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Uniform rectifiability of brittle fractures in linear elasticity

Camille Labourie

Abstract

We prove the uniform rectifiability of brittle fractures in arbitrary dimension. The existing approach for the Mumford-Shah functional, which relies on separation-type properties of the singular set, faces serious obstacles in the Griffith setting due to the lack of coarea formula for the symmetric gradient. We present an alternative route to uniform rectifiability for free-discontinuity problems by proving that cracks have ``plenty of big projections''.

Uniform rectifiability of brittle fractures in linear elasticity

Abstract

We prove the uniform rectifiability of brittle fractures in arbitrary dimension. The existing approach for the Mumford-Shah functional, which relies on separation-type properties of the singular set, faces serious obstacles in the Griffith setting due to the lack of coarea formula for the symmetric gradient. We present an alternative route to uniform rectifiability for free-discontinuity problems by proving that cracks have ``plenty of big projections''.
Paper Structure (6 sections, 15 theorems, 150 equations)

This paper contains 6 sections, 15 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. The property of projections
  4. Fine estimate on holes through a projection
  5. Plenty of big projections
  6. An auxiliary lemma on affine maps

Key Result

Proposition 1

A closed set $E \subset \mathbb{R}^N$ is uniformly rectifiable if and only if there exists a constant $C \geq 1$ such that for all $x \in E$ and $0 < r < \mathrm{diam}(E)$, and there exists a compact subset $A \subset \mathbb{R}^{N-1}$ and a mapping $f : A \to \mathbb{R}^N$ such that and

Theorems & Definitions (26)

  • Definition
  • Proposition
  • Theorem 1.1: Uniform rectifiability of brittle fractures
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3: Ahlfors-regularity
  • Proposition 2.4: Carleson estimate
  • proof
  • Corollary 2.5
  • proof
  • ...and 16 more