Some properties of Γ-limits of integral functionals

Carbone, Luciano; Sbordone, Carlo

doi:10.1007/bf02411687

Cited by 107 publications

(72 citation statements)

References 12 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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“…In this context, Spagnolo [23], with the G-convergence theory, and Murat & Tartar [25], [20], with the H-convergence theory, proved the compactness of the sequence F n , when A n is assumed to be both equicoercive and equibounded. A few times later, Buttazzo & Dal Maso [10] and Carbone & Sbordone [12] extended the result of compactness by only assuming that the sequence A n is bounded and equiintegrable in L 1 (Ω) 2×2 . At the same period, Fenchenko & Khruslov [15] showed that the equiintegrability condition cannot be relaxed since high conductivity regions in three dimension may induce nonlocal effects which correspond to a lack of compactness in the homogenization process (see also [2], [9], [4] for different approaches).…”

Section: Introductionmentioning

confidence: 99%

Asymptotic behaviour of equicoercive diffusion energies in dimension two

Briane

Casado–Díaz

2007

Calc. Var.

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In this paper, we study the asymptotic behaviour of a given equicoercive sequence of diffusion energies F n , n ∈ N, defined in L 2 (Ω), for a bounded open subset Ω of R 2 . We prove that, contrary to the three dimension (or greater), the Γ-limit of any convergent subsequence of F n is still a diffusion energy. We also provide an explicit representation formula of the Γ-limit when its domains contains the regular functions with compact support in Ω. This compactness result is based on the uniform convergence satisfied by some minimizers of the equicoercive sequence F n , which is specific to the dimension two. The compactness result is applied to the period framework, when the energy density is a highly oscillating sequence of equicoercive matrix-valued functions. So, we give a definitive answer to the question of the asymptotic behaviour of periodic conduction problems under the only assumption of equicoerciveness for the two-dimensional conductivity.

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“…When N > 2, convergence (3.21) holds when A n is bounded in L ρ (Ω) M ×N with ρ > (N − 1)/2. This condition is stronger than the equi-integrability of A n in L 1 (Ω) M ×N , which leads to a compactness result in the scalar case of [19] (M = 1). The proof of the scalar case is based on the maximum principle which does not hold for systems (M > 1).…”

Section: A Counterexamplementioning

confidence: 99%

“…In fact, the situation in three-dimensional linear elasticity is much more intricate since the closure set of equations is very large as shown in [18], while it is limited by the Beurling-Deny representation formula [4] in the conductivity case [17]. In view of the compactness result of [19] versus the nonlocal effects obtained in [27,29,3,16,17] and naturally connected with the Beurling-Deny formula by [34], the good assumption to avoid any loss of compactness in the homogenization process seems to be, at least in the scalar case and in any dimension, the equi-boundedness and the equi-integrability in L 1 of the sequences of coefficients.…”

Section: Introductionmentioning

confidence: 99%

A new div–curl result. Applications to the homogenization of elliptic systems and to the weak continuity of the Jacobian

Briane

Casado–Díaz

2016

Journal of Differential Equations

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A new div-curl result. Applications to the homogenization of elliptic systems and to the weak continuity of the Jacobian Abstract In this paper a new div-curl result is established in an open set Ω of R N , N ≥ 2, for the product of two sequences of vector-valued functions which are bounded respectively in L p (Ω) N and L q (Ω) N , with 1/p + 1/q = 1 + 1/(N − 1), and whose respectively divergence and curl are compact in suitable spaces. We also assume that the product converges weakly in W −1,1 (Ω). The key ingredient of the proof is a compactness result for bounded sequences in W 1,q (Ω), based on the imbedding of W 1,q (S N −1 ) into L p ′ (S N −1 ) (S N −1 the unit sphere of R N ) through a suitable selection of annuli on which the gradients are not too high, in the spirit of [26,32]. The div-curl result is applied to the homogenization of equi-coercive systems whose coefficients are equi-bounded in L ρ (Ω) for some ρ >It also allows us to prove a weak continuity result for the Jacobian for bounded sequences in W 1,N −1 (Ω) satisfying an alternative assumption to the L ∞ -strong estimate of [8]. Two examples show the sharpness of the results.

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Some properties of Γ-limits of integral functionals

Cited by 107 publications

References 12 publications

Deterministic homogenization of integral functionals with convex integrands

Deterministic homogenization of integral functionals with convex integrands

Asymptotic behaviour of equicoercive diffusion energies in dimension two

A new div–curl result. Applications to the homogenization of elliptic systems and to the weak continuity of the Jacobian

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