Existence of Minimizers in the Variational Approach to Brittle Fracture

Abstract We review the variational approach to crack growth of Francfort and Marigo, based on incremental free discontinuity problems in linearized elasticity, and the successive techniques used to establish the existence of minimizers for such problems. In particular, we mention the crucial Korn-Poincaré inequality for functions with small jump set (proved with S. Conti and G. Francfort) which is at the root of many subsequent important results. In the second half of the article we give, in the much simpler scalar case, an overview of the proofs of a Poincaré inequality with small jump set.