Iterative blow-ups for maps with bounded 𝒜-variation: A refinement, with application to BD and 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>BD</m:mi> </m:math> {\mathrm{BD}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>BV</m:mi> </m:math> {\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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:mi>p</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mi>D</m:mi> <m:mi>s</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mi>p</m:mi> </m:mrow> </m:mrow> </m:math> {Dp=D^{s}p} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi mathvariant="script">ℰ</m:mi> <m:mo>⁢</m:mo> <m:mi>p</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mi mathvariant="script">ℰ</m:mi> <m:mi>s</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mi>p</m:mi> </m:mrow> </m:mrow> </m:math> {{\mathcal{E}}p={\mathcal{E}}^{s}p} – at which such linearization fails.