Illinois Journal of Mathematics

Aronsson’s equations on Carnot–Carathéodory spaces

Changyou Wang and Yifeng Yu

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Abstract

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 \[ \mathcal{A}^{X}[u]:= X(H(x,Xu(x)))\cdot H_p(x, Xu(x)) =0 \quad\hbox{in } \Omega. \]

Article information

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

Dates
First available in Project Euclid: 1 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1254403713

Digital Object Identifier
doi:10.1215/ijm/1254403713

Mathematical Reviews number (MathSciNet)
MR2546006

Zentralblatt MATH identifier
1175.49025

Subjects
Primary: 35J 49L

Citation

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. https://projecteuclid.org/euclid.ijm/1254403713


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References

  • G. Aronsson, Minimization problem for the functional $\sup_xF(x,f(x),f'(x))$, Ark. Mat. 6 (1965), 33–53.
  • G. Aronsson, Minimization problem for the functional $\sup_xF(x,f(x), f'(x))$. II, Ark. Mat. 6 (1966), 409–431.
  • G. Aronsson, Minimization problem for the functional $\sup_x F(x, f(x), f'(x))$. III, Ark. Mat. 7 (1969), 509–512.
  • N. Barron, R. Jensen and C. Y. Wang, The Euler equation and absolute minimizers of $L^\infty $-functionals, Arch. Ration. Mech. Anal. 157 (2001), 255–283.
  • T. Bieske, On $\infty$-harmonic functions on the Heisenberg group, Comm. Partial Differential Equations 27 (2002), 727–761.
  • T. Bieske, Lipschitz extensions on generalized Grushin spaces, Michigan Math. J. 53 (2005), 3–31.
  • T. Bieske and L. Capogna, The Aronsson–Euler equation for absolutely minimizing Lipschitz extensions with respect to Carnot–Carathéodory metrics, Trans. Amer. Math. Soc. 357 (2005), 795–823 (electronic).
  • T. Champion and L. De Pascale, A principle of comparison with distance function for absolute minimizers, J. Convex. Anal. 14 (2007), 515–541.
  • M. Crandall, An efficient derivation of the Aronsson equation, Arch. Ration. Mech. Anal. 167 (2003), 271–279.
  • M. Crandall, L. Evans and R. Gariepy, Optimal Lipschitz extensions and the infinity laplacian, Cal. Var. PDE 13 (2001), 123–139.
  • M. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992), 1–67.
  • M. Crandall and P. L. Lions, Viscosity solutions of Hamilton–Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), 1–42.
  • M. Crandall, C. Y. Wang and Y. Yu, Derivation of the Aronsson equation for $C^1$-Hamiltonians, Trans. Amer. Math. Soc. 36 (2009), 103–124.
  • B. Franchi, R. Serapioni and F. Serra Cassano, Meyers–Serrin type theorems and relaxation of variational integrals depending on vector fields, Houston J. Math. 22 (1996), 859–890.
  • K. Friedrichs, The identity of weak and strong extensions of differential operators, Trans. Amer. Math. Soc. 55 (1944), 132–151.
  • R. Gariepy, C. Y. Wang and Y. Yu, Generalized cone comparison, Aronsson equation, and absolute minimizers, Comm. Partial Differential Equations 31 (2006), 1027–1046.
  • N. Garofalo and D. Nhieu, Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot–Carathéodory spaces, J. Anal. Math. 74 (1998), 67–97.
  • R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient, Arch. Ration. Mech. Anal. 123 (1993), 51–74.
  • P. Juutinen, Minimization problems for Lipschitz functions via viscosity solutions, Dissertation, University of Jyväskulä, Jyväskulä, 1998; Ann. Acad. Sci. Fenn. Math. Diss. 115 (1998), 53 pp.
  • P. Juutinen and N. Shanmugalingam, Equivalence of AMLE, strong AMLE, and comparison with cones in metric measure spaces, Math. Nachr. 279 (2006), 1083–1098.
  • A. Nagel, E. Stein and S. Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), 103–147.
  • P. Pansu, Carnot–Caratheodory metrics and quasi-isometries of rank-one symmetric spaces, Ann. of Math. (2) 129 (1989), 1–60.
  • C. Y. Wang, The Aronsson equation for absolute minimizers of $L\sp\infty$-functionals associated with vector fields satisfying Hörmander's condition, Trans. Amer. Math. Soc. 359 (2007), 91–113 (electronic).