Injectivity in second-gradient Nonlinear Elasticity

TL;DR

This paper investigates conditions ensuring injectivity of second-gradient nonlinear elastic models, establishing when a deformation is a homeomorphism based on integrability and zero Jacobian set properties.

Contribution

It introduces optimal conditions on integrability and Jacobian behavior that guarantee injectivity in second-gradient nonlinear elasticity models.

Findings

Proves that under certain conditions, the Jacobian cannot change sign.

Identifies maximal Hausdorff dimension of the critical set where Jacobian vanishes.

Provides counterexamples demonstrating sharpness of the conditions.

Abstract

We study injectivity for models of Nonlinear Elasticity that involve the second gradient. We assume that $\Omega\subset\mathbb{R}^n$ is a domain, $f\in W^{2,q}(\Omega,\mathbb{R}^n)$ satisfies $|J_f|^{-a}\in L^1$ and that $f$ equals a given homeomorphism on $\partial \Omega$. Under suitable conditions on $q$ and $a$ we show that $f$ must be a homeomorphism. As a main new tool we find an optimal condition for $a$ and $q$ that imply that $\mathcal{H}^{n-1}(\{J_f=0\})=0$ and hence $J_f$ cannot change sign. We further specify in dependence of $q$ and $a$ the maximal Hausdorff dimension $d$ of the critical set $\{J_f=0\}$. The sharpness of our conditions for $d$ is demonstrated by constructing respective counterexamples.