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
$\mathscr L(\xi)$ (
$\xi=(\xi_1,\dots,\xi_m)$) 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.
UDC:
517.946
MSC: 35N05 Received: 10.05.1973
Fulltext:
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