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Mat. Sb. (N.S.), 1980, Volume 113(155), Number 2(10), Pages 302–323 (Mi msb2793)  

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

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.

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English version:
Mathematics of the USSR-Sbornik, 1982, 41:2, 249–267

Bibliographic databases:

UDC: 517.946
MSC: Primary 35J25, 35B40; Secondary 35P20
Received: 25.12.1979

Citation: S. M. Kozlov, “Asymptotics of fundamental solutions of second-order divergence differential equations”, Mat. Sb. (N.S.), 113(155):2(10) (1980), 302–323; Math. USSR-Sb., 41:2 (1982), 249–267

Citation in format AMSBIB
\by S.~M.~Kozlov
\paper Asymptotics of fundamental solutions of second-order divergence differential equations
\jour Mat. Sb. (N.S.)
\yr 1980
\vol 113(155)
\issue 2(10)
\pages 302--323
\jour Math. USSR-Sb.
\yr 1982
\vol 41
\issue 2
\pages 249--267

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    2. Druskin V., “The Uniqueness in the Resistivity Soundings Inversion for a Piece-Wise Constant Conductivity Structure”, no. 1, 1982, 72–75  isi
    3. Kozlov S., “Ground-States of Quasi-Periodic Operators”, 271, no. 3, 1983, 532–536  mathscinet  zmath  isi
    4. Kawazu K., Kesten H., “On Birth and Death Processes in Symmetric Random Environment”, J. Stat. Phys., 37:5-6 (1984), 561–576  crossref  mathscinet  zmath  adsnasa  isi
    5. Druskin V., “The Uniqueness of the 3-Dimensional Inversion of Ground Data for a Stationary Or Monochromatic-Field Source”, no. 3, 1985, 63–69  mathscinet  isi
    6. Greven A., “Symmetric Exclusion on Random Sets and a Related Problem for Random-Walks in Random Environment”, Probab. Theory Relat. Field, 85:3 (1990), 307–364  crossref  mathscinet  zmath  isi
    7. M. Avellaneda, Fang Hua Lin, “Lp bounds on singular integrals in homogenization”, Comm Pure Appl Math, 44:8-9 (1991), 897  crossref  mathscinet  zmath
    8. Khanin K., “Random Walks in a Random Potential: Loop Condensation Effects”, Int. J. Mod. Phys. B, 10:18-19 (1996), 2393–2404  crossref  mathscinet  zmath  adsnasa  isi
    9. Ramm A., “Fundamental Solutions to Some Elliptic Equations with Discontinuous Senior Coefficients and an Inequality for These Solutions.”, Math. Inequal. Appl., 1:1 (1998), 99–104  mathscinet  zmath  isi
    10. Dungey N., ter Elst A., Robinson D., “On Second-Order Almost-Periodic Elliptic Operators”, J. Lond. Math. Soc.-Second Ser., 63:Part 3 (2001), 735–753  crossref  mathscinet  zmath  isi
    11. Alexopoulos G., “Random Walks on Discrete Groups of Polynomial Volume Growth”, Ann. Probab., 30:2 (2002), 723–801  crossref  mathscinet  zmath  isi
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    14. X. Blanc, F. Legoll, A. Anantharaman, “Asymptotic Behavior of Green Functions of Divergence form Operators with Periodic Coefficients”, Applied Mathematics Research eXpress, 2012  crossref  mathscinet
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  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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