INTRINSIC REGULAR GRAPHS IN HEISENBERG GROUPS VS. WEAK SOLUTIONS OF NON LINEAR FIRST-ORDER PDES FRANCESCO BIGOLIN AND FRANCESCO SERRA CASSANO Abstract. We continue to study H- regular graphs, a class of intrinsic regular hyper- surfaces in the Heisenberg group H n = C n × R ≡ R 2n+1 endowed with a left- invariant metric d∞ equivalent to its Carnot- Carath´ eodory metric. Here we investigate their rela- tionships with suitable weak solutions of nonlinear 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. 1. Introduction and statement of the main results A fundamental problem of geometric analysis is the investigation of the interplay be- tween a surface of a given manifold and its normal. Typically this investigation consists of the study of suitable PDEs once a system of coordinates for the surface has been fixed. Following this strategy, the present paper deals with relationships between weak solutions of nonlinear first order PDEs and H- regular intrinsic graphs. The H- regular intrinsic graphs are a class of intrinsic regular hypersurfaces in the setting of the Heisenberg group H n = C n × R ≡ R 2n+1 , endowed with a left- invariant metric d ∞ equivalent to its Carnot- Carath´ eodory (CC) metric. Given an intrinsic graph S = Φ(ω) ⊂ H n (see Definition 2.6 and (1.8)) where φ : ω ⊂ R 2n → R, we will study the relationships between S and φ so that S is an H- regular surface (see Definition 2.5) and φ is a suitable solution of the system (1.1) ∇ φ φ = w in ω, being ∇ φ the family of the first order differential operators defined in (1.12) and (1.13), w ∈ C 0 (ω; R 2n−1 ) prescribed. In the first Heisenberg group H 1 (1.1) reduces to the classical Burgers’ equation. The system (1.1) geometrically is a prescribed normal vector field PDE for the intrinsic graph S . In [1] ∇ φ φ has been recognized as intrinsic gradient of φ in a suitable differential structure as we will define later. The notion of intrinsic graph has been introduced in [18] in the setting of a Carnot group and deeply studied in the setting of H n in [1], although it was already implicitly used in [15]. Date : June 29, 2009. F.B. is supported by MIUR, Italy, GNAMPA of INDAM and University of Trento, Italy, the project GALA (Sixth Framework Programme, NEST, EU). F.S.C. is supported by MIUR, Italy, GNAMPA of INDAM and University of Trento, Italy, the project GALA (Sixth Framework Programme, NEST, EU). 1