Periodic solutions and homogenization of non linear variational problems

Marcellini, Paolo

doi:10.1007/bf02417888

Cited by 125 publications

(87 citation statements)

References 3 publications

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“…The main references for functionals of type (1.7) are Marcellini [17] and Carbone-Sbordone [7]. Apart from [1], all the works cited above have two common points : (1) they have been studied under the periodicity hypothesis on the fast variable y; (2) the convergence method used is that of Γ-convergence [12].…”

Section: Introductionmentioning

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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Section: Introductionmentioning

confidence: 99%

Deterministic homogenization of integral functionals with convex integrands

Nguetseng

Nnang

Woukeng

2010

Nonlinear Differ. Equ. Appl.

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“…Section 3 generalizes 2 to the nonlinear case. In the periodic setting, we give a new proof of the F-convergence of convex energies (see [31], [16], [17]), and we revisit the homogenization of monotone operators [42]. On the contrary of 2 and 3, 4 deals with an example of homogenization where typical two-scale phenomena appear.…”

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confidence: 99%

Homogenization and Two-Scale Convergence

Allaire¹

1992

SIAM J. Math. Anal.

1,958

2,637

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Abstract. Following an idea of G. Nguetseng, the author defines a notion of "two-scale" convergence, which is aimed at a better description of sequences of oscillating functions. Bounded sequences in L2(f) are proven to be relatively compact with respect to this new type of convergence. A corrector-type theorem (i.e., which permits, in some cases, replacing a sequence by its "two-scale" limit, up to a strongly convergent remainder in L2 (12)) is also established. These results are especially useful for the homogenization of partial differential equations with periodically oscillating coefficients. In particular, a new method for proving the convergence of homogenization processes is proposed, which is an alternative to the so-called energy method of Tartar [10], [40] for details). This. method is very simple and powerful, but unfortunately is formal since, a priori, the ansatz (0.3) does not hold true. Thus, the two-scale asymptotic expansion method is used only to guess the form of the homogenized operator L, and other arguments are needed to prove the convergence of the sequence u to u. To this end, the more general and powerful method is the so-called energy method of Tartar [42]. Loosely speaking, it amounts to multiplying equation (0.1) by special test functions (built with the solutions of the cell equation), and passing to the limit as e-0. Although products of weakly convergent sequences are involved, we can actually pass to the limit thanks to some "compensated compactness" phenomenon due to the particular choice of test functions.Despite its frequent success in the homogenization of many different types of equations, this way of proceeding is not entirely satisfactory. It involves two different steps, the formal derivation of the cell and homogenized equation, and the energy method, which have very little in common. In some cases, it is difficult to work out the energy method (the construction of adequate test functions could be especially tricky). The energy method does not take full advantage of the periodic structure of the problem (in particular, it uses very little information gained with the two-scale asymptotic expansion). The latter point is not surprising since the energy method was not conceived by Tartar for periodic problems, but rather in the more general (and more difficult) context of H-convergence. Thus, there is room for a more efficient homogenization method, dedicated to partial differential equations with periodically oscillating coefficients. The purpose of the present paper is to provide such a method that we call two-scale convergence method.This new method relies on the following theorem, which was first proved by Nguetseng [36].

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“…(i) The functional Λ may be compared with the homogenized functional of the calculus of variations (see, for example, [20,21,31,66]). However, the latter concerns the stationary setting, whereas Λ applies to processes.…”

Section: And Only If (Denoting By B * the Transpose Of The Tensor B) mentioning

confidence: 99%

Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl—Reuss model of elastoplasticity

Visintin

2008

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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This paper deals with processes in nonlinear inelastic materials whose constitutive behaviour is represented by the inclusionhere we denote by σ the stress tensor, by ε the linearized strain tensor, by B(x) the compliance tensor and by ∂ϕ(·, x) the subdifferential of a convex function ϕ (·, x). This relation accounts for elasto-viscoplasticity, including a nonlinear version of the classical Maxwell model of viscoelasticity and the Prandtl-Reuss model of elastoplasticity. The constitutive law is coupled with the equation of continuum dynamics, and well-posedness is proved for an initial-and boundary-value problem. The function ϕ and the tensor B are then assumed to oscillate periodically with respect to x and, as this period vanishes, a two-scale model of the asymptotic behaviour is derived via Nguetseng's notion of two-scale convergence. A fully homogenized single-scale model is also retrieved, and its equivalence with the two-scale problem is proved. This formulation is non-local in time and is at variance with that based on so-called analogical models that rest on a mean-field-type hypothesis.

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Periodic solutions and homogenization of non linear variational problems

Cited by 125 publications

References 3 publications

Deterministic homogenization of integral functionals with convex integrands

Deterministic homogenization of integral functionals with convex integrands

Homogenization and Two-Scale Convergence

Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl—Reuss model of elastoplasticity

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