Propagation of singularities for weak KAM solutions and barrier functions

Dynamic and asymptotic behavior of singularities of certain weak KAM solutions on the torus

For mechanical Hamiltonian systems on the torus, we study the dynamical properties of the generalized characteristics semiflows associated with certain Hamilton-Jacobi equations, and build the relation between the $ω$-limit set of this semiflow and the projected Aubry set.

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。

最优输运中的奇异性及其传播

Singularities and their propagation in optimal transport

本文研究最优输运中势能泛函的奇异性，证明弱KAM解的奇异性全局传播，并揭示沿割迹由不规则拉格朗日半流控制的解的存在性。

弱KAM解的选择原理：基于不变测度的Freidlin-Wentzell大偏差原理

A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures

本文从弱KAM视角重释Freidlin-Wentzell大偏差速率函数，通过环面不可逆扩散过程刻画Peierls势垒与Mather集，并利用自洽边界数据选择唯一弱KAM粘性解，作为随机过程的全局能量景观。

Weak KAM for commuting Hamiltonians

For two commuting Tonelli Hamiltonians, we recover the commutation of the Lax-Oleinik semi-groups, a result of Barles and Tourin ([BT01]), using a direct geometrical method (Stoke's theorem). We also obtain a "generalization" of a theorem of Maderna ([Mad02]). More precisely, we prove that if the phase space is the cotangent of a compact manifold then the weak KAM solutions (or viscosity solutions of the critical stationary Hamilton-Jacobi equation) for G and for H are the same. As a corrolary we obtain the equality of the Aubry sets, of the Peierls barrier and of flat parts of Mather's $α$ fu…

Aubry-Mather and weak KAM theories for contact Hamiltonian systems. Part 1: Strictly increasing case

This paper is concerned with the study of Aubry-Mather and weak KAM theories for contact Hamiltonian systems with Hamiltonians $H(x,u,p)$ defined on $T^*M\times\mathbb{R}$, satisfying Tonelli conditions with respect to $p$ and $0<\frac{\partial H}{\partial u}\leqslant λ$ for some $λ>0$, where $M$ is a connected, closed and smooth manifold. First, we show the uniqueness of the backward weak KAM solutions of the corresponding Hamilton-Jacobi equation. Using the unique backward weak KAM solution $u_-$, we prove the existence of the maximal forward weak KAM solution $u_+$. Next, we analyse Aubry s…