 Okay, so first of all, thank you very much to the organizers. I think it's wonderful to celebrate Poincaré. Of course, there are some areas of mathematics where the contribution of Poincaré has been very well known and advertised and so forth. And I think on the other side that there are areas of mathematics like partial differential equations where Poincaré has not been celebrated enough and one of them is really partial differential equation and that's what I would like to convince you today. So here is the plan of my talk. I will explain how Poincaré became my hero. And I mean hero from the point of view of partial differential equation. And it was fairly, fairly recently that this happened. And then I will explain why, according to Poincaré, one should study partial differential equations. Really there is a text where Poincaré has a prophetic insight into partial differential equations and really very modern answers to the question why one should study a partial differential equation. And then I'm going to go into specifics about the contribution of Poincaré in partial differential equations. First of all, the Dirichlet principle. And there I am going to talk and I don't think I exaggerate about the conspiracy of silence concerning Poincaré's contribution to the Dirichlet principle. And then I'll talk about the spectrum of the Laplacian. Poincaré's inequality I'll say go very briefly because we had just the previous talk by Michel Ledoux on this subject. And then I'll talk about a non-linear PDE solved by Poincaré. When I was a student in this building at the Institut Poincaré in the 60s, I was taking graduate courses. In this very room, Laurent Schwarz was teaching. I was taking his class C-star algebra, the no contribution of Poincaré as far as I know there, Poincaré was never mentioned. The god in this course was Gelfand of course. Then in the other room in the other anti-theatre, Darbou was taking the course of Gustave Chauquet. This was about extreme points, convex set, extreme point, integral representation, and so forth. I don't think Poincaré was ever mentioned there. And I was taking a class of Jacques-Louis Lyons in numerical analysis. There was nothing in partial differential equation. And he was in exile. This was not good enough. Partial differential equation was not good enough. At that time, Bourbaki was the king here. Not good enough to be taught in the Institut Poincaré. And he was in exile. Jacques-Louis Lyons was in exile in the Institut. Bless Pascal devoted to a numerical analysis. And that's probably where I heard for the first time Poincaré's inequality. But that's the period. Then I went to Pisa, then the Courant Institute. Always Poincaré's inequality, the inequality of Poincaré. And so for that period, no mention ever, never, ever anything else about Poincaré. And the situation changed, at least for me, about 15 years ago. I was asked to write a survey of PDEs in the 20th century. And I consulted Felix Browder, who is an expert on history of mathematics. And he accepted to join me on this project. And indeed, so here it is, our paper. And I was sure that we would start the 20th century in partial differential equation with Hilbert. And the famous, the big Hilbert and his famous solution of the Dirichlet principle. That's what I thought we would start. And as I'm going to explain, well, story is more complicated than that. And in fact, what we discovered is that before, just before Hilbert, there was major, major contribution of Poincaré in the 10 years of preceding Hilbert and his famous program. Now, you all know the story about the Dirichlet principle that in the period between 1830 and 1850, people realized that there was correspondence between the solution of the Laplace equation, the Dirichlet principle, the Laplace equation with a given boundary condition. And the minimization of the so-called Dirichlet integral. So those people were involved were Green, Gauss, Lord Kelvin, Riemann. And Riemann wrongly assumed, you know the story, that the minimum exists made a big mistake. He thought everything, every functional has a minimizer, it's bounded below. And it was a Weierstrass who caught this mistake in 1869. And so the existence of a solution for the Laplace equation became an open problem. And the standard folklore says that the problem was open until the big Hilbert solved this and gave a rigorous proof that the minimum was achieved and so gave a rigorous proof by consequence that the Dirichlet problem had a solution. So when we started this paper, we went to look at the Hilbert paper of 1900. Also it's a round date, good date. So 1900 Hilbert, the solution of the Dirichlet problem. And I must confess that I was under shock when we look when we saw this Hilbert papers. First of all, what I had in mind was that Hilbert, how do you prove today that the minimum is achieved of the Dirichlet integral? You take a minimizing sequence and you prove that it's a Cauchy sequence by the standard argument. You know the same argument that one uses today to prove the existence of a projection on a convex set in the Hilbert space. I thought that this was the proof that Hilbert had discovered. Of course, he could not have discovered this proof because this would imply Cauchy sequences would mean that he knew that the space L2 was complete, which was not known in 1900. It's only Fisher who proved this in 1906. And I don't know. In my dreams, I thought, well, maybe he invented weak topology and that he passed the limit on the minimizing sequence using weak topology. No, absolutely not. And so let me show you what we discovered. This is the Hilbert paper. In fact, it's in two parts. There is an announcement in 1900 and a more detailed paper in 1904. So this is the more detailed version. And I'm just going to flash it. It's to flash it, just a few pages. I must say it's really a paper which I find very confusing. Just look at the typography. That's not the kind of mathematics, nice mathematics that you want to read. So here is one page. Look at this, completely sloppy. Look, I mean, this is Hilbert's solution of very, very confusing, really totally obscure. In fact, what he does, I mean, in a few words, takes a minimizing sequence for the Dirichlet integral. Then he regularizes it. And then he passes to a limit along the subsequence using Ascoli, because that was the kind of tool, or kind of as a version of Ascoli, kind of tool that one knew at the time. And but I must say this, I considered Hilbert's contribution more as a program. And that he said, let's try to go back to the idea of Riemann and prove rigorously that it has a minimizer. And the program was indeed a very successful program, as it was the beginning of the modern calculus of variation. So this is Hilbert. And so we're very disappointed. And we continued searching. And here is something that we found a little later, Adamard on the Dirichlet principle. And this is a very interesting paper from 1906 in the Boulton-la Société Mathématique de France. Very short, three pages, four pages. And obviously, as you are going to see, he is not happy. He's really unhappy with Hilbert. But he is the only person in the world who has the guts to say it clearly, that the king is naked. You had to be Adamard to be willing to say this. And here are a few sentences that want to extract from Adamard's paper. Monsieur Hilbert has found a method. Did not really have indicated a method allowing the possibility to solve the Dirichlet problem by the method which has been used by Riemann that is minimizing directly the Dirichlet integral. And he gives the assumption. And he says the assumptions of Hilbert, in fact, is that the boundary should be analytic. And the boundary condition, the boundary of the domain, should be analytic. And the boundary conditions should also be analytic. OK, so that's the restriction, he says, Hilbert needs to. And then he says, well, look at this sentence. Someone knows that the existence has been established by other methods, assuming only the continuity of the boundary data. Very mysterious. Who has established it? No mention. I think this might have been the beginning of what I call conspiracy of silence. And so we didn't know who were the other people I'd established just under continuity of the boundary condition. And then he goes on, and he says, this is a very serious restriction indeed. And Hilbert, for example, by his method, could not solve it for a general continuous function. Hilbert's method would never work. And then he gives an example. You see, it's in a clickable. And then he goes on and gives an example. In fact, it was not the first time that such an example. There were similar examples given before by Prim, 20 years, or 30 years ago, before that, of an example of a boundary condition, which is continuous, but not in modern language, not h1h. So it's not the trace of a function of finite energy on the domain. And he says, with such a boundary condition, which is continuous, you cannot solve the problem by the method of Hilbert. And here as a concluding word, nous voyons que la méthode de Hilbert peut être inapplicable, alors même que le problème de Dirichlet a effectivement une solution, period. Really unhappy, he says the solution exists, but Hilbert's method would not give anything. He has not the guts to say exactly that the method is so-so, I don't understand, but he says anyway, this is not a method that is completely always useful. So I was, well, we were fascinated by this sentence in the paper of Adama saying one knows that the existence has been established by others, by other people, of the existence of a solution of the Dirichlet problem. And wanted to know who, and we went backwards now, and what we discovered, that's how we discovered the fundamental contributions of Poincaré on PDEs. So here is a list of four papers which are absolutely remarkable by Poincaré. And I'm going to discuss them. Look also, look where the papers are published. The first one is published in the American Journal of Math. The second one is in the Rendez-Conti di Palermo. Third one, Acta Mathematica, and the fourth one in Journal de... So there are four countries, that says already a lot about the international vision of Poincaré. The US, Italy, Sweden, France. That's quite remarkable. If you look at, for example, where the German mathematicians were publishing, mostly in German journals. So there's something special already about Poincaré and his international collaboration. Look also where the first paper is published. In the American Journal, in 1890, the US was not a very super mathematical superpower that it is today. I don't know, it would be like if you have a good paper and you send it, no offense, but say you send it to a Korean journal or something. That was what the American Journal of Math was in 1890. Okay, then there is other contributions by Poincaré, for example, on the telegraph equation. And here I refer to... I'm not going to talk about this. I refer to a very interesting paper of Jean Mawin, who is here on the Poincaré and telegraph equation. I'm going also to give you a few references of papers about, not by Poincaré, but about Poincaré. And just to show you that there has been very... For a long, long time, there has been very little analysis of Poincaré's contribution to PDEs. There is an extra Mathematica in 1921. The volume is dedicated to Poincaré and there is a long article of an analysis by Poincaré about his own work. And there are a few pages about where he explains what he did about PDEs. Then in the same volume, there is a long article by Adam Maw, where you find also just a few pages about Poincaré and PDEs. And Laurent Schwarz has also in the... published in the collected work of Poincaré in 1955, has also a few pages about Poincaré and PDEs. And then there is a big, big gap, and I think it's only in the past 15 years or so that people have put their focus on Poincaré and PDEs. And I really recommend to you, if you want to learn more details, I'll make a sketch in my talk. If you want to read more details, there are really beautiful papers of Jean Mawin on Poincaré and partial differential equations. I've learned myself a lot about that. I also recommend the book on the uniformisation des surfaces de Riemann, published by the École Normale de Lyon under its collective work. And again, here another paper, a recent paper by Jean Mawin, and also I refer to the paper by Grégoire Allaire on the Sobolevine Equality in Mataphi. But you see, that's all very recent, kind of recent discovery of what Poincaré did in PDEs. Okay, let's start with the first paper, the one in the Korean journal. And I want to go quickly over the first few pages because they are fascinating. And the first thing he does, he gives a list of some partial differential equations coming from physics. So first you have the Laplace equation, and he calls this a Dirichlet problem, like everyone calls, and we're together with a boundary condition. And then he goes on and talks about the heat equation and then the wave equation. That's interesting because it's already a fairly modern presentation of PDEs, elliptic, parabolic and hyperbolic. This is a classification which was then made more general by Adamach in the 20s. But it's already present here, this classification, and it's been followed, if you look, for example, in the three volumes of Lyon-Small-Genes, same thing. And I think Poincaré was the first to give this order elliptic Laplace equation, heat equation, and wave equation that has been followed since by lots of people. Okay, so here I come to what I call, there are a few lines in the introduction of this American journal paper. It's, since it's in French, I'll read it in French, but you have there the English translation of some of the passages. First of all, it's this quick review of the various parts of mathematics that has convinced us that all these problems, despite the extreme variety of limiting conditions and even differential equations, can be considered a family error. So family resemblance between the heat, Laplace, et cetera. It's impossible to recognize. But we have to wait to find a number of properties common, a number of common properties to all partial differential equations. And look at these sentences. Malheureusement, la première des propriétés communes à tous ces problèmes, c'est leur extrême difficulté. Interesting. Not seulement on ne peut le plus souvent les résoudre complètement, complètement, he means explicitly, there are no explicit solutions for the, mais ce n'est qu'au prix des plus grands efforts qu'on peut en démontrer rigoureusement la possibilité. You see that he worked, he himself acknowledges that he worked hard to prove the existence of a solution for the Laplace equation and others. And then he says something that I like very much. Cette démonstration est-elle nécessaire? Why work so hard? And he says, la plupart des physiciens ont fait un bon marché. Because you know that, anyway, the solution corresponds to some physical phenomenon and the physicist will tell you the phenomenon is there. Why prove the, why bother proving the existence of a solution? Okay. And then he says, in any case, even if we bother, should we do it with the same rigor as we do other part of analysis? And then he says, les équations différentielles auxquelles obéissent les phénomènes physiques n'ont été souvent établis que par des raisonnements peu rigoureux. On ne les regarde comme des approximations. Anyway, those equations, they've been derived with lots by physicists doing lots of approximations. So why do we need to work so hard and prove this rigorously? So why make so much effort? And I must say, this is a question that we hear all the time. These days, not so much from the physicists, but people who are doing scientific computing, they will tell you we can compute a solution. So it's there, why bother to give a proof? And then he gives the answer. Néanmoins, toutes les fois que je le pourrais, je viserais à la rigueur absolue, absolute rigor for two reasons, two reasons. First of all, il est toujours dur pour un géomètre, means mathematician, d'aborder un problème sans le résoudre complètement. That's beautiful. You feel the joy of the mathematician. You cannot leave a problem until you solve it. It's like a drug. When you start on a problem, you don't leave it until you completely solve it. And then he says something, and that's what I call the prophetic insight of point arrêt. Les équations que j'étudierais sont susceptibles non seulement d'applications physiques, mais encore d'applications analytiques. Analytique means within mathematics. C'est sur la possibilité du problème de diriclet que Riemann a fondé sa magnifique théorie de l'habilien. He says, look, Riemann has used, even if there was a flaw, I mean the proof was defective, he has used the solution of the Laplace equation. So we're in his theory of the Habilien function. So we need a proof because this is a pure, this is really a problem in mathematics. We really need a proof for that. And then he says, depuis d'autres géomètres, on fait d'importants applications principaux partis les plus fondamentales de l'analyse pure. Est-il encore permis de se contenter d'une demi-rugueur? Et qui nous dit que les autres problèmes de la physique mathématique ne seront pas un jour comme la déjà été le plus simple d'entre eux appelés à jouer en analyse un rôle considérable? This is absolutely prophetic. If you think, for example, of Cortevec de Frise and the solitons, the role that it had it started in fluid mechanics, but then the impact on algebraic geometry. And then if you think of Young Mills, the instantons coming from physics and its impact on low-dimensional topology. And of course the most beautiful example is the Ricci flow. It's a flow, it's something like, which started like a variant of the heat equation, I would say, and or distant cousin of the heat equation and the beautiful solution using the Ricci flow of the Poincaré conjecture. So maybe he had already a kind of prophetic insight that eventually the Poincaré conjecture would be solved via partial differential equation. Okay, now let's go to the papers. So here is the first theorem that already alluded to. In 1890, in the same American journal paper, Poincaré gave a proof of the theorem. You take a domain of R3, a smooth bounded domain in R3, but for that matter, any domain in Rn, smooth domain in Rn would also have the same kind of proof which could be applied. And given any continuous boundary condition, you have a unique solution of the Laplace equation. Laplacian u equals 0, that is finally the boundary condition, u equals 5 on the boundary. And of course there were some predecessors, and Poincaré is very careful to mention them. Hermann Schwartz in 1969 had a solution in R2, and especially Neumann in 1978 gave a solution, but just for convex domain. And it is in this paper that Poincaré invents the so-called balayage method. I'm not going to explain what it is. I just want to say that it relies on three ingredients which were known at the time of Poincaré. The maximum principle, the Poisson formula, which is a solution of the Dirichlet problem of the ball, in fact an explicit solution integral. You can write the solution of the Laplace equation in a ball with a given boundary condition. You can write it as an integral. That was known at the time of Poincaré. And then the Harnack principle, which was also known, that is if you have a sequence, it's related to the so-called Harnack inequality also. If you have a monotone increasing sequence of harmonic functions, and you know that at one point the sequence is bounded, then it will be bounded everywhere, and you have a limit converges to a harmonic function. And if you look at the, either at the American Journal of Maths article, or you can look at his book, published a few years later in 1999, Theorie du Potentiel Newtonian, which was from, of course, at the Sorbonne, in between 1994 and 1995, and really look at just a few pages. I'm going to flash a few pages, and I'm doing this on purpose just to show you the contrast with Hilbert's paper. Just this is beautiful. First of all, he starts the resolution of the problem of Dirichlet. So he states, Lenon said the problem of the Dirichlet, you have a domain, and he assumes that it's connected. Okay, but it's not necessarily simply connected. You have the continuous data on the boundary, and you want to construct a function which is harmonic inside and assumes the boundary conditions. Very clearly stated, and he states the problem of Dirichlet and you can't admit it's a solution. We already saw this, it seems unprincipled, uniqueness of the solution, and he says what we are going to prove is that it admits a solution. And look at the presentation. Look, just, I mean, I wish I had attended this course in PDE by Poir-Carpet, and just beautiful, very clear presentation, notation, everything is well explained. There are pictures, enormously crystal clear. Look at that, just look at that. Okay, and in fact, I want to mention here a comment, I don't know, it's a little small, it's a footnote in a paper by Adama which I discovered, and I like very much this footnote. In Poincaré, the first idea, so this is Adama, who writes this, the first idea of a research is always made in evidence with a wonderful clarity. We are far from always finding the same degree at the highest level. I don't know if he has in mind him or not, but probably. It means, look at this sentence, it means that the accusation of obscurity, which is sometimes against him, seems to us, at least, from a reader's point of view, often expressing the contrary of the truth. I say that's what one has in mind. Many people have in mind Hilbert, clear, everything transparent, when Poincaré makes mistakes and so forth. No, not true, this is completely wrong and should be, this is distortion of truth and I think everyone agrees, I hope everyone agrees with, okay, so that's about the paper in the American Journal. And then a few years later, and that's interesting, he comes back and give another proof, so this is in the Acta Mathematica paper, six years later, and he said, I'm giving you another method, even though, we already have proof of this, but even if this method is very good as a demonstration procedure, no problem with the proof, it is inferior as a calculation procedure of Neumann's. He had already in mind, probably, and I don't know, there are other sentences here and there where it's clear that Poincaré, was concerned with doing numerical computations. So, he said, and so here is another proof, Neumann had proved it, as he says, for convex surfaces. So I'm extending, I'm using the same method as Neumann, but now for general domains. Okay. Now, as I said, the Aftermath of the Poincaré's theorem is a bit said, I think, and I wonder if that's the right name, a conspiracy of silence or not, but Poincaré's name is never mentioned in modern textbooks on partial differential equation as the father of the solution of the Dirichlet problem. Never, ever. He remains alive in potential theory, because of the Balayage method, but PDE takes for example the textbook of Craig Evans on partial differential equation, and I challenge you if you see a textbook in partial differential equation where Poincaré is mentioned except for the Poincaré's inequality and we're going to come back to this, please send me an email and I'll correct my my statement. Now, why is that so? I have some two suppositions. First of all, Poincaré's method is indeed a bit restrictive, but for example, he uses the maximum principle and so it's restricts to second-order elliptic equations, and so for example the Baila-Plasien, this is excluded. That's where you really need the beauty of the energy method à la Hilbert to prove the existence of a solution. So, wider class, indeed, but still. And then he uses this explicit Poisson formula on the ball so that's also might be restricted to the La Plasien and one would have more difficulties adapting this for example to variable coefficient operator. Okay, I don't know. I don't know. Those are just suppositions. I have a suspicion that the bad guy here is Courant, Richard Courant who was a student of Hilbert and I think it's not, and you know, the Courant Institute in the 50s became the temple of partial differential equations. So that's where everyone and I went there in the late 60s to study partial differential equations. That's where all of the activity in partial differential equations was more or less concentrated and here is the book that Richard Courant wrote on the Dirichlet principle. There were earlier versions, maybe in German or so, on the Dirichlet principle in, this one was in English published in 1950. And again, let me flash a few pages first, so this is in page 2 of the introduction. So he tells the story about the Dirichlet problem, again the same story that already mentioned. Gals Thompson, Lord Kelvin wanted to solve the Laplace equation in the plane with a given boundary condition and then Riemann used it under the Dirichlet principle and then Weierstrasse found the flaw came as a shock to the mathematical world and there were efforts in later years after 69 to save Dirichlet principle and their efforts remained unsuccessful time and again attempts at the rigorous proof were made finally, look at this sentence 50 years after Riemann, Hilbert succeeded that's in a famous publication he established the existence theorem proving directly and so forth. No mention of Poincare is a book on the Dirichlet principle not in the text not in the footnotes not in the references this is almost unbelievable, unheard of so it's time to I think it's really time to break this conspiracy of science, oh by the way by the way, in this book if you really want to understand what was the content of the original idea of Hilbert, you can find it and how you can make a rigorous proof after Hilbert's idea, you can find this in the book. Now this book was written in 1950 at that time people knew very well had the Cauchy sequences and so forth so there were trivial methods for proving directly the existence of a minimizer that's not what he does in his book. What he does he takes Hilbert's ideas, the one that were very confusing and he takes those ideas and he explains how they can really work. That's how a good student should behave with his teacher. That's to defend his teacher and the reputation of his teacher. Anyway I have done my mea culpa on Poincaré. This is taken the page from my book Analyse fonctionnelle in 1983 where I talk about the Dirichlet principle and I credit Dirichlet Riemann Hilbert. This was in 1983 and look at this, this is my page from my book on functional analysis from two years ago same Dirichlet principle and now Dirichlet Riemann Poincaré Hilbert so I've done my share. Second big result of Poincaré on partial differential equation is about the spectrum of the Laplacian so this is from 1894. What he proves is you take a domain in R3 but could be Rn there exists a sequence lambda k going to infinity of eigenvalues and corresponding eigenfunction of the Laplacian under Dirichlet boundary condition. In fact he even had the more general boundary condition like what's called today the robin or third type of boundary condition. Here again there were predecessor Schwarz, Hermann Schwarz in 1985 had proved the existence of the first eigenvalue and picked out just a year before Poincaré had proved the existence of a second second eigenvalue. That's it but Poincaré did the big work of proving the existence of an infinite sequence of eigenvalues. It is absolutely spectacular achievement. I must say just to be absolutely honest that there is one deficiency in the, I mean not a deficiency but what he did not prove, what he did not establish there is that what one called when we say today that the eigenfunction form a complete system that is a Hilbert basis. Of course this kind of concept was discovered later so that he was not really worried but I must say maybe I don't know if he tried to prove something like this anyway it's not there and it's a bit missing in this paper and I'm saying this because one of the reasons I think that Poincaré was interested in getting all the eigenvalues of the Laplacian was that he wanted to solve the heat equation the wave equation by the Fourier method by Fourier expansion and for this if you wanted to have a complete proof that it works via the Fourier expansion of course you would need to know that the eigenfunctions form a Hilbert basis but still it's an absolutely spectacular achievement and I think his name should be mentioned, remembered celebrated because he paved the way in the subsequent years to the work of Fredon absolutely Fredon continued this work and the Hilbert and of course needless to say how much of spectral theory played an important role in the 20th century and that's a major example of spectral theory okay I'll be very brief on Poincaré's inequality because in the previous lecture Michel Le Duc did a very excellent job and so here you have the Poincaré inequality for the convex domains with the constant which is like a constant depending only on the dimension divided by the diameters word and I must say that in fact he was not after the Poincaré inequality that was not his goal but he used it as a tool in order to get the previous result the existence of an infinite sequence of eigenvalues that's where how he discovered this inequality almost by accident I don't think he was after that and here it's needless to add the tremendous importance of this inequality it has a huge descendants the Sobo-Levin equality where instead of L2 you have this LP norm and of course the Gagliardot Nirenberg inequality the whole business of best constant in the Sobo-Levin equality Michel Le Duc mentioned all those geometric applications of the Poincaré inequality but of course with the Sobo-Levin equality you have a major direction of applications also which we are not mentioned about the Yamabe problem solution of the Yamabe problem and by Thierry O'Branden and other people the use of the best constant in Poincaré inequality that's a major major role in the Yamabe in the final solution of the Yamabe problem what is a bit there is some ironical part that is the Poincaré inequality the way it's often so this is so called Poincaré virtingers inequality and what people call today the Poincaré inequality is very often another form it's for functions which vanish on the boundary in modern terminology H1 0 this is inequality but apparently he never considered this it was not Poincaré I don't know exactly to whom one should attribute this but this is really ironic Poincaré's inequality that everyone mentions and everyone teaches and that you hear over and over again is not due to Poincaré on the other hand major results which he really proved are hardly ever mentioned are not attached to his name okay and finally say a few words about another contribution of Poincaré in PDE this is this time to non-linear PDE and what Poincaré did he considered this problem minus Laplacian U plus theta of X E to the U equals K on the surface on S and S is the Riemann surface so two-dimensional with a Riemann metric G Laplacian is the Laplace-Beltramie operator on S theta of X which appears here is a given smooth function which is positive on S and K is a constant now this kind of problem was of interest to Poincaré because it arises and a basic ingredient in the uniformization theorem and here I could mention the excellent reference of the book Uniformisation des surface de Riemann in the published by this team of collective work published at the Ecole Normale and soon going to be published in English and of course it's also related to the search of a conformal metric on S with constant negative Gauss curvature so here is a result of Poincaré from 98 and also we're competing or complementing work of Peter on this problem this problem star has a unique solution given any function theta positive function theta and any positive constant K proof is a bit complicated in fact it's well explained in the reference in this book Uniformisation it's a bit complicated it has one original ingredient which is the continuation method which unfortunately is always attributed to Bernstein in 1906 but Poincaré did this founded eight years earlier and the idea is to embed the problem the problem double star that you want to solve in a one parameter family of partial differential equations P lambda and P0 the first one you start with some easy problem which you solve with little effort and at the end of the interval for lambda equals one you have the problem the tough problem that you want to solve and you solve it step by step going from zero from zero to one here I should say I must say that I have a little bit of mixed about this theorem on the one hand as you all know I'm working in a nonlinear partial differential equation so I'm happy I have prestigious ancestry Poincaré was studying nonlinear partial differential equations so I can claim great great great grandfather who was already working on a nonlinear partial differential equation on the other hand I must say that today there are much simpler methods which are more powerful much more powerful simpler to solve this equation double star well you are going to tell me that's not fair it's hundred years after Poincaré you tell me that there are new methods well that's not a big deal well let me make a point there are at least three methods which you can use today to solve this the first method is the so-called method of sub and super solution I'm going to say something about this in a minute it goes back to Perot in the 30s then convex minimization that's also possible you can use convex minimization but that kind of thing came later that was in some sense Hilbert's program and so Poincaré could not have thought of this when he did this nonlinear PDE and of course certainly not the monotone operators of minty, browser, etc which started only in the 60s which you can use and get very easily the resistance of the solution but I want to emphasize the method of sub and super solution because it's very simple very robust and uses only tools that Poincaré knew that is just basically the maximum principle and he had used it over and over again and Poincaré maximum principle plus a very simple monotone iteration gives the solution of this nonlinear PDE not one to be critical of course of Poincaré but still just to say that he might have discovered it probably fairly easy so here is the just for those of you who are not familiar with the statement so here is a statement extremely general you want to solve a nonlinear PDE minus Laplace in U equals F of X in U which is smooth in X and U just to make life simple and you have to pair of two functions a sub and a super solution sub solution means that you have this inequality super solution means you have the reverse inequality then that's it this is sufficient to prove the existence of a solution in between the sub and the super sub and super solution and just by you start with the sub solution for example you solve by iteration monotone iteration we listen to just a few lines and if you want to solve the problem which I mentioned, double star which I mentioned on the previous page you can just choose as sub solution minus a constant a large constant C should be a large constant and a super solution plus C, a positive large constant and you see immediately that you have those inequalities this is a sub solution which does not appear so all you need is this inequality theta e to the minus c because k is positive this is obviously satisfied when c is very large minus c plus c you take c very large positive and you have immediately a super solution now a final comment about this non-linear PDE you have to watch very carefully the sign and this method of sub and super solution had nothing to do with the exponential you can if you want to take any non-linearity the exponential of the exponential exponential of u to a power etc. it would work exactly the same way just a final comment that if you replace and this is very important the sign in front of the Laplacian the way I wrote minus Laplacian of u plus theta e to the u equals the right hand side if you replace this minus sign by a plus sign then this is a completely different story this is extremely difficult problem it's a so-called Nuremberg problem not fully solved when there is existence of solution under what conditions under function theta etc. and this is very active field of research it's still quite open for many open questions about this simple looking equation even in two dimensions okay so I stopped here and I really would like you to go back home and tell when you teach a course in PDE please or tell your colleagues not to forget to mention for example what's possible whatever the famousness would be well it's a good question I don't know how it was possible I don't know how it was possible if you look if you look in the there were two papers of Hilbert the paper, the big paper that everyone in the announcement in 1900 he says I don't think I have it here he says people have considered earlier the Laplace, the Dirichlet problem and he says Hermann Spass Neumann Poincare he doesn't see what they proved Poincare together okay that was in the the short paper from 1900 there the name Poincare appeared four years later when he wrote this big paper in 1904 Poincare had disappeared I don't know why this is probably I don't know you know have you heard that France and Germany were not terribly friendly to each other and I don't know that might be that might have been the reason of tension maybe there was some I don't know maybe there was some personal tension I'm not sure there was really personal tension because later on I think in 1908, 1909 Hilbert invited Poincare he invited I don't know that's a very very good question why Hilbert anyway he passed this he passed this virus to Courant and Courant said that he had also to hide that's more embarrassing and I must say I'm talking to people at the Courant Institute Louis Nirenberg have you heard that Poincare had never heard so it's clear that there was a problem connected with the Courant Institute which was what's more amazing and this I don't know why 1906 there's other people it's known there's a curse on the name of Poincare I don't know I don't know why for example does not mention the name of Poincare and certainly in France even he was not mention I asked Jacques to be yours who do you know about Poincare and PDE finally yes sir several remarks best of all I think that may be discrepant are you going to bring in the Russians who proved this before in parallel the Russians too? to say that oh, client client that was a general attitude to Poincare possibly the other remark is that you mentioned Bernstein in his paper 1906 continuation message if I remember correctly yes that's quite possible indeed indeed and the secondary market is just a question a good question, I don't know who was the first a good question maybe he knows, yeah I don't know I don't know maybe Jean-Blaain has the big expert probabilistic proof who was the first one to give a probabilistic proof what? now maybe Vinner as you said maybe a good question I'm sorry, I cannot talk concerning the relation between Poincare and Hilbert, there is a fact so I think they were competing to be the best mathematician of the world in this time and I think Hilbert was much more concerned with this competition that Poincare was not that was not in his character and an example the Boliai prize was attributed for the first time in 1905 to the best mathematician of the world of the time whatever it means and so there was a committee and Klein was in the committee and the competitors were of course Poincare and Hilbert and Klein was supposed to write the report on the winner and the winner was Poincare and Klein refused to write the report so the report was written by Radosh not so famous mathematician from Hungary because the second Boliai prize was attributed a few years later then again for the best mathematician of the time of course Poincare already has got it he was in the committee of the prize and the prize was attributed to Hilbert and Poincare wrote the report he was a nice guy he was a very nice guy in addition to everything that's good to say it any other question I could just add a comment to that concerns the invitation from Hilbert to Poincare to give these six lectures in Göttingen the first two lectures are on PDEs and Fredholm's work and it caused consternation in Göttingen they were really upset they felt patronized this was their work and somehow Poincare was telling them do it differently and when we get to the second Boliai prize Poincare praises Hilbert throughout until he gets to PDEs and he says but of course the real breakthrough is the work of Fredholm so I think actually there is a little more competition from the French side here understandably but also this may be part of the answer to your question that Poincare has found in Fredholm a worthy successor and in some sense being a generous person prepared to say well the real way to do this is Fredholm so he himself passes the crown to another person and that line of work is more consonant with the way Hilbert also wrote the theory it seems to me so maybe that's part of the answer to your question that Poincare being a modest person absents himself from the the story thank you by the way you say that Hilbert is not crediting Poincare for the Dirichlet problem but Poincare himself is crediting Schwarz Poincare himself in his paper is more or less saying that all the ideas are in Schwarz's paper no no no no he doesn't say that no no no I think you misunderstood what I said there are two papers of Poincare the first from 1890 is the Balayage method and Poincare says I mean it's very clear that this is an original method and that it works in general domain he says Neumann method which was completed therefore he attributes the credit to Neumann for solving the Laplace equation and the Dirichlet principle in convex domain he says that that a few years later that's the second paper that I mentioned he says now I am coming back to the method which was not the original it was not my original method I'm coming back to the method of Neumann because I like it and I think that historically if we were to compute the solution this would be more efficient than the Balayage method on the other hand he says also so he gives a proof he gives a proof of existence of the solution using the idea of Neumann but on the other hand he also says very clearly in the second proof this is not really a second proof because inside the proof he inserts the fact that I already know the existence so it's not really a second existence I would call it it's more like efficient method for computing the solution and there he gives credit to Neumann nevertheless the Balayage method is some kind of generalization of the alternating method of Schwarz so to go from Schwarz to Calgary is not so and the alternating method of Schwarz was already very powerful yeah, yeah, yeah, yeah okay we'll organize a conference of Hermann Schwarz someday or maybe suggest suggest to people in Germany where was he also in Göttingen that they should organize a conference there will be conference in the Hilbert 100th and it's here in about 40 or 30 something years