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Geometric second derivative estimates in Carnot groups and convexity. (English) Zbl 1387.35073
Summary: We prove some new a priori estimates for \(H _{2}\)-convex functions which are zero on the boundary of a bounded smooth domain \(\Omega \) 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 \(\partial \Omega \). 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 \(\partial \Omega \) one has \[ \sum \limits _{i,j=1} ^m \int \limits_\Omega ([X_i,X_j]u)^2 \, dg \, \leq \, \frac{4}{3} \int \limits_{\partial \Omega} \mathcal H\;|\nabla_H u|^2\, d\sigma_H. \]

MSC:
35B45 A priori estimates in context of PDEs
35E10 Convexity properties of solutions to PDEs and systems of PDEs with constant coefficients
35H20 Subelliptic equations
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