Document Type
Article
Publication Date
2-1-2005
Publication Title
Transactions of the American Mathematical Society
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 ε Lip(∂Ω) on ∂Ω, where Ω ⊂ G is an open subset of a Carnot group, ∇ 0u denotes the horizontal gradient of u: Ω ℝ 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.
Keywords
Absolute minimizers, Sub-riemannian geometry
Volume
357
Issue
2
First Page
795
Last Page
823
DOI
10.1090/S0002-9947-04-03601-3
ISSN
00029947
Recommended Citation
Bieske, Thomas and Capogna, Luca, "The Aronsson-euler Equation for Absolutely Minimizing Lipschitz Extensions with Respect to Carnot-Carathéodory Metrics" (2005). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/146
Comments
Archived as published. Open access paper.