Ennio De Giorgi

Ennio De Giorgi (born February 8, 1928 in Lecce , † October 25, 1996 in Pisa ) was an influential Italian mathematician. He made decisive contributions in the field of minimal areas, the calculus of variations and partial differential equations. Among other things, he is known for his contributions to solving Hilbert's 19th problem .

Life and Scientific Work

De Giorgi attended the University of Rome from 1946, where he first began studying engineering, but then switched to mathematics. In 1950 he received his diploma (acquisition of the laureate) and received his doctorate from Mauro Picone , whose assistant at the Castelnuovo Institute he became. In 1958 he became professor of analysis at the University of Messina and in 1959 at the Scuola Normale Superiore in Pisa . He was active in research until his death. De Giorgi was very religious. From 1966 to 1973 he taught once a year for one month at the nun-led University of Asmara in Eritrea . He was also a human rights advocate and an active member of Amnesty International .

De Giorgi credits Picone with a great influence on his academic career, which he describes as extremely liberal in scientific dialogue but respectful of the academic customs of his time. De Giorgi is described by his students and colleagues as a happy and open person who took great care of his students. He had a significant influence on Italian mathematics. His students include Giovanni Alberti , Luigi Ambrosio , Andréa Braides , Giuseppe Buttazzo , Gianni Dal Maso and Paolo Marcellini .

De Giorgi's early work dealt mainly with the geometric theory of dimensions. Already during his studies he heard lectures on this subject from Renato Caccioppoli . Among his most important achievements are the precise definition of the edge by Borel-Mengen and his work on minimal surfaces (partly in collaboration with Enrico Bombieri ). In 1960 he proved the regularity of these surfaces in a large class of cases. One of his most outstanding achievements is his contribution to the complete solution of the amber problem. Sergei Natanowitsch Bernstein had shown around 1914 that in the Euclidean space of two dimensions a complete minimal surface (graph of a function ) is a hypersurface (affine function ). The problem of whether the theorem also applies to higher dimensions was known as the Bernstein problem of differential geometry ( Wendell Fleming ). De Giorgi proved that the theorem also holds for and d = 3 and Frederick Almgren for d = 4. James Simons extended the sentence to all dimensions in 1968 . In 1969 De Giorgi, Bombieri and Enrico Giusti showed that this statement is wrong for all spatial dimensions (the counterexample, the Simons cone, had already been provided by James Simons). ${\ displaystyle d}$ ${\ displaystyle f \ colon \ mathbb {R} ^ {2} \ to \ mathbb {R}}$ ${\ displaystyle f (x) = nx + m}$ ${\ displaystyle f \ colon \ mathbb {R} ^ {d} \ to \ mathbb {R}}$ ${\ displaystyle d \ leq 7}$ ${\ displaystyle d \ geq 8}$

In 1955 de Giorgi gave the first example of ambiguity of the initial value problem for linear parabolic partial differential equations with regular coefficients.

In 1957, De Giorgi contributed significantly to the solution of the 19th Hilbert problem - the question of the analyticity of minimizers in the calculus of variations - as it occurs, for example, in the variation of the action function in physics (variation of a multiple integral of an analytical function with a convexity condition for the function) . De Giorgi proved the analyticity (continuity and differentiability of the solutions) independently and at about the same time as John Nash . He proved the following statement: Every solution of an elliptic differential equation of the second order with bounded coefficients is Hölder continuous . In 1971, together with L. Cattabriga, he proved the existence of analytical solutions of elliptic partial differential equations with constant coefficients in two dimensions.

He made a significant contribution to the calculus of variations in 1973 with the introduction of Γ-convergence , a special concept of convergence for functionals. This has a large number of uses for problems such as B. the reduction of dimensions or the transition from discrete (atomic) to continuous models in physics.

Together with Ferruccio Colombini and Sergio Spagnolo , in 1978/79 he showed the existence of solutions for hyperbolic partial differential equations with analytic coefficients and gave an example of the non-existence of a solution for non-analytic coefficients.

In the 1980s, de Giorgi dealt increasingly with the applications of the geometric theory of dimensions. He introduced the space of functions, the special functions of limited variation and, in collaboration with Michele Carriero and Antonio Leaci, proved the existence of weak solutions of the Mumford Shah functional in space . This functional - introduced by David Mumford and Jayant Shah - is of considerable importance in the theory of image processing. ${\ displaystyle SBV}$ ${\ displaystyle SBV}$

Awards

1969 Premio Caccioppoli
1973 Italian State Prize from the Accademia dei Lincei
1990 Wolf Prize

De Giorgi also received honorary doctorates from the Sorbonne (1983) and the University of Lecce. He was a member of the Accademia dei Lincei , the papal , Turin and Lombard academies, the Académie des sciences and the National Academy of Sciences (USA, since 1995).

In 1966 he was invited speaker at the International Congress of Mathematicians in Moscow (Hypersurfaces of minimal measure in pluridimensional euclidean spaces) and in 1983 in Warsaw (G-operators and Gamma-convergence).

Important writings

De Giorgi wrote 149 papers, most of which were published in Italian.

Un teorema di unicità per il problema di Cauchy, relativo ad equazioni differentenziali lineari a derivate parziali di tipo parabolico. Ann. Mat. Pura Appl. (4) 40, 371-377, 1955.
Un esempio di non unicità della soluzione di un problema di Cauchy, relativo ad un'equazione differential lineare di tipo parabolico. Rend. Mat. E Appl. (5) 14, 382-387, 1955.
Sull'analiticità delle estremali degli integrali multipli. Atti Accad. Naz. Lincei Rend. Cl. Sci. F sharp. Mat. Nature. (8) 20, 438-441, 1956.
Una estensione del teorema di amber. Ann. Scuola Norm. Sup. Pisa (3) 19, 79-85, 1965.
with E. Bombieri and E. Giusti: Minimal cones and the Bernstein problem. Invent. Math. 7, 243-268, 1969,
with L. Cattabriga: Una dimonstratzione diretta dell esistenza di soluzione analitiche nel piano reale di equazioni a derivate partiali a coefficienti constanti, Boll. U.N. Mat. Ital., Vol. 4, 1971, 1015-1027
with S. Spagnolo: Sulla convergenza degli integrali dell'energia per operatori ellittici del secondo ordine. Boll. U.N. Mat. Ital. (4) 8, 391-411, 1973.
with T. Franzoni: Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. F sharp. Mat. Nature. (8) 58, 842-850, 1975.
Gamma-convergenza e G-convergenza. Boll. U.N. Mat. Ital. (5) 14-A, 213-220, 1977.
with F. Colombini and S. Spagnolo: Existence et unicité des solutions des équations hyperboliques du second ordre à coefficients ne dépendant que du temps. CR Acad. Sci. Paris Sér. A 286, 1045-1048, 1978.
with F. Colombini and S. Spagnolo: Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6, 511-559, 1979.
with M. Carriero and A. Leaci: Existence theorem for a minimum problem with free discontinuity set. Arch. Rational Mech. Anal. 108, 195-218, 1989.

Collections of articles

De Giorgi: Selected Papers. Springer-Verlag 2006.

literature

Andrea Parlangeri, Uno Spirito Puro. Ennio De Giorgi, genio della matematica, Edizione Millela Lecce 2015
Obituary by Jacques-Louis Lions, Francois Murat, Notices AMS, October 1997, pdf

Web links

credentials

↑ ^a ^b ^c Interview with Ennio de Giorgi (PDF file; 105 kB)
↑ De Giorgi biography
^ Mathematics Genealogy Project
↑ De Giorgi, Una estensione del teorema di Bernstein, Ann. Scuola Normale Superiore Pisa, Volume 19, 1965, pp. 78-85, digitized
^ Bernstein Problem, Encyclopedia of Mathematics, Springer

personal data
SURNAME	Giorgi, Ennio De
BRIEF DESCRIPTION	Italian mathematician
DATE OF BIRTH	February 8, 1928
PLACE OF BIRTH	Lecce
DATE OF DEATH	October 25, 1996
Place of death	Pisa

[Emmer-1] Interview with Ennio de Giorgi (PDF file; 105 kB)

[2] De Giorgi biography

[3] Mathematics Genealogy Project

[4] De Giorgi, Una estensione del teorema di Bernstein, Ann. Scuola Normale Superiore Pisa, Volume 19, 1965, pp. 78-85, digitized

[5] Bernstein Problem, Encyclopedia of Mathematics, Springer