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Homogeneization theory revisited

UMPA Publication

Published on : July 4, 2016

A beautiful combination of mathematics and photography

Albert_Fathi_300x245.jpg
Albert Fathi's artistic portrait was created through a homogenization process. Photo CNRS.

In a recent paper published jointly in Inventiones Mathematicae and Mathematische Zeitschrift, Albert Fathi, mathematician for UMPA (and his coauthors Andrea Davini, Renato Iturriaga and Maxime Zavidovique revisit a fundamental result in homogenization theory, obtained in 1987 by Lions, Papanicolaou and Varadhan.

To simplify things: homogeneization phenomena describe situations where a medium which is chaotic at small scales gives rise to an organized behavior at large scales. This is exemplified by this picture, where Albert Fathi's face emerges from a multitude of apparently unrelated smaller pictures. Using a elaborate machinery developed in particular by Albert Fathi under the name of weak KAM theory, the authors manage to solve, in certain cases, a problem that was left open in the seminal work of Lions, Papanicolaou and Varadhan concerning the uniqueness of the corrector, which describes the discrepancy between the "idealized" homogenized medium, and the true, "imperfect" one.

Zoom on Albert's eye. All the pictures used to create this mosaïc were provided by the CNRS Image bank. To see better and admire the details click here CNRS). 

Using a elaborate machinery developed in particular by Albert Fathi under the name of weak KAM theory, the authors manage to solve, in certain cases, a problem that was left open in the seminal work of Lions, Papanicolaou and Varadhan concerning the uniqueness of the corrector, which describes the discrepancy between the "idealized" homogenized medium, and the true, "imperfect" one.

More information

Réferences : A. Davini, A. Fathi, R. Iturriaga and M. Zavidovique, Convergence of the solutions of the discounted Hamilton- Jacobi equation, Inventiones Mathematicae, juin 2016 .
A. Davini, A. Fathi, R. Iturriaga and M. Zavidovique, Convergence of the solutions of the discounted equation : the discrete case, Mathematische Zeitschrift, juin 2016


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