Note on injectivity in second-gradient Nonlinear Elasticity

Let $q>1$, $(1-\frac{1}{q})a\ge 1$ and let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be Lipschitz domain. We show that planar mappings in the second order Sobolev space $f\in W^{2,q}(\mathrm{\Omega },{\mathbb{R}}^2)$ with $\mathrm{\mid }{J}_f{\mathrm{\mid }}^{-a}\in L^1(\mathrm{\Omega })$ are homeomorphism if they agree with a homeomorphism on the boundary. The condition $(1-\frac{1}{q})a\ge 1$ 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 ${\mathbb{R}}^n$.