Invertibility and a Topological Property of Sobolev Maps
Stefan G. Müller, Scott J. Spector, Qi Tang · SIAM Journal on Mathematical Analysis · 1996
Let $\Omega $ be a bounded domain in $\mathbb{R}^n $, let ${\bf d}:\bar \Omega \to \bar \Omega $ be a homeomorphism, and consider a function ${\bf u}:\bar \Omega \to \mathbb{R}^n $ that agrees with ${\bf d}$ on $\partial \Omega $. If ${\bf u}$ is continuous and injective then ${\bf u}(\Omega ) = {\bf d}(\Omega )$. Motivated by problems in nonlinear elasticity the relationship between ${\bf u}(\Omega )$ and ${\bf d}(\Omega )$ is analyzed when the continuity and invertibility assumptions are weakened. Specifically maps that are continuous on almost every line and maps that lie in the Sobolev space $W^{1,p} $ with $n - 1 < p < n$ are considered.