ENNIO DE GIORGI and Γ-CONVERGENCE Gianni Dal

Home , Ennio de Giorgi, Gianni Dal Maso, Mauro Picone

DISCRETE AND CONTINUOUS doi:10.3934/dcds.2011.31.1017 DYNAMICAL SYSTEMS Volume 31, Number 4, December 2011 pp. 1017–1021

ENNIO DE GIORGI AND Γ-CONVERGENCE

Gianni Dal Maso SISSA, via Bonomea 265, 34136 Trieste, Italy

Abstract. Γ-convergence was introduced by Ennio De Giorgi in a se- ries of papers published between 1975 and 1983. In the same years he developed many applications of this tool to a great variety of asymptotic problems in the calculus of variations and in the theory of partial diﬀerential equations.

1. The starting point. Several problems in applied mathematics lead to the study of partial differential equations with rapidly oscillating coefficients. A typical example is the study of the mechanical properties of composite materials. At a mi- croscopic level the material is highly heterogeneous: the elasticity coefficients are discontinuous and oscillate between the different values characterizing each compo- nent. When the components are intimately mixed, the solutions to the elasticity equations become more and more complex. On the other hand, at a macroscopic level the material behaves like an ideal homogeneous material. The mathematical explanation of this phenomenon is that the solutions to the elasticity equations of the composite material converge to the solution corresponding to a homogenized material as the parameter that describes the fineness of the mixture tends to zero. It is remarkable that, in general, the coefficients of the homogenized equation are different from the limit, in some weak topology, of those corresponding to the composite material. This can be seen in simple examples in dimension one with strongly oscillating coefficients. This subject, called homogenization theory, was widely studied in the ’70s and ’80s, mainly in France, in Italy, in the Soviet Union, and in the United States. The notion of G-convergence for elliptic operators, introduced by Spagnolo [22, 23] in 1967-68 developing De Giorgi’s ideas, provides the mathematical framework to study the problems considered above when the underlying equations are elliptic or parabolic. The name means convergence of Green’s functions; it is defined as the weak convergence, in suitable function spaces, of the sequence of the inverse operators. In 1973 De Giorgi and Spagnolo [20] showed the variational character of G- convergence and its connection with the convergence of the corresponding energy functionals. Operators and differential equations were never used in [4], a paper dedicated to Mauro Picone in 1975 where De Giorgi studied the convergence of energies in a purely variational framework. Instead of a sequence of differential equations, he considered here a sequence of minimum problems for functionals of the calculus of variations. In particular, he studied integrals whose integrands have linear growth in the gradient of the unknown function, like the area functional in Cartesian form. Without writing the corresponding Euler operators, De Giorgi determined what is

1017 1018 GIANNI DAL MASO to be considered as the variational limit of this sequence of problems, and obtained also a compactness result. This was the starting point of Γ-convergence. The formal definition of Γ-convergence for a sequence of functions defined on a topological space appeared in a paper with Franzoni [16] few months later (see also [17]). It included the old notion of G-convergence as a particular case, and provided a unified framework for the study of many asymptotic problems in the calculus of variations. 2. The developments of Γ-convergence. In the decade 1975-85 De Giorgi developed the theoretical framework of Γ-convergence and explored the multifarious applications of this tool. In this period he stimulated the activity of a lively research group, introducing fruitful ideas and original techniques, whose developments were often left to students and collaborators. In [5] De Giorgi introduced the definition of the so called multiple Γ-limits, i.e., Γ- limits for functions depending on more than one variable, and presented the notion of G-convergence in a very general abstract framework. These notions have been the starting point for the applications of Γ-convergence to the study of the asymptotic behaviour of saddle points in min-max problems and of solutions to optimal control problems. In [8,9] he formulated the theory of Γ-limits in a very general abstract setting, starting from the more elementary notion of operators of type G, based only on the order relation. He also explored the possibility of extending these notions to complete lattices. This project was accomplished in [11, 18], written in collaboration with Buttazzo and Franzoni, respectively. The former paper also contains some general guide-lines for the applications of Γ-convergence to the study of limits of solutions of ordinary and partial differential equations, including also optimal control problems. The main research lines connected with the applications of Γ-convergence were illustrated by De Giorgi in [6], together with the main results obtained by his school in these fields. He also formulated some interesting conjectures that have had a fruitful influence in this area for many years (see [1, 21]). This paper contains also the first exposition of what was called the localization method for the study of Γ-limits of integral functionals, which had already been used in [4] in an implicit form. In [19], written in collaboration with Modica, Γ-convergence was used to con- struct an example of nonuniqueness for the Dirichlet problem for the area functional in Cartesian form on the circle. A different application of Γ-convergence was considered in [7, 13], the latter with Dal Maso and Longo. These papers deal with the asymptotic behaviour of the solutions to minimum problems for the Dirichlet integral with unilateral obstacles. Given an arbitrary sequence of obstacles, satisfying a very weak bound from above, a subsequence is selected for which the solutions of the corresponding minimum problems converge weakly to the solution of a limit problem, obtained through the localization method for Γ-limits. It is remarkable that, in some critical situations, the limit problem is not an obstacle problem, but involves a new integral functional, that may be considered as a relaxed obstacle. In a paper with Dal Maso [12] and in his lecture [10] at the 1983 International Congress of Mathematicians De Giorgi gave an account of the main results on Γ- convergence and of its most significant applications to the calculus of variations. ENNIO DE GIORGI AND Γ-CONVERGENCE 1019

The papers [10, 14, 15] develop an abstract framework for the study of Γ-limits of random functionals, in order to attack stochastic homogenization problems with Γ-convergence techniques. The papers by Dal Maso and Modica [2,3] had solved these problems in the case of equi-coercive functionals, using the fact that the space of these functionals is metrizable and compact with respect to Γ-convergence. Therefore, the convergence of sequences of probability laws on the space of integral functionals can be studied by means of the ordinary notion of weak convergence of measures. When the random functionals are not equi-coercive, the space of functionals can still be equipped with a topology related to Γ-convergence, but this topology is no longer metrizable and, although it satisfies the Hausdorff separation axiom, it exhibits some pathological properties, that imply that the only continuous functions on this space are constant. In [10] De Giorgi proposed several notions of convergence for measures defined on the space of lower semicontinuous functions, and formulated some problems whose solution would be useful to identify the most suitable notion of convergence for the study of Γ-limits of random functionals. This notion of convergence was pointed out and studied in detail in [14, 15], written in collaboration with Dal Maso and Modica. 3. Personal memories. I started to interact with Ennio De Giorgi in 1975, when I became one of his students. At that time he was already regarded as one of the most important mathematicians of the twentieth century, for his achievements on minimal surfaces and on the regularity theory for elliptic equations, and not only for that. I was impressed by the contrast between his modest appearance and his deep ideas. In every mathematical discussion with Ennio De Giorgi there was always a surprise. He often looked absent minded while listening, but at the end he used to interrupt with a remark or a question, which showed not only that he had grasped the main features of the problem, but also that he was able to propose his own solution, and to give suggestions on how to overcome the remaining difficulties. In the 70’s Ennio De Giorgi was the leader of a large scientific school, composed of his students and of many experienced collaborators, including senior scientists. In that period he was developing Γ-convergence and was exploring all possible applications of this tool. He always promoted research in collaboration. Scientific discussion was very important for him. He believed that the progress of mathematics is generated by the scientific discussion within a group of friends that share a deep interest in the same mathematical problems. He used to compare written literature with scientific discussion. A written report is certainly important, because it allows you to know the results even if you are not in direct contact with the authors. However, the scientific discussion is much more effective to spread new ideas. He used to reverse the Latin proverb “Verba volant, scripta manent”, which became for him “Scripta volant, verba manent”, meaning that writing in a scientific journal has not always a great impact, while scientific discussion has a deep and permanent effect on the scientific community. Ennio De Giorgi liked to discuss not only with experienced mathematicians, which often came to Pisa to meet him, but also with young students. I remember that on several occasions he asked me unexpectedly: “Do you have five minutes?” And then, in front of a blackboard, he explained me his latest ideas on a problem he was studying. Of course, I learned a lot from these discussions. It was really 1020 GIANNI DAL MASO a great privilege for me to have such an advisor and to continue my collaboration with him when I left Pisa. From the scientific point of view, the main legacy of Ennio De Giorgi are his fundamental results, which have been crucial for the developments of the calculus of variations and of the theory of partial differential equations. However, from a personal point of view, for me the main legacy of Ennio De Giorgi is the joy of sharing our mathematical ideas with our collaborators.

REFERENCES

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