53rd Midwest PDE Seminar March 19-21, 2004 Department of Mathematics University of Missouri

Nicola Garofalo Purdue University
The Busemann-Feller-Alexandrov theorem for Carnot groups

Abstract

I will discuss some recent joint work with D.Danielli, D.M.Nhieu and F.Tournier. Our main result states that in any Carnot group of step two a weakly H-convex function admits second horizontal derivatives at a.e. point. This result generalizes the classical result for convex functions due to Busemann-Feller-Alexandrov. Here, the notion of weak H-convexity is that recently introduced in joint work with Danielli and Nhieu. Since such notion is much weaker than the classical one, it is necessary to develop new tools to make-up for the lack of "global" control of the function itself. One of the main steps in the proof consists in showing that the commutator of a weakly H-convex function is locally in L², hence in particular it is a Radon measure.