Geodesic
On some properties of the set of boundary value problems
Sbornik. Mathematics, Tome 49 (1984) no. 2, pp. 367-377 
A. Kh. Mamyan. On some properties of the set of boundary value problems. Sbornik. Mathematics, Tome 49 (1984) no. 2, pp. 367-377. http://geodesic.mathdoc.fr/item/SM_1984_49_2_a5/
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     author = {A. Kh. Mamyan},
     title = {On some properties of the set of boundary value problems},
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     volume = {49},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1984_49_2_a5/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

For linear differential operations the author studies the connection between the general notion of a correctly posed boundary value problem, given by Hörmander, and its description in terms of the boundary conditions. It is shown that, knowing one correctly posed problem and the kernels of operators that are maximal for the original and adjoint operation, it is possible to describe all correctly posed problems. Examples of explicit realization of this construction are presented. For operators with constant coefficients in a compact domain the author establishes the existence of correctly posed problems with poor regularity properties for the solutions, as well as problems in whose graph there is no dense set of infinitely differentiable functions. Bibliography: 8 titles.

[1] Khërmander L., K teorii obschikh differentsialnykh operatorov v chastnykh proizvodnykh, IL, M., 1959

[2] Dezin A. A., “Prosteishie razreshimye rasshireniya dlya ultragiperbolicheskogo i psevdoparabolicheskogo operatorov”, DAN SSSR, 148:5 (1963), 1013–1016 | MR | Zbl

[3] Dezin A. A., “Operatory s pervoi proizvodnoi po vremeni i nelokalnye granichnye usloviya”, Izv. AN SSSR. Seriya matem., 31 (1967), 61–86 | MR | Zbl

[4] Mamyan A. X., “Postroenie razreshimykh rasshirenii v parallelepipede dlya lineinykh differentsialnykh operatorov s postoyannymi koeffitsientami”, Diff. uravneniya, 6:2 (1970), 358–370 | Zbl

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

[6] Trev F., Lektsii po teorii differentsialnykh uravnenii v chastnykh proizvodnykh, Mir, M., 1965 | Zbl

[7] Dezin A. A., Obschie voprosy teorii granichnykh zadach, Nauka, M., 1980 | MR | Zbl

[8] Akhiezer N. I., Glazman I. N., Teoriya lineinykh operatorov v gilbertovom prostranstve, Nauka, M., 1966 | MR | Zbl