A remark on a stability criterion for the radial cavitating map in nonlinear elasticity

TL;DR

This paper investigates an integral functional related to radial cavitating maps in 3D nonlinear elasticity, establishing a key inequality that supports the local minimality of such maps under certain conditions.

Contribution

It confirms a previously identified stability criterion for radial cavitating maps by proving an integral inequality on a subclass of admissible maps.

Findings

Established the inequality $I(w) \,\geq\, I(id)$ for certain maps

Confirmed the local minimality criterion for radial cavitating maps

Supports previous theoretical results in nonlinear elasticity

Abstract

An integral functional $I(w) = \int_{B} \left|\adj \nabla w \frac{w}{|w|^{3}}\right|^{q}$ defined on suitable maps $w$ is studied. The inequality $I(w) \geq I(\mathbf{id})$, where $\mathbf{id}$ is the identity map, is established on a subclass of the admissible maps, and as such confirms in these cases a criterion for the local minimality of the radial cavitating map in 3 dimensional nonlinear elasticity. The criterion was first identified by J. Sivaloganathan and S. Spector in, "Necessary conditions for a minimum at a radial cavitating singularity in nonlinear elasticity. Ann. I. H. Poincare 25 (2008) no.1, 201-213".