Properties of $\mathcal{X}$-convex functions and $\mathcal{X}$-subdifferential

Properties of $\mathcal{X}$-convex functions and $\mathcal{X}$-subdifferential

Thu, 24/11/2011
12:30 		Federica Dragoni (Cardiff University) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Properties of $ \mathcal{X} $-convex functions and $ \mathcal{X} $-subdifferential
In the first part of the talk I will introduce a notion of convexity ($ \mathcal{X} $-convexity) which applies to any given family of vector fields: the main model which we have in mind is the case of vector fields satisfying the Hörmander condition. Then I will give a PDE-characterization for $ \mathcal{X} $-convex functions using a viscosity inequality for the intrinsic Hessian and I will derive bounds for the intrinsic gradient and intrinsic local Lipschitz-continuity for this class of functions.
In the second part of the talk I will introduce a notion of subdifferential for any given family of vector fields (namely $ \mathcal{X} $-subdifferential) and show that a non empty $ \mathcal{X} $-subdifferential at any point characterizes the class of $ \mathcal{X} $-convex functions. As application I will prove a Jensen-type inequality for $ \mathcal{X} $-convex functions in the case of Carnot-type vector fields. (Joint work with Martino Bardi).