 1 + p 2 . ( 20
)
In particular A(p)(1 + p 2 ) → 0 as |p| → +∞.

Proof To obtain the characterization of A, we explicitly solve Equation [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF]. Due to the homogeneity properties of [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF] with respect to M , we immediately get A(p) = F (p, 1). We rewrite [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF] with M = 1. We consider the coefficient of ψ y and obtain, recalling the characterization (18) of χ, g(y)(p

+ χ y ) 1 + (p + χ y ) 2 - 2χ yy (p + χ y ) (1 + (p + χ y ) 2 ) 2 = p + χ y 1 + (p + χ y ) 2 g(y) 1 + (p + χ y ) 2 - 2χ yy 1 + (p + χ y ) 2 = 3g(y)(c(p) -G(y)).
The last two terms on the right-hand side of (15) coincide with

2χ yy χ p (p + χ y ) (1 + (p + χ y ) 2 ) 2 - g(y)χ p (p + χ y ) 1 + (p + χ y ) 2 = -3g(y)(c(p) -G(y))χ p .
Then, using the explicit formula for χ yp deduced from ( 18), ( 15) can be rewritten as

ψ yy + χ yp + 3g(y)(c(p) -G(y)) 1 -(c(p) -G(y)) 2 (ψ y + χ p ) = F (p, 1) 1 -(c(p) -G(y)) 2 - c ′ (p) (1 -(c(p) -G(y)) 2 ) 3 2 Note that 3g(y)(c(p)-G(y)) 1-(c(p)-G(y)) 2 = [log(1 -(c(p) -G(y)) 2 ) 3
2 )] y . Therefore we obtain

1 -(c(p) -G(y)) 2 3 2 (ψ y + χ p ) ′ = F (p, 1) 1 -(c(p) -G(y)) 2 -c ′ (p)
and integrating we get, for some constant d(p),

1 -(c(p) -G(y)) 2 3 2 (ψ y + χ p ) = d(p) + F (p, 1) ˆy 0 1 -(c(p) -G(s)) 2 ds -c ′ (p)y.
Then

ψ y = -χ p + d(p) + F (p, 1) ´y 0 1 -(c(p) -G(s)) 2 ds -c ′ (p)y (1 -(c(p) -G(y)) 2 ) 3 2
.

We look for a periodic solution ψ, then we impose that ψ y (0) = ψ y (1). Recalling the formula for c ′ (p) and χ p , we get χ p (0) = χ p (1) = 0, so ψ y (0) =

d(p) (1-(c(p)-G(0)) 2 ) 3 2
and

ψ y (1) = d(p) + F (p, 1) ´1 0 1 -(c(p) -G(s)) 2 ds -c ′ (p) (1 -(c(p) -G(1)) 2 ) 3 2
.

Then, since G is periodic, we obtain the condition F (p, 1) ´1 0 1 -(c(p) -G(y)) 2 dy = c ′ (p), which gives the desired characterization (19) of A(p). In particular, recalling the definition of c ′ (p) we get that A(p) > 0 for every p.

We prove now that A(p)(1 + p 2 ) ≤ 1. By definition of c(p) and Hölder inequality we get

|p| ≤ ˆ1 0 c(p) -G(y) 1 -(c(p) -G(y)) 2 dy ≤ ˆ1 0 (c(p) -G) 2 1 -(c(p) -G(y)) 2 dy 1 2 . Therefore 1 + p 2 ≤ ˆ1 0 1 1 -(c(p) -G(y)) 2 dy. (21)
By Holder inequality we get

ˆ1 0 1 1 -(c(p) -G(y)) 2 dy ≤ ˆ1 0 1 [1 -(c(p) -G(y)) 2 ] 3 2 dy 2 3 = ´1 0 1 [1-(c(p)-G(y)) 2 ] 3 2 dy -1 3 c ′ (p) . ( 22
)
Again by Jensen and Holder inequalities we obtain

1 ´1 0 1 -(c(p) -G(y)) 2 dy ≤ ˆ1 0 1 1 -(c(p) -G(y)) 2 dy ≤ ˆ1 0 1 [1 -(c(p) -G(y)) 2 ] 3 2 dy 1 3 . ( 23
)
Finally, recalling the definition (19) of A(p) (21), ( 22) and ( 23) give 1 + p 2 ≤ 1 A(p) . Finally we prove [START_REF] Lunardi | Analytic semigroups and optimal regularity in parabolic problems[END_REF]. From inequalities ( 21) and ( 22), we get

c ′ (p) ≤ 1 1 + p 2 3 2
.

We define the function

h(p) := ( ˆ1 0 1 -(c(p) -G(y)) 2 dy) -1 .
Notice that h ′ (p) = c ′ (p)ph -2 (p) and then h is decreasing in (-∞, 0) and increasing in (0, +∞). Moreover

lim p→+∞ h(p) = ( ˆ1 0 1 -(1 + m -G(y)) 2 dy) -1 lim p→-∞ h(p) = ( ˆ1 0 1 -(1 -m -G(y)) 2 dy) -1 .
We define the constant 2 dy .

K g := max 1 ´1 0 1 -(1 + m -G(y)) 2 dy , 1 ´1 0 1 -(-1 + m -G(y))
Note that K g > 0 depends only on g and that it explodes as g → 0. So we conclude, recalling [START_REF] Lions | Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications[END_REF]. 2

Remark Note that if g ≡ 0, then c(p) = p √ 1+p 2 and

A(p) = c ′ (p) ´1 0 1 -p 2 1+p 2 dy = 1 + p 2 (1 + p 2 ) 3 2 = 1 1 + p 2 .
Moreover, it is clear that the constant K g in ( 20) necessarily explodes as g → 0.

The convergence result

In this section we study the asymptotic behaviour as ε → 0 of the solutions u ε of the singularly perturbed equations

u ε t (t, x) = tr I - Du ε ⊗ Du ε 1 + |Du ε | 2 D 2 u ε + 1 ε g x ε 1 + |Du ε | 2 , ( 24
)
for (t, x) ∈ (0, +∞) × R n , with initial data u ε (0, x) = u 0 (x), where

u 0 (x) = q • x + v 0 (x)
with v 0 bounded and uniformly continuous.

(

) 25
We show that the functions u ε converge locally uniformly to a function u, which is a continuous viscosity solution to the effective quasilinear parabolic equation

u t (t, x) = tr A(Du(t, x))D 2 u(t, x) in (0, +∞) × R n (26)
with initial datum u(0, x) = u 0 (x), where the differential operator F (p, M ) = tr [A(p)M ] is the one defined in Lemma 3.2. Moreover u is the unique viscosity solution of (26) in the class

L q := {u ∈ C([0, +∞), R n ) s.t. u(t, x) -q • x ∈ C b ([0, +∞) × R n )},
where C b ([0, +∞), R n ) is the space of bounded continuous functions in [0, +∞) × R n . We also discuss the geometric counterpart of this result and some consequences for a related prescribed curvature problem. We start recalling two comparison principles for solutions of degenerate parabolic equations, which we will apply to the singularly perturbed and to the effective problem.

Theorem 5.1 Let w ε , v ε be respectively an upper semicontinuous subsolution and a lower semicontinuous supersolution to (24) in [0, +∞) × R n . Assume that there exists a constant k > 0 such that w ε (t,x) 1+|x| k , v ε (t,x) 1+|x| k → 0 as |x| → +∞ uniformly with respect to t ∈ [0, +∞), and that w ε (0, x) ≤ u 0 (x) ≤ v ε (0, x) for every x ∈ R n , where u 0 satisfies (25). Then

w ε (t, x) ≤ v ε (t, x) for every (t, x) ∈ [0, +∞) × R n .
Moreover, there exists a unique continuous viscosity solution u ε in L q to (24), with initial datum u 0 .

The proof of this comparison principle is given in [5, Theorem 2.1], while the existence of a unique solution to (24) can be done (with easy modifications) as in [START_REF] Barles | Uniqueness results for quasilinear parabolic equations through viscosity solutions methods[END_REF]Cor 2.1].

Notice that, if the initial datum u 0 is of class C 2+α (R n ) for some α ∈ (0, 1), then parabolic regularity theory [START_REF] Lunardi | Analytic semigroups and optimal regularity in parabolic problems[END_REF] gives that the solutions u ε in Theorem 5.1 are uniformly of class

C 1+α/2,2+α ([0, +∞)× R n ).
Theorem 5.2 Let w, v be respectively a bounded upper semicontinuous subsolution and a bounded lower semicontinuous supersolution to u t = tr[ Ã(Du)D 2 u] in [0, +∞) × R n , where Ã(p) is a symmetric n × n matrix, which depends smoothly on p and such that Ã(p) ≥ 0 for any p. Assume that w(0, x) ≤ v(0, x) for every x ∈ R n . Then w(t, x) ≤ v(t, x) for every (t, x) ∈ [0, +∞) × R n .

For the proof we refer to [START_REF] Crandall | Users guide to viscosity solutions of second order partial differential equations[END_REF].

We will apply this result to the limiting problem (26), letting Ã(p) := A(p + q) and changing the solution u(t, x) in ũ(t, x) = u(t, x)q • x. Since we look for solutions in L q , we actually have to deal with bounded solutions ũ.

However, we need to show that the effective differential operator F (p, M ) is regular and degenerate elliptic. Surprisingly the degenerate ellipticity of F is not known a priori, and we obtain it in a step of the convergence proof. We point out here that the degenerate ellipticity is expected as a consequence of results of Alvarez, Guichard, Lions and Morel [START_REF] Alvarez | Axioms and fundamental equations of image processing[END_REF] (see also [START_REF] Biton | Nonlinear monotone semigroups and viscosity solutions[END_REF]). Indeed, Equation (24) defines a monotone semi-group (in the sense that the solution u ε depends on u 0 or v 0 in a monotone way by the comparison principle) and the limiting semigroup is also expected to be monotone, hence, by the results of [START_REF] Alvarez | Axioms and fundamental equations of image processing[END_REF] or [START_REF] Biton | Nonlinear monotone semigroups and viscosity solutions[END_REF], it is certainly associated to a parabolic equation. We finally remark that we can see this monotonicity as a geometric "inclusion principle" and use as well the geometric version of the above results by Souganidis and the first author [START_REF] Barles | A new approach to front propagation problems : theory and applications[END_REF].

We now state our main result.

Theorem 5.3 Let u ε be the unique continuous viscosity solution to (24) with polynomial growth and initial datum u 0 , which satisfies (25) for some q ∈ R n . Then u ε (t, x) converges locally uniformly to the unique function u in the class L q which solves (26) in the viscosity sense, with initial datum u(0, x) = u 0 (x).

Proof The argument of the proof is an appropriate adaptation of the perturbed test function method, introduced by Evans in [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF] (see also [START_REF] Craciun | Effective motion of a curvature-sensitive interface through a heterogeneous medium[END_REF], [START_REF] Cardaliaguet | A discussion about the homogenization of moving interfaces[END_REF]). As we already pointed out above, we emphasize the fact that we do not know a priori that the limiting equation is degenerate parabolic. The proof is divided into five steps.

Step 1 (Local equiboundedness of u ε .) The existence and uniqueness of u ε is assured by Theorem 5.1. Actually, it is possible to show that u ε inherits the same growth of the initial data u 0 . We consider the solution χ of ( 13) with p = q and such that χ(0; q) = 0. Then the function w ε (t, x) = εχ x ε ; q +q •x+ v 0 ∞ +ε χ ∞ , where v 0 is the function appearing in (25), is a stationary solution to (24) with w ε (0, x) ≥ u 0 (x), for ε > 0. Analogously v ε (t, x) = εχ x ε ; q + q • xv 0 ∞ε χ ∞ is a stationary solution to (24) with v ε (0, x) ≤ u 0 (x), for ε > 0. Then, by the comparison principle (Theorem 5.1) we obtain that v ε (t, x) ≤ u ε (t, x) ≤ w ε (t, x), which gives in particular that

|u ε (t, x) -q • x| ≤ v 0 ∞ + 2ε χ ∞ for any ε > 0 and (t, x) ∈ [0, +∞) × R n .
Step 2 (Relaxed semilimits of u ε .) We define the relaxed semilimits (see [START_REF] Crandall | Users guide to viscosity solutions of second order partial differential equations[END_REF])

u(t, x) := lim inf (ε,t ′ ,x ′ )→(0,t,x) u ε (t ′ , x ′ ), u(t, x) := lim sup (ε,t ′ ,x ′ )→(0,t,x) u ε (t ′ , x ′ )
for (t, x) ∈ [0, +∞)×R n . Observe that u, u, due to the previous step, satisfy

|u(t, x)-q•x| ≤ v 0 ∞ , |u(t, x) -q • x| ≤ v 0 ∞ for any (t, x) ∈ [0, +∞) × R n .
We are going to prove that u (resp. u) is a viscosity subsolution (resp. supersolution) to (26).

We describe the argument just for u, since for u it is completely analogous. We consider a smooth function φ and we assume that uφ has a strict maximum at (t, x); we have to prove that

φ t (t, x) ≤ tr[A(Dφ(t, x))D 2 φ(t, x)]. (27)
We define the perturbed test function

φ ε (t, x) := φ(t, x) + εχ x ε ; Dφ(t, x) + ε 2 ψ x ε ; Dφ(t, x), D 2 φ(t, x) , (28)
where χ(y; Dφ(t, x)) is the periodic solution to [START_REF] Crandall | Users guide to viscosity solutions of second order partial differential equations[END_REF] with p = Dφ(t, x) and χ(0; p) = 0, and ψ(y; Dφ(t, x), D 2 φ(t, x)) is the periodic solution to [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF] with p = Dφ(t, x), M = D 2 φ(t, x) and ψ(0; p, M ) = 0. Observe that lim sup

(ε,t ′ ,x ′ )→(0,t,x) (u ε (t ′ , x ′ ) -φ ε (t ′ , x ′ )) = u(t, x) -φ(t, x).
By a standard compactness argument, there exist subsequences ε n → 0 and (t n , x n ) → (t, x) such that u εn (t n , x n )φ εn (t n , x n ) → u(t, x)φ(t, x) and (t n , x n ) is a strict maximum of u εnφ εn . Since u εn is a subsolution to (24), at (t n , x n ) we have

φ εn t ≤ tr I - Dφ εn ⊗ Dφ εn 1 + |Dφ εn | 2 D 2 φ εn + 1 ε n g x n ε n 1 + |Dφ εn | 2 .
Recalling the definition of φ ε in (28) and using standard asymptotic expansion arguments, we can rewrite the r.h.s. of the previous inequality as

1 ε n tr I - (Dφ + Dχ)) ⊗ (Dφ + Dχ) 1 + |Dφ + Dχ| 2 D 2 χ + g x n ε 1 + |Dφ + Dχ)| 2 + tr I - (Dφ + Dχ) ⊗ (Dφ + Dχ) 1 + |Dφ + D y χ| 2 D 2 ψ + D 2 φ + 2D 2 py χD φ -2 (Dφ + Dχ) T D 2 χ Dψ + (D p χ) T D 2 φ 1 + |Dφ + Dχ| 2 + 2 (Dφ + Dχ) T D 2 χ(Dφ + Dχ) (Dφ + Dχ) • (Dψ + (D p χ) T D 2 φ) (1 + |Dφ + Dχ| 2 ) 2 + g x n ε (Dφ + Dχ) • (Dψ + (D p χ) T D 2 φ) 1 + |Dφ + Dχ| 2 + R(ε n )
where R(ε n ) → 0 uniformly as ε n → 0. Using the characterization of χ and ψ as solutions of ( 13) and ( 15) respectively, and the regularity of φ, χ, ψ, A, we obtain that

φ t (t n , x n ) ≤ tr (A(Dφ(t n , x n ))) D 2 φ(t n , x n ) + R ′ (ε n )
for some R ′ (ε n ) → 0 uniformly as ε n → 0. Letting n → +∞, we obtain (27).

Step 3 (Degenerate ellipticity of the limiting operator.) We prove the following Proof It is sufficient to show that the matrix A(p) is nonnegative definite for any p ∈ R n . We consider the unique solution u ε to (24) with polynomial growth and initial data u 0 (x) = p • x (see Theorem 5.1) and the solution χ(y; p) of ( 13) with χ(0; p) = 0. Then, for all ε > 0, the function

w ε (t, x) = εχ x ε ; p + p • x + ε χ ∞ is a stationary solution to (24) with w ε (0, x) ≥ u 0 (x). Analogously v ε (t, x) = εχ x ε ; p +p•x-ε χ ∞
is a stationary solution to (24) with v ε (0, x) ≤ u 0 (x), for ε > 0. By the comparison principle (Theorem 5.1) we then obtain that v ε (t, x) ≤ u ε (t, x) ≤ w ε (t, x), which gives in particular lim ε→0 u ε (t, x) = p • x, locally uniformly.

According to Step 2, if (t, x) is a strict maximum of u(t, x)φ(t, x) = p • xφ(t, x) then (27) holds. For small δ > 0, we choose φ(t, x) := p • x + (q • x) 2 + δ(|x| 2 + (t -1) 2 ) and observe that (1, 0) is a strict maximum of u(t, x)φ(t, x) for any δ; moreover (27) reads

-A(p)q • q = -tr[A(p)(q ⊗ q)] ≤ o(δ) ,
where o(δ) → 0 as δ → 0. Since the inequality A(p)q • q ≥ 0 holds for any p and q, the thesis follows letting δ → 0.

2

Step 4 (u(x, 0) = u(x, 0) = u 0 (x).) Let v 0 be the function appearing in (25). For every δ > 0, it is possible to find

v δ + , v δ -∈ C ∞ (R n ) ∩ W 2,∞ (R n ) such that v δ -≤ v 0 ≤ v δ + and that v δ + - v 0 ∞ , v δ --v 0 ∞ ≤ δ (
this can be done by using standard mollification arguments). For any fixed δ > 0, we consider the functions

w ε,δ ± (t, x) := v δ ± (x) + q • x + εχ ± x ε ± C δ t + ε χ ± ∞
where χ + (x) = χ x; q + Dv δ + (x) is the solution to (13) with p = q+Dv δ + (x), χ -(x) = χ x; q + Dv δ -(x) is the solution to (13) with p = q + Dv δ -(x), and χ ± (0) = 0. Choosing C δ > 0 sufficiently large, it is easy to see that w ε,δ ± are respectively a super and a subsolution to (24). Moreover w ε,δ -(0, x) ≤ u 0 (x) ≤ w ε,δ + (0, x). So, by the comparison principle (Theorem 5.1), we get that w ε,δ -(t, x) ≤ u ε (t, x) ≤ w ε,δ + (t, x), for every δ > 0. Passing to the relaxed semilimits, we then obtain

v δ -(x) + q • x ≤ u(0, x) ≤ u(0, x) ≤ v δ + (x) + q • x.
Letting δ → 0, this gives u(x, 0) = u(x, 0) = u 0 (x).

Step 5 (Uniform convergence.) Let us define v(t, x) := u(t, x)-q•x and v(t, x) := v(t, x)-q•x.

It is easy to show that v and v are bounded, satisfy v(0, x) = v 0 (x) = v(0, x) and are respectively a sub and a supersolution of v t (t, x) = tr A(q + Dv(t, x))D 2 v(t, x) in (0, +∞) × R n .

(29)

Then, by the comparison principle (Theorem 5.2), we obtain that v(t, x) ≤ v(t, x) in (0, +∞) × R n , and therefore v(t, x) = v(t, x) in (0, +∞) × R n , since the opposite inequality holds by definition of the semilimits. In particular, we get that v(t, x) := v(t, x) = v(t, x) is the unique bounded viscosity solution to (29) with initial data v 0 . This implies that u(t, x) = u(t, x) = u(t, x) is the unique continuous viscosity solution to (26) in the class L q , with initial datum u 0 . Moreover, using the definition of relaxed limits, we obtain that u ε (t, x) → u(t, x) locally uniformly.

2

We observe that Theorem 5.3 has also a geometric counterpart. Namely, given a solution u ε to (24) we consider its graph Γ ε (t) := {(x, x n+1 ) ∈ R n+1 : x n+1 = u ε (x, t)}. Then, Γ ε (t) evolves in time accordingly to (2). In the limit ε → 0, we then obtain the geometric evolution

V = α(ν)H (30)
where the function α is defined as

α(ν) := A (-ν x /ν n+1 ) ν 2 n+1 ≥ 0 ν n+1 = 0.
Note that, when n = 1, due to (20), we have 0 < α(ν) ≤ K g |ν 2 |, which implies in particular lim ν2→0 α(ν) = 0. As discussed for instance in [START_REF] Novaga | Bump solutions for the mesoscopic Allen-Cahn equation in periodic media[END_REF], the asymptotic limit of ( 2) is strictly related to the existence of compact embedded solutions to the following prescribed curvature problem:

H + g = 0 . (31)
Indeed, the existence of a compact solution to (31) implies by rescaling the existence of compact solutions to H + 1 ε g x ε = 0, which can be used as barriers for the evolution (2). It would then follow that the solutions to (2) converge, as ε → 0, to a stationary hypersurface, thus implying α ≡ 0 in (30), that is, A(p) = 0 for all p ∈ R n . However, if g has zero average and does not depend on x n+1 , from Remark 3 we know that A(0) > 0 if g Lip < δ; moreover we know from (20) that A(p) > 0 for all p, if n = 1 and g satisfies [START_REF] Craciun | Effective motion of a curvature-sensitive interface through a heterogeneous medium[END_REF]. As a consequence, in such cases we conclude that there are no compact embedded solutions to (31).

We point out that this result cannot be expected for a generic function g, which is periodic and of zero average, but which depends also on x n+1 . Indeed, in [START_REF] Novaga | Bump solutions for the mesoscopic Allen-Cahn equation in periodic media[END_REF] it has been proved that we can always find a sequence g n → g in L 1 (R n ), where the functions g n are all periodic, of zero average, uniformly bounded, and satisfy [START_REF] Cesaroni | Curvature flow in heterogeneous media[END_REF], such that there exist compact embedded solutions to (31) with g replaced by g n .

3 2 dy . 2 Proposition 4 . 1

 241 Under the standing assumptions and for n = 1, the function A(p) in Lemma 3.2 is given by

Lemma 5 . 1

 51 The differential operator F (p, M ) = tr [A(p)M ] defined in Lemma 3.2 is degenerate elliptic.

Acknowledgements. The second and the third authors wish to thank the University of Tours and the Research Institute le Studium for the kind hospitality and support.


Eva Esteban Cardeñosa

Pascual Bolufer Gilabert

Inmaculada Juan Jimenez

Sarai Palanca Suela

Eva Barragán González

Ángel Isabel Chirivella González

Segura Huerta

Carmen Guillén Ponce

Eduardo Martínez De Dueñas

Inmaculada De

Juan Jimenez

Eva Barragan

Isabel Chirivella González

Ángel Segura Huerta

Broad and mutational spectrum and high incidence of recurrent and novel mutations in the eastern Spain population

ou non, émanant des établissements d'enseignement et de recherche

Dear Editor

It is well known that the pathogenic mutations in BRCA1 or BRCA2 genes are detected in 5-10% of total breast (BC) and ovarian cancer (OC) and in 20-30% of BC/OC found in families with strong history of cancer [START_REF] Wooster | Breast and Ovarian Cancer[END_REF]. However, the incidence of the BRCA1 and BRCA2 pathogenic mutations depends on the criteria adopted to select the families to be studied [START_REF] Malone | BRCA1 mutations and breast cancer in the general population: analyses in women before age 35 years and in women before age 45 years with first-degree family history[END_REF] and the mutation spectrum varies considerable due to the influence of the ethnic groups [START_REF] Szabo | Population genetics of BRCA1 and BRCA2[END_REF]. Additionally, ethnicity could substantiate the appearance of founder mutations coming from an old ancestor such as the recurrent mutations detected among Ashkenazim [START_REF] Simard | Common origins of BRCA1 mutations in Canadian breast and ovarian cancer families[END_REF][START_REF] Neuhausen | Haplotype and phenotype analysis of six recurrent BRCA1 mutations in 61 families: results of an international study[END_REF].

The Program of Genetic Counselling in Cancer of Valencia Community was launched in March 2005 and we reported the results of the first 147 non-related families included, describing the preliminary mutational spectrum [START_REF] Esteban | Mutaciones de BRCA1 y BRCA2 en familias estudiadas en el Programa de Consejo Genético en el Cáncer de la Comunidad Valenciana[END_REF] and novel mutations [START_REF] Esteban | Twenty-three novel BRCA1 and BRCA2 sequence alterations in breast and/or ovarian cancer families of Eastern Spain[END_REF]. However, since then, the number of subjects studied for BRCA1 and BRCA2 mutations has BRCA1 and BRCA2 mutations were detected by amplifying all the exons and the exonintron boundaries of both genes by polymerase chain reaction (PCR) using the primer pairs and the PCR conditions reported in the Breast Cancer Information Core (BIC) [11]. To identify the genetic variations, we carried out a pre-screening of the heteroduplexes of the PCR products by conformation sensitive gel electrophoresis (CSGE) [12] followed by the sequencing of the PCR products in which heteroduplexes were identified.

In BRCA1 we found 26 different deleterious mutations in 62 IPs (17 frameshift, three splicing, four nonsense and two missense mutations; Table 1). Sixteen of these mutations have been included in our previous reports [START_REF] Esteban | Mutaciones de BRCA1 y BRCA2 en familias estudiadas en el Programa de Consejo Genético en el Cáncer de la Comunidad Valenciana[END_REF][START_REF] Esteban | Twenty-three novel BRCA1 and BRCA2 sequence alterations in breast and/or ovarian cancer families of Eastern Spain[END_REF]. The mutation with the highest recurrence was c.5242C>A, missense mutation in exon 18, detected in nine families, followed by the frameshift mutations c.2080delA of exon 11 and c.187_188delAG of exon 2 found in seven and six families, respectively. These three mutations represent 35.5% (22/62) of all the mutations detected in this gene. If to these we add the missense c.330A>G and the splicing c.331+1G>A mutations, both in exon 5, each found in three families, the five mutations represent the 45.2% (28/62) of the mutations in this gene.

Besides, we found two novel frameshift pathogenic mutations, c.4424_4425delCT and c.4476delG of exon 13, both present in single families (Table 1). The first one causes a stop at codon 1439 while the c.4476delG mutation causes the loss of the last nucleotide (G) of exon 13 and a stop at codon 1455. The FruitFly software [21] predicts that the latter mutation presumably affects the donor-splicing site.

In BRCA2, we found 36 different deleterious mutations in 71 IPs (30 frameshift, one splicing and five nonsense; Table 2), 13 of these mutations had been previously reported [START_REF] Esteban | Mutaciones de BRCA1 y BRCA2 en familias estudiadas en el Programa de Consejo Genético en el Cáncer de la Comunidad Valenciana[END_REF][START_REF] Esteban | Twenty-three novel BRCA1 and BRCA2 sequence alterations in breast and/or ovarian cancer families of Eastern Spain[END_REF]. The recurrent mutations with highest incidence were all located in exon 23 such as the two frameshift c.9254_9258delATCAT and c.9206_9219del14 found in 14 and 6 families, respectively, and the nonsense c.9246C>A found in 5 families.

Likewise, the c.3492insT mutation of exon 11 showed a high recurrence and was identified in five families. These four mutations cover 42.2 % (30/71) of the families with the pathogenic mutations detected in this gene.

Besides in BRCA2, we found three novel mutations, two frameshift, c.1874_1877delAGGA in exon 10 and c.6118delA in exon 11 and the splicing mutation c.8860+2T>G in intron 20 (Table 2). The c.1874_1877delAGGA, creates a stop at The current mutational spectrum of our population has broadened with regard to our previous report, recurrent mutations have emerged and five novel mutations have been identified. Thus, the mutational spectrum for BRCA1 and BRCA2 increased from the 16 and 13 mutations in our first reports [START_REF] Esteban | Mutaciones de BRCA1 y BRCA2 en familias estudiadas en el Programa de Consejo Genético en el Cáncer de la Comunidad Valenciana[END_REF][START_REF] Esteban | Twenty-three novel BRCA1 and BRCA2 sequence alterations in breast and/or ovarian cancer families of Eastern Spain[END_REF] We observed greater diversity and higher incidence for BRCA2 mutations than for Letter 10 F: frameshift; I: intronic; N: nonsense; M: missense; S: splicing. * Mutations included in our previous reports [START_REF] Esteban | Mutaciones de BRCA1 y BRCA2 en familias estudiadas en el Programa de Consejo Genético en el Cáncer de la Comunidad Valenciana[END_REF][START_REF] Esteban | Twenty-three novel BRCA1 and BRCA2 sequence alterations in breast and/or ovarian cancer families of Eastern Spain[END_REF]. c.9246C>A* p.Y3006X N 5 [START_REF] Esteban | Mutaciones de BRCA1 y BRCA2 en familias estudiadas en el Programa de Consejo Genético en el Cáncer de la Comunidad Valenciana[END_REF]11,13] c.9254_9258delATCAT* Stop3015 F 14 [START_REF] Esteban | Mutaciones de BRCA1 y BRCA2 en familias estudiadas en el Programa de Consejo Genético en el Cáncer de la Comunidad Valenciana[END_REF]11,14,15] 25 c.9694C>T* p.Q3156X N 1 [START_REF] Esteban | Twenty-three novel BRCA1 and BRCA2 sequence alterations in breast and/or ovarian cancer families of Eastern Spain[END_REF] F: frameshift; I: intronic; N: nonsense; M: missense; S: splicing. * Mutations included in our previous reports [START_REF] Esteban | Mutaciones de BRCA1 y BRCA2 en familias estudiadas en el Programa de Consejo Genético en el Cáncer de la Comunidad Valenciana[END_REF][START_REF] Esteban | Twenty-three novel BRCA1 and BRCA2 sequence alterations in breast and/or ovarian cancer families of Eastern Spain[END_REF]. 17.

BRCA1

  increased considerably having reached the 704 families at the end of 2008. This large number of individuals prompted us to update our series and to establish a consistent mutational spectrum of the population of Eastern Spain identifying the recurrent mutations and relevance of the novel mutations. We studied 704 index patients (IP) (689 females and 15 males) who enrolled in the Program of Genetic Counselling in Cancer of the Valencian Community from March 2005 to December 2008. The IPs were selected by the Units of Genetic Counselling in Cancer (UGCC) according to the inclusion criteria established in the Program of Genetic Counselling in Cancer for familial BC/OC [8, 9]. All the individuals signed a written consent elaborated by the Consellería de Sanitat (Valencia Community) in accordance with the Helsinki Declaration (1964, amended in 1975 and 1983) [10].

  codon 556 with the consequent truncation of the translation of the BRCA2 protein. The c.6118delA causes the interruption of the synthesis of the BRCA2 protein at codon 2003. The splicing mutation mutation c.8860+2T>G affects the second nucleotide of intron 20, suppressing the donor site as predicted by the FruitFly software [21].

  to the 26 and 36 mutations, respectively of the present report. Eight of the mutations detected in BRCA1 were also observed in recent studies carried out in the Spanish population [13-15]. However, our results are not as concordant as those expected with the larger study carried out in the Spanish population [13]. Only seven of the 35 different mutations (20%) observed in the mentioned study, c.187_188delAG, c.330A>G, c.331+1G>A, c.2080delA, c.5242C>A, c.5272+5G>A and c.5273-1G>A, were detected in our population. These mutations represent 48% (30/62) of the pathogenic mutations detected in our study. However, only two of them, c.187_188delAG and c.330A>G, were recurrent in both studies. For BRCA2 there is a coincidence of 11 mutations with other reports performed in the Spanish population [13-15]. The report of Diez et al 2003 [13] is concordant with our study in 10 out of the 22 different mutations (45%), c.1538_1541delAAGA, , which represent 70% (37 out of 53) of the families reported by said study [13] and 58% of the families (41 out of 71) of this study. The frameshift mutation c.9254_9258delATCAT, the most recurrent mutation of the present study, showed also high incidence in the report of Diez et al 2003 [13]. This recurrent mutation is considered characteristic of Eastern Spain. However, for most of BRCA2 recurrent mutations no correspondence was observed between both studies either.

Table 2 .

 2 -Pathogenic mutations detected in BRCA2

	Exons	Mutation	Protein	Type N. Families	References
	3	c.373G>T c.489_490delCT	p.E49X Stop99	N F	1 2	[11] [11]
	5	c.598delA	Stop135	F	1	[11]
		c.1354delT	Stop398	F	1	[11]
		c.1538_1541delAAGA	Stop458	F	1	[11,13]
		c.1594_1595delGA	Stop460	F	1	[11]
	10	c.1676_1677delAGinsTTAC Stop513	F	1	[11]
		c.1835insT*	Stop542	F	2	[7, 13]
		c.1873_1876delAGGA	Stop 556	F	1	Novel
		c.2070dupT	Stop615	F	1	[11]
		c.2929delC	Stop 903	F	1	[16]
		c.3036_3039delACAA*	Stop958	F	2	[6, 11, 13-15]
		c.3492insT*	Stop1098	F	5	[6, 11, 14]
		c.4075_4076delGT	Stop1284	F	1	[11]
		c.4150G>T*	p.E1308X	N	3	[6, 11, 13, 15]
		c.5025delT*	Stop1616	F	3	[6, 7]
		c.5164_5167delGAAA*	Stop1668	F	1	[6, 11, 15]
	11	c.5340_5343delAATA c.5514_5515delTC	Stop1710 Stop1766	F F	1 1	[11] [11]
		c.5804_5807delTTAA	Stop1862	F	2	[11, 13, 14]
		c.5946_5949delCTCT	Stop1909	F	1	[11]
		c.6118delA	Stop2003	F	1	Novel
		c.6299delA	Stop2039	F	1	[11]
		c.6503_6504delTT*	Stop2099	F	2	[6, 11, 14, 15]
		c.6722delT*	Stop 2167	F	1	[7]
		c.6857_6858delAA	Stop2223	F	1	[11]
	14	c.7399insA c.7462insG	Stop2411 Stop2413	F F	1 1	[11] [11]
	17	c.8091T>A*	p.Y2621X	N	1	[11]
	18	c.8270_8271delCA	Stop2691	F	1	[11]
	I-20 c.8860+2T>G		S	1	Novel
		c.9206_9219del14*	Stop2975	F	6	[6, 13]
	23	c.9216_9218delATAinsTT	Stop3248	F	1	[11]

ACKNOWLEDGMENTS

We should express our gratitude to the ''Fundación para la Investigación del Hospital Universitario La Fe'' for granting two contracts for Inmaculada de Juan and Sarai Palanca (Bch Sc Specialist in Clinical Analysis), what made possible their participation in the present study.

We also should support given by the PI060505 and Generalitat Valenciana Grants AP-042/07, EVES 100/2008