\(L^{\infty}\) variational problems and weak KAM theory. (English) Zbl 1115.49025

Summary: In the first part of this paper, we extend several results in M. G. Crandall, L. C. Evans and R. F. Gariepy [Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Differ. Equ. 13, No. 2, 123–129 (2001; Zbl 0996.49019)] and M. G. Crandall and L. C. Evans [A remark on infinity harmonic functions, Electron. J. Differ. Equ. Conf. 6, 2001] to absolute minimizers of more general \(L^{\infty}\) functionals. In the second part, we present some interesting connections between \(L^{\infty}\) variational problems and weak KAM theory. As an application, we will advance the main result in A. Fathi and A. Siconolfi [Existence of \(C^1\) critical subsolutions of the Hamilton-Jacobi equation Invent. Math. 155, No. 2, 363–388 (2004; Zbl 1061.58008)], i.e, the existence of a \(C^{1}\) subsolution of the Hamilton-Jacobi equation. Moreover, we will propose a possible approximation of the projected Aubrey set by a variational approach that was first used by L. C. Evans in [Some new PDE methods for weak KAM theory, Calc. Var. Partial Diff. Equ. 17, No. 2 , 159–177 (2003; Zbl 1032.37048)].


49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
49N60 Regularity of solutions in optimal control
Full Text: DOI


[1] Aronsson, Ark Mat 6 pp 33– (1965)
[2] Aronsson, Ark Mat 6 pp 409– (1966)
[3] Aronsson, Ark Mat 6 pp 551– (1967)
[4] Aronsson, Ark Mat 7 pp 509– (1969)
[5] Aronsson, Bull Amer Math Soc (NS) 41 pp 439– (2004)
[6] Barron, Arch Ration Mech Anal 157 pp 255– (2001)
[7] ; A principle of comparison with distance functions for absolute minimizers. Available online at: http://cvgmt.sns.it/papers/chadep04/
[8] A visit with the -Laplace equation. UC Santa Barbara Math: Preprint Series 2006-26. Available online at: http://www.math.ucsb.edu/crandall/papers/visit.pdf
[9] Crandall, Arch Ration Mech Anal 167 pp 271– (2003)
[10] ; A remark on infinity harmonic functions. Proceedings of the USA-Chile Workshop on Nonlinear Analysis (Viña del Mar-Valparaiso, 2000), 123–129; Electron J Differ Equ Conf, 6, Southwest Texas State Univ., San Marcos, TX, 2001.
[11] Crandall, Calc Var Partial Differential Equations 13 pp 123– (2001)
[12] Crandall, Bull Amer Math Soc (NS) 27 pp 1– (1992)
[13] ; ; Derivation of the Aronsson equation for C1 Hamiltonians. UC Santa Barbara Math: Preprint Series 2006-28. Michael G. Crandall, Changyou Wang, and Yifeng Yu. Available online at: http://www.math.ucsb.edu/crandall/papers/conearon.pdf.
[14] Evans, Calc Var Partial Differential Equations 17 pp 159– (2003)
[15] ; C1, {\(\alpha\)} regularity for infinity harmonic functions in two dimensions. In preparation.
[16] The weak KAM theorem in Lagrangian dynamics. Cambridge Studies in Advanced Mathematics, 88. Cambridge University Press, Cambridge, 2003.
[17] Fathi, Invent Math 155 pp 363– (2004)
[18] ; Existence of solutions for the Aronsson-Euler equation. Preprint, 2004.
[19] Generalized solutions of Hamilton-Jacobi equations. Research Notes in Mathematics, 69. Pitman, Boston-London, 1982. · Zbl 0497.35001
[20] ; : Homogenization of Hamilton-Jacobi equations. Unpublished manuscript, 1988.
[21] Savin, Arch Ration Mech Anal 176 pp 351– (2005)
[22] Yu, Arch Rational Mech Anal 182 pp 153– (2006)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.