Calculus of Variations and Geometric Measure Theory

V. Magnani

Contact equations, Lipschitz extensions and isoperimetric inequalities

created by magnani on 14 Jan 2009
modified on 10 Jan 2012


Published Paper

Inserted: 14 jan 2009
Last Updated: 10 jan 2012

Journal: Calc. Var. Partial Differential Equations
Volume: 39
Number: 1-2
Pages: 233–271
Year: 2010


We characterize locally Lipschitz mappings and existence of Lipschitz extensions through a first order nonlinear system of PDEs. We extend this study to graded group-valued Lipschitz mappings defined on compact Riemannian manifolds. Through a simple application, we emphasize the connection between these PDEs and the Rumin complex. We introduce a class of 2-step groups, satisfying some abstract geometric conditions and we show that Lipschitz mappings taking values in these groups and defined on subsets of the plane admit Lipschitz extensions. We present several examples of these groups, called Allcock groups, observing that their horizontal distribution may have any codimesion. Finally, we show how these Lipschitz extensions theorems lead us to quadratic isoperimetric inequalities in all Allcock groups.

Keywords: Carnot groups, Isoperimetric inequalities, Lipschitz extensions