Abstract
Questions related to the homogenization of the Dirichlet problem for nondivergence elliptic equations of second order with rapidly oscillating \(\varepsilon\)-periodic coefficients are studied. Error estimates of homogenization in the Sobolev spaces of order \(\sqrt\varepsilon\) are obtained. Asymptotic homogenization methods are applied.
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REFERENCES
M. Sh. Birman and T. A. Suslina, “Periodic differential operators of second order. Threshold properties and averagings,” St. Petersburg Math. J. 15 (5), 639–714 (2004).
V. V. Zhikov, “On operator estimates in homogenization theory,” Dokl. Akad. Nauk 403 (3), 305–308 (2005).
V. V. Zhikov and S. E. Pastukhova, “Operator estimates in homogenization theory,” Russian Math. Surveys 71 (3 (429)), 417–511 (2016).
V. V. Zhikov and M. M. Sirazhudinov, “On \(G\)-compactness of a class of nondivergence elliptic operators of second order,” Math. USSR-Izv. 19 (1), 27–40 (1982).
V. V. Zhikov and M. M. Sirazhudinov, “The averaging of nondivergence second order elliptic and parabolic operators and the stabilization of solutions of the Cauchy problem,” Math. USSR-Sb. 44 (2), 149–166 (1983).
V. V. Zhikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators (Fizmatlit, Moscow, 1993) [in Russian].
V. V. Yurinskii, “Averaging of a diffusion in a random environment,” Trudy Inst. Mat. Sibirsk. Otdel. AN SSSR 5 (2), 75–85 (1985).
M. I. Freidlin, “Dirichlet’s Problem for an Equation with Periodical Coefficients Depending on a Small Parameter,” Theory Probab. Appl. 9 (1), 121–125 (1964).
S. E. Pastukhova and V. V. Zhikov, “Homogenization estimates of operator type for an elliptic equation with quasiperiodic coefficients,” Russ. J. Math. Phys. 22 (2), 264–278 (2015).
M. M. Sirazhudinov, “An asymptotic method for homogenization for generalized Beltrami operators,” Sb. Math. 208 (4), 546–567 (2017).
S. E. Pastukhova, “Estimates of homogenization for the Beltrami equation,” J. Math. Sci. (N.Y.) 219 (2), 226–235 (2016).
V. S. Vinogradov, “A boundary value problem for a first order elliptic equation on the plane,” Differ. Uravn. 7 (8), 1440–1448 (1971).
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer-Verlag, Berlin, 1983).
O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Elliptic Quasilinear Equations (Academic Press, New York, 1968).
G. O. Cordes, “Über die erste Randwertaufgabe bei quasilinearen Differentialgleichungen zweiter Ordnung in mehr als zwei Variablen,” Math. Ann. 131, 278–312 (1956).
S. Campanato, “Un risultato relativo ad equazioni ellittiche del secondo ordine di tipo non variazionale,” Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 21 (4), 701–707 (1967).
N. Dunford and J. T. Schwartz Linear Operators (Interscience Publ., New York–London, 1958), Vol. 1.
O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type (Amer. Math. Soc., Providence, RI, 1968).
A. A. Samarskii, R. D. Lazarov, and V. L. Makarov, Difference Schemes for Differential Equations with Generalized Solutions (Vyssh. Shkola, Moscow, 1987) [in Russian].
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This work was supported by the Russian Foundation for Basic Research under grant 16-01-00508-a.
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Sirazhudinov, M.M. Homogenization Estimates of Nondivergence Elliptic Operators of Second Order. Math Notes 108, 250–271 (2020). https://doi.org/10.1134/S0001434620070263
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DOI: https://doi.org/10.1134/S0001434620070263