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In this paper we consider the notion of commutation for a pair of continuousand convex Hamiltonians, given in terms of commutation of their Lax- Oleiniksemigroups. This is equivalent to the solvability of an associated multi- timeHamilton-Jacobi equation. We examine the weak KAM theoretic aspects of thecommutation property and show that the two Hamiltonians have the same weak KAMsolutions and the same Aubry set, thus generalizing a result recently obtainedby the second author for Tonelli Hamiltonians. We make a further step byproving that the Hamiltonians admit a common critical subsolution, strictoutside their Aubry set. This subsolution can be taken of class C^{1,1} in theTonelli case. To prove our main results in full generality, it is crucial toestablish suitable differentiability properties of the critical subsolutions onthe Aubry set. These latter results are new in the purely continuous case andof independent interest.

Weak KAM theoretic aspects for nonregular commuting Hamiltonians