Generalized Besicovitch spaces and applications to deterministic homogenization

Abstract

Abstract. The purposes of this paper is to introduce a framework which enables us to study nonlinear homogenization problems. The starting point for this work is the theory of algebras with mean value. Very often in physics, from very simple experimental data, one gets sometimes complicated structure phenomena. These phenomena are represented by functions which are permanent in mean, but complicated in detail, which functions are subject to verify a functional equation often nonlinear. The problem is therefore to give an interpretation of these phenomena by functions having the following qualitative properties: they are functions that represent a phenomenon on a large scale, which vary irregularly, undergoing many nonperiodic oscillations. In this work we study the qualitative properties of spaces of such functions, say generalized Besicovitch spaces, and we prove general compactness results related to these spaces. We then apply these results 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 [26] (Homogenization of nonlinear pde's in the Fourier-Stieltjes algebras, SIAM J. Math. Anal., Vol. 41, No. 4, pp. 1589-1620 to know whether there exist or not ergodic algebras that are not subalgebras of the Fourier-Stieltjes algebra, and further we show that the theory of homogenization algebras by Nguetseng in [36] (Homogenization structures and applications I, Zeit. Anal. Anw., 22 (2003) 73-107) cannot handle weakly almost periodic homogenization problems.

Homogenization in algebras with mean value

“…It should be noted that the proof of the above result in the ergodic case relies heavily on the ergodicity assumption made on A, see for instance the papers [15,30,34,42]. In the general situation that we consider in this work, the proof is based on a de Rham type result formulated as follows:…”

“…Suppose that (6.27) is satisfied. It can be shown (as in [42,Proposition 4.5]) that the function (x, t, y, τ , ω) → b(y, τ , Ψ(x, t, y, τ , ω)), denoted below by b(·, Ψ), belongs to B(Ω; C(Q T ; B p ′ ,∞ A )) N ×N ; assumption (6.27) is crucially used in order to obtain the above result. Likewise, the function (x, t, y, τ , ω) → g k (y, τ , ψ 0 (x, t, ω)) (for ψ 0 ∈ B(Ω; C(Q T ; (A) N ))) denoted by g k (·, ψ 0 ), is an element…”

“…#A is just the one defined in [34,42] as the completion of B ). We will therefore just give the outlines of the proof without any detail.…”

“…Preliminaries. Let A be an algebra with mean value (algebra wmv, in short) on R N [26,15,42,63], that is, A is a closed subalgebra of the C*-algebra of bounded uniformly continuous functions BU C(R N ) which contains the constants, is closed under complex conjugation (u ∈ A whenever u ∈ A), is translation invariant (u(· + a) ∈ A for any u ∈ A and each a ∈ R N ) and is such that each element possesses a mean value in the following sense: (MV ) For each u ∈ A, the sequence (u ε ) ε>0 (where u ε (x) = u(x/ε 1 ), x ∈ R N ) weakly * -converges in L ∞ (R N ) to some constant function M (u) ∈ C (the complex field) as ε → 0, ε 1 = ε 1 (ε) being a positive function of ε tending to zero with ε. It is known that A (endowed with the sup norm topology) is a commutative C*-algebra with identity.…”

Homogenization of linear hyperbolic stochastic partial differential equation with rapidly oscillating coefficients: The two scale convergence method

“…In 1994, the two scale convergence was further extended from the periodic to the random setting by Bourgeat, Mikelić and Wright [12] under the name of "Stochastic two-scale convergence". Recently two scale convergence has been generalized to homogenization problems on nonperiodic algebras, see for instance [29,30,46] and [48]. We also note the newly introduced unfolding method by Cioranescu, Damlamian and Griso in [14,15].…”