Fabien Soulier

Lionel Gouyet

Guy Cathébras

Serge Bernard

David Guiraud

Yves Bertrand

CONSIDERATIONS ON IMPROVING THE DESIGN OF CUFF ELECTRODE FOR ENG RECORDING Geometrical Approach, Dedicated IC, Sensitivity and Noise Rejection

Keywords: Multipolar cuff electrode, regular tessellation, electroneurogram, action potential, Laplacian, selectivity, ASIC, multi-input differential amplifier

Cuff electrodes have several advantages for in situ recording ENG signal. They are easy to implant and not very invasive for the patient. Nevertheless, they are subject to background parasitic noise, especially the EMG generated by the muscles. We show that the use of cuff electrodes with large numbers of poles can to increase their sensitivity and their selectivity with respect to a efficient noise rejection. We investigate several configurations and compare the performances of a tripolar cuff electrode versus a multipolar one in numerical simulation. One the other hand the use of cuff electrodes leads to the recording of the sum of the signals generated by all the axons within the nerve. This puts in evidence the need of signal separation techniques that require a great quantity of information. Again, we show that multipolar electrodes can solve this problem since poles can be switched one to another, provided that they are distributed along a regular tessellation. Finally, we present the structure of an ASIC preamplifier processing a spatial filtering to obtain the Laplacian of the potential rejecting low-frequency noise.

INTRODUCTION

In a context of neural system pathologies such as spinal cord injury, Functional Electrical Stimulation (FES) techniques are the possible alternatives to restore lost sensory or motor abilities. These techniques consist in generating artificial contraction by electrical stimulation. In FES system a direct opened loop control doesn't allow efficient stimulation. In order to provide a loopback control we need sensory information (force, contact. . . ) [START_REF] Djilas | Interpretation of ENG signal for FES closed-loop control[END_REF]). An attractive solution consists in using the natural sensors. The sensory information is propagated by associated afferent fibers. But unfortunately, in peripheral nerves the complete nerve activity due to the large number of axons makes the extraction of the studied signal particularly hard. Moreover the sensory signal seen through the nerve is a very low amplitude signal compared with the amplitude of parasitic signals. For instance, on a monopolar recording, EMG created by muscle activity have amplitude about three orders of magnitude higher than the ENG. In this context, the two main objectives to be able to exploit natural sensors are:

• to find a solution to separate the useful information from the complete ENG signal; • to reject the parasitic external signals.

The classical solution consists in using multipolar electrodes, but from tripole [START_REF] Ramachandran | Multipolar cuff electrodes with integrated pre-amplifier & filter to interface peripheral nerves for FES application[END_REF] to nine pole electrode [START_REF] Winter | Improved spatial filtering of ENG signals using a multielectrode nerve cuff[END_REF][START_REF] Taylor | Multipleelectrode nerve cuffs for low-velocity and velocityselective neural recording[END_REF], the selectivity of the neural information is not efficient enough to be suitable in closed loop FES system. To achieve both a better sensitivity and efficient background noise rejection we propose a new configuration of the cuff electrode with a large number of poles regularly distributed onto the cuff. In this configuration, a group of poles can behave, with suitable low level analog signal processing, like a kind of a directive antenna. Moreover, the large number of poles will allow enough channels in order to apply source separation signal processing on the ENG. Of course, the directivity of the sensor relies on the quality of the subsequent low-level analog signal processing.

In this paper, we first show how to generalize the preprocessing operations on the recorded signal from tripolar to multipolar configuration using the Laplacian formalism. Then we discuss on the optimal pole placement around the nerve regarding tessellation methods. Both the electrode configuration and the associated preprocessing circuit result from this pole distribution and must be taken into account. We particularly focus on the hexagonal seven-pole electrode, presenting the associated seven input preamplifier and preliminary simulation results.

EMG NOISE REJECTION

Cuff electrodes have been the most used in the last ten years [START_REF] Haugland | Skin contact force information in sensory nerve signals recorded by implanted cuff electrodes[END_REF][START_REF] Jensen | Improving signal reliability for on-line joint angle estimation from nerve cuff recordings of muscle afferents[END_REF][START_REF] Andreasen | Signal strength versus cuff length in nerve cuff electrode recordings[END_REF]. They are relatively easy to implant, they are not invasive for the nerve and implantation is very stable and thus allows chronic experiments. ENG can be recorded as the potential dif-

a a V 2 V 0 V 1 2R Figure 1: Tripolar electrode cuff model.
ference created on the electrodes by the charges associated to the action potentials (AP) propagating along the nerve fibers. Fig. 1 shows a typical tripolar cuff electrode. When recording with this kind of electrode, a classic method to reject parasitic signals consists in calculating the average of the potential differences between the central pole and each of the outer poles [START_REF] Struijk | Tripolar nerve cuff recording: stimulus artifact, EMG and the recorded nerve signal[END_REF][START_REF] Pflaum | Performance of alternative amplifier configurations for tripolar nerve cuff recorded ENG[END_REF]:

V rec = (V 0 -V 1 ) + (V 0 -V 2 ) 2 = V 0 - V 1 + V 2 2 (1)
The last expression shows that this operation consists in:

1. averaging the signal on the outer poles, i.e. applying a low-pass spatial filter. 2. subtracting the result to the signal of the central pole, keeping only the high spatial frequencies. Therefore, the recorded V rec signal can be considered as spatial high-pass filtered.

More precisely, this filter is a second-order one considering that the expressions

1 a (V 2 -V 0 ) and 1 a (V 0 -V 1 ) (2)
evaluate the first derivative dV dx . Thus the difference 1 a

(V 2 -V 0 ) a - (V 0 -V 1 ) a = 2 a 2 (V 1 + V 2 ) 2 -V 0 (3) denotes the second derivative d 2 V
dx 2 that is the onedimensional Laplacian of the potential. We can identify in the last expression the equation (1) without the known constant factor -2a -2 .

Laplacian filters can reject both homogeneous potentials and linearly varying ones like those created by far EMG sources. The purpose of this new design is to build two-dimensional Laplacian using more poles to obtain isotropic rejection.

POSITIONING THE POLES

A tripolar cuff electrode [START_REF] Demosthenous | An adaptive ENG amplifier for tripolar cuff electrodes[END_REF] provides only one recording which is the superposition of all action potentials "seen" by the electrode at a given moment. The use of several poles on a cuff electrode (see Fig. 2) could allow us to record more signals, thus increase the quantity of neural data and facilitate the signal post-processing on the recording system.

In order to achieve optimal placement of poles, we must pay attention to three constraints:

1. The electrodes have to be placed all around the nerve, thus the poles have to be distributed onto the whole surface of the cuff.

2. The poles have to be equally spaced to simplify electronics in charge of analog signal preprocessing (weight coefficients in Laplacian preamplifier).

3. They have to be able to be substituted one to each other, so we take benefits of the maximum measurement locations, allowing powerful signal processing.

Since the cylindric shape of the cuff results from the wrapping of an initially plane device, these conditions imply to look for a regular tessellation of the plane as the positions of the poles or, more precisely, tessellations composed of regular polygons symmetrically tiling the plane. It is well known that there are exactly three type of regular tessellations [START_REF] Weisstein | Tessellation. From MathWorld-A Wolfram Web Resource[END_REF]. They can be specified using the Schläfli symbols: {3, 6}, {4, 4} and {6, 3}. The first symbol in the Schläfli notation denotes the shape of the patch (triangle, square or hexagon). On the figure 3, each vertex corresponds to an pole. Each of them being surrounded by a number of equidistant poles given by the second Schläfli symbol, respectively 6, 4 and 3.

{6,3} {4,4} {3,6}
From the previous tessellations, one can build three kinds of electrodes by selecting one central pole and its closest neighbors. Namely, we can define a mesh of:

• triangular 4-pole electrodes,

• squared 5-pole electrodes,

• hexagonal 7-pole electrodes.

These candidates can be seen on the figure 4 and the resulting expressions for the Laplacian are:

V rec = V 0 - 1 3 3 ∑ i=1 V i for {6, 3} (4)
V rec = V 0 - 1 4 4 ∑ i=1 V i for {4, 4} (5)
V rec = V 0 - 1 6 6 ∑ i=1 V i for {3, 6} (6)
One can notice that the {4, 4} configuration correspond to the 2D Laplacian filter used in image processing [START_REF] Gonzales | Digital Image Processing[END_REF].

ENG AMPLIFIER

Because of the very low level of processed signals we propose to perform the maximum of signal processing as close as possible to the nerve. The more complex operations to be considered are those with the hexagonal electrode. For this purpose, we have designed a seven channels ASIC. Each channel compute a weighted difference between the measurement point and the six closest surrounding points. This is done in the analog domain using the preamplifier shown on figure 5. This preamplifier is build around a differential pair whose negative input transistor was split into six transistors (six times smaller, of course). It has a voltage gain that is about 100 and it is followed by an instrumentation amplifier whose gain is configurable between 6 dB and 80 dB. Each channel is composed of one preamplifier followed by an instrumentation amplifier.

{6,3} V 3 V 0 V 1 V 2 {6,3} {4,4} V 4 V 2 V 6 V 0 V 1 V 1 V 0 V 2 V 3 V 4 V 5 V 6 {3,6}
This circuit was designed to give an input-referred noise below 1 µV rms , a CMMR above 60 dB and a sufficient gain, i.e greater than 60 dB ; all these parameters in the bandwidth of interest (1 Hz ≤ f ≤ 3 kHz). The performances expected for this amplifier are given in Table 1 (the noise is measured in the band 1 Hz-3 kHz).

A microphotography of the fabricated circuit is

NUMERICAL RESULTS

Action Potential modeling

In order to evaluate the performances of multipolar electrodes, we need a model for the extracellular electric field created by an action potential. Let us consider a 10 µm diameter myelinated axon. Its Ranvier nodes are 1 µm long, while their diameter is 6 µm and their spacing is 1 mm. Let us call Ω the center of the Ranvier node. When the AP is present at this node, we can model it as a 6 µm diameter circle, perpendicular to the axon axis, with a positive charge +q at its center (Ω) and a negative charge -q spread on the circle. The potential created at a point M of the space by this AP can be approximated by:

V (M) = qa 2 8πε 0 ε r r 3 1 - 3 2 sin 2 ψ (7)
In this expression, a is the radius of the Ranvier node (3 µm), r is the distance between Ω and M, while ψ is the angle between the axe of the axon and --→ ΩM. This approximation, valid for r ≫ a, is in good accordance with measurements. In particular, we can see that V (M) is negative for ψ = π/2 (Stein, 1980, page 81). Last, q can be easily estimated from the characteristics of the Ranvier node. For this study, we took q ≃ 20 fC and ε r ≃ 80.

The model given by equation 7 was used to evaluate the sensitivity of the electrodes to action potentials occurring inside the nerve. For the evaluation of the rejection of parasitic signals, we must first recall that EMG are also action potentials, creating the same kind of electric field. But, in this case, we cannot make any assumption on the value of ψ. So, to evaluate the external sensitivity of electrodes, we chosen to use only a 1/r 3 model, unable to give voltages, but sufficient to compare the sensitivities of various electrodes.

We have limited the numerical study to the comparison of the classical tripolar cuff with the heptapolar (hexagonal shape {3, 6}) electrode. We have also studied the effect of the wrapping to the performance of the heptapolar electrode: we consider a plane electrode and then a cylinder-wrapped one.

Given the position of a single AP we can easily calculate the induced potential on each pole of the cuff, since they are very small. For the tripolar cuff, we need to average the potential on each ring. This lead to an elliptic integral we have solved using numerical methods.

Tripolar and heptapolar electrodes models

In the following, we compare a tripolar cuff electrode, whose diameter is 2R = 3 mm and ring spacing is a = 4R, with one patch of the hexagonal cuff. To get comparable results, this hexagonal cuff has the same diameter (2R = 3 mm) and the spacing between poles is d = R. Since this patch is partially wrapped around the nerve, we considered also another patch perfectly flat.

For all the calculations, the coordinates were fixed as follow: the origin O is at the center of the cuff electrode. The Ox axis is the axis of the nerve (and, obviously, of the cuff). The Oy axis passes by the center of the considered patch (which is perpendicular to this axe). Last the Oz axe is placed to form a direct trihedron with Ox and Oy.

Internal sensitivity

Figure 7 shows the radial sensitivities of the three electrodes (tripolar cuff, planar hexagonal patch and wrapped hexagonal patch) that we compare. The vertical axis is the value of V rec (in dB µV) calculated for an AP placed on the Oy axis, at abscissa yR. The graph shows clearly that while the sensitivity of the tripolar cuff is quasi constant on the section of the nerve, the sensitivity of the hexagonal patch is far higher (up to 30 dB) when considering an AP located between the center of the patch and the center of the cuff. Figures 8 and9 show the longitudinal sensitivities of the three considered electrodes. On figure 8, the sensitivity is computed for an AP placed on the Ox axe, while, on figure 9, the AP is placed on a line, parallel to Ox, cutting Oy at abscissa 0.8R. On this later figure, we can see an increase of sensitivity of the tripolar cuff in the vicinity of the rings, but this remains far lower than the sensitivity of any of the hexagonal patches. The figure 10 show the external sensitivities of our three electrodes for an AP placed on the Ox or on the Oz axis of the electrode. As stated above, the quantity plotted is not a voltage, but is homogeneous to the reciprocal of the cube of a distance. Nevertheless, we can see on these two graphs that the hexagonal patches exhibit a better rejection of parasitic signals than the tripolar cuff. This improvement is of 32 dB for Ox and 20 dB for Oz.

External sensitivity

The same study conducted along the Oy axis (figure 11) shows that, while the planar patch continues to have the better rejection of parasitic signals, the wrapped hexagonal patch has a sensitivity decreasing slowly along this Oy axis. In fact, the bent hexagonal patch only begins to have larger sensitivity than the tripolar cuff for action potentials placed at more than fifty times the radius of the cuff, corresponding to approximately 7 cm. At this distance, the parasitic signal could be neglected in comparison to ENG signal.

CONCLUSION AND PERSPECTIVES

We have presented here a method to build multipolar cuff electrodes and how to extract useful informations from the multiple channels. Although numerical investigations are still necessary to an exhaustive comparison of multipolar structures, the comparison between the classical tripolar cuff electrode for ENG recording and a multipolar electrode has shown that this new type of the design is very promising. In every simulation, multipolar electrodes prove to be more sensitive to sources located inside the nerve, and in almost every case they show better far source rejection. We hope the improvement of the recorded signal given by this new design will allow the use of signal processing techniques such as source separation. Then, multipolar configurations could make it possible to estimate more precise parameters like the speed and the direction of propagation of the AP [START_REF] Taylor | Multipleelectrode nerve cuffs for low-velocity and velocityselective neural recording[END_REF][START_REF] Rieger | Very lownoise ENG amplifier system using CMOS technology[END_REF].

Figure 2 :

 2 Figure 2: Multipolar electrode cuff model.

Figure 3 :

 3 Figure 3: There are exactly three regular tessellations composed of regular polygons symmetrically tiling the plane.

Figure 4 :

 4 Figure 4: Three possible configurations of electrodes.

Figure 5 :

 5 Figure 5: Seven input preamplifier.

Figure 6 :

 6 Figure 6: Microphotograph of the seven-channel prototype.

Figure 7 :

 7 Figure 7: Radial sensitivities of (A) a tripolar cuff electrode, (B) a planar hexagonal patch and (C) a bent hexagonal patch. The vertical axis is in dB µV and the unit for the horizontal axis is the radius R of the electrode.

Figure 8 :

 8 Figure 8: Longitudinal sensitivities on the axe of (A) a tripolar cuff electrode, (B) a planar hexagonal patch and (C) a bent hexagonal patch. The vertical axis is in dB µV and the unit for the horizontal axis is the radius R of the electrode.

Figure 9 :

 9 Figure9: Longitudinal sensitivities on an off-center (80 % of R) axis of (A) a tripolar cuff electrode, (B) a planar hexagonal patch and (C) a bent hexagonal patch. The vertical axis is in dB µV and the unit for the horizontal axis is the radius R of the electrode.

Figure 10 :

 10 Figure 10: External relative sensitivity along Ox and Oz axes for (A) a tripolar cuff electrode, (B) a planar hexagonal patch and (C) a bent hexagonal patch. The vertical axis is in dB and the unit for the horizontal axis is the radius R of the electrode.

Figure 11 :

 11 Figure 11: External relative sensitivity along the two halves of the Oy axis for (A) a tripolar cuff electrode, (B) a planar hexagonal patch and (C) a bent hexagonal patch. The vertical axis is in dB and the unit for the horizontal axis is the radius R of the electrode.

Table 1 :

 1 Amplifier characteristics (simulation).

	Active area (7 channels)	1.16 mm 2
	Supply voltage	3.3 V
	DC Current (Preamp)	20 µA
	Voltage gain (Preamp)	100 (40 dB)
	CMRR (Preamp)	80 dB (10 kHz)
	Voltage gain (Inst amp)	2 ≤ G ≤ 10 000
	CMRR (Full amp)	80 dB (10 kHz)
	Input-ref. noise (Preamp)	0.672 µV RMS
	Input-ref. noise (Full amp) 0.677 µV RMS
	Bandwidth (Full amp)	76 kHz
	presented Fig. 6. This circuit was designed in CMOS
	AMS 0.35-µm technology.


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

Tim Laux
email: tim.laux@hcm.uni-bonn.de

MULLINS-SEKERKA AS THE WASSERSTEIN FLOW OF THE PERIMETER

Keywords: Gradient flows, Wasserstein distance, sets of finite perimeter, Mullins-Sekerka, free boundary problems, Hele-Shaw cell Math. Subject Classification: 35A15, 35R37, 49Q20, 76D27, 90B06, 35R35

We prove the convergence of an implicit time discretization for the one-phase Mullins-Sekerka equation, possibly with additional non-local repulsion, proposed in [F. Otto, Arch. Rational Mech. Anal. 141 (1998) 63-103]. Our simple argument shows that the limit satisfies the equation in a distributional sense as well as an optimal energy-dissipation relation. The proof combines arguments from optimal transport, gradient flows & minimizing movements, and basic geometric measure theory.

Introduction

The Mullins-Sekerka equation, see ( 1)-( 4) below for its exact formulation with additional non-local repulsion, is a well-studied mathematical model which, among other phenomena, describes a Hele-Shaw cell: A viscous ferro-fluid is confined to a thin region between two parallel horizontal plates. Applying a strong magnetic field in the vertical direction leads to two opposing forces: (i) due to surface tension, the fluid wants to decrease its surface area; (ii) the probe becomes magnetized by the field and the particles repel each other due to the induced magnetic field. These two competing effects lead to the formation of intriguing patterns.

In this paper we construct weak solutions using an implicit time discretization proposed by F. Otto in [START_REF] Otto | Dynamics of labyrinthine pattern formation in magnetic fluids: A mean-field theory[END_REF] for the Mullins-Sekerka equation, possibly with nonlocal repulsion. Because of the gradient-flow structure of the equation, it is natural to consider minimizing movements, an implicit time discretization which comes as a sequence of variational problems [START_REF] De | Movimenti minimizzanti[END_REF]. The effective energy consists of two terms, (i) an attractive term due to surface tension, the total surface area of the lateral boundary of the region occupied by the fluid, and (ii) a non-local term due to the magnetic repulsion of the particles; see [START_REF] De | Movimenti minimizzanti[END_REF] below. In [START_REF] Otto | Dynamics of labyrinthine pattern formation in magnetic fluids: A mean-field theory[END_REF] it has been observed that the dissipation functional may be modeled by the Wasserstein distance, which arises in optimal mass transport; see [START_REF] Chen | Global asymptotic limit of solutions of the Cahn-Hilliard equation[END_REF]. The Wasserstein distance plays a crucial role for many diffusion equations as was pointed out by Jordan, Kinderlehrer, and Otto in the seminal work [START_REF] Jordan | The variational formulation of the Fokker-Planck equation[END_REF], see also [START_REF] Jordan | Free energy and the Fokker-Planck equation[END_REF].

The main theorem of the present work is a refined version of the announced result [START_REF] Otto | Dynamics of labyrinthine pattern formation in magnetic fluids: A mean-field theory[END_REF]Theorem 1], for which a detailed proof was not provided. Our simple proof establishes the convergence of the approximations obtained from the minimizing movements scheme to a weak solution. We derive the Mullins-Sekerka equation ( 1)-( 4) in a distributional form, and using De Giorgi's variational interpolations [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF], we show that the limit satisfies an optimal energy-dissipation relation. The convergence of the 1 energies as h → 0, a well-known assumption known from the more difficult case of mean curvature flow [START_REF] Luckhaus | Implicit time discretization for the mean curvature flow equation[END_REF], is not necessary in our case. In fact, our proof is much simpler and no regularity theory of almost minimal surfaces is needed. Dropping the assumption of energy convergence, however, comes at the price of a weaker solution concept involving varifolds instead of sets of finite perimeter. We need of course to assume in addition the convergence of the time-integrated energies in order to recover the expected formulation for sets of finite perimeter. It may be expected that this solution concept, at least in the case of energy convergence, satisfies a weak-strong uniqueness principle similar to the ones in the forthcoming works by Fischer, Hensel, Simon, and one of the authors for multiphase mean curvature flow [START_REF] Fischer | The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions[END_REF] and for the simpler two-phase Mullins-Sekerka equation [START_REF] Fischer | Weak-strong uniqueness for the Mullins-Sekerka equation[END_REF].

There has been continuous interest in the Mullins-Sekerka equation and similar gradient flows, so we only briefly point out some of the most relevant results related to the present work. Weak solutions to the two-phase Stefan problem have been constructed by Luckhaus [START_REF] Luckhaus | Solutions for the two-phase Stefan problem with the Gibbs-Thomson law for the melting temperature[END_REF]. In particular, Luckhaus discovered a hidden variational principle satisfied by his approximations, which allows to verify the convergence of the energies as h → 0. Luckhaus and Sturzenhecker [START_REF] Luckhaus | Implicit time discretization for the mean curvature flow equation[END_REF] constructed weak solutions of mean curvature flow and the two-phase Mullins-Sekerka equation conditioned on the convergence of the energies. Röger [START_REF] Röger | Existence of weak solutions for the Mullins-Sekerka Flow[END_REF] was able to remove the assumption in the case of this two-phase Mullins-Sekerka equation by showing that the assumption may only be violated along flat parts of ∂E. In the case of mean curvature flow, the assumption can be verified in very particular cases, like convex sets [START_REF] Caselles | Anisotropic curvature-driven flow of convex sets[END_REF], graphs [START_REF] Logaritsch | An Obstacle Problem for Mean Curvature Flow[END_REF], and mean convex sets [START_REF] De | Implicit time discretization for the mean curvature flow of mean convex sets[END_REF]. For generalizations to the anisotropic case, which for mean curvarture flow has already been introduced by Almgren, Taylor, and Wang [START_REF] Almgren | Curvature-driven flows: a variational approach[END_REF], we refer the interested reader to Garcke and Schaubeck [START_REF] Garcke | Existence of weak solutions for the Stefan problem with anisotropic Gibbs-Thomson law[END_REF] and Kraus [START_REF] Kraus | The degenerate and non-degenerate Stefan problem with inhomogeneous and anisotropic Gibbs-Thomson law[END_REF]. A variant relevant for image denoising has been introduced by Carlier and Poon [START_REF] Carlier | On the total variation wasserstein gradient flow and the TV-JKO scheme[END_REF] who relax the constraint χ E ∈ {0, 1}, which leads to the total variation flow. However, it seems that the convergence can only be proven under an additional assumption on the density. Glasner [START_REF] Glasner | A diffuse interface approach to Hele-Shaw flow[END_REF] introduced a phase-field approximation to the one-phase Mullins-Sekerka equation and studied its convergence by formal asymptotic expansions. While the analysis of the non-degenerate Cahn-Hilliard equation to the two-phase Mullins-Sekerka equation is by now well-understood, see for example the work of Chen [START_REF] Chen | Global asymptotic limit of solutions of the Cahn-Hilliard equation[END_REF] and Alikakos, Bates, and Chen [START_REF] Alikakos | Convergence of the Cahn-Hilliard equation to the Hele-Shaw model[END_REF], there seems to be no result for this degenerate version. Recently, also the computationally efficient thresholding scheme by Merriman, Bence, and Osher [START_REF] Merriman | Diffusion generated motion by mean curvature[END_REF][START_REF] Merriman | Motion of multiple junctions: A level set approach[END_REF] has been reinterpreted as a minimizing movements scheme by Esedoglu and Otto [START_REF] Esedoglu | Threshold dynamics for networks with arbitrary surface tensions[END_REF], which allowed one of the author together with Otto to prove conditional convergence results to multiphase mean curvature flow [START_REF] Laux | Convergence of the thresholding scheme for multi-phase meancurvature flow[END_REF][START_REF] Laux | Brakke's inequality for the thresholding scheme[END_REF]. Most recently, Jacobs, Kim, and Mészáros [START_REF] Jacobs | Weak solutions to the muskat problem with surface tension via optimal transport[END_REF] introduced an interesting thresholding-type approximation for the Muskat problem and proved a similar (conditional) convergence result for their scheme.

The paper is organized as follows: In §2 we recall the minimizing movements scheme and state our main result, Theorem 1, which will be proved in the following sections: §3 establishes the compactness; in §4 we recover the distributional equation for the limit and the optimal energy-dissipation relation; and §5 contains a simple nonlinear interpolation inequality and its proof.

Acknowledgement

The authors would like to warmly thank Prof. Felix Otto, Max Planck Institute (MPI), Leipzig, for mentioning this problem, and for numerous discussions and suggestions. This work was started during Mathematisches Forschungsinstitut Oberwolfach's (MFO) workshop #1746 and completed during MFO's workshop #1904, as well as visits to MPI Leipzig. The hospitality and support of these institutions is also acknowledged. This project was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy -EXC-2047/1 -390685813.

Statement of the main result

The physical model under consideration is described by the following system of equations for an evolving set (E(t)) t∈(0,∞) and a velocity field u :

R d × (0, ∞) → R d : The interface ∂E(t) is transported by the fluid (1) V = u • ν on ∂E(t)
(throughout, ν denotes the outer unit normal to ∂E(t)); the fluid is incompressible

(2) div u = 0 in E(t); the flow is irrotational, i.e., (3)
there exists p such that u = -∇p in E(t);

and on the interface, the following balance-of-forces condition holds

(4) p = H + 2k * χ E(t) on ∂E(t),
where H denotes the mean curvature of ∂E(t) with the sign convention H > 0 for convex E(t) and k is a non-negative, symmetric, and normalized convolution kernel k ≥ 0, k(-z) = k(z), and ´k = 1.

Since the velocity field u is divergence-free, any smooth solution E(t) is volumepreserving d dt |E| = 0. More importantly, combining (1)-( 4), we see that any such evolution is energy dissipating

(5) d dt H n-1 (∂E(t)) + ˆE(t) k * χ E(t) dx = - ˆE(t) |u| 2 dx ≤ 0.
More precisely, the above set of equations have a gradient-flow structure.

Here, the metric tensor ´E |u| 2 dx defined on divergence-free vector fields u : E → R d is less degenerate than the one of the mean curvature flow ´∂E V 2 dS defined on normal velocities V , but more degenerate than the one of the two-phase Mullins-Sekerka problem ´Rd |u| 2 dx, in which the ferro-liquid is assumed to be surrounded by another liquid of the same viscosity.

For more physical motivation, we refer to the introduction of [START_REF] Otto | Dynamics of labyrinthine pattern formation in magnetic fluids: A mean-field theory[END_REF] and the references therein.

Let us recall the implicit time discretization introduced by F. Otto in [START_REF] Otto | Dynamics of labyrinthine pattern formation in magnetic fluids: A mean-field theory[END_REF]: Given a time-step size h > 0, and initial conditions E 0 ⊂ R d , for n ≥ 1, find E n solving [START_REF] Caselles | Anisotropic curvature-driven flow of convex sets[END_REF] min

E 1 2h W 2 2 (χ E , χ En-1 ) + P (E) + ˆE k * χ E dx .
Here P (E) := sup{-´E div ξ dx : sup |ξ| ≤ 1} denotes the perimeter of E ⊂ R d and ( 7)

W 2 2 (χ E , χ F ) = inf ˆE |x -T (x)| 2 dx = min ¨|x -y| 2 dγ(x, y)
denotes the squared Wasserstein distance, where the infimum runs over all transport maps, i.e., volume preserving diffeomorphisms T : R d → R d such that T χ E (x)dx = χ F (y)dy, and the minimum runs over all transport plans, i.e., finite measures γ in R d × R d with marginals χ E (x)dx and χ F (y)dy. Note that the Wasserstein distance in the minimization term automatically enforces the volume constrained

|E n | = |E n-1 |,
just like the system of partial differential equations ( 1)-( 4) above.

We denote the total energy by ( 8)

E(E) := P (E) + ˆE k * χ E dx,
and let E h (t) := E [t/h] for t ≥ 0. Our standing assumption on the initial conditions is ( 9)

P (E 0 ) < ∞ and ˆE0 (1 + |x| 2 ) dx < ∞.
In particular |E 0 | < ∞ and w.l.o.g. by scaling we may assume that

|E 0 | = 1.
The main result is the following construction of solutions.

Theorem 1. Let E 0 ⊂ R d be initial conditions satisfying [START_REF] De | Implicit time discretization for the mean curvature flow of mean convex sets[END_REF] and let E h (t) be constructed as above. Then there exists a subsequence h ↓ 0, an L 1 -continuous one-parameter family of finite perimeter sets E(t) satisfying (9), and a vector field

u ∈ L 2 (R d × (0, +∞); R d ) such that lim h↓0 sup t∈[0,T ] |E h (t) E(t)| = 0 for all T < +∞ and (10) - ˆ+∞ 0 ˆE(t) (∂ t ζ + u • ∇ζ) dx dt = ˆE0 ζ(0) dx for all ζ ∈ C ∞ 0 (R d × [0, +∞))
, and E(t) satisfies the optimal energy dissipation rate

(11) E(E(T )) + ˆT 0 ˆE(t) |u(x, t)| 2 dx dt ≤ E(E 0 ) for all T > 0.
The measures

µ h := δ ν E h (t) ⊗ ∇χ E h (t) dt converge, µ h µ = µ t dt, to an oriented integral varifold µ, i.e., a non-negative measure on (ν, x, t) ∈ S d-1 × R d × [0, +∞), which satisfies the compatibility condition (12) -∇χ E(t) = ν E(t) |∇χ E(t) | = ˆSd-1 ν µ t (dν,
• ) in the sense of measures and in particular ∇χ E(t) ≤ ´Sd-1 µ t (dν, • ). Here and throughout, ν

E(t) = - ∇χ E(t) |∇χ E(t) |
denotes the (measure theoretic) outer unit normal of E(t). Furthermore, the tuple (E, u, µ) satisfies the distributional equation

- ˆ∞ 0 ˆE(t) u • ξ dxdt = ˆ∞ 0 ˆˆ(div ξ -ν • Dξ ν) dµ t (ν, x) dt + 2 ˆ∞ 0 ˆk * χ E(t) ξ • ν E(t) ∇χ E(t) dt (13) for all ξ ∈ C ∞ 0 (R d × (0, +∞), R d )
with div ξ = 0, as well as the optimal energy dissipation relation

µ T (S d-1 × R d ) + ˆE(T ) k * χ E(T ) dx + 1 2 ˆT 0 ˆE(t) |u(x, t)| 2 dx dt + ˆT 0 ˆˆ(div ξ -ν • Dξ ν) dµ t (ν, x) dt + 2 ˆT 0 ˆk * χ E(t) ξ • ν E(t) ∇χ E(t) dt - 1 2 ˆT 0 ˆE(t) |ξ| 2 dx dt ≤ E(E 0 ) (14)
for almost all T < +∞ and all ξ ∈ C ∞ 0 (R d × (0, +∞), R d ) with div ξ = 0. Remark 1. The system of equations derived in the theorem is indeed a weak form of the free boundary problem ( 1)-( 4) provided the sets E(t) are (essentially) open:

(i) The continuity equation ( 10) encodes ( 1) & (2) as well as the initial conditions E 0 . (ii) Under the assumption that no hidden boundary lies inside of E(t), i.e., spt µ t ∩ E(t) = ∅, Equation (13) encodes both (3) (since [START_REF] Garcke | Existence of weak solutions for the Stefan problem with anisotropic Gibbs-Thomson law[END_REF] says that u is orthogonal to divergence-free fields in E(t)) and the balance of forces (4) on the free boundary. (iii) The last three left-hand side terms involving the test vector field ξ in ( 14) can be viewed as a fractional Sobolev norm of H + 2k * χ E , and ( 14) is a type of De Giorgi inequality, which for a smooth gradient flow characterizes the solution.

Remark 2. Note also that we may replace the sum of the first two left-hand side terms in [START_REF] Glasner | A diffuse interface approach to Hele-Shaw flow[END_REF] by the (smaller) energy E(E(T )). Let us assume for a moment that the energies converge as h → 0, which in view of the continuity of the non-local term and the lower semi-continuity of the perimeter is equivalent to saying that the perimeters do not drop down as h → 0, i.e., lim sup h↓0

ˆT 0 P (E h (t)) dt ≤ ˆT 0 P (E(t)) dt.
Then we may replace the measure µ t by the BV -version δ ν E(t) ⊗ ∇χ E(t) in all terms appearing in ( 13) & [START_REF] Glasner | A diffuse interface approach to Hele-Shaw flow[END_REF]. Indeed, in that case, the convergence of the curvature term

lim h↓0 ˆT 0 ˆ div ξ -ν E h (t) • Dξν E h (t) |∇χ E h (t) | = ˆT 0 ˆ div ξ -ν E(t) • Dξν E(t) |∇χ E(t) |
follows directly from Reshetnyak's continuity theorem, see e.g. [START_REF] Maggi | Sets of finite perimeter and geometric variational problems: an introduction to Geometric Measure Theory[END_REF]Theorem 20.12].

Remark 3. The more precise structure of the varifold µ t is not clear. The integrality (or even rectifiability) of µ t away from supp |∇χ E(t) | does not simply follow from curvature bounds and the control in time on the sets E h (t). Indeed, it is easy to construct counterexamples for which µ t oscillates in time but both estimates are valid.

Remark 4. The optimal energy-dissipation rate (here in form of ( 11) or ( 14)) plays a crucial role in recent weak-strong uniqueness proofs and does not follow from the weak formulation ( 10) & ( 13).

In the following we write A B if there exists a generic constant C = C(d) such that A ≤ C B.

Compactness

Lemma 1 (Compactness). Suppose E 0 satisfies [START_REF] De | Implicit time discretization for the mean curvature flow of mean convex sets[END_REF] and let E h be constructed by the scheme as above. Then

(15) W 2 (χ E h (t) , χ E h (s) ) E(E 0 ) 1 2 (t -s) 1 2
and

(16) |E h (t) E h (s)| E(E 0 ) 3 4 (t -s) 1 4
for all t > s ≥ 0 with t -s ≥ h. Therefore, there exists a subsequence h ↓ 0 and a one-parameter family of finite perimeter sets (E(t)) t≥0 such that for any T < +∞ [START_REF] Jordan | The variational formulation of the Fokker-Planck equation[END_REF] sup

t∈[0,T ] |E h (t) E(t)| → 0 as h ↓ 0.
Furthermore, the limit satisfies

(18) W 2 (χ E(t) , χ E(s) ) E(E 0 ) 1 2 (t -s) 1 2
and

(19) |E(t) E(s)| E(E 0 ) 3 4 (t -s) 1 4
for all t > s ≥ 0.

Proof. Using E n-1 as a competitor in (6) yields 1 2h W 2 2 (χ En , χ En-1 ) + E(E n ) ≤ E(E n-1 ), so that after summation in n and telescoping [START_REF] Laux | Brakke's inequality for the thresholding scheme[END_REF] h

2 n1 n=n0+1 W 2 (χ En , χ En-1 ) h 2 + E(E n1 ) ≤ E(E n0 ).
In particular, for any pair of integers n 1 > n 0 ≥ 0, we have

W 2 (χ En 0 , χ En 1 ) ≤ (n 1 -n 0 )h n1 n=n0+1 1 h W 2 (χ En , χ En-1 ) 2 1 2 ≤ (n 1 -n 0 )h 2(E(E n0 ) -E(E n1 )),
which implies [START_REF] Jacobs | Weak solutions to the muskat problem with surface tension via optimal transport[END_REF]. The L 1 estimate ( 16) then follows from [START_REF] Jacobs | Weak solutions to the muskat problem with surface tension via optimal transport[END_REF] in conjunction with the interpolation inequality in Corollary 1 in Section 5 below and Jensen's inequality in the form of

W 1 (χ, χ) ≤ W 2 (χ, χ).
The energy estimate (20) also yields a uniform bound on the perimeter, hence

ˆ|χ E h (t) (x + z) -χ E h (t) (x)| dx ≤ |z|P (E h (t)) ≤ |z|E(E 0 ),
i.e., we have a uniform modulus of continuity in space. Together with the uniform modulus of continuity in time [START_REF] Jordan | Free energy and the Fokker-Planck equation[END_REF], which is valid down to scales h, this allows us to apply the Riesz-Kolmogorov compactness theorem in L 1 ([0, T ] × K) for any compact set K ⊂ R d and any T < +∞. A diagonal argument yields χ

E h → χ E in L 1 loc ([0, ∞)× R d ).
But since ´|x| 2 χ E h (t) (x)dx < ∞, which follows from ( 9) & ( 15), this implies the L 1 -convergence globally in space, and locally in time. Eventually, an Ascoli-Arzelà type argument, together with the estimate in Corollary 1, allows to deduce the local uniform convergence in time, i.e., [START_REF] Jordan | The variational formulation of the Fokker-Planck equation[END_REF]. The continuity estimates ( 18) & ( 19) then follow immediately.

Convergence

Proof of Theorem 1.

Step 1: Construction of u and verification of [START_REF] Esedoglu | Threshold dynamics for networks with arbitrary surface tensions[END_REF]. By Kantorovich duality

1 2h W 2 2 (χ En , χ En-1 ) = sup φ(x)+ψ(y)≤ |x-y| 2 2h ˆEn φ(x) dx + ˆEn-1 ψ(y) dy.
This supremum is reached at (φ n , ψ n ) such that

Φ n (x) = |x| 2 2 -hφ n (x), Ψ n (y) = |y| 2 2 -hψ n (y)
are convex conjugates and (∇Φ n ) χ En = χ En-1 solves the optimal transportation problem ( 7) defining the distance W 2 (χ En , χ En-1 ), see [START_REF] Brenier | Polar factorization and monotone rearrangement of vector-valued functions[END_REF]Theorem 1.3] or [29, Theorem 2.12]. Hence

(21) 1 2h W 2 2 (χ En , χ En-1 ) = h ˆEn |∇φ n (x)| 2 dx = h ˆEn-1 |∇ψ n (y)| 2 dy
and by ( 20)

(22) h n1 n=n0+1 ˆEn |∇φ n (x)| 2 dx ≤ E(E n0 ) -E(E n1 ) ≤ E(E 0 ),
that is, if we set u n := χ En ∇φ n and u h (t) := u [t/h] , then u h is uniformly bounded in L 2 . Let u = u(x, t) be a weak limit. Since (Id -h∇φ n ) χ En = χ En-1 , for η(x, t) a smooth test function

1 h ˆRd (χ En -χ En-1 ) η dx = 1 h ˆRd χ En (x) (η(x) -η(x -h∇φ n (x))) dx.
Using Taylor's theorem in the form η(x) -η(x -hξ) -hξ • ∇η(x) ≤ h 2 2 |ξ| 2 sup x |∇ 2 η|, we can replace the right-hand side by ˆRd ∇η(x) • ∇φ n (x)χ En (x)dx at the expense of the error

1 h h 2 2 sup |∇ 2 η| ˆRd |∇φ n | 2 χ En dx = sup |∇ 2 η| 1 2h W 2 2 (χ En , χ En-1 ).
After integration in time, this error term vanishes as h ↓ 0 because of (20) and we may pass to the limit in the time-integrated version of the above identity to obtain the continuity equation in form of (10).

Step 2: De Giorgi's interpolation and argument for [START_REF] Fischer | The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions[END_REF]. De Giorgi's variational interpolation [START_REF] Luckhaus | Implicit time discretization for the mean curvature flow equation[END_REF] Ẽh ((n -1)h + t) ∈ arg min

E 1 2t W 2 2 (χ E , χ En-1 ) + E(E)
satisfies the identity

h 2 W 2 (χ En , χ En-1 ) h 2 + 1 2 ˆnh (n-1)h W 2 (χ Ẽh (t+(n-1)h) , χ En-1 ) t 2 dt ≤ E(E n-1 ) -E(E n ). (24)
Although the proof is contained-in a more general context-in [3, Theorem 3.1.4], we repeat it here for the reader's convenience. W.l.o.g. we may assume n = 1; for notational convenience we also drop the index h for this short argument. Defining momentarily

f (t) := 1 2t W 2 2 (χ Ẽ(t) , χ E0 ) + E( Ẽ(t))
to be the minimal value in the variational problem [START_REF] Luckhaus | Implicit time discretization for the mean curvature flow equation[END_REF], we may compute for s < t, using the minimality of Ẽ(t),

f (t) -f (s) ≤ 1 2t W 2 2 (χ Ẽ(s) , χ E0 ) + E(E(s)) - 1 2s W 2 2 (χ Ẽ(s) , χ E0 ) -E(E(s)) = s -t 2st W 2 2 (χ Ẽ(s) , χ E0 ).
Since s < t, this implies

f (t) -f (s) t -s ≤ - 1 2st W 2 2 (χ Ẽ(s) , χ E0 ) → - 1 2t 2 W 2 2 (χ Ẽ(t) , χ E0 ) as s ↑ t.
The analogous reverse inequality may be obtained by using s > t in the above argument with the roles of s and t interchanged. Hence f is locally Lipschitz in (0, h]

with d dt f (t) = - 1 2t 2 W 2 2 (χ Ẽ(t) , χ E0 )
for almost every t ∈ (0, h). For ε > 0, integrating this inequality from t = ε to t = h, and then using lower semicontinuity w.r.t. the L 1 convergence E(ε) → E 0 yields [START_REF] Maggi | Sets of finite perimeter and geometric variational problems: an introduction to Geometric Measure Theory[END_REF]. Summing [START_REF] Maggi | Sets of finite perimeter and geometric variational problems: an introduction to Geometric Measure Theory[END_REF] over n from n 0 + 1 to n 1 and telescoping the right-hand side we obtain the sharp energy dissipation inequality:

h 2 n1 n=n0+1 W 2 (χ En , χ En-1 ) h 2 + 1 2 ˆn1h n0h W 2 (χ Ẽh (t) , χ E h (t) ) t -h[t/h] 2 dt ≤ E(E n0 ) -E(E n1 ). (25)
By [START_REF] Logaritsch | An Obstacle Problem for Mean Curvature Flow[END_REF] we have

h 2 N n=1 W 2 (χ En , χ En-1 ) h 2 = h 2 N n=1 ˆ|χ En ∇φ n | 2 dx, which implies (26) ˆT 0 ˆE(t) |u(x, t)| 2 dx dt ≤ lim inf h→0 h 2 N n=1 W 2 (χ En , χ En-1 ) h 2 since χ En ∇φ n = u n and u h u in L 2 .
Following the same strategy as in Step 1, we can show that ũh := χ Ẽh (t) ∇ φh (t) with

x-(t-h[t/h])∇ φh (x, t) optimal in W 2 (χ Ẽh (t) , χ E h (t) )-after passage to a subsequenceweakly converges to the same limit u = w -lim h↓0 u h . In particular, as before,

1 t -h[t/h] W 2 2 (χ Ẽh (t) , χ E h (t) ) = (t -h[t/h]) ˆ|∇ φ| 2 dx
so that after division by (t -h[t/h]) and integration in t

1 2 ˆT 0 W 2 (χ Ẽh (t) , χ E h (t) ) t -h[t/h] 2 dt = 1 2 ˆT 0 ˆ|∇ φh | 2 dx dt = 1 2 ˆT 0 ˆ|ũ h | 2 dx dt,
which is again lower semi-continuous. This concludes the argument for [START_REF] Fischer | The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions[END_REF] for a.e. T > 0. Now let T > 0 be arbitrary and let T n → T such that [START_REF] Fischer | The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions[END_REF] holds for each

T n . Since E(T n ) → E(T ) in L 1
, the first term on the left-hand side of ( 11) is lowersemicontinuous. The second one is clearly continuous in T so the inequality holds for all T > 0. We will prove the refined energy dissipation inequality [START_REF] Glasner | A diffuse interface approach to Hele-Shaw flow[END_REF] later on in Step 4.

Step 3: Derivation of (13). The Euler-Lagrange equation of the minimization problem ( 6) reads ( 27) -

ˆEn ∇φ n • ξ dx = ˆ(div ξ -ν En • Dξν En + 2k * χ En ξ • ν En ) |∇χ En |
for all smooth test vector fields ξ with div ξ = 0, where ν En = -

h = µ h t dt = δ ν E h (t) (x) ⊗ ∇χ E h (t)
dt are bounded (with moreover µ h (S d-1 × R d × I) ≤ P (E 0 )|I| for any I ⊂ (0, +∞) measurable), by Banach-Alaoglu, they have a weak- * limit µ = µ t dt (after passage to a subsequence). Hence we can identify the limit of the first right-hand side term in [START_REF] Otto | Dynamics of labyrinthine pattern formation in magnetic fluids: A mean-field theory[END_REF] as well:

lim h↓0 ˆ∞ 0 ˆ(div ξ -ν E h • Dξ ν E h )|∇χ E h | dt = lim h↓0 ˆ∞ 0 ˆˆ(div ξ -ν • Dξ ν) dµ h t (ν, x)dt = ˆ∞ 0 ˆˆ(div ξ -ν • Dξ ν) dµ t (ν, x)dt for any test vector field ξ ∈ C ∞ 0 (R d × (0, +∞), R d ).
Step 4: Proof of the optimal energy dissipation relation [START_REF] Glasner | A diffuse interface approach to Hele-Shaw flow[END_REF]. The local slope of E, defined via

|∂E(E)| := lim sup F →E (E(E) -E(F )) + W 2 (χ E , χ F ) ,
where the convergence of the sets F → E is to be understood with respect to W 2 , satisfies

|∂E|( Ẽ((n -1)h + t)) ≤ W 2 (χ Ẽ((n-1)h+t) , χ En-1 ) t ,
cf. [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF]Lemma 3.1.3]. Applying this to (25) yields the sharp energy dissipation inequality

h 2 n1 n=n0+1 W 2 (χ En , χ En-1 ) h 2 + 1 2 ˆn1h n0h |∂E| 2 ( Ẽh (t)) dt ≤ E(E h (n 0 h)) -E(E h (n 1 h)). ( 28
)
Our goal is to pass to the limit in [START_REF] Röger | Existence of weak solutions for the Mullins-Sekerka Flow[END_REF]. We have already done this for the metric term in Step 2. Rewriting the surface energy as

P (E h (t)) = µ h t (S d-1 × R d )
, it is clear that the right-hand side terms converge to the desired limits. It remains to show

ˆT 0 ˆ(div ξ -ν • Dξ ν) dµ t (ν, x)dt + ˆT 0 ˆ2k * χ E(t) ξ • ν E(t) |∇χ E(t) | dt - 1 2 ˆT 0 ˆE(t) |ξ| 2 dx dt ≤ lim inf h→0 1 2 ˆT 0 |∂E| 2 ( Ẽh (t)) dt (29)
for all test vector fields ξ with div ξ = 0. Given an arbitrary set of finite perimeter E satisfying (9), any smooth divergence free vector field ξ provides a one-parameter family of candidates for the lim sup in the definition of the local slope |∂E|(E) at E via the inner variations ∂ s χ Es + ξ • ∇χ Es = 0. Using the elementary relation 0

≤ 1 2 ( a b -b) 2 = 1 2 ( a b ) 2 -a + 1 2 b 2 , this yields 1 2 |∂E| 2 (E) ≥ lim s→0 1 2 1 s (E(E) -E(E s )) + 1 s W 2 (χ E , χ Es ) 2 ≥ lim s→0 1 s (E(E) -E(E s )) + - 1 2s 2 W 2 2 (χ E , χ Es ).
On the one hand, since ξ is divergence free, it generates one particular volumepreserving flow from E to E s . More precisely, the rescaled field sξ solves ∂ s χ E s + div(sξχ E s ) = 0 and transports E to E s in one unit of time and hence provides a particular candidate for the minimum problem in W 2 :

W integrating in t, and taking the limit h → 0 yields (29). Proof. We first consider, f, g smooth with compact support, a symmetric mollifier ρ, and σ > 0. We have ˆf g dx = ˆg (ρ σ * f ) dx + ˆg (f -ρ σ * f ) dx.

An interpolation inequality

For the first integral we use |∇(ρ σ * g)| ≤ ( ∇ρ L 1 /σ) g L ∞ , hence by symmetry of ρ:

ˆg (ρ σ * f ) dx = ˆf (ρ σ * g) dx ≤ C 1 σ f W -1,1 g L ∞ ,
with C 1 = ∇ρ L 1 . For the second integral, we write that for

f ∈ BV (R d ) ∩ C 1 (R d ), ˆ|f -ρ σ * f |dx = ˆ ˆˆσ 0 ρ(ξ)ξ • ∇f (x -tξ) dt dξ dx ≤ C 2 σ ∇f L 1 ,
where C 2 = ´|ξ|ρ(ξ)dξ. We deduce that ˆg(f -

ρ σ * f ) dx ≤ C 2 σ g ∞ |Df |(R d ).
Hence for any σ > 0

ˆf g dx ≤ g L ∞ C 1 σ f W -1,1 + C 2 σ|D(u -v)|(R d )
so that (minimizing the right-hand side with respect to σ > 0)

ˆf g dx ≤ 2 g L ∞ C 1 C 2 f W -1,1 |Df |(R d ).
Choosing g a mollification of sign f and passing to the limit it follows that

ˆ|f | dx ≤ 2 C 1 C 2 f W -1,1 |Df |(R d ).
This extends to f ∈ BV (R d ) such that f -1,1 = sup |∇g|<1 ´f g dx < +∞ and ´|x||f (x)|dx < +∞, by approximation.

Since χ E -χ F W -1,1 = W 1 (χ E , χ F ) we have the following immediate consequence of Lemma 2.

Lemma 2 .

 2 There exists C > 0 such that for u, v ∈ BV (R d ) with ´(|u| + |v|)|x|dx < +∞, one has u -v 2 L 1 ≤ C u -v W -1,1 |D(u -v)|(R d ).

Corollary 1 .

 1 For any sets E, F ⊂ R d with finite perimeter and with ´E∪F |x|dx < +∞ |E F | P (E) + P (F ) W 1 (χ E , χ F ).

  E h (t) ξ • ν E h (t) ∇χ E h (t) dt = -ˆ∞ 0 ˆ2k * χ E h ξ • ∇χE h converges since ∇χ E h → ∇χ E weakly as measures, and since k * χ E h (t) converges uniformly, which follows from the strong L 1 convergence χ E h → χ E and the observation that sup |k * χ -k * χ| ≤ ˆ|χ -χ| dx for any two characteristic functions χ, χ for which only the integrability ´k = 1 and non-negativity k ≥ 0 are needed. Since the measures µ

	∇χ En |∇χ En | denotes the
	outer normal.
	The non-local term ˆ∞ 0 ˆ2k * χ

  2 (χ E , χ Es ) ≤ ˆ1 0 ˆEs |sξ| 2 dx ds = s 2 ˆE |ξ| 2 dx + o(s 2 ). = -s ˆ(div ξ -ν E • Dξ ν E ) |∇χ E | -2s ˆk * χ E ξ • ν E |∇χ E | + o(s).Therefore, for any E satisfying (9), and any test vector field field ξ with div ξ = 0 it holds 1 2|∂E| 2 (E) ≥ -ˆ(div ξ -ν E • Dξ ν E ) |∇χ E | -2 ˆk * χ E ξ • ν E |∇χ E | -1 2 ˆE |ξ| 2 dx.(30)Replacing ξ by -ξ and applying this argument to E h (t), writing the first integral in terms of µ h t asˆ div ξ -ν E h (t) • Dξ ν E h (t) |∇χ E

	On the other hand, we have
	(E(E) -E(E s )) + ≥ E(E) -E(E s )
	= -s	d ds s=0	E(E s ) + o(s)

h (t) | = ˆ(div ξ -ν • Dξ ν) dµ h t (x, ν),