De Giorgi's first work was ingeometric measure theory, on the topic of the sets of finite perimeters which he called in 1958Caccioppoli sets, after his mentor and friend. His definition applied some important analytic tools and De Giorgi's theorem for the sets established a new tool for set theory as well as his own works.[citation needed] This achievement not only brought Ennio immediate recognition but displayed his ability to attack problems using completely new and effective methods which, though conceived before, can be used with greater precision as shown in his research works.
De Giorgi's earliest work aimed to develop a regularity theory for minimal hypersurfaces, changing how we view the advanced theory of minimal surfaces andcalculus of variations forever. The proof required De Giorgi to develop his own version of geometric measure theory along with a related key compactness theorem. With these results, he was able to conclude that a minimal hypersurface is analytic outside a closed subset of codimension at least two.[citation needed] He also established regularity theory for all minimal surfaces in a similar manner.
De Giorgi solved19th Hilbert problem on the regularity of solutions ofelliptic partial differential equations. Before his results, mathematicians were not able to venture beyond second-order nonlinear elliptic equations in two variables. In a major breakthrough, De Giorgi proved that solutions of uniformly elliptic second-order equations of divergence form, with only measurable coefficients, were Hölder continuous. His proof was proved in 1956/57 in parallel withJohn Nash's, who was also working on and solved Hilbert's problem. His results were the first to be published, and it was anticipated that either mathematician would win the 1958Fields Medal, but it was not to be. Nevertheless, De Giorgi's work opened up the field of nonlinear elliptic partial differential equations in higher dimensions which paved a new period for all of mathematical analysis.
Almost all of his work relates to partial differential equations, minimal surfaces and calculus of variations; these notify the early triumphs of the then-unestablished field ofgeometric analysis.[citation needed] The work ofKaren Uhlenbeck,Shing-Tung Yau and many others have taken inspiration from De Giorgi which have been and continue to be extended and rebuilt in powerful and effective mannerisms.
De Giorgi's conjecture for boundary reaction terms in dimension ≤ 5 was solved byAlessio Figalli and Joaquim Serra, which was one of the results mentioned in Figalli's 2018 Fields Medal lecture given byLuis Caffarelli.
De Giorgi's work on minimal surfaces, partial differential equations and calculus of variations earned him huge and lasting fame in the mathematical community, and he was awarded many honours for his contributions, including theCaccioppoli Prize in 1960, the National Prize of Accademia dei Lincei from the President of the Italian Republic in 1973, and theWolf Prize in 1990. He was also awarded Honoris Causa degrees in Mathematics from theUniversity of Paris in 1983 at a ceremony at the Sorbonne and in Philosophy from theUniversity of Lecce in 1992. He was elected to many academies including theAccademia dei Lincei, thePontifical Academy of Sciences, the Academy of Sciences of Turin, the Lombard Institute of Science and Letters, the Académie des Sciences in Paris, and theNational Academy of Sciences of the United States. At theInternational Congress of Mathematicians he was invited to be the plenary speaker in 1966 in Moscow[1] and was an invited speaker in 1983 in Warsaw.[2]
De Giorgi was associated for many years with the Scuola Normale Superiore in Pisa, leading one of the brilliant schools of analysis in Europe at that time. He corresponded with many leading mathematicians of his time, such asLouis Nirenberg, John Nash,Jacques-Louis Lions and Renato Caccioppoli. He is largely responsible for leading and driving the Italian school of mathematical analysis in the second half of the 20th century to an international level.
De Giorgi was also a person of deep human, religious and philosophical values; he once noted that mathematics is the key to discovering the secrets of God. His work withAmnesty International in the 70s greatly extended his already-immense fame within and outside of his scientific career. He also taught mathematics at theUniversity of Asmara,Eritrea from 1966 to 1973. He died on 26 October 1996 at the age of 68.[citation needed]
De Giorgi, Ennio (1953), "Definizione ed espressione analitica del perimetro di un insieme" [Definition and analytical expression of the perimeter of a set],Atti della Accademia Nazionale dei Lincei, Rendiconti della Classe di Scienze Fisiche, Matematiche e Naturali, 8 (in Italian),14:390–393,MR0056066,Zbl0051.29403. The first note published by De Giorgi on his approach to Caccioppoli sets.
De Giorgi, Ennio; Colombini, Ferruccio; Piccinini, Livio (1972),Frontiere orientate di misura minima e questioni collegate [Oriented boundaries of minimal measure and related questions], Quaderni (in Italian),Pisa: Edizioni della Normale, p. 180,MR0493669,Zbl0296.49031. An advanced text, oriented to the theory ofminimal surfaces in the multi-dimensional setting, written by some of the leading contributors to the theory.
^De Giorgi contributed a plenary paper but did not go to Moscow — his paper was read in Moscow byEdoardo Vesentini. reference: E. De Giorgi: Hypersurfaces of minimal measure in pluridimensional Euclidean space, Proc. Internat. Congr. Math., (Moscow, 1966), Izdat. “Mir”, Moscow, 1968, 395–401. 38-2646 (review byFrederick J. Almgren Jr.)
^De Giorgi, Ennio (1984). "G-operators and Γ-convergence".Proceedings of the International Congress of Mathematicians, 1983, Warsaw. Vol. 1. pp. 1175–1191.
De Cecco, Giuseppe; Rosato, Maria Letizia, eds. (2000),Ennio De Giorgi. Hanno detto di lui... [Ennio De Giorgi: they have said of him...], Quaderni del Dipartimento di Matematica dell'Università del Salento (in Italian), vol. 5, Lecce:Coordinamento SIBA dell'Università di Lecce, p. 195,ISBN88-8305-019-3 e-ISBN88-8305-020-7. A collection of almost all commemorative papers, transcriptions of commemorative addresses on Ennio De Giorgi and personal reminiscences of pupils and friends, collected jointly with some philosophical papers of De Giorgi himself.
