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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
\begin{equation} \mathscr L(D)u=f,\qquad\mathscr W^*(D)u=g, \end{equation}
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
\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.

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

Bibliographic databases:

UDC: 517.946
MSC: 35N05
Received: 10.05.1973

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}


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