Unique solvability of certain matrix partial differential equations

S. A. Lomov

Matematičeskie zametki, Tome 21 (1977) no. 4, pp. 525-530.

Résumé

A class of matrix-valued functions is picked out, invariant relative to the operator $\mathscr L\sum_{i=1}^n\lambda_i(x)\frac\partial{\partial t_i}-A(x)$, where $t=(t_1,\dots,t_n)$ are complex variables, $x$ is a real parameter, $A(x)$ is a matrix, $\{\lambda_i(x)\}_1^n=\sigma(A(x))$. It is shown that the operator $\mathscr L$ is normally solvable in the class picked out and a uniqueness theorem is proved for the solution of a nonstandard problem: the desired matrix-valued function $Z(x,t)$ is known only at a point and $\partial Z/\partial x\perp\operatorname{Ker}\mathscr L^*$. Such problems arise naturally when developing the general theory of singular perturbations.