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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.

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