Illinois Journal of Mathematics

Aronsson’s equations on Carnot–Carathéodory spaces

Changyou Wang and Yifeng Yu

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Let $(\mathbf{R}^n, d_X)$ be a Carnot–Carathéodory metric space generated by a family of smooth vector fields $\{X_i\}_{i=1}^m$ satisfying Hörmander's finite rank condition, and $\mathcal{H}_X= \{(x, \sum_{i=1}^m a_i X_i(x))| x\in\mathbf{R}^n, (a_i)_{i=1}^m\in\mathbf{R}^m\}$ be the horizontal tangent bundle generated by $\{X_i\}_{i=1}^m$. Assume that $H=H(x,p)\in C^1(\mathcal{H}_X)$ is quasiconvex in $p$-variable. We prove that any absolute minimizer $u\in W^{1,\infty}_X(\Omega)$ to $F_\infty(v,\Omega)=\operatorname{ess \sup}_{x\in\Omega} H(x, Xv(x))$ is a viscosity solution of the Aronsson equation \[ \mathcal{A}^{X}[u]:= X(H(x,Xu(x)))\cdot H_p(x, Xu(x)) =0 \quad\hbox{in } \Omega. \]

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Illinois J. Math., Volume 52, Number 3 (2008), 757-772.

First available in Project Euclid: 1 October 2009

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Primary: 35J 49L


Wang, Changyou; Yu, Yifeng. Aronsson’s equations on Carnot–Carathéodory spaces. Illinois J. Math. 52 (2008), no. 3, 757--772. doi:10.1215/ijm/1254403713.

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