Homogenization 2001, Proceedings of the First HMS2000 International School and Conference on Ho- Mogenization

in \Homogenization 2001, Proceedings of the First HMS2000 International School and Conference on Ho- mogenization. Naples, Complesso Monte S. Angelo, June 18-22 and 23-27, 2001, Ed. L. Carbone and R. De Arcangelis, 191{211, Gakkotosho, Tokyo, 2003". On Homogenization and Γ-convergence Luc TARTAR1 In memory of Ennio DE GIORGI When in the Fall of 1976 I had chosen \Homogénéisationdans les équationsaux dérivéespartielles" (Homogenization in partial differential equations) for the title of my Peccot lectures, which I gave in the beginning of 1977 at Collègede France in Paris, I did not know of the term Γ-convergence, which I first heard almost a year after, in a talk that Ennio DE GIORGI gave in the seminar that Jacques-Louis LIONS was organizing at Collègede France on Friday afternoons. I had not found the definition of Γ-convergence really new, as it was quite similar to questions that were already discussed in control theory under the name of relaxation (which was a more general question than what most people mean by that term now), and it was the convergence in the sense of Umberto MOSCO [Mos] but without the restriction to convex functionals, and it was the natural nonlinear analog of a result concerning G-convergence that Ennio DE GIORGI had obtained with Sergio SPAGNOLO [DG&Spa]; however, Ennio DE GIORGI's talk contained a quite interesting example, for which he referred to Luciano MODICA (and Stefano MORTOLA) [Mod&Mor], where functionals involving surface integrals appeared as Γ-limits of functionals involving volume integrals, and I thought that it was the interesting part of the concept, so I had found it similar to previous questions but I had felt that the point of view was slightly different. I thought that the idea could be useful for questions like surface tension, but I would have preferred to consider that in a dynamical situation, of course, and although there was no direct minimization of functionals in Ennio DE GIORGI's approach, it had for me the same limitations that I had observed in others, who clung to their obviously wrong belief that Nature minimizes energy. What I had taught in my Peccot lectures, contained extensions of some work that I had done with Fran¸coisMURAT, on a slightly more general approach than G-convergence, which he later called H-convergence, and on the notion of Compensated Compactness. I thought that it was clear from my lectures that H- convergence and Compensated Compactness were two aspects of the same question, which is to understand what kind of oscillations (which one often calls microstructure nowadays) are compatible with a given system of partial differential equations, and what effective equations could be derived, for describing the macroscopic behaviour of a few interesting quantities; in some way it is this global point of view which should be called Homogenization, although for simplicity Homogenization has been first identified with the simpler aspect of H-convergence (or G-convergence in some cases), but then the term seems to have lost its original meaning due to the limitations of those who were using it and who lost track of any goal by concentrating on too many similar examples; similarly the limitations of those using Γ-convergence has made it lose some of the power that Ennio DE GIORGI had put in the concept. Although I had borrowed the term Homogenization from Ivo BABUSKAˇ , who had been interested in questions with periodic structures, in the spirit of what Henri SANCHEZ-PALENCIA had also done, I had clearly set up a much more general framework, which should have been found natural to anyone who understood a little about Continuum Mechanics. I would have been greatly puzzled if I had been told at the time that some people to whom I had explained that elastic materials do not minimize their potential energy would still stick to that fake physical principle twenty years after, after having conscientiously misled gener- ations of students about that. I could hardly have understood either that it was possible that some people mistake Homogenization and Γ-convergence; certainly, a much better insight could be gained by using Ennio DE GIORGI's possibility of using general topologies, for example by considering the topology according to what my own approach of Homogenization/Compensated Compactness suggested. For various reasons, there are different groups of people who insist in attributing my ideas to their friends or themselves, and they do not seem aware that every good mathematician can observe that they do not understand well the methods that they use, and this casts a doubt on the fact that they could have 1 Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA. 1 initialized them; I usually explain in detail who had contributed about all the ideas that I know, and why the ones which I introduced myself had become natural to me, and some of these people even go down to insult me because I explain what are the ideas that I had introduced and the mistakes that some reasonably good mathematicians have made. Some of these people often pretend to answer questions of Continuum Mechanics by using Γ-convergence, but their limited understanding of Continuum Mechanics is too obvious to be missed, and although what they do is nonsense from the point of view of Continuum Mechanics it does not mean that one cannot use Γ-convergence for doing something useful in Continuum Mechanics, but for that one has to be more inventive concerning the topology that one chooses. Ennio DE GIORGI was a giant, and his ideas have had an important impact on some questions of Analysis and Geometry; as I have mentioned elsewhere, he gave me the feeling that he was interested in Mechanics but as he had not learned much in this direction I thought that he was trying to reinvent the field; his ideas have been important concerning the regularity of solutions of elliptic or parabolic partial differential equations and convergence effects related to minimization, but one cannot ignore the fact that most Continuum Mechanics or Physics is not about minimization, and that many equations are actually hyperbolic. Ennio DE GIORGI should have told his followers to learn a little about Continuum Mechanics, and as I have oriented my own research work towards overcoming the challenges coming from Continuum Mechanics or Physics, I want to offer all my contributions to his memory, hoping that it would deter many from propagating their erroneous views about Mechanics while using the name of Ennio DE GIORGI as a shield, because misleading was too opposed to his character, and he was a man of utmost integrity whom I admired for his religious approach to life, although a little different from mine. Those who feel the urge to attribute my ideas to others who have not done much and do not even understand them could then instead attribute them to Ennio DE GIORGI, as a token of appreciation of his mathematical contributions. Homogenization is a theory about partial differential equations, which may be elliptic, parabolic, hyperbolic or neither, and although unphysical minimization processes may well be useful for technical reasons (as a way to prove existence of some solutions, for example), one should observe that most partial differential equations are not about minimizing anything. However, Fran¸coisMURAT and I were actually led to these questions by starting from an academic minimization problem, and I have described the chronology of this approach in [Tar1]. Γ-convergence is a theory about functionals, and the order relation of the real line plays a role and there are plenty of small minimization problems hidden in the definition. As a first step towards appreciating the differences between the two theories, I think that it is useful to describe a few basic facts about Continuum Mechanics and Physics. As more space would be needed for explaining some technical questions related to my subject, I do plan to write more articles later. Does Nature minimize or conserve energy? In 1848, STOKES [Sto1] explained a discrepancy which CHALLIS [Cha] had noticed concerning some solutions of the equations of compressible gas dynamics, which POISSON [Pois] had obtained in an implicit form in 1808, by showing that solutions could approach a discontinuity in finite time; by using conservation of mass and conservation of momentum he had then correctly derived the jump conditions that discontinuous solutions must satisfy. These jump conditions are now interpreted as meaning that the discontinuous func- tions satisfy a partial differential equation in the sense of distributions, as developped by Laurent SCHWARTZ, following the pioneer work of Sergei SOBOLEV and of Jean LERAY. After STOKES,RIEMANN [Rie] derived independently the jump conditions (for isentropic motions) in his thesis in 1860, but instead of being called the Stokes{Riemann conditions, the jump conditions are now named after RANKINE [Ran] and HUGONIOT [Hug]. However, when STOKES edited his complete works in 1880 [Sto2], he did not reproduce there his 1848 proof of the jump conditions, and instead he apologized for having made a mistake, because he had been (wrongly) convinced by Lord RAYLEIGH and THOMSON (later to become Lord KELVIN), that his discontinuous solutions were not physical: they did not conserve energy. So in the third part of the 19th Century, good physicists were adamant: energy is conserved! KELVIN,RAYLEIGH and STOKES must have understood later that heat is a form of energy, and that the missing energy in STOKES's discontinuous solutions of (isentropic) gas dynamics is transformed into heat, which does make the temperature of the gas increase, but since that possibility is not allowed in the 2 mathematical model this energy is \apparently lost"; in other terms, these three great scientists had not grasped yet why one needs a notion of \internal energy", although it seems that WATT and CARNOT had understood much earlier (and independently) that mechanical energy may be transformed into heat and heat may be transformed into mechanical energy, but with inherent limitations about the proportion of mechanical energy which can be recovered after it has been transformed into heat.

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