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

Bieske, Thomas; Capogna, Luca

doi:10.1090/S0002-9947-04-03601-3

The Aronsson-Euler equation for absolutely minimizing Lipschitz extensions with respect to Carnot-Carathéodory metrics
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by Thomas Bieske and Luca Capogna PDF

Trans. Amer. Math. Soc. 357 (2005), 795-823 Request permission

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 \[ \begin {cases} \inf ||\nabla _0 u||_{L^{\infty }(\Omega )}, u=g\in Lip(\partial \Omega ) \text { on }\partial \Omega , \end {cases} \] 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.

References

Gunnar Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6 (1967), 551–561 (1967). MR 217665, DOI 10.1007/BF02591928
Gunnar Aronsson, On the partial differential equation $u_{x}{}^{2}\!u_{xx} +2u_{x}u_{y}u_{xy}+u_{y}{}^{2}\!u_{yy}=0$, Ark. Mat. 7 (1968), 395–425 (1968). MR 237962, DOI 10.1007/BF02590989
E. N. Barron, R. R. Jensen, and C. Y. Wang, The Euler equation and absolute minimizers of $L^\infty$ functionals, Arch. Ration. Mech. Anal. 157 (2001), no. 4, 255–283. MR 1831173, DOI 10.1007/PL00004239
André Bellaïche, The tangent space in sub-Riemannian geometry, Sub-Riemannian geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 1–78. MR 1421822, DOI 10.1007/978-3-0348-9210-0_{1}
Thomas Bieske, On $\infty$-harmonic functions on the Heisenberg group, Comm. Partial Differential Equations 27 (2002), no. 3-4, 727–761. MR 1900561, DOI 10.1081/PDE-120002872
Thomas Bieske, Viscosity solutions on Grushin-type planes, Illinois J. Math. 46 (2002), no. 3, 893–911. MR 1951247
T. Bieske, Lipschitz extensions on generalized Grushin-type spaces., preprint available at www.math.lsa.umich.edu/˜tbieske.
L. Capogna, D. Danielli, and N. Garofalo, Subelliptic mollifiers and a characterization of Rellich and Poincaré domains, Rend. Sem. Mat. Univ. Politec. Torino 51 (1993), no. 4, 361–386 (1994). Partial differential equations, I (Turin, 1993). MR 1289895
J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), no. 3, 428–517. MR 1708448, DOI 10.1007/s000390050094
T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
Michael G. Crandall, An efficient derivation of the Aronsson equation, Arch. Ration. Mech. Anal. 167 (2003), no. 4, 271–279. MR 1981858, DOI 10.1007/s00205-002-0236-3
M. G. Crandall, L. C. Evans, and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Differential Equations 13 (2001), no. 2, 123–139. MR 1861094, DOI 10.1007/s005260000065
Donatella Danielli, Nicola Garofalo, and Duy-Minh Nhieu, Notions of convexity in Carnot groups, Comm. Anal. Geom. 11 (2003), no. 2, 263–341. MR 2014879, DOI 10.4310/CAG.2003.v11.n2.a5
Donatella Danielli, Nicola Garofalo, and Duy-Minh Nhieu, On the best possible character of the $L^Q$ norm in some a priori estimates for non-divergence form equations in Carnot groups, Proc. Amer. Math. Soc. 131 (2003), no. 11, 3487–3498. MR 1991760, DOI 10.1090/S0002-9939-03-07105-3
Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), no. 2, 161–207. MR 494315, DOI 10.1007/BF02386204
G. B. Folland and Elias M. Stein, Hardy spaces on homogeneous groups, Mathematical Notes, vol. 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. MR 657581
Bruno Franchi, Raul Serapioni, and Francesco Serra Cassano, Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields, Houston J. Math. 22 (1996), no. 4, 859–890. MR 1437714
N. Garofalo, Analysis and Geometry of Carnot-Carathéodory Spaces, With applications to PDE’s, Birkhäuser, book in preparation.
Nicola Garofalo and Duy-Minh Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. 49 (1996), no. 10, 1081–1144. MR 1404326, DOI 10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A
Nicola Garofalo and Duy-Minh Nhieu, Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces, J. Anal. Math. 74 (1998), 67–97. MR 1631642, DOI 10.1007/BF02819446
Mikhael Gromov, Carnot-Carathéodory spaces seen from within, Sub-Riemannian geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 79–323. MR 1421823
Piotr Hajłasz, Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), no. 4, 403–415. MR 1401074, DOI 10.1007/BF00275475
Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917, DOI 10.1007/978-1-4613-0131-8
Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI 10.1007/BF02392081
Robert Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal. 123 (1993), no. 1, 51–74. MR 1218686, DOI 10.1007/BF00386368
Fritz John, Partial differential equations, 4th ed., Applied Mathematical Sciences, vol. 1, Springer-Verlag, New York, 1991. MR 1185075
Petri Juutinen, Absolutely minimizing Lipschitz extensions on a metric space, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 1, 57–67. MR 1884349
G. Lu, J. Manfredi and B. Stroffolini, Convex functions on the Heisenberg group, to appear in Calc. Var. PDE.
John Mitchell, On Carnot-Carathéodory metrics, J. Differential Geom. 21 (1985), no. 1, 35–45. MR 806700
Alexander Nagel, Elias M. Stein, and Stephen Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), no. 1-2, 103–147. MR 793239, DOI 10.1007/BF02392539
Linda Preiss Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3-4, 247–320. MR 436223, DOI 10.1007/BF02392419
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
C. Wang, The Euler equation of absolutely minimizing Lipschitz extensions for vector fields satisfying Hörmander’s condition, preprint (2003).