A New Kind of Lax-Oleinik Type Operator with Parameters for Time-Periodic Positive Definite Lagrangian Systems
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.
- Publication:
-
Communications in Mathematical Physics
- Pub Date:
- February 2012
- DOI:
- 10.1007/s00220-011-1375-x
- arXiv:
- arXiv:1011.2244
- Bibcode:
- 2012CMaPh.309..663W
- Keywords:
-
- Viscosity Solution;
- Cohomology Class;
- Lagrangian System;
- Uniform Limit;
- Extremal Curve;
- Mathematics - Dynamical Systems;
- 37J50
- E-Print:
- We give a new definition of Lax-Oleinik type operator