Geodesic
Degenerate operator equations
Sbornik. Mathematics, Tome 43 (1982) no. 3, pp. 287-298 
A. A. Dezin. Degenerate operator equations. Sbornik. Mathematics, Tome 43 (1982) no. 3, pp. 287-298. http://geodesic.mathdoc.fr/item/SM_1982_43_3_a0/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The differential-operator equation $$ [-D_tt^\alpha D_t-D_tA-P]u=f $$ is studied, where $D_t\equiv\frac d{dt}$, $t\in[0,b]$, $\alpha\geqslant0$, and the operators $A,P\colon\mathscr H\to\mathscr H$, which commute with $D_t$, act in a Hilbert space $\mathscr H$ and satisfy appropriate (quite strong) requirements formulated in terms of resolvent, or spectral, properties. The character of the boundary conditions with respect to $t$ (at $t=0,b$), which are imposed on the equation and ensure existence and uniqueness of the solution, is elucidated, and properties of the solution depending on $\alpha$ and on the properties of the operators $A$ and $P$ are investigated, as well. Bibliography: 7 titles.

[1] Keldysh M. V., “O nekotorykh sluchayakh vyrozhdeniya uravnenii ellipticheskogo tipa”, DAN SSSR, 77:2 (1951), 181–183

[2] Vishik M. I., “Kraevye zadachi dlya ellipticheskikh uravnenii, vyrozhdayuschikhsya na granitse oblasti”, Matem. sb., 35(77) (1954), 513–568 | Zbl

[3] Smirnov M. M., Vyrozhdayuschiesya ellipticheskie i giperbolicheskie uravneniya, Nauka, M., 1966 | MR

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

[5] Sobolev S. L., Nekotorye prilozheniya funktsionalnogo analiza v matematicheskoi fizike, izd-vo LGU, L., 1950

[6] Nikolskii S. M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1960 | MR

[7] Danford N., Shvarts Dzh. T., Lineinye operatory, T. 1, IL, M., 1962