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Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks. (Solutions à flux limité pour les équations de Hamilton-Jacobi quasi-convexes posées sur des réseaux.) (English. French summary) Zbl 1382.35075

Summary: We study Hamilton-Jacobi equations on networks in the case where Hamiltonians are quasi-convex with respect to the gradient variable and can be discontinuous with respect to the space variable at vertices. First, we prove that imposing a general vertex condition is equivalent to imposing a specific one which only depends on Hamiltonians and an additional free parameter, the flux limiter. Second, a general method for proving comparison principles is introduced. This method consists in constructing a vertex test function to be used in the doubling variable approach. With such a theory and such a method in hand, we present various applications, among which a very general existence and uniqueness result for quasi-convex Hamilton-Jacobi equations on networks.

MSC:

35F21 Hamilton-Jacobi equations
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35B51 Comparison principles in context of PDEs
35R02 PDEs on graphs and networks (ramified or polygonal spaces)
35D40 Viscosity solutions to PDEs
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure