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imported>Luis |
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| Homogenization refers to the phenomenon (and the corresponding method of analysis) where solutions of a class of equations with highly oscillatory coefficients behave approximately like the solutions of a regular (say translation invariant) equation (''effective equation'') the approximation becoming more accurate as the oscillation of the coefficients is higher.
| | #REDIRECT [[Extremal operators]] |
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| Typically, one needs some assumption in the way the oscillations organize across space, the most common in increasing order of generality include periodic coefficients, quasi-periodic coefficients, and stationary ergodic coefficients.
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| An illustrative elementary example is given by the family of linear elliptic equations:
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| \[(\epsilon)\;\;\;\; \left \{ \begin{array}{rl} a_{ij} \left (\frac{x}{\epsilon} \right ) u_{ij}^{(\epsilon)} = f \left (\frac{x}{\epsilon} \right ) & x\in \Omega \\
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| u^{(\epsilon)} = g(x) & x \in \partial \Omega \end{array}\right. \]
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| Where the functions $a_{ij},f$ are assumed to be $\mathbb{Z}^n$-periodic, $\Omega \subset \mathbb{R}^n$ is a smooth bounded domain and $g: \partial \Omega \to \mathbb{R}$ is continuous. Of course the matrix $a_{ij}$ is assumed to be uniformly elliptic.
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| In this setting, homogenization refers to the fact that as $\epsilon \to 0$, the unique solution $u^{(\epsilon)}$ of the problem above converges ''uniformly'' to a function $\bar u: \Omega\to\R$ which is the unique solution of
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| \[ \left \{ \begin{array}{rl} \bar a_{ij} \bar u_{ij} = \bar f & x\in \Omega \\
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| \bar u = g(x) & x \in \partial \Omega \end{array} \right. \]
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Latest revision as of 18:15, 7 June 2011