On the method of orthogonal extension of overdetermined systems

I. S. Gudovich

Sbornik. Mathematics, Tome 22 (1974) no. 3, pp. 456-464

Résumé

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.