Level-set convex Hamilton-Jacobi equations on networks
Résumé
The paper deals with Hamilton-Jacobi equations on networks with level-set convex (in the gradient variable) Hamiltonians which can be discontinuous with respect to the space variable at vertices. First, we prove that imposing a general \emph{vertex condition} is equivalent to imposing a specific one which only depends on Hamiltonians and an additional free paremeter, the \emph{flux limiter}. Second, a general method for proving comparison principles for \emph{flux-limited vertex conditions} is introduced. This method consists in constructing a \emph{vertex test function} to be used in the doubling variable approach. With such a theory and such a method in hand, a very general existence and uniqueness results is derived for Hamilton-Jacobi equations on networks. It also opens many perspectives for the study of these equations in such a singular geometrical framework. To illustrate this fact, we derive for instance a homogenization result for networks.
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