Asymptotics of the solution of the Dirichlet problem for an equation with rapidly oscillating coefficients in a rectangle

S. A. Nazarov

Sbornik. Mathematics, Tome 73 (1992) no. 1, pp. 79-110

Résumé

A complete asymptotic expansion is found for the solution of the Dirichlet problem for a second-order scalar equation in a rectangle. The exponents of the powers of $\varepsilon$ in the series are (generally speaking, nonintegral) nonnegative numbers of the form $p+q_1\alpha_1\pi^{-1}+\dots+q_4\alpha_4\pi^{-1}$, where $p$, $q_j=0,1,\dots$, and $\alpha_j$ is the opening of the angle which is transformed into a quarter plane under the change of coordinates taking the Laplace operator into the principal part of the averaged operator at the vertex $O_j$ of the rectangle. The coefficients of the series for rational $\alpha_j\pi^-1$ may depend in polynomial fashion on $\log\varepsilon$. It is shown that the algorithm also does not change in the case of a system of differential equations or in the case of a domain bounded by polygonal lines with vertices at the nodes of an $\varepsilon$-lattice. The spectral problem is considered; asymptotic formulas for the eigenvalue $\lambda(\varepsilon)$ and the eigenfunction are obtained under the assumption that $\lambda(0)$ is a simple eigenvalue of the averaged Dirichlet problem.

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