Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



On some problems of homogenization

Author: J. M. Burgers
Journal: Quart. Appl. Math. 35 (1978), 421-434
MSC: Primary 35A35; Secondary 65P05
DOI: https://doi.org/10.1090/qam/479821
MathSciNet review: 479821
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Abstract: This article concerns a type of partial differential equations in which coefficients occur that are periodic functions of the basic independent variables or coordinates $ {x_1}$, $ {x_2}$ (we restrict ourselves to two-dimensional problems). The period length $ H$, here assumed to be the same in both directions, is supposed to be small in comparison with the scale of the general field, and interest is directed to what happens when $ H \to 0$. An example is the equation

$\displaystyle \Sigma \frac{\partial }{{\partial {x_j}}}\left( {{a_{ij}}\frac{{\partial u}}{{\partial {x_i}}}} \right) = f\left( {{x_1},{x_2}} \right)$

with the boundary condition $ u = 0$ on a given closed curve $ C$ in the $ {X_1}$ , $ {x_2}$-plane. Size and shape of $ C$ are independent of $ H$, while the coefficients $ {a_{11}}$ , $ {a_{12}}$ , $ {a_{21}}$ , $ {a_{22}}$ are periodic functions of auxiliary variables $ {\xi _i} = {x_1}/H$ with period 1 in $ {\xi _1}$, $ {\xi _2}$ . The problem of interest is whether the unknown function $ u$, which must react upon all the fluctuations of the $ {a_{ij}}$ , can be related to a function $ U\left( {{x_1}, {x_2}} \right)$ which is determined by a partial differential equation with constant coefficients, and how these coefficients can be obtained from the $ {a_{ij}}$ . This is called the problem of "homogenization".

References [Enhancements On Off] (What's this?)

  • [1] I. Babuška. Homogenization and its applications; mathematical and computational problems, in Numerical solutions of partial differential equations III, B. Hubbard, ed., Academic Press, 1976
  • [2] I. Babuška, Homogenization approach in engineering, in Proc. symposium on computing methods in engineering, Dec. 15-19, 1945, Versailles, France
  • [3] G. Duvaut, Analyse fonctionelle et mécanique des milieus continus, application à l'etude des matériaux composites elastiques à structure périodique-homogénéisation, in: Theoretical and Applied Mechanics, Preprints IUTAM Congress, W. T. Koiter, ed., Aug. 30-Sept. 4, 1976, Delft, Netherlands, pp. 110-132
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DOI: https://doi.org/10.1090/qam/479821
Article copyright: © Copyright 1978 American Mathematical Society

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