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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A linear homogenization problem with time dependent coefficient


Author: Maria Luisa Mascarenhas
Journal: Trans. Amer. Math. Soc. 281 (1984), 179-195
MSC: Primary 45A05; Secondary 35B99, 45M05, 73F15
MathSciNet review: 719664
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Abstract: We consider: the homogenization problem

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {(\partial u\varepsilon /\partia... ...0,} \\ {{u_\varepsilon }(x,0) = \phi (x), \hfill} & {} \\ \end{array} } \right.$

where $ \beta $ is a strictly positive bounded real function, periodic of period $ 1$, and $ {\beta_\varepsilon }(x) = \beta (x/\varepsilon )$; the equivalent integral equation

$\displaystyle {u_\varepsilon }(x,t) + \int_0^t {{\beta_\varepsilon }(x)\,{u_\varepsilon }(x,s)\;ds = \phi (x)}; $

and the homogenized equation

$\displaystyle {u_0}(x,t) + \int_0^t {K(t - s)\,{u_0}(s)\,ds = \phi (x)}, $

where $ K$ is a unique, well-defined function depending on $ \beta $. We study this problem for a time dependent $ \beta $, and characterize a two-variable function $ K(s,t)$ satisfying

$\displaystyle {u_0}(x,t) + \int_0^t {K(s,t - s)\,{u_0}(x,s)\;ds = \phi (x)} $

and study its uniqueness.

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

  • [1] A. Kolmogorov and S. Fomin, Eléments de la théorie des fonctions et de l'analyse fonctionnelle, "Mir", Moscow, 1974.
  • [2] A. Korányi, Note on the theory of monotone operator functions, Acta Sci. Math. Szeged 16 (1955), 241–245. MR 0086110 (19,126b)
  • [3] Enrique Sánchez-Palencia, Nonhomogeneous media and vibration theory, Lecture Notes in Physics, vol. 127, Springer-Verlag, Berlin-New York, 1980. MR 578345 (82j:35010)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0719664-3
PII: S 0002-9947(1984)0719664-3
Keywords: Homogenization, convolution, kernel, integral equation
Article copyright: © Copyright 1984 American Mathematical Society