A continuous dependence estimate for viscous Hamilton–Jacobi equations on networks with applications

We study continuous dependence estimates for viscous Hamilton–Jacobi equations defined on a network Γ . Given two Hamilton–Jacobi equations, we prove an estimate of the C2 -norm of the difference between the corresponding solutions in terms of the distance among the Hamiltonians. We also provide two applications of the previous estimate: the first one is an existence and uniqueness result for a quasi-stationary Mean Field Games defined on the network Γ ; the second one is an estimate of the rate of convergence for homogenization of Hamilton–Jacobi equations defined on a periodic network, when the size of the cells vanishes and the limit problem is defined in the whole Euclidean space.