 Non è chiuso, basta che mi resti... Ok, ok, non so come la taglio, non la aiuto più. Ok, quindi questa è la ultima lectura da Enrico che va a parlare di questa cosa. Prima di tutto ho bisogno di scegliere alcune domande che erano durante il break del caffè. Quindi, in realtà questa foto è perché diversi persone scegliano di parlare della mia foto di Blackbird. Well, I was going to blame Francesco only endowed me with the two dimensional blackboard, but since he also has a very powerful computer screen I can make better pictures. So the main complaint is that my picture on the blackboard are always too naive because the blackboard is too dimensional. And so last time I was speaking about graphs that are also cones. So if I have the structure of graphs plus cones in 2D is very bad in the sense that I say the cone passes through the origin. And then I look at what happens corresponding to the point plus 1 and say the cone passes through here. And then there is a straight line that joins this point to the origin. And then if I look at what happens at minus 1, well there are only two possibilities, right? Either I am above this straight line or I am below say that I am above. Then what happens to me is that if I have a cone, which is also a graph in 2D dimensions, well it ends up being say above this straight line and being a convex set which means that this cone obviously cannot be a nice minimal cone because it's convex. And whenever you compare its fractional curvature with the curvature of the plane, well both the curvatures have to be zero, but one set is strictly contained in the other and so there is no way that the cone has a mean curvature zero. Nevertheless, this is just a problem of pictures in the blackboard because in more dimensions you can have things like this. So this is for instance the graph of mod x minus mod y. So you have a cone in one direction, a cone in the transversal directions but the two cones have opposite convexity. And so in principle you cannot repeat this kind of convexity argument for three-dimensional or more-dimensional pictures. Okay, another thing that people asked me. So last time I was almost completing the proof of this result with Alberto Farina saying that if you have a nice minimal cone, which is also a graph in rm plus 1, and then I was asking the boundary of e to be, I think, c to beta with beta bigger than s, then e is a half space. Okay, and so some people said how comes you are not putting any dimension of restriction, that sounds too good to be true, because we are used to counter examples in classical case. Well, basically we only have two counter examples to think about. One is a sort of Simon's cone, which is a cone done like this, and the other is this Bombieri de Giorgio Giusti surface in which the Simon's cone, since you can think it's r8, r8 is the product of r4 times r4 r8 is r9, and you have the subtle shape surface constructed by Bombieri de Giorgio Giusti or something. So, of course, a result like this, it's an ugly picture. No, yeah, since we're talking about pictures, I wanted to make a remark. So the Simon's cone, you draw it like that because it's misleading the picture. Because it's rotationally invariant. Because it's wrapping around the origin for every direction, right? Otherwise you could kill it easily. You will never be a minimizer, right? You're just a shop. So you cannot never separate enough spaces the origin from the rest of the cone. That's the thing. Okay, so this is a nice reminder. Let's go deeper in Francesco. It's really, you never can move an half plane to cut the origin outside of the rest of the cone, right? So you're saying that I could do this and reduce the energy, right? That's the picture, right? Right. So, in fact, this is a bad picture because this is in R4 and this is in R4 and the picture is rotational invariant so I have to think that I'm rotating through here, but I'm also rotating through here. So I don't know how to describe it. Should I put a rotation like this, a rotation like this? It's not really... I cannot really... Yeah. Yes, if I imagine it in R3 I can rotate this, for instance, but I should also rotate the other part and that makes the picture only valid in R4 or more. Yeah. And that's it. But maybe we have time... Let's see if we have time to re-discuss this picture in the fractional case because Francesco's proof of just cutting this edge will not work in the fractional case. So while removing this triangle here makes, obviously, the picture gain perimeter. In the fractional case, it's not obvious because whenever you remove this piece, well, this piece speaks with all the pieces and it's kind of a mess to take into account all the contributions. Okay, but... So let me say that this theory in itself does not rule out the possibility that objects like these two also exist in the fractional case. They are not known, so it is not known that the analogue of the Simon's cone or the Bombier de Giorgio Giusti surface exist for the fractional picture. But nevertheless, it's not ruled out by this and the reason is because it's true that in this theorem I'm not asking anything on the dimension but I'm asking two hypotheses on the geometry. So I'm asking both that it's a cone and it's a graph. So Simon's cone is a cone but it's not a graph. And the Giorgio Bombier de Giusti surface is a graph but it's not a cone. And it's interesting to remark that this surface grows to infinity and it grows more than it grows cubically. So when you blow down this picture you in fact obtain a cone you obtain a cone like this so if you do a zoom out basically you obtain something like this you have the Cartesian product between Simon's cone and R. So basically you have Simon's cone repeated at any slice. This picture makes sense. So I wanted to say you have the Simon's cone and then you repeat it at any slice. And so this guy is now a cone but it's not a graph anymore. So Ok, the further time asking two geometric things that are somehow incompatible makes the theorem valid in any dimension. And so as for the proof the proof was based on this fact that we patiently computed the linearized equation of the fractional curvature prescription. So we said that E was the graph of some function U and we took one derivative of U any derivative and then we obtain the linearized equation for V which was something like this it was F prime of X plus theta minus U of X divided by theta times V prime of X plus theta V prime of X plus theta minus V prime of X divided by theta to the N plus 1 plus S D theta Ok Well I was writing it in two pieces let's say with the plus theta plus the path with minus theta if you prefer you can write it like this thinking that you have a principal value here the fact is that for me it's easier to write it without the principal value because when I take derivatives I want to be careful that I can take the derivative inside the kernel so I thought it was more clean to write it without the principal value so that I can take the derivative inside the kernel which becomes an integrable object rather than exchanging the derivative with the principal value but I'm quite sure that one can also do that Ok Now before finishing the proof can I answer another question ah sorry this was not S it was alpha in the notation and one question was the following so Chavi always says that when alpha converges to one one should recover the classical minimal surfaces but one of the main objections that I received is this is not correct because alpha going to one means S going to one half so somehow please ah sorry it's V, it's V Thank you Sorry, sorry, to many derivatives I mean V is the derivative of U and so basically the objection is that alpha equal to corresponds to the square root of the Laplaccia, not to the Laplaccia so it doesn't look classical what's wrong here well I can give you two answers one is something like functional analysis answer that is thinking that the alpha perimeter is somehow related for instance suppose that E is a smooth bounded set this is somehow up to constant the semi norm of the characteristic function of E in H alpha say to the power 2 if you want now it turns out that this thing is finite only when alpha is less than one half in the sense that this is just sobolev embedding if you like when alpha is bigger than one half the characteristic functions of a set is not finite fractional sobolev norm and this is a possible answer one half is already the critical exponent in this setting this is not completely convincing for me because it looks too abstract but I think there is a nicer motivation here because I know that the kernel of the fractional Laplaccia is one over say theta to the n plus 2s right so here it seems that I have half alpha but I'm integrated in a n so I have to think that this is a sort of Laplace Beltrami written in this way it's n plus one plus alpha I write it as one plus alpha over two times two and here one plus alpha over two is my new big s and so when alpha goes to one this big s goes to one as well and so somehow alpha goes to one corresponds to I don't know the Laplace Beltrami on the surface equal to one so this is somehow a more geometric interpretation ok then someone asked me what happens when alpha goes to zero this is a little bit more delicate in the sense that I have almost no idea so basically this I think it's one of the main hard things to understand what happens is that in a sense maybe nothing bad happens because for most of the data that you can put outside the sets will like just to disappear or to occupy all the domain at its disposal so if you have a set like this for instance outside and you ask what is the s minimal set or the alpha minimal set well when alpha is equal to one in the classical case as Francesco says you go like a straight line but when alpha decreases the straight line is not good anymore because we can use somehow this convexity argument and compare with this half plane a point here we will see at infinity less mass than the half plane so this picture is bad so somehow this point will tend to move closer to the boundary so to form a set that locally sees more set than before so I don't know if the picture is clear but if you have a portion of a set like this this is the zooming of this point this new point near this point you see more set than before then far away you see less and these two things have to compensate but what happens if you do a computation careful enough then you see that if s becomes very small in this picture it's not possible anymore to compensate the mass coming from infinity that becomes more and more important when s is close to zero and so somehow after a threshold s zero when s becomes less than s zero then the set sticks completely at the boundary and so the s minimal set inside the ball becomes empty in this picture when s is small so this is a hint that maybe nothing really bad really bad happens inside the domain in the sense that the only bad thing that happens is at the boundary the set sticks at the boundary but inside there are no real issues the fact is that this is just an example there might be more crazy boundary data that may create other problems so the real thing that one does not know how to exclude is that somehow maybe the mass of the set is getting even completely empty but with a small mass one creates a lot of small particles which are negligible in terms of mass but which have a very bad regularity property even in the plane somehow what happens is that when s goes to zero the size of the regularity estimates and the size of the density estimates degenerates so in principle this is a picture which is kind of hard to exclude because when s goes to zero these little objects are not ruled out by density of regularity reasons and so one needs a new perspective to understand whether or not this is a real picture and it's actually detecting a new phenomenon in the non-local case or it's a wrong picture and one can exclude it by a very good argument ok so let me go back to the proof of the theorem as I was saying yesterday at this point the proof of the theorem is actually finished because u is also a cone so up to now we didn't use much on u we use enough regularity to take derivatives and we will remove this assumption later on and we use the graph property but we never use that it's a cone so we use it now because u is homogeneous of degree 1 which means that v is homogeneous of degree 0 which means that I can take for instance a big M to be the maximum on the sphere so and this is actually a global max for v because if you have at any point where you just projected on the sphere v is homogeneous of degree 0 so the norm of x to the 0 is just 1 and this guy is less than M so let's say that this is attained at some point x bar so if you argue like this you have this expression v of x if I compute it at x bar as a sign it doesn't really matter which sign is but so x bar is a maximum so all this guy is negative now f prime was the integral function but it doesn't really matter because so f was the integral function f prime was just a positive function f is positive now this is an integral of an object which is negative less or equal than 0 which means that the integrant has to be identically 0 for all theta which means that in particular since this part is strictly positive it means that v of x bar plus theta minus v of x bar has to be 0 for all theta which means that v is constant but this means that the gradient of u is constant because this is valid for n e i and this means that u is an affine function so this proves this result now as I was saying this result can be somehow generalized because you don't really need c2 beta is enough that the boundary of e is locally leaf sheets how much time do I have? noon maybe we started five minutes earlier so maybe it's worth to do the argument that says that the boundary of e is locally leaf sheets because it's a general argument based on blow-up and blow-down so basically the blow-up and blow-down results for non-local minimal surfaces were established by Caffarelli, Toggiof and Savin let me say if you want to have a compact reading of them you can also look at section 3 of paper I have with Alessio Figlialli the idea is basically the following so suppose that you have e which is alpha minimal oh sorry this also was supposed to be alpha so an alpha minimal set and suppose up to a translation that 0 belongs to the boundary then you can define for any R bigger than 0 eR to be 1 over R times e so you take any point of e and you multiply it by R now if you send R to 0 this eR will converge to e0 which is called blow-up cone and when R goes to plus infinity you converge to some e infinity which is called blow-down so what is obvious is that this e0 is somehow a zoom in of the picture so for e0 we are looking closer and closer to a point so we look at the picture closer and we look at the picture from far what is much less obvious is well first of all that e0 and e infinity are alpha minimal so the alpha minimal is preserved and what is very tricky is the fact that these objects are cones because you can have nasty counter examples so you can have in principle a picture which has locally many oscillations and so when you do for instance a blow-up or a blow-down these oscillations persist for whatever reason and of course one does not really believe that this is a minimizer but to rule them out one needs an object which Charlie reminded that was called monotonicity formula and one of the tricky part of this business is that these monotonicity formulas are only established with an extension method so this makes Francesco very unhappy because of course monotonicity formula is something that one should see without extension tricks right but no matter what I mean the fact is that these monotonicity formulas are very very precise so in a sense too precise to be true in general monotonicity formula is something that is so rigid that is already a miracle that works in some cases but this somehow poses a philosophical problem whether or not this is somehow the right way to proceed based on monotonicity formula that are so difficult to establish or there must be another way to obtain blow-up and blow-down cones improve that they are cones without monotonicity formula but this is I would say a major issue in a sense ok nevertheless these cones are useful once you know that they are cones and they are minimal they are useful because they can tell us everything we want to know on the regularity or on the flatness of the original surface so the fact is this is a half space it follows that e is c infinity near the origin I mean there is a small neighborhood near the origin for which u is a c infinity graph and this fact is by product of the theory of Caffelli, Rochefort and Savin and the result we did with Bagonia Barrios and Alessio Figlialli basically the idea is that if e0 is a half space it means that when you look the picture very close the picture is getting flatter and flatter but then you have an improvement of flatness here and that tells you that if you are flat enough in fact you are c1alpha and then you have a bootstrap type regularity that tells you that c1alpha ok, so it's no mystery that looking at e0 gives what happens near zero because it's somehow the right scale of the problem the other very useful thing is that if infinity is a half space then e the half space this is actually very strong because somehow it tells you that you started with something flat already and the idea for this is that again if the picture from far is flat enough the picture near the origin has to be flatter so if you write this in a careful way then since you have c1alpha regularity the regularity that you get near the origin has a better scale factor than the one at infinity so you are saying I'm flat at infinity which is an L infinity estimate then I have c1alpha regularity near the origin so somehow the scales gets better and better at any iteration so you will see that the flatness at infinity implies that your flatness at zero is already good enough to say that you are completely flat and you are a half space ok, this is very good because as I was saying you can suppose in this result that the boundary of e is not sorry, local elliptics I mean outside the origin enough to say outside the origin I think before I ask the boundary of e to be c2better outside the origin because it's a cone, you don't want to put regularity assumptions on the vertex and the reason for which it's enough to look at local elliptics around outside the origin is that you have your set e which is a minimal cone and the graph so I draw it like this but it has other directions ok, maybe then what happens if it is not smooth for instance at this point well if it is not smooth at this point it means that in these two other directions I am not flat so locally here I have something like a corner on the horizontal set but this means that I can now sit on this point and make a blow up around this point and when I do this blow up I basically erase whatever it's below right because suppose that I do a blow up here basically I erase and this becomes an infinite point an infinite cone then basically you slice it transversally to this and you just look at this cone and then you have a cone which is singular at the origin now if this cone is only singular at the origin you use the theorem that you had before because the Lipschitz graph is preserved and this is a cone which is now singular only at the origin so you can use the previous version and get that this is a half space but if this is a half space this means that the original set was going to be smooth at this point while you said that it was not smooth and so this is a contradiction then you said well the only possibility is that you cannot use your previous theorem which means that you have another point here suppose that this is again in higher dimension so I have another point here around which I'm not smooth well then I passionately repeat the argument so I sit down here and I blow up again the picture now unfortunately my dimensions on the blackboard are already over but suppose that you start with the very high number of dimensions this means that anytime the picture by one degree of freedom and so after a finite amount of time I finish my directions and so this means that sooner or later I can apply the theorem and be fine so somehow this local Lipschitz assumption is is nice to have it and we don't know how to remove it it's very useful to have something local Lipschitz because the local Lipschitz graph is preserved by the blow up while the class of graphs that are not Lipschitz are a little bit more difficult to treat so that's why we put the assumption so with that theorem we can actually give a very short proof of two results that we obtained with Alessio Figali with other methods so the first result is that suppose that e is alpha minimal in b1 with the boundary of e be local Lipschitz then the boundary of e in b1 in 50 well and this proof is exactly the one I did before because it's exactly the same proof so I can start with my local Lipschitz thing and look at the blow up and use this theorem to classify the blow up at any point in which I'm not smooth the second result is somehow the baseline type result well but local Lipschitz means that I am a local Lipschitz graph so sorry when I say that the surface is in some class of regularity I always mean that there is a direction along which it is a graph with that regularity so somehow up to a rotation I take the origin and near the origin I'm a local Lipschitz graph so when I blow it up I get a cone which is the only trick is that to use this assumption I need that the origin is the only singular point but I do the same I did before at all the other points and so up to a dimensional reduction I can always deduce to this situation and so I have to be seen infinity because of the classification of the blow up this simplifies the proof the other proof was cute though because it was easy cute but a little bit more involved I agree so the other result is the baseline type result let me state it in a strange way so the baseline type result that we had con Alessio with Alessio sounds like this so suppose that E is an alpha minimal graph and so the alpha minimal graph in Rn plus 1 and suppose that in one dimension less things are nice so suppose also that in Rn there are no singular cones this implies that E is a half space ok and the proof of number 2 is again a consequence of this result now you can use instead of the blow up the blow down you start with a minimal graph and you take the blow down cone if you know that the blow down cone is a half space you are done so you only have to say that the blow down cone is a half space ok but when you do the blow down argument then you basically end up with something that has to be a graph and you can avoid vertical parts just by sliding the graph up and down and then use this result to classify the blow down cone which is the original cone the original set to be half space to start with ok now what is the the main idea and many of the non local proofs it's a little bit hard to convince you in very few minutes but what I wanted to point out is that in senso whenever you can do a non local proof that works nicely probably you are taking advantage of the fact that the non local structure charges you for points far away so for instance this is something that we saw in the proof of this result if you remember this proof you basically end up with an integral equation and you are saying you have an integrande which has a sign and so I get an information everywhere so in the local case it would be just a local information but since you are non local this information spreads out so at some point I wrote v of x bar plus theta minus v of x bar is equal to 0 for all theta in a n and this is an advantage in whichever you have an information it somehow diffuses everywhere and this is a kind of philosophy that I think is very nicely explained by this picture unfortunately the scan is not very good but there is a non local effect that has some nice consequences and to explain another situation in which this thing works very quickly just to have you give you a flavor of an idea in which I think it's nice the idea comes from a paper with the video solving in which we obtain the regularity of alpha minimal sets in the plane or if you prefer the classification of alpha minimal cons in R2 ok so don't have time to give the full proof but the main idea is that one can compare what happens to a minimizer and what happens to the translation of the minimizer so somehow one has a minimizer let me always draw the nice picture like this then I translate this minimizer in one direction I draw the picture like this I translate in the other direction and draw the picture like this and then I match the boundary data far away ok and my size of far away it's called R then when I do this I can compute the energy with respect to this R so I make a translation which is say of small order or order 1 doing at size R and so I have this was say U this was U plus R and this is U minus R just to remember that in this direction I'm translating with plus a vector here I'm translating with minus a vector and then I can compute the energy in a ball of values R of 2R now if you do like this you can compute it in the sense that we learn somehow how to compute things in the non local case and there is basically no precise way to do it just be patient and do the computation so in the same way one can be patient and do the computation and checks the energy in some ball of U plus R plus the energy of U minus R minus twice the energy of U ok, this guy is positive because U is a minimizer so these two things are bigger than twice this but it's not that much positive in the sense that it's smaller in general than a constant times R to the N minus 2 minus alpha so in the particular case in which N is equal to 2 no, ok so the first time we did this we did it in the extension in the extension the computation is not that difficult because U plus R is just the thermophism of U so the only thing one has to do is to compute the gradient of the thermophism to change variable so basically U plus R is U of some phi of X I have to compute what well I have to compute the gradient of this guy square it and change variable so in two or three steps I can just compute this and then I factor exactly you translate the harmonic extension and what you have is that you have a common factor which is the gradient of U square times whatever comes from this guy from the determinant of the Jacobian now the first order simplifies and this is the reason for which one takes the plus here and the minus here exactly to simplify the first order because well this phi r is a phi r plus which takes into account this direction and so since direction is exactly the opposite of this the first order of the two will compensate and so you end up with a quadratic term and then the R comes from scaling because you have the integral of nabla U square and which is a cone and by homogeneity you can compute that and ok so in a sense this is a way to do this computation then we discover that you can actually do it without the extension this is a little bit more tedious because you have a kernel and you have to expand the kernel nevertheless I think yesterday we discussed how to expand kernels when we spoke about the master equation right so somehow expanding the kernels one can do similar computations and obtain the same strategy without the extension so if one does that it's almost in good shape because the proof ends like this so I take the minimum and the maximum ah so sorry n equals 2 ok C R to the minus alpha ok it's a trivial step but construction is very important because when n is large R is going to diverge so this factor is going to diverge but when n is equal to 2 this factor is going to help you so this is a great advantage or small dimension and here to conclude quickly the proof is that I am taking the sup and the inf of this translation so if you want, if you are looking at sets the union and the intersection but roughly speaking I will take V for instance to be the max the pointwise maximum between U and U plus R W to be the pointwise minimum between U and U plus R and the fact is that it is not possible that both V and W are minimal let me do a sort of picture of proof by picture because so I am forgetting about U minus for instance I am looking at U plus so if I look at the minimum I have this if I look at the maximum I have this ok, so if both of them were minimizers then I would have constructed minimizers that touch at the point and I would contradict my curvature equation or maximum principle so in a sense I have to find at scale 1 a better candidate for the energy of one of the two let's say for V for instance so I take some V star which improves the energy of V by a universal delta that does not depend on R, so I do this at scale 1 and then V is equal to V star outside B1 so all the contribution that takes me to BR are the same and now I just use what is written here so let's see I have this I write it as ER of U plus plus ER of U plus ER of U minus and then I think that it's minus 3 ER of U so I just added ER of U then if I look at the sum of the energy of two functions so the minimum and the maximum it's the same because point wise the integral will see one or the other so here I can put V or W but now I know information that V for instance is bigger or equal than V star plus delta very important not to miss this sign ok so this is estimated from below by this now what do I have so this guy is bigger or equal than ER of U this guy is bigger or equal than ER of U minus 3 ER of U so it's 2 minus 3 minus ER of U but now this guy is also bigger than ER of U but now I reach a contradiction because this delta was universal and these are I erase the importance that n is equal to 2 so now this factor here is going to 0 and so I get a contradiction and I prove the result so my time is over unfortunately but a small sentence to say that this idea of looking at translation is actually useful also in other context because also the universal perimeter estimate that Ciavi was mentioning is based on this kind of ideas basically one can look again at the thermomorphism back and forth and compute this object now it's not obvious but when you disobject explicitly you see that it remains from below a sort of volume term this is somehow the very different aspect of the non-local structure that picks up these small volume terms and this volume term is exactly somehow taking into account the norm of the translations of the characteristic functions so when you divide this by the size of the translation you get exactly the b v norm and this is why the perimeter estimate works in the non-local case and is not known and it doesn't work in the classical case so the non-local case gives an additional information because it picks up mass from everywhere and maybe with this philosophical sentence I should better stop and let's go to lunch