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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
Article

Abstract

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.

Keywords

Viscosity Solution Cohomology Class Lagrangian System Uniform Limit Extremal Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

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

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