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The Aronsson-Euler Equation for Absolutely Minimizing Lipschitz Extensions with respect to Carnot-Carathéodory Metrics

By Thomas Bieske and Luca Capogna


Abstract. We derive the Euler-Lagrange equation (also known in this setting as the Aronsson-Euler equation) for absolute minimizers of the L ∞ variational problem { inf ||∇0u| | L ∞ (Ω), u = g ∈ Lip(∂Ω) on ∂Ω, where Ω ⊂ G is an open subset of a Carnot group, ∇0u denotes the horizontal gradient of u:Ω → 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. 1

Year: 2013
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