Generalized Besicovitch spaces and applications to deterministic homogenization

Abstract

The purpose of the present work is to introduce a framework which enables us to study nonlinear homogenization problems. The starting point is the theory of algebras with mean value. Very often in physics, from very simple experimental data, one gets complicated structure phenomena. These phenomena are represented by functions which are permanent in mean, but complicated in detail. In addition the functions are subject to the verification of a functional equation which in general is nonlinear. The problem is therefore to give an interpretation of these phenomena using functions having the following qualitative properties: they are functions that represent a phenomenon on a large scale, and which vary irregularly, undergoing nonperiodic oscillations on a fine scale. In this work we study the qualitative properties of spaces of such functions, which we call generalized Besicovitch spaces, and we prove general compactness results related to these spaces. We then apply these results in order to study some new homogenization problems. One important achievement of this work is the resolution of the generalized weakly almost periodic homogenization problem for a nonlinear pseudo-monotone parabolic-type operator. We also give the answer to the question raised by Frid and Silva in their paper [35] [H. Frid, J. Silva, Homogenization of nonlinear pde’s in the Fourier–Stieltjes algebras, SIAM J. Math. Anal, 41 (4) (2009) 1589–1620] as regards whether there exist or do not exist ergodic algebras that are not subalgebras of the Fourier–Stieltjes algebra.