Communications in Mathematical Physics

, Volume 309, Issue 3, pp 663–691 | Cite as

A New Kind of Lax-Oleinik Type Operator with Parameters for Time-Periodic Positive Definite Lagrangian Systems

  • Kaizhi WangEmail author
  • Jun YanEmail author


In this paper we introduce a new kind of Lax-Oleinik type operator with parameters associated with positive definite Lagrangian systems for both the time-periodic case and the time-independent case. On one hand, the family of new Lax-Oleinik type operators with an arbitrary \({u \in C(M, \mathbb{R}^1)}\) as initial condition converges to a backward weak KAM solution in the time-periodic case, while it was shown by Fathi and Mather that there is no such convergence of the Lax-Oleinik semigroup. On the other hand, the family of new Lax-Oleinik type operators with an arbitrary \({u \in C(M, \mathbb{R}^1)}\) as initial condition converges to a backward weak KAM solution faster than the Lax-Oleinik semigroup in the time-independent case.


Viscosity Solution Cohomology Class Lagrangian System Uniform Limit Extremal Curve 
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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.School of MathematicsJilin UniversityChangchunChina
  2. 2.School of Mathematical SciencesFudan UniversityShanghaiChina

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