A new characterization of convexity in free Carnot groups
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- by Andrea Bonfiglioli and Ermanno Lanconelli PDF
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Abstract:
A characterization of convex functions in $\mathbb {R}^N$ states that an upper semicontinuous function $u$ is convex if and only if $u(Ax)$ is subharmonic (with respect to the usual Laplace operator) for every symmetric positive definite matrix $A$. The aim of this paper is to prove that an analogue of this result holds for free Carnot groups $\mathbb {G}$ when considering convexity in the viscosity sense. In the subelliptic context of Carnot groups, the linear maps $x\mapsto Ax$ of the Euclidean case must be replaced by suitable group isomorphisms $x\mapsto T_A(x)$, whose differential preserves the first layer of the stratification of $\operatorname {Lie}(\mathbb {G})$.References
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Additional Information
- Andrea Bonfiglioli
- Affiliation: Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato, 5, 40126 Bologna, Italy
- Email: bonfigli@dm.unibo.it
- Ermanno Lanconelli
- Affiliation: Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato, 5, 40126 Bologna, Italy
- Email: lanconel@dm.unibo.it
- Received by editor(s): October 13, 2010
- Received by editor(s) in revised form: March 30, 2011
- Published electronically: January 30, 2012
- Communicated by: Matthew J. Gursky
- © Copyright 2012 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 140 (2012), 3263-3273
- MSC (2000): Primary 31C05, 26B25, 43A80; Secondary 35J70
- DOI: https://doi.org/10.1090/S0002-9939-2012-11180-3
- MathSciNet review: 2917098