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

Yu, Yifeng

Commun. Pure Appl. Math. 60, No. 8, 1111-1147 (2007).

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)].

MSC:

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

Keywords:

absolute minimizers; Hamilton-Jacobi equation

Citations:

Zbl 0996.49019; Zbl 1061.58008; Zbl 1032.37048

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