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Dive into the research topics where Francesco Serra Cassano is active.

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Featured researches published by Francesco Serra Cassano.


Journal of Geometric Analysis | 2003

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

Bruno Franchi; Raul Serapioni; Francesco Serra Cassano

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.


Journal of Geometric Analysis | 2006

Intrinsic regular hypersurfaces in Heisenberg groups

Luigi Ambrosio; Francesco Serra Cassano; Davide Vittone

We study the ℍ-regular surfaces, a class of intrinsic regular hypersurfaces in the setting of the Heisenberg group ℍ n = ℂ n × ℝ = ℝ2n+1 endowed with a leftinvariant metric d∞ equivalent to its Carnot-Carathéodory (CC) metric. Here hypersurface simply means topological codimension 1 surface and by the words “intrinsic” and “regular” we mean, respectively notions involving the group structure of ℍ n and its differential structure as CC manifold. In particular, we characterize these surfaces as intrinsic regular graphs inside ℍ n by studying the intrinsic regularity of the parameterizations and giving an areatype formula for their intrinsic surface measure.


Advances in Calculus of Variations | 2010

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

Francesco Bigolin; Francesco Serra Cassano

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.


Potential Analysis | 1998

Irregular Solutions of Linear Degenerate Elliptic Equations

Bruno Franchi; Raul Serapioni; Francesco Serra Cassano

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.


Advances in Calculus of Variations | 2014

Graphs of bounded variation, existence and local boundedness of non-parametric minimal surfaces in Heisenberg groups

Francesco Serra Cassano; Davide Vittone

Abstract In the setting of the sub-Riemannian Heisenberg group ℍn, we introduce and study the classes of t- and intrinsic graphs of bounded variation. For both notions we prove the existence of non-parametric area-minimizing surfaces, i.e., of graphs with the least possible area among those with the same boundary. For minimal graphs we also prove a local boundedness result which is sharp at least in the case of t-graphs in ℍ1.


Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze | 2015

BV minimizers of the area functional in the Heisenberg group under the bounded slope condition

Andrea Pinamonti; Francesco Serra Cassano; Giulia Treu; Davide Vittone

We consider the area functional for t-graphs in the sub-Riemannian Heisenberg group and study minimizers of the associated Dirichlet problem. We prove that, under a bounded slope condition on the boundary datum, there exists a unique minimizer and that this minimizer is Lipschitz continuous. We also provide an example showing that, in the first Heisenberg group, Lipschitz regularity is sharp even under the bounded slope condition.


Mathematische Annalen | 2001

Rectifiability and perimeter in the Heisenberg group

Bruno Franchi; Raul Serapioni; Francesco Serra Cassano


Calculus of Variations and Partial Differential Equations | 2001

Surface measures in Carnot-Carathéodory spaces

Roberto Monti; Francesco Serra Cassano


Communications in Analysis and Geometry | 2003

Regular hypersurfaces, intrinsic perimeter and implicit function theorem in Carnot groups

Francesco Serra Cassano; Bruno Franchi; Raul Serapioni


Advances in Mathematics | 2007

Regular submanifolds, graphs and area formula in Heisenberg groups

Bruno Franchi; Raul Serapioni; Francesco Serra Cassano

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Luigi Ambrosio

Scuola Normale Superiore di Pisa

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