Skip to content
Accessible Unlicensed Requires Authentication Published by De Gruyter May 4, 2013

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

Francesco Serra Cassano and 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.

Funding source: INDAM

Award Identifier / Grant number: GALA

Funding source: INDAM

Award Identifier / Grant number: MIUR

Funding source: INDAM

Award Identifier / Grant number: GNAMPA

Funding source: University of Trento, Italy

Funding source: University of Padova, Italy

Funding source: Fondazione CaRiPaRo

Award Identifier / Grant number: Project “Nonlinear Partial Differential Equations: models, analysis, and control-theoretic problems”

We thank P. Yang for an example of solution of (3.1) with low regularity which helped us in Section . We are grateful to G. P. Leonardi for a suggestion in Example 4.10 as well as to R. Monti for fruitful discussions on the subject. We also thank J. H. Cheng, A. Malchiodi, M. Ritoré, C. Rosales and P. Yang for useful discussions on the topic during the RIM Conference on Mathematics held in Hong Kong in December 2007. We are grateful to G. Alberti, L. Ambrosio and S. Delladio for valuable discussions concerning Corollary 1.6.

Received: 2013-3-15
Accepted: 2013-4-8
Published Online: 2013-5-4
Published in Print: 2014-10-1

© 2014 by De Gruyter