Uniform rectifiability of brittle fractures in linear elasticity

TL;DR

This paper establishes the uniform rectifiability of brittle fractures in any dimension within linear elasticity, overcoming previous methodological obstacles by demonstrating that cracks have many large projections.

Contribution

It introduces a novel approach to proving uniform rectifiability for free-discontinuity problems without relying on the coarea formula, applicable to the Griffith fracture model.

Findings

Proves uniform rectifiability of brittle fractures in arbitrary dimensions.

Develops an alternative method based on big projections of cracks.

Overcomes limitations of previous approaches relying on separation properties.

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''.