非正则交换哈密顿量的弱KAM理论方面
Weak KAM theoretic aspects for nonregular commuting Hamiltonians
AI总结：
研究连续凸哈密顿量交换性（通过Lax-Oleinik半群交换定义）的弱KAM理论，证明它们共享相同的弱KAM解和Aubry集，并存在公共临界次解。
摘要：
本文考虑一对连续凸哈密顿量的交换概念，该概念由它们的Lax-Oleinik半群的交换性给出。这等价于相关的多时间Hamilton-Jacobi方程的可解性。我们研究了交换性质的弱KAM理论方面，并证明这两个哈密顿量具有相同的弱KAM解和相同的Aubry集，从而推广了第二作者最近针对Tonelli哈密顿量得到的结果。我们进一步证明，哈密顿量存在一个公共的临界次解，在其Aubry集之外是严格的。在Tonelli情形下，该次解可以取为C^{1,1}类。为了完全一般性地证明我们的主要结果，建立临界次解在Aubry集上的适当可微性质至关重要。这些后者的结果在纯连续情形下是新的，并且具有独立意义。
英文摘要：
In this paper we consider the notion of commutation for a pair of continuous and convex Hamiltonians, given in terms of commutation of their Lax- Oleinik semigroups. This is equivalent to the solvability of an associated multi- time Hamilton-Jacobi equation. We examine the weak KAM theoretic aspects of the commutation property and show that the two Hamiltonians have the same weak KAM solutions and the same Aubry set, thus generalizing a result recently obtained by the second author for Tonelli Hamiltonians. We make a further step by proving that the Hamiltonians admit a common critical subsolution, strict outside their Aubry set. This subsolution can be taken of class C^{1,1} in the Tonelli case. To prove our main results in full generality, it is crucial to establish suitable differentiability properties of the critical subsolutions on the Aubry set. These latter results are new in the purely continuous case and of independent interest.
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