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## Homogenization of non-linear variational problems with thin low-conducting

Citations: | 1 - 0 self |

### Citations

438 | Homogenization and two-scale convergence.
- Allaire
- 1992
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Citation Context ...nductivity are surrounded by higherconductivity sets. One of the first mathematical studies of the double-porosity model is due to Arbogast, Douglas and Hornung [1], subsequently revisited by Allaire =-=[3]-=- and extended in various ways in the linear case [5], [23], and in the non-linear case [22], [8]. If the higher-conductivity set has more than one connected component, the general form of the limit ef... |

178 |
Homogenization of Multiple Integrals
- Braides, Defranceschi
- 1998
(Show Context)
Citation Context ...nclusion Eε ⊂ E of (2.25) clearly implies that G0 ≤ G defined by (2.28). Note that G is the Γ-limit of the functional GE ε defined by (2.27) with the fixed periodic set E, according to [10] (see also =-=[9]-=- Chapter 20). Let us prove the inverse inequality G ≤ G0. Let v ∈ Lp (Ω; Rm ) and let vε ∈ Lp (Ω; Rm ) be a recovering sequence which strongly converges to v and such that Gε(vε) → G0(v). If G0(v) = +... |

176 | Γ-convergence for beginners
- Braides
- 2002
(Show Context)
Citation Context ...s Eε i (see Proposition 2.6 below). The convergence of the energy Fε then has to be understood in the sense of the Γ-convergence for the topology τ. Let us recall the definition of Γ-convergence (see =-=[6, 16]-=-): Definition 2.2 Let Fεbe a sequence of functionals defined on a vector space H and let F be a functional defined on a vector space K. Let τ be a topology defined by the convergence of sequences of H... |

103 |
Perturbations singulieres dans les problemes aux limites et en controle optimal,
- Lions
- 1973
(Show Context)
Citation Context ...iformly bounded with respect to ε. 82.4 Statement of the result In the classical framework, namely without zero-order interaction between the different conducting phases, it is known (see e.g. [24], =-=[21]-=- and [18]) that thin lowly conducting layers may modify the homogenized conductivity in certain regimes. In the present context we want to avoid such a phenomenon, which would modify the bulk part of ... |

92 | Derivation of the double porosity model of single phase via homogenization theory
- Arbogast, Douglas, et al.
- 1990
(Show Context)
Citation Context ...orosity models, where regions of low conductivity are surrounded by higherconductivity sets. One of the first mathematical studies of the double-porosity model is due to Arbogast, Douglas and Hornung =-=[1]-=-, subsequently revisited by Allaire [3] and extended in various ways in the linear case [5], [23], and in the non-linear case [22], [8]. If the higher-conductivity set has more than one connected comp... |

25 |
An introduction to Γ-convergence.
- G
- 1993
(Show Context)
Citation Context ...→ 0 we obtain the desired estimate. (iii) Since Rm×N is separable, there exists a subsequence of ΦK ε (not relabelled) which Γconverges to some functional ΦK 0 in Rm×N (see e.g. [6] Proposition 1.42, =-=[16]-=- Theorem 8.5) for any positive integer K, by using a diagonal extraction. On the other hand, the local Lipschitz condition of (2.23) implies that the sequence ΦK ε is equi-continuous on D with respect... |

21 |
On an extension of the method of two-scale convergence and its applications.
- Zhikov
- 2000
(Show Context)
Citation Context ... models induced by low-conducting regions. One of the precursors of this kind of homogenization problem has been Khruslov [19], [17], [20], whose works have also been extended in different ways [11], =-=[25]-=-, [12], [13], [4] and [14]. The usual setting for such problems is a fixed periodic microstructure, scaled by a small parameter ε, in which the coefficients of the energy are scaled differently with r... |

7 |
Asymptotic analysis of a double porosity model with thin cracks
- Pankratov, Rybalko
(Show Context)
Citation Context ... of the first mathematical studies of the double-porosity model is due to Arbogast, Douglas and Hornung [1], subsequently revisited by Allaire [3] and extended in various ways in the linear case [5], =-=[23]-=-, and in the non-linear case [22], [8]. If the higher-conductivity set has more than one connected component, the general form of the limit effective energy involves a multi-phase description. The mat... |

6 |
Homogenization in some weakly connected domains
- Briane
- 1998
(Show Context)
Citation Context ...vector models induced by low-conducting regions. One of the precursors of this kind of homogenization problem has been Khruslov [19], [17], [20], whose works have also been extended in different ways =-=[11]-=-, [25], [12], [13], [4] and [14]. The usual setting for such problems is a fixed periodic microstructure, scaled by a small parameter ε, in which the coefficients of the energy are scaled differently ... |

5 |
Nonlinear “double porosity” type
- Pankratov, Piatnitski
(Show Context)
Citation Context ...s of the double-porosity model is due to Arbogast, Douglas and Hornung [1], subsequently revisited by Allaire [3] and extended in various ways in the linear case [5], [23], and in the non-linear case =-=[22]-=-, [8]. If the higher-conductivity set has more than one connected component, the general form of the limit effective energy involves a multi-phase description. The mathematical approach of the double-... |

4 |
Chiadò Piat V., Dal Maso G., Percivale D.: An extension theorem from connected sets, and homogenization in general periodic domains. Nonlinear Anal
- Acerbi
- 1992
(Show Context)
Citation Context ...s the limit of Eε according to (2.25). g(y, λ + Dv) dy : v ∈ W 1,p 0 (KY ; R m ) } , (2.29) Remark 2.7 For a fixed ε > 0 the existence of a bounded extension operator satisfying (2.26) is a result of =-=[2]-=-. Here we need the assumption that Pε is uniformly bounded with respect to ε. 82.4 Statement of the result In the classical framework, namely without zero-order interaction between the different cond... |

4 |
Homogenization of a soft elastic material reinforced by fibers, Asymptot
- Bellieud, Bouchitté
- 2002
(Show Context)
Citation Context ... low-conducting regions. One of the precursors of this kind of homogenization problem has been Khruslov [19], [17], [20], whose works have also been extended in different ways [11], [25], [12], [13], =-=[4]-=- and [14]. The usual setting for such problems is a fixed periodic microstructure, scaled by a small parameter ε, in which the coefficients of the energy are scaled differently with respect to ε in th... |

4 |
Homogenization of nonlinear media with soft and stiff
- Braides, Garroni
- 1995
(Show Context)
Citation Context ... i=1 Ω f hom ∫ i (Dui) dx + Φ(u1, . . . , uN) dx, for ui ∈ W Ω 1,p (Ω; R m ). The function f hom i is obtained by the usual homogenization process on the set Ei defined as the limit of Eε i (see [7], =-=[10]-=-). In the present Γ-convergence process the convergence of the sequence uε −→ (u1, . . . , uN) may be understood as lim ε→0 N∑ ∫ i=1 Ω∩E ε i |uε − ui| p dx = 0. The interaction energy density Φ is def... |

3 |
Optimal conditions of convergence and effects of anisotropy in the homogenization of non-uniformly elliptic problems, Asymptot
- Briane
- 2001
(Show Context)
Citation Context ...s induced by low-conducting regions. One of the precursors of this kind of homogenization problem has been Khruslov [19], [17], [20], whose works have also been extended in different ways [11], [25], =-=[12]-=-, [13], [4] and [14]. The usual setting for such problems is a fixed periodic microstructure, scaled by a small parameter ε, in which the coefficients of the energy are scaled differently with respect... |

3 |
Homogenization in general periodically perforated domains by a spectral approach
- Briane
(Show Context)
Citation Context ...ducting regions. One of the precursors of this kind of homogenization problem has been Khruslov [19], [17], [20], whose works have also been extended in different ways [11], [25], [12], [13], [4] and =-=[14]-=-. The usual setting for such problems is a fixed periodic microstructure, scaled by a small parameter ε, in which the coefficients of the energy are scaled differently with respect to ε in the high an... |

2 |
A general double porosity model
- Bourgeat, Goncharenko, et al.
- 1999
(Show Context)
Citation Context .... One of the first mathematical studies of the double-porosity model is due to Arbogast, Douglas and Hornung [1], subsequently revisited by Allaire [3] and extended in various ways in the linear case =-=[5]-=-, [23], and in the non-linear case [22], [8]. If the higher-conductivity set has more than one connected component, the general form of the limit effective energy involves a multi-phase description. T... |

2 |
V Chiadò Piat, Remarks on the homogenization of connected media
- Braides
- 1994
(Show Context)
Citation Context ... N∑ ∫ i=1 Ω f hom ∫ i (Dui) dx + Φ(u1, . . . , uN) dx, for ui ∈ W Ω 1,p (Ω; R m ). The function f hom i is obtained by the usual homogenization process on the set Ei defined as the limit of Eε i (see =-=[7]-=-, [10]). In the present Γ-convergence process the convergence of the sequence uε −→ (u1, . . . , uN) may be understood as lim ε→0 N∑ ∫ i=1 Ω∩E ε i |uε − ui| p dx = 0. The interaction energy density Φ ... |

2 |
Chiadò Piat V., Piatnitski A., “A variational approach to doubleporosity problems”, Asymptotic Anal
- Braides
(Show Context)
Citation Context ...he double-porosity model is due to Arbogast, Douglas and Hornung [1], subsequently revisited by Allaire [3] and extended in various ways in the linear case [5], [23], and in the non-linear case [22], =-=[8]-=-. If the higher-conductivity set has more than one connected component, the general form of the limit effective energy involves a multi-phase description. The mathematical approach of the double-poros... |

2 |
Étude de transmission à travers des inclusions minces faiblement conductrices de codimension 1 : Homogénéisation et optimisation des structures
- Hnid
- 1990
(Show Context)
Citation Context ...ounded with respect to ε. 82.4 Statement of the result In the classical framework, namely without zero-order interaction between the different conducting phases, it is known (see e.g. [24], [21] and =-=[18]-=-) that thin lowly conducting layers may modify the homogenized conductivity in certain regimes. In the present context we want to avoid such a phenomenon, which would modify the bulk part of the limit... |

1 |
Increase of dimension by homogenization
- Briane
(Show Context)
Citation Context ...ced by low-conducting regions. One of the precursors of this kind of homogenization problem has been Khruslov [19], [17], [20], whose works have also been extended in different ways [11], [25], [12], =-=[13]-=-, [4] and [14]. The usual setting for such problems is a fixed periodic microstructure, scaled by a small parameter ε, in which the coefficients of the energy are scaled differently with respect to ε ... |

1 |
Khruslov E.Ya., “Asymptotic of solution of differential equations with strongly oscillating and degenerating matrix of coefficients”, Dokl
- Fenchenko
- 1980
(Show Context)
Citation Context ...gs to the class of homogenization problems leading to homogenized vector models induced by low-conducting regions. One of the precursors of this kind of homogenization problem has been Khruslov [19], =-=[17]-=-, [20], whose works have also been extended in different ways [11], [25], [12], [13], [4] and [14]. The usual setting for such problems is a fixed periodic microstructure, scaled by a small parameter ... |

1 |
The asymptotic behavior of solutions of the second boundary value problems under fragmentation of the boundary of the domain
- Ya
- 1979
(Show Context)
Citation Context ...the class of homogenization problems leading to homogenized vector models induced by low-conducting regions. One of the precursors of this kind of homogenization problem has been Khruslov [19], [17], =-=[20]-=-, whose works have also been extended in different ways [11], [25], [12], [13], [4] and [14]. The usual setting for such problems is a fixed periodic microstructure, scaled by a small parameter ε, in ... |

1 |
Comportement limite d’un problème de transmission à travers une plaque mince et faiblement conductrice
- E
- 1970
(Show Context)
Citation Context ... is uniformly bounded with respect to ε. 82.4 Statement of the result In the classical framework, namely without zero-order interaction between the different conducting phases, it is known (see e.g. =-=[24]-=-, [21] and [18]) that thin lowly conducting layers may modify the homogenized conductivity in certain regimes. In the present context we want to avoid such a phenomenon, which would modify the bulk pa... |