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13
CONVEXITY AND HORIZONTAL SECOND FUNDAMENTAL FORMS FOR HYPERSURFACES IN CARNOT GROUPS
"... Abstract. We use a Riemannian approximation scheme to give a characterization for smooth convex functions on a Carnot group (in the sense of Danielli–Garofalo– Nhieu or Lu–Manfredi–Stroffolini) in terms of the positive semidefiniteness of the horizontal second fundamental form of their graph. 1. ..."
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Abstract. We use a Riemannian approximation scheme to give a characterization for smooth convex functions on a Carnot group (in the sense of Danielli–Garofalo– Nhieu or Lu–Manfredi–Stroffolini) in terms of the positive semidefiniteness of the horizontal second fundamental form of their graph. 1.
Mannucci; Comparison principles for subelliptic equations of Monge-Ampère type
- Boll. Unione Mat. Ital. 9
"... Sunto. – We present two comparison principles for viscosity sub- and supersolutions of Monge-Ampère-type equations associated to a family of vector fields. In particular, we obtain the uniqueness of a viscosity solution to the Dirichlet problem for the equation of prescribed horizontal Gauss curvatu ..."
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Sunto. – We present two comparison principles for viscosity sub- and supersolutions of Monge-Ampère-type equations associated to a family of vector fields. In particular, we obtain the uniqueness of a viscosity solution to the Dirichlet problem for the equation of prescribed horizontal Gauss curvature in a Carnot group.
Convexity and semiconvexity along Vector Fields
"... 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 pro ..."
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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. 1 Introduction. Consider a smooth vector field X in R n and its trajectories, i.e., the solutions of ˙x(t) = X(x(t)). It is natural to say that a function u: Ω → R, Ω ⊆ R n open, is convex along the vector field X if its restriction to each trajectory of X is convex, that is, t → u(x(t)) is convex for all x(·). If u is smooth one computes
Characterizations of differentiability for H–convex functions in stratified groups
, 2010
"... Abstract. Using the notion of h-subdifferential, we characterize both first and second order differentiability of h-convex functions in stratified groups. Besides some new results involving the h-subdifferential of h-convex functions, we show that at all h-differentiability points of an h-convex fu ..."
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Abstract. Using the notion of h-subdifferential, we characterize both first and second order differentiability of h-convex functions in stratified groups. Besides some new results involving the h-subdifferential of h-convex functions, we show that at all h-differentiability points of an h-convex function, the existence of a second order expansion coincides with a suitable differentiability of its horizontal gradient.
On Hessian measures for non-commuting vector fields
, 2005
"... Previous results on Hessian measures by Trudinger and Wang are extended to the subelliptic case. Specifically we prove the weak continuity of the 2-Hessian operator, with respect to local L 1 convergence, for a system of m vector fields of step 2 and derive gradient estimates for the corresponding k ..."
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Previous results on Hessian measures by Trudinger and Wang are extended to the subelliptic case. Specifically we prove the weak continuity of the 2-Hessian operator, with respect to local L 1 convergence, for a system of m vector fields of step 2 and derive gradient estimates for the corresponding k-convex functions, 1 ≤ k ≤ m.
Subdifferential and Properties of Convex Functions with respect to Vector Fields
"... We study properties of functions convex with respect to a given family X of vector fields, a notion that appears natural in Carnot-Carathéodory metric spaces. We define a suitable subdifferential and show that a continuous function is X-convex if and only if such subdifferential is nonempty at every ..."
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We study properties of functions convex with respect to a given family X of vector fields, a notion that appears natural in Carnot-Carathéodory metric spaces. We define a suitable subdifferential and show that a continuous function is X-convex if and only if such subdifferential is nonempty at every point. For vector fields of Carnot type we deduce from this property that a generalized Fenchel transform is involutive and a weak form of Jensen inequality. Finally we introduce and compare several notions of X-affine functions and show their connections with X-convexity.
Regularity properties of H-convex sets
"... Abstract. We study the first- and second-order regularity properties of the boundary of H-convex sets in the setting of a real vector space endowed with a suitable group structure: our starting point is indeed a step two Carnot group. We prove that, locally, the noncharacteristic part of the boundar ..."
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Abstract. We study the first- and second-order regularity properties of the boundary of H-convex sets in the setting of a real vector space endowed with a suitable group structure: our starting point is indeed a step two Carnot group. We prove that, locally, the noncharacteristic part of the boundary has the intrinsic cone property and that it is foliated by intrinsic Lipschitz continous curves that are twice differentiable almost everywhere. 1.
Lipschitz estimates for convex functions with respect to vector fields
- Bruno Pini Mathematical Analysis Seminar
, 2012
"... Abstract. We present Lipschitz continuity estimates for a class of convex functions with respect to Hörmander vector fields. These results have been recently obtained in collaboration with M. Scienza, [22]. Sunto. Presentiamo alcuni recenti risultati ottenuti in collaborazione con M. Scienza, [22], ..."
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Abstract. We present Lipschitz continuity estimates for a class of convex functions with respect to Hörmander vector fields. These results have been recently obtained in collaboration with M. Scienza, [22]. Sunto. Presentiamo alcuni recenti risultati ottenuti in collaborazione con M. Scienza, [22], riguardanti la determinazione di stime quantitative sulla continuita ̀ lipschitziana di funzioni convesse rispetto a campi vettoriali di Hörmander.
A NEW CHARACTERIZATION OF CONVEXITY IN FREE CARNOT GROUPS
, 2012
"... A characterization of convex functions in 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 h ..."
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A characterization of convex functions in 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 G when considering convexity in the viscosity sense. In the subelliptic context of Carnot groups, the linear maps x ↦ → Ax of the Euclidean case must be replaced by suitable group isomorphisms x ↦ → TA(x), whose differential preserves the first layer of the stratification of Lie(G).
