
Journal of Convex Analysis 19 (2012), No. 2, 403452 Copyright Heldermann Verlag 2012 Characterization of the Multiscale Limit Associated with Bounded Sequences in BV Rita Ferreira F.C.T./C.M.A. da U.N.L., Quinta da Torre, 2829516 Caparica, Portugal ragf@fct.unl.pt Irene Fonseca Dept. of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A. fonseca@andrew.cmu.edu [Abstractpdf] The notion of twoscale convergence for sequences of Radon measures with finite total variation is generalized to the case of multiple periodic length scales of oscillations. The main result concerns the characterization of $(n+1)$scale limit pairs $(u,U)$ of sequences $\{(u_\varepsilon{{\cal L}^N\!}_{\lfloor\Omega}, {Du_\varepsilon}_{\lfloor\Omega})\}_{\varepsilon>0}\subset {\cal M}(\Omega;\mathbb{R}^d)\times {\cal M}(\Omega; \mathbb{R}^{d\times N})$ whenever $\{u_\varepsilon\}_{\varepsilon>0}$ is a bounded sequence in $BV(\Omega;\mathbb{R}^d)$. This characterization is useful in the study of the asymptotic behavior of periodically oscillating functionals with linear growth, defined in the space $BV$ of functions of bounded variation and described by $n\in\mathbb{N}$ microscales, undertaken in another paper of the authors [``Reiterated homogenization in $BV$ via multiscale convergence'', submitted]. Keywords: BVvalued measures, multiscale convergence, periodic homogenization. MSC: 28B05, 26A45, 35B27 [ Fulltextpdf (372 KB)] for subscribers only. 