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

## TL;DR

This paper studies the convergence of solutions to the Lax-Oleinik semi-group towards weak K.A.M. solutions, establishing weak $C^2$ convergence and providing examples illustrating different convergence behaviors.

## Contribution

It introduces upper Green regular solutions and proves weak $C^2$ convergence for various approximations, advancing understanding of second derivative convergence in Hamilton-Jacobi theory.

## Key findings

- Weak $C^2$ convergence holds for several classes of approximated solutions.
- Convergence in measure of second derivatives is achieved under certain conditions.
- An example shows $C^1$ convergence without second derivative convergence in measure.

## Abstract

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 (see [DFIZ16]); (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.