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The Aronsson equation for absolute minimizers of $ {L^\infty}$-functionals associated with vector fields satisfying Hörmander's condition

Author: Changyou Wang
Journal: Trans. Amer. Math. Soc. 359 (2007), 91-113
MSC (2000): Primary 35J20
Published electronically: June 9, 2006
MathSciNet review: 2247884
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Abstract: Given a Carnot-Carathéodory metric space $ (R^n, d_{\textrm{X}})$ generated by vector fields $ \{X_i\}_{i=1}^m$ satisfying Hörmander's condition, we prove in Theorem A that any absolute minimizer $ u\in W^{1,\infty}_{\textrm{X}}(\Omega)$ to $ F(v,\Omega)= \textrm{ess\,sup}_{x\in\Omega}f(x,Xv(x))$ is a viscosity solution to the Aronsson equation

$\displaystyle -\sum _{i=1}^{m} X_{i}(f(x,Xu(x))) f_{p_{i}}(x,Xu(x)) = 0,$$\displaystyle \text { in } \Omega , $

under suitable conditions on $ f$. In particular, any AMLE is a viscosity solution to the subelliptic $ \infty$-Laplacian equation

$\displaystyle \Delta _{\infty }^{(X)} u: =-\sum _{i,j=1}^{m} X_{i} u X_{j} u X_{i} X_{j} u = 0,$$\displaystyle \text { in } \Omega . $

If the Carnot-Carathéodory space is a Carnot group $ {\mathbf{G}}$ and $ f$ is independent of the $ x$-variable, we establish in Theorem C the uniqueness of viscosity solutions to the Aronsson equation

$\displaystyle A(Xu, (D^{2}u)^{*}):= -\sum _{i, j=1}^{m} f_{p_{i}}(Xu)f_{p_{j}}(Xu)X_{i} X_{j} u$ $\displaystyle = 0,$$\displaystyle \text { in } \Omega,$    
$\displaystyle u$ $\displaystyle = \phi ,$$\displaystyle \text { on } \partial \Omega ,$    

under suitable conditions on $ f$. As a consequence, the uniqueness of both AMLE and viscosity solutions to the subelliptic $ \infty$-Laplacian equation is established on any Carnot group $ {\mathbf{G}}$.

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Additional Information

Changyou Wang
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506

Received by editor(s): July 7, 2003
Received by editor(s) in revised form: July 30, 2004, and October 4, 2004
Published electronically: June 9, 2006
Article copyright: © Copyright 2006 American Mathematical Society

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