Regularity of weak KAM solutions and Mañé’s Conjecture
Rifford, Ludovic1
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), Exposé no. 27, 22 p.

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.

DOI : 10.5802/slsedp.22
Rifford, Ludovic 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 
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Rifford, Ludovic. Regularity of weak KAM solutions and Mañé’s Conjecture. Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exposé no. 27, 22 p. doi : 10.5802/slsedp.22. http://archive.numdam.org/articles/10.5802/slsedp.22/

[1] P. Bernard. Smooth critical sub-solutions of the Hamilton-Jacobi equation. Math. Res. Lett., 14(3):503–511, 2007. | MR | Zbl

[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. | Numdam | MR | Zbl

[3] J.-B. Bost. Tores invariants des systèmes dynamiques hamiltoniens. In Séminaire Bourbaki, Vol. 1984/85 Astérisque, 133-134:113–157, 1986. | Numdam | MR | Zbl

[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. | MR | Zbl

[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. | Numdam | MR | Zbl

[6] F. Clarke. A Lipschitz regularity theorem. Ergodic Theory Dynam. Systems, 27(6):1713–1718, 2007. | MR | Zbl

[7] F. Clarke. Functional Analysis, Calculus of Variations and Optimal Control. To appear.

[8] G. Contreras and R. Iturriaga. Convex Hamiltonians without conjugate points. Ergodic Theory Dynam. Systems, 19(4):901–952, 1999. | MR | Zbl

[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. | MR | Zbl

[10] A. Fathi. Solutions KAM faible conjuguées et barrières de Peierls. C. R. Acad. Sci. Paris Sér. I Math., 325(6):649–652, 1997. | MR | Zbl

[11] A. Fathi. Orbites hétéroclones et ensemble de Peierl. C. R. Acad. Sci. Paris Sér. I Math., 326(10):1213–1216, 1998. | MR | Zbl

[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. | MR | Zbl

[13] A. Fathi. Weak KAM Theorem and Lagrangian Dynamics. Cambridge University Press, to appear.

[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. | MR | Zbl

[15] A. Fathi and A. Siconolfi. Existence of C 1 critical subsolutions of the Hamilton-Jacobi equation. Invent. math., 1155:363–388, 2004. | MR | Zbl

[16] A. Figalli and L. Rifford. Closing Aubry sets I. Preprint, 2010.

[17] A. Figalli and L. Rifford. Closing Aubry sets II. Preprint, 2010.

[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. | MR | Zbl

[19] R. Mañé. Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity, 9(2):273–310, 1996. | MR | Zbl

[20] J. N. Mather. Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z., 207:169–207, 1991. | MR | Zbl

[21] J. N. Mather. Variational construction of connecting orbits. Ann. Inst. Fourier, 43:1349–1386, 1993. | Numdam | MR | Zbl

[22] J. N. Mather. Examples of Aubry sets. Ergod. Th. Dynam. Sys., 24:1667–1723, 2004. | MR | Zbl

[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. | MR | Zbl

[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. | Numdam | MR | Zbl

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