Uniform rectifiability and quantitative properties of Griffith minimizers

Following [Francfort, Marigo, 1998], the formation of a crack in a solid $\Omega\subseteq{\mathbb R}^N$ results from the

minimization of the Griffith functional

$\displaystyle F(u,K):=\int_{Ω\setminus K}{\mathbb A} e(u) : e(u) dx + {\mathcal H}^{N −1}(K),$

where $e(u) := (\nabla u + \nabla u^T )/2$, over pairs $(u, K)$ such that $K$ is a relatively closed subset of $\Omega$ of dimension $N − 1$ (the crack) and

$u : \Omega\setminus K\to{\mathbb R}^N$ is a displacement field satisfying a Dirichlet condition along $\partial\Omega$. In this talk, we

will present recent regularity results for Griffith minimizers, in particular the uniform rectifiability of

fractures. Joint works with: Antoine Lemenant (Université de Lorraine), Lorenzo Lamberti (Université de

Lorraine), Yana Teplitskaya (Université Paris-Saclay).