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 Mat. Sb. (N.S.), 1974, Volume 93(135), Number 3, Pages 451–459 (Mi msb3426)

On the method of orthogonal extension of overdetermined systems

I. S. Gudovich

Abstract: 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
$$\mathscr L(D)u=f,\qquad\mathscr W^*(D)u=g,$$
where $\mathscr L(\xi)$ ($\xi=(\xi_1,…,\xi_m)$) 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
$$\mathscr V(D)f=0,$$
where $\mathscr V(\xi)$ is a matrix of row vectors forming a basis in the cokernel of $\mathscr L$.
The system
$$\mathscr L(D)u+v^*(D)p=f,\qquad\mathscr W^*(D)u=g,$$
which is called an orthogonal extension of the original system, is introduced into consideration.
Bibliography: 13 titles.

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English version:
Mathematics of the USSR-Sbornik, 1974, 22:3, 456–464

Bibliographic databases:

UDC: 517.946
MSC: 35N05

Citation: I. S. Gudovich, “On the method of orthogonal extension of overdetermined systems”, Mat. Sb. (N.S.), 93(135):3 (1974), 451–459; Math. USSR-Sb., 22:3 (1974), 456–464

Citation in format AMSBIB
\Bibitem{Gud74} \by I.~S.~Gudovich \paper On the method of orthogonal extension of overdetermined systems \jour Mat. Sb. (N.S.) \yr 1974 \vol 93(135) \issue 3 \pages 451--459 \mathnet{http://mi.mathnet.ru/msb3426} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=388459} \zmath{https://zbmath.org/?q=an:0292.35010} \transl \jour Math. USSR-Sb. \yr 1974 \vol 22 \issue 3 \pages 456--464 \crossref{https://doi.org/10.1070/SM1974v022n03ABEH002169}