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Asymptotics of fundamental solutions of second-order divergence differential equations
S. M. Kozlov
Abstract:
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.
UDC:
517.946
MSC: Primary
35J25,
35B40; Secondary
35P20 Received: 25.12.1979
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