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.
