×

Existence of \(C^{1,1}\) critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds. (English. French summary) Zbl 1133.35027

Summary: We offer a simple proof of the existence of a \(C^{1,1}\) solution of the Hamilton-Jacobi equation in the context of Mather theory. We derive some dynamical consequences of this result. We also prove that the solution can be obtained strictly outside of the Aubry set.

MSC:

35F25 Initial value problems for nonlinear first-order PDEs
58J99 Partial differential equations on manifolds; differential operators
49L99 Hamilton-Jacobi theories
35D05 Existence of generalized solutions of PDE (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI arXiv Numdam EuDML

References:

[1] Bernard P., The dynamics of pseudographs in convex Hamiltonian systems, J. Am. Math. Soc. , in press. · Zbl 1213.37089
[2] Cannarsa P. , Sinestrari C. , Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control , Progress in Nonlinear Differential Equations and Their Applications , Birkhäuser , 2004 . Zbl 1095.49003 · Zbl 1095.49003
[3] Contreras G. , Iturriaga R. , Paternain G.P. , Paternain M. , Lagrangian graphs, minimizing measures and Mañé’s critical values , Geom. Funct. Anal. 8 ( 5 ) ( 1998 ) 788 - 809 . MR 1650090 | Zbl 0920.58015 · Zbl 0920.58015
[4] Fathi A., Weak KAM Theorem in Lagrangian Dynamics, book in press. · Zbl 0885.58022
[5] Fathi A. , Siconolfi A. , Existence of \({C}^{1}\) critical sub-solutions of the Hamilton-Jacobi equation , Invent. Math. 155 ( 2 ) ( 2004 ) 363 - 388 . Zbl 1061.58008 · Zbl 1061.58008
[6] Fathi A. , Siconolfi A. , PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians , Calc. Var. Partial Differential Equations 22 ( 2005 ) 185 - 228 . Zbl 1065.35092 · Zbl 1065.35092
[7] Lasry J.M. , Lions J.L. , A remark on regularization in Hilbert spaces , Israel J. Math. 55 ( 3 ) ( 1986 ) 257 - 266 . MR 876394 | Zbl 0631.49018 · Zbl 0631.49018
[8] Massart D., Sub-solutions of time-periodic Hamilton-Jacobi equations, Ergodic Theory Dynam. Systems , in press. Zbl 1121.37049 · Zbl 1121.37049
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.