Viscosity subsolutions of Hamilton–Jacobi equations and invariant sets of contact Hamilton systems

Abstract

This paper aims at providing a dynamic perspective of viscosity subsolutions to contact Hamiltonian–Jacobi equations H(x, ∂xu, u) = 0 on connected, closed manifold M. Based on implicit variational principles, we focus on the connection between positive invariant sets under Hamiltonian flow and viscosity subsolutions. We apply the connection to give a new necessary and sufficient condition for the existence of viscosity solutions of the stationary equation from the dynamical view. Finally, we discuss the multiplicity of viscosity solutions and give several illustrative examples.