Critical points for Sobolev homeomorphisms

Abstract

We consider a class of homeomorphisms $f:\Omega \subset {\mathbb{R}}^2\stackrel{\text{onto}}{⟶}{\Omega }^{\prime }\subset {\mathbb{R}}^2$ of the Sobolev space ${\mathcal{W}}_{\mathrm{loc}}^{1,1}(\Omega ;{\mathbb{R}}^2)$ whose Jacobian may vanish on a set of positive measure but cannot be zero a.e. in $\Omega$. This class is defined by the bi-Sobolev condition

and reveals useful also in the theory of changes of variables for Sobolev functions.