A note on the Hausdorff dimension of the singular set of solutions to elasticity type systems

TL;DR

This paper establishes partial regularity results for solutions to elasticity systems with p-growth, providing bounds on the Hausdorff dimension of their singular sets, which aids in modeling brittle fracture with damage and plasticity.

Contribution

It offers the first estimate on the Hausdorff dimension of the singular set for nonlinear elasticity minimizers in dimensions three and higher.

Findings

Hausdorff dimension of singular set is less than n - (p* ∧ 2)

In the autonomous case, the singular set's dimension is less than n-2

Results facilitate existence proofs for Griffith fracture models with nonlinear effects

Abstract

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 $3$.