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Uniform concentration property for Griffith almost-minimizers

Camille Labourie, Antoine Lememant

TL;DR

The authors prove that the Hausdorff limit of Griffith almost-minimizers is again a Griffith almost-minimizer, establishing a robust limiting theory for free-discontinuity problems with symmetric gradients. A novel uniform concentration method, independent of the coarea formula, underpins the compactness and lower-semicontinuity results and enables a general blow-up framework. The work yields applications to existence and structure of blow-up limits, a 2D classification of global minimizers with cone cracks, and a dimension bound for the singular set in terms of the integrability of the symmetric gradient. Overall, the paper advances the regularity theory for Griffith-type fracture models and provides concrete tools for constructing global minimizers via blow-ups.

Abstract

We prove that a Hausdorff limit of Griffith almost-minimizers remains a Griffith almost-minimizer. For this purpose, we introduce a new approach to the uniform concentration property of Dal Maso, Morel and Solimini which does not rely on the coarea formula, non available for symmetric gradient. We then develop several applications, including a general procedure to obtain global minimizers via blow-up limits.

Uniform concentration property for Griffith almost-minimizers

TL;DR

The authors prove that the Hausdorff limit of Griffith almost-minimizers is again a Griffith almost-minimizer, establishing a robust limiting theory for free-discontinuity problems with symmetric gradients. A novel uniform concentration method, independent of the coarea formula, underpins the compactness and lower-semicontinuity results and enables a general blow-up framework. The work yields applications to existence and structure of blow-up limits, a 2D classification of global minimizers with cone cracks, and a dimension bound for the singular set in terms of the integrability of the symmetric gradient. Overall, the paper advances the regularity theory for Griffith-type fracture models and provides concrete tools for constructing global minimizers via blow-ups.

Abstract

We prove that a Hausdorff limit of Griffith almost-minimizers remains a Griffith almost-minimizer. For this purpose, we introduce a new approach to the uniform concentration property of Dal Maso, Morel and Solimini which does not rely on the coarea formula, non available for symmetric gradient. We then develop several applications, including a general procedure to obtain global minimizers via blow-up limits.
Paper Structure (15 sections, 22 theorems, 302 equations)

This paper contains 15 sections, 22 theorems, 302 equations.

Table of Contents

  1. Introduction
  2. Definitions and statement of the main result
  3. Preliminaries on limits
  4. Standard properties
  5. A partial limiting property
  6. Fine lower density bound for quasiminimizers
  7. Initialization of the jump
  8. Size of holes through a projection.
  9. Proof of Proposition 4.1
  10. Uniform concentration property
  11. Applications
  12. Existence of blow-up limits
  13. Equivalent definitions of the singular part
  14. Dimension of the singular part
  15. Auxiliary lemmas about affine maps

Key Result

Theorem 1.1

For each constant $\varepsilon \in (0,1)$, there exist constants $\varepsilon_0 > 0$ and $C_0 \geq 1$ (depending on $N$, $\mathbb{C}$, $\varepsilon$) such that the following holds. Let $(u,K)$ be a topological Griffith almost-minimizer with any gauge $h$ in $\Omega$. For all $x_0 \in K$ and for all where $\omega_{N-1}$ is the measure of the $(N-1)$-dimensional unit disk.

Theorems & Definitions (50)

  • Theorem 1.1: Uniform concentration property
  • Remark 2.1
  • Definition 2.2: Quasiminimizers
  • Remark 2.3: Standard rescaling of quasiminimizers
  • Definition 2.4: Almost-minimal sets
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Remark 3.1
  • Lemma 3.2
  • ...and 40 more