Geodesic
On the general theory of boundary value problems
Sbornik. Mathematics, Tome 29 (1976) no. 2, pp. 147-155 
A. A. Dezin. On the general theory of boundary value problems. Sbornik. Mathematics, Tome 29 (1976) no. 2, pp. 147-155. http://geodesic.mathdoc.fr/item/SM_1976_29_2_a0/
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In a bounded domain $V$ in $n$-dimensional Euclidean space each formal, linear, partial differential operator $L(D)$ with constant coefficients may be connected with so-called minimal $L_0$ and maximal $\widetilde L$ operators in the Hilbert space $\mathscr L^2(V)$. The operator $L$ is said to be proper if $L_0\subset L\subset\widetilde L$ and the equation $Lu=f$ has a unique solution for any $f\in\mathscr L^2(V)$. Using the complete description of proper operators that we obtain for $n=1$, in this article we discuss problems connected with the description of proper operators in the general case when $n>1$. Bibliography: 8 titles.

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

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

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

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

[5] A. A. Dezin, “Opisanie pravilnykh operatorov”, DAN SSSR, 219:1 (1974), 27–30 | MR | Zbl

[6] Yu. M. Berezanskii, Razlozhenie po sobstvennym funktsiyam samosopryazhennykh operatorov, izd-vo «Naukova dumka», Kiev, 1965 | MR

[7] L. Gording, Zadacha Koshi dlya giperbolicheskikh uravnenii, IL, Moskva, 1961

[8] M. A. Naimark, Lineinye differentsialnye operatory, izd-vo «Nauka», Moskva, 1969 | MR