Theorems on the complete set of isomorphisms in the $L_2$-theory of generalized solutions of boundary value problems for a Petrovskii parabolic equation

N. V. Zhitarashu

Sbornik. Mathematics, Tome 56 (1987) no. 2, pp. 447-471

Résumé

The general boundary value problem is studied for a parabolic equation in spaces of insufficiently smooth and generalized functions. Starting from Green's formula, the generalized solution of a boundary value problem is defined, and two families (scales) of spaces are constructed in which the boundary value problem is studied: the spaces of solutions $\widetilde{\mathscr H}^s(\Omega)$, and the sapces of right-hand sides $\mathscr K^s(\Omega)$. It is proved that the closure with respect to continuity of the boundary value problem operator establishes an isomorphism of the spaces $\widetilde{\mathscr H}^s(\Omega)$ and $\mathscr K^s(\Omega)$ for $-\infty<s<\infty$. Bibliography: 35 titles.

Bibliographie

[1] Agranovich M. S, Vishik M. I., “Ellipticheskie zadachi s parametrom i parabolicheskie zadachi obschego vida”, UMN, 19:3 (1964), 53–161 | Zbl

[2] Solonnikov V. A., O kraevykh zadachakh dlya lineinykh parabolicheskikh sistem differentsialnykh uravnenii obschego vida, Tr. MIAN, 83, 1965 | MR

[3] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967

[4] Eidelman S. D., Parabolicheskie sistemy, Nauka, M., 1964 | MR | Zbl

[5] Lions J. L., Magenes E., Problèmes aux limites non homogenes et applications, v. I, II, III, Dunod, Paris, 1968

[6] Mikhailov V. P., “Reshenie smeshannoi zadachi dlya parabolicheskoi sistemy metodom potentsialov”, DAN SSSR, 132:2 (1960), 291–294

[7] Mikhailov V. P., “O zadache Dirikhle dlya parabolicheskogo uravneniya”, Matem. sb., 61(103) (1963), 40–63 ; 62(104), 140–169

[8] Ivasishen S. D., “Pro korektnu rozv'yaznIst zagalnykh parabolichnykh kraiovykh zadach u negativnikh prostorakh Gëldera”, Dop. AN URSR, ser. A, 1977, no. 5, 396–399

[9] Ditkin V. A., Prudnikov A. P., Operatsionnoe ischislenie, Vysshaya shkola, M., 1966 | Zbl

[10] Zhitarashu N. V., “O korrektnoi razreshimosti parabolicheskikh granichnykh zadach v prostranstvakh Soboleva–Slobodetskogo $H_{2xt}^{s\,s/2\beta}(\Omega)$”, Izv. AN MoldSSR. Ser. fiz.-tekh. i matem. nauk, 1981, no. 2, 13–19 | MR | Zbl

[11] Zhitarashu N. V., “Teoremy ob izomorfizmakh v $L_2$-teorii slabykh reshenii parabolicheskikh zadach”, DAN SSSR, 260:5 (1981), 1054–1058 | MR | Zbl

[12] Roitberg Ya. A., “Ellipticheskie zadachi s neodnorodnymi granichnymi usloviyami i lokalnoe povyshenie gladkosti vplot do granitsy obobschennykh reshenii”, DAN SSSR, 157:4 (1964), 798–801 | MR | Zbl

[13] Roitberg Ya. A., “O znacheniyakh na granitse oblasti obobschennykh reshenii ellipticheskikh uravnenii”, Matem. sb., 86(128) (1971), 248–267 | MR | Zbl

[14] Roitberg Ya. A., “Teorema o polnom nabore izomorfizmov dlya ellipticheskikh po Duglisu–Nirenbergu sistem”, Ukr. matem. zhurn., 27:4 (1975), 544–548 | MR | Zbl

[15] Roitberg Ya. A., Serdyuk V. A., Ellipticheskie zadachi s parametrom v $L_2$-prostranstvakh obobschennykh funktsii dlya obschikh sistem uravnenii, Preprint 82.30, In-t matem. AN USSR, 1982 | Zbl

[16] Roitberg Ya. A., Sheftel Z. G., “Optimalnoe upravlenie sistemami, opisyvaemymi obschimi ellipticheskimi zadachami”, Differents. uravneniya, 15:6 (1979), 1075–1087 | MR | Zbl

[17] Roitberg Ya. A., “O granichnykh znacheniyakh obobschennykh reshenii ellipticheskikh po Duglisu–Nirenbergu sistem”, Sib. matem. zhurn., 18:4 (1977), 848–860 | MR

[18] Gelfand I. M., Shilov G. E., Obobschennye funktsii, t. I–III, Nauka, M., 1968

[19] Vladimirov V. S., Obobschennye funktsii v matematicheskoi fizike, Nauka, M., 1979 | MR

[20] Sobolev S. L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Izd-vo LGU, 1950; Некоторые применения функционального анализа в математической физике, Новосибирск, 1962

[21] Slobodetskii L. N., “Obobschennye prostranstva S. L. Soboleva i ikh prilozheniya k kraevym zadacham dlya differentsialnykh uravnenii v chastnykh proizvodnykh”, Uch. zap. Leningr. gos. ped. in-ta, 187 (1958), 54–112

[22] Volevich L. R., Paneyakh B. P., “Nekotorye prostranstva obobschennykh funktsii i teoremy vlozheniya”, UMN, 20:1 (1965), 3–74 | MR | Zbl

[23] Petre Zh., “O novom podkhode k granichnym zadacham dlya ellipticheskikh uravnenii”, Sb. per. inostr. statei, 7:1 (1963), 43–65

[24] Khërmander L., Lineinye differentsialnye operatory s chastnymi proizvodnymi, Mir, M., 1965 | MR

[25] Burbaki N., Topologicheskie vektornye prostranstva, IL, M., 1959

[26] Volevich L. R., Gindikin S. G., “Metod energeticheskikh otsenok v smeshannoi zadache”, UMN, 35:5 (1980), 53–120 | MR | Zbl

[27] Babenko K. I., “O sopryazhennykh funktsiyakh”, DAN SSSR, 62:2 (1948), 157–160 | MR | Zbl

[28] Stein E. M., “Note on singular integrals”, Proc. Amer. Math. Soc., 8 (1957), 250–254 | DOI | MR | Zbl

[29] Eskin G. I., Kraevye zadachi dlya ellipticheskikh psevdodifferentsialnykh uravnenii, Nauka, M., 1973 | MR

[30] Vishik M. I., Eskin G. I., “Parabolicheskie uravneniya v svertkakh v ogranichennoi oblasti”, Matem. sb., 71(113) (1966), 162–190 | Zbl

[31] Viner N., Peli R., Preobrazovanie Fure v kompleksnoi oblasti, Nauka, M., 1964 | MR

[32] Berg I., Lëfstrem Ya., Interpolyatsionnye prostranstva. Vvedenie, Mir, M., 1980 | MR

[33] Tribel Kh., Teoriya interpolyatsii, funktsionalnye prostranstva, differentsialnye operatory, Mir, M., 1980 | MR

[34] Agranovich M. S, Sukhorutchenko V. V., “O neobkhodimosti algebraicheskikh uslovii parabolichnosti nestatsionarnykh zadach”, Matem. zametki, 2:6 (1967), 615–625 | Zbl

[35] Browder F. E., “Estimates and existence theorems for elliptic boundary value problems”, Proc. Nat. Acad. Sci. USA, 45:3 (1959), 365–372 | DOI | MR | Zbl