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Publication 10-CNA-18

Characterization of the Multiscale Limit Associated with Bounded Sequences in BV

Rita Ferreira
I.C.T.I. - Carnegie Mellon | Portugal, F.C.T./C.M.A. da U.N.L.
Quinta da Torre
2829.516 Caparica, Portugal,

Irene Fonseca
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213, USA

Abstract:The notion of two-scale 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{\mathcal{L}^N\!}_{\lfloor\Omega},
M}(\Omega;\mathbb{R}^d)\times \mathcal{ 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 on the space $ BV$ of functions of bounded variation and described by $ n\in\mathbb{N}$ microscales, undertaken in [10].

([10] R. Ferreira, I. Fonseca, Reiterated Homogenization in $ BV$ via Multiscale Convergence, in preparation)

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