关于$C^1$和$C^2$ -收敛于弱K. A.M.解
On the $C^1$ and $C^2$-convergence to weak K.A.M. solutions
摘要：
我们引入了上格林正则解的概念，定义在一个闭流形的$C^0$函数集上的Lax-Oleinik半群，通过Tonelli拉格朗日。然后我们证明了一类近似解的弱$C^2$收敛结果：（1）折扣解（见[DFIZ 16]）;（2）$C^0$函数在Lax-Oleinik半群上的象;（3）扰动上同调类的弱K. A.M.解.这种收敛性意味着二阶导数的依测度收敛。此外，我们提供了一个例子，这不是上格林正则的，我们有$C^1$收敛，但不依测度收敛的二阶导数。
英文摘要：
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.
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