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Hausdorff dimension of the singular set for Griffith almost-minimizers in the plane

Manuel Friedrich, Camille Labourie, Kerrek Stinson

TL;DR

This work analyzes the planar Griffith fracture energy and proves that the crack set of minimizers is locally uniformly rectifiable, leading to a Hausdorff dimension bound for the singular subset K^* with dim_H(K^*)<1. By coupling epsilon-regularity with a porosity framework, the authors show that K∖K^* is a smooth curve (C^{1,1/2}) and derive higher integrability for the displacement gradient e(u) in the minimizer case, also extending to almost-minimizers. The methodology hinges on a refined piecewise Korn–Poincaré inequality in GSBD^p, estimates on rigid-motion differences via the p-elastic energy, and a David–Semmes-type uniform rectifiability argument to control the crack geometry. Overall, the paper bridges geometric measure theory and variational fracture modeling to achieve sharp structural regularity results and gradient estimates, with implications for Mumford–Shah-type conjectures in two dimensions.

Abstract

We consider regularity of the crack set associated to a minimizer of the Griffith fracture energy, often used in modeling brittle materials. We show that the crack is uniformly rectifiable which in conjunction with our previous epsilon-regularity result allows us to prove that the singular set has dimension strictly less than $1$. This size estimate also applies to almost-minimizers. As a byproduct, we prove higher integrability for the gradient of local minimizers of the Griffith energy, providing a positive answer to the analog of De Giorgi's conjecture for the Mumford--Shah functional.

Hausdorff dimension of the singular set for Griffith almost-minimizers in the plane

TL;DR

This work analyzes the planar Griffith fracture energy and proves that the crack set of minimizers is locally uniformly rectifiable, leading to a Hausdorff dimension bound for the singular subset K^* with dim_H(K^*)<1. By coupling epsilon-regularity with a porosity framework, the authors show that K∖K^* is a smooth curve (C^{1,1/2}) and derive higher integrability for the displacement gradient e(u) in the minimizer case, also extending to almost-minimizers. The methodology hinges on a refined piecewise Korn–Poincaré inequality in GSBD^p, estimates on rigid-motion differences via the p-elastic energy, and a David–Semmes-type uniform rectifiability argument to control the crack geometry. Overall, the paper bridges geometric measure theory and variational fracture modeling to achieve sharp structural regularity results and gradient estimates, with implications for Mumford–Shah-type conjectures in two dimensions.

Abstract

We consider regularity of the crack set associated to a minimizer of the Griffith fracture energy, often used in modeling brittle materials. We show that the crack is uniformly rectifiable which in conjunction with our previous epsilon-regularity result allows us to prove that the singular set has dimension strictly less than . This size estimate also applies to almost-minimizers. As a byproduct, we prove higher integrability for the gradient of local minimizers of the Griffith energy, providing a positive answer to the analog of De Giorgi's conjecture for the Mumford--Shah functional.

Paper Structure

This paper contains 12 sections, 18 theorems, 127 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basic notions and main results
  3. Basic notions
  4. Main results
  5. Preliminaries
  6. Epsilon-regularity theorem
  7. Estimates on the difference of rigid motions via the $p$-elastic energy
  8. Piecewise Korn-Poincaré inequality
  9. Uniform rectifiability: Proof of Theorem 2.7
  10. Dimension of the singular set of K: Proof of Theorem 2.4
  11. Porosity and proof of the statement
  12. Initialization of the jump

Key Result

Theorem 1

Let $\Omega\subset \mathbb{R}^2$ be a bounded Lipschitz domain, and let $(u,K)$ be a minimizer of the Griffith energy (eq:energy). Then, the crack $K$ is (locally) uniformly rectifiable.

Figures (1)

  • Figure 1: The above picture helps to clarify the variety of sets introduced in the proof of Theorem \ref{['thm: david']}.

Theorems & Definitions (28)

  • Theorem
  • Theorem
  • Theorem 2.1: Regularity of almost-minimizers
  • Theorem 2.2: Uniform rectifiability
  • Remark 3.1: Normalization
  • Remark 3.2: Scaling and shifting properties for $\beta$
  • Theorem 3.3: Epsilon-regularity
  • Lemma 3.4: Balls of different size
  • Remark 3.5: Varying centers
  • Lemma 3.6
  • ...and 18 more