Averaging principles for eguations with rapidly oscillatory coefficients

G. V. Sandrakov

Sbornik. Mathematics, Tome 68 (1991) no. 2, pp. 503-553

Résumé

Elliptic equations of arbitrary order with smooth, rapidly oscillating coefficients are considered. An algorithm is set forth for constructing a formal asymptotic expansion of solutions of such equations. The algorithm consists in the successive solution of a number of periodic problems. The solvability conditions for these problems lead to an averaged equation (system) with constant coefficients. It is proved that if the solution of the equation is bounded and converges to some limit in a suitable sense, then the limit function (vector) satisfies the averaged equation (system). An asymptotic expansion of solutions of an equation of divergence form of arbitrary order is constructed. This makes it possible to obtain for such equations estimates of the form $$ \|u_\varepsilon-u_s^0\|_s\leqslant C\sqrt\varepsilon, $$ where $2s$ is the order of the equation, $u_\varepsilon$ is a solution of the equation, and $u_s^0$ comprises $s$ terms of the asymptotic expansion. Bibliography: 22 titles.