The Aronsson equation for absolute minimizers of <IMG BORDER="0" SRC="/tran/2007-359-01/S0002-9947-06-03897-9/gif-title0/img1.gif" >-functionals associated with vector fields satisfying Hörmander's condition期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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The Aronsson equation for absolute minimizers of -functionals associated with vector fields satisfying Hörmander's condition

Authors:

Changyou Wang

Institution:

Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506

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

. 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

is independent of the

-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

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

Keywords:

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