On the $C^1$ and $C^2$-convergence to weak K.A.M. solutions - Avignon Université Accéder directement au contenu
Article Dans Une Revue Communications in Mathematical Physics Année : 2022

On the $C^1$ and $C^2$-convergence to weak K.A.M. solutions

Résumé

We introduce a notion of upper Green regular solutions to the Lax-Oleinik semi-group that is defined on the set of $C^0$ functions of a closed manifold via a Tonelli Lagrangian. Then we prove some weak $C^2$ convergence results to such a solution for a large class of approximated solutions as: 1) the discounted solution ; 2) the image of a $C^0$ function by the Lax-Oleinik semi-group; 3) the weak K.A.M. solutions for perturbed cohomology class. This kind of convergence implies the convergence in measure of the second derivatives. Moreover, we provide an example that is not upper Green regular and to which we have $C^1$ convergence but not convergence in measure of the second derivatives.

Dates et versions

hal-02025882 , version 1 (19-02-2019)

Identifiants

Citer

Marie-Claude Arnaud, Xifeng Su. On the $C^1$ and $C^2$-convergence to weak K.A.M. solutions. Communications in Mathematical Physics, In press. ⟨hal-02025882⟩
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