Weak KAM theoretic aspects for nonregular commuting 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.

💡 Research Summary

The paper investigates the notion of commutation for a pair of continuous, convex Hamiltonians H and G, defined through the commutation of their Lax–Oleinik semigroups. This definition is equivalent to the existence of a viscosity solution for the multi‑time Hamilton–Jacobi system

 ∂ₜu + H(x,Du)=0, ∂ₛu + G(x,Du)=0

with a Lipschitz initial datum. The authors work in a very general setting: the Hamiltonians are only required to be continuous in (x,p), convex (strictly convex when needed) in the momentum variable, and super‑linear. No smoothness or symplectic flow is assumed.

The main achievements are threefold. First, they prove that if H and G commute, then they share the same critical value c (the minimal constant for which the stationary Hamilton–Jacobi equation admits subsolutions), the same set of critical (or weak‑KAM) solutions, and the same Aubry set. This extends a recent result of the second author that was limited to Tonelli (smooth, strictly convex, super‑linear) Hamiltonians. The proof avoids any use of Hamiltonian flows; it relies solely on properties of the Lax–Oleinik operators and viscosity solution theory.

Second, the authors establish a new differentiability result for critical subsolutions. They construct a “uniqueness set” D⊂M (M being either ℝⁿ or the flat torus) such that any critical subsolution is differentiable on D and its gradient there is independent of the particular subsolution. Moreover, D is a uniqueness set for the critical equation: if two weak‑KAM solutions coincide on D, they are identical everywhere. This result, presented in Section 4, is new for purely continuous Hamiltonians and is crucial for the first theorem.

Third, they show that commuting Hamiltonians admit a common critical subsolution that is strict outside the common Aubry set. In the Tonelli case this subsolution can be taken of class C¹,¹. This improves earlier work where such a common strict subsolution was known only under stronger regularity assumptions.

The paper is organized as follows. Section 2 collects notation, the standing assumptions (continuity, convexity, super‑linearity) and recalls basic facts about Hamilton–Jacobi equations, subsolutions, and the action semidistances Sₐ. Section 3 gives a brief overview of weak‑KAM theory for non‑regular Hamiltonians, with technical proofs deferred to Appendix A. Section 4 contains the differentiability analysis of critical subsolutions, proving the existence of the set D and its properties. Section 5 applies these results to the commutation problem, establishing Theorems 1.1 and 1.2. Appendices B and C contain auxiliary lemmas and the equivalence between the Lax–Oleinik commutation and the Poisson‑bracket condition {H,G}=0 in the smooth Tonelli setting.

Overall, the work demonstrates that the essential structures of weak‑KAM theory—critical value, Aubry set, weak‑KAM solutions—are preserved under the very weak notion of commutation defined via Lax–Oleinik semigroups, even when the Hamiltonians lack smoothness. This opens the way to apply weak‑KAM ideas in optimal control, homogenization, and other areas where only continuity and convexity are available.