Weak KAM theoretic aspects for nonregular commuting Hamiltonians

TL;DR

This paper explores the weak KAM theory for continuous convex Hamiltonians that commute, showing they share solutions and Aubry sets, and establishing the existence of common critical subsolutions with specific regularity properties.

Contribution

It generalizes weak KAM results to nonregular Hamiltonians, proving shared solutions, Aubry sets, and the existence of common critical subsolutions with regularity in the continuous case.

Findings

Hamiltonians have the same weak KAM solutions and Aubry set

Existence of a common critical subsolution strict outside the Aubry set

New differentiability properties of critical subsolutions on the Aubry set

Abstract

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.