Homogenization and Monster Pucci operator: Difference between pages

From nonlocal pde
(Difference between pages)
Jump to navigation Jump to search
imported>Russell
No edit summary
 
imported>Luis
(Redirected page to Extremal operators)
 
Line 1: Line 1:
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]]
 
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.
 
An illustrative elementary example is given by the family of linear elliptic equations:
 
\[(\epsilon)\;\;\;\; \left \{ \begin{array}{rl} a_{ij} \left (\frac{x}{\epsilon} \right ) u_{ij}^{(\epsilon)} = f \left (\frac{x}{\epsilon} \right ) & x\in \Omega \\
u^{(\epsilon)} = g(x) &  x \in \partial \Omega \end{array}\right. \]
 
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.
 
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
 
 
\[ \left \{ \begin{array}{rl} \bar a_{ij} \bar u_{ij} = \bar f & x\in \Omega \\
\bar u = g(x) &  x \in \partial \Omega \end{array} \right. \]

Latest revision as of 18:15, 7 June 2011

Redirect to: