Iterative blow-ups for maps with bounded $\mathcal{A}$-variation: a refinement, with application to $\mathrm{BD}$ and $\mathrm{BV}$

Abstract

We refine the iterated blow-up techniques. This technique, combined with a rigidity result and a specific choice of the kernel projection in the Poincaré inequality, might be employed to completely linearize blow-ups along at least one sequence. We show how to implement such argument by applying it to derive affine blow-up limits for $\mathrm{BD}$ and $\mathrm{BV}$ functions around Cantor points. In doing so we identify a specific subset of points - called totally singular points having blow-ups with completely singular gradient measure $Dp=D^{s}p$, $\mathcal{E}p={\mathcal{E}}^{s}p$ - at which such linearization fails.