DE GIORGI, ENNIO

(b. Lecce, Italy, 8 February 1928; d. Pisa, Italy, 25 October 1996), calculus of variations, partial differential equations, foundations of mathematics.

De Giorgi was the greatest Italian mathematician of the second half of the twentieth century. His activity ranged from the calculus of variations to the theory of partial differential equations, with a strong interest in foundations of mathematics.

De Giorgi was born in Lecce, in southern Italy in 1928. After his high school studies in Lecce, he moved to Rome in 1946 to begin his university studies in engineering. The following year he switched to mathematics, and graduated in 1950 under the direction of Mauro Picone. After a fellowship at the Istituto per le Applicazioni del Calcolo, he became an assistant of Picone at the University of Rome in 1951.

The Theory of Perimeters and the 19th Hilbert Problem . In the 1953–1955 period, De Giorgi obtained his first important mathematical results on the theory of perimeters, a notion of (n-1)-dimensional measure for oriented boundaries of n-dimensional sets introduced by Renato Caccioppoli, who had a great influence on his mathematical formation. These results led to the proof of the isoperimetric inequality for arbitrary measurable sets, published in 1958.

De Giorgi in 1955 obtained a counterexample to the uniqueness for regular solutions of the Cauchy problem for a linear partial differential equation with smooth coefficients, a question that had been open for about half a century. De Giorgi’s paper had a remarkable impact on the mathematical community and inspired other counterexamples to uniqueness, constructed by Jean Leray in 1966.

The most important result obtained by De Giorgi is the proof of the Hölder continuity of the solutions of elliptic equations with bounded measurable (possibly discontinuous) coefficients. This result, obtained in 1955 and published in complete form in 1957, was the last— and perhaps the hardest—step in the solution of the 19th problem posed by Hilbert in 1900: whether the solutions of regular minimum problems of the calculus of variations for multiple integrals are smooth, and even analytic if the data are analytic. Indeed, every minimizer satisfies a quasilinear elliptic equation, called euler equation, and De Giorgi’s theorem provides the tools that allow to study these solutions by using well known results on elliptic equations.

This theorem had an enormous impact in the theory of nonlinear elliptic equations. The same result for parabolic equations was proved independently by John F. Nash in the same period by totally different methods. It is remarkable that a few years later, in 1968, De Giorgi himself showed by a counterexample that the same result does not hold for uniformly elliptic systems with discontinuous coefficients.

Results on Minimal Hypersurfaces . In 1958, De Giorgi became professor of mathematical analysis at the University of Messina. After one year Alessandro Faedo invited him to the Scuola Normale of Pisa, where De Giorgi had the Chair of Algebraic and Infinitesimal Mathematical Analysis for almost forty years. In 1960, he was awarded the Caccioppoli Prize, established in the same year by the Italian Mathematical Union.

During the 1960s, De Giorgi’s scientific activity was mainly devoted to the theory of minimal hypersurfaces. His main success was the proof of the analyticity almost everywhere of minimal boundaries in any number of spatial dimensions. This is a striking example of the use of the theory of perimeters in the calculus of variations. He considered this result as a victory in one of his most audacious scientific challenges. His technique was immediately adapted by William K. Allard and Frederick J. Almgren to obtain partial regularity results for more general geometrical objects and has become widely known and used in contexts quite far from the initial one: nonlinear equations and systems of elliptic and parabolic type, harmonic maps, geometric evolution problems, and so forth.

In 1965, De Giorgi obtained an extension of Bernstein’s theorem to dimension three: The only solutions of the minimal surface equation defined on the whole three dimensional space are necessarily affine. This result was soon extended to dimensions up to seven by James Simons, who also constructed what is called a locally minimizing cone in dimension eight. De Giorgi then proved, in 1969, with Enrico Bombieri and Enrico Giusti, that Simons’s cone is also globally minimizing. Furthermore, using this cone, they constructed a solution of the minimal surface equation that is not affine and is defined in the whole Euclidean space of dimension eight. This striking result shows that Bernstein’s theorem cannot be extended to dimensions larger than seven. In the same year De Giorgi proved, with Enrico Bombieri and Mario Miranda, the analyticity of the solutions of the minimal surface equation in any space dimension.

In 1971, together with Lamberto Cattabriga, De Giorgi proved that any partial differential equation with constant coefficients and real analytic right-hand side has a real analytic solution in dimension two, while in dimensions larger than two there are examples, as simple as the heat equation, for which there is no analytic solution.

Gamma-Convergence . In the 1973–1985 period, De Giorgi developed the theory of Gamma-convergence, designed to give a unified answer to the following question, which arises in many problems of applied mathematics: Given a sequence F_k of functionals, defined on a suitable function space, does there exist a functional F such that the solutions of the minimum problems for F _k converge to the solutions of the corresponding minimum problems for F?

In 1973, the Accademia dei Lincei awarded him the Prize of the President of the Italian Republic. That year, with Sergio Spagnolo, De Giorgi showed the variational character of the notion of G-convergence of elliptic operators, introduced by Spagnolo in 1967–1968, and its connection with the convergence of the energy functionals associated with the elliptic operators. In an important paper published in 1975, De Giorgi passed from the “operational” notion of G-convergence to a purely “variational” one. Instead of a sequence of differential equations, he considered a sequence of minimum problems for functionals of the calculus of variations. Without writing the corresponding Euler operators, he established what is to be considered as the variational limit of this sequence of problems and also obtained a compactness result. This is the starting point of Γ-convergence.

The formal definition of this notion, together with its main properties, appeared a few months later in a paper with Tullio Franzoni. In the next ten years, De Giorgi was engaged in the development of the applications of this theory to many asymptotic problems of the calculus of variations, like homogenization problems, dimension reduction, phase transitions, and so on. De Giorgi, usually very temperate when speaking of his results, was very proud of this creation and considered it a conceptual tool of great importance.

Gradient Flows and Free Discontinuity Problems . At the beginning of the 1980s, in a series of papers with Antonio Marino and Mario Tosques, De Giorgi proposed a new method for the study of gradient flows, which can be applied to many problems with nonconvex nondifferentiable constraints. In 1983, De Giorgi gave a plenary lecture on Γ-convergence at the ICM in Warsaw. On that occasion he publicly expressed one of his deepest beliefs, declaring that the human thirst for knowledge was, in his opinion, the “sign of a secret desire to see some ray of the glory of God.” In the same year, during a ceremony at the Sorbonne, De Giorgi received the degree honoris causa in mathematics of the University of Paris.

De Giogi proposed, in a 1987 paper with Luigi Ambrosio, a very general theory for the study of a new class of variational problems characterized by the minimization of volume and surface energies. In a later paper he called this class “free discontinuity problems,” referring to the fact that the set where the surface energies are concentrated is unknown and can often be represented as the set of discontinuity points of a suitable auxiliary function. Surprisingly, in the same period David Mumford and Jayant Shah proposed, in the framework of a variational approach to image analysis, a minimum problem for which this theory is perfectly suited. The existence of solutions to this problem was proved by De Giorgi in 1989, in collaboration with Michele Carriero and Antonio Leaci. In 1990, De Giorgi was awarded the prestigious Wolf Prize in Tel Aviv.

From the mid-1970s, De Giorgi also worked on foundations of mathematics, adopting a nonreductionist point of view. For this work the University of Lecce awarded him the degree honoris causa in philosophy in 1992.

Personal Attributes . De Giorgi had a striking mathematical intuition, combined with the prodigious ability to obtain from it a complete proof, with all minor details. He had a very large number of students. His activity had a tremendous influence on the developments of the calculus of variations and of the theory of partial differential equations. Although he was surrounded by the deep admiration of his colleagues, friends, and students, he remained a very modest person. His office was always open to people who wanted to discuss with him some mathematical problem. On these occasions he often seemed inattentive, but he was always able to grasp the heart of the matter and to suggest new approaches—which worked.

De Giorgi was very active in the protection of human rights. He was a deeply religious man. His attitude toward a continuous search, his natural curiosity, and his open-mindedness to all ideas made it easy for him to have a constructive dialogue with others. From 1988, De Giorgi began to experience health problems. In September 1996, he was taken to a hospital in Pisa and, after undergoing surgery, he died on 25 October.

BIBLIOGRAPHY

WORKS BY DE GIORGI

Selected Papers. Published with the support of Unione Matematica Italiana and Scuola Normale Superiore. Edited by L. Ambrosio, G. Dal Maso, M. Forti, M. Miranda, and S. Spagnolo. Berlin: Springer, 2006.

“Su una teoria generale della misura (r-1)-dimensionale in uno spazio ad r dimensioni.” Annali di Matematica Pura ed Applicata (4)36 (1954): 191–212.

“Un esempio di non unicità della soluzione di un problema di Cauchy, relativo ad un’equazione differenziale lineare di tipo parabolico.” Rendiconti di Matematica e delle sue Applicazioni (5)14 (1955): 382–387.

“Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari.” Memorie dell’Accademia delle Scienze di Torino. Parte Prima, Classe di Scienze Fisiche, Matematiche e Naturali (3)3 (1957): 25–43.

“Sulla proprietà isoperimetrica dell’ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita.” Atti dell’Accademia Nazionale dei Lincei, Memorie della Classe di Scienze Fisiche, Matematiche e Naturali, Sezione I (8)5 (1958): 33–44.

“Un esempio di estremali discontinue per un problema variazionale di tipo ellittico.” Bollettino dell’Unione Matematica Italiana (4)1 (1968): 135–137.

With E. Bombieri and M. Miranda. “Una maggiorazione apriori per le ipersuperficie minimali non parametriche.” Archive for Rational Mechanics and Analysis32 (1969): 255–267.

With E. Bombieri and E. Giusti. “Minimal Cones and the Bernstein Problem.” Inventiones Mathematicae7 (1969): 243–268.

With L. Cattabriga. “Una dimostrazione diretta dell’esistenza di soluzioni analitiche nel piano reale di equazioni a derivate parziali a coefficienti costanti.” Bollettino dell’Unione Matematica Italiana (4)(1971): 1015–1027.

With S. Spagnolo. “Sulla convergenza degli integrali dell’energia per operatori ellittici del secondo ordine.” Bollettino dell’Unione Matematica Italiana (4) 8 (1973): 391–411.

“G-operators and Γ-convergence.” Proceedings of theInternational Congress of Mathematicians, no. 2 (Warsaw, 1983), 1175–1191. Warsaw and Amsterdam: Pwn and North-Holland, 1984.

With M. Carriero and A. Leaci. “Existence Theorem for a Minimum Problem with Free Discontinuity Set.” Archive for Rational Mechanics and Analysis 108 (1989): 95–218.

OTHER SOURCES

Ambrosio, L., G. Dal Maso, M. Forti; et al. “Ennio De Giorgi.” Bollettino dell’Unione Matematica Italiana (8) 2-B (1999): 1–31. Includes a complete list of De Giogi’s works.

Gianni Dal Maso