On Homogenization of Non-Divergence Form Partial Difference Equations
Ian F. Pilizzotto (University of Michigan, USA)
Abstract
In this paper a method for proving homogenization of divergence form elliptic equations is extended to the non-divergence case. A new proof of homogenization is given when the coefficients in the equation are assumed to be stationary and ergodic. A rate of convergence theorem in homogenization is also obtained, under the assumption that the coefficients are i.i.d. and the elliptic equation can be solved by a convergent perturbation series.
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Pages: 125-135
Publication Date: June 9, 2005
DOI: 10.1214/ECP.v10-1141
References
1. V.V. Anshelevich, K.M. Khanin and Ya.G.Sinai. Symmetric random walks in random environments. Comm. Math. Phys. 85 (1982), 449-470. Math Review 84a:60082
2. E. Bolthausen and A.S. Sznitman. Ten lectures on random media. DMV Seminar 32, Birkhauser Verlag, Basel, 2002. 116 pp. Math Review 2003f:60183
3. J.G. Conlon and A. Naddaf. On homogenization of elliptic equations with random coefficients. Electron. J. Probab. 5 (2000), 58pp. Math Review 2002j:35328
4. R. Durrett. Probability theory and examples. Second Edition, Duxbury Press, Belmont CA. 1996. 503 pp. Math Review 98m:60001
5. D. Gilbarg and N.S. Trudinger. Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics, Springer-Verlag, Berlin 2001. 517 pp. Math Review 2001k:35004
6. H.J. Kuo and N. S. Trudinger. Linear elliptic difference inequalities with random coefficients. Math. Comp. 55 (1990), 37-53. Math Review 90m:65176
7. G. Lawler. Weak convergence of a random walk in a random environment. Comm. Math. Phys. 87 (1982/82), 81-87. Math Review 84b:60093
8. G.C. Papanicolaou and S.R.S. Varadhan. Diffusions with random coefficients. Statistics and probability: essays in honor of C.R. Rao, pp. 547-552, North-Holland, Amsterdam, 1982. Math Review 85e:60082
9. M. Reed and B. Simon. Methods of modern mathematical physics I. Functional analysis. Academic Press, New York-London, 1972, 325 pp. Math Review 58#12429a
10. V.V. Zhikov, S.M. Kozlov and O.A. Oleinik. Homogenization of Differential Operators and Integral Functionals. Translated from the Russian by G.A. Yosifian. Springer-Verlag, Berlin, 1994. 570 pp. Math Review 96h:35003b
11.V.V. Zhikov and M.M. Sirazhudinov. G compactness of a class of second-order nondivergence elliptic operators. (Russian) Izv. Akad. Nauk SSSR Ser.Mat. 45 (1981), 718-733. Math Review 83f:35041