Intrinsic regular graphs in Heisenberg groups vs. weak solutions of non-linear first-order PDEs

We continue to study H-regular graphs, a class of intrinsic regular hypersurfaces in the Heisenberg group H-n = C-n x R R2n+1 endowed with a left-invariant metric d(infinity) equivalent to its Carnot-Caratheodory metric. Here we investigate their relationships with suitable weak solutions of non-linear first-order PDEs. As a consequence this implies some of their geometric properties: a uniqueness result for H-regular graphs of prescribed horizontal normal as well as their (Euclidean) regularity as long as there is regularity on the horizontal normal.