We study the Cauchy problem for time-periodic Hamilton–Jacobi equations with
Tonelli Hamiltonians. It is well known that the Cauchy problem admits a unique
bounded viscosity solution. We provide a more precise description of the boundedness
of the viscosity solution. We introduce the notion of asymptotic bounds of the viscosity
solution of the Cauchy problem. An asymptotic bound is a 1-periodic viscosity solution
of the Hamilton–Jacobi equation. We show how to obtain the optimal asymptotic
bounds, i.e., minimal asymptotic upper bound and maximal asymptotic lower bound.
Our method relies upon Mather theory and weak KAM theory on Lagrangian
dynamics.
Keywords
Hamilton–Jacobi equations, viscosity solutions, Cauchy
problem, asymptotic bounds, weak KAM theory
