Weak KAM theory for subriemannian Lagrangians

依赖未知函数的次黎曼拉格朗日量的弱KAM定理

Weak KAM theorems for subriemannian Lagrangians depending on the unknown function

研究将弱KAM理论结果扩展到特定次黎曼拉格朗日量，该拉格朗日量仅在水平分布定义且依赖未知函数。

Time-periodic Evans approach to weak KAM theory

We study the time-periodic version of Evans approach to weak KAM theory. Evans minimization problem is equivalent to a first oder mean field game system. For the mechanical Hamiltonian we prove the existence of smooth solutions. We introduce the corresponding effective Lagrangian and Hamiltonian and prove that they are smooth. We also consider the limiting behavior of the effective Lagrangian and Hamiltonian, Mather measures and minimizers.

离散弱KAM理论中的严格子解和Mañe势

Strict sub-solutions and Mañe potential in discrete weak KAM theory

在本文中，我们解释了弱KAM理论的离散情况下的一些事实。在该设置中，拉格朗日被替换为成本$c:X\times X \to \mathbb{R}$，在“合理”空间$X$上。这包括例如周期性的时间依赖拉格朗日的情况。众所周知，在这种情况下，可以适应大多数弱KAM理论。一个主要的区别是临界子解不一定是连续的。我们将展示如何定义Mañe势。与拉格朗日情形相反，这种势是不连续的。我们将从Mañe势的连续点集合以及临界子解中恢复Aubry集。

Caustics of weakly Lagrangian distributions

We study semiclassical sequences of distributions $u_h$ associated to a Lagrangian submanifold of phase space $\lag \subset T^*X$. If $u_h$ is a semiclassical Lagrangian distribution, which concentrates at a maximal rate on $\lag,$ then the asymptotics of $u_h$ are well-understood by work of Arnol'd, provided $\lag$ projects to $X$ with a stable Lagrangian singularity. We establish sup-norm estimates on $u_h$ under much more general hypotheses on the rate at which it is concentrating on $\lag$ (again assuming a stable projection). These estimates apply to sequences of eigenfunctions of integra…

图的Lax-Oleinik半群

The Lax-Oleinik semi-group on graphs

考虑图上的Tonelli Lagrangian，定义了弱KAM解，它们恰好是Lax-Oleinik半群的不动点，并证明了它们的唯一性集为Aubry集，给出了一个表示公式.主要结果是Lax Oleinik半群的长时间收敛性。弱KAM解是黏性解，在[CS]中的程函型Hamilton算子的情况下，我们证明了相反的情形成立.

弱KAM对与Monge-Kantorovich对偶

Weak KAM pairs and Monge-Kantorovich duality

拉格朗日系统的全局最小化轨道的动力学可以使用障碍函数来研究，就像马瑟第一次做的那样，或者使用Fathi引入的弱KAM解对。本文的中心观察是，Fathi弱KAM对正是Kantorovich问题的对偶Monge运输问题的障碍函数为成本的容许对。我们利用这一观察恢复障碍函数和一组弱KAM对之间的关系，在公理化和基本的方式。