The vertex test function for Hamilton-Jacobi equations on networks

A general method for proving comparison principles for Hamilton-Jacobi equations on networks is introduced. It consists in constructing a vertex test function to be used in the doubling variable technique. The first important consequence is that it provides very general existence and uniqueness results for Hamilton-Jacobi equations on networks with Hamiltonians that are not convex with respect to the gradient variable and can be discontinuous with respect to the space variable at vertices. It also opens many perspectives for the study of these equations in such a singular geometrical framework; to illustrate this fact, we show how to derive a homogenization result for networks from the comparison principle. AMS Classification: 35F21, 49L25, 35B51.