Characterization of the Multiscale Limit Associated with Bounded Sequences in BV

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 $\{(u_\varepsilon{{\cal L}^N\!}_{\lfloor\Omega}, {Du_\varepsilon}_{\lfloor\Omega})\}_{\varepsilon>0}\subset {\cal M}(\Omega;\mathbb{R}^d)\times {\cal M}(\Omega; \mathbb{R}^{d\times N})$ 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].