Note on injectivity in second-gradient Nonlinear Elasticity

TL;DR

This paper establishes sharp conditions under which planar mappings in second-order Sobolev spaces are homeomorphisms, based on integrability conditions of the Jacobian determinant and boundary agreement, with implications for the measure of singular sets.

Contribution

It proves sharp conditions for injectivity of Sobolev mappings in nonlinear elasticity and introduces a new result on the measure of the zero Jacobian set projection.

Findings

Mappings with integrable inverse Jacobian are homeomorphisms under boundary conditions.

The condition $(1-rac{1}{q})a extgreater= 1$ is proven to be sharp.

A new sharp estimate on the measure of the projection of the zero Jacobian set.

Abstract

Let $q>1$, $(1-\frac{1}{q})a\geq 1$ and let $\Omega\subset \mathbb{R}^2$ be Lipschitz domain. We show that planar mappings in the second order Sobolev space $f\in W^{2,q}(\Omega,\mathbb{R}^2)$ with $|J_f|^{-a}\in L^1(\Omega)$ are homeomorphism if they agree with a homeomorphism on the boundary. The condition $(1-\frac{1}{q})a\geq 1$ is sharp. We also have a new sharp result about the $\mathcal{H}^{n-1}$ measure of the projection of the set $\{J_f=0\}$ in $\mathbb{R}^n$.