Topological characteristics of non-smooth functionals

V. S. Klimov

Izvestiya. Mathematics , Tome 62 (1998) no. 5, pp. 969-984.

Résumé

We establish infinite-dimensional variants of the Poincare–Hopf theorem for many-valued vector fields generated by operators of monotonic type. We suggest conditions for the stabilization of the homology groups of closed subsets of a Banach space when approximated by finite-dimensional sections. Emphasis is given to the study of topological characteristics of Lebesgue sets of Lipschitz functionals defined on a closed convex subset of a reflexive space.

Bibliographie

[1] Pokhozhaev S. I., “O razreshimosti nelineinykh uravnenii s nechetnymi operatorami”, Funktsion. analiz i ego prilozh., 1:3 (1967), 66–73 | MR | Zbl

[2] Skrypnik I. V., Metody issledovaniya nelineinykh ellipticheskikh granichnykh zadach, Nauka, M., 1990 | MR

[3] Browder F. E., “Nonlinear elliptic boundary value problems and the generalized topological degree”, Bull. Amer. Math. Soc., 76:5 (1970), 999–1005 | DOI | MR | Zbl

[4] Borisovich Yu. G., “Ob otnositelnom vraschenii kompaktnykh vektornykh polei v lineinykh prostranstvakh”, Tr. semin. po funkts. analizu, no. 12, Izd-vo Voronezhskogo un-ta, Voronezh, 1969, 3–27

[5] Borisovich Yu. G., Gelman B. D., Myshkis A. D., Obukhovskii V. V., “Topologicheskie metody v teorii nepodvizhnykh tochek mnogoznachnykh otobrazhenii”, UMN, 35:1 (1980), 59–126 | MR | Zbl

[6] Krasnoselskii M. A., Zabreiko P. P., Geometricheskie metody nelineinogo analiza, Nauka, M., 1975 | MR

[7] Klark F., Optimizatsiya i negladkii analiz, Nauka, M., 1988 | MR | Zbl

[8] Milnor Dzh., Uolles A., Differentsialnaya topologiya. Nachalnyi kurs, Mir, M., 1972 | MR | Zbl

[9] Ioffe A. D., Tikhomirov V. M., “Dvoistvennost vypuklykh funktsii i ekstremalnye zadachi”, UMN, 23:6 (1968), 51–116 | MR | Zbl

[10] Bobylev N. A., “O topologicheskom indekse ekstremalei mnogomernykh variatsionnykh zadach”, Funktsion. analiz i ego prilozh., 20:2 (1986), 8–13 | MR | Zbl

[11] Bobylev N. A., Klimov V. S., Metody nelineinogo analiza v zadachakh negladkoi optimizatsii, Nauka, M., 1992 | MR | Zbl

[12] Zaslavskii A. Ya., “O kriticheskikh tochkakh lipshitsevykh funktsii na gladkikh mnogoobraziyakh”, SMZh, 22:1 (1981), 87–93 | MR

[13] Klimov V. S., “Minimaksnye kriticheskie znacheniya negladkikh funktsionalov”, SMZh, 33:3 (1992), 91–100 | MR

[14] Klimov V. S., Senchakova N. V., “Ob otnositelnom vraschenii mnogoznachnykh potentsialnykh vektornykh polei”, Matem. sb., 182:10 (1991), 1393–1407 | MR

[15] Senchakova N. V., “Vraschenie mnogoznachnogo vektornogo polya i resheniya variatsionnykh neravenstv”, Kachestvennye metody issledovaniya operatornykh uravnenii, Izd-vo Yarosl. un-ta, Yaroslavl, 1988, 70–78

[16] Skrypnik I. V., Nelineinye ellipticheskie uravneniya vysshego poryadka, Nauk. dumka, Kiev, 1973 | MR

[17] Massi U., Teoriya gomologii i kogomologii, Mir, M., 1981 | MR

[18] Stinrod N., Eilenberg S., Osnovaniya algebraicheskoi topologii, IL, M., 1958

[19] Lions Zh. L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR

[20] Dmitruk A. V., Milyutin A. A., Osmolovskii N. P., “Teorema Lyusternika i teoriya ekstremuma”, UMN, 35:6 (1980), 11–46 | MR | Zbl

[21] Dold A., “Fixed point and fixed point theorem for Euclidean neighbourhood retracts”, Topology, 4 (1965), 1–8 | DOI | MR | Zbl

[22] Dold A., Lektsii po algebraicheskoi topologii, Mir, M., 1976 | MR

[23] Ladyzhenskaya O. A., “O nakhozhdenii minimalnykh globalnykh attraktorov dlya uravnenii Nave–Stoksa i drugikh uravnenii s chastnymi proizvodnymi”, UMN, 42:6 (1987), 26–60 | MR

[24] Klimov V. S., “O chisle reshenii kraevykh zadach nelineinoi teorii obolochek”, DAN SSSR, 241:1 (1979), 57–60 | MR

[25] Klimov V. S., “Pochti periodicheskie resheniya evolyutsionnykh zadach mekhaniki vyazkoplasticheskikh sred”, Izv. VUZov. Ser. Matematika, 1994, no. 1, 17–24 | MR | Zbl

[26] Mosolov P. P., Myasnikov V. P., Mekhanika zhestkoplasticheskikh sred, Nauka, M., 1981 | MR | Zbl