Notions of Convexity in Carnot Groups

A new characterization of convexity in free Carnot groups

“…Ω if u is L-subharmonic with respect to all sub-Laplacians L on G. As we said in the introduction, this definition can be compared to other remarkable definitions available in the literature (as that of horizontal convexity; see [10]). Thanks to the remarks made a few paragraphs above, we explicitly observe that u is v-convex if and only if it is L A -subharmonic for every A ∈ A, where L A is as in (2.7).…”

“…Due to the relevance of the notion of convexity in theoretical and applied areas of mathematics, several notions of convexity have been recently proposed in the context of Carnot groups. The papers by Danielli, Garofalo and Nhieu [10] and by Lu, Manfredi and Stroffolini [22] opened the way for appropriate definitions of convexity in this context: they respectively introduced the classes of h-convex (horizontally convex) and v-convex (viscosity convex) functions. On the Heisenberg groups, these notions coincide (Balogh and Rickly [1]), whereas, generally, the v-convex functions are the u.s.c.…”

Geometric second derivative estimates in Carnot groups and convexity

“…The estimates that we obtain are of interest in connection with the study of the geometric notion of convexity in Carnot groups recently introduced in [DGN1], see also [LMS], [GM1], [GT], [DGNT], [Wa1], [Wa2], [Ma]. They also play a central role in establishing a priori bounds in L 2 for the (horizontal) second derivatives of solutions of non-variational operators with rough coefficients.…”

“…The matrix ∇ 2 H u plays a central role in the study of convexity in Carnot groups. It was in fact proved in [DGN1], [LMS] that a function u ∈ Γ 2 is H-convex (see definition (1.17) below) if and only if ∇ 2 H u ≥ 0. Our first result is an integral identity which connects an interesting fully nonlinear subelliptic operator to the geometry of the ground domain through the H-mean curvature H of its boundary (for the latter notion, see Definition 2.2 below).…”

“…We recall the following geometric notion of convexity introduced in [DGN1]. One should also see [LMS] where a notion of convexity in the viscosity sense was set forth.…”