 you kind of some numerical simulations of this thresholding scheme so that you get a better intuition of it. So here again is the scheme in its simplest version. So it's a time discretization of mean curvature flow and it proceeds by kind of looking at the characteristic function of the set, the boundary of which evolves by mean curvature and the time step size is h like what I did on the blackboard. So you're given the characteristic function of the set at time step n minus one, you convolve it with the heat kernel at time h which means I could also say you solve the heat equation with these initial data for time h so now you get a smooth function and then you look where this resulting function is less or larger than one half and that defines a new characteristic function or a new set so that's the that's the scheme and here is here is a numerical simulation so oops so here is a dumbbell shaped set at time step n minus one you blur the picture by convolving you find the level set of where the level set of the function of one half and that defines the new set and you can detect that it's slightly behaving as it should be kind of this this thin part has thickened a little bit as you would expect for mean curvature flow and as I mentioned last time that's a low complexity scheme because convolution can be done very efficiently after spatial discretization with almost optimal complexity and with not difficult programming by the fast Fourier transform that's what I mentioned last time that it satisfies the comparison principle and that's the basis for convergent proofs and here is the multi-phase version and and in a certain sense that's what makes this interesting that exactly essentially the same idea the same idea kind of works for multi-phase versions so you're describing a partition of your space by the corresponding characteristic functions that sum up to one you do exactly the same thing as before you do a convolution step where you convolve all the characteristic functions and then you ask where which of the characteristic functions is largest in a given point and and here you see kind of a purely academic numerical simulation where you you have this fairly irregular partition you modify it you ask for which which of the these three functions dominates you get a new arrangement and so on and you can already see that after a single time step you get the right angle condition for free which here in this case is just that all three angles have to be equal and and that makes it so in fact mean curvature was introduced in material science and and this is a kind of the fact that you can use this for many many phases makes it interesting for what's called grain growth in material science so in three-dimensional version of networks and and these three authors have demonstrated that you can really kind of simulate several thousands of grains in in three space dimensions and can do statistics on and but the multi-phase case in a certain sense I mean there's only this recent result by last week's speakers and this week's participant that that at least in the sense of bracket a solution exists okay so this this is was let me briefly recap what we did yesterday so this was I started with telling you about this minimizing movement work of the Georgie which by the right type of interpolation recovers the the exact or the energy the dissipation inequality without loss and and then I formulated two results the first one definitely I want to give you the argument for that this minimizing movement scheme is indeed sorry that the thresholding scheme is indeed a minimizing movement scheme and that this minimizing movement scheme has to do with mean curvature flow in the sense in the one sense I mean before you before proving a convergence result in the sense that the energy functional converges to the perimeter so the area of the interface written in this bv type fashion so those were the two results which I already stated last time so let me give you the argument for at least the first one so so let's look at this functional one has to minimize and so that's the energy e h plus one over two h square d h square u chi n minus one and let's plug in the definitions so that's one over square root of h the integral of u one minus u times g h convolved with u plus one over square root of u u minus chi g h u minus chi so that's I just copied what's what's on the blackboard and now you have to now there is a simple algebraic transformation which you see best by realizing that this expression here defines a symmetric bilinear form so let me just use this notation so this is one minus u comma u plus u minus chi comma u minus chi and if you use kind of elementary linear algebra you see that this can be rewritten as minus u two chi minus one plus chi chi I hope that's correct so this term is sitting here the quadratic term in u cancels with the quadratic term in u here the quadratic term in chi is sitting here and the mixed term between u and chi where the minus sign is sitting here so that's just linear algebra so what does that mean that means we have minus the integral of u g h convolved with two chi minus one plus chi g h convolved with chi this second term is completely irrelevant for this variation problem since it does not depend on you only the first term is relevant for the variation problem in fact it's a you know linear term and using the fact that the Gaussian has integral one I can rewrite this as the convolution of two times g h minus one if if we do this we see two things we see that so we want to minimize this expression so we want to minimize this expression which means we want to maximize this expression which means we can do that if we choose u to be equal to one where this is positive and equal to zero where this is negative so in this case we get that this is equal to minus two times g h chi minus one if u is equal to the characteristic function of the set where the convolution is larger than one-half but but that's really the smallest value this expression can have for any u remember that u is a function which assumes the values between zero and one that's really kind of the the the smallest value it can ever assume so it's this is equal to this value but it's always at least as large as this value and therefore we see that this function here indeed is the minimizer and that was just thresholding so so by very simple calculation you see that that thresholding indeed is is has the interpretation of solving this this almost trivial variational problem so so that's that's certainly very very easy argument and now kind of the work consists in in using this structure and the first the first part of the work kind of consists in understanding the energy functional and that's proposition one which tells you why the energy functional indeed has to do something with with what you would expect for mean curvature flow which is the gradient flow of the interfacial energy because this energy functional as this parameter h converges to zero converges to the parameter function and and this this lemma essentially sorry this proposition essentially relies on let's say for the most interesting part relies on the following lemma so that should be lemma three which i want to state now so for any characteristic function we have the following three statements we have that this functional this this energy functional which is defined up there indeed is always bounded by a buff by the parameter functional and we have monotonicity in the sense that as this parameter so this holds in a point wise point wise in the configuration for any natural number n so so if this of course is not really full monotonicity in this parameter n here because it's some kind of integer monotonicity but in a certain sense what these two what these two inequalities tell you is that for fixed configuration these energy functional monotonically converge to the parameter functional and and that's as people who are a little bit familiar with gamma convergence will know is already kind of a main step for the for the gamma convergence essentially that's that's essentially all all it relies I mean one needs one needs the consistency one also needs to know that for fixed chi e of h chi actually converges to this value as h goes to zero that requires a kind of a little bit of the approximation is a classical argument but in fact these two inequalities are the most interesting part and then there is a third property which I'm going to use later which I want to state here that again for an arbitrary configuration if you look at the l1 norm of the gradient of the convolution it's estimated up to some constant by so the energy even for finite h is at least strong enough to control the bv norm of the convolved configuration so those are all three statements for fixed configuration let me let me let me show you the proof at least of of the first two ones and then we'll see how much you want to see so so let's start with with the first the first inequality so I'm going to write down the definition of the energy functional and I'm using something which I already used over there that this is kind of symmetric in these two arguments so I can also write it like this and now I'm using for beanie in the sense that I this is an integral in x and if I'm not saying anything it's always in x when it's integrated in x it's always over the unit cell but now there is a second integral in here which is which comes from the convolution and I'm exchanging the orders of integration and pull the convolution integral outside so therefore I get the integral over the entire rd so integrals over z are always to be supposed implicitly supposed to be over the entire rd and now here we have the unit cell and we get 1 minus chi times x chi times x minus z plus chi times x 1 minus chi x minus z dx dz and I'm just I've just rewritten the the functional and now you realize that this expression because chi is a characteristic function this expression is nothing else than the the difference the modulus of the difference of chi at x and chi at x minus z and now because of periodicity so here you're looking you're looking in at an l1 modulus of continuity of your characteristic function chi you're looking at the l1 difference between chi and shifted chi and therefore by if you want the mean value property you can estimate that by the l1 norm of the gradient and if you think a little bit about it what you get is as an intake as an integrand you get the direction in which you shift times the normal vector so nu is like always in the bv context the measure theoretic normal and so that's the estimate which you get and now you I mean here you here we used for beanie and kind of exchanging and pulling the x integration outside now you pull the z integration outside like it was originally sorry the so we pulled the z integration outside now we pulled in and you see that you get so I should write it like this we get this expression and so now we have to look at we have to look at this at this integral here so first of all by rotational symmetry of the Gaussian you realize that it does not depend on in which on which direction this unit vector points so could just take the without loss of generality I could take the first the first direction so that's using the rotational symmetry now I'm changing variables recalling that the Gaussian of variance h was nothing else than the Gaussian of variance 1 rescaled by square root of h so if I change variables and I bring this in here you realize that this is just the integral of the unit Gaussian times the modulus of the first component and now you use the fact that the Gaussian in fact is something which is tensorial so it's the one-dimensional Gaussian in the z1 variable times the d minus one-dimensional Gaussian in the other variables and you use for beanie here you write that as an integral in z1 integral and d z prime and the integral in in z prime that is equal to one because of the normalization of the Gaussian so this in fact is let me continue here is just a one-dimensional integral of the 1d Gaussian of unit variance z1 d z1 and I forgot somewhere the factor of one-half which now is which came in here which now is important because I can write this by symmetry of the Gaussian as the integral from 0 to infinity of this expression so the 1d Gaussian is equal to 1 over square root of 2 pi exponential minus z1 square over 2 and if you multiplied with z1 I'm sure you have often seen or you have seen this calculation this is nothing else than the derivative of the 1d Gaussian so therefore up to a minus sign so therefore this integral is let me continue here this integral is is easy it's just the 1d Gaussian evaluated at the origin and now you look once more at the formula and you see that this is equal to 1 over 2 pi square root which we called c0 so so that shows that in a certain sense explains why in this gamma convergence result this constant c0 which is 1 over square root 1 over the square root of 2 pi comes up it's already present in the simple in equal sharp inequality which holds for any characteristic configuration okay let me also because that's that's perhaps more interesting let me also show you this monotonicity property here which uses essentially a similar trick so now I'm sorry so so let's let's think of chi as being chi as being smooth the so we're looking at this integral here now it's over the period cell but it's easier to think about the whole space and so you write this here as the integral from 0 to 1 gradient chi x minus tz dot z dt and now you use jensen in the dt integral and you get that this is equal to the integral gradient chi x minus tz dot z dx dt now that by translation variance of the lebesgue integral I can get rid of this translation and and this here I can write as the integral of z chi over modulus of chi times grad chi and that's the that's exactly the formula and now now you do an approximation argument so chi in fact is a characteristic function of a cacciopoly of b so you use the fact that that you can approximate bv functions by smooth functions and that in this case this this classical expression turns into the measure theoretic normal okay so let's look at this monotonicity so let me start with the left hand side and let me use this representation which we already had here and so this would be one over n square h square root the integral g n h square of z the integral of chi of x minus z minus chi of x dx dz let's do another change of variables which you already did over there to get the unit Gaussian here so this gives one over n one over square root of h g1 z so I should call it z hat but x minus n times square root of h z minus chi of x dx dz so so far this is an exact identity and now it's clear what you're doing you're just using the triangle inequality in l1 to write this here as the sum going from n middle n from 1 to infinity chi x minus n square root of h z minus chi x minus n minus 1 square root of h times z dx and you use translation in variants of the integral on the torus to see that all these terms are actually the same term and so with the sum it's just n times that term and now you see that these two n's cancel and indeed you get that this is by by the same formula that this is less than the energy of chi so so so this this monotonicity is is a very robust property and in fact it also holds nicely for the multi-phase case you just need a triangle inequality for your different interfacial surfaces interface as you would expect so so as I said everything here generate I mean I'm always presenting the results and the proofs in the case of just two phases or a single boundary but things generate to generalize to the general general case okay so now I guess I'm going to ask you what what other proofs you want to see before starting on the proof so there's a second lemma which I need in the order to prove a compactness result which is what it finishes the first chapter of preliminaries so the so I'm giving you lemma 4 in second so the compactness statement reads so let chi zero the initial data be such that the parameter energy is finite and let chi n n and n the solution well in fact I could even write the solution of the thresholding scheme with initial data chi zero and time step size h consider the piecewise linear interpolation so we're not yet doing the george's interpolation so piecewise linear interpolation means that chi superscript h at some time t this characteristic function is equal to chi n the nth step for t between n h and n plus one h so the if this is the time axis zero h two h three h you put chi zero here chi one there and so on chi two there then chi h then as the time step size goes to zero chi h is compact the l one but then also any lp topology in time space so that's the torus that's the time interval and since I can't really go with to infinite times I have to put a lock here or a finite time horizon so so this fifth lemma tells you that we can always there's always a limit we can select a subsequence I mean for any for any sequence of time step sizes that go to zero we can select a subsequence of the outcome of the scheme and it will converge to an evolving set I mean the strong convergence here is important for the limit to be a characteristic function again so so that of that's always in a certain sense an easy part in this type of analysis in probability theory you would call it tightness and and then of course all the work consists in identifying the limit and in order to prove this we need a fourth lemma which is just just three statements on on the relation between the energy functional and the metric so the first relation is that if you're interested in the difference between the convolved and the unconvolved function you can control this by by the energy this is little of square root of h if the energy stays bounded there's a second statement that if you look at two configurations u and u prime then the l2 difference the squared l2 difference is estimated by something which involves the the distance I'm looking for the right pre-factor 1 over 2 squared of h dh square u u prime plus this term for the both energies and then it's also convenient to have that the if you look at the differences of energies for two configurations and you look at the square then this is estimated by 1 over 2 square root of h to the power 3 or h to the power 3 halves times dh square u u prime and this last inequality is we'll need it at some point for the variational interpolation and but at this stage it tells you that for fixed h e is indeed a continuous functional even ellipse it's continuous functional with respect to the metric dh okay and the and the the idea is now that we're going to prove lemma 5 if you want lemma 5 based on on this here and on these two things so that that would be kind of now getting more into details alternatively I could start telling you about the convergence proof which is kind of the next would be the next section so so now you have the choice of either seeing kind of a bit more details I mean getting some getting some intuition on these on these functionals and some of easy manipulations and how you get a semi interesting result this compactness result or moving onwards who's in favor of seeing the details who is in favor of moving onwards that's pretty equal sir so um uh so for a certain class and that's something selim is a W has worked out for a certain class of anisotropies you keep kind of good both I mean you can you can get that from a thresholding scheme with a kernel which has kind of both good properties namely of being non-negative in real space and non-negative in four-year space so in a certain sense for this for this minimizing movement interpretation the non-negativity in four-year space is perhaps even more important because only that guarantees you that you can take kind of a square root which is important for for the metric so uh so if one's willing to give up more I think I think essentially you can do things for all for all anisotropies but they're nicer anisotropies where definitely everything should be should go through we didn't and in the work with tim laux we didn't really look at the anisotropic case we treated the multi-phase case and the case where every kind of grain boundary may have a different isotropic interfacial tension and that's that in this case there are no practically no changes and in the other case there would be some changes but my feeling is that for this class of zonoedal and isotropies as selium calls them it should be fine more questions yes yes so so even if it would slightly deviate from one half uh on the natural on the physical timescale your set would shrink or invade the entire space I mean if if the so in fact that's something which tim laux worked out that if um uh if you perturb it by around one half with uh I mean if you if you twist one half by something that's of the order of I guess square root of h uh or h square now I forgot but if you I mean if you if you modify it in a very controlled way uh then you get motion by mean curvature plus kind of a constant speed so so and in fact and but this this was already known before you can also get the motion by mean motion by area motion by volume preserving mean curvature flow by modifying the this uh thresholding scheme by insisting that you choose the threshold so that your new answer has exactly the same volume as the old answer and and also in this case you get uh things can be uh can be can be adapted plus a constant velocity oh I think there is no uh there is no problem of of uh kind of deciding a number whether a number is larger less than one half to machine precision and then of course if if square if h now is smaller than the machine precision then things might go wrong but I I don't think that whenever operates uh operates in that and that regime so I mean threat I mean uh so uh I mean stability with respect to spatial discretization errors and so that of course is important but uh but that I mean thresholding I think yes yes yes I mean so so the deviation would have to be larger than some power of h some moderate power of h now forgot h square or square root of h and typically that I mean that's uh that's not the machine precision more questions okay so now I got a split vote on uh on on what to uh what to present you so uh I don't know is it so I mean I wrote down I wrote down the proofs by hand is it possible to scan that and put it on the web uh and then uh and then you uh you can look at it and uh and then you can still ask me uh ask me tomorrow by tomorrow I mean if is it possible to get them on onto the web today okay then uh then you can tell me uh can tell me tomorrow whether you want to see whether you want to see some of the proofs and the other situation might be that I run out with what I've what I've prepared and then and then I will use the remaining time to give you give you the proofs okay so so so I'm going to erase these statements now anyway they're also I have them also on the screen here is around these two things and uh and now let me start with the uh with the second chapter which contains the main results and uh and the connection to the work by Andrin Taylor and Wong and uh and the uh the bracket and notion so and and and by the way I mean so you I can also give proofs after I mean I have a second session um this um this afternoon so if you come to me and tell me I really want I really want to see uh some of these proofs I can be can be easily convinced so uh main result okay so uh so let me start with the uh with an informal observation so it is kind of well appreciated by uh by by the mean curvature flow community that mean curvature flow in fact is a gradient flow formally what you would expect uh if I use the bv notation I write it like this but of course classically it's just the uh hostile measure of the uh of the boundary of the set and uh why why why is not surprising that something gives like this gives rise to curvature because the first variation and so let me use this notation for the first variation so I had one notation for metric slope and one notation for a first variation so if you look at the first variation of uh of the interfacial area in direction of a vector field xi so xi will always in my case denote a time or time independent vector field uh so here which of course lives on the same domain uh is uh as uh many of you will know is given by integrating uh the weak formulation is given by integrating the uh tangential divergence which I can write as the ambient divergence minus the uh normal normal part uh which uh classically is just the uh integrating the mean curvature um times uh nu xi so in my in my notation the mean curvature is a scalar and uh that's the mean curvature vector so uh that's the first variation so the first variation brings up mean curvature so uh mean curvature flow is a gradient flow of the area functional uh uh if uh you uh if you consider the metric so with respect to uh the infinitesimal inner product of the uh our two space on the on the boundary so as it stands this is just almost tautological and uh um and the first problem uh with this type of uh if now so if now you wanted to use this insight to write down a minimizing movement scheme uh you run into the following problem uh so namely the problem that the uh distance function which comes from this infinitesimal metric from this infinitesimal from this Romanian structure is degenerate induced distance function this formal Romanian structure is degenerate in this sense and that's uh I think something which uh was which many people were aware of but uh which Michaud and uh Mumford pointed out uh not that long ago and therefore if you want to write down a minimizing movement scheme uh without knowing the thresholding algorithm you have to come up with a proxy so hence for uh minimizing movement scheme a la de georgie proxy or substitute something which acts a little bit like what you would expect the uh induced distance to be um for the induced distance or induced metric and that's exactly what uh uh what Almgren, Taylor and Wang proposed in 93 by saying that Chi N minimizes i in fact there is uh for for um for technical reasons it's convenient to look at mean curvature flow up to a factor of two so the normal velocity two times the normal velocity is given by uh by mean curvature then things come out nicely with thresholding so v always denotes the normal velocity and h the mean curvature in the sense that it's the trace of the exterior curvature I mean the vine garden map and uh so Chi N minimizes uh the parameter functional plus uh one over two h and then comes the following expression uh you take uh the uh um non-negative so the unsigned distance function in uh rd to uh your previous um uh to the boundary of your previous configuration and you integrate this uh distance function over the symmetric difference between the new set and uh the previous set and I think here you will need a factor of four which uh comes from uh from this tool here so uh so that's the uh that's the famous Almgren, Taylor, Wang scheme so you minimize if I would write this geometrically you minimize the surface area of the boundary and keeping keeping track of this this type of distance to the previous time step and uh so um uh in the original paper they had a had a kind of uh a very conditional convergence proof of this and that was uh to my understanding improved but still uh uh being conditional by uh lookhouse and Sturzenhecker 95 and the statement is uh is as follows so uh let uh consider a sequence of time steps that goes to zero and uh a limiting configuration uh which satisfies that uh the uh piecewise linear uh sorry piecewise constant interpolation converges in l1 to chi let's say over a finite time horizon up to capital T and uh so this is something which is a benign assumption because uh well I mean here we have to work more but uh kind of compactness is uh is is also not a problem in this setting but now comes the uh which only makes it a conditional convergence result namely that um uh if you look at the parameter of the uh approximating sequence so the sequence which comes out of the scheme then this should converge to uh the parameter of the limit and uh and that is indeed uh uh kind of uh uh a non-trivial assumption which cannot typically not be checked uh at least presently not be checked for the scheme so let me just comment on it there is one direction which you get for free here so by lower semi-continuity you always get for free that the uh time integrated um bv norm of the limit is uh less that of the approximating sequence so uh in assuming this uh the main assumption you're making is that the parameter does not drop in in the limit and what this does for you is that it morally rules out ghost interfaces so it shouldn't be the case that on the positive xh level uh you have two uh two interfaces which get closer and closer they don't have to be straight and in the uh limit so here is chi is equal to one chi is equal to one and chi is equal to zero and in the limit it's gone because you're looking at it not from a very full point of view uh where you count multiplicity but you look at it from uh from a bv point of view where you disregard multiplicities and uh and that's the situation I think that is this that is essentially what you're ruling out by this assumption but if you make this assumption then um then you're fine uh so then uh there exists a normal velocity v uh which is square integrable with respect to the interfacial measure in the sense of uh that it's really uh uh it it's it's it's the normal velocity of the characteristic function in the distributional sense so that means that uh v times zeta v times grad chi uh plus the integral of dT zeta times chi is equal to minus the integral at t is equal to zero of chi zero and here I need uh spacetime integrals for any test function zeta so there exists there exists such a normal velocity and uh and uh we have uh the normal velocity is equal to the mean curvature in this sense which is uh suggested by uh by this weak formulation of mean curvature that means uh we have a gather spacetime integral divergence chi minus nu d chi nu minus two v chi nu chi dT is equal to zero for any test vector field psi so that's uh that's uh that's their uh their result uh so uh this um greentailer-wong scheme uh which is based on the idea of minimizing movements but historically it might even have been the other way around that um greentailer-wong wrote down the scheme and then the georgey uh kind of embedded this uh uh in in the more general theory of minimizing movements so anyway so uh that this type of minimizing movement scheme for mean curvature flow uh converges in this conditional sense in the sense that if you make this uh uh this extra assumption of no ghost interfaces then for any uh for all times it converges to weak solution of of mean curvature flow in uh in in the specified sense and uh and that is a pretty involved work I mean it uses to some extent uh uh regularity theory for minimal surfaces to get uh uh to get that these sets are nice behave not uniform in age but for a fixed age and uh um and in a certain sense the most delicate term is to recover this uh uh this metric term in the limit and now what we did but that's not the result which I'm which I wanted to focus on is essentially the same result uh for the uh for the thresholding scheme so uh so this here is the piecewise constant interpolation of greentailer-wong and now our theorem which was published uh two years ago uh is exactly the same thing just with the only uh uh with the with the following difference so um so uh now this is the uh piecewise constant interpolation thresholding here we have to take the moral equivalent so uh we're integrating these energy functionals e h over time and we want them to converge to the limit so that's uh that's uh the uh kind of the moral uh substitute in our framework uh of the uh assumption of Lukhausen-Sturzenhecker and it's essentially the same type of assumption then we get the same result and in fact I would say our proof is uh because uh the thresholding scheme involves less subtle geometry like looking at distance functions to boundaries of set our proof is uh is perhaps uh less technical more robust so we definitely don't use I mean we just use well I mean a little bit ideas of kind of excess and but then essentially structure theorem for for bv functions and again the term one has to work most in both proofs is is to pass to the limit here okay so that uh that is uh that is uh what's uh what's around what has been around so far but then we ask the question um so since both uh since both kind of convergence results start from minimizing movements so start from natural discretizations of gradient flows either in this academic way of the arm grain Taylor-Wong scheme or in kind of this numerically relevant way of thresholding at least the limit should satisfy the dissipation inequality so the question is is it true that for the limit if you look at the uh rate by which the interfacial energy decreases and I have to put a factor of two here because of this factor of two uh do I get that this is at least as an inequality given by integrating the square over the mean curvature and uh and that's not I mean this uh this in a certain sense is not strong enough to derive that uh dissipation inequality so the notion this notion of uh weak solution is not strong enough and and this is a little bit uh dissatisfactory right because uh you start from gradient flow the dissipation inequality is hardwired into gradient flow you're using kind of a natural time discretization for a gradient flow but uh you're deriving a kind of a notion of a limit that doesn't even allow for for this basic property of a gradient flow and uh uh and therefore so so that was uh that was dissatisfactory and therefore uh we um uh uh uh well we wanted to get this and we were kind of uh uh we're aware that um this issue of getting the right dissipation is in fact at the basis of uh Brachy Brachy's notion of uh um mean curvature flow so uh this is so the dissipation inequality in a localized version is uh is just uh is is is what uh what Brachy's notion is built on but let me remind you what uh uh what this means so uh uh and but let me still use this bv notation so given a localizing function zeta which uh is uh let's say smooth and of course periodic and non-negative you want to monitor the rate by which um uh by which this localized surface measure changes and uh and so by purely kinematic uh calculation or classical calculation you see that uh there are um now more than one configuration um contribution there is what you would expect namely the localized uh the localized term uh which comes from the localized term which which looks looks like this one here but then there must be another it's clear that there must be another term because even uh um even if uh uh uh even you know I mean even if nothing dramatic would happen your uh interface could move out of the region where this function is looking at so there is also transport term and sorry I so I wanted to write down the purely kinematics formulation so this would be normal velocity times uh mean curvature but then also the um you're moving out of uh the um what you see by the cutoff function so you get a transport term like this so that's a curvature effect and that's a transport effect and uh so this one you clearly see that uh so if zeta is constant then you uh then you kind of end up with the classical inside that the um uh that the um first variation of the area functional is given by mean curvature so that the rate of change is given by normal velocity times mean curvature and uh but now since due to the cutoff function due to the non constant and the non constancy of zeta you get such a transport term and therefore uh the the notion of mean curvature flow is encoded uh by requiring now let me put in the factor of two again this here is lesser equal to minus zeta h square plus gradient zeta dot new times h red chi for all non negative test function and it's uh in a certain sense at the basis of uh of bracker's uh notion of solution that for smooth evolutions this here really characterizes despite the fact that you just have an inequality sign characterizes mean curvature that's uh that's the uh that's the inside then of course he used very folds to formulate that in a kind of more robust way that's something I will not be using I wish we could but that's something which I will not be using here uh so I will stick to this bv type notation and the bv setting but uh but even before that the main idea is that you can encode uh an equality pde by inequality and that's something which uh you know I mean clearly is also present in the general gradient flow theory of uh of um uh um the georgi but uh is is present in in many other situations so for instance this notion of uh weak solutions to compressible Euler equation for instance edward fire riser is kind of one of the main uh people working in that area also works with inequality so it's not unusual that uh that pde's can be characterized by inequalities and that's um that's one instance okay so so therefore we were right away a little bit a little bit more ambitious and asked the question well can we instead of just getting this global dissipation inequality can we get this entire family of dissipation inequalities and that's indeed the result and that's kind of the result which I would like to explain a bit better in this course so that's theorem three that we posted it last year but now there is kind of a new version and again that works in the multi-phase case so uh so the assumption is very much like there so let me put it again for completeness so uh for uh chi zero uh for initial condition with finite uh parameter uh let uh chi n uh denote the solution of the thresholding scheme at uh for time uh with initially with these initial conditions and time step size h which i'm suppressing here in the notation let uh chi h denote the piecewise linear interpolation the sort of piecewise constant so what what i defined in lemma five uh let uh chi be such that for a subsequence of h going to zero we have exactly these two statements that the uh interpolation of the thresholding scheme converges uh in uh l one on time space and that uh we have not just an inequality here but uh inequality so that's the unfortunate assumption we we need uh but then we're fine so then there exists uh mean curvature the mean curvature which is square integrable on the boundary in time space and it's characterized in the usual weak way which is almost sitting here so uh it holds that uh zero t uh time integral of the tangential divergence of a test vector field uh minus h times the normal component of this test vector field integrated over the boundary integrated over time is equal to zero for any test vector field well let's take compact support in time but that's not necessary uh such that uh mean curvature flow in brackets and in form of brackets inequality is satisfied so uh by this uh i mean uh um now do i what how do i go i'm going to write it down let me write it down right away uh like uh like this so so that the gradient uh that zeta times the gradient chi at some point t plus the integral from zero to t the integral over space of h square then i guess i need a minus sign here and probably i forgot a two no i didn't uh that's okay uh minus but then i need the two here minus uh gradient zeta uh gradient sorry yeah gradient zeta dot nu times h uh gradient chi uh integrated over time is lesser equal to two times gradient chi zero and this is uh zeta and this holds for all zeta uh which uh which is smooth and periodic of course which are non-negative and it holds for almost every t okay so that's uh that's the uh that's if i didn't mess up with signs uh that's exactly the time into the time integrated version of this formal inequality okay so uh so that's the uh that's the that's the result i would like to uh i would like to explain and let me uh let me comment that this the proof of this is much easier than the proof of this so this is a much softer result and that's uh thanks to the tools by the georgie so uh so here uh uh here is here is where kind of the set of ideas comes together uh on the one hand uh this numerical scheme this thresholding scheme which uh which is uh you know real practical and successful numerical scheme for mean curvature flow uh this uh this very intelligent notion of encoding mean curvature flow as an inequality which is due to bracket and which is a localized which kind of uses the localized energy dissipation inequality and therefore is clearly related to the gradient flow structure and the tools by the georgie on minimizing movements so this is where we're kind of these three areas which i said in the beginning come together okay so uh how much what's when i do have to sir more or less over okay probably that that's a friendly way of saying it's over time already okay okay so right so that's the uh that's in a certain sense the those are that's really now the main statement of what i want to do in the rest and the rest of in the rest of the time and now you can you can either tell me before this afternoon you want to see some of the uh ideas from the easier statements i gave before in order to get a little bit of better feeling of what these kind of objects really do uh if you don't give me this uh strong impression we can post this uh tonight and you can have a look at that and uh and then if so if i don't get this this feedback i would start telling you uh how to uh go from the very first lemma by the georgie uh with the variational interpolation uh to uh to this inequality and uh and the main step is is again a very simple observation on the thresholding scheme namely that the thresholding scheme doesn't just satisfy a single um minimizing movement hasn't hasn't does not just have a single minimizing movement property but it has kind of a whole family of localized minimizing movement properties and it's this nice feature of of the thresholding scheme uh which uh which we use to localize to get to this kind of localized statement and here i forgot the zeta so uh so that's that would be uh that would be for uh for this afternoon if you uh if you don't tell me otherwise okay sorry for being over time and uh we'll continue over over over