Invertibility and a topological property of sobolev maps

Let Omega be a bounded domain in R(n), let d : <(Omega)over bar> --> <(Omega)over bar> be a homeomorphism, and consider a function u : <(Omega)over bar> --> R(n) that agrees with d on partial derivative Omega. if u is continuous and injective then u(Omega) = d(Omega). Motivated by problems in nonlinear elasticity the relationship between u(Omega) and d(Omega) 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,W-p with n - 1 < p < n are considered.