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Geometric second derivative estimates in Carnot groups and convexity

Authors:

Nicola Garofalo

Affiliation:

(1) Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

Abstract:

We prove some new a priori estimates for H ₂-convex functions which are zero on the boundary of a bounded smooth domain Ω in a Carnot group $${mathbb{G}}$$

. Such estimates are global and are geometric in nature as they involve the horizontal mean curvature $${mathcal{H}}$$

of ∂Ω. As a consequence of our bounds we show that if $${mathbb{G}}$$

has step two, then for any smooth H ₂-convex function in $$Omega subset {mathbb{G}}$$

vanishing on ∂Ω one has

$sum limits _{i,j=1} ^m int limits_Omega ([X_i,X_j]u)^2 , dg , leq , frac{4}{3} int limits_{partial Omega} mathcal H |nabla_H u|^2, dsigma_H$

. Supported in part by NSF Grant DMS-07010001.

Keywords:

Mathematics Subject Classification (2000) 35B45 35E10 35H20

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