Multi-scale Young measures

Abstract

Abstract. We introduce multi-scale Young measures to deal with problems where multi-scale phenomena are relevant. We prove some interesting representation results that allow the use of these families of measures in practice, and illustrate its applicability by treating, from this perspective, multi-scale convergence and homogenization of multiple integrals.

Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting

“…Following [37] and [2], the family {ν (x,y) } (x,y)∈Ω×Y is called two-scale W 1 L φ -(gradient) Young measure associated to {u n } at scale ε, while the Young measure generated by {∇u ε } ⊂ W 1 L Φ (Ω; R d ) will be simply called W 1 L Φ -gradient Young measure, to emphasize that they are extensions to the Orlicz-Sobolev setting of the notions of gradient Young measures introduced in [26,27], while the two-scale W 1 L Φ -gradient Young measures can be seen as an extension of [37,Definition 2.4] to sequences of gradients of fields in W 1 L φ . More precisely we have the following definition.…”

“…The macroscopic description of this material may be understood by an asymptotic as ε → 0, computed as the Γ-limit of (1.1) with respect to the L Φ (Ω; R d )-topology, (equivalently W 1 L Φ -weak if Ω is, for instance, Lipschitz). Since the method employed in [20,24,23,22] to compute the Γ-limit of functionals of the type (1.1), both under convexity and without convexity assumptions on f (x, x ε , •), has been two-scale convergence (introduced in [34,1] in the Sobolev setting and later extended to Sobolev-Orlicz spaces in [19]) (or periodic unfolding method introduced in [9,10,11], applied to homogenization of integral functionals in [7,8] and later extended to the Orlicz framework in [21,22]), we want to relate these tools to the one of Young measures following the ideas in [36,35,37]. Indeed, Young measures are an important tool for studying the asymptotic behavior of solutions of nonlinear partial differential equations as emphasized in the pioneering work [13].…”

“…The special properties of Young measures generated by sequences of gradients of Sobolev functions have been studied by Kinderlherer & Pedegral [27,26] and are relevant in the applications to nonlinear elasticity. Two-scale Young measures, which have been introduced in [15] and [37] to study periodic homogenization of nonlinear transport equations and integral functionals, contain some information on the amount of oscillations and extend Nguetseng's notion of two-scale convergence, thus entailing to get multiscale homogenization results for (1.1) in the classical Sobolev setting, e.g. the results in [2] where two-scale (W 1,p )-gradient Young measures have been exaustively studied and applied.…”

Weak lower semicontinuity by means of anisotropic parametrized measures

“…This section is devoted to a new tools which might be seen as a multiscale oscillation/concentration measures. It is a generalization of the approach introduced in [30] where only oscillations were taken into account. We also wish to mention that if {u k } k∈N is bounded in W 1,p (Ω ; R m ) for 1 < p < ∞ then (at least for a nonrelabeled subsequence) the Young measure generated by the pair…”

Loss of polyconvexity by homogenization: a new example

“…A map ν ∈ Y(Ω × , M d×s ) is a two-scale gradient Young measure if there exists a sequence u h in W 1,1 (Ω, R s ) such that ∇u h is bounded in L 1 (Ω, M d×s ) and generates ν in two-scale. For a complete characterization we refer to [6] (see also [19]).…”