Deterministic homogenization of non-linear non-monotone degenerate elliptic operators

Woukeng, Jean Louis

doi:10.1016/j.aim.2008.07.003

Cited by 10 publications

(18 citation statements)

References 9 publications

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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: 93%

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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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Section: Abstract Homogenization Resultsmentioning

confidence: 93%

“…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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“…As opposed to what is usually done in the deterministic homogenization theory, we present here a new approach based on the generalized Besicovitch type spaces (see Section 2), which widely opens the scope of application of our main homogenization result, Theorem 4.9, as we will see it therein. In particular we will work out the almost periodic homogenization problem without further assumption on the functions a i , as was usually the case in all the previous papers dealing with deterministic homogenization theory; see for instance hypothesis (4.34) in Remark 4.6 (which was fundamental in all the previous works [40,41,42,46], and which is purely dropped here).…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…In all the previous papers dealing with deterministic homogenization theory (see for instance [40,41,42,46]) the almost periodic homogenization problem were stated by combining hypothesis (4.33) above with the following one For each ( 0 ; ) 2 AP (R N y R ) N +1 and each (x; t) 2 Q we have sup k2Z N +1 R k+Y T (j a i (x; t; y ; ; 0 (y; ); (y; )) a i (x; t; y; ; 0 (y; ); (y; )) j p 0 dyd ) ! 0 as j j + j j !…”

Section: Y;mentioning

confidence: 99%

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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“…In that direction we refer, e.g., to the papers [15,30,32,33,34,54,58,59,56,31] in which only ergodic algebras are considered. In this paper we show how one can derive general homogenization results in algebras with mean value through the theory of strongly continuous groups of transformation.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Homogenization in algebras with mean value

Woukeng¹

2015

Banach J. Math. Anal.

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Abstract. In several works, the theory of strongly continuous groups is used to build a framework for solving stochastic homogenization problems. Following this idea, we construct a detailed and comprehensive theory of homogenization. This enables to solve homogenization problems in algebras with mean value, regardless of whether they are ergodic or not, thereby responding affirmatively to the question raised by Zhikov and Krivenko [V.V. Zhikov, E.V. Krivenko, Homogenization of singularly perturbed elliptic operators. Matem. Zametki, 33 (1983) 571-582 (english transl.: Math. Notes, 33 (1983) 294-300)] to know whether it is possible to homogenize problems in nonergodic algebras. We also state and prove a compactness result for Young measures in these algebras. As an important achievement we study the homogenization problem associated with a stochastic Ladyzhenskaya model for incompressible viscous flow, and we present and solve a few examples of homogenization problems related to nonergodic algebras.

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Deterministic homogenization of non-linear non-monotone degenerate elliptic operators

Cited by 10 publications

References 9 publications

Deterministic homogenization of integral functionals with convex integrands

Deterministic homogenization of integral functionals with convex integrands

Generalized Besicovitch spaces and applications to deterministic homogenization

Homogenization in algebras with mean value

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