Justifying the two-side approximate method for eigenvalues of a second-order elliptic operator (Q2714026)

The spectral problem for the second-order elliptic equation NEWLINE\[NEWLINE-\sum_{i,j = 1}^2\frac{\partial}{\partial x_i}\left( a_{ij}\frac{\partial u}{\partial x_j}\right) = \lambda u,\qquad u|_{\gamma} = 0NEWLINE\]NEWLINE is studied in a domain \(D\) where \(\gamma = \partial D\). To solve the problem, the author uses the two-side approximate method based on the fictitious-domain method applied to an auxiliary problem for the so-called conjugate-factorized operator, and simultaneously realizes a justification of this approach for a difference analog of the conjugate-factorized operator. As a result, the author obtains a power-series expansion by a small parameter for eigenvalues to the auxiliary problem and establishes a priori best possible estimates for the two-side approximations.