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Ennio De Giorgi Selected Papers Ennio De Giorgi (Courtesy of Foto Frassi, Pisa) Ennio De Giorgi Selected Papers Ennio De Giorgi Selected Papers Ennio De Giorgi (Courtesy of Foto Frassi, Pisa) Ennio De Giorgi Selected Papers Published with the support of Unione Matematica Italiana and Scuola Normale Superiore ABC Editors Luigi Ambrosio Mario Miranda SNS, Pisa, Italy University of Trento, Italy Gianni Dal Maso Sergio Spagnolo SISSA, Trieste, Italy University of Pisa, Italy Marco Forti University of Pisa, Italy Library of Congress Control Number: 2005930439 ISBN-10 3-540-26169-9 Springer Berlin Heidelberg New York ISBN-13 978-3-540-26169-8 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is con- cerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, re- production on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965,in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the editors and TechBooks using a Springer LATEX macro package Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper SPIN: 11372011 41/TechBooks 543210 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. We selected 43 papers, out of 152, and for 17 of them we decided to keep the 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. We warmly thank Luis Caffarelli and Louis Nirenberg for their contributions to this volume. Pisa, March 2005 Luigi Ambrosio Gianni Dal Maso Marco Forti Mario Miranda Sergio Spagnolo Contents 1 Biography 1 1.1Prizesandacademicawards..................... 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 ThedevelopmentsofΓ-convergence............ 27 2.4.2 Semicontinuityandrelaxationproblems.......... 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 Minimizingmovements.................... 32 2.5.4 Minimal surfaces in metric spaces . ............. 33 2.6TheworkonFoundationsofMathematics............. 33 2.6.1 The Seminar on Foundations . ............. 33 2.6.2 The Free Construction Principle . ............. 35 2.6.3 TheFrameTheoryandtheAmpleTheory......... 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 LouisNirenberg............................ 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. On a general (r−1)-dimensional measure theory in a r-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 partialdifferentialequationsofparabolictype........... 100 4. An example of non-unique solution to the Cauchy problem, for a partialdifferentialequationofparabolictype........... 107 5. New theorems on (r − 1)-dimensional measures in r-dimensional space.................................. 111 5a. Nuovi teoremi relativi alle misure (r−1)-dimensionali in uno spazio ad r dimensioni............................ 128 6. Some applications of a K-dimensional measure theory to the calculusofvariations......................... 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 differenziabilit`ae l’analiticit`a delle estremali degli integrali multipliregolari............................ 167 8. On the isoperimetric property of the hypersphere in the class of sets whose oriented boundary has finite measure ......... 185 8a. Sulla propriet`a 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-dimensionalspace......................... 212 10. Area-minimizing oriented boundaries ................. 231 11.AnextensionofBernsteintheorem.................. 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. Un esempio di estremali discontinue per un problema variazionale di tipo ellittico ............................ 288 15.MinimalConesandtheBernsteinProblem............. 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. Una dimostrazione diretta dell’esistenza di soluzioni analitiche nel piano reale di equazioni a derivate parziali a coefficienti costanti 350 19. Convergence of the energy integrals for second order elliptic operators ............................... 361 19a. Sulla convergenza degli integrali dell’energia per operatori ellittici del secondo ordine .......................... 380 20.Ontheconvergenceofsomesequencesofarea-likeintegrals.... 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.AframingtheoryfortheFoundationsofMathematics....... 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 flowbymeancurvature....................... 690 33.Newproblemsonminimizingmovements.............. 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.TechnicalcomplementstothelectureofE.DeGiorgi......
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