ASYMPTOTICS OF THE SOLUTION OF THE DIRICHLET PROBLEM FOR AN EQUATION WITH RAPIDLY OSCILLATING COEFFICIENTS IN A RECTANGLE

Abstract

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 in the series are (generally speaking, nonintegral) nonnegative numbers of the form , where , , and 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 of the rectangle. The coefficients of the series for rational may depend in polynomial fashion on . 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 -lattice. The spectral problem is considered; asymptotic formulas for the eigenvalue and the eigenfunction are obtained under the assumption that is a simple eigenvalue of the averaged Dirichlet problem.