Science China Mathematics

, Volume 57, Issue 2, pp 343–352 | Cite as

The convergence of Lax-Oleinik semigroup for time-periodic Lagrangian

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Abstract

In this article, the convergence of so-called Lax-Oleinik semigroup is studied for time-periodic Lagrangian systems when the degree of freedom is greater than 2. Under certain conditions, we show that the Lax-Oleinik semigroup converges if the rotation vector is completely irrational. Removing such conditions, we will give another kind of convergence of the sequence F c ((x, s), (x′, s′ +T n )), the convergence of which is closely related to the Lax-Oleinik semigroup.

Keywords

Lax-Oleinik semigroup Lagrangian systems Hamilton-Jacobi equation 

MSC(2010)

65P10 49L25 
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References

  1. 1.
    Bernard P. Connecting orbits of time dependent Lagrangian systems. Ann Inst Fourier, 2002, 52: 1533–1568CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bernard P. The dynamics of pseudographs in convex Hamiltonian systems. J Amer Math Soc, 2008, 21: 615–669CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Contreras G, Iturriaga R, Sanchez-Morgado H. Weak solutions of the Hamilton-Jacobi eqution for time periodic Lagrangians. PreprintGoogle Scholar
  4. 4.
    Fathi A. Sur la convergence du semigroupe de Lax-Oleinik. C R Acad Sci Paris Sér I Math, 1998, 324: 267–270CrossRefMathSciNetGoogle Scholar
  5. 5.
    Fathi A, Mather J. Failure of convergence of the Lax-Olenik semigroup in the time-periodic case. Bull Soc Math France, 2000, 128: 473–483MATHMathSciNetGoogle Scholar
  6. 6.
    Fathi A. Weak KAM Theorems in Lagrangian Dynamics. Seventh Preliminary Version. Pisa, 2005Google Scholar
  7. 7.
    Fathi A, Siconolfi A. Existence of C 1 critical subsolutions of the Hamilton-Jacobi equation. Invent Math, 2004, 155: 363–388CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Lions P L, Papanicolaou G, Varadhan S R S. Homogenization of Hamilton-Jacobi equations. Unpublished, Circa, 1988Google Scholar
  9. 9.
    Mather J. Action minimizing invariant measures for positive definite Lagrangian systems. Math Z, 1991, 207: 169–207CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Mather J. Variational construction of connecting orbits. Ann Inst Fourier (Grenoble), 1993, 43: 1349–1386CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Mather J. Differentiablity of the minimal average action as a function of the rotation number. Bol Soc Bras Mat, 1990, 21: 59–70CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Massart D. On Aubry sets and Mather’s action functional. Israel J Math, 2003, 134: 157–171CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Massart D. Subsolutions of time-periodic Hamilton-Jacobi equations. Ergodic Theory Dyn Syst, 2007, 27: 1253–1265CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Massart D. Vertices of Mather’s beta function. II. Ergodic Theory Dyn Syst, 2009, 29: 1289–1307CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Wang K, Yan J. A new kind of Lax-Oleinik type operator with parameters for time-periodic positive definite Lagrangian systems. Commun Math Phys, 2012, 309: 663–691CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.School of Mathematics and PhysicsSuzhou University of Science and TechnologySuzhouChina

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