 OK, so let me reconstruct what we had on the blackboard, which is the main theorem that starting from the thresholding scheme, which is kind of a practical numerical scheme, and using a minimizing movements interpretation, one can show that under this additional convergence assumption, which is a little bit the state of the art in this field, one can show that one converges to a solution of mean curvature flow in the sense of Brackers inequality, but formulated not in terms of very false, but formulated in terms of PV functions. And that essentially amounted to a localized dissipation inequality, where in a time-integrated way, where for any time t, you have that the localized surface energy plus the time-integrated dissipation, localized dissipation, which was of this form integrated over the boundary, and integrated over time, is estimated by the energy of the initial data. And so this holds for all test functions, which here are just depending on space and are smooth on the torus. And of course, they have to be non-negative and for almost every time t. So the Brackers localized energy dissipation inequality holds. OK, so that's what I want to explain. Other questions with respect to the statement? Or other questions? No questions? Everything is clear, or you're already still in lunch mode? Yes, yeah? Now your question is whether this condition is similar, or it has to be, what was your question, or? Yeah, yeah, the analog of this condition. Yeah, so in all cases, there is this condition. I mean, if you work with thresholding, there is this condition. If you work with Armgren-Taylor-Wong, there is the condition where you have both the parameter on both sides. So we don't know how to have to speak up. I'm half deaf. So I formulated them for the two-phase case, but they all hold for the multi-phase case. In particular, also the monotonicity holds in the multi-phase case. And what you need is essentially it's a consequence of the triangle inequality for. So I mean, the proof is users then, so now you have kind of different phases. And for every pair of phases, sigma i, sigma j, you have a surface tension. And the main condition which goes into the monotonicity is next to the obvious symmetry is the triangle inequality, which in a certain sense rules out complete wedding. So ij has to be less than or strictly less than ik plus sigma kj. This type of condition enters at a very precise, I mean the proof is of course more complicated, but it's essentially holds. Two-phase, yeah. I think that's more the form of this functional. So one didn't, for that part, one doesn't use much properties of the kernel. So the fact that the kernel comes from the heat equation is the heat kernel didn't play a role in that part. So the semi-group property didn't play a role. No, I don't think so. So I don't think that this has to do with the, that this monotonicity has to do with the, well, OK. So I mean, I'm using that the kernel is non-negative. So in that sense, yeah. OK, in that sense it has to do with the ordering, yeah. More questions, yes? No, because so the restriction coming there, I mean, there came from the fact that they use kind of regularity theory for almost minimal surfaces. And our proof is, I mean, both. So there was theorem two and theorem three. I mean, this is very soft. And theorem two is also fairly soft. We don't use any regularity theory. So there's no dimensional restriction. More questions? OK, so that's the statement. And now remember that the question was really to relate these three things, the Braque inequality, which is there, the thresholding scheme, which this is a statement on and the Georgie's minimizing moves, which will now play a role, and particularly the first lemma. But for this, we need kind of a new, a different connection to what I showed you this morning, the minimizing movement interpretation of thresholding. We need something, a localized version of that one where something which holds for every zeta. And that's the next lemma, which now has lemma six. So that's a localized version, lemma two. So we're given such a non-negative localizing function on the torus. Since the torus is compact, I can omit this. Then the statement is the thresholding scheme as the following minimizing movement property. So it satisfies another kind of minimizing movement property. So chi n minimizes a localized version of the energy plus a localized version of the metric among all u and x. And that was the space of unit interval value measurable functions on the torus. And now there is the tilde. So these two objects now mean something different than before. So I have to give them to you and put this here. So let me start with a simpler object, which is the metric contribution. So that's really what you would expect as the localized version from what we had before. So before we had using the semi-group property and the non-negativity, I mean the semi-group property, we rewrote this as gh over 2. So the heat kernel at time step h over 2 convolved with the difference of these two configurations squared. And now the only thing I'm doing here, because I don't want to destroy the structure of this term, is I put the localizing function there. So this is clearly still a distance function. And provided I put the square on both sides. And so that's the metric. And now for the energy functional, I guess I need more space while I will fit here. One is tempted to put exactly to follow the same strategy and just localize the energy functional we had before, which was this one. But that's not correct. So you're making an error here. So there is an additional term. Now I have to look up the sign. That's where the plus. So there is a term, which is difficult. So plus u minus chi times the commutator of multiplying with the function zeta and convolving with the heat kernel applied to the function 1 minus chi. That's not everything. But let's first look at this term. So this is, as I said in words, that's the commutator of multiplying with zeta and convolving with gh over 2, and it acts on what's coming here. So in general, the notation, I think many of you will be familiar with this. If you have two operators, this here is ab minus ba. And clearly, if zeta happened to be a constant function, then this term wouldn't be there. So we would end up with the old functional if zeta were constant. So that's the price to pay for smuggling in the localizing function here. But then we also have to pay a price from smuggling in the localizing function here. And that's the second term here, which we have to put into the energy, because we don't want to destroy this nice norm structure, squared norm structure of the metric. So there is another term, which is slightly more complicated, which has this shape. So those are the two terms. Additional terms one gets from localization. And this one will be important. Ultimately, this term here will be responsible for the transport term, which is kind of the interesting term in Bracken's formulation or localization, whereas the last term will never play a role. It's always higher. It will always turn out to be higher order or negligible. And that will connect to the transport term. Whereas the first term is the one which if you want, more or less, we can connect to the leading order dissipation, because it's just localizing the energy. So now it looks even stranger. I mean, and I missed the square here, we press something in the minimizing movement formulation. But the price we have to pay is now that these things, these objects are even more complicated than before. So here it's GH, and here it's GH over 2. So that's the statement. And again, I can show you that this is really a simple algebraic observation that this minimization problem is true. And then provided Zeta is strictly positive, also this space is a compact metric space or fixed positive H so that we can and will eventually use the Georgie's interpolation. OK, so that's kind of the localized version of the minimizing movement scheme that we'll need. And it's on this one that we apply to Georgie. So let me, this variation interpolation, so let me put this up again. And the way we're going to use the Georgie is, so perhaps now I can erase this. So we use lemma 1 in the following form. So it has these two parts. Here was the dissipation. In fact, typically it will be an identity, this dissipation identity into which we plug in the metric slope. And now we plug this in into both expressions. So then it turns into e of u of t plus 1 half metric slope squared of u of t plus 1 half times the integral from 0 to t metric slope squared u of s ds is less than e of chi. That's the form we're going to use it where you have both, I mean the metric slope in both places. And the way we're going to use it is now on this minimizing movement scheme. So our case, this turns into the following identity. So I take now this energy functional where I freeze. OK, so I should, in fact, as you see, my notation wasn't very good because this energy functional also depends on chi. So it's now kind of a two variable object, and that will, of course, be important. Yeah? Right, yeah, yeah. So here we have chi n minus 1 plus chi n minus 1. So it's the structure we had before. Now with a little twist that the energy itself depends on the previous step because that's the energy functional here, which clearly depends on chi, which here plays the role of the previous step. So that's now, in a certain sense, a little bit of departure of a more specific minimizing movement scheme. But in a certain sense, we just use it, I mean we use this variational interpolation and only between two time steps. So we don't care that the energy functional depends on the previous time step. OK, so let me write this down. So here I have to put in the previous time step. And here I put the variational interpolation at h time units later, so that will give me chi n. So that's this term plus 1 half. So here I take the metric slope of the modified energy functional in the active variable with the second variable frozen in at chi n minus 1 squared at chi n. That's this term. And then I have this integral, 1 half, the integral from 0 to h, dE h tilde, or let me rather write the integral from n minus 1 h to nh. And here I'm looking at the metric slope of the same functional, but now it's evaluated at the variational interpolation. So here I have to use another letter. So I use uHT and dt. And here we have the previous time step, e chi n minus 1 and where uH of t, t from n minus 1 h to nh, is the variational interpolation. So the Georgian's variational interpolation of chi n minus 1 to chi n. That is, it solves, it minimizes, or rather I should say, uH t minus t plus n minus 1 times h minimizes the functional which comes up there. So it minimizes e tilde h u chi n minus 1 plus 1 over 2h t tilde h u not 1 over t chi n minus 1. So in a certain sense now we have two types of interpolations. We have the piecewise constant interpolation, which we looked already before, for which we have the compactness. So here you have the value chi 0, chi 1, chi 2, chi 3, and that's the function chi of h. And now we have the variational interpolation, which typically will be rather a smooth connection between these things, and that's u of h. But clearly we expect both of them, and we will show that both of them converge to the same limit. Just we need this variational interpolation at this place here, because this allows us to recover that term. OK, so that's a kind of completely mechanical application of the Georgian. And the only twist is, OK, our energy functional depends on h. It depends on this other configuration, which is the first time step. The metric also depends on h, but we don't care. I mean, we can just apply it as it is, and we get this inequality. And now the idea, of course, is to sum this up. So let me do that in order to contrast it with what we have here. Let me make it look the same. Let me put the 1 half here. And let me put a c0 in front of everything, which was this constant for which we have gamma convergence of the functionals. And now how do I want to relate this? Let me call this here capital T. And how do I want to relate this? So in order to see the kind of how terms come out, term by term, I have to take care of the fact that I have the second argument here. And the way to do this is by summing over n. So I want to write this here as e h tilde chi capital N. That's not quite the term which I have there. Plus 1 half. Let me combine these two terms here in the integral from 0 to N h, rather from 0 to N h of the first term, which is chi h of t, chi h t plus h. That's just this term here, brought under the integral. There's something wrong in terms of h. Right, it's already wrong here. He is still 1t. If you look at the lemma, which here is an h, now it's correct. So there is this. And there is the metric slope, still at the same second argument, but now evaluated at the variational interpolation. And both times it's the squared of the slope, e t. So that takes care of these two terms. But now here I made an error. And that's the integral from 0 to t, 1 over h e h tilde chi t plus h comma chi t minus e tilde h chi t chi t. No, actually I want to write it. Let me look it up, because I want to write it slightly differently. t plus h minus t plus h. And that's the, there is a t here. And this is less than e tilde twiddle of chi 0 comma chi 0. So that should be this inequality summed up over time steps and bringing this here under the integral. And using the definition of the piecewise constant and the piecewise linear interpolation. So nothing really analytical has happened at this stage. It's just that, right. So now one should, so eventually what we have to show is that it's this term here that gives rise to the transport term. And it's this term here that gives rise to the main dissipation term. And of course this here turns into that. So individually the goal will be to show this type of convergence where we have, at least here, we have an equality which goes the right way and the other terms. So that's the strategy. So we write down this, we take the minimizing movement scheme, which is one scheme, but we write down a different, sorry, we take the thresholding scheme, which is a given scheme. We write down a different minimizing movement interpretation which contains Georgie's interpolation on that function zeta to derive kind of a preversion of Braque's inequality. And but then there is a one to one relationship between Braque's inequality and the terms in the Georgie. So that's the connection between Braque and the Georgie's minimizing movement scheme. Are there questions besides indices or things which you can't read? Sure, thank you. You're saving me. The definition of the etilda is here. I just forgot the tilde. So I mean I can give the argument for this. This is again essentially a simple algebraic manipulation as we had before. I mean the etilda in a certain sense is almost designed to provide us with this variant of the minimizing movement interpretation. But it turns out that it's exactly the etilda which gives kind of both contributions in the Braque definition, in the Braque inequality. If I had kept the original e, I would have gotten the simple dissipation inequality with the right constant. I mean that's what I said the first day. You get this when you have a minimizing movement scheme. You always get this a priori estimate which misses the dissipation inequality by a factor of 2. And it's exactly this term here which involves the variational interpolation, this tool by the Georgie that gives the missing factor, the missing term here. And so it would turn into kind of the global dissipation inequality. But in a certain sense, you also get this entire family of local dissipation inequalities from the same type of argument which I wouldn't know how to get otherwise, at least not from this BV notion of solution, because that doesn't even allow you to get the global one. More questions? So what do you want to see now? So I can return to the proofs of this morning. I can give you now the proof of this modified minimizing movement lemma. I can continue with writing down the main ingredients into these convergences. So who is for returning to the kind of somewhat simpler statements of this morning? Who is in favor of now seeing the proof of lemma 6? So the first option is I return to this kind of lemmas which I was stating this morning and give some of the proofs of that one. Second, and there was just one person in favor of that, and you understood it correctly, right? So there was one person in favor of that. The second option is to now prove lemma 6. And the third option is to continue with lemmas, I mean with a statement of lemma 7 and lemma 8. The senior people don't count. That was almost the same. OK, so since I deferred some of the proofs this morning, let me at least give you the argument for that. And then there should be when do I have to stop? At latest 3.15? Yeah, so I probably will have the time to give you also the statements of lemma 6 and 7. So then let me tell you why this strange variational principle is still true. So the thresholding scheme can be looked upon under many different angles. And it satisfies many minimizing movement properties. And that's another one. This morning I derived one. And now this is another one, which is slightly more involved. But essentially it's the same argument. So proof of lemma 6. So we just start with spelling out what this is using these two definitions. So the modified energy functional in some configuration u with some fixed configuration chi, which doesn't have to be an integer, sorry, a characteristic function, although I use the notation as if it were. So that's the object we have to look at. And now I'm just plugging in the definitions. And rearrange them slightly. So let me start with the metric term. And there is this pre-factor in front of everything. So big bracket. There is zeta gh over 2 convolved with u minus chi. That's this term. And now I'm taking this term, which as many of you will have, or some of you till around, will have noticed kind of clearly is related to that one. Because it has the heat kernel at time h over 2 at half the time. So that's the first contribution. So let's keep a little bit of free space here. And then there is this. zeta u gh 1 minus u plus u minus chi zeta gh convolved 1 minus chi. And then I have to close this bracket. OK, so that's just plugging in the definitions. And now by definition of the commutator and the semi-group property, this is gh convolved for the first part of this commutator minus gh over 2 convolution with a multiplication operator, and then again the convolution operator. So now I claim that these two terms cancel. Why is that the case? Because the second term here, so which comes with a minus sign, I can use the symmetry of the convolution operator to build this convolution operator from what comes after it to what comes before it. And if I do this, I'm left with gh over 2 convolved with u minus chi. Big bracket multiplied with zeta gh over 2 convolved with u minus chi. And now you see that this is exactly this term. So this term drops out, and this entire first line gives rise to something which looks a bit simpler, namely zeta u minus chi gh convolved with u minus chi, which is another way of writing the old metric term multiplying with zeta. OK, so that's the first term. And now, so that's the first line, what's left over from the first line. And for the second line, we use again the definition of the commutator. So this is that here minus gh zeta 1 minus chi. And let me see. Again, on the second term with the minus sign, I want to use the symmetry of the convolution operator. So to write this here as after integration as minus zeta 1 minus chi, gh convolved with u minus chi. So that's this term. So now if we collect all terms, and I think I need more space, but now I think I can erase the definition because we've used that. So what are we left with? So we have this 1 over square root of h in front of everything. We have sigma now multiplying all remaining terms. The remaining terms are u minus chi gh u minus chi. That comes from the metric term. And then we have three terms from the energy, u gh 1 minus u plus u minus chi gh 1 minus chi minus 1 minus chi gh u minus chi. And then this bracket goes closer. And now we should see that this here simplifies a bit. And so this entire integrand should be equal to minus u gh 2 times gh chi minus 1. And then there should be terms which do not depend on u. So let's see whether this is the case. So here in this term, we have expressions that are quadratic in u. Like for instance, this one and this one, those are the only ones that are quadratic in u, but they cancel because of the minus sign here. So the quadratic terms, so no term, no term, that is quadratic in u. That's already correct. Then there are terms which are linear in u. There is u and the convolution and the convolved chi with a minus sign. We have one of them here. There is chi with the convolved u with a minus sign, but that could should cancel with this term here. We have, well, chi with itself. We're not interested in this. We have u times 1, which is here. We have u times 1, which, u times 1, what does this go? Times 1, yeah, right, u times 1 is here. What else do we have? We have this term, but that one I said goes away. Am I getting things correctly here? I'm a bit unsure. OK. I mean, let's check. So we have terms, we have this term here. So now it's not, I can't use the symmetry anymore because I have this kind of function here. So we have this term, we have this term, we have this term. We have this term, we have this term, this minus u chi plus chi 1, which I don't care for, minus plus 1 plus chi chi. And here we have 1 u minus 1 u plus 1 chi plus chi u minus chi chi. And here what I want is minus 2 times u chi plus u 1. And I seem to have u 1, 1 u 1 too much, concerned about. And I have 1 u. Oops, I didn't make this too fast. And did I copy this term correctly? Yes, I did. OK, now I'm a bit confused because in the break I try to boil down kind of the multi-phase case in which situation terms are, in a sense, a little bit more symmetric to the single-phase case. And I might have made a mistake there. If I don't see something here, then it has to wait till tomorrow and I have to adjust perhaps the formula. Presently I think I did copy that correctly. There is the metric term, there is the energy term, and then there are two terms coming from the localization, which in a certain sense are where just chaffle, and if Zeta would not be there, these two terms would cancel. But it doesn't look immediately that I get this term here. But if that were true, and now I have to, I'm not quite sure, then we would be in the same situation as in lemma 2 that this expression here is minimized if U is the characteristic function of the set where this here is positive, which is just thresholding. So if this here were true, then now same argument, but now I'm a bit confused by the algebra. So it might be that I kind of reduce the multi-phase case in a little bit too simplistic way here. I have to check that. OK, so let's assume that this here were correct. Then we proved lemma 6, and now I can tell you about these two ingredients, which essentially are behind these converges. And let me erase this. So there are two main lemmas. So there is lemma 8, lemma 7. So lemma 7 is about the localization on level of the first variation. So it's perhaps not too surprising that the metric slope in the end will be related to the first variation. And on the level of the first variation, we have that if we look at the modified energy functional, I'm going to write down the definition in a second, which still depends on the second function. And we take the first variation in the first function in direction in some configuration U in direction of a vector field psi. I remind you of the definition of the first variation in second. Then this is very close to the first variation of the original energy functional, which just depends on one argument, but in direction of the localized vector field. And this is nicely estimated by h 1 quarter dh u chi over h. So let me remind you what I mean. So recall the first variation of the functional E in U along psi. How is this defined? You're modifying your configuration U by solving the transport equation involving the vector field psi and setting this at the parameter s equal to 0 to be your original configuration. So this gives you smooth deformation of your configuration. And if this is a characteristic function, it essentially means flowing, looking at the deformation of the set. So psi is a vector field as always. And then you define the first variation of this functional in the point U in direction of psi to be the first variation of the function. And so what are we doing here? We're comparing the first variation of this weird energy functional, the definition of which I erased, which essentially was this term. And then there were a bunch of correction terms to the first variation of our original energy functional, the non-localized one. So this one here has the zeta. And the other one doesn't. And let me reconstruct the definition here that was plus U minus psi zeta gh1 minus i plus a third term, which never plays a role. And so the statement is that on the level of the first variation, what I did here really acts like the localization. So kind of looking at the first variation of the localized functional in direction of a vector field psi is the same thing as looking at the first variation of the unlocalized functional in direction of the localized vector field. So localizing the functional and localizing the vector field commute. That's the main statement of this estimate. And this is a higher order term because by the typical energy estimate, this here should be of order 1 in an L2 sense in time. And there is still a small h power in front of it. So this will allow us to kind of relate the first variation of the localized energy functional to the first variation of the unlocalized energy function. So that's the first ingredient, which will play a role already for this term. And then there is a second ingredient, which has a little bit of a similar spirit. But now it's more related to the fact that we have this term here, which now kind of uses the fact that there is a second argument here. And so there the statement is, if we look at this difference, so u chi minus e of u u h tilde. So in the nonlocalized energy functional, that wouldn't make a difference because the original energy functional does not depend on the second argument. But now there is a dependence. And there is a non-trivial dependence in the sense that this gives rise exactly to the first variation of the nonlocalized metric. Indirection of the gradient of zeta. This expression is nicely estimated. So when I mean this here, I mean only up to constants which depend on higher order norms of zeta and of xi. Here there's only zeta. And we get a similar error term. Now in error term, we get the square of the same term. And we get a third term, which has to do with the energy h 1 half plus h 1 half of u. So again, all these terms here will be higher order. In the end, this will be of order h 1 quarter. This also will be, well, this should be then of order h 1 half, and this is of order h 1 half. So these error terms vanish in the limit. And here we take the first variation of the unperturbed metric term. So why does that make the connection between these two expressions? Because so this, let me rather put it here, this makes the connection to lemma 8 makes the connection to the transport term in bracket. Why? Because by the Euler Lagrange equation of the first of the unlocalized minimizing movement interpretation of thresholding, so that was lemma 2, we have that the first variation, at least if we plug in what we're interested in. So if here we take the previous time step and we look at the first variation in the actual time step in direction of some vector field, then with a minus sign, this is equal to the first variation of the original energy, the actual configuration in direction of this vector field. So also this term here connects to the first variation of the unperturbed energy functional. But now we're going to use that for, let me use the same color, we're going to use that by lemma 8, we have to use that not for an arbitrary vector field psi, but for our gradient field. So we're let automatically to look at the first variation of the energy functional in direction of the gradient of the cutoff function. Well, and that exactly gives this term. The first variation gives the curvature, and here's the gradient of the cutoff function. So the metric term comes in a very natural way. And now these two lemmas are a little bit like the lemmas, which I didn't talk so much about this morning, or which I didn't give you a proof of this morning. They hold for general configurations. These are just lemmas for any u and chi, and both of them need to be characteristic functions. And any vector field psi, with vector field psi. OK, so now at least I hope you start seeing, you start seeing the connection between this localized energy dissipation inequality, which according to Barker, characterizes mean curvature flow. And this localized minimizing movements interpretation of thresholding via these two ingredients. So what it's not yet so clear is how actually does the metric slope, which is a first variation, so differential, cotangent vector, how does it relate to the first variation of a functional, which is a cotangent vector, if you want, how does the first variation of a functional relate to the metric slope, which is a norm of the gradient. So there will be a metric term that converts the functional into the norm. And this is where we have to kind of maximize or minimize or optimize over these vector field psi. And that will give this, in the end, this inequality here. And but ultimately everything then relies on the fact that not only the energy functionals converge by assumption, even in the localized setting, but also that the first variations converge. And that's something classical in kind of this type of asymptotic limits of sharp end interface models. I mean, I know it as kind of the trick of Reshetnyak. And I know it from the paper of Lukhaus and Mordekar, that if you make this convergence assumption on the interfacial energy, then also automatically kind of the first variation converges. So the convergence of the first variation will be something that we'll get, essentially, from our assumption. OK, so that's the kind of the strategy. So next time, well, so I can give you, I mean, first of all, I have to see whether I missed something or just didn't see the right algebra on the blackboard. Or whether I need to modify a little bit the definition because I boiled it down in a naive way. And then if you want, I can give you kind of more details of the proofs. And I guess I stop here, then there's even nominally time for three minutes, time for questions or so.