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

Abstract We continue to study ℍ-regular graphs, a class of intrinsic regular hypersurfaces in the Heisenberg group endowed with a left-invariant metric d ∞ equivalent to its Carnot–Carathéodory 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.