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

Marco Caroccia, Nicolas Van Goethem

TL;DR

The paper develops an iterative blow-up framework for maps of bounded ${\mathcal{A}}$-variation, combining a rigidity principle with a kernel-projection in a Poincaré inequality to obtain affine blow-ups along at least one sequence. When applied to ${\mathrm{BV}}$ and ${\mathrm{BD}}$ functions, the method yields affine blow-ups at Cantor-type points outside aTotally Singular set ${\mathrm{TS}}(u)$ and clarifies that linearization cannot be completed at those TS points. The work relies on a general theory of ${\mathcal{A}}$-variation, tangent-measure concepts, and stability under iteration to facilitate relaxation and integral representation results for energies depending on ${\mathcal{A}}u$ and $u$, extending previous BV/BD analyses to a broader operator setting. By focusing on blow-ups of the function rather than solely on ${\mathcal{A}}u$, and by incorporating a general center-symmetric kernel projection ${\mathcal{R}}$, the framework broadens applicability to homogenization and relaxation problems in first-order variational contexts.

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 $D p=D^s p$, $\mathcal{E} p=\mathcal{E}^s p$ - at which such linearization fails.

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

TL;DR

The paper develops an iterative blow-up framework for maps of bounded -variation, combining a rigidity principle with a kernel-projection in a Poincaré inequality to obtain affine blow-ups along at least one sequence. When applied to and functions, the method yields affine blow-ups at Cantor-type points outside aTotally Singular set and clarifies that linearization cannot be completed at those TS points. The work relies on a general theory of -variation, tangent-measure concepts, and stability under iteration to facilitate relaxation and integral representation results for energies depending on and , extending previous BV/BD analyses to a broader operator setting. By focusing on blow-ups of the function rather than solely on , and by incorporating a general center-symmetric kernel projection , the framework broadens applicability to homogenization and relaxation problems in first-order variational contexts.

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 and 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 , - at which such linearization fails.

Paper Structure

This paper contains 14 sections, 14 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notations
  4. Maps of bounded ${\mathcal{A}}$-variation
  5. Ellipticity and Cancelling properties
  6. Poincaré inequality for ${\mathbb C}$-elliptic operators
  7. Structure of maps of bounded ${\mathcal{A}}$-variation maps
  8. Trace and Gauss-Green theorems
  9. Radon-Nikodým decomposition of ${\mathcal{A}} u$
  10. Annihilators and structure property of the polar of $\mu={\mathcal{A}} u$
  11. Iterated strict blow-ups
  12. Application
  13. Affine blow-ups in $\mathrm{BV}$
  14. Affine blow-ups in $\mathrm{BD}$

Key Result

Theorem 1

Let $n\geq 2$ and $u\in \mathrm{BV}(\Omega;\mathbb R^m)$. Let $K$ be a center-symmetric convex set. Then for $|D^c u|$-a.e. $x\in \Omega\setminus \mathrm{TS}(u)$ there exists a vanishing sequence $\varepsilon_i \downarrow 0$ such that

Theorems & Definitions (30)

  • Theorem : Affine blow-ups for $\mathrm{BV}$ functions - Theorem \ref{['thm:mainBV']}
  • Theorem : Affine blow-up for $\mathrm{BD}$ functions - Theorem \ref{['thm:blupSel']}
  • Definition 2.1: Elliptic
  • Definition 2.2: Cancelling
  • Theorem 2.3: Gagliardo-Nirenberg-Sobolev, see vanSchaftingen2013
  • Definition 2.4: $\mathbb{C}$-Elliptic
  • Proposition 2.5: Poincaré-Sobolev inequality
  • Remark 2.6
  • Theorem 2.7: Compactness Theorem
  • Proposition 2.8: Poincaré inequality
  • ...and 20 more