Regularity of Finite Energy Deformations

We review results of {Š}verák, and Goldstein-Hajłasz-Pakzad on how to show the continuity of functions in a critical Sobolev space with positive Jacobian. In the final chapter we expand on the theory of $VMO$ functions, showing a version of the change of variables theorem in this regularity, which generalizes the first step in {Š}verák’s proof to a co-dimension 1 case.