Generalized Besicovitch spaces and applications to deterministic homogenization

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...

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Vydáno v:	Nonlinear analysis Ročník 74; číslo 2; s. 351 - 379
Hlavní autoři:	Sango, Mamadou, Svanstedt, Nils, Woukeng, Jean Louis
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Amsterdam Elsevier Ltd 2011 Elsevier
Témata:	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
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Popis
Shrnutí:	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.
Bibliografie:	ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23
ISSN:	0362-546X 1873-5215
DOI:	10.1016/j.na.2010.08.033