Invertibility of Sobolev maps through approximate invertibility at the boundary and tangential polyconvexity

We work in a class of Sobolev $W^{1,p}$ maps, with $p > d-1$, from a bounded open set $\Omega \subset \mathbb{R}^{d}$ to $\mathbb{R}^{d}$ that do not exhibit cavitation and whose trace on $\partial \Omega$ is also $W^{1,p}$. Under the assumptions that the Jacobian is positive and the deformation can be approximated on the boundary by injective maps, we show that the deformation is injective. We prove the existence of minimizers in this class for functionals accounting for a nonlinear elastic energy and a boundary energy. The energy density in $\Omega$ is assumed to be polyconvex, while the energy density in $\partial \Omega$ is assumed to be tangentially polyconvex, a new type of polyconvexity on $\partial \Omega$.