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Register a new userGeneralized multi-scale Young measures
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This paper is devoted to the construction of generalized multi-scale Young measures, which are the extension of Pedregals multi-scale Young measures [Trans. Amer. Math. Soc. 358 (2006), pp. 591-602] to the setting of generalized Young measures introduced by DiPerna and Majda [Comm. Math. Phys. 108 (1987), pp. 667-689]. As a tool for variational problems, these are well-suited objects for the study (at different length-scales) of oscillation and concentration effects of convergent sequences of measures. Important properties of multi-scale Young measures such as compactness, representation of non-linear compositions, localization principles, and differential constraints are extensively developed in the second part of this paper. As an application, we use this framework to address the $Gamma$-limit characterization of the homogenized limit of convex integrals defined on spaces of measures satisfying a general linear PDE constraint.
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We give two characterizations, one for the class of generalized Young measures generated by $mathcal A$-free measures, and one for the class generated by $mathcal B$-gradient measures $mathcal Bu$. Here, $mathcal A$ and $mathcal B$ are linear homogeneous operators of arbitrary order, which we assume satisfy the constant rank property. The characterization places the class of generalized $mathcal A$-free Young measures in duality with the class of $mathcal A$-quasiconvex integrands by means of a well-known Hahn--Banach separation property. A similar statement holds for generalized $mathcal B$-gradient Young measures. Concerning applications, we discuss several examples that showcase the rigidity or the failure of $mathrm{L}^1$-compensated compactness when concentration of mass is allowed. These include the failure of $mathrm{L}^1$-estimates for elliptic systems and the failure of $mathrm{L}^1$-rigidity for the two-state problem. As a byproduct of our techniques we also show that, for any bounded open set $Omega$, the inclusions [ mathrm{L}^1(Omega) cap ker mathcal A hookrightarrow mathcal M(Omega) cap ker mathcal A, ] [ {mathcal B uin mathrm{C}^infty(Omega)} hookrightarrow {mathcal B uin mathcal M(Omega)}, ] are dense with respect to area-functional convergence of measures
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