Calculus of Variations and Geometric Measure Theory
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M. Bardi - F. Dragoni

Convexity and semiconvexity along vector fields

created by dragoni on 04 Jun 2009
modified by bardi on 27 Aug 2012


Published Paper

Inserted: 4 jun 2009
Last Updated: 27 aug 2012

Journal: Calc. Var. Partial Differential Equations
Volume: 42
Pages: 405--427
Year: 2011


Given a family of vector fields we introduce a notion of convexity and of semiconvexity of a function along the trajectories of the fields and give infinitesimal characterizations in terms of inequalities in viscosity sense for the matrix of second derivatives with respect to the fields. We also prove that such functions are Lipschitz continuous with respect to the Carnot-Carathéodory distance associated to the family of fields and have a bounded gradient in the directions of the fields. This extends to Carnot-Carathéodory metric spaces several results for the Heisenberg group and Carnot groups obtained by a number of authors.


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