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We introduce a notion of upper Green regular solutions to the Lax-Oleiniksemi-group that is defined on the set of $C^0$ functions of a closed manifoldvia a Tonelli Lagrangian. Then we prove some weak $C^2$ convergence results tosuch a solution for a large class of approximated solutions as (1) thediscounted solution (see [DFIZ16]); (2) the image of a $C^0$ function by theLax-Oleinik semi-group; (3) the weak K.A.M. solutions for perturbed cohomologyclass. This kind of convergence implies the convergence in measure of thesecond derivatives.  Moreover, we provide an example that is not upper Green regular and to whichwe have $C^1$ convergence but not convergence in measure of the secondderivatives.

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