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Geometric second derivative estimates in Carnot groups and convexity

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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

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

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Authors and Affiliations

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

    Nicola Garofalo

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  1. Nicola Garofalo
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Correspondence to Nicola Garofalo.

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Supported in part by NSF Grant DMS-07010001.

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Garofalo, N. Geometric second derivative estimates in Carnot groups and convexity. manuscripta math. 126, 353–373 (2008). https://doi.org/10.1007/s00229-008-0182-y

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  • DOI: https://doi.org/10.1007/s00229-008-0182-y

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