 So this is the first lecture of Professor Otto. OK, so let me start by reminding us on the slides where we are. So that's the main theorem I was mentioning last time. So I can write down a couple of things. So the thresholding scheme is as follows. So you define chi n to be the characteristic function the indicator function of the set where the convolution of the previous time step is larger than 1 half. And that's minimizing the energy functional, which is given by looking at u by solving the heat equation with initial data u up to time t and looking how much of the mass has escaped into the neighboring in the other phase and dividing by one of a square root of h. And the distance function, and it's better to define it right away how it appears in the thresholding, is as this square of a norm u minus u prime gh convolved u minus u prime, which I can also by the semi-group property really write as this expression. So that's what you wanted to see. So those are the two definitions. And so the statement is the following. So we have an initial configuration for the thresholding scheme E0, which is a characteristic function as finite parameter. That's this condition. And then we consider the solution which is given to us by the thresholding scheme. We look at the piecewise constant interpolation. So chi h of t always is equal to chi n for t between n h and n plus 1 h. So chi h is the piecewise constant interpolation of what you get out of the thresholding scheme. And now the statement is, suppose you're given any subsequence of time steps that goes to 0 and a configuration chi, which is the limit in the sense that you have strong convergence in L1. So chi is, again, a characteristic function. And you have convergence of the time-integrated energies. And that was kind of the, this is kind of a benign assumption by the compactness result I stated last time. But this is an assumption which doesn't come for free. But, however, it has a tradition in this type of results. And if we make this additional assumption, we get convergence towards a Bracke solution. In the sense there exists a curvature, which is a square integrable and a curvature in this usual weak sense, where here you look at the tangential divergence on the interface. So it's defined by this integration by parts formula on the interface. And equipped with this curvature, we have that the limit satisfies mean coverage flow in the sense that the family of Bracke's inequalities is satisfied. So in other words, that this family of localized dissipation inequalities is satisfied. So for any localizing function zeta, for any non-negative function zeta, you look at the local parameter. And now that's the dissipation inequality in a time-integrated fashion, which means the localized energy at time capital T plus the integral of the dissipation is bounded by the localized energy of the initial data. And so this is a good factor of 1 half, which has to do with the two here. So this is exactly with the right pre-factor Bracke's inequality. And I remind you that there are these two terms. There is the first term, which would also be there if you would just look at the non-localized problem, where the dissipation rate is given by the mean curvature square. But then there is the second term, this transport term, which comes up because of the localization and involves the gradient of the localizing function. So that's the main statement, that there is this conditional convergence towards a Bracke's solution. And it's also true in the multi-phase case, although here I'm just formulating everything in the single-phase case. OK, so that was the main theorem, which I stated last time. And the idea is to get it from the tools of the Georgie, which I introduced in my first lecture. So these tools for gradient flows or minimizing movements in metric spaces, namely the tool of a variational interpolation and the tool of metric slope, which allow you to get the right dissipation inequality with the right factor. And the way we use it here is by replacing the distance function in both places by the metric slope. And that's the way we're going to use it. So there are two types of interpolations. There is the piecewise constant interpolation. There is the variational interpolation in between. And now if you take both into account, you do get already on this time discreet level the right energy dissipation inequality in the sense that the energy at a later time is controlled by the energy at an earlier time plus the time integral of the metric slope squared. 1 half plus 1 half is equal to 1, as it should be. And it's kind of the merit of the variational interpolation to get this second term. And now the most, yeah, OK. So then that was also Lemma, which I stated. And I kind of messed up the proof. But one of you found out that there was no reason to get anxious. The proof is OK. I can finish it today if you want. So in order to prove this convergence result, in order to prove these localized energy inequalities, we don't work with this minimizing movement structure. But we use a second. I mean, the fact that thresholding is also minimizing movement satisfies an entire family of minimizing movements. So for any non-negative localizing function which is sufficiently smooth, so C infinity is not important, but it has to be a couple of times differentiable, as you will see from the proof, we have that thresholding satisfies minimizing movements with respect to a different structure where it's easier to first write down the metric. So now I put, to distinguish it from the standard structure, I put a tilde on the H. And the only change here is that I smuggle in the localizing function there. And besides that, this is unchanged. So I'm still the square of a norm. A norm provided zeta is strictly positive and not just non-negative. So that's the definition and the notation. So the only difference is that I smuggled in the cutoff function here, and I put the tilde here. But for this to be true, I have to modify the energy more substantially. So it has to depend on the previous step. So it now depends not just on you, but also on a second configuration, chi. Of course, I put the twiddle. And the first term follows the same principle as here. It's just localizing the energy from before. But then there are two correction terms which involve the commutator between multiplying by the localizing function and convolving. That's an important correction term. And then there is a correction term, which always in the end will turn out not to affect anything, which comes from smuggling in the, from in a certain sense, from the error. I mean, in principle, you would like to smuggle in the cutoff function here. But you have to do it here and the difference between these two expressions brings in another commutator, which involves the kernel at half the time. So that's the statement of lemma 6. And the proof is very much like this here was lemma 2, of which I gave the full proof. And I started giving you the proof of lemma 6 last time, but I messed up a little bit. But everything seems fine. So that's the starting point. We have, if you want, a localized minimizing movement interpretation of the thresholding scheme. And now the next slide, and I kind of alluded to it already last time, is really perhaps the central idea. So let's, or kind of gives the central structure of the proof. So this again is the correct, including the factor, the correct energy dissipation inequality for minimizing movements, which you get thanks to the variational interpolation of the Georgie. So that was the main merit of what the Georgie did. He kind of recovered the missing one half in front of the square gradient term by introducing this adapted interpolation. So let me, well, I remind you of it in a second. And now we want to apply this to our situation, which is a little bit more complicated, because the energy functional depends on the previous time step. So there is this additional dependence on chi n minus 1. And the energy functional is not named e, but e tilde h, but that's just a different name. But if you think a bit about it for a second, that doesn't constitute any, that's not a problem, because you're just using this inequality for one time step. So that your functional depends on the previous time step is no problem at all in applying to Georgie's idea. So we can just copy this line and give the symbols their more specific meaning. So we now use this localized energy functional with fixed second argument. We get the metric slope of that functional in the first argument at place chi n, the metric slope of this functional, again, with fixed second argument in the variational interpolation. And let me remind you the definition of the variational interpolation. So u of n minus 1 times h plus t, and I put a twiddle here and an h superscript, is the minimizer or is a minimizer of exactly what belongs to this structure. So e tilde h u chi n minus 1 plus 1 over 2 h d tilde square h u chi n minus 1. Among all, u in x, and remember x was the space of non-necessarily characteristic functions. So this variational interpolation typically will not be a characteristic function, because it has to satisfy this variational problem. Sorry, and here is a t. There's no reason why this variational problem is minimized by a characteristic function. In general, it will not. But we don't care. We know that it will stay close to a characteristic function. We'll use that as stay close to a characteristic function. But in fact, it's not a characteristic function. So this variational interpolation kind of leaves a little bit the geometric world by taking this variational structure by the letter. OK, and I put a tilde on it because it's really the variational interpolation which belongs to this structure and not to the simplest structure. So of course, they would also give you different ways, different interpolations. But again, we don't care. So again, I mean, this was the abstract result by the Georgie where the big thing is to get this red term. And now we just stupidly, mechanically apply it to our situation. And now of course, in order to get kind of the dissipation inequality in the limit, you want to sum up this inequality. You want to sum up the time steps of this inequality. And now because you have kind of an energy functional which has the second argument, you realize you don't have a telescoping structure anymore. So if you take this energy inequality or this dissipation inequality and you sum it up, then you have this telescoping property that kind of these terms cancel. Now this is no longer the case because here you have n, n minus 1, and here n minus 1, n minus 1. So it wasn't a problem going from here to here, but it now becomes a problem when we want to sum up. So we do sum up. And because of this mismatch, in order to still have something like the telescoping effect, we get an error term, which is this red term, which exactly kind of monitors the difference of the energy functional in the second, if you want, artificial or additional argument. So this red term here exactly is a consequence of the fact that telescoping here fails. So you get an additional term. But now this is, in a certain sense, already exactly the right structure. We have the green term, which is kind of, in a certain sense, the original to Georgie term, which will converge, at least in a lower semi-continuous way, to the main curvature dissipation term. And it's this term here, which will converge to the transport term with the right prefactors. So the remaining proof is really now really using very much the philosophy of Georgie that, in a certain sense, this inequality calls for using lower semi-continuity methods. And indeed, on the green term, we just have one kind of convergence in the right direction. We're using a lower semi-continuity argument to go from here to here. In the other terms, by our assumption, we actually have convergence. So that's really the structure of the proof. We just go from this discrete version of the dissipation inequality to the continuum version of the dissipation inequality. So I hope the structure of the proof is clear. And now I'll ask you for questions. But now I want to give you kind of the main two arguments why the green terms converge or converge in terms of an inequality and why the red term converges. So that's what I want to explain now. Do you have questions? No questions? I can't believe that. So any definition you miss or anything you want me to write on the blackboard? OK. So before kind of giving you more details, let me tell you what now the strategy is in going from here to here. And that now uses kind of the lower semi-continuity property built into the metric slope. So let me recall how the metric slope was defined. So very generally, I mean the definition by the Georgians, you have a functional. And you're interested in the metric slope in a configuration u. And that was defined as a limb soup considering all other configurations which approach you in your metric. And then you're looking at the difference quotient of u minus uv plus divided by du dv. And in fact, it's a matter of simple algebra that and that's more in a natural to look at. If it's more natural, I mean from the point of view dissipation inequality, it's more natural to look at the squared metric slope and it's convenient to put the factor one-half in there. And it's an easy consequence that that one and we just need this inequality can also be written in a more quadratic term and I don't need to put the plus here minus one-half d square uv. So if this here were linear form then this is just simple linear algebra and you just plug this estimate in here So that's just a little rewriting and now here comes the main idea in this lower semi-continuity argument that since we're just interested in this type of inequality we are completely free to choose in which way we approach our configuration u here in this sequence. We're allowed to take anyway but we may take something that's convenient for our purpose and what's convenient for our purpose is to kind of approach the configuration u which now you should think of a characteristic function in a smooth way like we do it in differential geometry or in the calculus variations namely by kind of deforming the set. So it's convenient to consider for a given vector field xi so I always call vector fields xi and let's assume that it's perfectly smooth. So for a given vector field xi we consider the flow and we move the set by the flow in other words we evolve the characteristic function by solving this transport equation. This initial data, so initial s is just a parameter a fake time with initial data given by u. So now we take this curve so instead of approaching this point u in configuration space in an arbitrary way we take this very specific way of approaching it and why do we do this because that leads to the concept of the first variation because by definition the first variation so here I use partial and here I use del the first variation of a functional in the point in the configuration u in direction of the vector field xi is given by taking the derivative at s is equal to zero along this curve and that typically in our case will be linear functional in xi so this is how that's one way I mean how you would define the first variation of the area functional so it's the right geometric notion of gradient of first variation and now you can just do what I said you can take this specific approach and if you do that you see that you get another lower bound where now you soup over all vector fields and what you put here is the first variation of your functional u xi and okay now I have to choose my metric so perhaps I should do this in the more specific case of the tilde problem chi so as on the blackboard and here I should put a dot square at u controls this quantity here and here we have a term which just comes which is the infinitesimal so now we're looking at our specific metric and we're approaching u along this nice curve and we're just interested at the infinitesimal part here and that's exactly the expression which I've written down there so there's the localizing function there is the convolution with h over 2 and there is xi dot grat applied to u squared so that's what saves the day this is the fact that in our situation the metric slope can be kind of bounded below by something that involves the geometric first variation of our energy functional which gamma converges to the parameter functional so that the first variation kind of will converge to the first variation of the area functional so the weak expression for mean curvature that saves the day and then we have to worry about this term but it will be a consequence of our convergence assumption that this term also converges to what it should converge to so here you see kind of really the Georgie's ideas at work that you can use inequalities the right way so we've passed from something which our priority looks kind of complicated namely the metric slope to something that we're much more familiar with in geometric analysis namely the first variation and if you want the metric slope controls the first variation is a linear form and the metric slope controls the norm of the linear form where the norm is taken with respect to this norm here of xi so that's perhaps the most important idea and that's exactly what you would hope from getting from the Georgie's approach that you get these kind of benevolent inequalities which work in your favor that's exactly I think what one hopes from this type of approach okay so and now the crucial ingredient is that when you look at the first variation now we have our strange modified localized energy functional which is the localized energy functional but then it has these two error terms that at least on the level of the first variation the localization just acts trivially so I mean if you want to use fancy words you might say that localizing the energy functional and taking the first variation commutes because here what's stated is that if you take the first variation of the localized energy functional the answer you get is the non localized energy functional but with the localized first variation so the xi kind of goes from sitting in the functional to sitting in the variation and it's not exactly true but it will be true in the limit because this expression here can be estimated by something which depends on high norms of our localizing function and on high norms on this vector field but that's again the nice thing about the soup formulation we can fix when we pass to the limit we can fix the vector field and then take the supremum at the end that's the basic idea how you prove lower semi-continuity of norms or for instance the BV norm so one's using the same trick here so there's no problem with the fact that this here depends on the localizing function and depends on the test vector field and what's important is that what you get here on the right hand side is something which you control by the energy estimate at least when you integrate in time times something that goes to zero with power one fourth but you don't worry goes to zero as h goes to zero so that's in a certain sense a consistency I mean if I were an americist I would say this is a certain consistency property of the scheme okay so this will help us to pass to the limit in the green term and you would, yeah, okay so now what's slightly more subtle is to pass to the limit in the other term in the red term perhaps I should kind of write this down somewhere so before taking the limit we have the h tilde functional evaluated excuse me at chi-ht second argument is chi-ht by the way if in this localized energy functional if both arguments are the same then these additional terms drop out and it's really just the localized energy functional so that's a good expression and then we had plus the integral from zero to t one half of the metric slope of e-h tilde with fixed chi-ht at chi-ht plus h the square plus one half metric slope of e-h tilde at chi-ht but now evaluated at the at the variational interpolation dt so that was the curvature term and then we had the term which will give rise to the transport term one over h e-h tilde chi-ht plus h chi-ht minus e-h tilde chi-ht plus h chi-ht plus h goes to this bracket dt is less or equal than e-h tilde chi-zero chi-zero so that was the inequality and we want to see that this goes to let me just write it in exactly the same order so in principle it goes to this here with the c-zero sitting so that's the main task and so the fact that we can lower bound this expression by fixing this test vector field by the first variation and that as I will point out we have good convergence of the first variation of our convergence assumption we're hopeful that we can go from here to here because we also understand the limit of this term again because of our convergence assumption so at least in this sense it will converge but what I now have to tell you or before getting into more details why we have the second convergence and so that's lemma 8 so here it is so again that's kind of a very general lemma which in a certain sense is a consistency result and again it takes it looks at two general configurations u and chi chi although I use the letter doesn't have to be a characteristic function and here we're looking exactly at this type of difference which is sitting here so we're looking at the difference quotient if you want of this localized energy functional in this additional argument so that's exactly what we need for here and we have exactly the same combination u, chi, u, u so that's clear that we will need something like this and what turns out is that this here can be related to the first variation of the metric the non localized metric so in fact this term is to leading order equal up to factor one half which I forgot last time is equal to the first variation of the metric expression in direction of the vector field which is given by the gradient of the localizing function so that's slightly more non trivial or more subtle than the previous lemma that one has this type of relationship and again there is an error which because by the standard a priori estimates we control this here in L2 in time this in L1 in time this in L infinity in time we get a certain order here which again is as bad as H1 fourth but we don't care and now how do we get the connection between this discrepancy term here which is sitting here and a curvature term which is where we want to get to to this term here where there in a certain sense all we have to use is the Euler Lagrange equation for this original variational principle for the original minimizing movement scheme for thresholding so the fact that thresholding satisfies this non localized minimizing movement scheme which we proved in lemma 2 because just taking the Euler Lagrange equation using the fact that it's stationary we get this relationship we get the relationship I forgot a minus sign here we get the relationship that the first variation of the metric of the metric term in direction of some arbitrary vector field psi is equal to minus times the first variation of the non localized energy functional and now that's exactly this is exactly what you plug in here and the only changes that you get the gradient of the cutoff function which plays the role of the first variation and that's exactly this term in the continuum limit that's exactly this term the gradient of the cutoff function plays the role of the vector field in which direction you're taking the first variation of the area functional so now you should see if you don't see ask me question you should see that it's not at all surprising that this term here which is related to the fact that the energy functional depends on this additional variable exactly gives rise to the transportation term so are there any questions with respect to the statements or the strategy? Yes, so in a certain sense so this is I mean that's the term which comes from the DH we have to pass to the limit too and now if you think about this how does it look like? So this is something which one has to do in the process of the proof so let me copy it here so I'm using kind of this operator type notation where this here acts on everything which comes afterwards but now let me put the brackets there so you can kind of massage this term a little bit to see that essentially it is given by the following expression so you can so this here is equal to the divergence of u times psi minus this term that term doesn't play a role in the limit so you're left with this one this one gets on you can put this in the convolution you can put this onto the g function and so what you get is this type of expression and okay and there's another argument which tells you that as h is small you can kind of pull this psi out and you would I forgot this I forgot the square there should be a square there's a square here you pull the psi out so it gets psi square and here you get this type of expression okay so so this is all I mean this can be controlled very well and just depends on the smoothness of psi and now if you look if you look for a moment at this term what is this term well I mean here here you have something u is something which approaches or is a characteristic function you're taking its convolution and then you're taking the gradient of the result so that will be something that's very steep near the interface but you're scaling it in the right way so that this here now I've treated I did another bad thing I've treated psi as a scalar but I should really write it like this so this here will in a certain sense converge to the normal so this thing here converges up to perhaps this constant c0 to this expression so what this what this metric term is picking up is this first variation of this infinitesimal version of the metric term is picking up is really the normal the altitude norm of the normal component of the vector field by which you flow and that's exactly the metric structure which you would expect for mean curvature flow that's the formal inner product the altitude norm on the interface now here's this twist that we did it not with respect to the original metric function but with respect to the localized metric function so therefore you have this cutoff function sitting here but the cutoff function is also sitting here by the lemma which I had before and that leads to the fact that the cutoff function is also sitting here so that's fine so that gives you that gives you perhaps a little bit of an intuition that this metric term at least in its infinitesimal version is related to what it should namely the L2 scalar product on normal velocities well I mean here you just so the advantage of this one you're not looking at the distance function in the large which has the same problem you're looking I mean you're just looking at the infinitesimal part of it I mean here that's you're not looking I mean when you do this lower bound you're not looking at the metric in the large but you look at the infinitesimal part of the metric by you know I mean this behaves like s square times that expression and so therefore you circumvent this problem with the fact that this metric in the large is badly behaved because in the end it's just the infinitesimal the metric tensor in the Riemannian sense that plays a role that wouldn't make much that wouldn't make much sense I mean this like in a certain sense like in the Andrian-Taylor-Wong scheme this metric term is a reasonable term only kind of close by and it would have kind of a different meaning kind of let's say non-physical meaning if you look at two two distant configurations I mean it's the same thing with Alan Kahn right Alan Kahn is the gradient flow of Ginzburg-Landau with respect to bulk L2 the bulk L2 in a product so that's the that's the Ginzburg-Landau functional and now if you look at the L2 bulk so let me stick to my periodic gradient flow of this you get the Alan Kahn equation and so the Alan Kahn equation has a gradient flow structure but it's not the right gradient flow structure of the limit because while the functional converges in the sense of gamma convergence to the perimeter you're changing the metric you're going from bulk L2 to surface L2 so also I mean I could write down a minimizing movement scheme for the Alan Kahn equation based on this metric and I would know by standard PDE theory that this is a good I mean that the scheme converges but now you know there is this interchange of limits and it wouldn't really make sense if you're interested in mean curvature flow in the end to give a lot of importance to this bulk L2 metric and it's I would say it's a little bit the same thing that's taking place here it's really the infinitesimal it's really how it acts on close by configurations that matters more questions so right so that's in a certain sense now those are the two main steps of the proof and I hope the structure the structure of the argument is clear and I hope you see that it really kind of uses the takes benefit of the ideas of the Georgie on gradient flows so now I can do several things I can kind of fix my proof of Lemma 6 and go back to where I started last time and close that but in the end it's a little bit a boring algebraic calculation very much as in very much in the same style as Lemma 2 so there is no mystery I mean it's just in a certain sense of course we did reverse engineering I mean we said well somehow we want to have this function and this metric what are the terms we have to put there so that it works out and the proof kind of just works the other direction so there's nothing deep it's just I mean in a certain sense I mean it's clear that these two terms which you add have to vanish when xi is constant when zeta is constant clear these commutators are zero when this is a constant function because multiplication with a constant function commutes with everything and in a certain sense these terms just just monitor how how much you have destroyed symmetry by smuggling in smuggling in kind of a cut-off function and which prevents you from kind of using symmetry in the way you want so I can do it but it's not inspiring I can go back to the proofs which I didn't give you on Tuesday or I can give you the proofs of these two lemmas or one of them so what's the vote who wants to see who wants me to go back to let's say the compactness proof and that kind of stuff one person who wants me to go I mean to finish this proof of lemma 6 two persons and who wants me to go on with 7 and 8 more than majority okay so then I I mean there might be time for the other stuff too okay so so let's go back to lemma 7 because that's a bit easier and gives you already the the main idea so what can I raise so essentially I need the definition of the first variation because it's a statement about the first variation and so by the way I mean I've so for those people who don't believe that I fixed the proof of lemma 6 I mean of course it might still be a mistake but we can post it if you want to so let me see how I wanted to do this lemma 7 so so there is there's first so proof of lemma 7 so there is the first step is just a representation of the quantity which we want to look at so the first first variation of the localized energy functional with fixed configuration chi at configuration u in direction of the vector field xi minus the first variation of the unlocalized energy functional in the configuration xi in direction of zeta times xi that's the quantity we want to we want to estimate and it can be written as 2 over square root of h gh over 2 u minus chi times zeta gh over 2 xi so that's the formula and then from this formula we get the estimate by estimating this remainder term so let's first let's first get this formula so so we have to look at we have to look at what these what these expressions are using the definition of the first variation and so let's start with the first term which is the first variation of this term so wherever there is a u we get a term so so we'll get term from here from here from here and from there is that correct so many terms I'm afraid so and then of course we get also terms from here so one over square root of h and everything is under the integral zeta so for the first term we always get whenever we plug this in the expression we get is the operator the directional derivative in direction xi of u with a minus sign but there is a minus sign sitting here so for the first term it's a plus sign for the second term we get a minus sign 1 minus u gh gret xi gret u and then we get one from here minus xi gret u zeta gh over 2 convolved with 1 minus chi and we get 2 from the last term minus xi gret u zeta gh over 2 gh over 2 u minus chi that's the first term and the last term is minus chi u minus chi gh over 2 gh over 2 xi gret u so nothing has happened I've just mechanically replaced replaced things and so that's the that's this term and now we have to take the first variation of of this energy that just gives us two terms the first term is I continue with the integral so the first term is zeta xi gradient u gh u that's nice because it would just cancel with this one and the second term is plus zeta no there's no zeta here 1 minus u gh zeta xi gret u and we're left with all the other terms so the fact that these two cancel is already pointing in the right direction and now we have to see that there is a substantial simplification with the other terms and that was a bit subtle and not deep so we have to use we have to use the kind of anti symmetry of a couple of these terms of the commutator so let me continue here and so let me see first of all whether I copied everything correctly this here this term I can right away combine two commutator this term here is the commutator of so I can write it as minus 1 minus u commutator of zeta gh xi right u right that's that's this term so the first term goes away and the difference of these two terms just gives rise to commutator so now we just have commutators and the idea is essentially to organize things in such a way that that these products I mean all these terms have the structure of function operator function and you organize the terms in such a way that the this term here this directional derivative is in the second argument and if you do that you have to switch the order here for instance and so you have to use that this operator here is anti-symmetric I mean the commutator of two symmetric operators is naturally anti-symmetric so when you switch the order in this product you change the sign and I hope that's what that was the right logic to give me for the first term here now the first term I'm happy with so I just copy it but there is a simplification between the first term and the second term why is that the case because let's do that for a second in a different color so if here I do exactly what I said I change the order 1 minus chi zeta g h over 2 psi grad u so I've done exactly what I said of course it's point wise it's not true but it's true after integration I see that there is a certain cancellation between these two terms and what I'm left with is the term u minus chi zeta g h 2 psi grad u so now we're done up to here and we're just left with these two other terms and now I think what I said should be okay I'm freeing everything I change the order here I'm already in the right order and if I do that I see that what I get has this structure so that was the term which we got from here here I'm switching order once more so I get the minus turns into a plus sign g h over 2 convolved with the commutator and here I'm I'm fine so I just copy that over 2 g h okay so now I have to see that this expression simplifies and indeed it does by kind of in a certain sense by spelling out what these commutators mean you see that this here is nothing else than two times the commutator which is sitting here two times g h over 2 zeta g h over 2 and then the last thing to do is to bring this here on the left-hand side which is what we did there okay so it's just this type of straightforward calculation which brings what you expect to be small into a nice form and and of course it's not it's not the right type of thing to do on the blackboard but now we can estimate this term that's so this is if you want algebra and now comes the analysis and the so let me erase this here so the second step is estimate of the left-hand side and this here was right-hand side of course I'm always getting confused with left and right so okay so what this certainly cries out for using Cauchy-Schwarz because here if we take the L2 norm of the first factor we get something which we control namely our distance function which we can afford so by Cauchy-Schwarz this is estimated by so I don't care for constants now so let's forget about the 2 gh over 2 u minus chi squared and the commutator gh over 2 xi ret u squared so that was just Cauchy-Schwarz and we're happy with the first term because this here is equal by our definition 1 over 2 squared of h the distance function the non-localized distance function between u and chi and now all we need is that this term is bounded controlled by some derivatives of the Cauchy function and the test vector field then I get the right powers of h here this is a square I'm taking a square root so I get the metric to the power 1 and I get 1 over h3 4 which I can write as h 1 quarter times 1 over h which is exactly the right hand side so the only thing is to convince ourselves that this that this term is bounded and now a priori this looks this looks a bit dangerous because I have a gradient on u and u is a characteristic function or close to a characteristic function and not only am I taking the l1 norm which would be okay I'm taking the l2 norm so certainly it's not a naive estimate which I can use here and so some of you will know that commutators have regularizing properties so if you have the commutator between a smooth function and a differential operator or some type of Fourier space operator then the commutator has better regularity properties has better smoothing properties than each of the terms individually and that's what's going to save us so let me but that's of course a very standard standard PDE PDE idea and in fact we can get a point wise estimate on this in fact we can even control the integrant in the l infinity norm so not just l2 but l infinity which is even better because we're always on the unit square thanks to our periodic boundary conditions so how do we see this we see this by using so now I'm looking at the I'm looking at this expression evaluated at a point x so let's forget for a moment that I'm applying the commutator to some function let's call this function v for a moment so what's the formula I get so for the first factor I'm taking the function I'm convolving it and then I'm multiplying so of course I can put the multiplication inside the integral so I can write this as g of h zeta of x leave a little bit of space x minus z and for the second part of the commutator I'm doing the same thing in the opposite order I first multiply this function with zeta and then I convolve so I take a minus x minus z here so that's the that's the expression and now you see just from this simple representation you see that there is something to gain the smoothness of the Kader function because you have a difference here over a small length scale because the kernel concentrates on scales squared of h so x and x minus z are very close and that's what we use and now we remind ourselves that the function v was actually xi x minus z times gradient u x minus z dz so so far we haven't done much besides rewriting the integral and now the key step is to get rid of this to get rid of this gradient because we can't afford the gradient on this function because we have no control on that so the idea of course is to do an integration by parts and to do an integration by parts in the z variable integrating in the z variable and which means if I do integration by parts the gradient will kind of affect everything here which depends on z so there are three places where the z is sitting so we get three terms so when the gradient falls here there's no minus sign from partial integration because of the minus z here so we get grad ghz that's a bad term because taking a derivative of this very pointed gaussian brings in huge slopes but luckily it's multiplied by this small expression and then also of course by that one so that's the first term now the derivative can fall on this term which gives us now I don't know about the sign so this is unaffected there's no minus sign from integration by parts there's no minus sign from differentiation here so it should be plus gradient zeta x minus z xi x minus z and then the gradient falls on this one here which is plus ghz zeta of x minus zeta of x minus z gradient xi x minus z and everything is multiplied with u of x minus z dz and here the sign should be minus sign okay we don't care we use the triangle inequality anyway so how can we estimate these things so here this term we can estimate by the infinity norm by the Lipschitz norm of the test function zeta by the infinity norm of the test vector field xi and u is a function between 0 and 1 so that's always estimated by 1 and we're left with the integral z gh gradient ghz so that's the first term the second term is estimated has the same pre-factor but instead of this integral you just have the that integral and the last term there we don't need this difference so we're being very cavalier here and estimate it like this and get again that and now this by normalization is equal to 1 and here the gradient is bad but the z saves you by scaling you figure out that this is exactly the same expression as if the scale was equal if you were looking at the standard Gaussian and that's of course finite and computable so we get that this entire thing here is controlled by the al-infinity norm of the gradient of xi and the al-infinity norm of xi and the vice versa so we need some regularity of these test objects but as I said in the beginning we don't care at all so that's the advantage of this setup so that proves this this lemma 7 can start with lemma 8 or I can answer questions that's a threat right either now you have to ask come up with a question or you have to suffer another proof it's over oh I'm sorry no I always thought that sorry my apologies so I went over time and there's no time for questions which is good so after lunch then I can prove I can tell you about the next lemma which is slightly more involved in analytic terms or I can answer questions or go back to lemma 6 or to the proofs from last time