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Iterative blow-ups for maps with bounded 𝒜-variation: A refinement, with application to BD and BV

  • Marco Caroccia ORCID logo and Nicolas Van Goethem ORCID logo EMAIL logo
Published/Copyright: August 29, 2025

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 BD and 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 D ⁢ p = D s ⁢ p , ℰ ⁢ p = ℰ s ⁢ p – at which such linearization fails.

MSC 2020: 49J45; 46E30

Communicated by Jan Kristensen


Funding statement: Marco Caroccia thanks the financial support of PRIN 2022R537CS “Nodal optimization, nonlinear elliptic equations, nonlocal geometric problems, with a focus on regularity” funded by the European Union under Next Generation EU. Nicolas van Goethem was supported by the FCT project UIDB/04561/2020.

Acknowledgements

The authors are deeply grateful to F. Gmeineder for the content of proposition 5.

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Received: 2024-06-18
Accepted: 2025-08-01
Published Online: 2025-08-29
Published in Print: 2025-10-01

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