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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.

Full text: PDF file (1764 kB)
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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
\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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    Citing articles on Google Scholar: Russian citations, English citations
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    2. Birman M., Suslina T., “Threshold Effects Near the Lower Edge of the Spectrum for Periodic Differential Operators of Mathematical Physics”, Systems, Approximation, Singular Integral Operators, and Related Topics, Operator Theory : Advances and Applications, 129, eds. Borichev A., Nikolski N., Birkhauser Verlag Ag, 2001, 71–107  mathscinet  zmath  adsnasa  isi
    3. M. Sh. Birman, T. A. Suslina, “Periodic differential operators of second order. Threshold properties and averagings”, St. Petersburg Math. J., 15:5 (2004), 639–714  mathnet  crossref  mathscinet  zmath
    4. Minoru Murata, Tetsuo Tsuchida, “Asymptotics of Green functions and Martin boundaries for elliptic operators with periodic coefficients”, Journal of Differential Equations, 195:1 (2003), 82  crossref
    5. Lohoue N., Alexopoulos G., “On the Large Time Behavior of Heat Kernels on Lie Groups”, Duke Math. J., 120:2 (2003), 311–351  crossref  mathscinet  zmath  isi
    6. Grégoire Allaire, Yves Capdeboscq, Andrey Piatnitski, Vincent Siess, M. Vanninathan, “Homogenization of Periodic Systems with Large Potentials”, Arch Rational Mech Anal, 174:2 (2004), 179  crossref  mathscinet  isi  elib
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    10. Sjoberg D., Engstrom C., Kristensson G., Wall D., Wellander N., “A Floquet-Bloch Decomposition of Maxwell's Equations Applied to Homogenization”, Multiscale Model. Simul., 4:1 (2005), 149–171  crossref  mathscinet  isi
    11. Sjoberg D., “Homogenization of Dispersive Material Parameters for Maxwell's Equations Using a Singular Value Decomposition”, Multiscale Model. Simul., 4:3 (2005), 760–789  crossref  mathscinet  isi
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    13. ERIC BONNETIER, FATEN KHAYAT, “INFLUENCE OF DISTORTION IN THE HOMOGENIZATION OF FIBER-REINFORCED COMPOSITES”, Math. Models Methods Appl. Sci, 16:11 (2006), 1861  crossref
    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
    16. Birman M.S., Suslina T.A., “Homogenization of Periodic Differential Operators as a Spectral Threshold Effect”, New Trends in Mathematical Physics, ed. Sidoravicius V., Springer-Verlag Berlin, 2009, 667–683  crossref  zmath  isi
    17. T. A. Suslina, “Homogenization of the Parabolic Cauchy Problem in the Sobolev Class $H^1(\mathbb{R}^d)$”, Funct. Anal. Appl., 44:4 (2010), 318–322  mathnet  crossref  crossref  mathscinet  zmath  isi
    18. Bunoiu R., Cardone G., Suslina T., “Spectral Approach to Homogenization of an Elliptic Operator Periodic in Some Directions”, Math. Meth. Appl. Sci., 34:9 (2011), 1075–1096  crossref  mathscinet  zmath  isi  elib
    19. E. S. Vasilevskaya, T. A. Suslina, “Homogenization of parabolic and elliptic periodic operators in $L_2(\mathbb R^d)$ with the first and second correctors taken into account”, St. Petersburg Math. J., 24:2 (2013), 185–261  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    20. X. Blanc, F. Legoll, A. Anantharaman, “Asymptotic Behavior of Green Functions of Divergence form Operators with Periodic Coefficients”, Applied Mathematics Research eXpress, 2012  crossref
    21. S. E. Pastukhova, “Approximations of the operator exponential in a periodic diffusion problem with drift”, Sb. Math., 204:2 (2013), 280–306  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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
    24. C.E.. Kenig, Fanghua Lin, Zhongwei Shen, “Periodic Homogenization of Green and Neumann Functions”, Commun. Pur. Appl. Math, 2013, n/a  crossref
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  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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