Commuting Hamiltonians and multi-time Hamilton-Jacobi equations

Abstract

We prove that if a sequence of pairs of smooth commuting Hamiltonians converge in the $C^0$ topology to a pair of smooth Hamiltonians, these commute. This allows us define the notion of commuting continuous Hamiltonians. As an application we extend some results of Barles and Tourin on multi-time Hamilton-Jacobi equations to a more general setting.

References in corpus (2)

• Weak KAM theorem on non compact manifolds
• On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows

Cited by in corpus (24)

• C^0-rigidity of Poisson brackets
• Quasi-states, quasi-morphisms, and the moment map
• On Decomposition Models in Imaging Sciences and Multi-time Hamilton-Jacobi Partial Differential Equations
• A C^0 counterexample to the Arnold conjecture
• Quasi-morphisms and quasi-states in symplectic topology
• On the (non) existence of viscosity solutions of multi-time Hamilton-Jacobi equations
• Hamiltonian pseudo-representations
• Viscosity solution of Hamilton-Jacobi equation by a limiting minmax method
• New energy-capacity-type inequalities and uniqueness of continuous Hamiltonians
• Weak KAM for commuting Hamiltonians
• Noether theorem in stochastic optimal control problems via contact symmetries
• Homological differential calculus
• Overcoming the curse of dimensionality for some Hamilton--Jacobi partial differential equations via neural network architectures
• Quasi-morphisms and the Poisson bracket
• C^0-rigidity of the double Poisson bracket
• Weak KAM theoretic aspects for nonregular commuting Hamiltonians
• Poisson brackets, quasi-states and symplectic integrators
• Non-convex Mather's theory and the Conley conjecture on the cotangent bundle of the torus
• Stochastic homogenization for variational solutions of Hamilton-Jacobi equations
• Rigidity of the L^p norm of the Poisson bracket on surfaces
• Poisson brackets and symplectic invariants
• Quasi-states and the Poisson bracket on surfaces
• Non-existence of global characteristics for viscosity solutions
• $C^0$-rigidity of Poisson diffeomorphisms