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On the second derivatives of convex functions on Hilbert spaces

Author: Nobuyuki Kato
Journal: Proc. Amer. Math. Soc. 106 (1989), 697-705
MSC: Primary 47H05; Secondary 46G05, 58C20
MathSciNet review: 960646
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Abstract: Let $ \phi $ be a proper l.s.c. convex function on a real Hilbert space $ H$. We show that if $ H$ is separable, then $ \phi $ is twice differentiable in some sense on a dense subset of the graph of $ \partial \phi $.

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  • [1] Richard Arens, Operational calculus of linear relations, Pacific J. Math. 11 (1961), 9–23. MR 0123188
  • [2] Hidekazu Asakawa, Restriction of maximal monotone operator to closed linear subspace, TRU Math. 23 (1987), no. 1, 97–116. MR 931763
  • [3] H. Asakawa, private communication.
  • [4] Hédy Attouch, Familles d’opérateurs maximaux monotones et mesurabilité, Ann. Mat. Pura Appl. (4) 120 (1979), 35–111 (French, with English summary). MR 551062,
  • [5] Jean-Pierre Aubin and Ivar Ekeland, Applied nonlinear analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 749753
  • [6] Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR 0390843
  • [7] H. Brezis, Operateurs maximaux monotones, North-Holland, Amsterdam, 1973.
  • [8] H. Brézis and F. E. Browder, Linear maximal monotone operators and singular nonlinear integral equations of Hammerstein type, Nonlinear analysis (collection of papers in honor of Erich H. Rothe), Academic Press, New York, 1978, pp. 31–42. MR 0513047
  • [9] J.-B. Hiriart-Urruty, A new set-valued second order derivative for convex functions, FERMAT days 85: mathematics for optimization (Toulouse, 1985) North-Holland Math. Stud., vol. 129, North-Holland, Amsterdam, 1986, pp. 157–182. MR 874365,
  • [10] F. Mignot, Contrôle dans les inéquations variationelles elliptiques, J. Functional Analysis 22 (1976), no. 2, 130–185 (French). MR 0423155
  • [11] J. L. Ndoutoume, Calcul differential generalise du second ordre, Publ. AVAMAC Université du Perpignan, vol. 2, 1986.
  • [12] R. T. Rockafellar, Maximal monotone relations and the second derivatives of nonsmooth functions, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 3, 167–184 (English, with French summary). MR 797269
  • [13] Gabriella Salinetti and Roger J.-B. Wets, On the relations between two types of convergence for convex functions, J. Math. Anal. Appl. 60 (1977), no. 1, 211–226. MR 0479398,
  • [14] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095

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Keywords: Convex function, subdifferential, second derivative, convergence in the sense of Mosco
Article copyright: © Copyright 1989 American Mathematical Society

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