Commuting Hamiltonians and Hamilton-Jacobi multi-time equations

The aim of this article is twofold. First, we show that the C^0-limit of a pair of commuting Hamiltonians commutes. This means, on the one hand, that if the limit of the Hamiltonians is smooth, the Poisson bracket of their limit still vanishes and, on the other hand, that we may define “commutation” for C^0-functions. The second part of this article deals with solving multi-time Hamilton-Jacobi equations using variational solutions. This extends the work of Barles and Tourin in the viscosity case to include the case of C^0-Hamiltonians, and it removes their convexity assumption, provided that we work in the setting of variational solutions.