Nonlocal Hamilton–Jacobi Equations on a network with Kirchhoff type conditions

Abstract

In this article, we consider nonlocal Hamilton–Jacobi Equations on networks with Kirchhoff type conditions for the interior vertices and Dirichlet boundary conditions for the boundary ones: we provide general existence and comparison results in the case when the nonlocal operator is an integro-differential operator of order strictly less than 1 . The main originality of these results is to allow these nonlocal terms to have contributions on several different edges of the network. The existence of Lipschitz continuous viscosity solutions is proved in two ways: either by using the vanishing viscosity method or by the usual Perron’s method. The comparison proof relies on arguments introduced by Lions and Souganidis. We also introduce a notion of flux-limited solution, nonlocal analog to the one introduced by Imbert and Monneau, and prove that the solutions of the Kirchhoff problem are flux-limited solutions for a suitable flux-limiter. After treating in details the case when we only have one interior vertex, we extend our approach to treat general networks.