We learned in the last class that the non-existence of Rank-1 connections in a set K is not enough to guarantee that a weakly converging sequence of gradients tending closer to K will converge strongly. We consider the question of if non-existence of Rank-1 connections in a set K is enough to guarantee improved regularity of the differential inclusion into K. A special case of this is when we have a finite set of matrices without Rank-1 connections, improved regularity would then have to imply the mapping is affine. This is what we call the finite gradient problem. After introducing this we sketch a proof the 1/0 law for Quasiregular mappings, this law says that either the determinant of the gradient is positive a.e. or the determinant of the gradient is zero a.e.
