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Mat. Sb. (N.S.), 1981, Volume 115(157), Number 2(6), Pages 204–222 (Mi msb2382)  

This article is cited in 27 scientific papers (total in 27 papers)

An asymptotic expansion of the solution of a second order elliptic equation with periodic rapidly oscillating coefficients

E. V. Sevost'yanova

Abstract: This paper studies the asymptotic behavior of the fundamental solution $K_\varepsilon(x,y)$ of the equation
$$ -\frac\partial{\partial x_i}(a_{ij}(\frac x\varepsilon)\frac\partial{\partial x_j}u_\varepsilon)=f(x), $$
specified on the whole space $\mathbf R^n$, $n>2$, as $\varepsilon\to0$. The coefficients $a_{ij}(y)$ are periodic functions which satisfy the conditions of ellipticity, symmetry, and infinite smoothness.
The main result is the construction of the asymptotics of $K_\varepsilon(x,y)$ in the form
$$ K_\varepsilon(x,y)=\sum^M_{s=0}\varepsilon^s\Phi_s(x-y,\frac x\varepsilon,\frac y\varepsilon)+\varepsilon^{M+1}R_M(x,y,\varepsilon), $$
where $M$ is an arbitrary positive integer, the $\Phi_s(x,y,z)$ are homogeneous of degree $-s-n+2$ in the first argument and periodic in the remaining arguments, and for the remainder term $R_M(x,y,\varepsilon)$ on the set $|x-y|>\delta$, $\delta>0$, the estimate
$$ |R_M(x,y,\varepsilon)|<\frac{C_M(\delta)}{|x-y|^{M+n-1}} $$
holds, where the constants $C_M(\delta)$ are independent of $x$, $y$, and $\varepsilon$.
Figures: 1.
Bibliography: 9 titles.

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English version:
Mathematics of the USSR-Sbornik, 1982, 43:2, 181–198

Bibliographic databases:

UDC: 517.946
MSC: Primary 35J15, 35B40; Secondary 35J05
Received: 28.03.1980

Citation: E. V. Sevost'yanova, “An asymptotic expansion of the solution of a second order elliptic equation with periodic rapidly oscillating coefficients”, Mat. Sb. (N.S.), 115(157):2(6) (1981), 204–222; Math. USSR-Sb., 43:2 (1982), 181–198

Citation in format AMSBIB
\by E.~V.~Sevost'yanova
\paper An asymptotic expansion of the solution of a~second order elliptic equation with periodic rapidly oscillating coefficients
\jour Mat. Sb. (N.S.)
\yr 1981
\vol 115(157)
\issue 2(6)
\pages 204--222
\jour Math. USSR-Sb.
\yr 1982
\vol 43
\issue 2
\pages 181--198

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    14. Murata M., Tsuchida T., “Asymptotics of Green Functions and the Limiting Absorption Principle for Elliptic Operators with Periodic Coefficients”, J. Math. Kyoto Univ., 46:4 (2006), 713–754  mathscinet  zmath  isi
    15. Allaire G., “Periodic Homogenization and Effective MASS Theorems for the Schrodinger Equation”, Quantum Transport, Lecture Notes in Mathematics, 1946, eds. Allaire G., Arnold A., Degond P., Hou T., Springer-Verlag Berlin, 2008, 1–44  crossref  mathscinet  zmath  isi
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    22. S. E. Pastukhova, “Approximations of the Resolvent for a Non–Self-Adjoint Diffusion Operator with Rapidly Oscillating Coefficients”, Math. Notes, 94:1 (2013), 127–145  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    23. Cardone G. Pastukhova S.E. Perugia C., “Estimates in Homogenization of Degenerate Elliptic Equations by Spectral Method”, Asymptotic Anal., 81:3-4 (2013), 189–209  crossref  zmath  isi
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  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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