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

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

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
\Bibitem{Koz80} \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 \mathnet{http://mi.mathnet.ru/msb2793} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=594840} \zmath{https://zbmath.org/?q=an:0483.35009|0469.35019} \transl \jour Math. USSR-Sb. \yr 1982 \vol 41 \issue 2 \pages 249--267 \crossref{https://doi.org/10.1070/SM1982v041n02ABEH002232} 

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3. Kozlov S., “Ground-States of Quasi-Periodic Operators”, 271, no. 3, 1983, 532–536
4. Kawazu K., Kesten H., “On Birth and Death Processes in Symmetric Random Environment”, J. Stat. Phys., 37:5-6 (1984), 561–576
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
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
7. M. Avellaneda, Fang Hua Lin, “Lp bounds on singular integrals in homogenization”, Comm Pure Appl Math, 44:8-9 (1991), 897
8. Khanin K., “Random Walks in a Random Potential: Loop Condensation Effects”, Int. J. Mod. Phys. B, 10:18-19 (1996), 2393–2404
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
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
11. Alexopoulos G., “Random Walks on Discrete Groups of Polynomial Volume Growth”, Ann. Probab., 30:2 (2002), 723–801
12. G. Allaire, I. Pankratova, A. Piatnitski, “Homogenization and concentration for a diffusion equation with large convection in a bounded domain”, Journal of Functional Analysis, 2011
13. Jun-zhi Cui, Wen-ming He, “The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$”, CPAA, 11:2 (2011), 501
14. X. Blanc, F. Legoll, A. Anantharaman, “Asymptotic Behavior of Green Functions of Divergence form Operators with Periodic Coefficients”, Applied Mathematics Research eXpress, 2012
15. C.E.. Kenig, Fanghua Lin, Zhongwei Shen, “Periodic Homogenization of Green and Neumann Functions”, Commun. Pur. Appl. Math, 2013, n/a
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