Maria de Maeztu Annual Workshop | Alberto Maione

e [Music] so welcome everybody and first of all thank you for the nice introduction and for the for the invitation to our answer the director Marcel so um today I'm going to present for the first time so perhaps there are a lot of mistakes in the slid sorry I finished them tonight and this is a completely new result uh about let's say theory of homogeneization on the mathematical pure anal iCal point of view that involves for the first time as far as I know no local linear operators in fractional Divergence form and this is done in collaboration with two young colleges Michael caponi and Alexandra carbotti um both Italian from different University and as much of my collaboration it was born during a coffee break in a conference just for fun and then discussing and so on at some point it became really a project um okay the plan of the talk is at the beginning to introduce let's say historical background on the topic As aransa Told Me few weeks ago like an analyst we are lucky because we can steal something from physics from engineering so we have something to discuss at the beginning to make it more interesting let's see and then sorry I have to go deeply Into the Heart setting which is the no local one and then I will present the main result and just a sketch of the proof but if you interested in more details please let me know so the results should still appear I hope by the summer break it will be online so everything is totally new and if you have ideas suggestions please tell me everything is welcome okay uh the starting point let's say we are in Paris 1970 somehow the school of Jac lons and in particular a um mathematical physicist I think it's the correct pronunciation who named Sanchez paleno he is born in Madrid but he's considered a French mathematician I don't know why and um he is the really starting point of the story and the story is this is the following one so so we are interested in the study of heterogeneous materials so think for instance the overlapping of different materials or mixture of different materials with elastic Properties or whatever it is and in physics the equs we are interested in are in Divergence form so we have let's say minus Divergence of a matrix a depending on X because we are interested in heterogeneous materials times the gradient of U equal to a thir term F and in most 99% of the application The Matrix should be symmetric but please if you know examples in which it is it cannot be symmetric let me know because we are looking for them for to to write a nice introduction and um um important role in this theory is played by the momentum which is the Matrix a times the gradient examples wellknown examples in physics or engineering are equation of electrostatics magnetostatics or time independent it transfer but also if we move to elasticity we have other examples so uh in all the presentation U is a scalar function so from RN to R this is important okay in the case in which we would study homogeneous materials then the Matrix a will not depend on X why is this important because problems appear when the Matrix a the coefficient of the Matrix a start oscillating a lot and from a numerical point of view it's really hard to approach this problem so the theory of the homogeneization is to approximate in some sense the study of this problem so the necessity of finding solutions to this problem by studying a limiting homogeneous case and then to study how far we are from the original Solutions and as I said in France in 1970 it was studied the so-called Astic expansions of solution which is the real mathematical starting point but we are not really interested in this just to make a precise introduction uh what we are going to introduce in instead what nowadays is called the H Convergence Theory the h Convergence Theory so now I go mostly into mathematics can work also for possibly non symmetric matrices which is um much more difficult to treat that satisfy let's say standard growth condition so in the case of symmetry we will have the uniform ticity and the civity condition and we are going to study problem of minus diver Matrix time the gradient equal to F the is problems so U equal to Z on the boundary of Omega Omega for us will be always a open and bounded subset of our n n greater than two greater or equal than two only because we are interested in matrices but once we move to fractional case we should be um delicate with the choice of the dimension and the fractional index so um here we defined capital M depending on two constants Lambda small Capital Lambda and Omega the class of such matrices why should we work with classes because all of this theory is given four classes so it's difficult to define the H convergence like a general statement like gamma convergence thatman um introduced before for the class um but we have to work with classes of matrices so we consider a sequence of elliptic problems in the vergence form and fixed uh SE G in this case we what we do is we look for we find we look for sorry a limit Matrix which belong to the same class up to different constants such that if we consider each solution to the original problem so I imagine that for standard reasons like m whatever you want any of this problem admits a unique weak solution in the sub space h10 of Omega if we consider all the sequence of solutions then we want to find a limit infinite still belonging to the same space such that up to subsequences we have weak convergence in the topology of H1 Z of Omega so this is the first requirement the second requirement which is very important in the nonsymmetric case because in the nonsymmetric case let's say if we perturbate here by skew symmetry matrices the Divergence don't doesn't see it so we have to ask something more to have also the convergence of the moment always in the we to topology n true and then what we ask is if the limit function W infinite is also a solution to a similar problem to the homogenized problem in the theory of homogenization yes yes this is always satisfied in the linear case yes exactly so this is let's say for free in the symmetric case while in the non symmetric one it's a bit delicate and I will not go too much into details but there is um important Theory called the compensated compactness Theory or div Carma is what so called which study let's say this part of the problem okay just few references as I told you everything starts from this works from Sanchez Palencia we are in the 70s let's say but at some point Sanchez Palencia I think during a conference met people from the Italian school which are ano deori and Sergio Spano that few years before were working in this more general theory of at that time was called G convergence nowadays H convergence h means homogeneization and uh so they spoke at some point and realized they were doing similar stuff in different ways and from that point also French people starting to develop um in particular for the non-symmetric case this Theory and the most relevant contribution are the one for from Luke Tartar and Fran mura uh that nowadays are let's say the my opinion the most relevant results in this uh topic there is also a contribution of the Russian school or colleagues here so the very difficult part is this we are in the 70s and French people write in French Italians in Italian Russian in russan so also for me when I studied for the first time this topping during my PhD thesis it was almost impossible to find some references they are published in unknown Journal that don't have online service impossible to find um but even if you find them they are in Russian or in French and for me was a bit tricky and um so my dream at some point is to write a survey or almost a book let's say to only to rec up all of this results because they are very interested in my opinion okay let me go a bit faster so now I introduce the no local part of the story I don't know if you ever work it with fractional app or things like that this is the main idea I just recall the essential things so don't worry and starting from classical I call sometimes local to distinguish between the nonlocal part local sub spaces and it has been introduced also the notion of fractional sub spaces also here I found a PhD Tes about 10 different ways to Define them and the equivalence between the definition in particular in the case in the albertian case P equal to 2 they are mostly all equivalent apart from the spectral one in which we as to have exactly the same again values of the laion but this is something different and the most known one are the one by means of the Gallardo semi Norm or by means of for transfer but now we have a problem which is our problem if you remind we wants to work with Divergence problem prob in Divergence form how can we characterize a Divergence form in the nonlocal setting this is a problem in particular because the space HS where s is our fractional parameter between Z and one uh does not have an evident distributional nature in particular the Gallardo semi Norm does not seems to be the L2 Norm of a weak gradient in any possible sense so what has been studied very recently by she Spector and then by Comey and Stephanie is a new notion of s fractional Divergence and S fractional gradient and we are lucky because in the case P equal to2 they somehow Define fractional sub space spaces that are equivalent to the one we had before these Notions are a bit technical but you can see are defined in all n like an integral of the scalar product between the the gradient it will be at the end and the points and here we have the the r kernel but what is more important apart from the definition is their properties that I will go to show you in a few moments uh the properties are this so first if we consider minus fractional Divergence of the fractional gradient of U for the same fractional index we recover exactly the fractional appli this is in for p equal to 2 this is important uh the second but maybe in my opinion the most important property is that these operators are dual dual in which sense that the integration by parts formula hold so in analysis this is maybe one of the most important results we have without this we cannot talk about uh weak definitions weak Solutions and whatever it is so once we have this we have done half of the work the other relevant properties are Al liess rule so the gradient s gradient of a product which is which has let's say a reminder no local term that we can control so so it's bad but not too bad we have also a pank type inequality and very very important A Relic theorem so we have a compact bending between the space h0s in L2 where h0s for us is the [Music] closure of regular functions in Omega by means of the norm in know RN depending on the nor L to Norm of you and L to Norm of the S gradient of you so this is the main novelty of this definitions okay this is all about this fractional setting all we need and as I said this theory was initiated by Shin Spector in 2015 so very very recent and is still underdeveloped studied by comei Stephanie so um and also some German colleges in RB and her students okay let me go to the main results main result is let's say one and some sketch of the proof so if you remind at the beginning we Define the class of matrices with the bounce between the two constants and this uh let me go back sorry very first this definition here makes sense inside Omega so for people working in fractional stuff you know that it is a problem when we have a bounded domain and we study forance the fractional laion because once we move to the boundary of the set we should know information also outside the set because the fractional appion the fractional operators are not local so we cannot localize inside the set so we have to extend the notion of this class of matrices and what we do this is just our choice it can be um modified improved so if you have suggestions please let me know we said first fix a matrix in the sense of the former definition so we have the control of the bounds but this is in all RN our main class will be given by matrices that outside of Omega coincide with a constant Matrix 0 okay so we belong to this class inside Omega outside we stop them some sense we look with a constant Matrix the symmetric part is not now relevant and now we are able to write finally the uh Divergence the fractional Divergence problem which are given by minus fractional Divergence of matrices belonging to this class times the fractional gradient of U equal to a fix data F inside Omega and U so what is the disly boundary condition in the fractional setting we should work in all the remaining of our n so we should impose U equal to zero not all in the boundary it's not somehow clear what is a boundary but in RN sometimes minus the closure of Omega or minus Omega in this case and for for us now at this point the uh Divergence behave like this after the integration by parts formula so the um first question is uh do this problems admit a solution is it unique can we give some bounds for the solution the answer is positive and for any age so for any problem we can find exactly one weak solution in the sense that this equality hold and we also have standard bounds like in the local case and like in the local case once we have all the construction of the second part of the presentation it is just an application of LX mam LMA or R fres in the symmetric case the main result oh sorry uh so the goal now is to find a limit Matrix at least for different constants such that we have H convergence but now we have to Define again what is h convergence because as I told you before it depends on the class of matrices and on the problems so we give uh this uh definition so this is the first time it is uh done so maybe it's not the best but we hope it is and we say that the sequence of matrices H converges to the Limit one if again we have convergence of solution we and the convergence of the momentum and the main result we achieved is not only we have H convergence but we also have compactness of the class meaning that the constants that control from above and below the limit matrices are exactly the same of the starting sequence otherwise they could be bigger the upper one and lower the um lower one so if we have few minutes I just give you a idea of the proof there are two important results that are used at the beginning of the proof which are given in the local setting by tartar the first one is if we reduce to the classical D problem with the local Divergence the class is compact again so we have H convergence and one results which which is um not so trivial is that if we consider the transpose matrices once we have each convergence for the starting matrices it also hold for the transpose one to the transpose of the limit so the idea now is to work like this first for any J we um only work inside Omega and we call BJ now this Matrix here and by the standard Theory we have we have a h limit so now we call what we hope will be the limit Matrix inside Omega the one given by the local uh Theory and 0 where a z was how we fixed the matrices outside Omega what happens outside of me and what we want to show is in fact there is H convergence in the fraction sense um let's say that if we consider the sequence of solutions so now we know there is one solution for any problem we consider the sequence of what solution by the bounds by the reflexivity of the spaces we have a limit but we don't know if this limit is also a solution of a Divergence problem we also consider the sequence of momenta they also are bounded in a reflexive space they also have a limit but we don't know anything about this limit either if it is something of the form Matrix time gradient so what we have to show which is the most complicated part is that this limit limits speak together themselves so that m is exactly the momentum of the limit by means of the Matrix we defined before so now I stop here with the sketch of the proof but just to let you know inside sorry outside Omega is trivia let's say it's just a by the definition of the limit matrices inside Omega is the hell we have to use local techniques all the properties we gave before for the fractional operators but the most complicated part which um took almost two years to write this um proof is that all the original Theory works for with localization arguments so we consider compact set inside Omega cut off functions and then we extend to all Omega and all this part of the theory cannot work with no local things and what helps us is I SK it before is the fact that this fraction Divergence and this fractional gradient can be seen like a composition of the local corresponding one and the isk potential with this attend we can also apply localization arguments and we are also surprised about that but it it seems to work everything so um future perspectives can we extend for p greater than two so as I told you for P equal to 2 everything is equivalent in the fractional setting for p greater than 2 we have let's say um inclusions between soble sets so it's not so easy to treat then what is done in the classical literature once you have the linear case is to move to the monoton one so where instead of having Divergence of a matrix times the gradient we have a carod fun function that its points and gradients and this is a totally open question the other one that we are now working in this is to extend the So-Cal div Carma and the compensated compactness Theory to this fractional setting and then it is mostly related to my former studies in my PhD pces to consider also other Frameworks like sub Manan manifolds for instance Heisenberg groups Caro groups and so on um just to conclude why I'm interested in fractional stuff because with my supervisor CH cab we are now working in extending uh well-known results given by consulant ra Morales to the fractional setting so this is nothing now that we I'm going to talk about but this is the project here the concrete project here at CRM and we are considering alen can nonlinearities which are very very important in the theory of partial differential equations and applications and we consider no boundary condition recently introduced by dier roton and valoi which are totally new so this is a very interesting class of problems we hope to study so okay um very few slides more so first I said this is a collaboration with two young Italian mathematician and it was possible because last December Michael one of the two guys was hosted here by the campus of the UAB thanks to the analysis seminars that happen every Thursday they have a little of budget so they uh helped us in hosting Michael for one week and just recently I discovered by a ranch that there is a great opportunity at CRM which is a sort of researching Pur project so I just tell you if you are interested I'm interested so I will apply soon that people let's say a minimum of two maximum of six I think is written somewhere yes can be hosted uh here like a co-funding if I understood correctly so there should be also some other findings uh background to work together in uh top quality I hope this is top quality research in um here at CRM and last advertisement is related so there was a spoiler by K before is this uh class I'm going I'm very happy to be able to give not here but at UB but only because people working on this topic live around UB mostly and this is uh the theory of gamma convergence so now I didn't speak about Gamma convergence but if you restrict to the symmetric case so you can move from the problem to the oiler lrange uh let's say to the energies sorry all this Theory can be formulated by in my opinion a better Theory which is the gamma convergence studied by the Georgie and this here is exactly 50 years story and I'm very happy to be able to give this class in November and December Tuesday and Thursday from 11 to1 so I will be very happy if you would join and just another thing today is the 23rd of May and this is exactly five years from my first talk like a PhD student so uh just coincidence but thank you very much [Music]