Asymptotics of fundamental solutions of second-order divergence differential equations

S. M. Kozlov

Sbornik. Mathematics, Tome 41 (1982) no. 2, pp. 249-267

Résumé

Let $K(x,y)$ be the fundamental solution of a divergence operator of the following form: $$ A=-\sum^n_{i,j=1}\frac\partial{\partial x_i}a_{ij}(x)\frac\partial{\partial x_j}. $$ Two types of asymptotics of $K(x,y)$ are considered in the paper: the asymptotic behavior at infinity, i.e. as $|x-y|\to\infty$, and the asymptotic behavior of $K(x,y)$ at $x=y$. In the first case, for operators with smooth, quasiperiodic coefficients the principal term of the asymptotic expression is found, and a power estimate of the remainder term is established. In the second case the principal term in the asymptotic expression for $K(x,y)$ as $x\to y$ is found for an operator $A$ with arbitrary bounded and measurable coefficients $\{a_{ij}(x)\}$. These results are obtained by means of the concept of the $G$-convergence of elliptic differential operators. Further, applications of the results are given to the asymptotics of the spectrum of the operator $A$ in a bounded domain $\Omega$. Bibliography: 13 titles.