A New Kind of Lax-Oleinik Type Operator with Parameters for Time-Periodic Positive Definite Lagrangian Systems |
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Authors: | Email author" target="_blank">Kaizhi?WangEmail author Email author" target="_blank">Jun?YanEmail author |
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Institution: | 1.School of Mathematics,Jilin University,Changchun,China;2.School of Mathematical Sciences,Fudan University,Shanghai,China |
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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. |
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