Differentiability of Intrinsic Lipschitz Functions within Heisenberg Groups

We study the notion of intrinsic Lipschitz graphs within Heisenberg groups, focusing our attention on their Hausdorff dimension and on the almost everywhere existence of (geometrically defined) tangent subgroups. In particular, a Rademacher type theorem enables us to prove that previous definitions of rectifiable sets in Heisenberg groups are natural ones.     

Regular submanifolds, graphs and area formula in Heisenberg groups

We describe intrinsically regular submanifolds in Heisenberg groups Hn. Low dimensional and low codimensional submanifolds turn out to be of a very different nature. The first ones are Legendrian surfaces, while low codimensional ones are more general objects, possibly non-Euclidean rectifiable. Nevertheless we prove that they are graphs in a natural group way, as well as that an area formula holds for the intrinsic Hausdorff measure. Finally, they can be seen as Federer-Fleming currents given a natural complex of differential forms on Hn.     

Intrinsic regular submanifolds in Heisenberg groups are differentiable graphs

We characterize intrinsic regular submanifolds in the Heisenberg group as intrinsic differentiable graphs. G. Arena is supported by MIUR (Italy), by INDAM and by University of Trento. R. Serapioni is supported by MIUR (Italy), by GALA project of the Sixth Framework Programme of European Community and by University of Trento.     

Intrinsic Lipschitz Graphs Within Carnot Groups

A Carnot group is a connected, simply connected, nilpotent Lie group with stratified Lie algebra. We study the notions of intrinsic graphs and of intrinsic Lipschitz graphs within Carnot groups. Intrinsic Lipschitz graphs are the natural local analogue inside Carnot groups of Lipschitz submanifolds in Euclidean spaces, where “natural” emphasizes that the notion depends only on the structure of the algebra. Intrinsic Lipschitz graphs unify different alternative approaches through Lipschitz parameterizations or level sets. We provide both geometric and analytic characterizations and a clarifying relation between these graphs and Rumin’s complex of differential forms.     

Irregular Solutions of Linear Degenerate Elliptic Equations

We give examples of discontinuous solutions and of unbounded solutions of linear isotropic degenerate elliptic equations. Discontinuous solutions exist even when both the maximum eigenvalue and the inverse of the minimum eigenvalue of the matrix of the coefficients are in the intersection of all the Lp spaces.     

On the structure of finite perimeter sets in step 2 Carnot groups

In this article we study codimension 1 rectifiable sets in Carnot groups and we extend classical De Giorgi ’s rectifiability and divergence theorems to the setting of step 2 groups. Related problems in higher step Carnot groups are discussed, pointing on new phenomena related to the blow up procedure. First author was supported by University of Bologna, Italy, funds for selected research topics; second and third authors were supported by MURST, Italy, and University of Trento, Italy.     

Degenerate parabolic equations and Harnack inequality

Sunto Viene risolto il problema di Cauchy Dirichlet relativo all'operatore parabolico degenere u/t–/x_i(a_ij(x, t) u/x_j), in opportune ipotesi di integrabilità per gli autovalori di a_ij(x, t). Vengono inoltre forniti controesempi circa l'impossibilità di risultati di regolarità per le soluzioni deboli mostrando in tal modo che operatori parabolici degeneri hanno un comportamento radicalmente differente da quello dei corrispondenti operatori ellittici degeneri.

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Both the authors were supported in part by a grant of the italian C.N.R.     

Rectifiability and perimeter in the Heisenberg group

In this paper, we fully extend to the Heisenberg group endowed with its intrinsic Carnot-Carathéodory metric and perimeter the classical De Giorgi's rectifiability divergence theorems. Received: 27 March 2000 / Revised version: 13 December 2000 / Published online: 24 September 2001