Weak KAM theoretic aspects for nonregular commuting Hamiltonians
For mathematicians studying Hamilton-Jacobi equations and weak KAM theory, this provides a generalization to nonregular Hamiltonians, though the advance is incremental over existing Tonelli results.
The paper extends weak KAM theory to nonregular commuting Hamiltonians, showing they share the same weak KAM solutions and Aubry set, and admit a common critical subsolution that is strict outside the Aubry set. This generalizes previous results for Tonelli Hamiltonians.
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.