离散弱KAM理论中$C^{1,1}$临界子解的存在性
Existence of $C^{1,1}$ critical subsolutions in discrete weak KAM theory
摘要：
在这篇文章中，继作者的第一个工作，我们研究临界下解的离散弱KAM理论。特别地，我们证明了如果光滑连通流形上的费用函数$c:M \times M\to \R{}$是局部半凹的，并且满足扭曲条件，则存在严格于极大集（即Aubry集之外）的$C^{1,1}$临界子解.我们还解释了如何适用于成本来自托内利拉格朗日。最后，以下的想法介绍了工作的法蒂-Maderna和马瑟，我们研究不变的费用函数，并将这项研究应用到某些覆盖空间，引入离散模拟马瑟的$α$功能上同调。
英文摘要：
In this article, following a first work of the author, we study critical subsolutions in discrete weak KAM theory. In particular, we establish that if the cost function $c:M \times M\to \R{}$ defined on a smooth connected manifold is locally semi-concave and verifies twist conditions, then there exists a $C^{1,1}$ critical subsolution strict on a maximal set (namely, outside of the Aubry set). We also explain how this applies to costs coming from Tonelli Lagrangians. Finally, following ideas introduced in the work of Fathi-Maderna and Mather, we study invariant cost functions and apply this study to certain covering spaces, introducing a discrete analogue of Mather's $α$ function on the cohomology.
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