Geodesic
On the method of orthogonal extension of overdetermined systems
Sbornik. Mathematics, Tome 22 (1974) no. 3, pp. 456-464 
I. S. Gudovich. On the method of orthogonal extension of overdetermined systems. Sbornik. Mathematics, Tome 22 (1974) no. 3, pp. 456-464. http://geodesic.mathdoc.fr/item/SM_1974_22_3_a6/
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In the article a description is given of Noether boundary value problems for overdetermined systems of partial differential equations with constant coefficients of the form \begin{equation} \mathscr L(D)u=f,\qquad\mathscr W^*(D)u=g, \end{equation} where $\mathscr L(\xi)$ ($\xi=(\xi_1,\dots,\xi_m)$) is an $N\times n$ matrix inducing a homomorphism $\mathscr L\colon\mathscr P^n\to\nobreak\mathscr P^N$ whose kernel and cokernel are assumed to be free modules ($\mathscr P^n$ is the module composed of all $n$-dimensional vectors with coordinates polynomially depending on $\xi$). The matrix $\mathscr W(\xi)$ is composed of column vectors forming a basis in the kernel of $\mathscr L$. A necessary condition for the solvability of (1) is \begin{equation} \mathscr V(D)f=0, \end{equation} where $\mathscr V(\xi)$ is a matrix of row vectors forming a basis in the cokernel of $\mathscr L$. The system \begin{equation} \mathscr L(D)u+v^*(D)p=f,\qquad\mathscr W^*(D)u=g, \end{equation} which is called an orthogonal extension of the original system, is introduced into consideration. Bibliography: 13 titles.

[1] I. S. Gudovich, S. G. Krein, “O nekotorykh kraevykh zadachakh, ellipticheskikh v podprostranstve”, Matem. sb., 84 (126) (1971), 595–606 | MR | Zbl

[2] V. A. Solonnikov, “Ob uslovii dopolnitelnosti dlya pereopredelennykh sistem”, Zapiski nauchnogo seminara LOMI, 14, 1969, 237–255 | MR | Zbl

[3] V. A. Solonnikov, “O pereopredelennykh ellipticheskikh kraevykh zadachakh”, DAN SSSR, 199:2 (1971), 279–281 | MR | Zbl

[4] E. S. Zukhovitskaya, “Kraevye zadachi dlya pereopredelennykh sistem differentsialnykh uravnenii”, DAN SSSR, 201:3 (1971), 523–526

[5] E. S. Zukhovitskaya, Kraevye zadachi dlya pereopredelennykh sistem differentsialnykh uravnenii, Kandidatskaya dissertatsiya, MGU, 1972

[6] M. I. Eskin, Kraevye zadachi dlya nekotorogo klassa pereopredelennykh ellipticheskikh sistem, Kandidatskaya dissertatsiya, RGU, 1972

[7] A. S. Dynin, “Ellipticheskie kraevye zadachi dlya psevdodifferentsialnykh kompleksov”, Funkts. analiz, 6:1 (1972), 75–76 | MR | Zbl

[8] I. S. Gudovich, “O kriterii bazisnosti sistemy vektorov odnogo modulya”, Trudy nauchno-issledovatelskogo instituta matematiki VGU, no. XI, 1973 | Zbl

[9] I. S. Gudovich, Metod ortogonalnogo rasshireniya v granichnykh zadachakh dlya pereopredelennykh sistem differentsialnykh uravnenii v chastnykh proizvodnykh, Kandidatskaya dissertatsiya, VGU, 1972

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

[11] L. R. Volevich, “Ob odnoi zadache lineinogo programmirovaniya, voznikayuschei v differentsialnykh uravneniyakh”, Uspekhi matem. nauk, XVIII:3(111) (1963), 155–162

[12] I. M. Kulikov, I. S. Tetievskaya (Gudovich), “Kraevye zadachi dlya operatora rot, ellipticheskie v podprostranstve”, Sb. rabot aspirantov po matematike i mekhanike, Voronezh, 1969, 38–42

[13] I. S. Gudovich, S. G. Krein, I. M. Kulikov, “Kraevye zadachi dlya uravnenii Maksvella”, DAN SSSR, 207:2 (1972), 321–324 | MR | Zbl