Abstract
We present a survey of homogenization methods for diffusion equations with periodic and random diffusivity.
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References
M. Avellaneda, T. Hou and G. Papanicolaou, “Finite difference methods for partial differential equations with rapidly oscillating coefficients,” Math. Modeling and Num. Analysis, 25: 693–710 (1991).
I. Babuska, “Solution of interface problems by homogenization I, II, III,” SIAM J. Math Analysis, 7: 603–634, 635-645, 1976.
I. Babuska, “Solution of interface problems by homogenization I, II, III,” SIAM J. Math Analysis, 8: 923–937 (1977).
N. Bakhvalov and G. Panasenko, Homogenization: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of Composites, Kluwer (1989).
A. Bensoussan, J. L. Lions and G. Papanicolaou, “Boundary layer analysis in homogenization of diffusion equations with Dirichlet conditions in the half space,” in: Proc. of Intern. Symp. SDE, Kyoto 1976, pp. 21–40, edited by K. Ito, J. Wiley, New York (1978).
A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis of Periodic Structures, North Holland, (1978).
R. Caflisch, M. Miksis, G. Papanicolaou and Lu Ting, “Effective equations for wave propagation in bubbly liquids,” J. Fluid Dynamics, 153: 105–120 (1985).
R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1, Wiley (1953).
G. Dal Maso, Introduction to Г-Convergence, Birkhäuser (1992).
E. DeGiorgi and S. Spagnolo, “Sulla convergenza degli integrali dell’ energia per operatori ellittici del secundo ordine,” Boll U.M.I., 8: 391–11 (1983).
B. Engquist and T. Hou, “Particle method approximation of oscillatory solutions to hyperbolic partial differential equations,” SIAM J. Num. Anal., 26: 289–319, (1989).
R. Figari, G. Papanicolaou and J. Rubinstein, The Point Interaction Approximation for Diffusion in Regions with Many Small Holes, Lecture Notes in Biomathematics, Vol. 70, Springer, 202-209 (1987).
H. Hasimoto, On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres, J. Fluid Dynamics, 5: 317–328 (1959).
E. I. Hruslov and V A. Marchenko, Boundary Value Problems in Regions with Fine-Grained Boundaries, Naukova Duma, Kiev (1974).
M. Kac, “Probabilistic methods in some problems of scattering theory,” Rocky Mountain J. of Math., 4: 511–537,(1974).
K. Kawazu, Y. Tamura and H. Tanaka, “Localization of diffusion processes in one-dimensional random environment,” J. Math. Soc. Japan, 44: 515–550 (1992).
J. B. Keller, “Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders,” Journal of Appl. Physics, 34: 991–993 (1963).
O. D. Kellogg, Foundations of Potential Theory, Dover (1953).
S. Kozlov, “The method of averaging and walks in inhomogeneous environments,” Russian Math Surveys 40: 97–128 (1985).
S. Kozlov, “Geometric aspects of averaging,” Russian Math. Surveys, 44: 91–144 (1989).
P. Kuchment, Floquet Theory for Partial Differential Equations, Birkhäuser, Basel (1993).
R. Landauer, “Electrical conductivity in inhomogeneous media,” in: Electrical Transport and Optical Properties of Inhomogeneous Media, J. C. Garland and D. B. Tanner (eds.), Amer. Ins. Physics (1978).
J. C. Maxwell, A Treatise on Electricity and Magnetism, Vol. 1, Clarendon Press, Oxford (1881).
O. Oleinik, Mathematical Problems in Elasticity and Homogenization, North Holland (1992).
S. Ozawa, “Point interaction potential approximation for (−Δ + U)−1 and eigenvalues for the Laplacian on wildly perturbed domains,” Osaka, J. Math, 20: 923–937 (1983).
G. Papanicolaou and S. R. S. Varadhan, “Boundary value problems with rapidly oscillating random coefficients,” Colloquia Mathematica Societatis Janos Bolyai 27, Random Fields, Esztergom (Hungary) 1979, North Holland, 835-873, (1982).
G. Papanicolaou and S. R. S. Varadhan, “Diffusion in regions with many small holes,” in: Proceedings of Vilnius Conference in Probability, B. Grigelionis ed., Springer Lecture Notes in Control and Information Sciences, vol. 25, 190-206 (1980).
J. W. S. Rayleigh, “On the influence of obstacles arranged in rectangular order upon the properties of the medium,” Phil. Mag., 34: 481–502 (1892).
A. S. Sangani and A. Acrivos, “The effective conductivity of a periodic array of spheres,” Proc. Roy. Soc. Lond. A, 386: 263 (1983).
Ya. Sinai, Introduction to Ergodic Theory, Princeton University Press (1977).
Ya. Sinai, “Limit behavior of one-dimensional random walks in random environment,” Theor. Prob. Appl., 27: 247–258 (1982).
M. Vogelius and G. Papanicolaou, “A projection method applied to diffusion in a periodic structure,” SIAM J. Appl Math., 42: 1302–1322 (1982).
K. Yosida, Functional Analysis, Springer (1979).
J. M. Ziman, Principles of the Theory of Solids, Cambridge (1972).
M. Zuzovski and H. Brenner, “Effective conductivity of composite materials composed of cubic arrangements of spherical particles embedded in an isotropic matrix,” ZAMP, 28: 979–992 (1977).
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Papanicolaou, G.C. (1995). Diffusion in Random Media. In: Keller, J.B., McLaughlin, D.W., Papanicolaou, G.C. (eds) Surveys in Applied Mathematics. Surveys in Applied Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-0436-2_3
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DOI: https://doi.org/10.1007/978-1-4899-0436-2_3
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