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Convexity and semiconvexity along vector fields

Bardi, Martino and Dragoni, Federica ORCID: 2011. Convexity and semiconvexity along vector fields. Calculus of Variations and Partial Differential Equations 42 (3-4) , pp. 405-427. 10.1007/s00526-011-0392-0

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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.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Publisher: Springer
ISSN: 1432-0835
Last Modified: 18 Oct 2022 14:26

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