Global invertibility of Sobolev maps

We define a class of Sobolev W1-p(Omega, R-n) functions, with p > n -1, such that its trace on de is also Sobolev, and do not present cavitation in the interior or on the boundary. We show that if a function in this class has positive Jacobian and coincides on the boundary with an injective map, then the function is itself injective. We then prove the existence of minimizers within this class for the type of functionals that appear in nonlinear elasticity.