An asymptotic expansion of the solution of a~second order elliptic equation with periodic rapidly oscillating coefficients

E. V. Sevost'yanova

Sbornik. Mathematics, Tome 43 (1982) no. 2, pp. 181-198

Résumé

This paper studies the asymptotic behavior of the fundamental solution $K_\varepsilon(x,y)$ of the equation $$ -\frac\partial{\partial x_i}\biggl(a_{ij}\biggl(\frac x\varepsilon\biggr)\frac\partial{\partial x_j}u_\varepsilon\biggr)=f(x), $$ specified on the whole space $\mathbf R^n$, $n>2$, as $\varepsilon\to0$. The coefficients $a_{ij}(y)$ are periodic functions which satisfy the conditions of ellipticity, symmetry, and infinite smoothness. The main result is the construction of the asymptotics of $K_\varepsilon(x,y)$ in the form $$ K_\varepsilon(x,y)=\sum^M_{s=0}\varepsilon^s\Phi_s\biggl(x-y,\frac x\varepsilon,\frac y\varepsilon\biggr)+\varepsilon^{M+1}R_M(x,y,\varepsilon), $$ where $M$ is an arbitrary positive integer, the $\Phi_s(x,y,z)$ are homogeneous of degree $-s-n+2$ in the first argument and periodic in the remaining arguments, and for the remainder term $R_M(x,y,\varepsilon)$ on the set $|x-y|>\delta$, $\delta>0$, the estimate $$ |R_M(x,y,\varepsilon)|\frac{C_M(\delta)}{|x-y|^{M+n-1}} $$ holds, where the constants $C_M(\delta)$ are independent of $x$, $y$, and $\varepsilon$. Figures: 1. Bibliography: 9 titles.