Graphical methods in first and second-order differentiability theory of integral functionals
Tóm tắt
We discuss several notions of first and second-order differentiability for integral functionals on a Hilbert space.
Tài liệu tham khảo
Aronszajn, N.: Differentiability of Lipschitz mappings between Banach spaces,Studia Math. LVII (1976), 147–190.
Attouch, H.: Familles d'opérateurs maximaux monotones et mesurabilité,Ann. Mat. Pura Appl. 120 (1979), 35–111.
Attouch, H., Ndoutoume, J. L., and Théra, M.: On the equivalence between epi-convergence of sequences of functions and graph convergence of their derivatives, to appear.
Attouch, H. and Wets, R. J.-B.: A convergence theory for saddle functions,Trans. Amer. Math. Soc. 280 (1983), 1–41.
Attouch, H. and Wets, R. J.-B.: Isometries for the Legendre-Fenchel transform,Trans. Amer. Math. Soc. 296 (1986), 33–60.
Aubin, J.-P. and Ekeland, I.:Applied Nonlinear Analysis, Wiley, New York, 1984.
Bangert, V.: Analytische Eigenschaften konvexer Funktionen auf Riemannschen Mannigfaltigkeiten,J. reine angew. Math. 307 (1979), 309–324.
Blot, J.: Démonstration des théorèmes de Vainberg et Skrypnik, Publications du Département de Mathématiques de l'Université de Limoges.
Borwein, J. M.: Asplund spaces are sequentially reflexive, to appear.
Borwein, J. M. and Fabian, M.: On convex functions having points of Gâteaux differentiability which are not points of Fréchet differentiability' to appear.
Borwein, J. M. and Fabian, M.: Generic second order Gâteaux differentiability of convex functions, to appear.
Borwein, J. M., Fabian, M., and Vanderwerff, J.: Locally Lipschitz functions and bornological derivatives, to appear.
Borwein, J. M. and Noll, D.: Second order differentiability of convex functions in Banach spaces,Trans. Amer. Math. Soc. to appear.
Borwein, J. M. and Preiss, D.: A smooth variational principle with applications to sub-differentiability and to differentiability of convex functions,Trans. Amer. Math. Soc. 303 (1987), 517–527.
Busemann, H.:Convex Surfaces, Interscience Publishers, New York 1955.
Deville, R., Godefroy, G., and Zizler, V.: A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions, to appear.
Do, C. N.: Generalized second-order derivatives of convex functions in reflexive Banach spaces,Trans. Amer. Math. Soc. to appear.
Dolecki, S., Salinetti, G., and Wets, R.: Convergence of functions: equi-semicontinuity,Trans. Amer. Math. Soc. 276 (1983), 409–429.
Ekeland, I.:Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, New York, 1989.
Fabian, M.: Lipschitz smooth points of convex functions and isomorphic characterization of Hilbert spaces,Proc. London Math. Soc. 751 (1985), 113–126.
Fitzpatrick, S. and Phelps, R. R.: Differentiability of the metric projection in Hilbert space,Trans. Amer. Math. Soc. 270 (1982), 483–501.
Hiriart-Urruty, J. B. and Seeger, A.: ‘Calculus rules on a new set-valued second derivative for convex functions,Nonlinear Anal. Theory Meth. Appl. 13 (1989), 721–738.
Kato, N.: On the second derivatives of convex functions in Hilbert space,Proc. Amer. Math. Soc. 106 (1989).
Loewen, P. D. and Zeng, H.: Epi-derivatives of integral functionals with applications, to appear.
Mignot, F.: Contrôle dans les inéquations variationelles elliptiques,J. Funct. Anal. 22 (1976), 130–185.
Moreau, J. J.: Proximité et dualité dans un espace Hilbertien,Bull. Soc. Math. France 93 (1965), 273–299.
Ndoutoume, J. L.: Calcul différentiel généralisé du second ordre,Publ. AVAMAC, Univ. de Perpignan, 1987.
Ndoutoume, J. L. and Théra, M.: Generalized second-order derivatives of convex functions in locally convex topological vector spaces, to appear.
Noll, D.: Second order differentiability of integral functionals on Sobolev spaces andL 2-spaces,J. reine angew. Math. 436 (1993), 1–17.
Noll, D.: Generalized second fundamental form for Lipschitzian hypersurfaces by way of second epi deivatives,Canadian Math. Bull. 35(4) (1992), 523–536.
Noll, D.: Directional differentiability of the metric projection in Hilbert space,Pacific J. Math., to appear.
Palais, R. S. and Smale, S.: A generalized Morse Theory,Bull. Amer. Math. Soc. 70 (1964), 165–172.
Rockafellar, R. T.:Conjugate Duality and Optimization, SIAM Publ., Philadelphia, 1974.
Rockafellar, R. T.: Maximal monotone relations and the second derivatives of nonsmooth functions,Ann. Inst. H. Poincaré, Anal. Non Linêaire 2 (1985), 167–184.
Rockafellar, R. T.: Generalized second derivatives of convex functions and saddle functions,Trans. Amer. Math. Soc. 322 (1990), 51–77.
Rockafellar, R. T.: Second order optimality conditions in non-linear programming obtained by way of epi derivatives,Math. Oper. Res. 14 (1989), 462–484.
Salinetti, G. and Wets, R. J.-B.: On the relation between two types of convergence for convex functions,J. Math. Anal. Appl. 60 (1977), 211–226.
Skrypnik, I. V.: On the application of Morse's method to nonlinear elliptic equations,Dokl. Akad. Nauk SSSR 202 (1972), 202ff.