A new kind of Lax-Oleinik type operator with parameters for time-periodic positive definite Lagrangian systems

Details A new kind of Lax-Oleinik type operator with parameters for time-periodic positive definite Lagrangian systems A new kind of Lax-Oleinik type operator with parameters for time-periodic positive definite Lagrangian systems

2010-11-10

arxiv.org/abs/1011.2244v2

Mathematics

Dynamical Systems

We give a new definition of Lax-Oleinik type operator; add some references

Scientific paper

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 new family of 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 new family of 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.

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