The Aronsson-Euler equation for absolutely minimizing Lipschitz extensions with respect to Carnot-Carathéodory metrics

Bieske, Thomas; Capogna, Luca

doi:10.1090/S0002-9947-04-03601-3

The Aronsson-Euler equation for absolutely minimizing Lipschitz extensions with respect to Carnot-Carathéodory metrics
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by Thomas Bieske and Luca Capogna

Trans. Amer. Math. Soc. 357 (2005), 795-823

DOI: https://doi.org/10.1090/S0002-9947-04-03601-3

Published electronically: September 23, 2004

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Abstract:

We derive the Euler-Lagrange equation (also known in this setting as the Aronsson-Euler equation) for absolute minimizers of the $L^{\infty }$ variational problem \[ \begin {cases} \inf ||\nabla _0 u||_{L^{\infty }(\Omega )}, u=g\in Lip(\partial \Omega ) \text { on }\partial \Omega , \end {cases} \] where $\Omega \subset \mathbf {G}$ is an open subset of a Carnot group, $\nabla _0 u$ denotes the horizontal gradient of $u:\Omega \to \mathbb {R}$, and the Lipschitz class is defined in relation to the Carnot-Carathéodory metric. In particular, we show that absolute minimizers are infinite harmonic in the viscosity sense. As a corollary we obtain the uniqueness of absolute minimizers in a large class of groups. This result extends previous work of Jensen and of Crandall, Evans and Gariepy. We also derive the Aronsson-Euler equation for more “regular" absolutely minimizing Lipschitz extensions corresponding to those Carnot-Carathéodory metrics which are associated to “free" systems of vector fields.

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