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Mathematics of the USSR-Sbornik, 1982, Volume 41, Issue 2, Pages 249–267
DOI: https://doi.org/10.1070/SM1982v041n02ABEH002232
(Mi sm2793)
 

This article is cited in 15 scientific papers (total in 15 papers)

Asymptotics of fundamental solutions of second-order divergence differential equations

S. M. Kozlov
References:
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.
Received: 25.12.1979
Bibliographic databases:
UDC: 517.946
MSC: Primary 35J25, 35B40; Secondary 35P20
Language: English
Original paper language: Russian
Citation: S. M. Kozlov, “Asymptotics of fundamental solutions of second-order divergence differential equations”, Math. USSR-Sb., 41:2 (1982), 249–267
Citation in format AMSBIB
\Bibitem{Koz80}
\by S.~M.~Kozlov
\paper Asymptotics of fundamental solutions of second-order divergence differential equations
\jour Math. USSR-Sb.
\yr 1982
\vol 41
\issue 2
\pages 249--267
\mathnet{http://mi.mathnet.ru//eng/sm2793}
\crossref{https://doi.org/10.1070/SM1982v041n02ABEH002232}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=594840}
\zmath{https://zbmath.org/?q=an:0483.35009|0469.35019}
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  • This publication is cited in the following 15 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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