Inserted: 26 apr 2018
Last Updated: 1 feb 2020
Journal: Commun. Contemp. Math.
We prove partial regularity for minimizers to elasticity type energies in the nonlinear framework with $p$-growth, $p>1$, in dimension $n\geq 3$. It is an open problem in such a setting either to establish full regularity or to provide counterexamples. In particular, we give an estimate on the Hausdorff dimension of the potential singular set by proving that is strictly less than $n-(p^*\wedge 2)$, and actually $n-2$ in the autonomous case (full regularity is well-known in dimension $2$).
The latter result is instrumental to establish existence for the strong formulation of Griffith type models in brittle fracture with nonlinear constitutive relations, accounting for damage and plasticity in space dimensions $2$ and $3$.