Homogenization by blow-up

Braides, Andrea; Maslennikov, M. V.; Sigalotti, Laura

doi:10.1080/00036810802555458

Cited by 33 publications

(35 citation statements)

References 12 publications

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“…In this section, we prove the Γ-lim inf inequality for Theorem 2.6 by adapting the blow-up method introduced by [25] (see also [7,Section 4.1] and [14]). In the present context, a subtle use of the corrector for the convex problem is needed.…”

Section: γ-Lim Inf Inequality By Blow-upmentioning

confidence: 99%

Stochastic Homogenization of Nonconvex Unbounded Integral Functionals with Convex Growth

Duerinckx

Gloria

2016

Arch Rational Mech Anal

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We consider the well-travelled problem of homogenization of random integral functionals. When the integrand has standard growth conditions, the qualitative theory is well-understood. When it comes to unbounded functionals, that is, when the domain of the integrand is not the whole space and may depend on the space-variable, there is no satisfactory theory. In this contribution we develop a complete qualitative stochastic homogenization theory for nonconvex unbounded functionals with convex growth. We first prove that if the integrand is convex and has p-growth from below (with p > d, the dimension), then it admits homogenization regardless of growth conditions from above. This result, that crucially relies on the existence and sublinearity at infinity of correctors, is also new in the periodic case. In the case of nonconvex integrands, we prove that a similar homogenization result holds provided the nonconvex integrand admits a two-sided estimate by a convex integrand (the domain of which may depend on the space variable) that itself admits homogenization. This result is of interest to the rigorous derivation of rubber elasticity from polymer physics, which involves the stochastic homogenization of such unbounded functionals. ContentsDate: August 4, 2018. 1 2 M. DUERINCKX AND A. GLORIA A.3. Ergodic Weyl decomposition 53 A.4. Measurability results 54 A.5. Approximation results 59 References 62 Proposition 3.2 (Γ-lim inf inequality). For all ω ∈ Ω 1 , all bounded domains O ⊂ R d , and all sequences (u εProof. Let O, (u ε ) ε , and u be as in the statement. Then, for all ω ∈ Ω 1 and all k ∈ N, using the Γ-lim inf result for J k ε (·, ω; O) towards J k (·; O) (see Section 3.2), and thatso that, by monotone convergence, lim inf ε↓0 J ε (u ε , ω; O) ≥ lim k↑∞ J k (u; O) = J(u; O).From this Γ-lim inf result, we deduce the locality of recovery sequences, if they exist.

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Section: γ-Lim Inf Inequality By Blow-upmentioning

confidence: 99%

Stochastic Homogenization of Nonconvex Unbounded Integral Functionals with Convex Growth

Duerinckx

Gloria

2016

Arch Rational Mech Anal

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“…Let Π n be a linear subspace of R N of dimension n satisfying (4), let Σ be defined by (5) and satisfy hypotheses (H) and 7. Let the energy F ε be defined by (8) and the surface tension ϕ be defined by (15). Then the Γ-limit of F ε with respect to the convergence (12) is given by…”

Section: It Is Sufficient Then To Prove That the (Characteristic Funcmentioning

confidence: 99%

Interfacial energies on quasicrystals

Braides

Causin

Solci

2012

IMA Journal of Applied Mathematics

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“…The proof of the lower bound will be achieved through the use of the blow-up technique of Fonseca and Müller [13] (see also [8] for an analysis of this method for homogenization problems). Let A be of finite perimeter; let {A ε } be a sequence of admissible sets such that A ε → A.…”

Section: Lower Boundmentioning

confidence: 99%

“…i.e., (8). Using the lower semicontinuity of the total measure and the positiveness of µ, we have lim inf…”

Section: Lower Boundmentioning

confidence: 99%