-convergence of nonlinear parabolic operators

Nguetseng, Gabriel; Woukeng, Jean Louis

doi:10.1016/j.na.2005.12.035

Cited by 35 publications

(58 citation statements)

References 7 publications

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“…Thanks to the identi…cation G 1 (B p A ) = L p ( (A)) we get more accurate convergence results as those obtained in the previous papers by Nguetseng et al [36,40,41,42]. In all that follow is an open subset of R N (integer N 1) and A is an algebra w.m.v.…”

Section: The Two-scale A-convergencesupporting

confidence: 65%

“…Remark 3.2. As it will be seen below, Theorems 3.6 and 3.7 extend all the results of all the previous papers dealing with deterministic homogenization theory [42,40,41] since all the H-algebras till encountered so far are ergodic algebras. We will also see that our results apply to particular algebra w.m.v.…”

Section: = (I) (Ii) + (Iii)supporting

confidence: 65%

“…We rely on the concept of vector valued algebras to prove a general compactness result (Theorem 3.6) which generalizes to all points of view, those already obtained in the papers [36,40,41,42]. In particular in the above-cited papers, this result is available in the only case when p = 2 in the context e.g.…”

Section: Introductionmentioning

confidence: 94%

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Generalized Besicovitch spaces and applications to deterministic homogenization

Sango

Svanstedt

Woukeng

2011

Nonlinear Analysis: Theory, Methods & Applications

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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.

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Section: The Two-scale A-convergencesupporting

confidence: 65%

Section: = (I) (Ii) + (Iii)supporting

confidence: 65%

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Generalized Besicovitch spaces and applications to deterministic homogenization

Sango

Svanstedt

Woukeng

2011

Nonlinear Analysis: Theory, Methods & Applications

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“…This being so, we observe that here we have two significant improvements: (1) firstly, the hypothesis (4.4) on the uniform equicontinuity is purely dropped; (2) secondly, the homogenization problem is stated here in general terms since the H-algebra is W 1,p -proper for any real p > 1 and not only for p = 2 as considered in the papers [21][22][23]25]. This is a true advance as the applications in the almost periodic setting are concerned.…”

Section: Abstract Homogenization Resultsmentioning

confidence: 95%

“…In almost all the previous papers dealing with deterministic homogenization theory (see for instance [21][22][23]25]) the almost periodic homogenization problem were stated by combining hypothesis (4.3) above with the following one:…”

Section: Abstract Homogenization Resultsmentioning

confidence: 99%

Deterministic homogenization of integral functionals with convex integrands

Nguetseng

Nnang

Woukeng

2010

Nonlinear Differ. Equ. Appl.

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Abstract. In order to widen the scope of the applications of deterministic homogenization, we consider here the homogenization problem for a family of integral functionals. The homogenization procedure tending to be classical, the choice focused on the convex integral functionals is made just to simplify the presentation of the paper. We use a new approach based on the Stepanov type spaces, which approach allows us to solve various problems such as the almost periodic homogenization problem and others without resorting to additional assumptions. We then apply it to obtain a general homogenization result and then we provide a number of physical applications of the result. The convergence method used falls within the scope of two-scale convergence. Mathematics Subject Classification (2000). 35B40, 45G10, 46J10.

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