Selected papers

897 Pages·2006·3.929 MB·English

by  Ennio De Giorgi, et al

Ennio DeGiorgi
SelectedPapers
Ennio DeGiorgi
(Courtesy of Foto Frassi, Pisa)
Ennio De Giorgi
Selected Papers
Published with the support of
Unione Matematica Italiana and
Scuola Normale Superiore
ABC
Editors
LuigiAmbrosio MarioMiranda
SNS,Pisa,Italy UniversityofTrento,Italy
GianniDalMaso SergioSpagnolo
SISSA,Trieste,Italy UniversityofPisa,Italy
MarcoForti
UniversityofPisa,Italy
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ISBN-10 3-540-26169-9SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-26169-8SpringerBerlinHeidelbergNewYork
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Preface
The project of publishing some selected papers by Ennio De Giorgi was under-
taken by the Scuola Normale Superiore and the Unione Matematica Italiana
in 2000. The main motivations for this project have been the desire to make
some of his classical papers, originally published in Italian, available to a large
public and to display the broad range of his achievements and his entire intel-
lectual path, as a problem solver and as a proponent of deep and ambitious
mathematical theories.
Weselected43papers, outof152, andfor17ofthemwedecidedtokeepthe
original Italian version as well, in order to give a feeling of De Giorgi’s original
style.
In the development of this long project we have been helped by several col-
laborators. In particular we wish to thank Diego Pallara and Emanuele Paolini,
for their many fundamental contributions, and Sisto Baldo, Giovanni Bellettini,
Andrea Braides, Piero D’Ancona, Massimo Gobbino, Giacomo Lenzi, Roberto
Monti, Maurizio Paolini, Paolo Tilli and Vincenzo Maria Tortorelli, who helped
us in the typing and the translations of the papers.
WewarmlythankLuisCaffarelliandLouisNirenbergfortheircontributions
to this volume.
Pisa, March 2005 Luigi Ambrosio
Gianni Dal Maso
Marco Forti
Mario Miranda
Sergio Spagnolo
Contents
1 Biography 1
1.1 Prizes and academic awards . . . . . . . . . . . . . . . . . . . . . 1
1.2 Teaching and academic engagements . . . . . . . . . . . . . . . . 2
1.3 The activities out of Pisa . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Civil, political, and religious engagement . . . . . . . . . . . . . . 4
2 The scientific work of Ennio De Giorgi 5
2.1 Complete list of De Giorgi’s scientific publications . . . . . . . . 5
2.2 From the juvenile papers to Plateau’s Problem and the solution
of Hilbert’s 19th Problem . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 The first articles . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.2 The development of the theory of perimeters . . . . . . . 20
2.2.3 The solution of Hilbert’s 19th Problem . . . . . . . . . . . 21
2.2.4 The Plateau Problem and the Bernstein Theorem . . . . 22
2.3 Partial differential equations and the foundations of Γ-convergence 23
2.3.1 The examples of nonuniqueness . . . . . . . . . . . . . . . 23
2.3.2 Gevrey classes and hyperbolic equations . . . . . . . . . . 24
2.3.3 Analytic solutions of equations with constant coefficients 25
2.3.4 The foundations of Γ-convergence . . . . . . . . . . . . . 26
2.4 Asymptotic problems in the Calculus of Variations . . . . . . . . 27
2.4.1 The developments of Γ-convergence . . . . . . . . . . . . 27
2.4.2 Semicontinuity and relaxation problems . . . . . . . . . . 29
2.4.3 Evolution problems for nondifferentiable functionals . . . 30
2.5 The most recent developments in the Calculus of Variations . . . 31
2.5.1 Free discontinuity problems . . . . . . . . . . . . . . . . . 31
2.5.2 Mean curvature evolution . . . . . . . . . . . . . . . . . . 31
2.5.3 Minimizing movements. . . . . . . . . . . . . . . . . . . . 32
2.5.4 Minimal surfaces in metric spaces. . . . . . . . . . . . . . 33
2.6 The work on Foundations of Mathematics . . . . . . . . . . . . . 33
2.6.1 The Seminar on Foundations . . . . . . . . . . . . . . . . 33
2.6.2 The Free Construction Principle . . . . . . . . . . . . . . 35
2.6.3 The Frame Theory and the Ample Theory . . . . . . . . . 36
2.6.4 The basic theories . . . . . . . . . . . . . . . . . . . . . . 36
2.6.5 The theories of 2000 . . . . . . . . . . . . . . . . . . . . . 37
VIII Contents
2.7 Remarks on some of the analytic works of Ennio De Giorgi, by
Louis Nirenberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.8 De Giorgi’s contribution to the regularity theory of elliptic
equations, by Luis Caffarelli . . . . . . . . . . . . . . . . . . . . . 42
3 Selected papers 49
1. Definition and analytic expression of the perimeter of a set . . . . 50
1a. Definizione ed espressione analitica del perimetro di un insieme . . 54
2. Onageneral(r−1)-dimensionalmeasuretheoryinar-dimensional
space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2a. Su una teoria generale della misura (r −1)-dimensionale in uno
spazio ad r dimensioni . . . . . . . . . . . . . . . . . . . . . . . . 79
3. A uniqueness theorem for the Cauchy problem relative to linear
partial differential equations of parabolic type . . . . . . . . . . . 100
4. An example of non-unique solution to the Cauchy problem, for a
partial differential equation of parabolic type . . . . . . . . . . . 107
5. New theorems on (r −1)-dimensional measures in r-dimensional
space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5a. Nuoviteoremirelativiallemisure(r−1)-dimensionaliinunospazio
ad r dimensioni . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6. Some applications of a K-dimensional measure theory to the
calculus of variations . . . . . . . . . . . . . . . . . . . . . . . . . 145
6a. Alcune applicazioni al calcolo delle variazioni di una teoria della
misura K-dimensionale. . . . . . . . . . . . . . . . . . . . . . . . 147
7. On the differentiability and the analyticity of extremals of regular
multiple integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7a. Sulla differenziabilita` e l’analiticita` delle estremali degli integrali
multipli regolari. . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
8. On the isoperimetric property of the hypersphere in the class of
sets whose oriented boundary has finite measure . . . . . . . . . 185
8a. Sulla proprieta` isoperimetrica dell’ipersfera, nella classe degli
insiemi aventi frontiera orientata di misura finita . . . . . . . . . 198
9. Complements to the (n−1)-dimensional measure theory in a
n-dimensional space . . . . . . . . . . . . . . . . . . . . . . . . . 212
10. Area-minimizing oriented boundaries. . . . . . . . . . . . . . . . . 231
11. An extension of Bernstein theorem . . . . . . . . . . . . . . . . . . 264
12. Hypersurfaces of minimal measure in pluridimensional Euclidean
spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
13. Removable singularities of minimal hypersurfaces . . . . . . . . . . 278
14. An example of discontinuous extremals for a variational problem
of elliptic type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
14a. Unesempiodiestremalidiscontinueperunproblemavariazionale
di tipo ellittico . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
15. Minimal Cones and the Bernstein Problem . . . . . . . . . . . . . 291
16. An a priori estimate related to nonparametric minimal surfaces . . 316
17. A representation formula for analytic functions in Rn . . . . . . . 331
Contents IX
17a. Una formula di rappresentazione per funzioni analitiche in Rn . . 335
18. A direct proof of the existence of analytic solutions in the real
plane of partial differential equations with constant coefficients . 339
18a. Unadimostrazionedirettadell’esistenzadisoluzionianalitichenel
piano reale di equazioni a derivate parziali a coefficienti costanti 350
19. Convergence of the energy integrals for second order elliptic
operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
19a. Sullaconvergenzadegliintegralidell’energiaperoperatoriellittici
del secondo ordine . . . . . . . . . . . . . . . . . . . . . . . . . . 380
20. On the convergence of some sequences of area-like integrals . . . . 399
20a. Sulla convergenza di alcune successioni d’integrali del tipo
dell’area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
21. Γ-convergence and G-convergence . . . . . . . . . . . . . . . . . . 429
21a. Γ-convergenza e G-convergenza . . . . . . . . . . . . . . . . . . . 437
22. On hyperbolic equations with coefficients depending on time only 445
23. Convergence problems for functionals and operators . . . . . . . . 487
24. On a type of variational convergence . . . . . . . . . . . . . . . . . 517
25. Evolution problems in metric spaces and steepest descent curves . 527
26. Generalized limits in Calculus of Variations . . . . . . . . . . . . . 534
27. A framing theory for the Foundations of Mathematics . . . . . . . 570
28. A self–reference oriented theory for the foundations of mathematics 583
29. Introduction to free-discontinuity problems . . . . . . . . . . . . . 634
30. Existence theorem for a minimum problem with free discontinuity
set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654
31. New functionals in Calculus of Variations . . . . . . . . . . . . . . 679
32. Conjectures on limits of some quasilinear parabolic equations and
flow by mean curvature . . . . . . . . . . . . . . . . . . . . . . . 690
33. New problems on minimizing movements . . . . . . . . . . . . . . 699
34. Lectio Magistralis: “The sapiential value of Mathematics” . . . . . 714
34a. Lectio Magistralis “Il valore sapienziale della matematica” . . . . 720
35. Fundamental principles of Mathematics . . . . . . . . . . . . . . . 726
36. Technical complements to the lecture of E. De Giorgi . . . . . . . 734
37. Introducing basic theories for the Foundations of Mathematics . . 745
37a. Una proposta di teoria base dei Fondamenti della Matematica . . 758
38. Anintroductionofvariablesintotheframeofthebasictheoriesfor
the Foundations of Mathematics . . . . . . . . . . . . . . . . . . 771
38a. Introduzione delle variabili nel quadro delle teorie base dei
Fondamenti della Matematica . . . . . . . . . . . . . . . . . . . . 783
39. General Plateau problem and geodesic functionals . . . . . . . . . 795
40. Conjectures about some evolution problems . . . . . . . . . . . . . 801
40a. Congetture riguardanti alcuni problemi di evoluzione . . . . . . . 814
41. Towards the axiomatic systems of the third millennium
in Mathematics, Logic, and Computer Science . . . . . . . . . . . 827
42. Truth and judgments within a new axiomatic framework . . . . . 846
X Contents
43. Overcoming set–theoretic reductionism in search of wider and
deeper mutual understanding between Mathematicians and
scholars of different scientific and human disciplines . . . . . . . 867
43a. Dal superamento del riduzionismo insiemistico alla ricerca di una
piu`ampiaeprofondacomprensionetramatematiciealtristudiosi
di altre discipline scientifiche ed umanistiche. . . . . . . . . . . . 878
Chapter 1
Biography
Ennio De Giorgi was born in Lecce on February 8, 1928. His mother, Stefania
Scopinich, came from a family of navigators from Lussino, while his father,
Nicola, professor of literature in the teachers’ training school of Lecce, was an
esteemed scholar in Arabic Language, History, and Geography. His father died
an untimely death in 1930; his mother, to whom Ennio was particularly bound,
lived until 1988.
In 1946, after his high school studies in Lecce, Ennio moved to Rome, where
he started his university studies in Engineering. The following year he switched
to Mathematics, and graduated in 1950 under the direction of Mauro Picone.
Soon afterwards he obtained a fellowship at the Istituto per le Applicazioni del
Calcolo,andin1951hebecameassistanttoPiconeattheMathematicalInstitute
“Guido Castelnuovo” of the University of Rome.
In1958hewasawardedtheChairofMathematicalAnalysisbytheUniversity
of Messina, where he started in this post in December. In the autumn of 1959,
following a proposal by Alessandro Faedo, he was hired by the Scuola Normale
of Pisa, where he held the Chair of Algebraic and Infinitesimal Mathematical
Analysis for almost forty years.
In September 1996 he was admitted to the hospital in Pisa. He underwent
surgical treatment, and passed away on October 25.
1.1 Prizes and academic awards
In 1960 the Italian Mathematical Union awarded him the Caccioppoli Prize,
founded in the same year. In 1973 the Accademia dei Lincei awarded him the
Prize of the President of the Republic. In 1990 he received the prestigious Wolf
Prize in Tel Aviv.
In1983,duringasolemnceremonyatSorbonne,hereceivedthedegreehono-
ris causa in Mathematics at the University of Paris. In 1992 the University of
Lecce awarded him the degree honoris causa in Philosophy, of which he was
particularly proud.