Spectral problems for pseudodifferential systems elliptic in the Douglis--Nirenberg sense, and their applications

A. N. Kozhevnikov

Sbornik. Mathematics, Tome 21 (1973) no. 1, pp. 63-90

Résumé

Pseudodifferential systems elliptic in the Douglis–Nirenberg sense on a compact manifold without boundary are studied. A theorem on the completeness of the generalized eigenvectors is proved. It is not assumed here that all orders of the operators of the system situated on the main diagonal are equal. The formula $N(\lambda)\overset{\text{def}}=\sum_{\operatorname{Re}\lambda_j\leqslant\lambda}1\sim C\lambda^{n/s}$ is obtained, where the $\lambda_j$ are the eigenvalues of the system taking account of the root multiplicity, $n$ is the dimension of the manifold, $\mu$ is the minimum order of the operators of the system situated on the main diagonal and $C$ is a constant expressed in terms of the symbol. This formula permits us to determine the asymptotic behavior of the eigenvalues for general elliptic boundary value problems containing $\lambda$ in the boundary conditions. Bibliography: 23 titles.