# Leonid Berlyand : Flux norm approach to finite-dimensional homogenization approximation with non-separated scales and high contrast

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Classical homogenization theory deals with mathematical models of strongly
inhomogeneous media described by PDEs with rapidly oscillating coefficients
of the form A(x/\epsilon), \epsilon → 0. The goal is to approximate this problem by a
homogenized (simpler) PDE with slowly varying coefficients that do not depend
on the small parameter \epsilon. The original problem has two scales: fine
O(\epsilon) and coarse O(1), whereas the homogenized problem has only a coarse
scale.
The homogenization of PDEs with periodic or ergodic coefficients and
well-separated scales is now well understood. In a joint work with H. Owhadi
(Caltech) we consider the most general case of arbitrary L∞ coefficients,
which may contain infinitely many scales that are not necessarily well-separated.
Specifically, we study scalar and vectorial divergence-form elliptic PDEs with
such coefficients. We establish two finite-dimensional approximations to the
solutions of these problems, which we refer to as finite-dimensional homogenization
approximations. We introduce a flux norm and establish the error
estimate in this norm with an explicit and optimal error constant independent
of the contrast and regularity of the coefficients. A proper generalization of
the notion of cell problems is the key technical issue in our consideration.
The results described above are obtained as an application of the transfer
property as well as a new class of elliptic inequalities which we conjecture.
These inequalities play the same role in our approach as the div-curl lemma
in classical homogenization. These inequalities are closely related to the issue
of H^2 regularity of solutions of elliptic non-divergent PDEs with non smooth
coefficients.

**Category**: Applied Math and Analysis**Duration**: 02:29:55**Date**: April 7, 2010 at 10:55 AM-
**Tags:**seminar, Applied Math And Analysis Seminar

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