Optimal regularity and structure of the free boundary for minimizers in cohesive zone models

TL;DR

This paper investigates the regularity and structure of free boundaries in cohesive zone models for fracture mechanics, establishing smoothness properties of minimizers and fracture sets under certain conditions.

Contribution

It provides new regularity results for minimizers and fracture sets in cohesive zone models, specifically showing $C^{1, 1/2}$ regularity of minimizers and $C^{1, eta}$ regularity of the fracture set.

Findings

Minimizers are $C^{1, 1/2}$ smooth.

Fracture set is $C^{1, eta}$ near non-degenerate points.

Results depend on smoothness assumptions of boundary conditions and energy density.

Abstract

We study optimal regularity and free boundary for minimizers of an energy functional arising in cohesive zone models for fracture mechanics. Under smoothness assumptions on the boundary conditions and on the fracture energy density, we show that minimizers are $C^{1, 1/2}$, and that near non-degenerate points the fracture set is $C^{1, \alpha}$, for some $\alpha \in (0, 1)$.