Florent Autrusseau

Larry Thibos

Steven K Shevell
email: shevell@uchicago.edu

Chromatic and Wavefront Aberrations: L-, M-and S-cone Stimulation with Typical and Extreme Retinal Image Quality

Keywords: Chromatic aberration, wave aberrations, optics, retinal image, spread light, point spread function

The first physiological process influencing visual perception is the optics of the eye.

The retinal image is affected by diffraction at the pupil and several kinds of optical imperfections. A model of the eye (Thibos & Bradley, 1999), which takes account of pupil aperture, chromatic aberration and wavefront aberrations, was used to determine wavelength-dependent point-spread functions, which can be convolved with any stimulus specified by its spectral distribution of light at each point. The resulting retinal spectral distribution of light was used to determine the spatial distribution of stimulation for each cone type (S, M and L). In addition, individual differences in retinal-image quality were assessed using a statistical model (Thibos, Bradley & Hong, 2002) for population values of Zernike coefficients, which characterize imperfections of the eye's optics. The median and relatively extreme (5 th and 95 th percentile) modulation transfer functions (MTFs) for the S, M and L cones were determined for equal-energy-spectrum (EES) 'white' light.

The typical MTF for S cones was more similar to the MTF for L and M cones after taking wavefront aberrations into account but even with aberrations the S-cone MTF typically was below the M-or L-cone MTF by a factor of at least 10 (one log unit). More generally, the model presented here provides a technique to estimate retinal image quality for the S, M and L cones for any stimulus presented to the eye. The model is applied to some informative examples.

INTRODUCTION

A single point of monochromatic light that enters the eye has a two-dimensional retinal image due to imperfections of the eye's optics. The retinal image can be characterized by a point spread function (PSF), which varies with the wavelength of light.

With broadband light, the retinal image is determined by decomposing the point of light into its spectral components; the PSF is applied separately at each wavelength [START_REF] Barnden | Calculation of axial polychromatic optical transfer function[END_REF][START_REF] Ravikumar | Calculation of retinal image quality for polychromatic light[END_REF]. The distribution of light on the retina is the superposition of the light distributions for each of the wavelengths. When an observer views a complete scene rather than a single point, each point in the scene is independently affected by the eye's optics; conceptually, the resulting retinal image at each wavelength is the superposition of the distribution of light from each point in the scene. Retinal image quality depends on both the PSF for each wavelength and the spatial and spectral distribution of the light in view.

Optical models of the eye have been sought for centuries for a variety of applications (Huygens, 1702;Listing, 1851;Emsley, 1952;[START_REF] Smith | Schematic eyes: history, description and applications[END_REF]. A relatively recent model [START_REF] Thibos | Modeling the refractive and neuro-sensor systems of the eye[END_REF] was used here to determine the spatial and spectral distribution of light on the retina; this distribution then was used to find the spatial distribution of light for each receptoral cone class, L, M and S.

The model used here has two advantages in comparison to [START_REF] Marimont | Matching color images: The effects of axial chromatic aberration[END_REF] well known model of retinal image quality. First, higher-order wavefront aberrations were considered explicitly here rather than as an implicit property of a wavelength-independent point spread function (Marimont and Wandell, 1994, p.3116).

Second, the Thibos and Bradley model-eye depends on specific parameters (Zernike coefficients) that characterize an individual eye, and these parameters have a known multivariate population distribution. The population distribution allowed estimates of individual differences in retinal image quality among people with normal corrected vision (Thibos, Bradley and Hong, 2002). While a general comparison of the retinal image quality given by the Marimont and Wandell model versus the one used here is not possible because the models depend on different assumptions, results from the two models are compared in the Discussion using a typical eye from the population distribution given by Thibos et al. (2002). The two models agree well in this special case (as discussed later).

The first part of this paper focuses on retinal image formation. The optical model is described and the calculated photoreceptor absorptions are explained. In the second part, the model is applied to broadband 'white' spectral stimuli to assess typical and extreme retinal contrast sensitivity in a normal population of human observers. The third part considers some specific cases to show how retinal image quality depends on particular features of a visual stimulus, and to demonstrate how the model may be applied to particular types of images.

PART 1: RETINAL IMAGE MODEL

The retinal image is determined by characterizing the eye's optics. Any optical system can be fully described mathematically over an isoplanatic area by its optical transfer function [START_REF] Williams | Introduction to the Optical Transfer Function[END_REF]) so calculating the retinal image involves determining the eye's optical transfer function (OTF) or the closely related point spread function (PSF). The PSF gives the retinal image of a monochromatic point source, taking account of the optics of the eye. The OTF is the Fourier transform of the intensity PSF.

A precise model of the eye includes various optical factors that affect the PSF at each wavelength. This section covers the main properties of the model eye used to determine the PSFs. Recall that the external stimulus pattern entering the eye is decomposed into multiple monochromatic stimulus patterns, and then the stimulus pattern at each wavelength is convolved with its wavelength-dependent PSF. This gives the retinal image I(x,y,λ) at each wavelength λ. Superposition of these monochromatic retinal patterns yields the retinal spectral distribution of light for each spatial location (x,y). These spectral distributions allow calculation of the spatial distribution I C (x,y) of cone excitation for each cone type C (C = L, M or S) by applying the appropriate cone spectral sensitivity function S C (λ) as a weighting factor:

I C (x, y) = S C (λ)I(x, y, λ) dλ ∫ . ( 1
)

Optics of the human eye

The human eye has three main optical components that affect retinal image quality: the pupil, the cornea and the lens. The pupil diffracts light entering the eye, resulting in a PSF with a central point surrounded by concentric rings [START_REF] Roorda | Human Visual System -Image Formation[END_REF][START_REF] Williams | Formation and acquisition of the retinal image[END_REF]. The cornea accounts for most of the eye's refraction (about 43 diopters) while the lens, which refracts light after passing through the pupil, adds more than 20 diopters in a young adult [START_REF] Roorda | Human Visual System -Image Formation[END_REF]. The cornea and lens are the primary contributors to wave aberrations, which degrade retinal image quality.

Longitudinal chromatic aberration (LCA) is a consequence of refraction by a dispersive medium: light of different wavelengths is brought into focus at different distances. The focal distance increases with wavelength. LCA is included in the model.

Transverse chromatic aberration is ignored here because of its weak marginal influence on image quality when LCA and wave aberrations are considered [START_REF] Ravikumar | Calculation of retinal image quality for polychromatic light[END_REF].

Similarly, the slight effect of higher-order chromatic aberrations on image quality is not included [START_REF] Nam | Wavelength adjustment using an eye model from aberrometry data[END_REF].

Retinal image quality depends on pupil size and the wavelength of light. The greatest loss of image quality from diffraction occurs with a small pupil and long wavelengths. On the other hand, the greatest loss from refractive elements occurs with a large pupil and short wavelengths. The sharpest retinal image, therefore, typically is at an intermediate pupil size near 3 mm, which balances the tradeoff between diffraction (worse at smaller pupil diameters) and the deleterious effects of wave aberrations (worse at larger diameters).

The best known wave aberrations caused by the eye's optics are defocus and astigmatism, which are ameliorated by standard corrective lenses. Higher-order wave aberrations (trefoil, coma, spherical, as well as other still higher-order aberrations) also reduce image quality [START_REF] Packer | Light, the retinal image, and photoreceptors[END_REF]. The imperfections captured by the higher-order wave aberrations include the irregularities in optical elements within the eye.

The eye's lower-and higher-order wave aberrations can be modeled accurately using Zernike polynomials; the first 15 Zernike mode numbers were used here for the polynomials [START_REF] Thibos | Modeling the refractive and neuro-sensor systems of the eye[END_REF]Thibos et al., 2002). Statistical models of Zernike aberration coefficients produce monochromatic PSFs that are representative of human eyes [START_REF] Thibos | Retinal image quality for virtual eyes generated by a statistical model of ocular wavefront aberrations[END_REF].

As mentioned above, each wavelength of light is affected differently by the optics of the eye so each wavelength has its own distinct PSF; we refer to the set of PSFs for all wavelengths in the visible spectrum as a hyperspectral PSF. Spectral sampling here was every 10 nm, a choice supported by the analysis of [START_REF] Ravikumar | Calculation of retinal image quality for polychromatic light[END_REF]. A hyperspectral PSF captures two important aspects of retinal image quality. First, an object with a single broadband chromaticity, such as equal-energy-spectrum 'white', does not necessarily produce on the retina an image at only that chromaticity because some wavelengths are more strongly dispersed than others. Second, spectral mixtures that are visually indistinguishable in color as large homogenous patches (color metamers) may not match at other spatial frequencies because the wavelengths composing each metamer are unequally affected by optics [START_REF] Poirson | Appearance of colored patterns: pattern-color separability[END_REF][START_REF] Marimont | Matching color images: The effects of axial chromatic aberration[END_REF]).

An implication of the second point is that the influence of optics on the retinal image can be determined from only the full spectral distribution of light entering the eye. In general, a trichromatic description of the light --for example CIE X,Y,Z tristimulus values or the excitations of the three types of cones --is not sufficient, except under exceptional circumstances (e.g. using only the R pixels in an RGB display) that allow the full spectral distribution to be determined from the trichromatic specification [START_REF] Barnden | Calculation of axial polychromatic optical transfer function[END_REF][START_REF] Ravikumar | Calculation of retinal image quality for polychromatic light[END_REF].

Cone quantal absorptions

Transduction of light at the photoreceptors establishes the neural responses that mediate vision. There are three classes of cone photoreceptor, labeled S, M and L, with peak sensitivity in the short-, middle-or long-wave part of the visible spectrum, respectively. The response of each cone type depends on the rate of quantal absorption.

The relative spectral sensitivity of each type of cone is known [START_REF] Smith | Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm[END_REF] so the rate of quantal absorption for S, M or L cones can be calculated directly from the spectral energy distribution of light at each point on the retina. (The spacing between adjacent cones in the retinal mosaic is ignored here.) This gives the spatial distribution of stimulation for each cone type. The retinal spatial distribution of excitation for each cone type, therefore, requires the full spectral distribution of light at each point within the viewed scene. This implies, for example, that a scene on a color display with each pixel specified by its R ('red'), G ('green') and B ('blue') components must be transformed into a pixel-by-pixel spectral energy distribution in order to determine the retinal image. For example equal-energyspectrum (EES) 'white' light, which by definition has all wavelengths in the visible spectrum at the identical energy, should not be used to calculate the retinal image of the metamer to EES on a color display because the display's weighted sum of the R, G and B components has a different physical spectral composition than true EES with all wavelengths at the same energy (compare dashed and solid lines, Fig. 1). Instead, calibration of the display is required to transform trichromatic coordinates of the display to the full spectral distribution of light at each point of the stimulus. This transformation is specific to each individual display; it differs among displays of the same manufacturer and model, and even for the same display over time.

PART 2: TYPICAL AND POPULATION EXTREMES OF RETINAL IMAGE QUALITY

Point spread functions

The effects of pupil size and wave aberrations on the shape of the PSF can be determined separately. The PSFs shown in Fig. 2 (Thibos et al., 2002). Figure 3 shows the first 15 mean Zernike coefficients for a pupil diameter of 3.0, 4.5 or 6.0 mm. These results are consistent with other population studies [START_REF] Porter | Monochromatic aberrations of the human eye in a large population[END_REF][START_REF] Castejon-Mochon | Ocular wave-front aberration statistics in a normal young population[END_REF][START_REF] Salmon | Normal-eye Zernike coefficients and root-meansquare wavefront errors[END_REF].

In general, values of the Zernike coefficients vary with both the characteristics of a particular eye and with pupil size. As mentioned above, the slight variation expected in higher-order aberrations with wavelength and the variation of the prismatic terms responsible for transverse chromatic aberration are ignored [START_REF] Nam | Wavelength adjustment using an eye model from aberrometry data[END_REF]. Point spread functions that include wave aberrations are shown in Fig. 4 for a 6 mm diameter pupil, which is close to the pupil size for 20-30 year-old observers at a luminance of about 100 cd/m2 [START_REF] Winn | Factors affecting lightadapted pupil size in normal human subjects[END_REF]. Analyses that follow use the same 6 mm diameter pupil. The mean Zernike coefficients from Fig. 3c were used for the PSFs in Fig. 4a but, as argued below, these PSFs are not characteristic of a typical human eye. The PSFs in Fig. 4b are based on Zernike coefficients for a "standard observer" described later.

Modulation transfer functions and phase shifts

The modulation transfer function (MTF) is used to assess the eye's optical quality.

Conceptually, the MTF can be determined for an EES light or any other spectral distribution by finding the contrast in the retinal image for sine waves at various spatial frequencies. coefficients for a "standard observer" (see text and Table 1, below).

L-, M-and S-cone MTFs were determined here for 100 different independent random samples of Zernike coefficients taken from the population distribution given by Thibos et al. (2002). The first 15 Zernike coefficients were used. The wavelength in focus was 570 nm. The spectral distribution of light was EES, and luminance was varied sinusoidally in the horizontal direction (i.e., a vertically oriented sine wave). With the PSFs determined for each visible wavelength, the retinal image at each wavelength was found by convolving the PSF for that wavelength with the visual stimulation at that wavelength.

Repeating this for each wavelength in the stimulus and then taking the superposition of all wavelengths gave the spectral distribution of light at each point on the retina resulting from a particular visual stimulus (for example, an EES sine-wave grating varied in luminance at 10 cycles per degree). Then, at each point on the retina, the amount of each wavelength was weighted by the [START_REF] Smith | Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm[END_REF] One sampled set of Zernike coefficients gave L-, M-and S-cone MTFs that are close to the median values (see lines connecting symbols, Fig. 5). This set of Zernike coefficients (Table 1) was used to define a "standard observer" for a 6 mm pupil; the MTFs from this set were consistent with an observer with about half the sample having better retinal cone contrast and half having worse contrast. While the standard observer's S-cone MTF is slightly better than the median (at worst the 65%ile instead of the median's 50%ile, at 10 cycles per degree), this set of sampled Zernike coefficients was selected for the standard observer because the set of coefficients gave values close to both the median MTFs (Fig. 5) and the median magnitudes of image displacement (computed as the ratio of absolute value of phase shift to spatial frequency) for all three types of cone. The absolute value of image displacements from the 100 random independent samples of Zernike coefficients is shown in Fig. 6. The median absolute values were close to the image displacements for the standard observer (compare solid lines, with and without symbols, Fig. 6). Of course, this particular set of Zernike coefficients is not unique in terms of giving MTFs and image displacements that are near the median for all three cone types. gave MTFs close to the median for the three cone types (Fig. 5), the standard observer was selected to closely represent both the median MTFs (Fig. 5) and median phase shifts (Fig. 6). To assess the reliability of the median and extreme MTFs, the 100 sampled sets of Zernike coefficients were randomly split into two groups of 50 samples each. Then the median, 5%ile and 95%ile MTFs for the L-, M-and S-cone were compared for the two halves. The comparison of the two groups showed good agreement (Fig. 7). A straightforward way to quantify the differences in MTFs among the L, M and S cones is to plot relative M/L and S/L contrast sensitivity for the median, 5%ile and 95%ile MTFs (Fig. 8). This shows that median contrast sensitivity is similar for the M and L cones.

Median S-cone contrast sensitivity is a log unit or more lower than L, except at the lowest spatial frequencies.

Zernike coefficients: Population means versus values for standard observer

There are substantial differences between the Zernike coefficients for the "standard observer" (Table 1, and gray bars in Fig. 9) and the mean Zernike coefficients for the population of human observers (from Thibos et al., 2002;diamonds, Fig. 9). For most purposes, the standard observer is more representative of a typical observer's image quality than calculations based on population means for Zernike coefficients. As noted by The first two parts of the paper considered retinal image quality and how it affects stimulation of the L, M and S cones for typical and relatively extreme normal human observers, assuming a 'white' stimulus with an EES spectral distribution. MTFs were derived using vertical sine-wave gratings. More generally, the model developed above may be applied to any visual stimulus. In this part, retinal image quality is determined for other types of stimuli, with a focus on some informative cases.

As discussed above, the eye's optical imperfections can be quantified by wavelengthdependent PSFs (Fig. 4), the effects of which are determined by decomposing the visual stimulus into its monochromatic components, applying each wavelength-dependent PSF, and then combining each monochromatic spatial light distribution by superposition. Thus, the effect of the eye's optics on the retinal spatial and spectral distribution of light depends on the spectral distribution of the stimulus.

The significance of the underlying spectral distribution of the stimulus can be missed by wrongly applying the color-vision principle of trichromacy, which holds that the color of any spectral distribution of light can be perfectly matched by a mixture of just three wavelengths (for example, 440 nm, 540 nm and 660 nm, which alone appear violet, yellowish-green and red, respectively). The basis for trichromacy is the three types of cone photoreceptors: L, M and S. Any given spectral distribution is perceived to be an exact match to an entirely different spectral distribution so long as both distributions have the identical rate of quantal absorption by the L, M and S cones; unequal spectral distributions that satisfy this matching criterion are called metamers. An example of two spectral distributions that appear identical is in Fig. 1. The flat energy distribution as a function of wavelength for EES 'white' (dashed line) results in the same stimulation of the L, M and S cones as the irregularly shaped spectral distribution (solid line), which is typical of a CRT display set to exactly match EES 'white'.

Two metameric spectral distributions of light, however, may not have exactly the same appearance when presented as complex patterns of light. The reason is that the underlying wavelength distributions that deliver identical rates of L-, M-and S-cone quantal absorptions for large uniform patches (as used for color matching) may be unequally affected by the eye's optics, because the effects of optics vary according to the wavelengths composing the two stimuli. For example, MTFs for the L and M cones for the standard observer are similar for true EES light and for its CRT metamer but contrast sensitivity for S cones is lower for true EES light compared to the metameric CRT chromaticity (compare open and filled symbols, Fig. 11). In general, retinal images of identical stimulus patterns composed of metameric lights may not be assumed to be equal.

Stimulus orientation and the retinal image

In the second part of the paper, MTFs were determined from luminance modulation of EES light varied in the horizontal direction (i.e., vertically oriented gratings). In the absence of wave aberrations, any orientation of the stimulus gives the same results

because PSFs are circular (Fig. 2). PSFs that include effects of wave aberrations, however, are not circular in shape (Fig. 4) so the retinal image of even a uniform bar of light can (i) be asymmetric with respect to the center of the uniform bar and (ii) depend on stimulus orientation.

Asymmetric distortion from wave aberrations can be illustrated with a square-wave EES grating of two cycles per degree with 100% luminance contrast (solid lines, Fig. 12); in the figure, the one-dimensional square-wave is a profile of a two-dimensional vertical square-wave grating. Retinal-image profiles at monochromatic wavelengths between 400 and 750 nm are shown for a 6 mm pupil (Fig. 12). In the top panel, only diffraction and LCA are considered (no wave aberrations). The wavelength-dependent distortion of the square-wave stimulus is symmetric around the center of the stimulus peaks and troughs.

By comparison, when wave aberrations for the standard observer are included (middle panel, Fig. 12), there is clear left-right asymmetry; moreover, the shape of the asymmetry is wavelength dependent. The asymmetry carries over to the spatial profile of stimulation for the L, M and S cones (bottom panel, Fig. 12).

The influence of stimulus orientation can be seen with the same EES, two cycle per degree square-wave grating. For the standard observer (6 mm pupil), the retinal-image profiles for the L, M and S cones are different for a vertically compared to horizontally oriented grating (see dotted and dashed lines, Fig. 13). In general, when wave aberrations are considered, the retinal image depends on the overall stimulus orientation as well as the spectral and spatial distribution of light. (Bottom) Relative S-cone stimulation.

The retinal image of an EES 'white' E

The loss in retinal image quality caused by the eye's optics can be visualized by the retinal image of the letter "E" of width 0.5 deg visual angle (cf. Williams & Hoffer, 2004).

An E presented to the eye as EES 'white' light is shown in the far left panel of Fig. 14.

The retinal image of the E, assuming only diffraction and LCA (no wave aberrations), is in the three top panels on the right, separately for stimulation of the L, M and S cones. A modest drop in image quality is apparent for the L-cone and M-cone retinal images, while the S-cone retinal image is far worse than for L or M. Adding wave aberrations of the standard observer (bottom three panels at right) further degrades the L-cone and M-cone retinal images but not the S-cone image, which actually appears a bit better than the Scone image without wave aberrations. Note, however, that the S-cone retinal image with wave aberrations (bottom right panel) is poorer than the L-and M-cone images that include wave aberrations (compare three bottom panels at right, Fig. 14). The effect of wave aberrations on the retinal image for each type of cone is taken up again in the Discussion, where previous studies are also considered.

Retinal image quality for typical R, G and B components of a color video display

The model developed here may be applied to a visual stimulus with any spectral For any known spectral distributions of light, the MTF can be determined for the L, M or S cones of the standard observer. The R, G and B spectra of the CRT were measured using a PhotoResearch PR-650 spectroradiometer. For each component (R, G or B), the L-, M-and S-cone MTFs are plotted together in a single panel in Fig. 15. There are two main points. First, the MTFs for R and G are higher than for B, for every type of cone.

Second, the MTFs for the R and G components are similar to each other for the L and M cones but not for the S cones, for which G has substantially higher contrast sensitivity than R. For S cones, in fact, contrast sensitivity for R is nearly as poor as for B.

DISCUSSION

The approach here used a comprehensive model of the eye (Thibos & Bradley, 1999) that incorporates pupil aperture, longitudinal chromatic aberration and wavefront aberrations. The model gives the spatial and spectral distribution of light on the retina from any image presented to the eye. Such distributions were used to determine typical and extreme (5 th and 95 th percentile) MTFs for each type of cone (L, M and S), by incorporating known individual differences for the eye's optics (Thibos et al., 2002).

A large number of random samples from the statistical distribution for individual differences also revealed median levels of contrast sensitivity, from which a "standard observer" was defined with retinal image quality near the central tendency of the normal human population. Typical image quality is better represented by this standard observer than by calculations based on population-mean Zernike coefficients because mean values tend to cancel out positive and negative coefficients and thus underestimate the loss of image quality, which is largely dependent on the coefficients' magnitudes regardless of sign.

Other investigators have examined related questions about retinal image quality for spectrally broadband images. Results from the approach here are compared to two well known studies below.

A comparison to the Marimont and Wandell model

As mentioned in the Introduction, [START_REF] Marimont | Matching color images: The effects of axial chromatic aberration[END_REF] consider a similar problem but with a model that has two important differences. First, the model here explicity considers higher-order wavefront aberrations. Second, the model here can be used to estimate individual differences based on the statistical population distribution for characteristics of the eye's optics (Thibos et al., 2002). Nonetheless, in special cases the two models should give similar results. In particular, when only diffraction and longitudinal chromatic aberration are included in the model here (no wave aberrations), the modeled retinal image should be somewhat better than the results from the Marimont and Wandell model, which incorporates an approximation for higher-order aberrations.

Also, the full model here with wave aberrations of the standard observer (Table 1) should give a retinal image similar to but different in shape than the Marimont and Wandell model, because introducing explicit wave aberrations results in asymmetries in the PSFs (Fig. 4) and thus in the retinal light profile (e.g., middle panel of Fig. 12).

Retinal light profiles for an inhomogeneous chromatic image (Fig. 16) were determined from the model here, both without and with wave aberrations (dotted and dash-dot lines, respectively), and from the Marimont and Wandell model (dashed line).

For this test case, the stimulus was composed of a square-wave vertical stripe of width 7.3 min of arc, centered within a 1 deg wide surround. Both the stripe and surround had the same chromaticity (0.665) in the l = L/(L+M) direction of the MacLeod and Boynton (1979) l,s cone-based coordinate system but differed in s = S/(L+M): the stripe had an s chromaticity of 2.50 so when viewed alone would appear decidedly bluish, while the surround had an s chromaticity of 0.99 so was virtually metameric to EES 'white'. The stripe and surround were equal in luminance. The full spectral distributions of the chromaticities were assumed to be from the CRT considered in Fig. 15.

In Fig. 16, the vertical axis is the light profile for the s chromaticity, derived from the retinal image for S, M and L and then applying the definition of s = S/(L+M). The square wave (solid line) is the image at the cornea. A technical point for calculation of s is selection of units for S, M and L; previous results for S-, M-and L-cone MTFs and light profiles were not dependent on units because plotted quantities were relative values. For Fig. 16, L and M were normalized so that L+M gave luminance (as for the [START_REF] Smith | Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm[END_REF] cone fundamentals), and S was scaled so that s=1.0 for the spectrally flat distribution of EES 'white'.

As expected, the model here without wave aberrations specified a light profile for s that that was sharper (better retinal image quality) than either the Marimont and Wandell model or the model here that includes wave aberrations. The light profiles from the Marimot and Wandell model and from the model here with wave aberrations were similar but only the latter captured the asymmetry in the distribution of light implicit in the asymmetric PSFs (as seen in Fig. 4). Note that the light profile with higher-order wave aberrations (dash-dot line) depends on the orientation of the stripe in the stimulus.

the optics of the standard observer (Table 1) gives results in accord with those from the Marimont and Wandell model. Comparing MTFs with versus without wave aberrations shows that introducing wave aberrations can make contrast sensitivity more similar for the three types of cone. For example at 20 cpd (see left panel, Fig. 17), with the wave aberrations of the standard observer the contrast sensitivity for L cones (and M cones) is about 20 times better than for S cones; without wave aberrations (only LCA and diffraction), L-cone (and M-cone) sensitivity is about 200 times better than S. These results confirm that wave aberrations can reduce the difference in contrast sensitivity among the three types of cone.

A related issue is whether wave aberrations fully (or nearly) eliminate the differences in contrast sensitivity for the L, M and S cones. [START_REF] Mclellan | Imperfect optics may be the eye's defence against chromatic blur[END_REF] report that they do:

the MTFs for all three cone types are nearly identical to each other, for each of their three observers p.175). To consider this question, the random sample of 100 sets of Zernike coefficients was searched for sets with the S-cone MTF near or above the L-and M-cone MTFs. For example, one sample (#14; middle panel, Fig. 17) had close L-, M-and S-cone MTFs (cf. the standard observer's MTFs in left panel of Fig. 17); the difference between L-and M-cone contrast sensitivity was about the same as between L and S sensitivity. At 20 cpd for #14, the contrast sensitivity for L cones was less than twice that of S cones. This set of MTFs is similar to the set for an observer reported by McClelland et al. (2002;their Fig. 3b). Another sample at 20 cpd (#92; right panel, Fig. 17) also had L-cone sensitivity less than two times higher than S, and at 40 cpd had S sensitivity fractionally better than L or M sensitivity (reminiscent of the MTFs for the observer in Fig. 3c of McClelland et al. (2002)). In sum, a few of the 100 samples were consistent with the view that wave aberrations can (nearly) eliminate contrast sensitivity differences among the three types of cone.

Recall, however, that the MTFs in Figs. 17b,c were selected from the sample of 100 sets because contrast sensitivity was similar for the L, M and S cones. To consider whether wave aberrations typically eliminate contrast-sensitivity differences among the three types of cone, the MTF for the S cones was compared to the MTFs for L and M cones at 10 and 20 cpd for all 100 randomly sampled sets of Zernike coefficients.

Because L-and M-cone MTFs tended to be similar, average L and M contrast sensitivity was compared to S sensitivity (that is, Average(L-sensitivity,M-sensitivity) relative to Ssensitivity). This sensitivity ratio for the 100 samples is plotted in a histogram in Fig. 18 (left panel, 10 cpd; right panel, 20 cpd). A value of zero indicates equal contrast sensitivity for S cones compared to the average for L and M; a positive [negative] value indicates L and M cones had higher [lower] sensitivity than S. The horizontal axis in Fig. 18 is a log scale so, for example, a value of +2.0 indicates lower S contrast sensitivity by a factor of 100.

Overall, S-cone contrast sensitivity nearly always was lower than L and M sensitivity (in 99% of cases at 10 cpd, and 97% of cases at 20 cpd). In the vast majority of cases, S contrast sensitivity was at least 10 times lower (> +1.0 on the log scale) than L and M sensitivity (73% of cases at 10 cpd, 77% of cases at 20 cpd). While S-cone sensitivity that is lower by a factor of 10 is a much smaller sensitivity difference than a factor of more than 200 for retinal image quality based on only diffraction and LCA (see arrows, (Right) 20 cpd. The value for the standard observer is 1.30, and for only diffraction and LCA (no wave aberrations) is 2.33.

Other applications for the model of retinal image quality

The model can be employed for many other purposes. Psychophysical experiments aiming to infer neural processes often depend on knowledge of an accurate retinal (not corneal) spatial distribution of light. The model can be used to determine a typical retinal image for the standard observer, given any external stimulus; moreover, a population range of retinal images can be estimated for, say, the central 90% of the population by finding the extreme L-, M-and S-cone MTFs for the best and worst 5% of the population.

This can be useful for excluding optical stimulus distortion as a viable account for experimental measurements, thereby implicating neural processes.

The pupil is largest, and retinal image quality often poorest, with dim illumination. A straightforward generalization of the model is to substitute the scotopic luminosity function V'(λ) for the cone spectral sensitivity function S C (λ) in Eq.( 1). This gives the spatial distribution I R (x,y) for rod excitation, from which a rod MTF or other characteristics of rod stimulation can be determined.

The model's ability to incorporate population variation in retinal image quality can be applied to several practical issues, including design decisions for instruments, spectral illumination and chromatically complex surfaces. An estimate of the retinal image for any light stimulus, including any object under any illuminant, can be useful for example to determine the legibility of text for, say, 99% of the population by assuming retinal image quality for the worst 1% of normal human observers. This may be particularly valuable for examining nighttime visibility (for example, for signage), when the pupil is largest.

More generally, standards can be developed for detection or discrimination that take account of losses in retinal image quality for the vast majority of the normal population (say 99%, again by using retinal image quality for the worst 1% of the normal population).

A related application is assessment of image quality (digital or otherwise). The ability to incorporate normal variation in the eye's optics permits perceived image-quality assessments to take account of the substantial variation in the retinal image, which depends on both the particular image presented to the eye and individual differences within the population of human observers.

Acknowledgement --Supported by NIH grants RO1 EY-05109 (LT) and RO1 EY-04802 (SS).

Figure 1 :

 1 Figure 1: Spectral power distribution of a true equal-energy-spectrum 'white' stimulus (dashed line) and from a typical color monitor displaying a light metameric to the equalenergy spectrum (solid line).

Figure 2 :

 2 Figure 2: Point spread functions due to only diffraction and longitudinal chromatic aberration (no wavefront aberrations) for monochromatic wavelengths 400, 450, 500, 550, 600, 650, 700 and 750 nm. Focus wavelength is 570 nm. The height and width of each panel is 1 deg of visual angle. The sum of the volume under the PSF in each panel is 1.0 but, for visual clarity, the maximum in each plot is scaled to appear white; the actual maximum in each plot, which varies by more than 2000:1, is shown in the bottom right of each panel. (a) 3 mm diameter pupil. (b) 6 mm diameter pupil.

Figure 3 .

 3 Figure 3. Mean Zernike coefficients for the first 15 Zernike modes, for pupil diameter (a) 3mm, (b) 4.5mm or (c) 6 mm (from the model of Thibos et al., 2002).

Figure 4 .

 4 Figure 4. Point spread functions due to diffraction, longitudinal chromatic aberration, and wavefront aberrations for monochromatic wavelengths 400, 450, 500, 550, 600, 650, 700 and 750 nm. The height and width of each panel is 1 deg of visual angle. The sum of the volume under the PSF in each panel is 1.0 but, for visual clarity, the maximum of each plot is white; the actual maximum in each plot, which varies by over 500:1, is shown in the bottom right of each panel. Focus wavelength is 570 nm; pupil diameter is 6 mm. (a) Based on mean population Zernike coefficients from Fig. 3c. (b) Based on Zernike

  cone spectral sensitivity function, separately for the L, M and S cones. This gave the relative quantal absorption for each cone type at that point on the retina. The MTF for each cone type then was determined by finding the retinal contrast as a function of the spatial frequency of the stimulus.The complete set of one hundred MTFs, determined separately for the L, M, and S cones, is shown in Fig.5(thin gray lines). The median (thick black line), 5%ile (dotted line) and 95%ile (dashed line) values are also shown. The figure reveals the large individual differences in the MTFs estimated to occur within a normal population. The Scone median MTF is well below the L-and M-cone median curves, though the population variation for each cone's MTF is large.

Figure 5 .

 5 Figure 5. Modulation transfer functions for 100 randomly sampled sets of Zernike coefficients for pupil diameter 6 mm (gray lines). The median contrast sensitivity among the 100 values at each spatial frequency is shown by the solid black line; the 5 th and 95 th percentile contrast sensitivity is shown by dotted and dashed lines, respectively. The solid black lines connecting the symbols are MTFs for the "standard observer" (see text). (Left) L-cone MTFs. (Center) M-cone MTFs. (Right) S-cone MTFs.

Figure 6 .

 6 Figure 6. Image displacement functions for 100 randomly sampled sets of Zernike coefficients (6 mm diameter pupil). Ordinate values are the ratio of |phase shift| / spatial frequency. The median, 5 th percentile and 95 th percentile shifts are indicated by solid, dotted and dashed black lines, respectively. Solid black lines connecting the symbols are image displacements for the standard observer whose Zernike coefficients are given in Table 1. (Left) L cone. (Middle) M cone. (Right) S cone.

Figure 7 .

 7 Figure 7. Modulation transfer functions for split halves of the 100 randomly sampled sets of Zernike coefficients for pupil diameter 6 mm. Each split half had 50 sampled sets of Zernike coefficients. The median, 5 th percentile and 95 th percentile contrast sensitivity for each split half are shown as a function of spatial frequency by the solid, dotted and dashed lines, respectively. (Left) L-cone MTFs. (Center) M-cone MTFs. (Right) S-cone MTFs.

Figure 8 .

 8 Figure 8. The ratio of contrast sensitivity for pupil diameter 6 mm as a function of spatial frequency, based on 100 randomly sampled sets of Zernike coefficients, for (left) M cones relative to L cones and (right) S cones relative to L cones. The median, 5 th percentile and 95 th percentile are shown in each panel by the solid, dotted and dashed line, respectively.

Figure 9 .

 9 Figure 9. The first fifteen Zernike coefficients for the "standard observer" (6 mm diameter pupil, gray bars), whose MTFs and image displacements are similar to the median values from the 100 randomly sampled sets of Zernike coefficients. Mean Zernike coefficients are also shown (diamonds, replotted from Fig. 3c)

Figure 10 .

 10 Figure 10. Modulation transfer functions based on 100 randomly sampled sets of Zernike coefficients for pupil diameter of 6 mm. The median contrast sensitivity among the 100 samples at each spatial frequency is shown by the thick solid line; 5 th and 95 th percentile contrast sensitivity is shown by dotted and dashed lines, respectively. Solid lines connecting open symbols are MTFs for the "standard observer"; solid lines connecting solid symbols are MTFs based on the mean Zernike coefficients in Fig. 3c. (Left) L-cone MTFs. (Center) M-cone MTFs. (Right) S-cone MTFs.

Figure 11 .

 11 Figure 11. Modulation transfer functions for the "standard observer" for vertical gratings composed of (i) EES 'white' light (open symbols) or (ii) the typical spectral distribution from a CRT display set to be metameric to EES 'white' (filled symbols). (Left) L-cone MTFs. (Center) M-cone MTFs. (Right) S-cone MTFs.

Figure 12 .

 12 Figure 12. Retinal-image profiles for an EES, two cycle per degree, 100% contrast square-wave grating. The solid line in each panel shows the square-wave stimulus. (Top and Middle) Profiles for wavelengths 400, 450, 500, 550, 600, 650, 700 and 750 nm (6mm pupil, 570 nm focus wavelength), with effects of only diffraction and longitudinal chromatic aberration (no wave aberrations, top) or with added effects of wave aberrations of the standard observer (middle). (Bottom) Retinal-image profiles for L-, M-and S-cone stimulation with wave aberrations of the standard observer.

Figure 13 .

 13 Figure 13. Retinal-image profiles for an EES, two cycle per degree, 100% contrast square-wave grating (6 mm pupil, 570 nm focus wavelength) including effects of wave aberrations for the standard observer. The solid line in each panel shows the square-wave stimulus profile. The dotted and dashed lines show the profiles for a vertically or horizontally oriented grating, respectively. (Top) Relative L-cone stimulation; (Middle) Relative M-cone stimulation;

  Figure 14. Retinal images of an EES letter "E" subtending in width 0.5 deg visual angle. Each panel is 1 deg square and rendered relative to the maximum light level in the stimulus before entering the eye (shown at the far left). (Right, top row) The retinal image for the L, M and S cones assuming only diffraction and LCA (no wave aberrations, 6 mm diameter pupil). (Right, bottom row) The retinal image for the L, M and S cones with the added effects of wave aberrations of the standard observer (6 mm diameter pupil).

Figure 15 .

 15 Figure 15. Modulation transfer functions for the standard observer (6 mm diameter pupil) for vertical gratings composed of a typical video display's (left) R component, (center) G component or (right) B component. The MTFs for the L, M and S cones are shown within each panel. Focus wavelength 570 nm.

Figure 16 .

 16 Figure 16. Comparison between the retinal light profiles for s=S/(L+M) from the model used here (focus wavelength 570 nm), without wave aberrations (dotted line) or with the wave aberrations of the standard observer (dash-dot line), and from the line spread function of Marimont and Wandell (dashed line). The corneal stimulus (solid line) is a vertical stripe of width 7.3 min within a 1 deg wide surround. The stripe and surround have different chromaticites (see text). Pupil diameter is 6 mm.

  Fig, 18), the results show that wave aberrations very seldom bring L-, M-and S-cone contrast sensitivity to comparable levels. The simulated retinal image of the letter E (Fig.14) is a demonstration of this point for the standard observer. The top three panels on the right of that figure include only diffraction and LCA; the image for S cones is far worse than for L and M. Wave aberrations, included in the images in the three bottom panels, further degrade the image for L and M cones but improve somewhat the image for S cones; nonetheless, even with wave aberrations the retinal image for S cones does not approach the quality of the image for L or M cones.

Figure 18 .

 18 Figure 18. Histogram of the contrast sensitivity ratio for the average for L and M cones compared to S cones, for the 100 randomly sampled sets of Zernike coefficients. Note the log scale on the horizontal axis. A positive value indicates lower S than L and M sensitivity. Pupil diameter is 6 mm and focus wavelength is 570 nm. (Left) 10 cpd. The value for the standard observer is 0.98, and for only diffraction and LCA (no wave aberrations) is 2.43.

TABLE 1 .

 1 Zernike coefficients for "standard observer" with 6 mm pupil

	Zernike Mode Number	Coefficient Value
	0	0.3243
	1	-0.5796
	2	0.7083
	3	-0.1115
	4	0.8638
	5	0.0148
	6	-0.1116
	7	-0.1644
	8	0.3053
	9	-0.0673
	10	-0.0273
	11	-0.0115
	12	0.2394
	13	0.0228
	14	0.0784
	The significance of spatial phase shifts in the image plane should not be ignored when
	wave aberrations are introduced. A circularly symmetric PSF due only to diffraction and
	longitudinal chromatic aberration (i.e., no wave aberrations) causes no shift in the phase of

the stimulus. Wave aberrations, however, can shift phase in the image plane. A PSF reflects both the MTF and phase shifts so a full description of the retinal image requires an accurate representation of both. While several other sampled sets of Zernike coefficients


Ariela Briani
email: ariela.briani@lmpt.univ-tours.fr

Antonin Chambolle
email: antonin.chambolle@cmap.polytechnique.fr

Matteo Novaga
email: novaga@math.unipd.it

Giandomenico Orlandi
email: giandomenico.orlandi@univr.it

On the gradient flow of a one-homogeneous functional

We consider the gradient flow of a one-homogeneous functional, whose dual involves the derivative of a constrained scalar function. We show in this case that the gradient flow is related to a weak, generalized formulation of a Hele-Shaw flow. The equivalence follows from a variational representation, which is a variant of well-known variational representations for the Hele-Shaw problem. As a consequence we get existence and uniqueness of a weak solution to the Hele-Shaw flow. We also obtain an explicit representation for the Total Variation flow in one dimension, and easily deduce basic qualitative properties, concerning in particular the "staircasing effect".

Introduction

This paper deals with the L 2 -gradient flow of the functional J k (ω) := A |dω| dx k ∈ {0, . . . , N -1} defined on differential forms ω ∈ L 2 (A, Ω k (R N )), where A ⊆ R N is an open set. We will focus on the particular case k = N -1: in that case, the dual variable is a scalar and this yields very particular properties of the functional J k and the associated flow. Notice that, when k = 0, the functional J 0 reduces to the usual total variation. When k = N -1 we can identify by duality ω ∈ L 2 (A, Ω N -1 (R N )) with a vector field u ∈ L 2 (A, R N ), so that J N -1 is equivalent to the functional

D(u) := A |div u| dx (1)
that is, the total mass of div u as a measure.

The gradient flow of D has interesting properties: we show in particular that it is equivalent to a constrained variational problem for a function w such that ∆w = div u. Moreover, under some regularity assumption on the initial datum u 0 , such a variational problem allows to define a weak formulation of the Hele-Shaw flow [START_REF] Elliott | A variational inequality approach to Hele-Shaw flow with a moving boundary[END_REF][START_REF] Gustafsson | Applications of Variational inequalities to a moving boundary problem for Hele Shaw flows[END_REF] (see also [START_REF] Kim | Homogenization of a Hele-Shaw problem in periodic and random media[END_REF] for a viscosity formulation). Therefore, it turns out that the flow of (1) provides a (unique) global weak solution to the Hele-Shaw flow, for a suitable initial datum u 0 . But our formulation allows us to consider quite general initial data u 0 , for which for instance div u 0 may change sign, or be a measure.

The plan of the paper is the following: in Section 2 we introduce the general functional we are interested in, we write the Euler-Lagrange equation for its Moreau-Yosida approximation and, in Section 2.1, we express it in a dual form that will be the base of our analysis.

In Section 3 we focus on the case k = 1 which is analyzed in this paper. We show many interesting properties of the flow: comparison, equivalence with a weak Hele-Shaw flow if the initial datum is smooth enough, and qualitative behavior when the initial datum is not smooth. In Section 4.1 we observe that, in dimension 2, the case k = N -1 also covers the flow of the L 1norm of the rotation of a vector field, which appears as a particular limit of the Ginzburg-Landau model (see [START_REF] Baldo | Convergence of Ginzburg-Landau functionals in 3-d superconductivity[END_REF][START_REF] Sandier | Vortices in the magnetic Ginzburg-Landau model[END_REF] and references therein).

Another interesting consequence of our analysis is that it yields simple but original qualitative results on the solutions of the Total Variation flow in dimension one (see also [START_REF] Bellettini | The total variation flow in R N[END_REF][START_REF] Bonforte | Total Variation Flow and Sign Fast Diffusion in one dimension[END_REF]). We show in Section 4.2 that the denoising of a noisy signal with this approach will, in general, almost surely produce a solution which is "flat" on a dense set. This undesirable artefact is the wellknown "staircasing" effect of the Total Variation regularization and is the main drawback of this approach for signal or image reconstruction.

Gradient flow

Given an initial datum ω 0 ∈ L 2 (A, Ω k (R N )), the general theory of [START_REF] Brézis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF] guarantees the existence of a global weak solution ω ∈ L 2 ([0, +∞), L 2 (A, Ω k (R N ))) of the gradient flow equation of J k :

ω t ∈ -∂J k (ω) t ∈ [0, +∞) , (2)
where ∂J k denotes the subgradient of the convex functional J k . Given ε > 0 and f ∈ L 2 (A, Ω k (R N )), we consider the minimum problem min

ω:A→R N J k (ω) + A 1 2ε |ω -f | 2 dx. (3)
Notice that min

ω:A→R N J k (u) + A |ω -εf | 2 2 dx = ε min ω:A→R N J k (u) + A 1 2ε |ω -f | 2 dx.
The Euler-Lagrange equation corresponding to (3) is

ε(f -ω) ∈ ∂J k (ω),
that is there exists a (k + 1)-form v with |v| = 1 such that v = dω/|dω| if dω = 0, and

ε(f -ω) = d * v in A and ( * v) T = 0 on ∂A. (4)

Dual formulation

Equation ( 4) is equivalent to

ω ∈ ∂J * k (ε(f -ω)),
where

J * k (η) := sup w:A→R N A η • w dx -J k (w) = 0 if η * ≤ 1 +∞ otherwise and η * = sup A η • w dx : J k (w) ≤ 1 . Note that J k (w) + J * k (η) ≥ A w • η dx
for all w, η. The equality holds iff A η • w dx = J k (w), and in such case we have η * ≤ 1.

Letting u be a minimizer of (3) and η = (fu)/ε we then get

A u • f -u ε dx = J k (u),
which implies

J * k f -u ε = 0 that is ε ≥ f -u * .
In particular, we showed the following (see also [START_REF] Meyer | Oscillating patterns in image processing and nonlinear evolution equations[END_REF] for the same result in the case of the Total Variation).

Proposition 2.1. The function u = 0 is a minimizer of (3) if and only if

ε ≥ ε c := f * . (5)
Note that η * < ∞ implies that

A ηw = 0
for all w such that dw = 0. By Hodge decomposition, this implies that η = d * g for some 2-form g, with g N = 0 on ∂A. It follows that

η * = sup A |dw|≤1 A d * g • w dx = sup A |dw|≤1 A g • dw dx + ∂A w ∧ * g N = sup A |dw|≤1 A g • dw dx. ( 6
)
We then get

η * = inf d * g=η g N | ∂A =0 g L ∞ (A) .
Indeed, it is immediate to show the ≤ inequality. On the other hand, by Hahn-Banach Theorem, there exists a form g ′ , with d * g ′ = d * g = η such that

η * = sup A |dw|≤1 A g • dw dx = sup A |ψ|≤1 A g ′ • ψ dx = g L ∞ (A) .
Fix now φ 0 such that d * φ 0 = η. We can write g = φ 0 + d * ψ, so that (6) becomes

η * = min ψ: (φ0+d * ψ)•ν A =0 φ 0 + d * ψ L ∞ (A) . (7)
The Euler-Lagrange equation of ( 7) is similar to the infinity laplacian equation

d ∞ (φ 0 + d * ψ) = 0.
By duality problem (7) becomes min

ψ∈W 1,∞ 0 (A) ∇ψ + φ 0 L ∞ (A) , (8)
and the corresponding Euler-Lagrange equation is

(∇ 2 ψ + ∇φ 0 )(∇ψ + φ 0 ), (∇ψ + φ 0 ) = 0. ( 9
)
3 The case k = N -1

In this case, we recall that we are considering the gradient flow of the functional (1), which is defined, for any u ∈ L 1 loc (A; R N ), as follows

D(u) = sup A -u∇v dx : v ∈ C ∞ c (A) , |v(x)| ≤ 1 ∀x ∈ A . (10)
This is finite if and only if the distribution div u is a bounded Radon measure in A. We now see it as a (convex, l.s.c., with values in [0, +∞]) functional over the Hilbert space L 2 (A; R N ): it is then clear from [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF] that it is the support function of

K = -∇v : v ∈ H 1 0 (A; [-1, 1])
and in particular p ∈ ∂D(u), the subgradient of D at u, if and only if p ∈ K and A p • u dx = D(u) = A |div u|:

∂D(u) = -∇v : v ∈ H 1 0 (A; [-1, 1]) , A -∇v • u dx = A |div u| .
We can define, for u ∈ dom D, the Radon-Nikodym density

θ div u (x) = div u |div u| (x) = lim ρ→0 B(x,ρ) div u B(x,ρ) |div u| ,
which exists |div u|-a.e. (we consider that it is defined only when the limit exists and is in {-1, 1}), and is such that div u = θ div u |div u|. We can also introduce the Borel sets

E ± u = {x ∈ A : θ div u (x) = ±1} .
Then, we have:

Lemma 3.1. ∂D(u) = -∇v : v ∈ H 1 0 (A; [-1, 1]) , v = ±1 |div u|-a.e. on E ± u . Proof. Consider v ∈ H 1 0 (A; [-1, 1]
). Then we know [START_REF] Adams | Functions spaces and potential theory[END_REF] that it is the limit of smooth functions

v n ∈ C ∞ c (A; [-1, 1]
) with compact support which converge to v quasi-everywhere (that is, up to a set of H 1 -capacity zero).

We recall that when u ∈ L 2 (A; R N ), the measure div u ∈ H -1 (A) must vanish on sets of H 1capacity 0 [1, §7.6.1]: it follows that v n → v |div u|-a.e. in A. Hence, by Lebesgue's convergence theorem,

- A ∇v(x)u(x) dx = lim n→∞ A v n (x)θ div u (x)|div u|(x) = A v(x)div u(x) .
It easily follows that if v = ±1 |div u|-a.e. on E ± u , -∇v ∈ ∂D(u) and conversely, that if v ∈ ∂D(u) then v = ±1 |div u|-a.e. on E ± u .

We now define, provided u ∈ dom ∂D (i.e., ∂D(u) = ∅),

∂ 0 D(u) = arg min A |p| 2 dx : p ∈ ∂D(u) :
it corresponds to the element p = -∇v ∈ ∂D(u) of minimal L 2 -norm. Using Lemma 3.1, equivalently, v is the function which minimizes A |∇v| 2 dx among all v ∈ H 1 0 (A) with v ≥ χ E + u and v ≤ -χ E - u , |div u|-a.e.: in particular, we deduce that it is harmonic in

A \ E + u ∪ E - u .
Let us now return to the flow [START_REF] Alter | A characterization of convex calibrable sets in R N[END_REF]. In this setting, it becomes

u t = ∇v u(0) = u 0 ( 11
)
where v satisfies |v| ≤ 1 and

D(u) + A u • ∇v = 0.
It is well know, in fact, that the solution of ( 11) is unique and that -∇v(t) = ∂ 0 D(u(t)) is the right-derivative of u(t) at any t ≥ 0 [START_REF] Brézis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF]. Given the solution (u(t), v(t)) of ( 11), we let

w(t) := t 0 v(s) ds,
which takes its values in [-t, t]. We have

u(t) = u 0 + ∇w(t). Theorem 1. Assume u 0 ∈ L 2 (A; R N ).
The function w(t) solves the following obstacle problem

min 1 2 A |u 0 + ∇w| 2 dx : w ∈ H 1 0 (A) , |w| ≤ t a.e. . (12)
Observe that in case we additionally have div u 0 ≥ α > 0, this obstacle problem is well-known for being an equivalent formulation of the Hele-Shaw problem, see [START_REF] Elliott | A variational inequality approach to Hele-Shaw flow with a moving boundary[END_REF][START_REF] Gustafsson | Applications of Variational inequalities to a moving boundary problem for Hele Shaw flows[END_REF].

Proof. Given u 0 ∈ L 2 (A; R N ), we can recursively define u n+1 ∈ L 2 (A; R N ) as the unique solution of the minimum problem min

u∈L 2 (A,R N ) D ε (u, u n ),
where

D ε (u, v) = D(u) + A 1 2ε |u -v| 2 dx.
Then, there exists v n+1 ∈ ∂D(u n+1 ) such that

u n+1 -u n -ε∇v n+1 = 0. ( 13
) It follows that v n+1 ∈ H 1 0 (A) minimizes the functional A |u n + ε∇v| 2 dx
under the constraint |v| ≤ 1. Let now

w n := ε n i=1 v i .
The from [START_REF] Kinderlehrer | The smoothness of the free boundary in the one phase Stefan problem[END_REF] we get

u n = u 0 + ∇w n , (14)
and w n minimizes the functional

A |u 0 + ∇w| 2 dx (15) under the constraint |w -w n-1 | ≤ ε. Notice that |w n -w n-1 | ≤ ε for all n implies |w n | ≤ nε. (16)
We now show that w n minimizes (15) also under the weaker constraint [START_REF] Baldo | Convergence of Ginzburg-Landau functionals in 3-d superconductivity[END_REF]. Indeed, letting ŵn be the minimizer of ( 15) under the constraint ( 16), we have

ŵn -ε ≤ ŵn+1 ≤ ŵn + ε,
which follows by noticing that min{ ŵn , ŵn+1 + ε} and max{ ŵn , ŵn+1 -ε} minimize (15), hence they are both equal to ŵn . It then follows w n = ŵn for all n.

Passing to the limit in n we get the corresponding result in the continuum case.

Remark 3.2. The previous proof also shows that for any initial

u 0 ∈ L 2 (A; R N ), u(t) = u 0 + ∇w(t) is the unique minimizer of A |div u| + 1 2t |u -u 0 | 2 dx.
We recall that obviously, such property does not hold for general semigroups generated by the gradient flow of a convex function. It is shown in [START_REF] Alter | A characterization of convex calibrable sets in R N[END_REF] to be the case for the Total Variation flow, in any dimension, when the initial function is the characteristic of a convex set.

Some properties of the solution

A first observation is that t → w(t) is continuous (in H 1 0 (A), strong), as follows both from the study of the varying problems [START_REF] Kim | Homogenization of a Hele-Shaw problem in periodic and random media[END_REF] and from the fact that the flow u(t) = u 0 + ∇w(t) is both continuous at zero and L 2 (A)-Lipschitz continuous away from t = 0 (and up to t = 0 if u 0 ∈ dom ∂D).

In fact, one can check that w is also L ∞ -Lipschitz continuous in time: indeed, it follows from the comparison principle that for any s ≤ t,

w(s) -t + s ≤ w(t) ≤ w(s) + t -s (17)
a.e. in A, hence w(t)w(s) L ∞ (A) ≤ |t -s|. The comparison ( 17) is obtained by adding the energy in [START_REF] Kim | Homogenization of a Hele-Shaw problem in periodic and random media[END_REF] of min{w(t), w(s) + t -s} (which is admissible at time t and hence should have an energy larger than the energy of w(t)) to the energy of max{w(t)-t+ s, w(s)} (which is admissible at time s), and checking that this sum is equal to the energy at time t plus the energy at time s. This is quite standard, see [START_REF] Caffarelli | The obstacle problem revisited[END_REF][START_REF] Kim | Homogenization of a Hele-Shaw problem in periodic and random media[END_REF].

In particular, we can define for any t the sets

E + (t) = { w(t) = t} and E -(t) = { w(t) = -t} , (18)
where w(t) is the precise representative of w(t) ∈ H 1 (A), defined quasi-everywhere by

w(t, x) = lim ρ→0 1 ω N ρ N B(x,ρ) w(t, y)dy (19)
(ω N is the volume of the unit ball). It follows from ( 17) and ( 19) that if w(t, x) = t, then for any s < t, x is also a point where w(s, x) is well-defined, and its value is s; similarly if w(t, x) = -t then w(s, x) = -s. Hence: the functions t → E + (t), t → E -(t) are nonincreasing. Also, if s < t, one has from ( 17)

1 ω N ρ N B(x,ρ) w(s, y)dy -t + s ≤ 1 ω N ρ N B(x,ρ) w(t, y)dy ≤ 1 ω N ρ N B(x,ρ) w(s, y)dy -s + t so that if x ∈ E + (s), 2s -t ≤ lim inf ρ→0 1 ω N ρ N B(x,ρ) w(t, y)dy ≤ lim sup ρ→0 1 ω N ρ N B(x,ρ)
w(t, y)dy ≤ t and sending s to t, we find that if x ∈ s<t E + (s), w(t, x) = t and x ∈ E + (t): hence these sets (as well as E -(•)) are left-continuous.

We define

E + r (t) = s>t E + (s) ⊆ E + (t) and E - r (t) = s>t E -(s) ⊆ E -(t) , (20)
as well as

E(t) = E + (t) ∪ E -(t), E r (t) = E + r (t) ∪ E - r (t).
Then, there holds the following lemma:

Lemma 3.3. If s ≤ t, then E -(t) ⊆ E -(s) and E + (t) ⊆ E + (s), E - r (t) ⊆ E - r (s) and E + r (t) ⊆ E + r (s).
Moreover, for t > 0, v(t) = ±1 quasi-everywhere on E ± r (t) and E ± u(t) ⊆ E ± r (t), up to a set |div u(t)|-negligible. In particular

div u(t) (E - r (t)) c ≥ 0, div u(t) (E + r (t)) c ≤ 0, div u(t) (E + r (t) ∪ E - r (t)) c = 0.
Here, for a Radon measure µ and a Borel set E, µ E denotes the measure defined by µ E(B) := µ(E ∩ B).

Proof. The first two assertions, as already observed, follow from [START_REF] Ring | Structural properties of solutions to total variation regularization problems[END_REF] and the definition of E ± r . We know that the solution of equation [START_REF] Gustafsson | Applications of Variational inequalities to a moving boundary problem for Hele Shaw flows[END_REF] satisfies

∂ + t u = -∂ 0 D(u(t)) = ∇v(t) for any t > 0, but the right-derivative of u = u 0 + ∇w(t) is nothing else as lim h→0 ∇[w(t + h) -w(t)]/h. We easily deduce that v(t) = lim h→0 [w(t + h) -w(t)]/h (which converges in H 1 0 -strong). Since when x ∈ E + r (t)
, w(t, x) = t and w(t + h, x) = t + h for h small enough, we deduce that v(x) = 1 on that set, in the same way v = 1 on E - r (t).

Observe that the Euler-Lagrange equation for [START_REF] Kim | Homogenization of a Hele-Shaw problem in periodic and random media[END_REF] is the variational inequality

A (u 0 + ∇w(t)) • (t∇v -∇w(t)) dx ≥ 0 , for any v ∈ H 1 0 (A; [-1, 1]). In other words since u(t) = u 0 + ∇w(t), - A u(t) • ∇ w(t) t ≥ - A u(t) • ∇v
for any |v| ≤ 1, and we recover that -∇w(t)/t ∈ ∂D(u(t)).

Hence (using Lemma 3.1),

E ± u(t) ⊆ E ± (t). Now, if ṽ ∈ H 1 0 (A; [-1, 1]
) with ṽ = ±1 on E ± r , one deduces that for any s > t,

- A ∇ṽ • u(s) dx = D(u(s)) . Sending s → t, it follows - A ∇ṽ • u(t) dx ≥ D(u(t)) ,
hence ṽ ∈ ∂D(u(t)). We deduce that E ± u(t) ⊆ E ± r (t), invoking Lemma 3.1.

Remark 3.4. We might find situations where |v(t)| = 1 outside of the contact set. For instance, assume the problem is radial, div u 0 is positive in a crown and negative in the center. Then one may have that E + is a crown (w should be less than t at the center) and E -is empty. In that case, v should be equal to one also in the domain surrounded by the set E + .

We show now another simple comparison lemma:

Lemma 3.5. Let u 0 and u ′ 0 in L 2 (A; R N ) such that div u ′ 0 ≤ div u 0 in H -1 (A).
Then for any t ≥ 0, w ′ (t) ≤ w(t), where w ′ (t) and w(t) are the solutions of the contact problem [START_REF] Kim | Homogenization of a Hele-Shaw problem in periodic and random media[END_REF], the first with u 0 replaced with u ′ 0 .

Proof. Let t > 0, ε > 0, and w ε be the minimizer of

min |w|≤t 1 2 A |∇w| 2 dx - A w(div u ′ 0 -ε)
which of course is unique. We now show that w ε ≤ w(t) a.e., and since w ε → w ′ (t) as ε → 0 the thesis will follow. We have by minimality

A |∇w(t)| 2 2 dx - A w(t)(div u 0 ) ≤ A |∇(w(t) ∨ w ε )| 2 2 dx - A (w(t) ∨ w ε )(div u 0 ) , A |∇w ε | 2 2 dx - A w ε (div u ′ 0 -ε) ≤ A |∇(w(t) ∧ w ε )| 2 2 dx - A (w(t) ∧ w ε )(div u ′ 0 -ε) ,
where we denote w(t) ∨ w ε := max{w(t), w ε } and w(t) ∧ w ε := min{w(t), w ε }. Summing both inequalities we obtain

A (w(t) ∨ w ε -w(t))div u 0 ≤ A (w ε -w(t) ∧ w ε )(div u ′ 0 -ε) ,
from which it follows ε A (w εw(t)) + dx ≤ 0, which is our claim.

Corollary 3.6. Under the assumptions of Lemma 3.5,

E -(t) ⊆ E ′ -(t) and E ′ + (t) ⊆ E + (t) , (21)
and it follows that v ′ (t) ≤ v(t), for each t > 0.

Proof. Eqn (21) follows at once from the inequality w ′ (t) ≤ w(t) (Lemma 3.5). We deduce, of course, that also E - r (t) ⊆ E ′ - r (t), and

E ′ + r (t) ⊆ E + r (t). Consider the function v = v ′ (t) ∧ v(t) = min{v ′ (t), v(t)}. As it is ±1 on E ′ ± r (t)
, it follows from Lemmas 3.3 and 3.1 that -∇v ∈ ∂D(u ′ (t)). In the same way,

v ′ = v ′ (t) ∨ v(t) = max{v ′ (t), v(t)} is such that -∇v ′ ∈ ∂D(u(t)). Since A |∇v| 2 dx + A |∇v ′ | 2 dx = A |∇v(t)| 2 dx + A |∇v ′ (t)| 2 dx , either A |∇v| 2 dx ≤ A |∇v(t)| 2 dx or A |∇v ′ | 2 dx ≤ A |∇v ′ (t)| 2 dx. By minimality (as -∇v(t) = ∂ 0 D(u(t))) it follows that v = v(t) and v ′ = v ′ (t).

The support of the measure div u

Throughout this section we will assume that div u 0 is a bounded Radon measure on A. Lemma 3.7. Let u 0 ∈ L 2 (A; R N ) ∩ dom D, δ > 0 and u = (I + δ∂D) -1 (u 0 ). Then for a positive Radon measure µ ∈ H -1 (A), the Radon-Nikodym derivatives of div u and div u 0 with respect to µ satisfy (div u/µ)(x) ≤ (div u 0 /µ)(x) for µ-a.e. x ∈ E + u , and (div u/µ)(x) ≥ (div u 0 /µ)(x) for µ-a.e. x ∈ E - u . In particular, div u << div u 0 and (div u) ± ≤ (div u 0 ) ± .

Remark 3.8. It follows from the Lemma that div u = θdiv u 0 (E + u ∪ E - u ) for some weight θ(x) ∈ [0, 1]. We can build explicit examples where θ < 1 at some point. Consider for instance, in 1D, A = (0, 1) and the function u 0 (x) = 0 if x < 1/3 and x > 2/3, and 2 -3x if 1/3 < x < 2/3. Then, one shows that u(t) is given by

u(t, x) =              3t if x < 1 3 1 -2 √ 3t if 1 3 < x < a(t) := 1 3 + 2t √ 3 2 -3x if a(t) < x < b(t) := 1 - √ 1+6t 3 √ 1 + 6t -1 if x > b(t) until t = 1 -2 √ 2/3. We have div u(t) = u(t) x = (1 -2 √ 3t -3t)δ 1/3 -3χ (a(t),b(t)
) for such t: E + u(t) = {1/3} stays constant for a while (and disappears suddenly right after t = 1 -2 √ 2/3), while the density of the measure div u(t) goes down monotonically until it reaches zero (notice that v(t) will jump right after 1 -2 √ 2/3), while E - u(t) = (α(t), β(t)) shrinks in a continuous way, and carries the constant continuous part of the initial divergence (-3).

Proof. We have u = u 0 + δ∇v with -∇v ∈ ∂D(u). Let x ∈ E + u . Recall that the precise representative of v is defined by ṽ

(x) = lim ρ→0 B(x,ρ) v(y) dy ω N ρ N ,
where ω N = |B(0, 1)|, and that this limit exists quasi-everywhere in A. We assume also that ṽ(x) = 1.

Then, for a.e. ρ > 0, one may write

B(x,ρ) div u = ∂B(x,ρ) u • ν dH 1 = ∂B(x,ρ) u 0 • ν dH 1 + δ ∂B(x,ρ) ∇v • ν dH 1 = B(x,ρ) div u 0 + δ ∂B(x,ρ) ∇v • ν dH 1 . (22)
Now, let f (ρ) = (1/ρ N -1 ) ∂B(x,ρ) v dH 1 (which is well-defined for any ρ). Then, since ṽ(x) = 1 and v ≤ 1 a.e., lim sup

ρ→0 f (ρ) = N ω N .
One can also show that for a.e. ρ > 0, f ′ (ρ) = (1/ρ N -1 ) ∂B(x,ρ) ∇v • ν dH 1 , in fact f is locally H 1 in some small interval (0, ρ 0 ). Since v ≤ 1 a.e., f (ρ) ≤ N ω N a.e., so that lim inf

ε→0 ρ ε 1 r N -1 ∂B(x,r) ∇v • ν dH 1 dr = lim inf ε→0 ρ ε f ′ (r) dr = lim inf ε→0 f (ρ) -f (ε) ≤ 0
for any ρ. If follows that for any ρ small, the set I + ρ = {r ∈ [0, ρ] : ∂B(x,r) ∇v • ν dH 1 ≤ 0} has positive Lebesgue measure, and for any r ∈ I + ρ , we deduce from (22) that B(x,r) div u ≤ B(x,r) div u 0 . Now consider µ a positive Radon measure: µ-a.e., we know that the limits

div u µ (x) = lim r→0 B(x,r) div u µ(B(x, r)) and div u 0 µ (x) = lim r→0 B(x,r) div u 0 µ(B(x, r))
exist. If moreover, as before, x ∈ E + u and ṽ(x) = 1 (which holds µ-a.e., since µ ∈ H -1 (A)), we can find a subsequence r n such that B(x,rn) div u ≤ B(x,rn) div u 0 for each n, and it follows (div u/µ)(x) ≤ (div u 0 /µ)(x).

The following corollaries follows: Corollary 3.9. Let t > s ≥ 0: then (div u(t)) ± ≤ (div u(s)) ± . In particular, E ± u(t) ⊆ E ± u(s) , |div u(s)|-a.e. in A.

Proof. Indeed: if t > s, then u(t) = (I + (ts)∂D) -1 (u(s)). We deduce that for quasi-every x ∈ E + u(t) , 1 = θ div u(t) (x) ≤ (div u(s)/(div u(t)) + )(x), and it follows (div u(t)) + ≤ (div u(s)/(div u(t)) + )(div u(t)) + ≤ (div u(s)) + . Corollary 3.10. We have that (div u(t)) ± * ⇀ (div u 0 ) ± as t → 0, weakly- * in the sense of measures. Moreover, E ± u0 ⊂ E ± r (0) (up to a |div u 0 |-negligible set), and div u 0 (E + r (0)) ≥ 0, div u 0 (E - r (0)) ≤ 0.

Proof. We know that as t → 0, u(t) → u 0 in L 2 (A; R N ), and thanks to the boundedness of div u(t) it follows that div u(t) * ⇀ div u 0 in the sense of measures. Now consider a subsequence (t k ) such that (div u(t k )) + * ⇀ µ, (div u(t k )) - * ⇀ ν. Since µν = div u 0 , it follows that µ ≥ (div u 0 ) + and ν ≥ (div u 0 ) -. The reverse inequalities follow from Lemma 3.7 and the first part of the thesis follows.

From the previous results we obtain that for each t, one can write

(div u(t)) + = θ t (x)(div u 0 ) +
The function θ t (x) = lim inf ρ→0 ( B(x,ρ) div u(t) + )/( B(x,ρ) (div u 0 ) + ) is well-defined on the set E + u0 which supports the measure (div u 0 ) + , and we find that θ t (x) ≤ 1 is nonincreasing in t. Hence there exists for all x ∈ E + u0 the limit lim t→0 θ t (x) = sup t>0 θ t (x), and this limit must be 1 (div u 0 ) + -a.e., otherwise this would contradict that (div u(t)) + * ⇀ (div u 0 ) + . It follows that up to a (div u 0 ) + -negligible set,

E + u0 ⊆ t>0 {x ∈ E + u0 : θ t (x) > 0}. Now, if x ∈ E + u0 and θ t (x) > 0, then x ∈ E + u(t) : indeed, B(x,ρ) div u(t) B(x,ρ) |div u(t)| = (div u(t)) + (B(x, ρ)) -(div u(t)) -(B(x, ρ)) (div u(t)) + (B(x, ρ)) + (div u(t)) -(B(x, ρ)) ρ→0 -→ 1 , since (div u(t)) -(B(x, ρ)) ≤ (div u - 0 )(B(x, ρ)) = o((div u + 0 )(B(x, ρ))) ≤ o((div u(t)) + (B(x, ρ)))
(the equality is because x ∈ E + u0 , the last inequality because θ t (x) > 0). It follows that

E + u0 ⊆ t>0 E + u(t)
and the conclusion follows from Lemma 3.3.

The regular case

Let us now assume that div u 0 = g ∈ L p (A), p > 1. The obstacle problem which is solved by w(t) can be written min

w∈H 1 0 :|w|≤t 1 2 A |∇w(x)| 2 dx - A g (x)w(x) dx .
Standard results show that w(t) ∈ W 2,p (A), (see Theorem 9.9 in [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]). In particular, we have that in the L p sense, -∆w(t) = gχ {|w(t)|<t} and, since u(t) = u 0 + ∇w(t), we deduce that in this case

div u(t) = div u 0 χ E(t) (23)
for any t > 0. In particular, formally, we deduce from [START_REF] Gustafsson | Applications of Variational inequalities to a moving boundary problem for Hele Shaw flows[END_REF] that

div u 0 ∂χ E(t) ∂t = ∆v(t) , (24)
and since ∆v(t) is the jump of the normal derivative of v(t) on ∂E ± (t), we find that these sets shrink with a normal speed |∇v(t)|/|div u 0 |. This can be written rigorously in the sense of distributions:

(E + , E -, v) are such that v ∈ L 1 ([0, T ); H 1 0 (A; [-1, 1])), v = ±1 on E ± for a.e. t

and x, and for any

φ ∈ C ∞ c ([0, T ) × A), A div u 0 (x)φ(0, x) dx + T 0 A div u 0 (x)χ E(t) (x) ∂φ ∂t (x, t) dx dt - T 0 A ∇v(t, x) • ∇φ(t, x) dx dt = 0. ( 25
)
We observe that the evolution equation ( 25) is reminiscent of the enthalpy formulation of the one-phase Stefan problem [START_REF] Rodrigues | Variational methods in the Stefan problem[END_REF]. We expect that with either the additional information that div u 0 is a.e. nonnegative on E + and nonpositive on E -, or that the maps E ± (t) are nonincreasing, then (25) characterizes the unique evolution [START_REF] Gustafsson | Applications of Variational inequalities to a moving boundary problem for Hele Shaw flows[END_REF]. On the other hand, without this additional assumption, then a timereversed evolution with will satisfy the same weak equation, with u 0 replaced with -u 0 . With both assumptions we can actually show the following result: Proposition 3.11. Let E + , E -be measurable subsets of A × [0, T ], and v ∈ L 1 ([0, T ); H 1 0 (A)) with |v| ≤ 1 a.e., v = ±1 a.e. on E ± , and satisfying (25). Assume in addition that ±div u 0 ≥ 0 a.e. on E ± , and E ± (t) ⊆ E ± (s) for a.e. t > s .

(26)

Then u(t, x) := u 0 (x) + ∇ t 0 v(s, x) ds is the unique solution of [START_REF] Gustafsson | Applications of Variational inequalities to a moving boundary problem for Hele Shaw flows[END_REF].

Proof. Let w(t) = t 0 v(s) ds. Thanks to (26), we have that |w(t, x)| ≤ t for a.e. x ∈ A, and w(t, x) = ±t for a.e. x ∈ E ± (t), for all t. We can approach test functions of the form χ [0,t] φ(x), φ ∈ H 1 0 (A), with smooth functions and pass to the limit to check that

A div u 0 φ dx - E(t) div u 0 φ dx = A ∇w(t) • ∇φ dx ,
for almost all t (up to a negligible set, which we can actually choose independently of φ, as H 1 0 (A) is separable).

If we choose φw(t, •) as the test function in this equation, we find

A div u 0 (x)φ(x) dx - A div u 0 (x)w(x, t) dx - E(t) div u 0 (x)(φ(x) -w(x, t)) dx = A ∇w(t, x) • ∇φ(x) dx - A |∇w(t, x)| 2 dx , = - 1 2 A |∇w(t, x) -∇φ(x)| 2 dx + 1 2 A |∇φ(x)| 2 dx - 1 2 A |∇w(t, x)| 2 dx .
If |φ| ≤ t, we have that -div u 0 (x)(φ(x)w(x, t)) ≥ 0 for a.e.

x ∈ E(t), so that w(t) is the minimizer of ( 12) and the thesis follows.

Remark 3.12. As mentioned above, it is a natural question whether assumption (26) is necessary to prove this result. For instance, in case E + and E -are closed sets in [0, T ) × A with E + (t) ∩ E -(t) = ∅ for any t > 0, and {div u 0 = 0} is a negligible set, then one can actually deduce (26) from (25). Indeed, using localized test functions φ(x)χ [s,t] , one shows first that v is harmonic in A \ E(t) for a.e. t, and then that E(s) div u 0 φ dx -E(t) div u 0 φ dx ≥ 0, and (26) follows.

Remark 3.13. When p > N/2, we can deduce some further properties of w from the regularity theory for the obstacle problem [START_REF] Caffarelli | The obstacle problem revisited[END_REF]. Indeed, letting Ψ ∈ H 1 0 (A) ∩ W 2,p (A) such that -∆Ψ = g, we have that w = w -Ψ ∈ H -equivalent to the support of (div u) + and since from the equation it follows u = u 0 a.e. on E ± (since v = ±1 a.e. on E ± , so that ∇v = 0 a.e., the problem being in general that this will not be true quasi-everywhere), we deduce that div u = div u 0 a.e. on E + ∪ E -= spt(div u).

Examples 4.1 The antiplane case in dimension 2

Let N = 2 and k = 1. We have

J(ψ) = |rot ψ|(A) = sup A ∇ ⊥ • ψ : v ∈ C ∞ c (A; [-1, 1]) where rot ψ = ∂ 1 ψ 2 -∂ 2 ψ 1 and ∇ ⊥ = (∂ 2 , -∂ 1 )
. Then, we check easily that in L 2 (A; R 2 ) the functional J is the support function of the closed convex set

K = ∇ ⊥ v : v ∈ H 1 0 (A; [-1, 1]) .
As we mentioned in the Introduction, this functional appears as limit of the Ginzburg-Landau model in a suitable energy regime [START_REF] Sandier | Vortices in the magnetic Ginzburg-Landau model[END_REF].

Letting ψ ⊥ = (ψ 2 , -ψ 1 ), we get J(ψ) = A |div ψ ⊥ |, so that the flow can be described as above.

Proposition 4.1. Let u 0 ∈ L 2 (A; R 2 ) with rot u 0 = g ∈ L p (A), p > 1. Then for t > 0 there exist nonincreasing left-continuous closed (and disjoint) sets E ± (t) ⊂ {±g ≥ 0}, such that rot u(t) = rot u 0 (χ E -(t)∪E + (t) ). Moreover, letting E ± = ∪ t≥0 {t} × E ± (t), there exists a function v(t, x) with v = ±1 a.e. on E ± such that (E + , E -, v) are the unique closed sets and function solution of the weak Hele-Shaw flow (25).

The one-dimensional Total Variation Flow

Let now N = 1, k = 0: the previous analysis also provides interesting qualitative information on the behavior of the flow of the Total Variation, in dimension 1. We consider u 0 ∈ L 2 ((a, b)), a < b, and the flow u(t) of the total variation J(u

) := sup{ b a uv ′ dt : v ∈ C ∞ c (a, b; [-1, 1])}.
Notice that in this situation, the function w minimizes [START_REF] Kim | Homogenization of a Hele-Shaw problem in periodic and random media[END_REF], being in

H 1 0 (a, b), is also in C 1/2 ([a, b]) with w(a) = w(b) = 0.
In particular, the sets E ± (t) defined in (18) are closed, disjoint sets compactly contained in (a, b).

We can state the following result. Moreover there exist nonincreasing, disjoint closed sets E ± (t) ⊂ (a, b) such that u(t) = u 0 a.e. on E ± (t), u 0 is nondecreasing on any interval contained in E + (t), nonincreasing on any interval contained in E -(t), and u(t) is constant on each connected component of (a, b)

\ (E + (t) ∪ E -(t)).
If u 0 is smooth enough, one can also characterize the speed of the boundary points of E ± (t) in term of u 0 and the size of the intervals of (a, b) \ (E + (t) ∪ E -(t)).

Proof. The first part of the thesis is a consequence of Remark 3.2. Then, if u 0 ∈ BV (a, b), the thesis is a consequence of Lemma 3.3. Indeed, for a.e. x on E ± (t), we have ∂ x w(t, x) = 0 and u(t, x) = u 0 (x) + ∂ x w(t, x) = u 0 (x). If I ⊂ E + (t) is an interval, since the measure Du(t) I must be nonnegative, u(t) is nondecreasing on I, but as u(t) = u 0 a.e. on I it follows that u 0 is nondecreasing on I.

If u 0 ∈ BV (a, b), we use the fact that for all ε > 0, u(ε) ∈ BV (a, b). Then the Proposition holds for t > ε, and we have u(t) = u(ε) a.e. on E ± (t), u(ε) is nondecreasing on any interval contained in E + (t), nonincreasing on any interval contained in E -(t), and u(t) is constant on each connected component of (a, b) \ (E + (t) ∪ E -(t)). The sets do not depend on ε, as they are defined as the contact sets in [START_REF] Kim | Homogenization of a Hele-Shaw problem in periodic and random media[END_REF]. Sending then ε → 0 we deduce the result.

We can deduce the following, quite interesting result -see also [START_REF] Ring | Structural properties of solutions to total variation regularization problems[END_REF][START_REF] Bonforte | Total Variation Flow and Sign Fast Diffusion in one dimension[END_REF][START_REF] Kielak | Almost classical solutions to the total variation flow[END_REF] for other results on the one-dimensional Total Variation flow and in particular [START_REF] Ring | Structural properties of solutions to total variation regularization problems[END_REF]Prop. 4] for a similar statement. Remark 4.4. The property that |Dn|(I) = +∞ for any interval I, almost surely, is satisfied for instance by the Wiener process (as its quadratic variation is positive a.s.). For a Gaussian stationary process, it will depend on the behaviour of the autocorrelation function and can be characterized by conditions on the power spectrum of the process, see for instance [START_REF] Yu | Continuity and Hölder's conditions for sample functions of stationary Gaussian processes[END_REF] for (non sharp) conditions.

Proof. We let A(t) = (a, b) \ (E + (t) ∪ E -(t)), and from the previous result we know that u(t) is constant on each connected component of A(t) while u = u 0 on (a, b) \ A(t). Now assume there is an interval I with I ∩ A(t) = ∅: without loss of generality we may assume that I ⊂ E + (t). Then u 0 must be nondecreasing on I, in particular there exists I ′ ⊂ I with |Du 0 |(I ′ ) < +∞. But this yields that |Dn|(I ′ ) < +∞, which is a.s. impossible.

Proposition 4 . 2 .J

 42 The function u(t) is the unique minimizer of min u

Corollary 4 . 3 .

 43 Let u 0 = ū0 + n where ū0 ∈ BV (a, b) and n is a stochastic process (a, b) with n ∈ L 2 (a, b) a.s. and such that |Dn|(I) = +∞ for any interval I ⊂ (a, b), almost surely. Let u(t) be the total variation flow starting from u 0 . Then almost surely, at t > 0, there is "staircaising" everywhere in the interval (a, b): u(t) is constant on each connected component of an open set A(t) which is dense in (a, b).

  Since p > N/2, we have w(t) ∈ C α (A), with α = 2 -N/p, so that E(t) = {|w(t)| = t} is a closed set. In this case, v(t) can be defined as the harmonic function in A \ E(t) with Dirichlet boundary condition v(t) = 0 on ∂A and v(t) = ±1 on E ± (t). Moreover, it is easy to check that -∇v(t) ∈ ∂ 0 D(u(t)), and v(t) is continuous out of the singular points of ∂A ∪ ∂E(t). Remark 3.14. If A = R N one can easily show easily by a translation argument that u 0 ∈ H 1 (A; R N ) ⇒ u(t) ∈ H 1 (A; R N ) with same norm, so that the H 1 -norm of u(t) is nonincreasing. In this case, E + u(t) is a.e.

	1 0 (A) solves the obstacle problem
	min -t-Ψ≤ w≤t-Ψ	1 2 A	|∇ w(x)| 2 dx.