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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Thomas Bieske and Luca Capogna
Journal: Trans. Amer. Math. Soc. 357 (2005), 795-823
MSC (2000): Primary 35H20, 53C17
Published electronically: September 23, 2004
MathSciNet review: 2095631
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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{displaymath}\begin{cases} \inf \vert\vert\nabla_0 u\vert\vert _{L^{\infty... ...g\in Lip(\partial\Omega) \text{ on }\partial\Omega, \end{cases}\end{displaymath}

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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Additional Information

Thomas Bieske
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Address at time of publication: Department of Mathematics, University of South Florida, Tampa, Florida 33620
Email: tbieske@umich.edu, tbieske@math.usf.edu

Luca Capogna
Affiliation: Department of Mathematics, University of Arkansas, Fayetteville, Arkansas 72701
Email: lcapogna@uark.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03601-3
PII: S 0002-9947(04)03601-3
Keywords: Absolute minimizers, sub-Riemannian geometry
Received by editor(s): November 11, 2002
Received by editor(s) in revised form: October 15, 2003
Published electronically: September 23, 2004
Additional Notes: The second author was partially supported by NSF CAREER grant No. DMS-0134318
Article copyright: © Copyright 2004 American Mathematical Society