Abstract

The aim of this review is to analyze the impact of a personal list of selected mathematical contributions of Jean Jacques Moreau. I start with Legendre's transform, Mandelbrojt's transform and Fenchel's conjugate to come to the concept of Moreau and Rockafellar of the conjugate as a function defined on the whole topological dual space taking on values in the extended real line. Then I discuss a large number of Moreau's fundamental ideas in modern convex analysis.

Author information

Contact details are reproduced from the original publication and may be historical.

Keywords

  • Biographical review
  • Jean Jacques Moreau
  • convex analysis
  • mechanical systems
  • sweeping process
  • differential inclusion
  • conjugate function

References

  1. J. J. Moreau: Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuellement polaires, C. R. Acad. Sci., Paris 255 (1962) 238–240.Google Scholar
  2. J. J. Moreau: Sur la naissance de la cavitation dans un fluide incompressible, Communication in International Congress of Mathematicians, Stockholm (1962).Google Scholar
  3. J. J. Moreau: Fonctions convexes duales, Séminaire de Mathématique, Faculté des Sciences de Montpellier (1962) 18 p.Google Scholar
  4. J. J. Moreau: Fonctions convexes duales et points proximaux dans un espace hilbertien, C. R. Acad. Sci., Paris 255 (1962) 287–289.Google Scholar
  5. J. J. Moreau: Propriétés des applications “prox”, C. R. Acad. Sci., Paris 256 (1963) 1069–1071.Google Scholar
  6. J. J. Moreau: Applications “prox”, Séminaire de Mathématique, Faculté des Sciences de Montpellier (1963) 20 p.Google Scholar
  7. J. J. Moreau: Inf-convolution, Séminaire de Mathématique, Faculté des Sciences de Montpellier (1963) 48 p.Google Scholar
  8. J. J. Moreau: Fonctions à valeurs dans [−∞, +∞]; notions algébriques, Séminaire de Mathématique, Faculté des Sciences de Montpellier (1963) 35 p.Google Scholar
  9. J. J. Moreau: Etude locale d’une fonctionnelle convexe, Séminaire de Mathématique, Faculté des Sciences de Montpellier (1963) 25 p.Google Scholar
  10. J. J. Moreau: Les liaisons unilatérales et le principe de Gauss, C. R. Acad. Sci., Paris 256 (1963) 871–874.Google Scholar
  11. J. J. Moreau: Sur la fonction polaire d’une fonction semi-continue supérieurement, C. R. Acad. Sci., Paris 258 (1964) 1128–1131.Google Scholar
  12. J. J. Moreau: Sur la naissance de la cavitation dans une conduite, C. R. Acad. Sci., Paris 259 (1964) 3948–3950.Google Scholar
  13. J. J. Moreau: Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. Fr. 93 (1965) 273–299.https://doi.org/10.24033/bsmf.1625Google Scholar
  14. J. J. Moreau: Quadratic programming in mechanics: dynamics of one-sided constraints, SIAM J. Control 4 (1966) 153–158.https://doi.org/10.1137/0304014Google Scholar
  15. J. J. Moreau: Fonctionnelles Convexes, 2nd Ed., Facoltà di Ingegneria, Università di Roma ”Tor Vergata”, Rome (2003).Google Scholar
  16. J. J. Moreau: Inf-convolution, sous-additivité, convexité des fonctions numériques, J. Math. Pures Appl., IX. Sér. 49 (1970) 109–154.Google Scholar
  17. J. J. Moreau: Sur l’évolution d’un système élastoplastique, C. R. Acad. Sci., Paris, Sér. A 273 (1971) 118–121.Google Scholar
  18. J. J. Moreau: Rafle par un convexe variable I, Sém. Anal. Convexe Montpellier, Exposé 15 (1971) 43 p.Google Scholar
  19. J. J. Moreau: Rafle par un convexe variable II, Sém. Anal. Convexe Montpellier, Exposé 3 (1972) 36 p.Google Scholar
  20. J. J. Moreau: Rétraction d’une multiapplication, Sém. Anal. Convexe Montpelllier, Exposé 13 (1972) 90 p.Google Scholar
  21. J. J. Moreau: On unilateral constraints, friction and plasticity, in: New Variational Techniques in Mathematical Physics (Bolzano, 1973), G. Capriz, G. Stampacchia (ed.), C.I.M.E., Edizioni Cremonese, Roma (1974) 173–322.Google Scholar
  22. J. J. Moreau: Multiapplications à rétraction finie, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 1 (1974) 169–203.Google Scholar
  23. J. J. Moreau: Sur les mesures différentielles des fonctions vectorielles à variation localement bornée, Sém. Anal. Convexe Montpellier, Exposé 17 (1975) 39 p.Google Scholar
  24. J. J. Moreau: Sur les mesures différentielles de fonctions vectorielles et certains problèmes dévolution, C. R. Acad. Sci., Paris, Sér. A 282 (1976) 837–840.Google Scholar
  25. J. J. Moreau: Evolution problem associated with a moving convex set in a Hilbert space, J. Differ. Equations 26 (1977) 347–374.https://doi.org/10.1016/0022-0396(77)90085-7Google Scholar
  26. J. J. Moreau: Liaisons unilatérales sans frottement et chocs inélastiques, C. R. Acad. Sci., Paris, Sér. II 296 (1983) 1473–1476.Google Scholar
  27. J. J. Moreau: Bounded variation in time, in: Topics in Nonsmooth Mechanics, J. J. Moreau et al. (ed.), Birkhäuser, Basel (1988) 1–74.Google Scholar
  28. J. J. Moreau: Unilateral contact and dry friction in finite freedom dynamics, in: Nonsmooth Mechanics and Applications, J. J. Moreau, P. D. Panagiotopoulos (eds.), CISM Courses and Lectures 302, Springer, New York (1988) 1–82.https://doi.org/10.1007/978-3-7091-2624-0_1Google Scholar
  29. J. J. Moreau: Jump functions of a real interval to a Banach space, Ann. Fac. Sci. Toulouse, V. Sér., Math. 1989, Suppl. (1989) 77–91.https://doi.org/10.5802/afst.689Google Scholar
  30. J. J. Moreau: Numerical aspects of the sweeping process, Comput. Methods Appl. Mech. Eng. 177 (1999) 329–349.https://doi.org/10.1016/S0045-7825(98)00387-9Google Scholar
  31. J. J. Moreau: Some basics of unilateral dynamics, in: Unilateral Multibody Contacts (Munich, 1998), F. Pfeiffer, C. Glocker (eds.), Kluwer, Dordrecht (1999) 1–14.https://doi.org/10.1007/978-94-011-4275-5_1Google Scholar
  32. J. J. Moreau: Contact et frottement en dynamique des systèmes de corps rigides, Rev. Eur. Élém. Finis 9 (2000) 9–28.https://doi.org/10.1080/12506559.2000.10511427Google Scholar
  33. J. J. Moreau: An introduction to unilateral dynamics, in: Novel Approaches in Civil Engineering, M. Frémond, F. Maceri (eds.), Springer, Berlin (2004) 1–46.Google Scholar
  34. H. Attouch: Variational Convergence for Functions and Operators, Applicable Mathematics Series, Pitman, London (1984).Google Scholar
  35. J. P. Aubin: Mathematical Methods of Game and Economic Theory, Dover, Mineola (2007).Google Scholar
  36. H. H. Bauschke, P. L. Combettes: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics, Springer, Berlin (2011).https://doi.org/10.1007/978-1-4419-9467-7Google Scholar
  37. G. Beer: Topologies on Closed and Closed Convex Sets, Kluwer, Dordrecht (1993).https://doi.org/10.1007/978-94-015-8149-3Google Scholar
  38. J. M. Borwein, J. D. Vanderwerff: Convex Functions. Constructions, Characterizations and Counterexamples, Cambridge University Press, Cambridge (2010).https://doi.org/10.1017/CBO9781139087322Google Scholar
  39. C. Castaing: Intégrales convexes duales, C. R. Acad. Sci., Paris, Sér. A 275 (1972) 1331–1334.Google Scholar
  40. C. Castaing: Intégrales convexes duales, in: New Variational Techniques in Mathematical Physics (Bressanone, 1973), G. Capriz et al. (ed.), CIME Summer Schools 63, Springer, Berlin (1973) 27–42.https://doi.org/10.1007/978-3-642-10960-7_2Google Scholar
  41. C. Castaing, M. Valadier: Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580, Springer, Berlin (1977).https://doi.org/10.1007/BFb0087685Google Scholar
  42. R. Courant, D. Hilbert: Methods of Mathematical Physics. Vol. 2, Interscience, New York (1962).Google Scholar
  43. W. Fenchel: On conjugate convex functions, Can. J. Math. 1 (1949) 73–77.https://doi.org/10.4153/CJM-1949-007-xGoogle Scholar
  44. W. Fenchel: Convex Cones, Sets and Functions, Lecture Notes Princeton University, Princeton, New Jersey (1951).Google Scholar
  45. J.-B. Hiriart-Urruty, C. Lemaréchal: Convex Analysis and Minimization Algorithms I & II, Grundlehren der Mathematischen Wissenschaften 305, 306, Springer, Berlin (1993).https://doi.org/10.1007/978-3-662-02796-7Google Scholar
  46. A. D. Ioffe, V. L. Levin: Subdifferentials of convex functions, Tr. Mosk. Mat. O.-va 26 (1972) 3–73 (in Russian).Google Scholar
  47. A. D. Ioffe, V. M. Tikhomirov: The duality of convex functions and extremal problems, Usp. Mat. Nauk 23 (1968) 51–116 (in Russian); Russ. Math. Surv. 23 (1968) 53–124 (in English).https://doi.org/10.1070/RM1968v023n06ABEH001251Google Scholar
  48. A. Jourani, L. Thibault, D. Zagrodny: Differential properties of the Moreau envelope, J. Funct. Anal 266 (2014) 1185–1237.https://doi.org/10.1016/j.jfa.2013.11.008Google Scholar
  49. A.-M. Legendre: Mémoire sur l’intégration de quelques équations aux différences partielles, Histoire de l’Académie Royale des Sciences (1787) 309–351.Google Scholar
  50. S. Mandelbrojt: Sur les fonctions convexes, C. R. Acad. Sci., Paris 209 (1939) 977–978.Google Scholar
  51. M. D. P. Monteiro Marques: Differential Inclusions in Nonsmooth Mechanical Problems. Shocks and Dry Friction, Birkhäuser, Basel (1993).Google Scholar
  52. R. R. Phelps: Convex Functions, Monotone Operators and Differentiability, 2nd Ed., Springer, New York (1993).Google Scholar
  53. R. T. Rockafellar: Convex Functions and Dual Extremum Problems, Thesis, Harvard (1963).Google Scholar
  54. R. T. Rockafellar: Extension of Fenchel’s duality theorem for convex functions, Duke Math. J. 33 (1966) 81–90.https://doi.org/10.1215/S0012-7094-66-03312-6Google Scholar
  55. R. T. Rockafellar: Conjugates and Legendre transforms of convex functions, Can. J. Math. 19 (1967) 200–205.https://doi.org/10.4153/CJM-1967-012-4Google Scholar
  56. R. T. Rockafellar: Convex Analysis, Princeton Mathematical Series 28, Princeton University Press, Princeton (1970).Google Scholar
  57. R. T. Rockafellar: Convex integral functionals and duality, in: Contributions to Nonlinear Functional Analysis, E. Zarantonello (ed.), Academic Press, New York (1971) 215–236.https://doi.org/10.1016/B978-0-12-775850-3.50012-1Google Scholar
  58. R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Grundlehren der Mathematischen Wissenschaften 317 Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  59. S. Simons: From Hahn-Banach to Monotonicity, 2nd Ed., Lecture Notes in Mathematics 1693, Springer, Berlin (2008).Google Scholar
  60. M. Valadier: Intégration de convexes fermés, notamment d’épigraphes, d’Inf-convolution continue, Rev. Franç. Inform. Rech. Opér. 4(R-2) (1970) 57–73.https://doi.org/10.1051/m2an/197004R200571Google Scholar
  61. M. Valadier: Convex integrands on Souslin locally convex spaces, Pac. J. Math. 59 (1975) 267–276.https://doi.org/10.2140/pjm.1975.59.267Google Scholar
  62. M. Valadier: Jean Jacques Moreau, Gazette des Mathématiciens 140 (2014) 75–77.Google Scholar
  63. W. L. Young: On classes of summable functions and their Fourier series, Proc. R. Soc. Lond., Ser. A 87 (1912) 225–229.https://doi.org/10.1098/rspa.1912.0076Google Scholar
  64. C. Zălinescu: Convex Analysis in General Vector Spaces, World Scientific, Singapore (2002).https://doi.org/10.1142/5021Google Scholar

Suggested citation

L. Thibault. “Jean Jacques Moreau. A Selected Review of his Mathematical Works.” Journal of Convex Analysis 23 (2016), No. 1, 5–22. https://doi.org/10.68381/jca23001

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