Regularity of weak KAM solutions and Mañé’s Conjecture

Ludovic Rifford[1]

  • [1] Université de Nice-Sophia Antipolis Labo. J.-A. Dieudonné, UMR CNRS 6621 Parc Valrose 06108 Nice Cedex 02 France & Institut Universitaire de France

Séminaire Laurent Schwartz — EDP et applications (2011-2012)

  • Volume: 2011-2012, page 1-22
  • ISSN: 2266-0607

Abstract

top
We provide a crash course in weak KAM theory and review recent results concerning the existence and uniqueness of weak KAM solutions and their link with the so-called Mañé conjecture.

How to cite

top

Rifford, Ludovic. "Regularity of weak KAM solutions and Mañé’s Conjecture." Séminaire Laurent Schwartz — EDP et applications 2011-2012 (2011-2012): 1-22. <http://eudml.org/doc/251159>.

@article{Rifford2011-2012,
abstract = {We provide a crash course in weak KAM theory and review recent results concerning the existence and uniqueness of weak KAM solutions and their link with the so-called Mañé conjecture.},
affiliation = {Université de Nice-Sophia Antipolis Labo. J.-A. Dieudonné, UMR CNRS 6621 Parc Valrose 06108 Nice Cedex 02 France & Institut Universitaire de France},
author = {Rifford, Ludovic},
journal = {Séminaire Laurent Schwartz — EDP et applications},
keywords = {weak KAM theory; Mañé conjeture; critical Hamilton-Jacobi equation; Aubry set; Tonelli Hamiltonian},
language = {eng},
pages = {1-22},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Regularity of weak KAM solutions and Mañé’s Conjecture},
url = {http://eudml.org/doc/251159},
volume = {2011-2012},
year = {2011-2012},
}

TY - JOUR
AU - Rifford, Ludovic
TI - Regularity of weak KAM solutions and Mañé’s Conjecture
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2011-2012
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2011-2012
SP - 1
EP - 22
AB - We provide a crash course in weak KAM theory and review recent results concerning the existence and uniqueness of weak KAM solutions and their link with the so-called Mañé conjecture.
LA - eng
KW - weak KAM theory; Mañé conjeture; critical Hamilton-Jacobi equation; Aubry set; Tonelli Hamiltonian
UR - http://eudml.org/doc/251159
ER -

References

top
  1. P. Bernard. Smooth critical sub-solutions of the Hamilton-Jacobi equation. Math. Res. Lett., 14(3):503–511, 2007. Zbl1125.37044MR2318653
  2. P. Bernard. Existence of C 1 , 1 critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds. Ann. Sci. École Norm. Sup., 40(3):445–452, 2007. Zbl1133.35027MR2493387
  3. J.-B. Bost. Tores invariants des systèmes dynamiques hamiltoniens. In Séminaire Bourbaki, Vol. 1984/85 Astérisque, 133-134:113–157, 1986. Zbl0602.58021MR837218
  4. P. Cannarsa and C. Sinestrari. Semiconcave functions, Hamilton-Jacobi equations, and optimal control. Progress in Nonlinear Differential Equations and their Applications, 58. Birkhäuser Boston Inc., Boston, MA, 2004. Zbl1095.49003MR2041617
  5. M. Castelpietra and L. Rifford. Regularity properties of the distance function to conjugate and cut loci for viscosity solutions of Hamilton-Jacobi equations and applications in Riemannian geometry. ESAIM Control Optim. Calc. Var., 16 (3):695–718, 2010. Zbl1201.35087MR2674633
  6. F. Clarke. A Lipschitz regularity theorem. Ergodic Theory Dynam. Systems, 27(6):1713–1718, 2007. Zbl1138.49032MR2371592
  7. F. Clarke. Functional Analysis, Calculus of Variations and Optimal Control. To appear. Zbl1277.49001
  8. G. Contreras and R. Iturriaga. Convex Hamiltonians without conjugate points. Ergodic Theory Dynam. Systems, 19(4):901–952, 1999. Zbl1044.37046MR1709426
  9. A. Fathi. Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens. C. R. Acad. Sci. Paris Sér. I Math., 324(9):1043–1046, 1997. Zbl0885.58022MR1451248
  10. A. Fathi. Solutions KAM faible conjuguées et barrières de Peierls. C. R. Acad. Sci. Paris Sér. I Math., 325(6):649–652, 1997. Zbl0943.37031MR1473840
  11. A. Fathi. Orbites hétéroclones et ensemble de Peierl. C. R. Acad. Sci. Paris Sér. I Math., 326(10):1213–1216, 1998. Zbl0915.58033MR1650195
  12. A. Fathi. Sur la convergence du semi-groupe de Lax-Oleinik. C. R. Acad. Sci. Paris Sér. I Math., 327(3):267–270, 1998. Zbl1052.37514MR1650261
  13. A. Fathi. Weak KAM Theorem and Lagrangian Dynamics. Cambridge University Press, to appear. Zbl0885.58022
  14. A. Fathi, A. Figalli and L. Rifford. On the Hausdorff dimension of the Mather quotient. Comm. Pure Appl. Math., 62(4):445–500, 2009. Zbl1172.37018MR2492705
  15. A. Fathi and A. Siconolfi. Existence of C 1 critical subsolutions of the Hamilton-Jacobi equation. Invent. math., 1155:363–388, 2004. Zbl1061.58008MR2031431
  16. A. Figalli and L. Rifford. Closing Aubry sets I. Preprint, 2010. Zbl1321.37061
  17. A. Figalli and L. Rifford. Closing Aubry sets II. Preprint, 2010. Zbl06412980
  18. Y. Li and L. Nirenberg. The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton-Jacobi equations. Comm. Pure Appl. Math., 58(1):85–146, 2005. Zbl1062.49021MR2094267
  19. R. Mañé. Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity, 9(2):273–310, 1996. Zbl0886.58037MR1384478
  20. J. N. Mather. Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z., 207:169–207, 1991. Zbl0696.58027MR1109661
  21. J. N. Mather. Variational construction of connecting orbits. Ann. Inst. Fourier, 43:1349–1386, 1993. Zbl0803.58019MR1275203
  22. J. N. Mather. Examples of Aubry sets. Ergod. Th. Dynam. Sys., 24:1667–1723, 2004. Zbl1090.37047MR2104599
  23. L. Rifford. On viscosity solutions of certain Hamilton-Jacobi equations: Regularity results and generalized Sard’s Theorems. Comm. Partial Differential Equations, 33(3):517–559, 2008. Zbl1134.70007MR2398240
  24. J.-M. Roquejoffre. Propriétés qualitatives des solutions des équations de Hamilton-Jacobi (d’après A. Fathi, A. Siconolfi, P. Bernard). In Séminaire Bourbaki, Vol. 2006/2007 Astérisque, 317:269–293, 2008. Zbl1162.35003MR2487737

NotesEmbed ?

top

You must be logged in to post comments.