Generalized Besicovitch spaces and applications to deterministic homogenization

• Published in: Nonlinear analysis Vol. 74; no. 2; pp. 351 - 379
• Main Authors: Sango, Mamadou, Svanstedt, Nils, Woukeng, Jean Louis
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Ltd 2011; Elsevier
• Subjects: Algebra; Algebras with mean value; Difference and functional equations, recurrence relations; Ergodic processes; Exact sciences and technology; Finite differences and functional equations; Functions of a complex variable; Generalized Besicovitch spaces; Homogenization; Homogenizing; Matematik; Mathematical analysis; Mathematical sciences; Mathematics; Nonlinearity; Numerical analysis; Numerical analysis. Scientific computation; Operators; Oscillations; Partial differential equations; Pseudo-monotone operators; Sciences and techniques of general use; Weakly almost periodic functions; secondary; Weakly almost periodic functions; Generalized Besicovitch spaces; Pseudo-monotone operators; Homogenization; Algebras with mean value; primary; 46L06; Nonlinear analysis; Functional equation; Fourier analysis; Periodic function; Nonlinear problems; Partial differential equation; primary 35B27; Non linear operator; 35K65; 46J10; Oscillation; 46T30; 35B40; Mathematical model; Data structure; 46N20; secondary 46Axx
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2010.08.033

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.