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

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

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
\Bibitem{Sev81} \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 \mathnet{http://mi.mathnet.ru/msb2382} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=622145} \zmath{https://zbmath.org/?q=an:0494.35019|0469.35024} \transl \jour Math. USSR-Sb. \yr 1982 \vol 43 \issue 2 \pages 181--198 \crossref{https://doi.org/10.1070/SM1982v043n02ABEH002444} 

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